journal of mathematics teacher education_6

EDITORIAL: JMTE AND PROFESSIONAL DEVELOPMENT

At the start of a new volume, the sixth in the life of the Journal ofMathematics Teacher Education and the second from the current editorialteam, I should like to begin with thanks to two groups of people. The firstconsists of authors who submit articles to JMTE on which the quality ofthe journal depends. The second is an increasing cohort of reviewers, inand beyond the JMTE Board, who offer unstintingly their time, energyand enthusiasm in writing cogent yet supportive reviews of submittedpapers. We are entirely dependent on both groups and thank them for theirongoing interest, support and work for JMTE. The review process is clearlycrucial in ensuring the highest quality in published papers. It is also, andperhaps even more importantly, a significantly formative process both indeveloping quality in the presentation of research and in developing a highstandard of reviewing. As editors, it seems important to us to communicatereviewers’ and editors’ advice on a paper to all reviewers who have workedon a paper. Many of our reviewers have suggested that such sharing ofreviews enables them to make balanced judgments and develop consist-ency in the review process. Authors, while suffering perhaps from a severecritique of their work, acknowledge the contribution the review processmakes to their professional development.

The work of a journal such as JMTE is highly complex depending oneffective management and communication. Sometimes this results in aslow and ponderous response to paper submission. Both publishers andeditors are working hard to streamline this process, so that papers canreach publication more quickly. However, our principal goal as editors isto produce the highest quality of journal, reflecting the strength, diversityand innovation of research in our field.

We seek to present a truly international exchange of ideas and perspec-tives, research that is rigorously argued and justified, and insights into newpractices and theories in teaching development and teacher education. Weask authors to refer to sources internationally rather than only those whichare home grown. Without compromising on quality, we want to supportnew researchers, and authors for whom English is not a first language.Members of the JMTE Board have agreed recently to offer support onone or both of these fronts where editors ask for it. This would occur

Journal of Mathematics Teacher Education 6: 1–3, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

2 EDITORIAL

when a paper has very clear academic value and interest to JMTE readersbut suffers from the inexperience of the author or its use of English. Weencourage, very positively, authors from non-English speaking countriesto make a contribution to JMTE.

In this issue we have papers from Spain, the United States and theNetherlands. These papers deal with issues in the education and develop-ment of student teachers or experienced teachers, of primary or secondaryteachers; they address teachers’ knowledge of mathematics and/or ofpedagogy and use quantitative or qualitative research methods. It is alwaysinteresting to me as an editor that, taking any three papers ready forpublication, it is always possible to find some theme in their content.Despite the differences noted, a common theme in these papers mightbe seen in the ways they address mathematics, teachers’ knowledge andunderstandings of mathematics and ways in which such understandingscontribute to teachers’ treatment of subject matter, pedagogy and the needsof their students.

The paper from Victoria Sanchez and Salvador Llinares focuses onstudent teachers’ understandings of functions: of their preferred represen-tations and images of functions, and their pedagogical reasoning withrespect to functions. The paper offers qualitative insights into howthe student teachers’ mathematical perceptions of functions guide theirthinking on working to promote their students’ learning of functions.

The paper from Wim Van Dooren, Lieven Verschaffel and PatrickOnghena explores future primary and secondary teachers’ approaches toand preferences in using arithmetic and algebraic approaches to solvingproblems. It links and distinguishes problem types, modes of solutionand the mathematical experience of the student teachers solving the prob-lems. The authors use a quantitative methodology to contrast these variousfactors and highlight usage and preferences at the beginning and at theend of the education programme. During their teacher education, students’problem solving skills evolved but not their strategy preferences. Thesefindings suggest implications for treatment of pupils’ transition fromarithmetic to algebra.

While these two papers deal with mathematical topics and processesper se, the paper from Paola Sztain goes beyond the mathematics to look atways in which teachers conceptualise mathematics in relation to the pupilsthey teach. A particular focus of this paper is the social circumstances ofthe pupils and ways in which the mathematics offered to different socialgroups is conceived by the teacher to address social issues. The paperaddresses issues of equity in mathematical provision and its potential forpupils’ mathematical growth.

EDITORIAL 3

All three papers address teachers’ perceptions of mathematics andthe potential of perceptions to influence what, and how, mathematics isoffered to pupils. The research reported challenges educators to considerhow student teachers can be encouraged to address their own prefer-ences and preconceptions of mathematics and to consider how they willmake decisions for their pupils’ learning. Together they challenge theeducational community to recognise issues in the teaching of experi-enced teachers and work on enabling student teachers to address someof the observed issues in mathematical knowledge and its interpretationinto classroom practices. We should like to see more papers reportingresearch into transitions of student teachers into experienced practice andaddressing associated critical questions for educators.

VICTORIA SÁNCHEZ and SALVADOR LLINARES

FOUR STUDENT TEACHERS’ PEDAGOGICAL REASONING ONFUNCTIONS

ABSTRACT. This study attempts to identify the influence of student teachers’ subjectmatter knowledge for teaching on the process of pedagogical reasoning. This influence isstudied through the way in which the concept of function is presented to pupils in teachingthrough the textbook problems. Our findings show that the four student teachers in ourstudy differed in their subject-matter knowledge for teaching both in the different aspects ofconcepts they emphasised and in the use of a representation repertoire to structure learningactivities. All of this conditioned the use of graphical and algebraic modes in their planningof subject matter to be presented to pupils. We explored also the influence of imagesof mathematics, teaching and learning on student teachers’ organisation of the subjectmatter for teaching, but found this only slight. Finally, regarding the relationship betweensubject matter knowledge and pedagogical content knowledge in student-teachers’ ways ofknowing the subject matter, we offer some implications of these findings for mathematicsteacher education programmes.

KEY WORDS: function concept, images, learning to teach, pedagogical reasoning,pedagogical content knowledge, subject matter knowledge

INTRODUCTION

Learning to Teach

Learning to teach is a complex process that is influenced by a range offactors. Over the last few years, research efforts have provided insight intothis process and the factors conditioning it (Carter, 1990; Borko & Putman,1996). Interest in the role of subject matter knowledge for teaching and inthe process of transformation of subject matter for the purpose of teachinghas grown, particularly since the work of Shulman and his colleagues(Shulman, 1986; Wilson, Shulman & Richert, 1987). These researchershave introduced the construct Pedagogical Content Knowledge to note “theways of representing and formulating the subject that make it comprehens-ible to others . . . [it] also includes an understanding of what makes thelearning of specific topics easy or difficult” (Shulman, 1986, p. 9). In thiscontext, we use pedagogical reasoning as a theoretical construct to portraythe transformation of content knowledge for the purposes of teaching.


6 VICTORIA SANCHEZ AND SALVADOR LLINARES

When we consider learning to teach as an active constructive process,one factor that influences the student teachers’ learning is what theybring to the teacher education programme (Ball, 1990; Even, 1993;Simon, 1993). Further, the complexity of the interrelationships betweenthe different components of student teachers’ knowledge is seen in thedifferent impact of mathematics education courses on their mathema-tical and pedagogical understanding (Wilson, 1994). This complexityshows the existence of several variables which influence student teachers’construction of the knowledge needed to teach and their developmentof pedagogical reasoning. For example, Even & Tirosh (1995) studiedthe influence of subject-matter knowledge and knowledge of learnerson student teachers’ instructional decisions. They found that the studentteachers’ organisation of the content for teaching was influenced by theirsubject matter knowledge and not so much by their knowledge of pupils’ways of thinking.

Likewise, the relationship between subject matter knowledge andpedagogical content knowledge has been considered as a key aspect inthe development of student teachers’ pedagogical reasoning (Even, 1998).Wilson (1994) described how a student teacher understood a function as apart of the mathematics that she would teach. He reported that the courseintegrating mathematical content and pedagogy had influenced her under-standing of function (subject-matter knowledge) but did not affect herapproach to teaching (beliefs). These studies help us to understand betterthe role played by the comprehension of a specific topic of subject matterin teacher learning (Cooney & Wilson, 1993), and point out the necessityfor deeper research into the relations between the components of teacherknowledge.

Ways of Knowing the Subject Matter and Images

In this study, we use the term “teacher’s ways of knowing the subjectmatter” to take into account the aspects of the mathematical contentemphasised, their explicit connections and the different uses of modes ofrepresentation that the teacher emphasises in teaching.

For example, a teacher who emphasises the concept of function as anaction can underline the meaning of function as a chain of operations. So,this teacher might put greater emphasis on algebraic modes and computa-tional activities in teaching. On the other hand, a teacher can emphasisethe function as a model for a real situation, using graphs. From theseapproaches, the activities of translating among different representationmodes (e.g., models of real-world situations using functions) can playa different role in teaching. In turn, the different aspects of the concept

FOUR STUDENT TEACHERS’ PEDAGOGICAL REASONING ON FUNCTIONS 7

emphasised, its explicit connections and the way in which the represen-tation modes are used will define the teacher’s goals. In the context oflearning to teach, the student teacher’s ways of knowing the subject-matterhelps us to understand how a student teacher might make sense of theactivities of teacher education programmes.

From other perspectives, other researchers have also emphasised therole that beliefs play in the process of learning to teach. Cooney et al.(1998) suggested that “the various ways in which the teachers structuredtheir beliefs helped account for the fact that some beliefs were permeablewhereas others were not” (p. 306). Grossman (1990) considered as oneof the components of pedagogical content knowledge the overarchingconception of what teaching a particular subject means. The overarchingconception reflects aspects of the student teachers’ beliefs that are morespecifically related to how they think about the mathematical content forthe pupils (what pupils should learn about mathematics) and the nature ofmathematics. Since affective issues seem to be integrated with the beliefs,we think that the term “image” could be appropriate to describe the studentteachers’ beliefs and attitudes towards mathematics, learning and teaching.Calderhead & Robson (1991) argue that “Images . . . represent knowledgeabout teaching but might also act as models for action and, in addition,they frequently contain an affective component, being associated withparticular feelings and attitudes” (p. 3). For example, student teachers canhold an image of mathematics as abstract, unreal or as a set of systematicprocedures. Likewise, the mathematical activity can be seen as a game ora mechanism that works properly. Johnston (1992) used the construct of“image” to identify the ways in which student teachers think about them-selves as teachers and how this relates to their teaching practice. In thissense, the images can influence the perspective and models for action takenby student teachers and permeate aspects of their experience in teachereducation programmes.

Pedagogical Reasoning and Teacher Learning

We have considered two aspects of teacher’s subject-matter knowledgefor teaching that seem to have some importance in the characterisationof teacher learning:

• a teacher’s ways of knowing the subject matter and• his/her images

From the perspective of teacher learning, it seems relevant to analyse theinfluence of these aspects on the ways in which student teachers struc-


ture learning activities (Grossman et al., 1989; McDiarmind et al., 1989).Influence on the organisation of mathematical content in teaching may beshown in the process of pedagogical reasoning, i.e., when student teacherstransform the subject matter for the purposes of teaching and give argu-ments about it. Wilson et al. (1987) consider that during this transformationthe ‘critical interpretation’ of subject matter becomes apparent, whichinvolves ‘reviewing instructional materials in the light of one’s own under-standing of the subject matter’ (p. 119). For us, this critical interpretationincludes the characteristics of the concept which are identified, the type ofproblem chosen and the order in which the different aspects of the conceptare presented by the student teachers.

Likewise, in the transformation, the teacher’s ‘representational reper-toire’ becomes evident in the sense of the different activities, assignments,examples, and so on, that ‘teachers use to transform the content forinstruction’ (Wilson et al., 1987, p. 120). We include here the modes ofrepresentation of subject matter that a student teacher emphasises anduses to convey something about the subject matter to the learner. Wilsonet al. (1987) also consider ‘the adaptation’ of the subject matter, whichinvolves fitting the transformation to the characteristics of the students ingeneral. Globally considered, the interpretation, representation and adapt-ation contribute to generate an action plan for teaching specific subjectmatter. Within this framework, we pose the following question:

How do the student teacher’s ways of knowing and their images about school mathematicsand mathematics learning/teaching influence the ways in which they think about presentingthe subject matter to pupils?

We focus on a specific topic, the concept of function, at secondaryschool level (pupils aged 14–16). We have chosen this particular domainbecause it is one of the most important curricular topics at this level. It isplaying an increasingly important role in the secondary curriculum and isrelated to other subjects (e.g., Physics).

METHODOLOGY

Participants and Context

The study involved four university graduates who volunteered to parti-cipate: Juan, Rafael, Alberto and José (all names are pseudonyms). Eachhad obtained a degree in mathematics or in another branch of science.Their ages ranged from 22 to 25. They presented no special characteristics


other than their interest in collaborating in anything that might help toimprove the design of mathematics teacher education programmes.

In order to obtain the credits necessary to teach mathematics atsecondary school level (pupils aged 12–18), they had to enroll on a post-graduate course. This course was focused on pedagogical, psychologicaland mathematics education issues (with only 30, out of 180 hours, focusedon mathematics education issues). There was also a practical component(student teaching) in which each student teacher carried out his/her studentteaching in a secondary classroom with a mathematics teacher as a tutorfor four weeks. Data for the study were collected at the beginning of thispost-graduate course. At this stage, the student teachers’ knowledge aboutfunctions and their images resulted from their previous experiences inschool and university mathematics courses and not from the educationaltheory of the postgraduate course.

The specific context of secondary mathematics teacher education inSpain has some particular characteristics. Student teachers receive widetraining in mathematics (and sometimes in other branches of science suchas physics, chemistry) for five years to obtain a degree in mathematics(or science). So, when the student teachers take the post-graduate coursethey have already a solid background in mathematics. This situation isdifferent from those countries in which teacher training programmes havemore or less fully integrated courses with mathematical subject contentand mathematics education content. Specific training in mathematics forsuch a long time (five years) may influence a student teacher’s imagesof mathematics and its teaching/learning. We believe it to be worthwhileto study the relationships between the ways of knowing and the imagesas references in the development of pedagogical reasoning processes. Onthe other hand, nowadays the secondary school curriculum in our country(Junta de Andalucía, Diseño Curricular Base, 1993) emphasises a model-ling approach to functions. This curricular approach points out the readingand interpretation of graphs (linked to a real-situation) as specific objec-tives. This view, which gives less prominence to formal definitions andalgebraic expressions, can be affected by the approach to the conceptadopted by the teacher.

Data Collection and Instruments

We designed four interviews to obtain information about the studentteachers’ ways of knowing the concept of function and their imagesas well as to study the relationships between the aspects of functionelicited and the mode of representation selected. We asked each student


teacher to complete all the interviews. The first interview, which wassemi-structured, was a general interview aimed at obtaining informationconcerning his/her biographical background related to mathematics andeliciting data regarding his/her images about mathematics, teaching andlearning.

For the second interview, we asked students to engage in practical taskswith associated textbook problems. In consideration of the literature onstudents’ understanding of functions (Vinner & Dreyfus, 1989; Leinhardtet al., 1990; Dubinsky & Harel, 1992) and on mathematical representations(Janvier, 1987; Kaput, 1991), we chose a set of 22 textbook problems thatdiffered in two dimensions:

(i) the mode of representation in which the function concept was shown(real-world, situation, algebraic mode, graphs and tables);

(ii) the hypothetical activity that the problem demanded from the problemsolver (Garcia & Llinares, 1995).

Each of these problems was written on a card to make the handlingeasier The two practical tasks designed with these problems were:

• ‘Classification’ task: Sort the 22 textbook problems relating to theconcept of function, written on the cards, and provide argumentsto explain the sets made. [Here, problems were named TextbookProblem 1, Textbook Problem 2, and so on.]

• ‘Textbook Problem Analysis’ task: Analyse the 10 textbook prob-lems (chosen from the 22 textbook problems written on the cards)from different perspectives. [Here, problems were named TextbookProblem A, Textbook Problem B, and so on.]

In this interview, we posed questions like:

❖ Describe this problem in your own words.❖ Do you think this task is necessary to teach functions?❖ What mathematical content might be learnt with this problem?❖ What objectives will you try to achieve?

We wanted to obtain information about the student teachers’ reasons forusing a specific problem in their teaching, and how they thought that alearner would solve it.

In the third interview we asked student teachers to use the text-book problems in the planning of a hypothetical teaching sequence forthe concept of function and provide arguments that might justify theirdecisions. The aim was to identify what was behind the presentation ofthe mathematical content in the planning prepared by each student teacher.


We considered that making them think about the presentation of the subjectmatter to pupils using textbook problems might generate mental activityfrom which we could identify the source of their choices and decisions.

For the fourth interview, we designed four hypothetical situations(Cases 1 to 4). The cases were constructed bearing in mind the resultsof the review of research into the learning of functions (Leinhardt et al.,1990). The content of the cases was

(i) misconceptions in the overall interpretation of graphs;(ii) the role of images that the pupils construct as a result of the type of

task that they usually carry out;(iii) the separation between visual and analytical processing of the infor-

mation;(iv) the difficulties created by the notion of variable.

Each case described a pupil’s response to a problem with functions andposed several questions. Some of these questions focused on diagnosingpupils’ thinking and asked student teachers to identify the causes for thepupil’s response; some others asked about the way in which the teachercould help the pupil.

Data Analysis

The four interviews were recorded and transcribed. Using the transcrip-tions, different analyses were performed. In the first step, from eachinterview we identified the arguments used by the student teachers. Theinformation obtained was categorised in relation to the two dimensionsof subject matter knowledge for teaching that we had considered: waysof knowing the subject matter and images. In each of the categoriesobtained, we identified data relevant for the transformation of subjectmatter for teaching from the different interpretations of subject matter, themodes of representation emphasised, and information about the process ofadaptation of the subject matter for the learner.

The results obtained have been organised into two subsections;

❖ Subject matter knowledge for teaching (a pre-service secondaryteacher’s ways of knowing and images);

❖ Processes of transformation of the subject matter described throughcritical interpretation, repertoire of representational modes and adapt-ation to pupils’ mathematical thinking.

We continue with a discussion of the results of our analysis and theirimplications for mathematics teacher education programmes.


Texbook problem A. Reading a graph

Let us consider the graph of the function

f(x)= 1/(1 + x2)

Could you answer the following questions?

I. What is the maximun, or supremun?

II. What is the minimun, or infimum?

III. In which interval is the function increasing?And decreasing?

IV. Is the function banded?

V. Is the function symmetric? If it is, with regardto what?

Textbook problem 7

In the hall of a high school there is a machine with cans of soft drinks. One day, themachine’s owner made a study of how many cans there were at each moment in themachine from 8 am to 8 pm.The results of the study are represented in the following graph.By looking at the graph carefully, try to answer these questions:

I. How many cans there were in the machine at 8 o’clock in the morning?

II. From which periods of time was no can consumed?

III. How many cans were consumed in the morning break, between 11am and 11:30 am?

IV. At what time was the machine filled?

V. From the graph, could you say at what time the afternoon school classes finish?

VI. When were more cans per hour sold, during the morning break or during the lunchbreak?

Figure 1. Three textbook problems and one of the hypothetical cases used in theinterviews posed for the student teachers.


Texbook problem 11

Juan wants to buy a car. He can choose between a petrol car and a diesel car. The firstcar costs 1,300,000 pesetas to buy, and for every 100 km it consume 8 litres of petrol.The second car costs 1,500,000 pesetas to buy, and for every 100 km it consumes 5 litresof diesel. The price of petrol is 90 pts/l and the price of diesel is 60 pts/l. How manykilometres do you have to drive before the second car become more economical? For thiscase, fill the following table with the total cost of each car plus combustibles, and draw therespective graphic representations.

Km 5000 10000 15000 20000 30000

Gasoline car

Kerosene car

Case 1

When introducing functions and graphs in a class of 14–15 years-old pupils, tasks wereused which consisted of drawing graphs based on a set of data contextualized in a situationand from equations. One day, when starting the class, the following graph was drawn onthe blackboard by the teacher, and the pupils were asked to find a situation to which itmight possibly correspond.

One pupil answered

“It may be the path of an excursion during which we had to climb up a hillside, then walkalong a flat stretch and then climb down a slope and finally go across another flat stretchbefore finishing”.

How could you respond to this pupil’s comment? What do you think may be the reasonsfor this comment?

Figure 1. Continued.

RESULTS

Subject Matter Knowledge for Teaching

Student teachers’ ways of knowing the subject matter. In order to describethe student teachers’ ways of knowing the concept of function we iden-tified the different aspects that they emphasised (see Figure 2) and theconnections between them, and the roles of different modes of representa-tion used.


Emphasised aspects Representative protocolsJuan Relation between elements from sets

Algebraic manipulationFunction shows the existence of a rule orlaw that may be symbolised with analgebraic expressionFunction as a succession of operations

“. . . mainly that a function leads me . . .

elements from one set into other. Then tobegin putting down a number of points andleading them on to others . . . (Juan,682–683, Case 1).“. . . instead of putting all these numbers andto each number we do that, well we put . . . ageneric number, an x, since x means all theelements in this set . . .” (Juan, 690–697,Case 1).

Rafael Function as a factoryRelationship of dependence andcausality between input/output

“In my opinion, for the case of a functionthere has always been the typical example ofa factory where . . . raw material goes in,undergoes a modification and another onecomes out . . . there is a relationship betweenone object and another . . .” (Rafael,133–144; General Interview).

Alberto Function as a model of a real situationStudy of the functional behaviour

“. . . in practice, when there is a real problemI’ll have to convert it into a mathematicalmodel and look at the behaviour of thefunction mathematically . . . this (TextbookA) is perhaps the “skeleton” for a realproblem. The real problem does not appear,but it is the “skeleton” of a possible realproblem” (Alberto, 1001–1004; TextbookProblem A analysis.)

Jose Relationship between variablesEmphasis on the mathematical languageand notationGraph displaying the functionalbehaviour

“They (the pupils) would see that . . . well, itis a correspondence of R into R, or which isthe abscissa axis and which is the ordinateaxis. The “y” can be represented . . . thefunctions can be represented graphically and. . . by means of tables of values” (Jose,880–890; Classification)“. . . one can draw conclusions from agraphical representation of a function asregards its behaviour” (Jose, 887–889;Classification)

Figure 2. Student teachers’ ways of knowing the concept of function.

All four student teachers in our study saw the concept of functionas a correspondence between sets, but they emphasised different aspectsof the mathematical concept for the purpose of teaching. Two of them,Juan and Rafael emphasised algebraic aspects and a view of functions asactions. For instance, for these students, the algebraic formulae y = 4x +3 was viewed as “four times a number plus three”. What they consideredimportant was for pupils to learn formal aspects of the algebraic manip-ulation. Another student teacher, Alberto, emphasised the meaning of therelationship between the variables in real-world situation problems. ForAlberto, the consideration of the function concept as a real situation modelwas an important aspect in teaching. Finally, the last student teacher,


José, considered from a balanced perspective both function as a modelin which graphs allow one to see how a phenomenon behaves and func-tion as an correspondence of R into R. The possibility of consideringsimultaneously the aspects of concept that they had emphasised, varioustypes of functions and specific properties of functions in the problemsturned out to be quite difficult for these student teachers. For instance,only Rafael used different characteristics jointly in his classification ofthe textbook problems: linear versus non-linear, relating to reality versusformally mathematical, geometric versus. non-geometric and continuousversus discrete domains. However, as is shown in the following quota-tion from the classification of the textbook problems, even Rafael found itdifficult to consider these characteristics simultaneously:

Although at first I thought that there could be many organisations [of the problems], thetruth is that, as I have been looking at the problems, . . . I thought that it was difficult tochange this [what I have written]. (Rafael, 1253–55; Classification).. . . the thread . . . is the linear or non linear behaviour of the functions as compared to thebehaviour . . . a geometric approach. (Rafael, 1623–64; Classification).

Finally, the four student teachers thought about the modes of representa-tion of functions (graphic, algebraic, and real situation) in a different way,linked to the different aspects of the concept that they had emphasised.Juan and Rafael considered the activities of reading and interpreting graphsand translation between different modes of representation as a complementof the activities with algebraic mode (for them, the key activities withfunctions were the ones where pupils had to handle algebraic symbols). Onthe other hand, Alberto and José extended the range of uses of the graphsas “instruments” for solving situations (that is an idea linked to the notionof function as a model). For example, in the classification of the prob-lems, Alberto mentioned three complementary perspectives of the graphs:forming part of the situation and transmitting information, an instrumentfor solving a problem, and representing relations.

Student teachers’ images. As mentioned in our theoretical framework,the student teachers’ images can mediate in the transformation processof concept of function from mathematical knowledge to its considerationas a teaching-learning object. Juan and Rafael viewed mathematics as aset of systematic procedures that may be used to solve problems and asknowledge organised in an accumulated way. For example, as they statedin the general interview:

So I have some information, which I have to use. Then it is clear that I have to go that way.Or rather, that I am supposed to do what the problem tells me . . . in a more systematic andreasoned way . . . (Juan, 531–535; General Interview).


In so far as it [mathematics] has been organised in an accumulated way, backing each other,perhaps it is a good way for dealing with the mathematical content that is being explained. . . (Rafael, 242–245; General Interview).

For Juan and Rafael, the logic that organises the mathematical knowledgeconstituted a referent for the introduction of teaching content. Further-more, these student teachers thought that mathematical content in teachingwas organised to solve the usual problems arising at the end of the chaptersin mathematical textbooks as application problems. That is why they said:“I am going to do what the problems tells me . . .”, and “So I have thatinformation, which I have to use”. On the other hand, José’s image aboutthe mathematics included two ideas: mathematics as an abstraction “math-ematics are somewhat unreal, in the sense that when you are dealing withwhat is abstract, you move away from reality” (José, 1211–1214; textbookproblem analysis), and mathematics as a set of useful instruments, “thepupil must realise that a mathematical instrument can help him/her to solvea real case” (José, 1206–1208; textbook problem analysis). This imageallowed him consider two approaches to the concept of function (i) puttingthe emphasis on the formal aspects in the algebraic context and (ii) usingthe graphs as an instrument for studying the functional behaviour of anyreal phenomenon. He considered these two approaches independently.

Alberto associated several terms with school mathematics: mathematicsas a game, that conveys the idea of something in which you participateand produces satisfaction; and mathematics as a unit composed by relatedparts. He said:

. . . the pupil recognises mathematics as a game, as a mechanism that is amusing and thatalways functions well (Alberto, 112–114; General Interview);. . . I like the fact that they [the pupils] see mathematics rather as a unit, and that everythingis related (Alberto, 940–942; Classification).

In addition, Alberto viewed the mathematics as a theory that helped toexplain real situations,

. . . the mathematical theory (linked to a problem that you are going to explain) probablycomes from real events . . . (Alberto, 350–352; General Interview).

For example, in his analysis of textbook problems, with respect to theTextbook Problem D,

Textbook Problem D

Draw the graphs for the following parabolas by locating the vertext first

y = x2 + 2; y = x2 – 8x + 16; y = x2 – 4x; y = –x2 – 5; y = x2 + 4x + 5; y = 2x2 + 2

Alberto stated:


. . . because it is probably the ‘skeleton’ of a real, practical problem, of a physical problem,of a problem of another type (Alberto, 1342; Textbook Problem D analysis).

Finally, student teachers’ images about learning were very similar. Forthem, learning is a question of knowing the information previouslyprovided by the teacher. From this perspective, the difficulties and mistakesof the pupils were associated with

. . . a lack of knowing the concept (José, 1301; Case 1);

. . . they just have a mistaken a priori idea of a function . . . They haven’t understood theconcept of function (Alberto, 1870–1873; Case 2).

This image about learning was related to the image of teaching asa process for transmitting information. All four student teachers viewedteaching as telling and learning as remembering. From this position, thesestudent teachers held the image that mathematics teachers had the knowl-edge and the responsibility for transmitting it, and their pupils wouldassimilate it without any difficulty, as is shown in the following assertionabout the most important thing for teaching mathematics:

The first thing is having something to teach. To know why it is important for me to teachthat. And then, to know how to transmit to the pupil all those things I think necessary forhim/her to learn, and to be able to transmit it so that he/she understands the idea adequately(Alberto, 59–62; general interview).

For these student teachers, the teacher is the one who decides what it isgood or bad and the one who decides on the correctness of the answers tothe problems.

Transformation of the Subject Matter

The hypothetical plans for teaching the concept of function that the studentteachers prepared and the reasons they provided were influenced by theirways of knowing the concept of function. During their transformation ofsubject matter for planning a hypothetical teaching sequence, three aspectsbecame apparent:

• their critical interpretation of the subject matter;• the representational repertoire they used; and• their adaptation of the subject matter.

Critical interpretation. In this section, we describe how the studentteachers’ ways of knowing, and their images, influenced the aspects of theconcept of function identified, the type of problems chosen and the orderin which the different aspects of the concept were presented.


Textbook Problem 1. Given the function f(x) = x2 – 4, calculate f(0), f(1), f(3).

For which values of x is f(x) = 0?

Textbook Problem 16. Given the function f(x) = 2x + 3.

a) Make a table and draw a graph.

b) Indicate the points cutting the graph in the x-axis.

Indication the point cuttign the graph in the y-axis.

c) For which values of x is f(x) = –9?

d) For which values of x is f(x) > 0?

Figure 3. Textbook problems 1 and 16.

Juan and Rafael considered it important for the pupils to know thisconcept as a relationship between variables, emphasising the meaning offunctions as a succession of operations. For example, Juan used the twoproblems shown in Figure 3 to start his hypothetical teaching sequence,which reflected the idea of a 1-1 correspondence and functions as asuccession of operations.

He justified the use of these problems in his teaching sequence saying:“They are useful problems to introduce what a function is, the points,how the images are calculated . . . they are easier because the function isseen directly”. For Juan, these were the problems that reflected the idea offunction most clearly.

Likewise, the starting point for the teaching sequence established byRafael is captured in the following statement: “what should be clearlyexplained is that a function is a univalent correspondence between twosets”. However, in order to achieve this objective, two characteristicsthat determined how he structured learning activities were evident in hisdecisions and curricular choices. The first characteristic was the iden-tification of the functional relationships as linear or non-linear (contentknowledge). The other characteristic was assigning a low difficulty level tothe problems related to linear functions. Rafael selected Textbook Problem11 for starting his hypothetical teaching sequence (see Figure 1). He said,

Initially, problems related to real-world situations, allowing the students’ acquisition ofintuitive concepts about what is going to be explained, are set out (Rafael, 1116–1118;Interview 3).

Rafael talked about the use of these problems in his teaching sequence:

To start off, with respect to what the most profound content is, I would go on explainingfunctions. Firstly, what a function is, then I would go on towards linear functions . . . .Within linear problems, the first things that I would use would be those that are related tothe construction of graphs from the “evaluation” of algebraic expressions that they alreadyknow about . . . (Rafael, 1080–1084; Interview 3).


What is relevant in this situation is that Juan and Rafael put theiremphasis on the relationship between variables in an algebraic context andconsidered this relationship important for pupils’ learning about functions.Although Rafael included problems describing real-world situations, theywere used exclusively to provide an intuitive view of the idea of relation-ship between variables and input-output pairs. These problems were notused to promote graph reading and interpretation activities, which couldshow the function concept as a real-world situation model.

The other student teachers, José and Alberto, used a dual approachbased on their ways of knowing the mathematical content and their images.José emphasised the idea of function as an abstract mathematical objectand its implications (the meanings of the ‘x’ and of the ‘y’ in the systemof Cartesian axes; calculation of images, domains, slopes. In the caseof parabola, the vertex, relationship between concavity and the sign ofthe coefficient of x2). In addition, and related to his images about math-ematics, José also indicated the need to show that mathematics can beuseful to solve a real situation, which was what justified the introductionof problems linked to real life for him. However, he made no mention ofthe possible relationship between the ‘formal’ aspect of the mathematicalcontent (with the emphasis placed on the algebraic mode) and the use ofgraphs in the interpretation of situations of the teaching sequence. It is as ifthere were two ‘worlds’ for the function concept that the teacher must usein the teaching sequence, but at no time are the relationships made betweenthe textbook problems for interpreting a situation using a graph and thosebased on the algebraic mode.

Like José, Alberto used a double criterion when interweaving “prob-lems of a technical nature with real problems”. For him, “technicalproblems” are problems without any real context that, in accord withhis ways of knowing the concept of function, can be the “skeleton” ofa possible real problem. The problems with a real context have a rolefor motivation, related to his image that considering the mathematics canhelp to explain real situations, “so that the individual can see the rela-tionship that exists between theoretical mathematics and reality” (Alberto,854–855; Classification). Alberto pointed out that this could occur aftermotivating the pupils through connecting the mathematical topic with reallife:

I would begin with this type of technical problems [but] always maintain the graphicrepresentation as the most important aspect, because with the latter you can display allthe study of increase, decrease, symmetry . . . (Alberto, 1102–1105; Classification).

That is to say, Alberto emphasised the study of the overall functional prop-erties through the visualisation allowed by the graphic representation of areal life situation.


Representational repertoire. Here we consider the alternative ways forrepresenting the concept of function that the student teachers used and thereasons provided.

The hypothetical teaching sequence that Juan and Rafael preparedhighlighted the relational view of the concept from the ‘point-image’perspective. These student teachers gave a priority role to algebraic repre-sentation. What it is important to emphasise here is not just the types ofproblems that the student teacher used, but also the objective that waspursued. Juan and Rafael considered the problems of reading and inter-preting graphs as application problems. On the other hand, for José andAlberto the graphs and the real situations played a key role in their teachingsequence. For example, José’s one objective was that pupils should becomefamiliar with a formal definition of the function concept as a univalentcorrespondence of R into R, and with some skills needed to perform thegraphs of the linear functions. So, he started his hypothetical teachingsequence with problems of graphic representations of linear functions.Likewise, Alberto complemented the use of the function as a model withthe algebraic mode. He gave greater significance to the graphs in theteaching sequence, taking into account the different uses of the graphs: forinstance, the graphs playing a complementary role to the algebraic expres-sions in order to study the overall properties of the functional relationshipsand as a means to understand a real situation.

Adaptation. Here we consider what characteristics of the pupils’ learningof the concept of function (the pupils’ prior knowledge and the mostfrequent mistakes) seem to be taken into account by student teachers.

Juan and Rafael considered a hypothetical level of difficulty of theproblems while José and Alberto took into account the idea of ‘motiva-tion’, but these ideas were always used in a general manner and withoutany more specification. Another idea that influenced the adaptation ofmathematical content to pupils was the meaning given by the studentteachers to the pupils’ ‘prior knowledge’. The four student teachers saw theprerequisite knowledge needed to solve the problems as the prior contentthat the teacher should have provided earlier. The textbook problems wereseen as an application of mathematical content that had been explained inadvance. The problems were seen as a means for the pupils to ‘practice’the procedures provided beforehand by the teacher. None of the studentteachers provided information regarding the pupils’ mathematical under-standing. The significance given to the pupils’ prior knowledge by thestudent teachers was compatible with their images about teaching andlearning as ‘telling and remembering’.


DISCUSSION AND IMPLICATIONS

This study is an attempt to identify the influence of student teachers’ waysof knowing the subject matter and images of mathematics, teaching andlearning on their hypothetical presentation of subject matter for teachingin the context of functions. Along the same lines as other studies (Wilson,1994; Even & Tirosh, 1995), our results point out the influence of subjectmatter knowledge for teaching regarding the way in which these studentteachers tried to represent the subject matter to the pupils. However, of thetwo dimensions considered in subject matter for teaching, ways of knowingthe subject matter and images of mathematics teaching and learning, theinfluence was different.

Ways of Knowing and the Use of Modes of Representation in Teaching

The student teachers in our study differed in their ways of knowing theconcept of function. In particular, they considered different aspects of theconcept and the modes of representation in a different way. This influ-enced their pedagogical reasoning. Juan and Rafael’s ways of knowing theconcept of function emphasised the operational aspect of functions and thealgebraic mode of representation. They considered the graphs as a comple-ment of the algebraic mode of representation. These student teachers gavea priority role to algebraic representation and computational activities overthe problems of reading and interpreting graphs. José and Alberto incor-porated the use of graphs as an ‘instrument’ for solving real situations.However, while José seemed to attribute two independent words (modeland correspondence) to the function concept in the organisation of subjectmatter for teaching, Alberto’s emphasis showed the complementary rolesof graphs and the algebraic mode. These emphases influenced these studentteachers’ organisation of content and the types of problems chosen in theteaching sequence. For all four student teachers, their ways of knowingthe concept of function as a teaching-learning object influenced what theyconsidered important for the learner and affected their use of the modes ofrepresentation in teaching, considered as teacher’s tools to obtain his/herteaching goals.

Even (1998) considered that flexibility in moving from one representa-tion to another is intertwined with flexibility in using different approachesto functions. In the same way, our findings indicate the importanceof analysing the relationship between different aspects of mathematicalcontent that student teachers emphasised and different use of modes ofrepresentation in their pedagogical reasoning. One implication of ourresults is that the mathematical content should become a context for the


contemplation of pedagogical issues, (e.g., discussion about different waysof representing specific mathematics topics, their strengths and limitationslinked to aspects of concept emphasised). In this context, student teacherscan discuss and evaluate the multiple representations linked to differentapproaches to a specific concept.

On the other hand, the limitations in student teachers’ knowledge ofpupils’ understanding seems logical in student teachers that have notattended teaching practice yet, and with an education mainly focused onmathematical content. Nevertheless, we think that one implication is thatthe assessment of different modes of representation should be coupled withdiscussion on specific knowledge of pupils’ understanding of particularconcepts during teacher education. Stacey et al. (2001) have shown theneed for teacher education to emphasise content knowledge that integ-rates different aspects of a topic and ‘pedagogical content knowledgethat includes a thorough understanding of common difficulties’ (p. 205).From the results of our study, a focus on the relationships among thedifferent aspects of the concept, modes of representation and knowledge ofstudents’ mathematical understanding of a particular topic has been shownas necessary.

Influence of Student Teachers’ Images

In relation to the influence of student teachers’ images on their pedagogicalreasoning the results obtained are not so clear. The four student teachersviewed mathematics as a set of systematic procedures for solving prob-lems. They held similar images about teaching and learning, such as the‘telling and remembering’, and ‘saw’ the ‘prior knowledge’ necessary toperform school problems as ‘prior content’ previously introduced. Twoof them made general references about motivation in their own learning tojustify some aspects on the lesson plans, using real-word problems as toolsto motivate pupils.

From these results, we can appreciate that when these student teachersthink about the content for teaching their initial decisions are closelyrelated to their ways of knowing the mathematical content. Possibly inthe student teaching experiences, where these student teachers reconstructtheir knowledge about the mathematical concepts as teaching-learningobjects (Jones & Vesilind, 1996), a better identification of the role ofimages is possible.

Implications for Teacher Education

If we recognise that the subject matter knowledge for teaching derivesfrom a variety of sources; our results point to the fact that these do


not all have the same influence when the student teachers think aboutthe subject matter with the purpose of teaching (pedagogical reasoning).Therefore, during the teacher education program the student teachersshould approach pedagogical content knowledge in more than one way.In particular, learning situations on the specific cognition of mathema-tical topics should be introduced in the method courses of mathematicseducation. Furthermore, mathematics teacher education should consideropportunities for the student teachers to design learning tasks, analyse themathematical field of these tasks and consider the curricular learning goalstheir engagement might support. All of this could be used in seeing themathematical content in practice, generating the possibility of analysis andreflection on the influence the different ways of knowing and images haveon decisions in the practice of teaching.

We believe that the use of mathematical knowledge in teaching isa concern of teacher education. Knowing more about the relationshipbetween subject matter knowledge for teaching and pedagogical reasoningprovides us with information to design activities in teacher educationprogrammes. Cases, critical incidents and interviews concentrating on thecognition of mathematical topics (Barnett, 1998; Markovits & Even, 1999)can allow us to pose questions of learning and to develop analyses andreflections about mathematics teaching and learning. If we consider thatthe use of knowledge in teaching (e.g., to examine and sequence differentmathematics problems to design a plan for teaching a particular concept)is different from knowing the mathematical content, we should take newdecisions in relation to teacher education. These decisions must allow usto prepare teachers who not only know content but make use of it to helpstudents to learn.

ACKNOWLEDGEMENT

We would like to thank to Mercedes García Blanco and anonymousreviewers for their helpful comments on earlier versions of this paper.Thanks also to Thomas Cooney and Barbara Jaworski for their suggestionsand interest in improving our English text.

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Departamento de Didáctica de las MatemáticasFacultad de Ciencias de la EducaciónUniversidad de SevillaAvenida Ciudad Jardín, 22, 4105 SevillaSpainE-mail: [email protected] (Victoria Sánchez)E-mail: [email protected] (Salvador Llinares)

WIM VAN DOOREN, LIEVEN VERSCHAFFEL and PATRICK ONGHENA

PRE-SERVICE TEACHERS’ PREFERRED STRATEGIES FORSOLVING ARITHMETIC AND ALGEBRA WORD PROBLEMS

ABSTRACT. In this study we investigate the arithmetic and algebra word problem-solving skills and strategies of future primary and secondary school teachers in Flanders(Belgium). Moreover, we describe the evolution of these skills and strategies from thebeginning to the end of their teacher education. The results show that future secondaryschool mathematics teachers preferred the use of algebra, even when an arithmetic solu-tion was more straightforward. The solutions of future primary school teachers were morediverse: one subgroup tended to apply exclusively arithmetic methods (which led to failureson the most difficult word problems), whereas another subgroup was more adaptive in itsstrategy choices. Finally, student teachers evolved in their problem-solving skills duringtheir teacher education, but not in their strategy preferences. The research findings indicatethat, in the education of pre-service primary and secondary school teachers, there is a needfor an explicit treatment of pupils’ transition from arithmetical to algebraic thinking.

INTRODUCTION

When pupils pass from primary to secondary school at the age of 12,the acquisition of algebraic ways of mathematical reasoning and problemsolving is one of the most important mathematical learning tasks (Filloy& Sutherland, 1996; Schmidt & Bednarz, 1997). In primary school, thereis an exploration of basic mathematical properties (such as equality,commutativity and associativity), the generalization of which lays thefoundations for algebraic reasoning (Carpenter & Levy, 2000; Carraher,Brizuela & Schliemann, 2000; Slavitt, 1999). Nevertheless, elementaryschool children’s solution strategies can be typified as “arithmetical”, in thesense that these strategies consist of consecutively performing operationson known numbers, while their meaning remains invariably connected (orcan be quite easily connected) to the original problem context (Kieran,1992; Linchevski & Herscovics, 1996). When entering secondary school,algebraic ways of thinking and problem solving are introduced: symbolsrepresenting unknown quantities are used to write equations, and operatingon these equations leads to the identification of unknowns and ultimately tothe answer of the problem (Reed, 1999). Typical for the algebraic approachto problem solving is that a significant part of the solution process consistsof manipulating symbolic expressions containing unknown values, while


28 WIM VAN DOOREN ET AL.

the concrete meaning of these manipulations in relation to the problemcontext is temporarily suspended (van Amerom, 2001).

For many pupils the introduction of algebraic problem solving, andof formal algebraic systems in general, creates a serious barrier (Kieran,1992). In the past decades research has been carried out on the difficultiespupils experience when learning algebra, the causes of these difficulties,and the ways in which they can be prevented and remedied through instruc-tion (e.g., Filloy & Sutherland, 1996; Herscovics & Linchevski, 1994;Kieran, 1992; Lee & Wheeler, 1989; Sfard & Linchevski, 1994). In thisrespect, several mathematics educators and researchers have suggestedthat, to a certain extent, the elementary mathematics curriculum shouldbe “algebrafied” (e.g., Ainley, 1999; Davis, 1985; Discussion Documentfor the Twelfth ICMI Study, 2000; Kaput, 1995; Swafford & Langrall,2000; Vergnaud, 1988). It is argued that early in mathematics education,the arithmetic activities can and should be infused gradually with algebraicmeaning, in order to bring out their inherent algebraic character. Withintheir arithmetical way of thinking and problem solving, pupils can developpre-algebraic skills (e.g., symbolizing, generalizing, reasoning about rela-tionships, representing unknowns and even operating on them), to end withformalizing their skills to a higher level of algebraic conception (e.g., equa-tion solving, functions). These authors stress that algebraic understandingdevelops slowly over many years, but one should not await adolescence totrigger its evolution.

The focus of this article is not on pupils and their algebra learning,but on mathematics teachers who have to stimulate and support thisalgebra learning process. We take this focus because there are manyindications that teachers’ content-specific knowledge, beliefs and attitudesinfluence pupils’ learning outcomes (e.g., Calderhead, 1996; Carpenter,Fennema, Peterson & Carey, 1988; Fennema & Loef, 1992; Shulman,1986; Thompson, 1992), and because several authors (e.g., Kieran, 1992;Nathan & Koedinger, 2000; Schmidt & Bednarz, 1997; van Amerom,2001) stress teachers’ crucial role in helping pupils to make the transitionfrom arithmetic to algebra.

Another focus of this article is that we only consider the issue ofalgebraic problem solving. Notwithstanding the many other interpretationsof what (school) algebra is or should be (e.g., generalizing, mathema-tical modeling, describing and analyzing functions), problem solving byconstructing and solving equations is a core activity in every algebracurriculum (van Amerom, 2001), and it requires a variety of skills,such as the ability to recognize relationships between quantities, expressthese relationships into one or more algebraic equations using symbols

PRE-SERVICE TEACHERS’ PREFERRED STRATEGIES 29

to represent unknown quantities, manipulate equations and operate onunknowns to identify their value, and interpret the outcomes to fit withinthe original problem (Schmidt, 1994, 1996).

The study described here is part of a larger investigation onstudent teachers’ content-specific knowledge, skills, beliefs and attitudesconcerning arithmetic and algebra, and on how these different character-istics of student teachers affect their didactical behaviour. In this article weonly report the first part of the larger investigation which aimed at iden-tifying student teachers’ own skills and strategies for solving arithmeticand algebra word problems. The second part of the investigation, whereinthe relation between future teachers’ content-specific knowledge, skills,beliefs and attitudes and their didactical behaviour is examined, is reportedelsewhere (Van Dooren, Verschaffel & Onghena, 2002).

BACKGROUND OF THE STUDY

The introduction of pupils into algebra is a long and complex process.This process makes an important demand on both primary school andsecondary school teachers’ professional knowledge and skills, althoughwe agree with Schmidt (1994, 1996; Schmidt & Bednarz, 1997) that thenature of this challenge is quite different for both groups of teachers.

Primary school teachers can be encouraged to develop in pupils arich knowledge base consisting of several basic mathematical concepts(such as part-whole relations, the symmetrical meaning of the equal-sign(=), commutativity and associativity principles) as well as several generalarithmetical skills (such as seeing relationships between two or moregiven quantities, choosing the correct operation(s) to identify an unknownvalue in a given context and executing arithmetical operations fluentlyand correctly) (Schmidt, 1994). These arithmetical concepts and skills arealso necessary foundations and starting points for algebraic thinking andproblem solving. To fulfill this instructional task, at least a minimal under-standing and mastery of algebraic thinking in primary school teachersseems necessary. This requirement becomes more pertinent if we takeinto account the claim of some mathematics educators and researchers,described earlier, to reconceptualize the mathematics in elementary gradesto provide more opportunities for (pre)algebraic thinking.

Secondary school teachers, first of all, can benefit from a good under-standing of the mathematical histories of pupils entering secondary educa-tion. Throughout mathematics lessons in primary school these pupils havedeveloped concepts, techniques and habits that are considered as “typicallyarithmetical”, and, consequently, as not always completely compatible


with or supportive of algebraic ways of thinking (Schmidt, 1994; Schmidt& Bednarz, 1997). For example, the equal-sign in arithmetic tasks atthe primary school level often has (merely) the meaning of a “resultssign” (i.e., a signal that indicates where the outcome of the arithmeticoperation(s) has to appear, leading to mathematically incorrect numbersentences such as the following: 54 + 14 = 54 + 10 = 64 + 4 = 68),while in algebraic equations the meaning of this sign is fully symmetricand transitive. Another example is that many primary school pupils applya “guess-and-check” strategy for solving number sentences and word prob-lems of the type “? + 23 = 41” or “? – 7 = 15”. In this strategy, thepupil guesses the answer, checks the correctness of the guess, and repeatsthis process of guessing and checking until (s)he finally arrives at thecorrect answer. When pupils are confronted with new algebraic ways ofthinking at the start of secondary education, they may apply these morefamiliar concepts, techniques, and habits spontaneously (Schmidt, 1994).Besides having good insights in both arithmetical and algebraic ways ofthinking, secondary school teachers also face the task of demonstrating totheir pupils the validity and the pertinence or necessity of algebraic waysof thinking as a powerful mathematical tool (Discussion Document forthe Twelfth ICMI Study, 2000; Schmidt & Bednarz, 1997, van Amerom,2001). Finally, they can be encouraged to strive to develop in their pupilsa disposition to apply arithmetical and algebraic strategies in a flexibleway, taking into account the characteristics of the problem to be solved(Schmidt, 1994, 1996).

To carry out these complex instructional tasks properly, it is importantthat primary and secondary school teachers understand, master and appre-ciate both arithmetical and algebraic problem-solving strategies them-selves, and have the capacity and the willingness to switch flexibly fromone strategy to another whenever necessary. Indeed, considering the indic-ations in the literature of the impact of teachers’ subject-matter knowl-edge and pedagogical content knowledge on pupils’ learning processes(see Calderhead, 1996; Carpenter et al., 1988; Fennema & Loef, 1992;Shulman, 1986; Thompson, 1992), we should encourage future primaryand secondary school teachers to master, at least to some extent, andappreciate algebraic as well as arithmetical problem-solving skills.

However, there are good reasons to suspect that a substantial numberof primary and secondary school teachers may need additional support toreach that goal. Schmidt (1994) has shown that many Canadian studentsat the beginning of their teacher education cannot switch spontaneouslyand flexibly between arithmetical and algebraic strategies. Nearly allstudent teachers who wanted to become remedial teachers (for primary and


secondary education) and about half of the future primary school teacherswere unable to apply algebraic strategies properly, or were reluctant to usethem. Consequently, they experienced serious difficulties when they wereconfronted with more complex mathematical problems. Many of thesebeginning student teachers perceived algebra as a difficult and obscuresystem based on arbitrary rules (Schmidt, 1994, 1996; Schmidt & Bednarz,1997). This finding contrasts with the preference of students who wanted tobecome secondary-school (mathematics) teachers. Most of these studentsused algebraic procedures exclusively, even for solving problems whichcould be easily solved arithmetically. In this group of beginning studentteachers, algebra was often seen as the only real “mathematical” problem-solving method, whereas arithmetic was considered as “inferior”, or asa form of “improvising” (Schmidt & Bednarz, 1997). Taking into accountthese results, Schmidt raised serious concerns about these student teachers’readiness for the complex instructional tasks with which they would befaced soon as professionals.

In the present study, which is partly a replication of Schmidt’sresearch, we investigated whether beginning student teachers in Flandershave similar problem-solving behaviour to beginning Canadian studentteachers. There is, however, a difference in the sense that the Flemishteacher education system has no special pre-service education programfor remedial teachers, so that only two instead of three groups of futureteachers were distinguished in our study: primary and secondary schoolstudent teachers. Besides the replication of Schmidt’s study, the presentresearch involved an important elaboration, which is, in our view, neces-sary to get an appropriate view of their preparedness for teaching mathe-matics. Schmidt’s (1994) study involved only pre-service teachers at thestart of their teacher education. This jeopardizes seriously the validity ofher (serious) concerns about student teachers’ readiness for the job at thetime they have finished their teacher education. In the present study, wealso included pre-service teachers near the end of their teacher educa-tion. This allowed us not only to derive more warranted conclusions aboutteachers’ readiness for their job at the start of their professional career, italso enabled documentation of the differences in future teachers’ strategiesfor, and attitudes towards, arithmetic and algebraic forms of thinking at thebeginning and at the end of their teacher education.


METHOD

Participants

The participants were 97 pre-service teachers of one teacher educationinstitute in Flanders. These 97 student teachers belonged to four differentgroups, depending on whether they were at the beginning of their firstor at the end of their last year of teacher education and on whether theywere intending to become primary or lower-secondary school teachers. Inthe group of future primary school teachers, there were 26 participantsin the first and 36 in the third year. In the group of future secondaryschool teachers, these groups contained 19 and 16 participants, respec-tively. While a study of this kind could benefit from a longitudinal designin which the same student teachers would be compared at the beginningand at the end of their education, we opted for a cross-sectional designpermitting data collection on one single moment and excluding unwantedeffects of repeated measurement.

In Flanders, the education of primary and lower-secondary schoolteachers takes place in departments of teacher education, which belongto non-university institutes for higher education, and which provide athree-year professional education to students, most of whom have justfinished secondary education. Although both types of education take placewithin the same non-university institutes, there is a difference in theaims, contents, structure and culture of the education of future primaryand lower-secondary school teachers (De Rijdt, 1999). Whereas primaryschool student teachers are prepared to teach all courses (including mathe-matics) to primary school children (aged 6 to 11), future lower-secondary-school teachers specialize in only two or three curricular domains (e.g.,mathematics and physics) to be taught to 12–15-year-olds. The threeyears of education for both primary and secondary school student teachersconsist of a mixture of general courses in psychology, educational andinstructional sciences, and content-specific courses wherein content andpedagogical content knowledge and skills are taught in an integratedmanner. These latter courses are heavily, if not exclusively, focused onthe level of the curriculum that these student teachers will have to addressin their future teaching. In addition, from the beginning of their teachereducation, the primary and secondary student teachers regularly visit theirrespective schools with specific observation and/or teaching tasks, andcomplete several short and longer terms of placement in the school levelfor which they are prepared.


Algebra problem Arithmetic problem

372 people are working in a large com-pany. There are 4 times as many labourersas clerks, and 18 clerks more than man-agers. How many labourers, clerks andmanagers are there in the company?

A primary school with 345 pupils has asports day. The pupils can choose betweenin-line skating, swimming and a bicycleride. Twice as many pupils choose in-line skating as bicycling, and there are30 fewer pupils who chose swimmingthan in-line skating. 120 pupils want togo swimming. How many chose in-lineskating and bicycling?

Figure 1. Schematization of an arithmetic and an algebra word problem from the wordproblem test.

Instrument

A paper-and-pencil test was collectively administered to the 97 parti-cipants. The test consisted of six arithmetic and six algebra word problemsoffered in randomized order to be solved individually by the parti-cipants within one hour. It was similar to Schmidt’s (1994) test, butwe used more word problems and we also divided them more equallyover the three different semantic categories distinguished in her rationaltask analysis, namely “unequal partition”, “transformation”, and “relationbetween quantities” (Schmidt, 1994). Figure 1 presents an example of anarithmetic and an algebra problem from the test, as well as the schematiza-tion of the structure of these problems that explains why the first item wasconsidered as an arithmetic problem and the second item as an algebraicone. Actually, these problems were presented in Dutch, the native languageof the participants. The English translations presented here are as literal aspossible.

The schematization of the problems in Figure 1 was based on theanalysis method of Bednarz and Janvier (1996). Like many others, theseauthors acknowledge that it is impossible to classify a word problemunequivocally as being arithmetical or algebraic. Nevertheless, they arguethat it is important, both from a theoretical and a practical point of view, to


distinguish between arithmetical and algebraic problems based on the kindof solution strategy that is most likely elicited by the problem. For instance,when we look at the arithmetic problem in Figure 1, it is clear that it can besolved by “undoing” the operations, one after the other. One can add 30 to120 to obtain 150, and then divide this intermediate result of 150 by 2, toobtain 75. So, this problem can be solved by doing one or more arithmeticoperations with the given numbers and with the intermediate outcomes ofthese operations (Bednarz & Janvier, 1996). For the algebraic problem inthis figure, such a straightforward arithmetic solution is not possible. Thesemantic structure of the problem is identical, but because there is onemore unknown value in a critical position, the process of making calcu-lations with the known values to generate new values that are needed forthe solution is no longer straightforward. A solution strategy in which allknown and unknown values are captured in one static representation – forinstance, an algebraic equation – is most efficient here.

While it is clear that an arithmetical problem can be solved algebra-ically and vice versa, the mathematical classification (by means of thenumber of known and unknown values and their relative position in thesemantic structure) presented here gives an indication of the most efficientsolution strategy to tackle a particular word problem. For someone whomasters both types of strategies, some problems are more efficiently solvedarithmetically, while for other problems an algebraic strategy is moreappropriate. Consequently, this classification also indicates which strategyis most likely (but not conclusively) elicited by that problem by an expertproblem solver. The capacity and willingness to adjust one’s approachto the characteristics of the task (environment) – or stated differently: tobe adaptive or flexible in one’s strategy choices – can be considered as abasic component of a genuine mathematical disposition (De Corte, Greer& Verschaffel, 1996; Koedinger & Tabacheck, 1994).

After having generated the 12 word problems in this way, they wereplaced in a randomized order in the test. Before starting, participants weretold they had one hour to solve the problems. No other instructions weregiven as to how the problems had to be solved. Furthermore, they wereinstructed to write down not only the answer, but also the underlyingreasoning and solution process (including all computations).

Participants’ solutions were scored in two ways. First, every answerwas scored as correct or incorrect. Second, all solutions (including theincorrect and incomplete ones) were scored in terms of the kind of strategyused. (When the answer sheet did not contain any response, the solutionwas categorized as “no answer”.) For this scoring we used an a prioriclassification scheme based on the work of several other researchers (Hall,


Figure 2. Classification used for the solutions on the word problem test, with exemplarysolutions for an algebraic and an arithmetical word problem.

Kibler, Wenger & Truxaw, 1989; Filloy & Sutherland, 1996; Kieran, 1992;Linchevski & Herscovics, 1996; Reed, 1999; Schmidt, 1994, 1996; Sfard,1991; Stacey & McGregor, 2000; Wolters, 1976). In Figure 2 we presentand illustrate the different categories from this classification scheme.

According to our classification scheme, both algebra and arithmeticword problems can be solved in three different ways: one algebraic and twoarithmetical. To be scored as an algebraic solution, the following criteriahad to be fulfilled: (a) the protocol contains at least one equation whereinknown and unknown values are related to each other and (b) the answeris found through transformations of the equation(s) and operating on the


unknowns (Filloy & Sutherland, 1996; Kieran, 1992; Reed; 1999; Wolters,1976). The arithmetical strategy called “manipulating the structure of theword problem” (hereafter called “manipulating the structure”) consists ofaltering or restructuring the problem so that it becomes solvable arith-metically, without algebra. As illustrated in Figure 2, this strategy can beused to solve algebra problems quite efficiently. For arithmetic problems,however, this method is inefficient and cumbersome, since the problemsolver does not take into account all known values and, therefore, makesunnecessary detours (Schmidt & Bednarz, 1997). The third strategy isalso arithmetical, and is named differently when applied to an algebraor to an arithmetic word problem. When applied to solve an arithmeticword problem, it is called “generating numbers”. In an arithmetic problemthe “starting number” is already known, and the missing value(s) can bedirectly generated from it by performing the correct arithmetic opera-tion(s). When applied to an algebra problem, this third kind of strategyis called “guess-and-check”. In fact, the method works completely parallelwith “generating numbers”, but since no “starting number” is available, thesolvers have to guess its value, to check the correctness of the guess, and torepeat this process of guessing and checking until they finally arrive at thecorrect value of the unknown (Filloy & Sutherland, 1996; Schmidt, 1994,1996). Within this strategy, one can either make subsequent “random”guesses, or apply a “try-and-improve” approach wherein the outcome ofprevious guesses are actively and reflectively used to make better guessesin next trials.

RESEARCH QUESTIONS AND HYPOTHESES

The first set of research questions and hypotheses concerns the solutionstrategies used by pre-service teachers to solve the problems on the paper-and-pencil test. We expected to find different solution strategy patternsfor the arithmetic and the algebra word problems. Since we generated theword problems using the analysis method of Bednarz and Janvier (1996),we predicted that the problems we had typified as algebraic would elicitmore algebraic solutions than the so-called arithmetic problems. Further,as in Schmidt’s (1994) study, we hypothesized different patterns of solu-tion strategies for the primary school and the secondary school studentteachers. More particularly, we predicted that the future secondary schoolteachers would solve the algebra word problems as well as the arithmeticproblems mainly with algebraic strategies. Furthermore, we predicted thatprimary school teachers would approach the arithmetic word problemsarithmetically, whereas for the algebra word problems we expected a


mixture of arithmetic and algebraic approaches. The reason for that latterprediction was that, in line with Schmidt’s (1994) findings, we expectedto find two subgroups of elementary-school student teachers, one withoutaffinity with algebra, and the other with a disposition to apply arithmeticand algebraic strategies adaptively.

The second set of research questions and hypotheses concerns thestudent teachers’ performance. We predicted a lower accuracy rate forthe algebra than for the arithmetic problems. Further, we anticipatedthat the future secondary school teachers would solve more word prob-lems correctly than future primary school teachers. More particularly, weexpected that the secondary school student teachers’ mastery of algebraicmethods would permit them to solve both the arithmetic and the algebraproblems quite successfully. For the primary school student teachers, wepredicted good performance on the (easy) arithmetic problems, but weakerperformance on the (difficult) algebra problems. Indeed, we expected thatthere would be a considerable number of primary school student teacherswho would not be able to solve the algebra problems because they wouldnot possess the required algebraic knowledge and because they would beunable to activate and/or apply one of the alternative arithmetic methods.

A third set of research questions and hypotheses concerns the differencebetween student teachers at the beginning and at the end of their teachereducation. We hypothesized a difference in the problem-solving strategiesand skills between the pre-service teachers who were at the beginning oftheir teacher education and those who were finishing it, for the followingreason. As in many other countries, a major characteristic of the courses inmathematics education in the departments of teacher education in Flandersis that pre-service teachers are stimulated to solve mathematical problemsin the way they will later have to teach it to their pupils (Van de Plas, 1995).In most instances in the mathematics education courses of pre-serviceprimary school teachers in Flanders, little or no attention is paid to algebra,whereas arithmetic solutions, and particularly the “manipulating the struc-ture” method, receive considerable instructional attention. In contrast, inthe mathematics education program for secondary school teachers, algebraand algebraic solution methods receive pivotal, if not exclusive, attention.Therefore, we predicted that the above-mentioned expected differencesin strategy use between primary and secondary school student teachers,would increase during the education. Furthermore, we hypothesized thatthese learning experiences during teacher education would foster betterproblem solving skills in both groups of pre-service teachers. For thisreason, we predicted that both groups of third-year students would performbetter on the test than the first-year students, and that this better perfor-


mance – especially on the more complex (algebra) word problems – wouldbe due mainly to an improvement in the mastery of the kind of strategythat was addressed prominently in their teacher education, i.e., “manipu-lating the structure” and “algebra” for, respectively, primary and secondaryschool student teachers.

RESULTS

In the first two sub-sections of the results section we present the findingsdealing, respectively, with the solution strategy and the performance data.The results dealing with the differences between first-year and third-yearstudent teachers are integrated in these two sub-sections. In a third sub-section, the relationship between strategy use and success rate is analyzed.

Solution Strategies

Every solution protocol was scored in two ways. First, it was categorizedas either correct or incorrect. Second, we determined the strategy thatwas used to obtain the solution to the problem, using the classificationscheme described earlier. In this sub-section, we describe the analysisof the latter characteristic. In order to investigate whether the differentgroups used different strategies to solve the two types of word prob-lems, a multivariate 2 × 2 × 2 repeated measures analysis of variancewas performed, with three independent variables: one within-subject vari-able, namely type of problem (arithmetic vs. algebra problems), and twobetween-subject variables, namely study branch (primary vs. secondaryschool student teachers) and year of course (first vs. third-year studentteachers). As dependent variable, we opted for the number of algebraicsolutions used by the participants. The use of this variable for the statisticaltests is in fact a reduction of the original data, since we distinguishedthree classes of solution strategies (one algebraic and two arithmeticalstrategies). However, since the main interest of the study is on the shiftbetween algebra and arithmetic (and vice versa), the number of algebraicstrategies used by a participant is a valid indicator for his or her strategyuse in general. The results of this ANOVA are summarized in Table I.

First, we found the expected main effect of the type of problem. Ashypothesized, the items that we considered as algebra problems elicitedmore algebraic strategies (3.65 on a total of 6) than the items that wereconsidered as arithmetic ones (1.75 on a total of 6). More importantly,however, there was a main effect of study branch, showing that futuresecondary school teachers used much more algebraic strategies (9.29 on


TABLE I

ANOVA-table for the analysis of the solution strategies (performed on the number ofalgebraic solutions)

Effect Testing statistic MSE p

Type of problem F(1, 93) = 106.77 159.95 < 0.00015

Study branch F(1, 93) = 11.33 31.34 = 0.0011

Year of course F(1, 93) = 1.30 1.84 = 0.2564

Type of problem × Study branch F(1, 93) = 0.00 0.00 = 0.9703

Type of problem × Year of course F(1, 93) = 0.07 0.11 = 0.7862

Year of course × Study branch F(1, 93) = 0.05 0.07 = 0.8276

Type of problem × Study branch × Year of course F(1, 93) = 0.01 0.01 = 0.9398

Figure 3. Percentage of use of the different solution strategies for the six arithmetic wordproblems by the primary and secondary school student teachers.

a total of 12) than future primary school teachers (3.21 on a total of12). Since no Study Branch × Type of Problem interaction effect wasfound, this difference between the future primary and secondary schoolstudent teachers was observed for both kinds of problems (arithmetic andalgebraic).

Figure 3 gives a more detailed overview of the solution strategies of thefuture primary and secondary school teachers for the arithmetic problems.The percentage of unanswered problems, which could not be classified interms of strategy use, is also indicated in these figures.

As expected, the solution strategies of future primary and secondaryschool teachers for the arithmetic problems were very different. Amongthe secondary school student teachers, using algebra was the most common


Figure 4. Percentage of use of the different solution strategies for the six algebra wordproblems by the primary and secondary school student teachers.

strategy for solving the arithmetic word problems: almost two thirds ofthe solutions could be characterized as algebraic, whereas only about onethird belonged to the arithmetic category “generating numbers”. Primaryschool student teachers, on the other hand, applied the arithmetic strategy“generating numbers” in almost 80% of the cases whereas they workedalgebraically in about 10% of the cases only. The other arithmetic strategy,“manipulating the structure”, was used rarely in both groups. This was notsurprising, because, as explained earlier, this is an awkward strategy forarithmetic word problems. Finally, arithmetic word problems were seldomleft unanswered in both groups, which suggests that most participantsconsidered these problems as easier.

Figure 4 shows the percentages of use of the different strategies for thealgebra word problems.

As shown in this figure, the solution strategies of the two groups ofparticipants differed again substantially. Future secondary school teacherssolved nearly all algebra word problems algebraically, whereas primaryschool student teachers’ solutions were more equally divided among thefour possible categories. About 40% of the protocols were categorized asalgebraic and the two arithmetic categories contained each about 20% ofthe solutions. In about 20% of the cases, the algebra problems remainedunanswered, which can be considered as an indication that these algebraproblems were generally experienced as difficult by some primary schoolstudent teachers.

Before discussing the results with respect to the effect of the studentteachers’ year of course (first-year vs. third-year), we report the resultsof a more fine-grained analysis of the primary school student teachers’


solution strategies for the algebra problems, aimed at figuring out whetherthe mixed approaches of the future primary school teachers are due to theexistence of two subgroups, namely a subgroup without any affinity toalgebraic solutions and a subgroup who switched adaptively between arith-metic and algebraic solutions. For each individual primary school studentteacher, we calculated the number of algebraic solutions for the six algebraproblems.

The finding that about half of the 62 primary school student teacherseither used algebra for none of the algebra problems (19 participants) orfor all six of them (12 participants), as well as the overall U-shaped formof the frequency distribution, can be considered as evidence in favor ofthe claim that within the group of future primary school teachers, twosubgroups could be distinguished: one subgroup that never applied algebra,and another subgroup that tried to solve most of the algebra problemsalgebraically.

Finally, returning to the ANOVA, we see that it revealed no main effectfor the variable year of course, indicating that, taken as a whole, thesolution strategies of the third-year student teachers were not statisticallydifferent from those of the first-year student teachers. More importantly,we did not find a significant Year of Course × Study Branch interactioneffect, implying that our hypothesis that three years of teacher educa-tion would strengthen the initial differences in strategy use between thefuture primary and secondary school teachers, was rejected. In otherwords, contrary to our expectation, there was no significant decrease inthe number of algebraic strategies from the first to the third year amongthe future primary school teachers, and in the group of future secondaryschool teachers there was no significant increase in the number of algeb-raic strategies from the first to the third year. The implications for teachereducation are discussed below.

Performance

As for the solution strategy data, a multivariate 2 × 2 × 2 repeatedmeasures analysis of variance was performed with the same independentvariables as in the previous analysis and with “correctness of answer” asthe dependent variable. The outcomes of this ANOVA are presented inTable II.

First, the ANOVA revealed a main type of problem effect. As expected,the average score on the arithmetic problems (5.19 out of a total of six)was considerably higher than on the algebra problems (3.69 out of a totalof six). A second main effect was found for the variable study branch. Theaverage score of the future secondary school teachers on the whole test was


TABLE II

ANOVA-table for the analysis of the performances (performed on the number of correctsolutions)

Effect Testing statistic MSE p

Type of problem F(1, 93) = 66.73 80.01 < 0.00015

Study branch F(1, 93) = 11.33 31.34 = 0.0011

Year of course F(1, 93) = 3.62 10.03 = 0.0601

Type of problem × Study branch F(1, 93) = 15.44 18.51 = 0.0002

Type of problem × Year of course F(1, 93) = 8.80 10.56 = 0.0038

Year of course × Study branch F(1, 93) = 0.34 0.95 = 0.5597

Type of problem × Study branch × Year of course F(1, 93) = 0.02 0.02 =0.8926

9.89 (on a maximum of 12), compared to an average of 8.31 for the futureprimary school teachers. This was a relatively small difference, but it wasin line with our expectations. The significant interaction effect betweentype of problem and study branch, indicates that as expected, the betterperformance of the secondary school student teachers was due to theirbetter performance on the more difficult algebra problems. For the arith-metic word problems the mean number of correct responses for the futureprimary and secondary school student teachers was, respectively, 5.11 and5.31; for the algebra word problems, on the other hand, these numberswere, respectively, 3.19 and 4.57. Thus, the relatively small differencein the scores of both groups on the whole test (consisting of about 1.5questions on a total of 12) were almost completely situated in differentscores on the algebra problems.

Third, since the ANOVA showed no significant main effect of yearof course, the overall performance of the first-year and the third-yearstudent teachers was largely the same. However, there was a significantYear of Course × Type of Problem interaction effect. The performanceon the arithmetic problems was largely the same for the first-year and thethird-year student teachers (5.22 and 5.15 respectively), which is possiblydue to a ceiling effect. But for the algebra word problems the third-yearstudents (with a mean score of 4.04) performed considerably better thanfirst-year students (3.29). Thus, the results were in line with our hypoth-esis that three years of teacher education would have a beneficial influenceon the student teachers’ problem-solving skills, especially with respect tothe more difficult algebra word problems. This beneficial influence wasfound for future primary school teachers as well as future secondary school


teachers, since the three-way Year of Course × Type of Problem × StudyBranch interaction effect was not significant.

Relation between Strategies and Performances

To perform a more fine-grained analysis of the solution patterns, we separ-ately calculated for each of the four groups of student teachers the ratiobetween the number of times a strategy was used successfully and thenumber of times this strategy was used in total. In general, the successrates for the arithmetic word problems were higher than for the algebraproblems. This echoes the difference in performance for these two problemtypes, reported earlier. More importantly, the overall high success rates forthe arithmetic problems indicate that whatever kind of strategy was usedby any group of student teachers, this resulted nearly always in a correctanswer. For the algebra word problems, we found greater accuracy differ-ences between the groups. A closer look at the success ratios provides adetailed explanation of these observed differences. The lower performanceof the first group was partly due to the higher number of unanswered prob-lems, but also to their lower success rate with “manipulating the structure”(especially first-year students) and “algebra”. “Guess-and-check” was, asexpected, also mainly used by primary school teachers, and it appeared asa rather successful strategy. However, one has to take into account that thismethod is rather cumbersome and time-consuming. Additional analyses(see Van Dooren, Verschaffel & Onghena, 2001) moreover have shownthat the rigorous arithmetic-users were responsible for the high number ofunanswered algebra problems.

Not only differences in performance between pre-service primaryand secondary school teachers are explained by the success rates forthe different solution strategies, but also the difference in performancebetween first and third-year students can be accounted for in this way.The results show that the better performance of third-year secondaryschool student teachers on algebra problems was the result of an improvedcapacity to use an algebraic strategy correctly, and not, for instance, theresult of a shift in the kind of strategy used. This increased efficacy in theuse of the algebraic method was possibly due to the fact that throughouttheir three-year education, these student teachers had ample opportunity toimprove their algebraic problem-solving skills. There was also a remark-able difference in the success of future primary school teachers using“manipulating the structure”, whereas the success rate for the two otherstrategies did not differ so much between the first- and third-year group.Again, as with the secondary school teachers, the better performance ofthe third-year primary school student teachers compared to the first-year


student teachers, was not due to a shift in the kind of the strategies used, butrather to an increase in the efficacy with which they applied “manipulatingthe structure”. This increase in efficacy is again not surprising if one takesinto account the nature of the mathematics education curriculum for futureprimary school teachers in Flanders (Van de Plas, 1995) which typicallyfocuses on such strategies for solving complex word problems.

Conclusion

First of all, our data confirm the validity of our test design: the problemsthat we considered as typically arithmetical elicited more arithmetic solu-tions, while the problems considered as algebraic were more frequentlysolved algebraically. Furthermore, as in Schmidt’s (1994) study, wepredicted that future secondary school teachers would solve both algebraand arithmetic problems mainly algebraically, whereas future primaryschool teachers would approach arithmetic word problems arithmeticallyand algebra word problems with a mixture of arithmetic and algebraicapproaches. Our results were in line with this prediction. Moreover, closeranalyses confirmed Schmidt’s (1994) finding concerning the existence oftwo subgroups of future primary school teachers: those who exclusivelyand stubbornly tried to apply arithmetic strategies for all problems in thetest (sometimes in vain), and those who chose flexibly between arithmeticand algebraic strategies depending on the nature of the problem.

With respect to student teachers’ performances, we anticipated that thefuture secondary school teachers would perform well both on the arith-metic and the algebra problems, whereas for the primary school studentteachers, we predicted a rather good performance on the arithmetic prob-lems but a relatively weak performance on the algebra problems. Thisprediction was also confirmed by the data. Moreover, we expected thatthere would be several primary school student teachers who would notbe able to solve the algebra problems because they would not masteralgebra and because they would have great difficulty in applying one ofthe arithmetic methods on these difficult problems. A detailed analysis ofthe primary school student teachers’ performance data provided evidencefor that latter prediction as well.

Finally, we predicted that the above-mentioned expected differencesin strategy use between primary and secondary school student teachers,would increase as a consequence of participation in the teacher educa-tion. This expectation was not supported by the findings, as no significantdifference in strategy use was observed between the first-year and third-year student teachers in each study branch. It is, however, possibly duethe cross-sectional design of our study, with a comparison of first-year


and third-year students, instead of a longitudinal one. We further hypothe-sized that third-year pre-service teachers would also be more competent insolving these problems, and that this better performance would be mainlydue to an improvement in the primary school student teachers’ mastery of“manipulating the structure” and in the secondary school student teachers’mastery of algebra (i.e., the strategies which are prominent in their teachereducation and in their future classroom practices). This prediction wasconfirmed by the last analysis of the relationship between the strategy andthe accuracy data.

DISCUSSION

At the end of her study of beginning Canadian future primary school,secondary school, and remedial teachers’ skills in and strategies for solvingarithmetic and algebra word problems, Schmidt (1994) raised seriousconcerns about the ability of these student teachers to support their futurepupils in the difficult transition from arithmetic to algebra. Many of theprimary school student teachers and remedial teachers did not spontan-eously use algebraic strategies when necessary, mainly because they didnot master algebra and/or did not understand it sufficiently, and/or becausethey perceived it as manipulating meaningless mathematical symbols.Many secondary school student teachers, on the other hand, stubbornlyapplied the algebraic method, whether efficient or not, and had little or noaffinity with other solution strategies.

The findings of the present study, which can be considered as a replic-ation and an elaboration of Schmidt’s study, confirm these concerns andexpand them to the Flemish context. At a general level, our results urgeus to raise these concerns even more strongly because it was shown thatpre-service teachers who had arrived at the end of their teacher educationstill continued to demonstrate problem-solving behaviour characterized by(some of) the problematic features of the student teachers who had juststarted their teacher education.

Considering our research findings, we would like to formulate thefollowing considerations about the implications for the pre-service andin-service education of teachers. With respect to the education of futureprimary school teachers, the following positive and promising observa-tions can be mentioned. First, a substantial number of student teacherscould be typified as quite successful and flexible problem solvers, beingable to solve not only arithmetic but also more difficult algebra problemsby adaptively applying the kind of strategy that was most appropriate fora particular problem. Second, the three years of teacher education led to


an increase in the problem-solving skills of some of these future primaryschool teachers, perhaps because during this education they learned tosolve more difficult mathematical application problems by means of asophisticated arithmetic method that we described as “manipulating thestructure”. However, besides this subgroup of future teachers of primaryschool children, there was also another, more problematic, subgroup offuture primary school teachers, even among those who had come at theend of their education. Student teachers in this latter subgroup were unableto solve the more difficult problems from the word problem test becauseof their lack of affinity with algebra and because they were incapable ofcorrectly applying one of the arithmetic methods. Like Schmidt (1994), weare concerned about the readiness of these future primary school teachersfor their job as a teacher of mathematics of children up to the age of 11because they might lack the proper disposition to establish in their pupilsthe necessary foundations for algebraic reasoning.

Based on the results of our study, our concerns with respect to futurelower secondary school teachers are, like those of Schmidt (1994), ofa different nature. These student teachers probably have no insufficien-cies in their mathematical content knowledge but the problem is ratherlocated in their habits and attitudes while problem solving. Many of thesefuture teachers tended to use an algebraic method for solving word prob-lems in a stereotyped way despite the fact that half of the items couldbe solved more easily and more quickly arithmetically. We were able toshow that this characterization of student teachers as possessing “routineexpertise” instead of “adaptive expertise” (Hatano, 1988) was not only truefor student teachers who just started their teacher education, but also forthose who were finishing it. Our main concerns here are to be seen in rela-tion to some developments in mathematics education, such as the generallyaccepted learning principles that pupils should construct new mathema-tical concepts and procedures starting from their available strategies andprior knowledge (Gravemeijer, 1994; Schoenfeld, 1992; Verschaffel, Greer& De Corte, 2000), and that being able to use different mathematicaltools and strategies in a flexible and reflective way is one of the majorgoals of mathematics education (De Corte et al., 1996; Gravemeijer, 1994;Koedinger & Tabacheck, 1994; NCTM, 2000). In this respect, we wonderwhether these future teachers will be empathic for pupils who comestraight from the primary school and who bring with them a strong arith-metical history. Will they be able to understand and to react adequately tothe arithmetical concepts, skills and habits of these pupils? Will they haveinsight into the difficulties that these pupils are confronted with in thistransitional stage, and will they be able to help these pupils in overcoming


them? And finally, will they allow pupils to use a diversity of (arithmeticaland algebraic) strategies when solving problems, depending on the specificproblem and on the pupils’ preferences? Possessing both algebraic andarithmetic strategies and being able to switch flexibly between them, isan important characteristic of skilled mathematical problem-solving beha-viour, and if we want pupils to become skilled in choosing the mostappropriate tool, it is crucial that their teachers possess this dispositionthemselves and that they act as an appropriate model in their classroom.

It should be acknowledged that, in this study, the pre-service teachers’problem-solving behaviour might be due partly to the experimental setting.It could be argued that the future teachers responded in a “test mode”instead of reacting to the problems as a prospective teacher. It would there-fore also be interesting to investigate how they would respond to the test ifthey were asked how they would propose to work with children on solvingthese problems or to ask what exactly they were thinking when doing theword problems test.

In this respect, however, there is another consideration. It seems evidentthat the problem solving behaviour of the student teachers is to a largeextent shaped by their particular experiences during three years of teachereducation. If, like in Flanders, primary school student teachers are nevertaught and/or stimulated to use (pre)algebraic approaches to problems,they will probably not tend to do this, even in a real classroom setting.Analogously, it is unlikely that future secondary school teachers willmake extensive use of arithmetical approaches if this is not cultivated intheir teacher education. While the content and the culture of their teachertraining experiences may somewhat accentuate the differences, both thefindings from Schmidt (1994) and those obtained in the present study showthat these differences are already present at the very beginning of teachereducation. So, the origin of the observed differences lies in the differenteducational backgrounds and in the intentions and task conceptions as afuture teacher that these student teachers bring to their teacher training.

IMPLICATIONS FOR TEACHER EDUCATION

Taking into account the above-mentioned considerations, we suggest thefollowing pathways for optimizing the practice of the pre-service educa-tion of both groups of teachers. Our suggestions are in line with theconstructivist and reflective perspectives on teacher education that promotethe development of flexibility of teacher thinking and stress the impor-tance of a well-organized body of professional knowledge in teachers(e.g., Brown, Cooney & Jones, 1990; Fennema & Loef, 1992; Grouws &


Schulz, 1996; Shulman, 1985). Moreover, these pathways are inspired byBoero, Dapueto and Parenti (1996), who propose several methodologiesto bring results of research in didactics of mathematics into pre-serviceteacher education. First, it seems valuable that pupils’ transition from anarithmetical to an algebraic way of thinking is treated explicitly in themathematics and mathematics education courses of pre-service primaryschool teachers. Besides attempts to improve the algebraic problem-solving skills, the courses for future primary school teachers could containmoments of discussion on the similarities and differences between arith-metical and algebraic approaches, and on the difficulties pupils mightexperience when making their first steps into the learning of algebra. Thediversity in solution strategies applied by the student teachers themselvescould be a meaningful starting point here. But also in the mathematicsand mathematics education courses of future secondary school teachers,the transition from arithmetic to algebra is a worthwhile topic of instruc-tion. It seems valuable to explore, compare and evaluate the efficiencyof different problem-solving approaches, including arithmetical ones, andthe didactical and conceptual similarities and dissimilarities between thesedifferent approaches should also be addressed. Teacher educators couldalso apply our research instrument in their lessons or even confront theirstudent teachers with the results of the present study, in order to make themmore conscious about the issue.

We end this article with an important reflection on the conclusionsderived from the study presented here. The implications described aboverely on the claim that student teachers’ content-specific knowledge, skillsand beliefs have a strong impact on the quality of their future teaching. Ingeneral, this claim is supported by the research literature (see, e.g., Calder-head, 1996; Carpenter et al., 1988; Fennema & Loef, 1992; Shulman,1986; Thompson, 1992) but the study reported in this article does notprovide direct empirical evidence for this assumed link between (future)teachers’ cognitions and their teaching behaviour. As mentioned before,the present study was complemented by a second one, wherein the samegroup of pre-service teachers was asked to evaluate different kinds of(arithmetical and algebraic) solution strategies of pupils for some of theproblems they had solved themselves in Study 1, and to justify theirevaluations. As argued in Van Dooren et al. (2002), the confrontationof participants’ reactions to both tasks yielded insights into the relationbetween different aspects of student teachers’ domain-specific subject-matter knowledge and pedagogical content knowledge, on the one hand,and an important aspect of their (future) teaching behaviour, i.e., the wayin which they appreciate pupils’ solutions, on the other. It appeared that


these depend on the frequency with which they use a particular approachthemselves, their success rate in applying that approach and their percep-tion of a solution strategy as superior or inferior for solving word problemsin a classroom context.

ACKNOWLEDGEMENTS

The authors want to thank the student teachers and the teacher educators ofthe participating institute for their cooperation on this research. Wim VanDooren receives a grant from the National Fund for Scientific Research –Flanders (F.W.O. – Vlaanderen)

REFERENCES

Ainley, J. (1999). Doing algebra-type stuff: emergent algebra in the primary school.Proceedings of the 23rd annual conference of the International Group for the Psychologyof Mathematics Education (Vol. 2, pp. 9–16). Haifa, Israel.

Bednarz, N. & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran& L. Lee (Eds.), Approaches to algebra: Perspectives for research and education(pp. 115–136). Dordrecht: Kluwer.

Boero, P., Dapueto, C. & Parenti, L. (1996). Didactics of mathematics and the profes-sional knowledge of teachers. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick &C. Laborde (Eds.), International handbook of mathematics education (pp. 1097–1121).Dordrecht: Kluwer.

Brown, S. I., Cooney, T. J. & Jones, D. (1990). Mathematics teacher education. In W. R.Houston (Ed.), Handbook of research on teacher education (pp. 639–656). New York:Macmillan.

Calderhead, J. (1996). Teachers: Beliefs and knowledge. In D. Berliner & R. Calfee (Eds.),Handbook of educational psychology (pp. 709–725). New York: Macmillan.

Carpenter, T. P., Fennema, E., Peterson, P. I. & Carey, D. A. (1988). Teachers’ pedagogicalcontent knowledge of students’ problem solving in elementary arithmetic. Journal forResearch in Mathematics Education, 19, 385–401.

Carpenter, T. P. & Levy, L. (2000). Developing conceptions of algebraic reasoning inthe primary grades. (Res. Rep. 00–2). Madison, WI: National Center for ImprovingStudent Learning and Achievement in Mathematics and Science. [available atwww.wcer.wisc.edu/ncisla]

Carraher, D., Brizuela, B. & Schliemann, A. (2000). Bringing out the algebraic characterof arithmetic: Instantiating variables in addition and subtraction. Proceedings of the24th annual conference of the International Group for the Psychology of MathematicsEducation (Vol. 2, pp. 145–152). Hiroshima, Japan.

Davis, R. (1985). ICME-5 Report: Algebraic thinking in the early grades. Journal ofMathematical Behavior, 4, 195–208.


De Corte, E., Greer, B. & Verschaffel, L. (1996). Learning and teaching mathematics. In D.Berliner & R. Calfee (Eds.), Handbook of educational psychology (pp. 491–549). NewYork: Macmillan.

De Rijdt, C. (1999). Instaptoets wiskunde voor aspirant-leerkrachten (wiskundedidactiek)[A Mathematics intake test for future primary-school teachers]. Unpublished master’sthesis, University of Leuven, Belgium.

Discussion document for the twelfth ICMI study (2000). Educational Studies in Mathe-matics, 42, 215–224.

Fennema, E. & Loef, M. (1992). Teachers’ knowledge and its impact. In D. A. Grouws(Ed.), Handbook of research on learning and teaching mathematics (pp. 147–164). NewYork: Macmillan.

Filloy, E. & Sutherland, R. (1996). Designing curricula for teaching and learning algebra.In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), Internationalhandbook of mathematics education (pp. 139–160). Dordrecht: Kluwer.

Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht, The Neth-erlands: Freudenthal Institute, University of Utrecht.

Grouws, D. A. & Schultz, K. A. (1996). Mathematics teacher education. In J. Sikula, T. J.Buttery & E. Guyton (Eds.), Handbook of research on teacher education (pp. 442–458).New York: Macmillan.

Hall, R., Kibler, D., Wenger, E. & Truxaw, C. (1989). Exploring the episodic structure ofalgebra story problem solving. Cognition and Instruction, 6, 223–283.

Hatano, G. (1988). Social and motivational bases for mathematical understanding. NewDirections for Child Development, 41, 55–70.

Herscovics, N. & Linchevski, L. (1994). The cognitive gap between arithmetic and algebra.Educational Studies in Mathematics, 27, 59–78.

Kaput, J. (1995). Transforming algebra from an engine of inequity to an engine of mathe-matical power by “algebrafying” the K-12 curriculum. Paper presented at the 1995NCTM meeting.

Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.),Handbook of research on learning and teaching mathematics (pp. 390–419). New York:Macmillan.

Koedinger, K. R. & Tabacheck, H. J. M. (1994, April). Two strategies are better than one:Multiple strategy use in word problem solving. Paper presented at the annual meetingof the American Educational Research Association. San Francisco.

Lee, L. & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathe-matics, 20, 41–54.

Linchevski, L. & Herscovics, N. (1996). Crossing the cognitive gap between arithmeticand algebra: operating on the unknown in the context of equations. Educational Studiesin Mathematics, 30, 38–65.

Nathan, M. J. & Koedinger, K. R. (2000). Teachers’ and researchers’ beliefs about thedevelopment of algebraic reasoning. Journal for Research in Mathematics Education,31, 168–190.

National Council of Teachers of Mathematics (2000). Principles and standards for schoolmathematics. Reston, VA: NCTM.

Reed, S. K. (1999). Word problems: Research and curriculum reform. Mahwah, NJ:Lawrence Erlbaum Associates.

Schmidt, S. (1994). Passage de l’arithmétique à l’algèbre et inversement de l’algèbre àl’arithmétique, chez les futurs enseignants dans un contexte de résolution de problèmes


[Future teachers’ transition from arithmetic to algebra in a problem solving context].Unpublished doctoral dissertation, Université de Québec à Montréal, Canada.

Schmidt, S. (1996). La résolution de problèmes, un lieu privilégié pour une articulationfructueuse entre arithmétique et algèbre [Problem solving as a privileged context for afruitful connection between arithmetic and algebra]. Revue de Sciences de l’Education,22, 277–294.

Schmidt, S. & Bednarz, N. (1997). Raisonnements arithmétiques et algébriques dans uncontexte de résolution de problèmes: difficultés rencontrées par les futurs enseignants[Arithmetical and algebraic reasoning in a problem-solving context: difficulties met byfuture teachers]. Educational Studies in Mathematics, 32, 127–155.

Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacogni-tion and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research onmathematics teaching and learning (pp. 334–370). New York: Macmillan.

Sfard, A. (1991). On the dual nature of mathematical conceptions. Reflections on processesand objects as different sides of the same coin. Educational Studies in Mathematics, 22,1–36.

Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of reification: the case of algebra.Educational Studies in Mathematics, 26, 191–228.

Shulman, L. S. (1985). On teaching problem solving and solving the problems of teaching.In E. A. Silver (Ed.), Teaching and learning mathematical problems solving: Multipleresearch perspectives (pp. 439–450). Hillsdale, NJ: Laurence Erlbaum.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educa-tional Researcher, 15(2), 4–14.

Slavitt, D. (1999). The role of operation sense in transitions from arithmetic to algebraicthought. Educational Studies in Mathematics, 37, 251–274.

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Thompson, A. (1992). Teachers’ beliefs, and conceptions: A synthesis of the research.In D. A. Grouws (Ed.), Handbook of research on learning and teaching mathematics(pp. 127–146). New York: Macmillan.

van Amerom, B. (2001). Reinvention of algebra: Developmental research on the transitionfrom arithmetic to algebra. Unpublished doctoral dissertation, University of Utrecht, theNetherlands.

Van de Plas, I. (1995). De inhoud van het vak Wiskunde in de opleidingsinstituten voorleerkrachten lager onderwijs in Vlaanderen: een exploratief onderzoek [The contentsof the mathematics course in the training institutes for primary-school teachers inFlanders: an exploratory study]. Unpublished master’s thesis, University of Leuven,Belgium.

Van Dooren, W., Verschaffel, L. & Onghena, P. (2001). Rekenen of algebra? Gebruik vanen houding tegenover rekenkundige en algebraïsche oplossingswijzen bij toekomstigeleerkrachten [Arithmetic or algebra? Student-teachers’ use of and attitudes againstarithmetical and algebraic problem solving strategies]. Leuven: University Press.

Van Dooren, W., Verschaffel, L. & Onghena, P. (2002). The impact of preservice teachers’content knowledge on their evaluation of students’strategies for solving arithmetic andalgebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351.


Vergnaud, G. (1988). Long terme et court terme dans l’apprentissage de l’algèbre.In C. Laborde (Ed.), Actes du premier colloque franco-allemand de didactique desmathématiques et de l’informatique (pp. 189–199). Paris: La Pense Sauvage.

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University of LeuvenCenter for Instructional Psychology and TechnologyVesaliusstraat 23000 LeuvenBelgiumE-mail: [email protected]

PAOLA SZTAJN

ADAPTING REFORM IDEAS IN DIFFERENT MATHEMATICSCLASSROOMS: BELIEFS BEYOND MATHEMATICS

ABSTRACT. The goals of this study are to understand elementary school teachers’ beliefsand practices and to unveil factors that influence the way teachers adapt mathematicsreform rhetoric when trying to adopt it. In the research, I searched for beliefs beyondmathematics that influence teachers’ decisions and choices for teaching mathematics.Working with children from different socioeconomic backgrounds, teachers interpretreform in different ways. Based on their concept of students’ needs, teachers select whichparts of the reform documents are appropriate for their students. While children from uppersocioeconomic backgrounds experience problem solving, those from lower socioeconomicbackgrounds undergo rote learning. Because not all children have the opportunity to learnthe same quality mathematics, the emerging concern of this study is the issue of equity inmathematics teaching.

KEY WORDS: equity, ideology, reform, teacher beliefs

Studies of teachers dealing with innovations in school mathematics suchas those described in the Curriculum and Evaluation Standards for SchoolMathematics (National Council of Teachers of Mathematics [NCTM],1989) have shown that, despite our long history of resistance to change(Cuban, 1993), mathematics teachers can successfully make modifica-tions in their practices (e.g., Fennema & Nelson, 1997; Ferrini-Mundy &Schram, 1997; Schifter & Fosnot, 1993; Wood, Cobb & Yackel, 1991). Thefavorable results reported are usually based on notions such as sustainedpartnership or leadership, on-going collaboration, community, and long-term support for teachers to embrace new recommendations. In thesestudies, teachers are often part of a wider change initiative, which includesat least one of many resources such as workshops with mathematicsspecialists (in different formats), connections with universities or researchgroups, and systemic endeavors of change.

However, everyday classroom reality for most teachers does not includesuch sustained, on-going, long-term support for change. Although 99% ofthe teachers participate each year in professional development opportu-nities (U.S. Department of Education, 1999), most of these opportunitieslast less than one day of work. Thus, teachers typically are left to face themany challenges of reform in mathematics by themselves. They have toput together, according to their own beliefs and interpretations of existing


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rhetoric, what they consider to be the best mathematics education for theirstudents.

In this article, case studies are presented of two elementary schoolteachers interpreting and implementing current recommendations forchange in mathematics teaching. These teachers are “typical” in that theyare not part of any project or major collaboration effort aimed at changingtheir knowledge, beliefs, or practices in mathematics. They are in theirclassrooms, at their schools, working hard to make sense of current callsfor change. They may exchange ideas with their colleagues, they mayread mathematics-related reform materials, but they rely on their ownknowledge and beliefs to implement reform changes.

The research conducted aimed at understanding these typical teachers’beliefs and practices, unveiling factors that shape how teachers adaptreform rhetoric when trying to adopt it. From a holistic perspective, Isearched for ways in which beliefs that go beyond mathematics, mathe-matics teaching and learning experiences influence teachers’ decisions andchoices for teaching this subject.

In the next sections, I present the assumptions that led the investigation,connecting reform issues and research on teachers’ beliefs in mathematicseducation. I describe the case studies, the participants, and the researchmethodology. For each participating teacher, I provide a brief overviewof her mathematics teaching and a summary of broader concerns thatguide her practice. I contrast teachers’ practices and discuss factors thatinfluence what they do in their classrooms. Finally, I analyze the waysin which teachers’ ideological vision of their students play a role in theirmathematics teaching.

The results reveal that, working in very different school settings anddealing with children from different socioeconomic backgrounds, theteachers in this study interpret reform in different ways. Based on theirvalue-laden concept of students’ needs, they select which parts of currentcalls for change are appropriate to their students. Due to their selections,these teachers instruct different mathematics to children of different socialand economic background. Thus, I conclude the paper by considering afundamental question that arose from the research: are current calls forchange moving mathematics education towards good quality mathematicsfor all children? The emerging concern of this study is the issue of equityin mathematics teaching.

BEYOND MATHEMATICS 55

BELIEFS, REFORM, AND MATHEMATICS EDUCATION

In the 1980s, the mathematics education community began to look atreform through a lens that captured a broader picture than the one framedin the 1960s. The earlier reform effort focused on revising the content ofschool mathematics to align it with fundamental concepts in the discip-line of mathematics (e.g., sets). Recommendations for change during the1980s addressed not only questions about what to teach, but also abouthow to teach it. The assumption was that to improve the state of schoolmathematics it was necessary to rethink the goals for mathematics taughtat schools and the approaches used to teach it. Therefore, the role andimportance of teachers in the implementation of reform ideas became ofinterest.

Studies of teachers’ beliefs gained attention, with increasing recogni-tion that beliefs play an important role in teaching (Clark & Peterson,1986; Thompson, 1984). Trying to understand the role beliefs play in theway mathematics teachers embrace change, researchers have considereddifferent theoretical positions. They have attempted to promote andsupport change through the modification of teachers’ beliefs about thenature of learning and the nature of mathematics, children’s mathema-tical thoughts and school mathematics content itself (Nelson, 1997). Thesestudies have focused on beliefs about mathematics (what to teach) and itsteaching and learning experiences (how to teach).

Mathematics educators recently have begun to examine other setsof beliefs that influence mathematics teaching practices.1 Skott (2001)showed how beliefs not directly related to mathematics teaching alsohelp one understand mathematics teachers’ practices. In his study, heconsidered micro-aspects of the social context of mathematics classrooms.He presented the teacher’s overarching concern about students’ self-esteem as justification for mathematics teaching episodes. My researchtakes into account the macro-structures of school mathematics and bringsideology into the picture. It shows that teachers’ beliefs about the needsof children they teach shape their mathematics teaching and their imple-mentation of reform changes. Students’ needs emerge in this research asa concept that influences and justifies the mathematics teachers choose toteach to children from different socioeconomic backgrounds.

According to Ernest (1991), when reform documents arrive inclassrooms, interpretations hamper changes in teachers’ practices. Inter-pretations of reform documents are problematic because readers interpretthe ideas promoted in the document according to their personal perspec-tives and ideological positions. Ernest defines ideology as “an overall,

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value-rich philosophy or world view, a broad inter-locking system of ideasand beliefs” (p. 111). This is the working definition of ideology used in thepresent study.

In education, the discussion of ideology has moved beyond the dicho-tomy between a true or false vision of the world. This important conceptcomprehends an understanding that ideas are not isolated and are notvalue-free. All our personal ideas transmit a vision of the world that isdirectly associated with the interests of different social groups. Within thisframework, an ideology “is not a homogenous, coherent set of ideas puttogether in a logically consistent construction” (Moreira & Silva, 1995,p. 25), although it is usually perceived as “the way things obviously are”by those who support it. Ideologies are complex sets of beliefs about theworld, forged through a person’s life experiences. Particularly, in schools,ideologies provide “the value system from which decisions about prac-tical educational matters are made. . . . [They include] beliefs about whatschools should teach, for what ends, and for what reasons” (Eisner, 1992,p. 302).

In mathematics education, Ernest (1991) connected different philo-sophies of mathematics with tacit belief-systems that underpin them,developing what he called a Model of Educational Ideology for Mathe-matics. Some of the primary elements that he used to characterize hismodel (in particular, theory of the child, theory of society, and educationalaims) offer the possibility of understanding mathematics teachers from aperspective that is broader than the one usually considered in studies onteacher beliefs. These elements allow one to consider beliefs that influ-ence one’s mathematics teaching but go beyond mathematics education.Ernest’s ideological elements served as the starting point for this researchproject.

METHODOLOGY AND PROCEDURES

Participating Teachers and Their Schools

To situate this study within the current discussion of reform in schoolmathematics, participating teachers had to be aware of recent reformproposals. Therefore, part of the selection criteria used in the project wasthat teachers know something about the Curriculum and Evaluation Stand-ards for School Mathematics (NCTM, 1989).2 At the same time, selectedteachers could not be part of any long-term, professional developmentchange initiative.


Teresa Walker,3 a third-grade teacher, graduated in 1974 with a majorin elementary education. She saw herself as a “language-arts person.” Forher re-certification, she took a three-week, summer-intensive, graduate,mathematics education course that focused on problem solving and the useof manipulatives to teach mathematics in elementary classrooms. Duringthe summer classes, the NCTM Standards were read and discussed. Imet Teresa when I was auditing this problem-solving course at a nearbyuniversity. We sat near each other and worked on a few tasks together. Iasked Teresa if I could come to her class to conduct my research project.I explained I wanted to see how she taught mathematics in light of thechanges proposed in the Standards.

Julie Farnsworth, a fourth-grade teacher, was a doctoral student inelementary education. She earned a B. S. degree in elementary education in1982 as well as a master’s degree in 1985. I met Julie through courses shetook in mathematics education at the same university as Teresa. Julie wasasked to participate in the project because she was trying to be innovativein her mathematics teaching. Julie said she was more of “a mathematicsand science person,” and she was aware of the NCTM Standards beforewe talked about the project. She was “embarrassed to confess” she hadnever really read the 1989 document, but as the research progressed, Juliereported she was reading it.

When Teresa and Julie first took a mathematics methods course aspart of their pre-service education, the Standards had not been published.Therefore, Standards-related ideas were not discussed in those courses.Teresa and Julie had been teaching for 9 years and as practicing classroomteachers they were trying to figure out how to align their ideas with newrecommendations for school mathematics. The fact that Teresa and Juliehad both taught elementary school for several years was important intheir selection as participants. They were experienced teachers who wereworking hard to offer their students what they thought was good mathe-matics education. Teresa and Julie worked in environments that supportedtheir classroom practices. Their school peers, superiors, and some parentsconsidered them to be good teachers.

Teresa and Julie taught in public schools in two small Midwesterntowns. In both Teresa and Julie’s classrooms, all students were Caucasianand reflected the population in each school. Teresa taught in a Kindergartenthrough grade 5 school with 300 students enrolled and had 19 studentsin her class. Julie’s taught in a grade 3 through grade 6 school with 500students enrolled and had 25 children in her class.

These two schools differed with respect to the socioeconomic back-ground of the children as measured by the percent of children on free and

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reduced lunches. At Teresa’s school, 40% of the children received free orreduced lunches, while at Julie’s school only 10% of the children qualifiedfor these programs. The educational level of the parents, based on the prin-cipals’ estimations and teachers’ opinions, were also very different in bothschools. While in Teresa’s school most of the parents were low income andhad more manual labor jobs, in Julie’s school the middle-income parentswere doctors, lawyers, and university professors.

Data Collection

I collected data through four weeks of classroom observation spread overa semester and five semi-structured interviews with each teacher. Otherdata collected included classroom handouts and teachers’ planning notes,as well as interviews with the principal, other teachers, and some parentsat each school.

Classroom observations. I explained to the teachers that I was interested inunderstanding their mathematics teaching with respect to what they knewabout current calls for change. Because I was not in the classroom to facil-itate the implementation of reform ideas, I did not help teachers plan orevaluate their lessons. In the classroom, I was mostly an observer. I tookextensive field notes, particularly during mathematics lessons and classdiscussions. I helped teachers by distributing and collecting materials,making copies, and grading papers.

When observing a teacher, I stayed in her classroom all day, every dayof the week. To develop a broader picture of the teacher and her practice,I did not restrict myself to mathematics classes; quite the contrary, I sawthe teachers teaching all subjects. I also stayed at the school during lunchand breaks, and I attended staff meetings. When I was in the classroom, Icollected copies of all the mathematics worksheets and assignments thestudents completed, as well as most of the materials from activities inother subjects. I collected all tests the teacher gave and a few samples ofstudents’ work. I also looked at the teacher’s lesson plans, grading books,and students’ report cards.

Ernest’s (1991) elements constituted my guideline for classroom obser-vations. When observing the teachers’ practices, I searched for indicatorsof their ideological positions by searching for evidence of what wouldcharacterize these teachers’ personal theories about the children theyworked with, the society in which they live and their educational aims andgoals as teachers. These indicators were probed during the interviews. Inthe classrooms and during the meetings, I also looked for factors that could


help me understand the ways in which the teachers were making sense ofreform-related proposals for school mathematics.

Semi-structured interviews. The first interview with each teacher addressedher educational background and answers to two preliminary beliefs ques-tionnaires. The other four interviews followed each week spent in theclassroom and dealt mainly with issues raised during the observations. Todevelop a set of tentative questions before each meeting, I consulted myfield notes and transcripts of previous meetings. During the interviews, Iasked teachers to justify some of their classroom actions, explaining whythey did what they did at certain observed teaching episodes. I also askedteachers to further clarify ideas we had talked about, but which I did notthink I had fully understood.

The study of these teachers’ beliefs was anchored strongly on the obser-vation of their practices. The intention was to “deduce” teachers’ beliefsfrom intensive classroom observation and discussion of practice. Neverdid the interviews make use of questions such as “what are your beliefsabout . . .” Rather, all questions were based on concrete situations observedin the classroom. During the interviews, I asked teachers to discuss theirmotivations and intents.4

Other interviews and observations. To gather more information about thecontexts in which the teachers worked, I interviewed their principals, otherteachers in the schools, and a few parents. In both schools I met twice withthe principals: the first time was an informal, introductory meeting duringthe first week of observation, and the second time was an audio-recordedinterview around the third week of observation. I asked similar questionsof both principals concerning their students’ socioeconomic background,the profile of parents’ types of jobs, their perceptions of the students’emotional and academic needs, and their vision of what would constitutegood mathematics teaching. In each school, I also observed two otherteachers, from the same grade level of the teacher in the study, teachinga mathematics lesson. Finally by phone, I talked to parents of five studentsin each class.

Data Analysis

Data collection and analysis in this research constituted part of aprogressive problem solving (Erickson, 1986). I was constantly checkingon the work done, gathering more information, looking at special cases,planning, trying out tentative solutions (interpretations or explanations)and checking back on the work done. The data analysis began with

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analytical notes I wrote while keeping field notes. Doubts that arosefrom these notes guided the next round of observations and interviews.I was constantly looking for different ways of revisiting with the teachersthemes that caught my attention, and for ways of asking the teachers forjustifications and more explanations concerning their practices.

A second phase of the data analysis process took place when I leftthe field. Although I had developed initial ideas about the teachers andtheir perspectives while collecting data, I did not begin to code the datasystematically until I had all my transcripts and field notes in hand. I usedseven major descriptive categories to code the data initially – the teacher,the students, the classroom, the school, parents, general educational goals,and mathematics. With this organization, I used data triangulation to“clarify meaning, verifying the repeatability of an observation or interpre-tation” (Stake, 1994, p. 241). I also used constant comparison procedures,searching for the meaning of every piece of information (Glaser & Strauss,1967; Strauss & Corbin, 1990). I compared all pieces of coded datawith each other. I looked for ways in which beliefs beyond mathematicsinfluence teachers’ adaptation of reform ideas into practice.

Looking between and within categories for each teacher, I developeda motto that characterized her teaching, coordinating her perspectiveon education in general and mathematics teaching in particular. I thenconstructed, for each teacher, a justified description of her practice, thatis, a summary of her teaching combined with her explanations for specificactions. I called these descriptions “portraits”. The portraits had themottoes as the connecting lines that organized the picture. Each teacherread her own portrait and agreed with the justified description provided forher practice.

Considering both teachers’ portraits, I compared and contrasted thetwo cases. I merged different descriptive categories into new themes thatseemed to guide both teachers’ adaptation of reform ideas into their prac-tices. All data was once again checked with these themes in mind. It was inthese searches that the concept of students’ needs emerged as an importantfactor in the teaching of these two teachers. This concept includes teachers’beliefs about children, society, and education. It is a concept that goesbeyond mathematics. The concept of student needs allows for the inter-pretation of the way teachers adapted proposed changes for mathematicsinstruction.


THE CASE STUDIES

Teresa M. Walker

Teresa’s mathematics teaching. For Teresa, the Standards are a list ofcontent topics for the mathematics program and it represents the mostcurrent version of the list teachers need to cover in mathematics classes.She says her school program has a list similar to the Standards in that itcontains about the same topics – which she tries to teach. Teresa believescurrent calls for changes in school mathematics do not bring any newdemand for her mathematics teaching because her school has alreadyadded topics like problem solving to the curriculum. For Teresa, teachersat her school, who follow the program and teach what they are supposed to,already have their practices aligned with NCTM recommendations. Teresafeels she does not need to be particularly worried about reform – she isalready implementing it.

Teresa claims that teachers have “stepping stones” for what childrenmust learn in mathematics at different grades, and she thinks it is theteachers’ duty to follow this path. In third-grade, mathematics has manyrules that students need to learn. Therefore, one important goal for Teresais to teach her students all these rules. She must also help them “remember”– a key word in her classroom. Teresa tells her students, for example, thatnumbers under 50 are hard ones when rounding to the nearest hundredbecause they need to “remember” these numbers go all the way to 0.Students also need to “remember” what to do when they have to add andcarry to the tens place or when they are borrowing in subtraction. Forexample, when Teresa had a list of subtraction problems written in verticalform on the board, including 60–34, she told students:

Now, okay class, let me remind you of a few important things that might be rusty. It isalright not to borrow when you do not need to. A mistake many of you make, like in thelast exercise [60–34], is to make zero minus 4 equals 4. Remember that we cannot changethe order of the numbers and make 4 minus zero. You need to take the bottom numberaway from the top number. If you can’t take away, you cannot take the top number fromthe bottom. You need to remember to borrow from the tens place to the ones place. (Fieldnotes)

Teresa works hard on what she sees as the teacher’s part in the students’learning process. She is careful about presenting in a “very clear way” thetopics to be studied; she explains examples and goes over the exerciseswith the children. She states the rules and helps students remember them.Teresa asks questions and guides the students through the concepts. Sheselects activities and worksheets from the book for her students, and shehelps them do the work. Not only does Teresa explain to the students what

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they need to do, but she is also always willing to repeat patiently anyexplanation as many times as needed.

In general, Teresa expects her students to work as hard as she doesand do their “personal best”. In mathematics, it is important for Teresathat her students try to solve exercises and problems, and that they persist,even if they make mistakes, because “learning comes with practice”. InTeresa’s class, practice takes two different forms. One of them is solvingword problems. Teresa says children develop their brains when they solveword problems. The second type of practice that Teresa gives her studentsin mathematics is drill. “Some people believe that drill isn’t necessary”,Teresa explains, “but in some things drill is necessary”. For Teresa, not alltopics in mathematics need to be memorized. The ones that do, however,demand a lot of drill, and efficient drill requires organization. Teresaexplains to her students that the human mind needs order to work properly.“Our brain”, she tells children, “cannot work in a cluttered environment”.

For Teresa, problem solving, critical thinking, and other higher-orderthinking skills are also important for her students. “I think that all educatorswould agree with that”, she notes. The difficulty in teaching analyticalthinking, Teresa says, is managing to teach and still control the disciplinein the room.

As a teacher, the things that I get very frustrated about when we try to work on the higherthinking skills [is that] (. . .) sometimes you have such a large range of levels that the kidsare dealing with, that when you go to do an activity . . . It’s like a third of them are withyou, a third of them maybe has an idea of what you are doing, and another third has no ideaof what you are doing. And you are lost. I think that’s probably the same kind of frustrationthat most educators are running into, the discipline of the kids. (Interview #2)

Situating Teresa’s mathematics teaching. Teresa values discipline andresponsibility, and she wants to teach her students to be well behavedand responsible. Good behavior and responsibility, according to Teresa,are necessary when one needs to get a job and function in society. Teresa’sclassroom is usually quiet, and she likes her students to be quiet and orga-nized. She wants them to feel they are in a productive working environmentwhere they can focus on their tasks, working hard to achieve their goals.Teresa believes this environment is different from the one her studentsexperience at home.

Some of my kids have such crappy and disordered home lives, I don’t want school to belike home for them. (. . .) I want my class to be pleasant, I want it to be safe, organized, Iwant them to feel good. But this isn’t home, and it shouldn’t be home. (Interview #3)

In general, Teresa believes structure and order are beneficial for allpeople – and mathematics is good for teaching this because it is structured


and has rules to be followed. For her students, in particular, Teresa thinksthis is even more fundamental because they live in a community that lackssuch organization. Teresa classifies the community as “mobile and tran-sient” due to the fact that these children come from “unstable families”,she says referring to parents’ marital status, personal relationships, and jobsituations.

The school principal substantiates Teresa’s view and indicates that over50% of the school children live in one-parent households or have gonethrough at least one divorce situation. “The percentage of our studentsliving with original moms and dads”, he says, “is very, very small”. Amongthe families where there are two parents at home, both parents work outsidethe home in 95% of the cases. Concerning the types of jobs parents have,the principal says:

Most of our population is the manual labor type population rather than the professionallabor. We have some schools in this system where, maybe 80% of your mothers and fathersare college educated, professional people. That is not our case here. I haven’t really goneback and figured out, but I bet you would be surprised at how high a percentage of ourparents are not college or even high school graduates. (Interview with principal)

The small number of children from two-parent households and thelack of parent participation in their children’s education are among themain problems Teresa sees at her school. According to her, family influ-ence is fundamental for children’s performance in formal education. ForTeresa, because most parents at this school are not really involved in theirchildren’s upbringing, teachers have a bigger task to accomplish. Although“all children are children”, what is part of a child’s education varies fromschool to school, depending on socioeconomic background. For the “poor”children, Teresa says, teachers “have to teach them a lot more” becausethey have to teach “a lot of the social skills too”.

I think all the children have the same needs. I mean, the kids need to be cared for, they needto be loved, they need to learn some things. Okay. So I think those needs are universal,whether you are talking about the lowest income kids or the very wealthy kids. But then Ithink these kids have . . . . Well, probably they still have the same needs. It is just that theirneeds that aren’t taken care of at home, we take care of them here at school. (Interview #3)

Teresa’s motto: Teaching to form responsible citizens. The overarchingissue that emerges from Teresa’s teaching, which serves as a connectingthread among her perceptions of society, schooling, and mathematicseducation is her concern with responsibility and order. She considers thisissue especially relevant for her students, due to her perception of theirbackground, their needs, and their future role in American society.

Teresa is a responsible person who sees teaching as a job in whichshe has defined goals and tasks. For Teresa, working hard and following

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guidelines that tell her what to do are the keys to fulfilling her duties. FromTeresa’s perspective, most individuals follow working patterns similar tohers – external goals, hard work, accountability. In her view, these students,probably, will need to deal with a similar working situation. Because she isconcerned with helping her students become responsible citizens, Teresateaches in a style that is compatible with the conditions under which sheworks. She sets goals and tasks for her students, she expects them towork hard and be accountable. This is true for every subject in Teresa’sclassroom, including mathematics.

For Teresa, her students have many needs and attending to these needsis her main duty. Using her personal values about family to judge herstudents’ background, Teresa says these children come from unstable,chaotic homes, which are not good environments for children. Moreover,parents are not willing to participate in the children’s education andthey do not provide the appropriate experiences children should havebefore coming to school. Teresa’s personal beliefs about her students’lives outside school are based on a cultural deficit perspective in whichcertain children are believed to bring less to school. Irvine (1990), forexample, describes a cultural deficit perspective as the assumption thatsome children, because of cultural and environmental differences, lack theadaptations and knowledge necessary to succeed in school. Teresa saysthat to educate children from the low socioeconomic background she hasto bridge their social gap. It is her duty to help students become more orga-nized and responsible. This social education is the most important issue forTeresa as a teacher, and she gives it much of her time and attention, oftentaking time from content-oriented teaching.

In mathematics, although Teresa believes higher-order thinking skillsare important, basic facts, drill, and practice are at the core of what sheperceives as her students’ needs. Mathematics, as Teresa sees it, plays animportant role in helping her students be more “mentally organized” andready for their role in the workplace. Even though Teresa knows about the“list” of topics that is in the NCTM Standards, she does not have timeto teach all the topics. Therefore, she gives up problem solving and otherhigher-order thinking skills to focus on the teaching and learning of mathe-matical rules and procedures that demand rote learning through practice.Most important to Teresa is the fact that learning and following rules ina responsible and organized way is what her students need in order tofind their place in society – an approach that would more closely resembleErnest’s (1991) pragmatist profile.


Julie Farnsworth

Julie’s mathematics teaching. When Julie joined the research project, shesaid all she knew about the NCTM Standards was that “they focus moreon process than on product”. Because she considered her mathematicsteaching process oriented, Julie hoped she was already incorporatinginto her practice some of the current reform ideas. For Julie, focusingon the process meant that teachers should pay attention to what theirstudents were doing instead of the final answers they gave. As the researchprogressed, Julie thought she could improve her teaching further by imple-menting some of the suggestions she was reading in the NCTM document.Nevertheless, she was satisfied to conclude that many of her teaching ideaswere aligned with the reform recommendations.

The Standards is really helping me re-focus my teaching. I think that I always knew thatI wanted to teach in a problem-solving, creative-type way. But in math I’ve always beenmore tied to the textbook than in any other subject. Just because of that belief in gettingthe basic facts down, which I still think is important. But I think that I really like the de-emphasis on, oh, doing thirty-five long-division problems and things like that. (Interview#3)

Julie thinks students need to be interested and happy in order to learn.Therefore, she tries to make mathematics fun and her students happyby incorporating manipulatives and problem solving in her mathematicslessons. She also teaches through projects. Beyond teaching mathematicscontent knowledge, Julie believes projects are empowering experiencesthat teach students responsibility, persistence, and collaboration. Duringthis study, Julie’s students worked on different projects such as construc-tion of toothpick skyscrapers and bridges; making and selling of tie-dyeT-shirts; tessellations; experiments for a science fair; problem-solvingbooks to exchange with another fourth-grade class. Each project lasted afew weeks, and sometimes the children worked on two or three projects atthe same time.

According to Julie, her students need to experience sustained problem-solving, “they need more things that are not just one-shot deals”. Julieclaims that problem solving is important for all children because a personwho is a problem solver is not afraid of facing challenges. “And mathnaturally lends itself to problem solving”, she explains. Julie says thatchildren have different ways of solving problems in mathematics. Shealways accepts children’s different solutions to a problem and asks studentsto share their ways with the class. With these actions, Julie expects thatstudents will learn to search for meaningful ways of solving problems.

In Julie’s class, even routine word problems generate debate amongstudents. Most children like to present their ways and tell the class how

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they solved a certain problem, how they carried on a specific mentalcomputation. For example,

Julie: Let’s look at this problem. Our class has sold 28 T-shirts so far,at $5 a shirt. How much money do we have?

Students: 140.Julie: That’s right, but how did you find it out? Moses?Moses: I did 20 times 5 is 100 and 8 times 5 is 40, so it’s 140.John: I only did 28 times 5.Lisa: I counted by 5s.Ann: You could also count by 28s, and add it 5 times together.Leah: If you’ll do that, you could just find 28 plus 28, you add that to

itself, and then you add 28 more.John: If you just multiply by 5 it’s faster.Julie: Well, that is the beauty of multiplication, isn’t it? But you all

solved the problem. (Field notes)

Although Julie likes to teach mathematics through projects and problemsolving, she believes it is also important that children know the basicfacts. “I still teach computation”, she says. Before children can work onmore advanced problems and explore richer mathematical situations, Juliebelieves it is to their advantage to automatically recall their facts; to learnfacts, Julie claims that children need to practice them. However, Julieobserves that basic facts are not a problem in the school where she nowteaches because children know them.

Situating Julie’s mathematics teaching. Julie believes that parents at thisschool have one main interest: they want to know their children are happyat school. Julie enjoys this expectation of parents because it is alignedwith her idea that, to learn, children must be happily involved in what theyare doing. Therefore, Julie feels that this school is especially receptive toher projects, the fun things she does in the classroom, and the interestingactivities she selects for her students.

Julie claims she has always managed to do some projects in her classes,in whichever school she taught. However, at this school, Julie noticesshe can spend a larger part of her time doing the type of activities sheenjoys because she has to deal with fewer problems, and worry less aboutteaching rote basics. “My students this year are quick to catch basic skills”,she observes. For Julie, children at this school learn their basic facts athome. “This is a dedicated group of parents”, she explains. Previously,Julie taught at schools where children came from lower socioeconomic


backgrounds, and she felt she had to spend more time teaching childrenbasic knowledge and skills.

When I taught at a school where children were poorer, [the kids] needed to become literateand numerate. I mean, really, that was a goal for me to, for them to be literate and numeratebecause their low socioeconomic background went hand in hand with their skills. So Ihad to work on all these things with those low kids. With wealthier students it’s differentbecause they have that background before school. (Interview #1)

Julie, like Teresa, believes that in schools where children come fromlower socioeconomic backgrounds, teachers have to teach a lot of socialskills. For both teachers, teaching social skills is a main role of schools.Julie says that responsibility, for example, is something she wants to teachall her students. However, in the school she is now teaching, most childrenlearn responsibility in their homes. Children are accustomed to having andfollowing due dates, to knowing that they need to complete their assign-ments and turn them in, and to doing what they are expected to do. Thesechildren do not need practice on skills or social teaching and, therefore,can explore more sophisticated content issues.

Julie acknowledges that there are many other differences betweenthis school and lower socioeconomic schools where she taught before.Concerning discipline, for example, Julie often mentions that her class thisyear has many behavioral problems. She thinks her students are spoiled,immature, and they do not respect others as much as they should. Never-theless, Julie explains that these discipline problems are minor and thatdealing with them is really not too time consuming or too serious.

The principal supports Julie’s view and says that she does not deal witha lot of behavioral problems.

Most of these children come from very good homes and they have already learned, by thetime they come to school, that one does not solve a problem by fighting. Rather, one talksand makes a good argument to defend a position. These children have learned to respecteach other. (Field notes from informal interview with Principal)

The principal estimates that 70% or more of the children at thisschool come from homes where parents have at least a two-year collegedegree. “We don’t collect information on parents, but this is a veryaffluent neighborhood, and many parents are doctors, lawyers, professors,administrators, and business people”.

For Julie, at this school, parents are involved in their children’s educa-tion and the children know it. All parents, mostly in couples, come toparent-teacher conferences. Parents volunteer to help in the classroom,participate in field trips and talk to teachers about their children’s educa-tion. The principal recognizes that this intense contact with parents is a

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characteristic of the school. She also says that parents have high expecta-tions for learning at school. These are professional parents who want theirchildren to have what they consider to be the best possible education. Theypush for higher-order thinking skills, claims the principal, because “theysee at their jobs that you don’t just make rote decisions”.

Julie’s motto: Teaching to promote the growth of happy individuals. Theoverarching issue that emerges when observing and talking to Julie is herstrong desire to see her students having a good time at school. Julie wantsher students to enjoy, and this idea guides her actions and her teaching.People strive to be happy, and happy people are better citizens of the world.Furthermore, happy people are better learners. To make sure her studentswill like being in her class, Julie tries to prepare interesting activities andorganize projects in which the children will have fun.

Julie has a clear understanding that the school where she works is animportant factor in her search for “happy teaching”. Julie feels that at thisschool, more than at others, she can concentrate on giving her class projectsand fun activities. Despite routine behavioral problems, her students comefrom environments where they learn how to be polite and responsible, andshe does not have to worry about teaching these values to her pupils. Juliealso does not have to teach these children the “basics” because they eitherknow them from home or, in case they do not, parents take the responsi-bility for seeing that their children learn them. Julie can concentrate on thecontent she needs to teach her students.

Julie’s perception of her teaching and of what she “can teach” tochildren from different socioeconomic backgrounds depends on what they“bring” to school with them. Like Teresa, Julie operates within the deficitperspective. However, because she currently works with children fromupper socioeconomic backgrounds, Julie considers that these childrenare ready to experience problem-solving projects developing higher orderthinking skills. In mathematics, Julie’s focus is on a more process-orientedinstruction, which is most compatible with Ernest’s (1991) vision ofa progressive mathematics educator. Julie’s view of mathematics as aprocess, however, relies on the assumption that knowing the “basics” arealready met.

Contrasts and Emerging Theme

Teresa and Julie thought their practices were aligned with current recom-mendations for change in school mathematics. Teresa’s view of reform wasthe newest list of mathematics topics teachers are supposed to follow, whileJulie’s perspective of reform placed a greater emphasis on mathematical


processes. However, both believed they were implementing reform ideasthat were appropriate to their students. In fact, they believed they hadalready been implementing such ideas before this study begun, and thisbelief did not change during the investigation.

To contrast Teresa’s and Julie’s different ways of teaching mathe-matics, and the beliefs that support their teaching, one could choose touse Raymond’s (1997) 5-level scales, which categorize teachers’ practicesand beliefs about mathematics, mathematics teaching, and mathematicslearning. On the one hand, Teresa would be at the traditional end of thescales, where mathematics is believed to be a collection of facts and rules,students are believed to passively receive mathematical knowledge fromteachers and teaching means following the book and structuring lessons.On the other hand, Julie would tend to the nontraditional side of the scales,where mathematics is problem driven, students learn through problem-solving explorations and teaching means selecting tasks based on students’interests and asking students to justify their ideas.

These classification categories do not adequately explain the similari-ties in beliefs between Teresa and Julie. Both teachers believe children canonly solve problems when they know the basic facts. They also believesome children bring “the basics” from home, while others do not. Further-more, those who do not bring “the basics” do not seem to bring necessarysocial skills from home either. Both Teresa and Julie operate within thedeficit model, in which they perceive students’ needs based on personalbeliefs and values. Therefore, to understand Teresa and Julie it is necessaryto consider beliefs that go beyond mathematics. Teresa and Julie teachdifferently because of the way they perceive students who come fromdifferent socioeconomic backgrounds.

As a teacher, Teresa aims at teaching her students the social valuesand norms their families are failing to provide them. Combining beliefsabout the needs of the children she works with, American society, andeducation, Teresa teaches to transform lower socioeconomic students intogood citizens. She believes what these children need is discipline, rulesand facts. Julie holds similar beliefs to Teresa, but because her studentscome from a higher socioeconomic background, she teaches higher-order thinking through educationally rich projects. Both Teresa and Juliecombine their value-laden visions of students, of parents, and of societyinto the emerging concept of students’ needs.

IDEOLOGY, REFORM, AND STUDENTS’ NEEDS

In a U.S. elementary school, students’ needs are part of teachers’ dailyconcerns, discussion and goals. After all, should not all teachers work

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toward fulfilling such needs? Defining what these needs are, however,is ideological in nature. Maybe due to the fact that elementary schoolteachers have a broad view of their students, they take different factorsinto consideration when defining what is necessary for each child. Thesefactors include what children know and what knowledge they are believedto lack, their previous experiences and what they bring to school, whatthey need to succeed later in life and what jobs they are likely to have.Teachers’ judgments of such factors are not value free; they are based onteachers’ own beliefs about what is valued in society, what children’s livesare like outside school and the appropriateness of their out of school exper-iences. These judgments are based on teachers’ value-rich philosophiesand world-views, that is, their ideologies.

In mathematics education, the concept of students’ needs provide a con-nection between one’s ideological vision of the world and what is selectedto be taught in elementary classrooms. It relates beliefs beyond mathe-matics to what is considered appropriate mathematics instruction. It is avenue through which personal values influence elementary mathematicsinstruction, combining Ernest’s (1991) primary ideological elements suchas teachers’ theory of the child, society, and education.

In an era of reform, teachers’ concepts of what their students needshape their adaptation to and adoption of reform rhetoric. While differenteducational ideas can be considered important, and different perspectiveson mathematics teaching can all be considered pertinent in general, whenit comes to prioritizing them in the classroom, teachers’ concepts of whattheir particular students need come into play. Ideological decisions, aboutwhat within the reform rhetoric fits particular children, are then made.There is enough vagueness in current reform documents to allow teachersto follow only the recommendations they consider appropriate to partic-ular teaching situations. Different interpretations of these documents arepossible and teachers do not need to change to believe they are teachingaccording to current reform visions. Teachers can teach very differentmathematics, with very different justifications, and think they are actingaccording to the current recommendations that apply to “their children”with “particular need”.

Reform documents such as the NCTM Standards, which intend toserve as models of what mathematics teaching should look like, do notchallenge teachers’ concept of what their students need when learningmathematics. Because the concept of students’ need is constructed uponideological beliefs beyond mathematics, challenging it would require oneto challenge a person’s vision of the world. Current calls for change donot transform teachers’ ideological visions; quite the contrary, teachers’


ideological visions shape the possibilities of interpreting and implementingreform.

Mathematics for All, but Which Mathematics?

The result of this decade of work, beginning with the publication of the Curriculum andEvaluation Standards for School Mathematics in 1989, is a national vision of a powerfulmathematics education for all students, regardless of color, ethnicity, socioeconomic status,primary language, or gender (Lappan, 1999, p. 576).

When I first started this project, I thought I would answer my questionsby describing how beliefs that go beyond mathematics influence “typical”elementary classrooms, discussing how the teachers implemented reformideas in mathematics. However, the emerging concept of students’ needs,as well as teachers’ value judgments used in its definition, brought tothe surface a more challenging matter that permeated my experiencesin these two classrooms: lack of equity in mathematics education. Thecases of Teresa and Julie provide an opportunity to consider the issue ofmathematics for all. Can reform documents promote such vision?

I want to make clear that Teresa and Julie are the heroes of the storiespresented – never the villains. They are not, in a Machiavellian way, tryingto hold students back, lessen their chances of going to college, or ensurethat they will remain where they are in the socioeconomic spectrum. Quitethe contrary, both of them use their best judgments when defining whattheir students need, and once they form this concept, they direct their workand all their efforts toward meeting these needs. Most teachers try to offerstudents what they believe is best for the children’s future. Their decisionsof what is best, however, are value-laden; they are mediated by the distinctsocial contexts in which teachers operate and the different children theyteach.

In the cases of Teresa and Julie, because their students come fromdifferent socioeconomic backgrounds, they are learning different mathe-matics. Mathematics can represent learning as order or learning as fun.Children are also learning different social behavior patterns throughmathematics. Mathematics can mean following rules or thinking criticallyor creatively. In these case studies, the differences observed in mathe-matics instruction went along socioeconomic lines.5 Teachers’ values andexpectations are connected to their perceptions of students’ lives and needsaccording to socioeconomic variables. Thus the emerging issue from thisresearch mirrors the current debate about equity in mathematics education.We believe that all children can learn mathematics, but which mathe-matics are children from different socioeconomic backgrounds having theopportunity to learn?

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Anyon (1981) reported that students from different social-class back-grounds were exposed to different types of mathematical knowledge.Working in five New Jersey schools (two working-class, one middle-class,one affluent professional, and one executive elite school), she concludedthat as students’ socioeconomic level increased, instruction went fromrote, to creative, to analytical. Oakes (1990) found that poorer childrenexperienced different mathematics and had less access to science andmathematics in their curriculum. Using data from the National ScienceFoundation’s 1985–1986 National Survey of Science and MathematicsEducation, she argued that, during the elementary grades, the experi-ences of children from low-income and from more affluent families differin “small but significant ways”. These experiences become “strikinglydifferent” by the time the students reach secondary schools. In partic-ular, Oakes reported that low-income students have less-extensive andless-demanding mathematics programs available in their schools.

Secada’s (1992) review of research dealing with children frompreschool to high school showed that students from low socioeconomicbackgrounds were the lowest achieving in mathematics. He observed that,many times, people seem to accept different achievement levels acrossdifferent socioeconomic backgrounds as being normal, natural, inevitable,explainable, or even acceptable.

In this research, I observed well-intentioned teachers who operatedwithin a deficit model, accepting different expectations for children fromlow socioeconomic backgrounds as natural. These teachers worked withincontexts that supported their practices and they were viewed as profes-sionals by their peers, administrators, and students’ parents. Althoughthere is no data to support the claim that others in these schools sharedTeresa’s and Julie’s ideological vision, teachers and administrators sharedtheir perceptions about children and their concerns for students’ needs.

Disputing claims that reform-oriented curricula may not be beneficial toall students, Boaler (2002) argued that teachers’ mediation is fundamentalto implementing equitable mathematics instruction. In her study, teachers’beliefs seemed particularly important. The teachers Boaler presenteddiffered from Teresa and Julie in that they believed that reform-oriented,open-ended mathematics could benefit all students. They believed it wastheir job to make this type of mathematics accessible to all. These teachersrefused to let others train them in the “pedagogy of poverty” (Haberman,1991); they refused to let children’s home lives impede the explorationof rich mathematics. As a result, they increased student achievement andreduced inequalities among youngsters.


Boaler’s results strengthen the importance of understanding teachers’beliefs and practices in the study of reform. They show that teachers whoare committed to the principle that all students can learn mathematics ata significant level can make it happen in their classrooms. For teacherslike Teresa and Julie, however, committing to such principle requires adifferent vision of the world, a different perception of what students need,and a different ideological interpretation of reform documents.

To promote mathematical power for all, mathematics educators need tomove beyond the production of reform documents, or at least the produc-tion of umbrella documents (Apple, 1992) under which all can fit. AsStigler & Hiebert (1999) remind us, teaching is not a simple skill but acomplex cultural activity. Features of teaching that are recommended inreform documents such as the NCTM Standards are easily misinterpretedwhen one lacks references to “the system of teaching in which they areembedded or the wider cultural beliefs that support them” (p. 108).

Our current reform rhetoric does not sound the same to all ears – andwe may discuss whether it ever will. Still, if we want to have a biggerchance of promoting changes in the school mathematics experience of allchildren, offering all of them what we see as best mathematics, we needto increase our efforts in the direction of clarifying some of the underlyingideological assumptions that support current calls for change in mathe-matics education. What are the ideological underpinnings of our currentvision of reform? This fundamental question will need to be addressed ifwe are to educate teachers who can successfully promote equal, powerfulmathematics truly for all children.

ACKNOWLEDGEMENTS

This research is based on the author’s doctoral dissertation conductedat Indiana University under the supervision of Dr. Frank K. Lester, Jr.The study was supported by a scholarship from Conselho Nacional deDesenvolvimento Científico e Tecnológico – CNPq, Brazilian Government.

The author would like to thank Frank K. Lester and Tom Cooney fortheir comments on a previous version of the manuscript.

NOTES

1 The relation between beliefs and practice is dialogical and does not have a one-waycausal direction. Examining the ways in which this relation functions, however, goesbeyond the scope of this article, which focuses instead on revealing different factors thatmake up this relation.

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2 From here on, when using the term Standards I am referring to the 1989 NCTM docu-ment. Although additional documents were published in 1991 and 1995, I did not requireparticipating teachers to know them. This research was conducted prior to the release ofthe 2000 Principles and Standards for School Mathematics.3 All names are pseudonyms.4 There is a lot of discussion about the consistency between beliefs and practice (teacherswho do not “walk their talk”). For me, beliefs and practice are consistent – if in a study wefind they are not, than I think we asked the wrong questions.5 Since this project focused only on the teachers, no comments can be made about thestudents’ assimilation or resistance to the reproductive structure in place at their schools.

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(Ed.), Handbook of research on teaching (3rd ed.) (pp. 255–296). New York: Macmillan.Cuban, L. (1993). How teachers taught: Constancy and change in American classrooms,

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Handbook of research on teaching (3rd ed.) (pp. 119–161). New York: MacmillanPublishing Company.

Ernest, P. (1991). The philosophy of mathematics education. Bristol, PA: Falmer Press.Fennema, E. & Nelson, B. S. (Eds.) (1997). Mathematics teachers in transition. Mahwah,

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Haberman, M. (1991). The pedagogy of poverty versus good teaching. Phi Delta Kappan,73, 290–294.

Irvine, J. J. (1990). Black students and school failure. New York: Praeger.Glaser, B. & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine.Lappan, G. (1999). Revitalizing and refocusing our efforts. Journal for Research in

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introdução. [trans. Sociology and curriculum critical theory: An introduction] In A. F.Moreira & T. T. da Silva (Eds.), Currículo, cultura e sociedade (2nd ed.) (pp. 7–37). SãoPaulo, Brazil: Cortez Editora.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.

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(Eds.), Mathematics teachers in transition (pp. 3–15). Mahwah, NJ: Lawrence ErlbaumAssociates.

Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and trackingopportunities to learn mathematics and science. Santa Monica, CA: Rand Corporation.

Raymond, A. (1997). Inconsistency between a beginning elementary school teacher’smathematics beliefs and teaching practices. Journal for Research in MathematicsEducation, 28, 550–576.

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University of GeorgiaMathematics Education105 Aderhold HallAthens, CA 30602-7124USA

BOOK REVIEW

Mason, John (2002). Researching your own practice: The discipline ofnoticing. Routledge Farmer. xii + 272 pages

Q: Which of the following describes John Mason’s new book ResearchingYour Own Practice: The Discipline of Noticing?

a. A practical guide to becoming a better teacherb. A commentary on research methodsc. An attempt to define a new field of research and practiced. A Zen meditation

A: all of the above, and more.Noticing is an ambitious book. As Mason writes,

This book is about transitions:from being a sensitive practitioner awake to possibilities, perhapsdissatisfied with the status quo;through reflective practices,to engaging in productive and effective personal professional devel-opment;through drawing on published research and colleagues’ experience,to contributing to the professional development of others;through being systematic and disciplined in recording,to undertaking research and participating in a research community(p. 5).

The core of the book is preceded by the usual front matter, but in Mason’scase the usual isn’t usual. For example, the Preface begins as follows:

Noticing is something we do all the time. For example, as you read this, you may becomeaware of the fact that you are holding a book, and also of the fact that you are reading it.You can also become aware of sensation in your hands and feet, and of noises around you. . . (p. xi).

Now, what does that have to do with professional development, or withresearch?

A lot, in fact. Professionals become professionals by developing per-ceptions and skills, and by routinizing them. We survive, we even thrive,


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thanks to our routines. They enable us to recognize and categorize situa-tions, and to react to them without having to think things through fromscratch. But routines also deaden us. When things seem familiar and wereact according to pattern or habit, we may not really be seeing what’sthere. That means that we may not be doing as well as we might. The artof noticing is to keep open to new perceptions while standing on the baseof skills, routines, and knowledge that enables us to function as well as wedo. The discipline of noticing is to keep such noticing productive – afterall, we wouldn’t be very effective teachers or researchers if we focusedon sensations in our hands and feet when we interacted with students! So,how does one open oneself up, focus on what counts, and reflect on itproductively to change for the better? And . . . hmmm . . . just what counts,and what does it mean to change for the better? Those too are objects forreflection. These issues, writ large, are at the core of Mason’s agenda.

The core of Noticing comes in six parts. Part I, entitled “Enquiry,”consists of a single chapter concerning “forces for development.” Masonmakes it clear that change is something intensely personal, something thatcomes from within; also that change is fostered by openness and reflec-tion. He will offer you, the reader, structured opportunities for growth andreflection, but you will need to pick up on his suggestions and make themyour own. It will not be easy – change is hard. In true Zen character, thebook begins with the following inscription:

I cannot change others;I can work at changing myself (p. v).

and ends with these words from the Tao Te Ching:

Knowing others is wisdom;Knowing the self is enlightenment.

Mastering others requires force;Mastering the self requires strength (p. 254: Tao Te Ching c.

300 BC).

“Enquiry” sets the stage with an informal discussion of some examples.Can you take an incorrect answer (for example, that 0.3 × 0.3 = 0.9) andcome up with some reasons why a student might produce it? Can youtake a standard teaching practice (e.g., repeating back what a student saysfor the whole class) and describe both its advantages and disadvantages?These are the first two of many “accounts” in the book, telegraphicdescriptions of experiences that can serve as grist for conversationbetween the author and the reader, or among readers. The goal of suchexercises isn’t necessarily to change anything, but to “problematize thenormal” – to reflect anew on things that might go unnoticed. Many of

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these exercises are abstract, but Mason also invites you, in clever ways,to refine your observations of your own practice. Pick nine students atrandom, he suggests, making a table as follows:

A B C

D E F

G H I

Consider the triples you get by considering rows, columns, and diagonals:(ABC), (ADG), (AEI), etc. Can you find a word that describes two of thestudents in the first triple, but not the third? (For example, might two ofthe students be diligent about bringing in homework, while a third is not?)Can you find a different word for the second triple, and so on? How doyour descriptions compare with those of others who know the students?How does your vocabulary compare? What do similarities or differencesin your descriptions suggest? “Forces for development” also discussesvarious personal and social forces on the individual, and ways to thinkabout them.

Part II of the book, called “Noticing”, consists of three chapters inwhich Mason introduces a series of distinctions and definitions that helpto sharpen and codify what one notices. The first chapter, called “Formsof Noticing”, introduces the technical terms intentional noticing, marking,and recording. Regarding intentionality: it’s one thing to observe thingsin passing, but it’s quite another to set oneself out to look for things, andcarry through. Mason gives some simple “try these soon” exercises to thereaders, such as “Set yourself to notice when you walk through doorways,and mark it each time by saying something like ‘I am walking througha doorway’ with a stress on the I ” (p. 31). Marking means singling outsomething you’ve noticed to the degree that you can remember it and“re-mark” upon it later to others. Recording is, of course, a tool for furtherreflection and exchange.

The second chapter, “Impartiality”, introduces further distinctions,specifically between “accounting of” and “accounting for”. The former is(as much as possible) a neutral description of observed phenomena – thatis, simple description “minimizing emotive terms, evaluation, judgements,and explanation”. Mason’s point is that people’s descriptions of situationsare often value-laden or judgmental. It is better, he argues, to separatethe description from the judgment. I agree. It is a fascinating exerciseto have a group of people watch a videotape of (say) a student-teacherinteraction and write down what they “see”. What they write is usuallyin the form of summary judgments, and those judgments often differ

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widely. These conflicting interpretations often lead to arguments, whichlead to going back to the data (the tape), which leads to re-examination(often many times!) and a more objective description of what took place.Training oneself to (try to) be objective is a valuable exercise. Accountingfor events can best take place, Mason argues, after the events have beenobjectively described. Of course, being objective doesn’t necessarily meanbeing boring. Through a series of “accounts”, Mason indicates how onecan provide brief-but-vivid descriptions of events, with enough detail toconvey the flavor of what happened. The third chapter in Part II, “BeingMethodical”, is a two-page discussion of the pluses and minuses of beingsystematic.

In Part III, “The Discipline of Noticing”, Mason begins to system-atize noticing. “Disciplined Noticing” elaborates a series of steps in the(effective) noticing process: keeping accounts, developing sensitivities,recognizing choices, preparing and noticing, and validating with others.Here the reader is being offered some sensitivity training. Can you stopif you feel your voice becoming harder and your words becoming ironic?What does that say to you about your reaction to the person you’re inter-acting with? What message does it send to that person? Can you pull upshort, consider alternatives, take a different path? Can you keep an eyeopen for things that might normally pass you by, and act on them? Can youbreak out of your well-developed professional habits to see new options?Can you react “in the moment”?

Of course, any individual’s systematic observation, reflection, andaction could be highly idiosyncratic; what I see may or may not correspondto what you see. Thus the need for the next chapter, “Validity”. “Validationof noticing and acting is based not on convincing others through rationalargument or through the weight of statistics or tradition, but rather throughwhether the other can recognize what is being described or suggested . . .

and whether they find that their own sensitivities to notice are enhanced”(p. 93).

Part IV, “Using aspects of the discipline of noticing”, moves to appli-cations. “Probing Accounts” looks at the ways that accounts (records ofthings noticed) can be used. Having recorded aspects of some event, howcan one profit from the account? How can it be used to raise issues to thinkabout, things to work on, things to try? This relatively long chapter (40pages, where quite a few are 10 pages or less) provides numerous sampleaccounts and ways to work with them – looking for underlying assump-tions, for familiar patterns (scripts), for issues of importance; and more.Almost two dozen accounts are given in this chapter – some discussed,

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some as stimuli. Here is account 49, which is given without comment.Nothing is truly representative, but this account is as representative as any:

Account 49: Imagining.T: Now boys and girls, shut your eyes and imagine a large fish tank half-

filled with water. Drop in five marbles: plop, plop, plop, plop, plop.Alright, the marbles are now at the bottom of the tank.

S: But my marbles are in the middle!T: [Rather impatiently, eager to get on with the lesson]: What do you

mean by ‘in the middle’?

I then realized that his marbles were somehow suspended in the water.I asked him whether he could try to push his marbles down. He said hewould try as he flashed his unmistakably winsome smile and shut hiseyes as if nothing had happened (p. 138).

The next chapter moves to issues of “Noticing and Professional Devel-opment”, PD being one major application of noticing. The chapter offerssome general precepts for working with others: “The classic source fordisturbances to professional practice [things to notice and discuss inprofessional development] is in watching colleagues at work (p. 140)”.“Real change also requires the support of a compatible group of peoplewhose presence can sustain individuals through difficult patches, and whoprovide both a sounding board and a source of challenge for observations,conjectures, and theories” (p. 144). There are suggestions for working withcolleagues both informally and formally.

In Part V, Mason moves “From Enquiry to Research”. “What isResearch?” breaks research down into a range of components (ques-tions asked; objects of study; methods; purpose of study; data; analysis;claims; products; validation) and gives a bird’s-eye view of each. In asense, the chapter offers a global orientation to research (noticing andrecording in ways that are important and convincing to others) – startingwith the observation that all observations and questions are grounded inone’s (often unspoken) theoretical assumptions, so that the real work ofresearch begins with an unpacking of those. The next chapter is called“Noticing IN research”. Mason hits the nail on the head in his chaptersummary: “. . . refining their noticing is what researchers do, and theyhave this in common with any expert” (p. 182). It is followed by “NoticingAS research”. Here Mason revisits the components of research discussedabove, showing how they play out in the Discipline of Noticing. Thequestions asked, for example, are “What am I attending to, moment bymoment? What choices are available to me moment by moment? Whatpossible acts could I initiate? What am I sensitised or attuned to notice

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and what other possibilities might there be?” The objects of study are “myown actions, my own choices, the structure of my attention” (p. 185). Thepurpose of the study is, in effect, self-enlightenment (in the language of the60s, consciousness-raising) and the support of others in doing the same.

There are, of course, issues in moving from what any one individualperceives to the production of robust, reliable, valid results. There are alsoissues of falsifiability. Here we enter the realm of meta-noticing:

Suppose an external observer fails to see what I claim to see. Suppose even that I am unableto persuade colleagues that something I am noticing is noticeable or fruitful, to them? . . .

There is always the possibility that colleagues are not oriented or in an appropriate state tonotice what I notice. It may be that I am deluding myself; it may be that my attempts topoint out what I see are not succeeding; it may be that such noticing is not appropriate forthem at this time (p. 189).

Of course, there are the standard methodological safeguards. One doeswhat one can – using accounting-of, seeking resonance, working from thebottom up as in grounded theory. While “there are no guarantees”, thereare some safeguards when “Researching From the Inside”. For example,here is one of Mason’s “things to try now”:

If possible, arrange with a colleague to observe each other, and to describe brief-but-vividmoments noticed. See whether you can reduce judgments and explanations and concentrateon specific phenomena of teaching. See whether you notice a heightened awareness whenthey are observing you, and whether that continues when they are not even there (p. 210).

Researching one’s experience is ultimately a form of soul-searching. Thusin this chapter we find questions such as

Is experience merely the story we tell, or is there more? (p. 211)What is referred to by ‘myself’? Is there a unique core self that is ‘me’? (p. 212)

Such questions are followed by exercises designed not to answer the ques-tions, but to provoke readers into noticing more about themselves. Forexample, the ‘myself’ question is followed by a number of questions,including these:

What are you like as a householder (cleaning, washing, preparing meals), as a professionalor as a consumer (shopping, seeking entertainment, . . .)? How do you change when youhear different people speaking on the other end of the telephone? How do you switch fromone to another? Can you catch the moment of transition? (p. 212)

Finally, the three chapters in Part VI raise and address “Questions andConcerns” about practitioner research, qualitative research, and noticingrespectively. An acute observer, Mason is also an acute doubter. In a seriesof very productive questions on pp. 227–228, he asks, for example: “Datais a construction by a researcher: How do I know I would construct the

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same record in the same situation again?” And, just as you settle intothinking about the question, Mason undermines you:

As soon as I set out to describe a situation, I am caught in the conceit that there is asituation to be described. Yet reflection on the nature of events, particularly after watchingvideo recordings, leads to questioning what it is that constitutes ‘the event’ (p. 228).

In short, Mason sets out to provoke, to disturb, to induce you as the readerto observe and reflect. No, that’s not quite right. Mason wants to incite youto act, observe, and reflect. This is a book about change. And while thebook’s motto is “I cannot change others; I can work at changing myself”,Mason tries to connect with readers in a way that they will try to change.That’s a difficult task face-to-face. It’s near-impossible in a book.

If anybody could succeed at this task, it’s John Mason. Twenty yearsago Mason co-authored the book I didn’t dare to write, Thinking Mathe-matically. For some years I had taught a problem solving course, live, inmy own idiosyncratic way. Much of what I did was very much “in themoment” – being in the moment is, as Mason notes, a large part of whatteaching is all about. How could one capture that in a book? ThinkingMathematically succeeded by engaging the reader in dialogue, by takinghuge risks and succeeding. Yes, teaching is “in the moment” – but it’salso about noticing, expertise, and routine. Each group of students in myproblem solving course is new, each student unique – but at the same time,their behavior as a collective is remarkably predictable. When I’ve founda problem that “works” and it becomes part of my repertoire, I can usuallyanticipate student reactions and work with them. [Have you ever seen thecalling card that says “Pick a number from 1 to 4” on one side? The otherside says “Why did you pick 3?” It hits the mark a very large percent ofthe time.] Likewise, there are some gambits in problem solving that havea good chance of hitting home. “Stuck? Try thinking about this . . .”.

Mason and his colleagues took this insight and made it work it inprint. The result, in Thinking Mathematically, was a non-standard andpowerful book. The book invited students to work a range of problems. Itengaged them in conversations about their work on the problems, helpingwith problem solving strategies; it encouraged them to reflect on the waysthey went about the tasks, and to develop an “internal monitor”. Masonand colleagues had found a successful mechanism for taking some of the“connecting” that takes place in live interactions and stimulating it via text.This means of communication – seeking resonance via the stimulationof shared experience – is very powerful when it works. Inviting/incitingsomeone to act and knowing enough about that person’s probable reactionto be able to have a meaningful conversation about it is an act of great skill.It is a major component of both teaching and professional development. It

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is the approach taken in Researching Your Own Practice: The discipline ofNoticing.

I admire Mason’s current enterprise, and I find much to think about inNoticing. Yet, I do not find this book as successful as I hoped it wouldbe. It may be because of the scope and ambition of the book itself. Acouple of hundred pages may not be nearly enough to achieve the book’sgrand ambitions. It may be that the space of possibilities is too large forMason’s method to work (that is, the method may be successful in thecase of problem solving because, if you can induce someone to work ona problem that you know inside-out, you may be able to guess what thatperson will do, and have a conversation in print. But the range of reactionsto Mason’s accounts of practice may be so large that the author’s hit rate ismuch lower). Or perhaps I’m simply not as receptive as I might be.

I want to start my commentary with an affirmation of Mason’s basicpoints:

• Noticing is a fundamental element of expertise;• Routines are both the basis of expertise and one of its greatest

potential pitfalls;• Seeking and taking advantage of “disturbances” (events that chal-

lenge the routine) is a powerful tool for personal and professionalgrowth.

I claim that this is the case in all aspects of my life – for myself as teacher,as researcher, as husband or parent, as reader, as producer or consumerof food (or just about anything else). As a teacher, for example, part ofmy effectiveness resides in my knowing what my students are likely todo in many circumstances, and being prepared for it. When a class of 20or 30 students is working on a problem that I’ve discussed with manystudents in previous years, the odds are that most of the comments thestudents make as they discuss what they’ve tried will be things I’ve heardbefore. Knowing what to expect, I’m ready with comments and sugges-tions. This, of course, is a double-edged sword. On the one hand, thisreadiness frees me up to think, to be attentive and responsive to nuances ofstudent behavior. On the other hand, there is always the danger that I’ll goon auto-pilot, depending so much on my routines that I don’t really hearwhat my students are saying.

As a researcher, my whole career is devoted to noticing (and to convin-cing others that there is substance to what I notice). My research group has– literally – spent more than a year analyzing one hour of videotape of aclassroom, trying to understand as best we could what was taking place.Looking, looking more, looking yet again; comparing and contrastingideas; theorizing; testing and rejecting ideas; these are all part of the enter-

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prise. As with teaching, they require discipline. As with teaching, hardwon professional knowledge and experience provide the perceptions androutines that comprised expertise. And as with teaching, one goes staleunless there is a constant push to see new things.1

The same is true in my personal life. As a family member, I have gottento the point where I can finish others’ sentences (and they mine), and pushtheir buttons (and they mine) – but the relationships only stay alive if theyare consistently renewed. As a reader, I find that I am often most engagedwhen the author has captured me with an observation, a turn of phrase, afine detail that makes the text spring to life. As a “foodie”, I pay a lot ofattention to flavors and textures. A new taste, a new combination, will openup possibilities and get me going. And so on. Mason is absolutely right –noticing is everywhere.

Let me now turn to issues of teachers’ professional development, a topicthat has been much on my mind of late. My recent experiences raise ques-tions for me regarding broad needs for professional development, aboveand beyond noticing. In the preceding paragraphs I argued that noticingis (or should be) everywhere. But the question is, how large a role can orshould it play in professional development?

I begin by providing a description of the American context. In theUnited States, what has come to be called the “Algebra for All” movementis aimed at helping as many students as possible to do well at algebra.The emphasis on helping all students learn mathematics comes from awish to address (better, redress) historical inequities associated with math-ematics instruction in the U.S., specifically the differential success ratesof various socioeconomic and ethnic groups in mathematics (Lee, 2002;National Research Council, 1990). Algebra has “gateway” status in theAmerican curriculum. For many years, students who have failed at algebra– a disproportionately high percentage of whom are poor and/or childrenof color – have been deprived of skills and credentials that are widelyconsidered passports to the American economic mainstream. The stakesare getting higher as many states institute standardized tests as require-ments for high school graduation. These tests typically contain a largeamount of algebra. Thus failing at algebra may mean failing to graduatefrom high school. Given the relatively high failure rates of African Amer-icans, Latinos, Native Americans, and poor children, access to high qualityalgebra instruction has become an issue of civil rights (Moses, 2001;Schoenfeld, 2002).

What makes the issue of professional development even more complexthan “simply” trying to reach more students is the fact that the algebracurriculum in the U.S. has evolved substantially over the past decade.

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Traditionally, algebra instruction focused on mastering a body of proce-dures (e.g., solving linear and quadratic equations; factoring) and thesolution of stereotypical classes of word problems (e.g., distance-rate-time problems, age problems, mixture problems). An emerging curricularview is much broader. The algebra standards in the National Council ofTeachers of Mathematics’ (2000) Principles and Standards, for example,focus on the following: understanding patterns, relations, and functions;representing and analyzing mathematical situations and structures usingalgebraic symbols; using mathematical models to represent and under-stand quantitative relationships; and analyzing change in various contexts.Many teachers, who learned the traditional curriculum, find themselves ill-prepared to help students deal with this different view of algebra. In short,teachers around the United States find themselves in need of professionalexperiences that would help them to develop a deeper conceptual under-standing of the content of “algebra” and to make that content accessible toa much broader range of students than typically succeed at it.

These issues lie at the core of the professional development effortsthat have concerned me as part of my involvement with a project calledDiversity in Mathematics Education (DiME). To help concretize thisdiscussion, I reproduce below an artifact of our work with the BerkeleyUnified School district. We produced it in response to a request to helpteachers think through the ways in which different instructional programsmight support all students in the learning of middle school mathematics,leading to the study of algebra in eighth grade. See Table I.

My hopes and expectations are that conversations about these issuesover the next few years will lead to significant professional growth formany of the teachers and to enhanced educational opportunities for theirstudents. Note that although this list of questions was generated withinthe context of a concern for “algebra for all”, they are more general. Withminor variations they would be appropriate for conversations about thecontent of mathematics instruction at any grade level. Note also that thequestions are only a starting point. In many ways, these questions dealwith values. Just what are our goals for our students? How do they meshwith the constraints the system (apparently) imposes on us? How can wereach as many students as possible? Do we count student disposition asan outcome of instruction? Also, the questions deal, at least implicitly,with issues of teacher knowledge. For example, what are the big ideas ofnth grade mathematics? To answer this question well, you have to knowwhere the mathematics is going – what are the big ideas in the curriculumas a whole, what does the mathematics at this grade level lead to? In theUnited States, elementary and middle school teachers tend to teach one

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TABLE I

Criteria for Program Evaluation

Content

• What are the big ideas of mathematics at this grade level? How well do thecandidate instructional programs address them?

• To what degree does the mathematics in the programs come across as coherent?

• Do the programs help students understand the material, or do they focus simply onprocedures? (How do we want to test for that?)

• Do the programs lay a solid foundation for the study of algebra? (What would welook for?)

Equity

• How does each program address the goal of mathematics for all? How does whatcan be done with the program fit with district-wide plans to support diversity?

• Making math accessible to all kids: Do the programs offer a large number of“ways in” (multiple entrance points) to the mathematics? Do they support differentlearning styles? A range of classroom activities, pedagogy?

• Does the program provide access and handholds for English Language learners?

• Are the materials accessible to parents and other helpers?

• What kinds of support structures, inside and outside the school, are necessary togive everyone a “fair chance?”

Assessment

• How would we know what the students have learned? What kind of assessmentswould reveal what the kids really know and can do?

• Program-based student assessment: Do the programs provide varied ways ofdemonstrating competence?

• Are students “ready” for the next math course? By what criteria?

Student reaction to program

• How do the students feel about mathematics, about the books/programs? Dothey find the mathematics meaningful? Interesting? Easy/hard to make sense of?Arbitrary? Coherent?

• Issues of disposition: What can we say about student persistence, sense of whatmath is all about, etc.?

Teacher support

• How do the programs support teachers – beginning teachers especially – in comingto a deeper understanding of the mathematics they teach? (that is, can teacherslearn form the programs?)

• How accessible are the big ideas of the curriculum to teachers? Are there ways tokeep a focus on the big ideas discussed?

• Are teachers given support in understanding what their students understand? Inhow to evaluate a lesson-in-progress?

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TABLE I

Continued

Systemic issues

• How do the programs line up with the state standards, national standards, bigideas? How can teachers become confident that students are getting what theyneed, that there’s room to build understanding and to do well on tests that willaffect kids’ and teachers’ lives?

Overall assessment

• What are the strengths of each program, and the weaknesses? How can they besupplemented?

• Given all of the above, is there strong reason to use one program over the other(appropriately supplemented), in any particular contexts? (e.g., for new teachers,or . . .?)

grade or one small grade band for many years. This can mean that theyknow less than they might about what their students have experiencedbefore entering their classrooms, or where their students will be goingmathematically. There are numerous issues surrounding assessment. Howdo you craft tasks that have the potential to reveal student understandings?How do you interact with students in ways that enable you to get a goodgrasp on what they understand? And more. In various ways, these issueslie at the core of our agenda for professional development.

To sum up in brief, some of the factors I see as necessary for successfulprofessional development are the following:

• Openness to noticing and help in developing it. Continually refiningone’s sense of vision is an essential aspect of professional growth.

• Focus. There are myriad things one might attend to in professional(or self-) development. Typically, teachers (whether individuallyor collectively) have relatively little time to invest in PD. Unlessdecisions are made to focus on a small set of centrally important activ-ities, the limited amount of time and energy available for professionaldevelopment can be squandered.

• Knowledge – at least enough knowledge to be able to focus on theright things, and to have a sense of what directions would lead toimprovement.

• Community and culture. It is difficult to sustain efforts at changein isolation. The right kinds of collegial cultures provide emotional,intellectual, and professional support.

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• Environmental stability. Productive change is a slow and evolutionaryprocess. Environments in which things change rapidly (and capri-ciously) make it extremely difficult for the individuals within themto work coherently toward change.

• And frequently, help in establishing some or all of the above.

A look at this list suggests that those concerned with professional develop-ment will need to be concerned with a lot more than noticing.

I want to state unequivocally that the teachers I know are talentedindividuals who are sincerely devoted to helping their students do well;I believe this is true of the teaching profession in general. And, I have saidthat noticing is important. But, if a group of teachers set on their own ona program of noticing, what might be the focus of their noticing? It’s hardfor me to say.

For example, it is not at all clear to me that many American teachers,when trying to decide about the relative merits of two proposed instruc-tional programs, would think on their own to raise questions such as:“What are the big ideas of mathematics at this grade level? How well dothe programs address them?” Part of the reason is that (broadly speaking)they are not socialized to do so – such questions have not been institu-tionalized as part of their professional discourse. Part is that, given theway the system works, the “answer” to what is important is handed downto teachers, de facto, by those who serve on curriculum framework andtextbook adoption committees. But thinking through what “counts” forone’s students is essential. If teachers do not raise and resolve such issuesfor themselves, but instead become the passive vehicles for implementingothers’ curricular decisions, our students are ill served.

One could raise similar issues regarding most aspects of professionalpractice. For example, how do the teachers think about issues of assess-ment, of understanding what the students understand? There are tools andtechniques for delving deeply into student understandings of importantmathematical ideas; these are accessible and can be used, IF they becomepart of the discourse. But will they, if the environment is not configured inways that induce teachers to grapple with them?

To cast this discussion in theoretical terms, let me note that issues oflearning and development are very much the same for adults as they arefor children (see, e.g., National Research Council, 2000, especially pages26–27, for an interesting commentary regarding professional developmentin this regard). The Vygotskean construct of zone of proximal development(ZPD; see Vygotsky, 1978) has gotten a great deal of play in recent years.But one point that is often overlooked is that “the ZPD” for a particularindividual at a particular point in time is not a single fixed entity; it is a

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function of the social environment in which the in individual finds him-or herself. That is, the environment might provide affordances for learningin particular ways, for thinking about some issues . . . or it may not. Noti-cing operates within the teacher’s zone of proximal development. What theteacher notices, and thus what he or she works on, may depend very heavilyon what the environment offers (or doesn’t). The sad fact is that the currentenvironment for many teachers is not likely to be very nourishing. Hencenoticing alone, without help in perceiving possible avenues of growth, maynot be sufficient for the kinds of professional growth we would like to see.

Having said this, let me return to Researching Your Own Practice: TheDiscipline of Noticing as a vehicle for self-enlightenment and, concomit-antly, for professional development. As indicated above, I think that Masonis right that noticing lies at the core of our being, and at the core of profes-sional development. Mason has zeroed in on something big. As noted, thetopic may simply be too big for one book, at least if one hopes for practicalends. There is so much that might be noticed, and so much that might bemissed! At the same time, the book is well worth reading. I jumped at thechance to review it, because (as I told the editor) anything by John Masonis worth digging into. Mason focuses on things that count. Given the formof interaction he is forced to adopt in text, it is inevitable that some of hisaccounts or descriptions would fail to resonate with me, or with any reader.But it’s not the “misses” that count in a book like this, it’s the “hits”. Masonhas a way of getting under my skin, of provoking me to think about issuesI might otherwise let slide by. And that’s a lot.

NOTE

1 There is a research component to the book that I cannot address, for lack of space. Letme say simply that enhanced sensitivity to the issues one examines is always a good thing.So is a doubting stance (in moderation). I find it valuable to spend most of my time workingas though my assumptions make sense – and then periodically stopping to examine themcarefully, and see if they do.

REFERENCES

Lee, J. (2002). Racial and ethnic achievement gap trends: Reversing the progress towardsequity? Educational Researcher, 31(1), 2–12.

Moses, R. P. (2001). Radical equations: Math literacy and civil rights. Boston MA: BeaconPress.

National Council of Teachers of Mathematics (2000). Principles and standards for schoolmathematics. Reston, VA: National Council of Teachers of Mathematics.

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National Research Council (1990). A challenge of numbers. Washington, DC: NationalResearch Council.

National Research Council (2000). How people learn: Brain, mind, experience, and school(expanded edition). Washington, DC: National Research Council.

Schoenfeld, A. H. (2002). Making Mathematics work for all children: Issues of standards,testing, and equity. Educational Researcher, 31(1), 13–25.

Vygotsky, L. S. (1978). Mind in society. Cambridge: Harvard University Press.

Alan H. SchoenfeldUniversity of California at Berkeley

KONRAD KRAINER

EDITORIALTEAMS, COMMUNITIES & NETWORKS

In its sixth year, the Journal of Mathematics Teacher Education launchesfor the first time an issue that is dedicated to a particular topic. Althoughit is not intended to have such issues on a regular basis, wheneverwe find a number of promising papers on an interesting topic we willconsider another special issue. The current issue “Teams, Communities &Networks” emerged in this way. This editorial sketches the background ofthe issue, reflects the increasing importance of its theme and introduces thefour contributions – two research papers, one paper dedicated to “TeacherEducation around the World” and a book review.

THE INCREASING AWARENESS OF THE SOCIALDIMENSION IN MATHEMATICS TEACHER EDUCATION

The growth of mathematics education as a scientific field can be regardedas a continuous process of having a deeper and deeper understanding ofthe complexity of learning and teaching. Such a growing awareness goeshand in hand with a shift of attention. At the beginning, a focus on mathe-matics and the belief in easy transferability of knowledge dominated. Laterthe active individual learner, the individual teacher as facilitator of activelearning and links to the “real world” became more and more important.Further, learning and teaching were increasingly regarded as interac-tion processes, embedded in social, cultural, organizational and politicalcontexts. Indicators for that movement are the emergence and usage of newtheories that go beyond cognitive views on learning (see e.g., Vygotsky,1978; Senge, 1990; Lave & Wenger, 1991; Ernest, 1994; Jaworski, 1994;Cobb & Bauersfeld, 1995; Brousseau, 1997; Willke, 1999) or discussionsof alternative modes of instruction and adequate learning environments.Although the social dimension (and I include here organizational andsystemic aspects) with emphasis on collaborative learning, sharing knowl-edge, designing didactical contracts, and negotiating norms, is reallyimportant, the confrontation of individual learners with challenging mathe-matics should be considered equally as an essential component of teaching.


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However, where the current practice of mathematics teaching is concerned,we seem still to be far away from taking up the challenge that working inteams, whole class discussions, projects and so on would be the dominantmode of instruction. In viewing mathematics as a powerful means forarguing, critical thinking, reasoning, and communicating, multiple socialcontexts where these processes can be practised are needed.

Increasingly, papers in teacher education refer to some kind of “com-munities”, for example “teacher inquiry groups” (Hammerman, 1997),“study groups” (Birchak et al., 1998), “communities of practice” (Wenger,1998) or “networks of critical friends” (Krainer, 2001). All these groupsare intended to facilitate joint reflection and to lead to an improvementof practice. Different kinds of research support these assumptions. Forexample, research on “successful” schools shows that such schools aremore likely to have teachers who have continual substantive interac-tions (Little, 1982) or that inter-staff relations are seen as an importantdimension of school quality (Reynolds et al., 2002). The latter study illus-trates, among others, examples of potentially useful practices, of whichthe first (illustrated by an US researcher who reflects on observations inother countries) relates to teacher collaboration and community building(p. 281):

Seeing excellent instruction in an Asian context, one can appreciate the lesson, but alsounderstand that the lesson did not arrive magically. It was planned, often in conjunctionwith an entire grade-level-team (or, for a first-year teacher, with a master teacher) in theteachers’ shared office and work area. [Referring to observed schools in Norway, Taiwanand Hong Kong:] . . . if one wants more thoughtful, more collaborative instruction, we needto structure our schools so that teachers have the time and a place to plan, share and think.

This demonstrates that “community” is always influenced by adminis-trative and organizational aspects such as support from administrators,time, space and other resources (see also the paper from Arbaugh inthis issue), by general conditions of the educational system (e.g., thepower of principals to hire teachers, the existence in schools of content-related departments which have specific responsibilities) or the culturaland societal character of a region or nation (e.g., the autonomy and repu-tation of teachers, a shared understanding in society about “good practice”at schools).

There are many reasons for aiming at more collaboration amongteachers. However, there are also potential negative effects that shouldnot be overlooked (see e.g., Noddings, 1996). For example, if a groupof teachers is not open to new ideas and new people, it may result inconformity and the exclusion of others. While collaboration and commit-ment inside the group increases, the collaboration with people outside

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decreases, thus creating “insiders” and “outsiders”. The gain in autonomyand power of one group, if not networked with others, might be a loss ofautonomy and power for the rest. Often, a lack in networking goes handin hand with a lack of reflection on one’s own practice (see e.g., Krainer,2001). This causes restrictive conditions for individual teachers’ profes-sional development, in particular for those who have little power (e.g.,novice teachers or teachers who change to a new school), but this mightalso have severe implications for the whole organisation. Interventions byteacher education programs into a school should therefore look not only atthe “target group” but also at other “relevant environments” like the wholedepartment, the principal, non-participating teachers etc.

TEAMS, COMMUNITIES AND NETWORKS

Co-operation and collaboration (the latter notion indicating a higher degreeof interaction, sharing of interests etc., see e.g., Peter-Koop et al., 2003)can be formal or informal, the members can be selected or take partvoluntarily, the goals and tasks can be pre-determined or negotiated.Referring to Allee (2000), I differentiate between “teams”, “communities”and “networks”. Teams (and project groups) are mostly selected by themanagement, have pre-determined goals and therefore rather tight andformal connections within the team. Communities are regarded as self-selecting, their members negotiating goals and tasks. People participatebecause they personally identify with the topic. Networks are loose andinformal because there is no joint enterprise that holds them together. Theirprimary purpose is to collect and pass along information. Relationships arealways shifting and changing as people have the need to connect.

Whereas teams have a long history in organizational development,communities and in particular networks are rather new. Xerox was oneof the first enterprises that promoted the establishment of “communities”in the 1980’s. Today “communities of practice” are deeply embedded intoXerox culture; also other companies like IBM or The World Bank use themas a means for further development. Wenger (1998, as cited in Allee, 2000,p. 7) describes three important dimensions of “communities of practice”:

• Domain: People organize around a domain of knowledge that givesmembers a sense of joint enterprise and brings them together.Members identify with the domain of knowledge and a joint under-taking that emerges from shared understanding of their situation.

• Community: People function as a community through relationshipsof mutual engagement that bind members together into a social

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entity. They interact regularly and engage in joint activities that buildrelationship and trust.

• Practice: It builds capability in its practice by developing a sharedrepertoire and resources such as tools, documents, routines, vocabu-lary, symbols, artifacts, etc., that embody the accumulated knowledgeof the community. This shared repertoire serves as a foundation forfuture learning.

This description does not seem to designate one a specific memberwithin the community, nor one coming from outside. A role like a“teacher” or an “external facilitator” (like a teacher educator) is notmentioned. A “community of practice” is more about autonomous andjoint learning by its members than about the teaching of others. In contrastto schools and universities, companies do not have a pre-determined canonof subjects, but a non-forseeable emergence of practical and interdisci-plinary problems that have to be dealt with. For example, mathematicseducation would never be a topic per se, but certain problems wouldneed mathematical considerations and would need informal mathematicseducation environments in which to explore solutions.

The success of “communities of practice” in organizational develop-ment can be explained by the fact that knowledge and learning havebecome central issues in the strategy of companies. Wenger’s book“Communities of Practice: Learning, Meaning and Identity” (1998) pro-vides a theory of learning that assumes learning as social participation(and is supplemented by a recently published book outlining models andmethods, see Wenger et al., 2002). How does this relate to those organiza-tions that seem to be primarily responsible for knowledge and learning – toschools and universities? Are they losing their monopoly for educationalaffairs? To what extent can an approach like “Community of practice” beapplied to learning at schools and universities? What can we learn from“learning enterprises”? What implication for research in teacher educationhas an approach that builds on “community of practice”? Similar questionsmight also be addressed concerning other approaches.

Mathematics educators are increasingly noticing the work of Lave andWenger (1991) and Wenger (1998), both with regard to learning mathe-matics and to mathematics teacher education (see e.g., Bohl & Van Zoest,2002; Graven, 2002). One challenge of translating Wenger’s approach intolearning in schools is his emphasis on learning, and in contrast to that,the minor role he ascribes to teaching. More about strengths and chal-lenges of Wenger’s book (1998) can be read in the book review by Gravenand Lerman in this issue, where further references to its application inmathematics (teacher) education teaching can be found.

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FACILITATING AND INVESTIGATING –BALANCING THE DOUBLE ROLE

When teacher educators aim at supporting teams, communities or networksof teachers, there is an implication of intervening into a social systemfrom outside, of dealing with concrete practice and of interactions withpeople within the system (internal experts like practitioners, administrativeleaders, etc.). Given that the teacher educators also aim at doing research,the intervention has to do with at least two different interests that have tobe reflected on and shared.

On the one hand, the internal experts expect that the interventionhas the intended outcome. When external experts are engaged, there arecertain expectations about effects and thus there is an interest in devel-opment: something (e.g., a new curriculum, teachers’ content knowledge,or collaboration among teachers) should be changed, improved, or furtherdeveloped. If possible, the relevant processes and products (in most casesthese two elements are narrowly interconnected) should be jointly expe-rienced and developed by the whole department and highly accepted byother responsible people. Of course it is in the interest of the externalexperts that these goals are achieved. In addition, the external experts areoften also interested in further developing their intervention model and incommunicating their model to a greater public, for example the scientificcommunity.

On the other hand, there is also an interest in understanding whichmight relate to getting more insight into either the specific practice orinto the specific intervention. In the first case, internal and external expertsoften share that interest. However, the interest might also aim at gettinginsights that go beyond the specific case and relate to the generation ofmore general scientific knowledge about the practice or to new theoreticalfoundations of interventions and their effects.

Taking these two interests into consideration, teacher educators havea delicate double role: on the one hand they are facilitators and aim atpromoting the further development of practice and the professional growthof practitioners, and on the other hand they are investigators and aim atincreasing their own understanding and theoretical knowledge in order toshare it in the scientific community. Among others, this means that theycope with the, often underestimated, challenge of balancing the “general”and the “particular”. Teachers are mostly more interested in a furtherdevelopment of their particular situation, whereas teacher educators oftenhave research interests and aim at looking at the situation from a moregeneral perspective (e.g., embedding the experiences into a larger body ofknowledge). However, going deep into the specific context, it seems that

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the process of “particularization” has priority over the process of “gener-alization”, both with regard to facilitating and to investigating. The doublerole of the teacher educators requires them to negotiate interests and totake ethical norms into consideration.

THE FUSION OF TEACHER EDUCATION AND RESEARCH

Working with teams, communities or networks of teachers and investi-gating their professional growth are activities where teacher education as afield of practice and as a field of research merge (see e.g., Cooney, 1994).These are the areas of “research in the context of teacher education”,“research on teacher education” and “research as teacher education” (seee.g., Krainer & Goffree, 1999). The result is neither a research where theinvestigators stand outside the practice (not having the improvement ofpractice as a goal) nor is it a research where teachers themselves inves-tigate their own practice in order to improve it (in the sense of actionresearch, see e.g., Altrichter et al., 1993). This is a type of research thatcombines intervention and research, and thus might be called interven-tion research (first ideas of that approach are described by Bammé, 2002;Heintel, 2002; Krainer, 2002). Intervention research does not only applyknowledge that has been generated within the university, but much more,it generates “local knowledge” that could not be generated outside thepractice. Thus this kind of research is mostly process-oriented and context-bounded, generated through continuous interaction and communicationwith practice. Intervention research tries to overcome the institutionaliseddivision of labour between science and practice. It aims both at balancingthe interests in developing and understanding, and at balancing the wish toparticularize and generalize.

Concerning the extent to which research is also a task of the practi-tioners, one could differentiate between participative, co-operative andcollaborative intervention research. Whereas in the first case the prac-titioners are informed about the goals, methodology and outcomes ofthe research and make data available, in the third case practitioners andacademic researchers jointly share all important tasks. Here, one adequatekind of division of labour is that practitioners investigate their own prac-tice (first order action research), and that academic researchers analyzeteachers’ findings (e.g., a comparative study with regard to teachers’ casestudies). In addition, they gather and analyze further data (e.g., a sampleof case studies about teacher’s professional development or a question-naire for all participants, the principal, students, etc.) or investigate theirintervention practice (second order research). In many cases, a mixture of

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participative and collaborative elements will occur and lead to differentkinds of “co-operative intervention research”.

CONTRIBUTIONS TO THIS ISSUE

The research paper by Andrea Lachance and Jere Confrey deals with aprofessional development intervention by a university team of two staffmembers and three graduate students into a secondary school’s entiremathematics faculty of thirteen teachers; five other teachers from neigh-boring schools also participated but the paper refers mainly to the firstgroup.

The initiative came from the university staff who presented a set ofmultimedia pre-calculus materials (as a means for helping teachers extendand deepen both their knowledge of mathematics and their technologicalskills) to responsible people of a school district. Whereas the researchersoriginally intended to get participants on a voluntary basis (and thus enthu-siastic teachers who disseminate their expertise), the district administratorand her staff convinced them of the need to integrate the interventioninto the district’s reform plan and its focus on vertical teams – teamsof teachers from a given high school and the middle and elementaryschools that feed that high school. This intervention strategy was basedon the assumption that joint work with the faculty of a single school isan effective means of encouraging change. Thus, the strategy includednegotiation of interests between researchers and district administrators,and is the result of combining the emphasis on teachers’ mathematical anddidactical knowledge and the emphasis on joint learning of teachers fromthe same organizational context. This was the seed of integrating contentand community.

Although all participants reported some reservation about taking partin the two-weeks-workshop, which was followed-up by a workshop daythree months later, and four of them even reported being “forced” by theprincipal to take the workshop and thus suffering significant financial loss,the university team apparently succeeded in creating a fruitful workshopexperience that contributed to the building of community within the depart-ment. Two indicators that demonstrate that effect are that the co-operationbetween the university team and the school, as well as the mathematicsteachers’ regular meetings continued. However, lack of proximity thatteachers have to each other, lack of support from administrators, lackof time and opportunities for professional exchange, staff turnover, largeclasses, and other constraints impede continued relationship building inthe practice context (outside the good working conditions of the workshop

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and support from external experts). Nevertheless, the study shows that eveninterventions that do not emerge as a wish of the “target group” can leadto considerable growth. However, strong elements of learning processesare needed, where the participants see considerable benefit for their ownsituation.

Maybe the most important contribution of that study is that it demon-strates that teachers’ joint mathematical investigations can act as a meansto build teacher community. Each kind of community building needs akernel, something important to be shared. It is much easier when thissharing happens “on equal footing” as one of the teachers expressed it.This means that dealing with new mathematical problems, including theuse of new technology, not only contributes to an individual’s mathema-tical content knowledge, but also enables them to experience the diversityof learning processes and to transfer that insight to the learning of theirstudents. The feeling of sitting “in the same boat” decreases fears, andopens the way for learning from each other. Without external expertscoming into the system, such processes of confidence building are moredifficult to achieve. Given the genesis of the intervention, the univer-sity team had to show a high level of sensitivity for the context of theteachers. They did not only bring in their expertise but also built stronglyon teachers’ expertise and autonomy (see e.g., one teacher’s assessment“where somebody was down to earth enough to tell you something andthen let you do it”). Since the goals of the intervention and the selec-tion of the participants were mostly pre-determined, the teachers couldbe seen, at the outset, as a team. However, after the workshop, the aspectof community-building grew stronger. The study shows clear features ofintervention research, starting from a participative version during the firsttwo workshops but with a good potential to increase the teachers’ interestand involvement in research matters.

The research paper by Frances Arbaugh shows features in commonwith the study above. In both cases, the intervention is directed towardssecondary teachers of a mathematics department in a US context, aims atcontributing to the further development of both content and communityissues, and the facilitator comes from a university. However, there aremany dimensions where the two cases differ considerably.

The initiative came from the mathematics department chair who askeda researcher from the university to support a group of teachers in workingon a new geometry curriculum that would be more student-centeredand inquiry based than the curriculum they were currently using. Thechair aimed at assistance in designing and implementing a collaborative,teacher-centered environment where the group works towards reaching

EDITORIAL 101

these goals. The researcher and the department chair decided jointly tobegin a “study group” that would focus on the teaching of geometry. Thegroup of nine teachers met ten times on a voluntary basis during one halfyear. Most meetings took place during school hours (with release time forthe participants), lasting about 2.5 hours on average. The meetings weretypically organized around two to four topics of discussion, initiated by theresearcher, the department chair and the teachers. The agenda for the nextmeeting was negotiated. In contrast to the intervention by Lachance andConfrey, the kernel of the group’s work was focused on their mathematicsteaching and not on their mathematics learning. Indicators for the successof the intervention are, among others, the fact that after the project theteachers continued to support collaboration, and that some members begananother study group focused on the teaching of algebra joined by newteachers at the school. Since the goals of the intervention were generatedwithin the practice context, the participation was self-selecting and theteachers had a clear joint enterprise that held them together, the group’sactivities can be regarded as those of a community of practice.

The study of Arbaugh also shows clear features of interventionresearch, aiming both at facilitating the group’s progress towards theirgoals and understanding better the effects of the study group approach.The research indicates some elements of co-operation, but they seem to bedeveloped mostly with regard to the relationship between the researcherand the department chair who has a delicate double role of leading thegroup but also participating in the process. It is not surprising that the studyconfirms Birchak’s et al. (1998) findings that teachers value participationin study groups because they contribute to building community and rela-tionships, to making connections across theory and practice, to supportingcurriculum reform, and to developing a sense of professionalism. However,the teachers’ voice gives them a really authentic meaning, and underlinesthe argument that the intervention increased participants’ confidence. Asin the case of Lachance and Confrey, the interactive and supportive style ofthe researcher contributed to a socially warm and cognitively challengingatmosphere of learning and collaborating. Even the reading and discussionof articles were seen as supportive elements and regarded as contributionsto the connection of theory and practice. In contrast to the interventionof Lachance and Confrey, the co-operation with the school seemed tobe based on better general conditions (e.g., the initiative came from theschool, release time for the teachers, smaller group). The participantsregarded the organizational aspects of study group meetings as elementsthat essentially influenced the success of the initiative.

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The contribution to “Teacher Education around the world” by RenukaVithal does not focus directly on the issue of teams, community andnetworking, and does not report on research. Nevertheless, some connec-tions to both are apparent.

Within their pre-service mathematics teacher education programme,ten student teachers voluntarily took part in a project where they taughtmathematics to “street children” in a particular urban institution in SouthAfrica. Each student teacher was expected to gather relevant data and towrite a portfolio. Intervention in their activity by their mathematics teachereducator, facilitating the activity of the student teachers, was related tostudent teachers’ reflections on their own intervention with the streetchildren from both a mathematical and social perspective. The reflectionsof the teacher educator were based on two student teacher portfolios andher own journal entries and lecture notes. These forms of (more or lesssystematic) reflection are similar to activities within the framework offirst and second order action research. Both teacher educator and studentteachers had roles of facilitators of learning processes and of investig-ators. Thus, this project can be seen also as kind of intervention research,although the generation of scientific knowledge is less marked than inthe other two cases. The focus is on mathematics learning, but it alsoreflects the innovative and challenging social context of these learningprocesses.

In this regard, all three papers combine content and community in agenuine and interesting way, and in all three cases the generation of newknowledge is strongly connected with local practice outside university.In contrast to the cases of Lachance and Confrey, and Arbaugh, whereteachers have stable membership within a department and aim at moreinnovative curricula as a joint enterprise, the student teachers in Vithal’scase have more informal connections to each other, and their joint enter-prise was restricted to a more or less loose common (voluntarily chosen)task. Thus this situation has features that come close to those of a networkwhere students take part because they find a specific topic attractive andare interested in sharing experiences they make in different contexts. Thestudents’ growth related to their learning about learners, about teachingand about relationships. As a further indicator of success, the programmewas expanded to other places, a community service programme was madea requirement for all prospective teachers.

All three papers report apparently successful intervention programmes.Although having a different genesis, general conditions, approaches tosupport (student) teachers’ learning and different ways to generate knowl-edge about the initiatives, the success of the programmes seem to have

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some important features in common. They all established learning environ-ments where participants not only planned and carried out meaningfulactions, but also were supported in reflecting deeply on their experiencesand sharing them with other colleagues. Apparently the three initiativescreated a dynamic balance between the dimensions autonomy, networking,action and reflection (see e.g., Krainer, 2001). In all three cases, gather-ing of data by the researchers both orally and in a written form wereinterventions into participants’ learning processes. They stimulated furtherreflections and networking among participants. It seems that the combi-nation of investigations with interventions and fostering participants’investigative attitude are important factors that bring theory and practiceof teacher education closer together.

The review of Wenger’s book “Communities of Practice: Learning,Meaning and Identity” (1998) by Mellony Graven and Stephen Lermanreflects the challenges that the work of Lave and Wenger opens for(mathematics) learning and teaching and teacher education. The reviewargues that Wenger provides a social theory of learning whose primaryfocus is neither the individual nor social institutions but “communities ofpractice”. Graven and Lerman problematize the low importance Wengerascribes to teaching, in comparison to the emphasis he pays to learning.In particular, in mathematics education, where the content-related knowl-edge of teachers is of great importance, the question is raised aboutthe role teachers should play in students’ learning process. The reviewsketches a current research project by Graven (2002) within the context ofa mathematics in-service teacher education programme in South Africa,where Wenger’s framework has been applied and extended. As in thepaper of Arbaugh, the phenomenon of confidence appeared as an addi-tional category that is worthy of further consideration. Graven and Lermanconclude by recommending the book to those interested in locatinglearning beyond individual cognitive development. The same recom-mendation might be offered with respect to the papers in this issue ofJMTE.

REFERENCES

Allee, V. (2000). Knowledge networks and communities of practice. OD Practitioner,Journal of the Organization Development Network, 32(4), 4–13.

Altrichter, H., Posch, P. & Somekh, B. (1993). Teachers investigate their work. Anintroduction to the methods of action research. London, New York: Routledge.

Bammé, A. (2002). Auf dem Wege zur Interventionswissenschaft. Unpublished paper.Klagenfurt: IFF.

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Birchak, B., Connor, C., Crawford, K.M., Kahn, L.H., Kaser, S., Turner, S. & Short, K.G.(1998). Teacher study groups: Building community through dialogue and reflection.Urbana, IL: National Council of Teachers of English.

Bohl, J. & Van Zoest, L. (2002). Learning through identity: a new unit of analysis forstudying teacher development. In A. Cockburn & E. Nardi (Eds.), Proceedings of the26th Annual Conference of PME, Vol. 2 (pp. 137–144). Norwich: University of EastAnglia.

Brousseau, G. (1997). Theory of didactical situations in mathematics (French originaltranslated and edited by N. Balacheff, M. Cooper, R. Sutherland & V. Warfield).Dordrecht: Kluwer.

Cobb, P. & Bauersfeld, H. (1995). The emergence of mathematical meaning: Interactionin classroom cultures. Hillsdale, NJ: Lawrence Erlbaum.

Cooney, T. (1994). Research and teacher education: In search of common ground. Journalfor Research in Mathematics Education, 25, 608–636.

Ernest, P. (Ed.) (1994). Mathematics, education and philosophy: An internationalperspective. London: Falmer Press.

Graven, M. (2002). Mathematics teacher learning, communities of practice and thecentrality of confidence. Doctoral Dissertation. Faculty of Science, University of theWitwatersrand, South Africa.

Hammerman, J.K. (1997). Leadership in collaborative teacher inquiry groups. Paperpresented at the Annual Meeting of the American Educational Research Association(Chicago, IL, March 24–28, 1997). (ERIC Document No. ED408249).

Heintel, P. (2002). Interventionsforschung (der Paradigmenwechsel der angewandtenSozialforschung). Unpublished paper. Klagenfurt: IFF.

Jaworski, B. (1994). Investigating mathematics teaching. A constructivist enquiry. London:Falmer Press.

Krainer, K.: Teachers’ growth is more than the growth of individual teachers: The caseof Gisela. In: F.L. Lin & T. Cooney (Eds.) (2001). Making sense of teacher education(pp. 271–293). Dordrecht, Boston, London: Kluwer.

Krainer, K. (2002). Ausgangspunkt und Grundidee von IMST2. Reflexion und Vernetzungals Impulse zur Förderung von Innovationen. In K. Krainer, W. Dörfler, H. Jungwirth,H. Kühnelt, F. Rauch & T. Stern (Eds.), Lernen im Aufbruch: Mathematik und Natur-wissenschaften. Pilotprojekt IMST2 (pp. 21–57). Innsbruck, Wien, München, Bozen:Studienverlag.

Krainer, K. & Goffree, F. (1999). Investigations into teacher education: Trends, futureresearch, and collaboration. In K. Krainer, F. Goffree & P. Berger (Eds.), Europeanresearch in mathematics education I.III. On research in mathematics teacher education(pp. 223–242). Osnabrück: Forschungsinstitut für Mathematikdidaktik.

Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. NewYork: Cambridge University Press.

Little, J.W. (1982). Norms of collegiality and experimentation: Workplace conditions ofschool success. American Education Research Journal, 19(3), 325–340.

Noddings, N. (1996). On community. Educational theory, 46(3), 245–266.Peter-Koop, A., Begg, A., Breen, C. & Santos-Wagner, V. (Eds.) (2003). Collaboration

in teacher education. Examples from the context of mathematics education. Dordrecht:Kluwer.

Reynolds, D., Creemers, B., Stringfield, S., Teddlie, C. & Schaffer, G. (Eds.) (2002). Worldclass schools. International perspectives on school effectiveness. London, New York:Routledge Falmer.

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Senge, P. (1990). The fifth discipline: The art and practice of the learning organization.New York: Currency Doubleday.

Vygotsky, L. (1978). Mind in society. Cambridge, MA: Harvard University Press.Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. New York:

Cambridge University Press.Wenger, E., McDermott, R. & Snyder, W. (2002). Cultivating communities of practice: A

guide to managing knowledge. Cambridge, MA: Harvard Business School Press.Willke, H. (1999). Systemtheorie II: Interventionstheorie (3rd ed.). Stuttgart: Lucius &

Lucius UTB.

ANDREA LACHANCE and JERE CONFREY

INTERCONNECTING CONTENT AND COMMUNITY: AQUALITATIVE STUDY OF SECONDARY MATHEMATICS

TEACHERS

(Accepted 7 February 2003)

ABSTRACT. The publication of the National Council of Teachers of Mathematics initialStandards (1989) has acted as a catalyst to begin reforming the way mathematics is taughtin the USA. However, the literature regarding reform movements suggests that changingour educational systems requires overcoming many barriers and is thus difficult to achieve.Reform in mathematics education, like reform movements in other areas of education,has thus been slow to take hold. One structure that has been shown to support educa-tional reform, particularly instructional reform, has been teacher community. This paperdiscusses a professional development intervention that attempted to start a professionalcommunity among a group of secondary mathematics teachers through in-service work onmathematical problem solving and technology. The results of this study suggest that theuse of mathematical content explorations in professional development settings provides ameans to help mathematics teachers build professional communities. Together, these twocomponents – mathematical content explorations and teacher community – provided thesesecondary mathematics teachers with a strong foundation for engaging in the reform oftheir mathematics classes.

KEY WORDS: mathematical content knowledge for teachers, problem solving andteachers, professional communities, professional development and mathematics teachers,secondary mathematics teachers, technology use and teachers

PREFACE

The largest study ever undertaken of the causes of crime and delinquency has found thatthere are lower rates of violence in urban neighborhoods with a strong sense of communityand values, where most adults discipline children for missing school or scrawling graffiti.In an article published last week in the journal Science (Sampson, Raudenbush, & Earls,1997), three leaders of the study team concluded, ‘By far the largest predictor of the violentcrime rate was collective efficacy’, a term they use to mean a sense of trust, common valuesand cohesion in neighborhoods. Dr. Felton Earls, the director of the study and a professorof psychiatry at the Harvard School of Public Health, said the most important characteristicof ‘collective efficacy’ was a ‘willingness by residents to intervene in the lives of children’.(F. Butterfield, New York Times, August 17, 1997)

The study cited above made national news, its results being reported onnetwork TV and major newspapers and radio programs throughout the


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country. And why not? A scientific study that claims a correlation between“collective efficacy” and reduced violent crime is good news. In a worldplagued by ethnic warfare, extreme poverty, and excessive materialism,such results give us hope that we can work with each other to effectpositive change not just for ourselves, but for our world.

INTRODUCTION

In educational systems, the power of working together as a community hasbeen recognized repeatedly. Consider the movement towards cooperativelearning as an instructional practice. Allowing children to work togetheras “communities of learners” has been promoted as an effective teachingtool for over two decades (Slavin, 1980; Johnson & Johnson, 1987).In addition, many of the educational reform movements currently beingundertaken place the creation of school community at their centers. Someof these movements look to create community among all stakeholders inthe school: administrators, teachers, students, parents and school neighbors(Sergiovanni, 1994). Other reform efforts concentrate on communityamong one segment of a school’s constituents – creating cohesion amongdifferent “houses” in a large school (Raywid, 1996) or bringing togetherparents and other community leaders to support curriculum changes insubject areas such as mathematics (Moses, Kamii, Swap & Howard, 1989).

Operating under the assumption that teachers occupy prominent posi-tions in classrooms, the development of teachers’ professional com-munities is seen as a means for promoting positive change in our schools(Westheimer, 1998). Given the broad reform of mathematics educationthat has been underway for the past ten years, the examination of teachercommunity and its relationship to reformed mathematics instructiondeserves attention. In this paper, we will discuss a professional devel-opment workshop that led to a budding professional community amongthe mathematics teachers who were its participants. Our qualitative studyof how this workshop provided the catalyst to develop this professionalcommunity among these secondary school mathematics teachers will bethe main focus of this discussion. But before getting to know how thisintervention worked, we must first explore why we believed examiningthe development of professional communities among these teachers was aworthy pursuit.

CONTENT AND COMMUNITY 109

WHY PROMOTE TEACHER COMMUNITIES INMATHEMATICS EDUCATION?

For nearly a decade, the initial publication of Curriculum and Evalu-ation Standards for School Mathematics (1989) by the National Councilof Teachers of Mathematics has acted as a catalyst for reform activity inmathematics education and provided the impetus for rethinking the waysin which mathematics is taught in this country. Yet, change has been slowto come. While new research suggesting that non-traditional methods areeffective in helping students learn and understand mathematics (Schoen,Fey, Hirsch & Coxford, 1999), it is still difficult for many teachers totransform their practice to align with reform mandates.

One structure that could motivate and support teachers to adoptStandards-based instructional practices is membership in a professionalcommunity. There is substantial research in the broader area of schoolreform that suggests that peer collaboration and support is a crucialprerequisite for teachers to be successful in restructuring their classroomsand their schools (Levine & Lezotte, 1995; Gilmore, 1995). For example,in a study of four “successful” and two “unsuccessful” schools, Little(1982) found that more successful1 schools had teachers who had continualand substantive interactions. These teachers “sustain shared expectations(norms) both for extensive collegial work and for analysis and evalu-ation of experimentation with their practices; continuous improvementis a shared undertaking in these schools, and these schools are the mostadaptable and successful of the schools we studied” (p. 338). Little foundteachers in successful schools more likely to be influenced by professionaldevelopment and more open to trying new ideas.

Examples from other studies support this same idea. In studying thecharacteristics of schools that support the professionalization and growthof teachers, McLaughlin and Yee (1988) named “collegiality” as animportant quality to support teacher achievement. Teachers they surveyedand interviewed pointed to peer interaction as an important source of feed-back and stimulation. Similarly, Rosenholtz (1989) found that in schoolswith high levels of teacher collegiality, teacher interaction focused moreon improving curriculum and practice; in environments where teacherswere more isolated, teacher interaction consisted of complaining aboutstudents.

In his discussion of the role of professional communities in teachers’professional development, Lord (1994) suggests that “critical colleague-ship” may act as a means to help teachers reform their practice. However,to attain “critical colleagueship,” teachers must be more than collegial.They must be willing to subject themselves to on-going critique:


For a broader transformation, collegiality will need to support a critical stance towardteaching. This means more than simply sharing ideas or supporting one’s colleagues inthe change process. It means confronting traditional practice – the teacher’s own and thatof his or her colleagues – with an eye toward wholesale revision (p. 192).

This type of interaction is quite different from what many teachers mightexpect from a professional community. But Lord (1994) argues that the“difference and conflict” which drive “critical colleagueship” is the keycomponent to instigating deep and meaningful change in teacher practice.

Along the same lines, Krainer (2001) suggests that not only are currentprofessional development efforts insufficient to help teachers substantiallyimprove their practice, but they focus too much on the individual. Krainer(2001) uses the case of Gisela, a secondary mathematics teacher with over20 years experience, to illustrate how the growth of a single teacher canhave little impact on the whole system. It is only when Gisela takes on anadministration post that her own professional transformation provides themotivation to help other teachers develop. From this case study, Krainer(2001) concludes that professional development activities need to aim formore of “a good balance between initiatives focusing on individual aspects,on organizational ones, and on those related to the whole educationalsystem” (p. 291).

In addition to being better for teachers, McLaughlin and Talbert (1993)point to the benefits for students from strong teacher communities. Theseresearchers claim that the “character of teachers’ professional community”is highly correlated with how teachers perceive students and student work.The authors argue that “supportive collegial communities, committed tothe success of all students, provide the necessary conditions to begin tomount a collective challenge to constraining myths (i.e. the kids can’t doit, etc.) as explanations for unsuccessful student outcomes or disappointingclassrooms” (p. 244).

In mathematics education, where we are attempting to get teachersto think about and teach mathematics in ways which they have neverexperienced as learners, the benefits of teacher community, cited in theschool change literature, have significant relevance. We need teachers tobe open to the new instructional methods promoted by the original Stand-ards (NCTM, 1989) and the recently published Principles and Standardsfor School Mathematics (NCTM, 2000). We need them to be willing tolook critically at their curriculum and make the necessary changes andimprovements. Most importantly, we need mathematics teachers to believethat with appropriate instruction, all students can learn and be successfulin mathematics. Thus, given the positive outcomes associated with teachercommunity in the context of school change, promoting such communities


among mathematics teachers may spell greater success in implementingreform of mathematics teaching.

CONTEXT AND DESIGN OF THIS INTERVENTION

In the spring of 1997, our research group approached the Mathe-matics Administrative Supervisor for the Forest (TX) Independent SchoolDistrict (FISD)2 with some ideas for professional development aimedat reforming mathematics instruction at the classroom level. Specifi-cally, the researchers presented a set of multimedia pre-calculus materials(described below) as a means for helping teachers extend and deepenboth their knowledge of mathematics and their technological skills. Theresearchers also believed that before teachers could reform their ownteaching, they must revisit and reconstruct their own understandings ofand attitudes towards mathematics (Schifter & Fosnot, 1993). Thus, ouroriginal goal in this professional development intervention was to buildteachers’ mathematical content knowledge and not necessarily to buildteacher community.

The district administrator, with whom our research group startednegotiations, was pleased to hear what resources we had to offer as sheand her staff were in the process of revamping the entire district’s mathe-matics curriculum. While reform was well underway at the elementaryand middle school level, it was somewhat stalled at the secondary level.The district personnel saw the lack of innovative curricula for secondarymathematics courses and the general resistance of secondary teachers tocurricular change as being significant barriers to reform. Thus, the admin-istrator and her staff saw a workshop aimed at improving and deepeningsecondary teachers’ understanding of mathematics as a good first step toreforming their instructional practice.

The one point upon which the administrators differed from researcherswas in the sample of teachers with whom this intervention would beconducted. The researchers had assumed that teachers would be from allover the district and would participate on a voluntary basis. In this way wecould be assured of enthusiastic participants and begin “sowing seeds” ofreform in a variety of schools. However, the district administrator and herstaff wanted to focus on the entire mathematics faculty of a single highschool along with the mathematics teachers from the middle schools thatfeed that high school.

Her reasons for this were two-fold. Firstly, working with such a groupwould be consistent with the district’s reform plan and its focus on verticalteams – teams of teachers from a given high school and the middle and


elementary schools that feed that high school. Secondly, and perhaps moreimportantly, the district administrator felt that conducting this workshopwith the faculty from a single school would be a more effective meansof encouraging change. The administrator believed that when individualteachers attend professional development workshops, they often struggleto implement change without support from peers who have had the sameprofessional development experience. Thus, the district leaders thoughtthat any changes that would take place as a result of our intervention wouldbe better sustained if an entire mathematics faculty participated together inour course.

Given what the researchers had to offer and what the district needed,the group decided that a set of mathematics teachers from the same verticalteam at the middle and high school level would make an appropriate targetfor a content-based professional development intervention. Because thedistrict had already adopted a standards-based middle school mathematicscurriculum (The Connected Mathematics Project) that was set to be imple-mented in grades 6, 7 and 8, the Algebra I and subsequent secondarymathematics courses3 seemed to be the next logical target for reform. Thecourse would use higher-level content materials (specifically the multi-media pre-calculus materials described below) with teachers so they couldgain the skills and experience with reformed content in order to engage indiscussions about what changes should be made in Algebra I and the otherhigh school mathematics courses they teach.

The district administrator’s insistence that the workshop be conductedwith the entire mathematics faculty from a given school led to our interestin developing teacher community. How would working on mathematicalcontent together affect the relationships of the teachers in this mathematicsdepartment? We thus broadened our focus to include examining the impactof this intervention on teacher community.

THE WORKSHOP AND ITS PARTICIPANTS

The intervention was planned as a three credit4 university-level mathe-matics education course aimed at high school mathematics teachers. Itconsisted of two weeks of full-time class sessions which took place June9–20, 1997, with a follow-up workshop day on September 29, 1997.Between the two-weeks of class sessions and the follow up day in thefall, teachers had to continue working on the problems that we explored inthe workshop. This included complete written solutions of the problems,with supporting explanations and computer generated tables and graphs.The multimedia pre-calculus materials mentioned above were used as the


content for the course. These materials emphasize problem solving andmodeling using families of functions and transformations as organizingconcepts. The activities themselves are derived from a pre-calculus coursetaught at the researchers’ university for the past eleven years. In their newform, the materials are presented in Netscape which allows for access tothe Internet and interactive diagrams as well as the use of multimediaresources including text, graphics, video, photographs, animations, andsound (Confrey & Maloney, in preparation).

In this professional development course, topics which overlap withmaterial covered in high school mathematics courses were emphasized.Those topics included: introduction to functions through sequences, linearfunctions, transformations, and quadratic functions. Over the two weeks,teachers worked together on the problems and on learning how to use thecomputer hardware and software as tools to solve problems.

The school selected for this intervention was Tree High School, aninner-city school that serves approximately 1720 students and employs101 teachers. The student body is somewhat diverse, consisting of approxi-mately 72% Hispanic students, 14% white students and 14% AfricanAmerican students. Approximately 52% of the students are eligible for freeor reduced cost lunch. An on-site daycare facility is available for studentswho are parents.

In all, eighteen teachers participated in the course. The core group ofthirteen of these teachers was from Tree High School. Two other teacherswere from Elm Middle School, one of three feeder middle schools toTree High School.5 The other three teachers were representatives fromtwo other high schools in the same district that applied to be the siteof the intervention but could not be accommodated because of limitedresources. The entire group consisted of eleven women and seven men.Seven teachers were from ethnic minority groups. Because this paper isfocused on the development of teacher collegiality among the faculty of aspecific mathematics department, this discussion will focus mainly on thethirteen participants from Tree High School.6

The subgroup of thirteen teachers from Tree High School had thefollowing characteristics: five were male, eight were female; six belongedto ethic minority groups (three males and three females); four of thesethirteen teachers were new hires who had not previously worked at theschool. Of these four, two were newly trained teachers who had neverbefore taught high school although one had considerable experienceteaching at the community college and college level. The other newly-trained teacher had worked for many years as a computer engineer butnever as a classroom teacher.


METHODS

Because outcomes in a study such as this are highly context dependentand because different participants in this study had different perspectiveson and experiences with this intervention, data collection and analysistook place within an interpretivist framework (Smith, 1993; Guba &Lincoln, 1989; Patton, 1990). As a paradigm for inquiry, interpretivismasserts that knowledge does not exist separate from the knower. In effect,interpretivism acknowledges that all knowledge is constructed and thatsuch constructions are influenced by the prior beliefs, knowledge andexperiences of the knower (Smith, 1993). As such, different people exper-iencing the same intervention will have different constructions of thatexperience. As a framework for research, interpretivism expects thesemultiple perspectives and encourages their solicitation and representationin research.

To uncover the various perspectives on this experience a multitude ofdata was collected. All participants, including district mathematics admin-istrators and the school’s principal, were interviewed prior to the work-shop and several months after the workshop. Each workshop participantcompleted informational surveys about their background and experiencebefore the start of the workshop and, on the last day of the workshop,filled out evaluation forms discussing their feelings about the intervention.In addition, each participant completed a portfolio consisting of samplesof work on various content problems and several written reflection pieces.All sessions of the workshop were videotaped.

The videotaped observations, documents, and interviews were analyzedthrough a process of coding and category building (Miles & Huberman,1994; Patton, 1990). Using this scheme, the researchers created a series ofcodes by which to mark the data. Codes in this case represent “bins” orcategories within which data can be “placed”. As the data were coded, theinitial codes changed, merged, or decayed until the resulting set of codescould accommodate, describe, and represent all the themes and issuespresent in the data.

DISCUSSION OF FINDINGS

Pre-Existing Conditions

Prior to the start of the workshop, each teacher participant was interviewed.Interviews were structured and lasted anywhere from 45–60 minutes inlength. The interview protocols focused on the several areas of interest


in this intervention. There were sets of questions relating to: Teachers’attitudes towards and beliefs about learning and teaching mathematics,reform in mathematics education, the use of technology in the classroom,and the faculty relationships in their department. This paper is based onthe data related to the last set of questions. A typical question in this setwas: Describe the types of interactions you typically have with other mathteachers. How satisfied have you been with these interactions?

From the description of faculty relationships given prior to the work-shop, the Tree mathematics department did not appear to be a verycohesive group. Department meetings were infrequent and generally wereonly called to deal with administrative tasks such as the administrationof standardized tests. There was little or no discussion of curricular ormathematical issues.

However, a core group of approximately six female members of thedepartment interacted on a regular basis. Three members of this core groupworked together to create a more discovery-oriented geometry curriculum,and, in addition, they often had lunch together and reported positivesocial relationships with other members of the core group. The rest of themembers of Tree’s mathematics department appeared to be somewhat onthe periphery of the core group’s interactions.

To the outside observer, it may seem that the existence of this “core”group made this mathematics faculty more prone to building a larger,department-wide community. The “core” group could act like a foundationupon which a larger community could be developed. However, the oppositeappeared to be true. In discussing the faculty’s relationships prior to theworkshop, there appeared to be an “us versus them” relationship betweenthe core group members and the other members of the department. To themembers of the core group, the other members of the department wereunwilling to collaborate or were anti-social. To the other members of thedepartment, the core group was seen as being something of a “clique” andnot welcoming of outsiders.

This phenomenon is what Noddings (1996) refers to as the “dark side”of community. While many researchers extol the virtues of community,Noddings’ work sounds a call of caution. She warns of community’s“tendencies toward parochialism, conformity, exclusion, assimilation,distrust (or hatred) of outsiders, and coercion” (p. 258). As she sees it,a commitment to community can sometimes support movements towarduniformity as opposed to those that celebrate difference. Thus, the coregroup’s existence could actually be seen as a barrier to building a faculty-wide community in this department. Indeed, four of the nine experiencedfaculty members in this department reported in pre-workshop interviews


that there was tension among individual members of the mathematicsfaculty.

Tree High School was chosen as the site of this intervention largely dueto its ability to get 100% of its mathematics faculty members to participatein the workshop. However, unbeknownst to the researchers when Tree waschosen as the intervention site, although all of Tree’s mathematics teachershad agreed on paper to participate in the workshop, several of them werenot very happy about having to attend this summer course. This added tothe tension that already existed in the department. Four members of thedepartment reported being “forced” to take the workshop by the principal,who was new to the school. Some of these teachers had summer jobs thatthey had to give up (at a significant financial loss) in order to participatein the two week intervention. Other teachers had to change travel plans. Inaddition, the majority of the teachers had little idea of what the workshopwould entail and thus were somewhat apprehensive about participating.All of the teachers reported some reservation about taking this workshop.

General Reaction to the Workshop

Despite the faculty tensions and the concerns the Tree mathematicsteachers had about participating in the workshop, the videotapes of theworkshop sessions, the workshop evaluations and the post-workshop inter-views suggest that all of Tree’s mathematics teachers had a positiveworkshop experience. The extent of the positive experience of the teacherworkshop is reflected in the workshop evaluation forms. These evaluationsconsisted of thirty-five Likert scale items (with a five point scale where 5= strongly agree and 1 = strongly disagree) and six open-ended items (seeAppendix A for some selected items from the workshop evaluation).

The Likert scale items in the evaluations covered the following sixareas: change in teachers’ beliefs about algebra (3 items), change inteachers’ feelings toward technology (10 items), change in teacher rela-tionships (4 items), change in teacher content knowledge (9 items),improvement of teachers’ understanding of student thinking (5 items), andchange in teachers’ attitudes toward reform (4 items). The overall averageresponse for all of the Likert scale items was 4.22 with a standard deviationof 0.63. When the items were analysed by topic, the average responsefor each topic was above 4.00. Such high average responses suggest anoverwhelmingly positive reaction to the workshop.

Change in Teacher Relationships

Interestingly enough, among the Likert scale items, the post course evalu-ation area that had the highest response average was the one on the change


in teacher relationships, which had an average of 4.53 with a standarddeviation of 0.75. These items focused on whether participants thoughttheir relationships with their colleagues had improved because of the work-shop experience. From their responses, we judge that the teachers believedstrongly that the workshop had led to improved faculty relationships.

The open-ended anonymous comments at the end of the evaluationsupported this Likert scale data. In response to the open-ended item thatasked teachers to describe the best part of the workshop, six of the thirteenTree teachers pointed to the opportunity to work with other members oftheir faculty. Here are some of their comments:

The big plus for me was getting to meet and get acquainted with the teachers who will bein my department – and feel comfortable with them!

Working with other teachers – especially those at Tree with whom I work on a regular basis[– was the best].

[The best aspect was] the development of a more open and purposeful department dynamic.

Interview data also revealed teachers’ perceptions that their facultyhad become closer as a result of their participation in the workshop.Approximately six weeks into the school year (1997–1998) followingthe workshop, teachers at Tree High School were interviewed abouttheir experiences in the workshop. These post-workshop interviews wereintended to parallel the interviews conducted before the workshop. Thus,they were of the same structure and duration and focused on the sameareas of interest as the pre-workshop interviews (teacher’s attitudesabout teaching mathematics, reform in mathematics education, the useof technology in the classroom, and the faculty relationships). While thepre-workshop interviews were intended to discover the existing conditionsprior to the workshop, the post-workshop interviews were intended todiscover the impact of the workshop as an intervention.

At this time, the mathematics department at Tree consisted of fourteenteachers. Twelve of those teachers had participated in the summer work-shop (one of the thirteen workshop participants from Tree took another joband left the faculty before the start of the school year). Of those twelve,four teachers were new to Tree that year and eight were experiencedteachers at the school.

Of these eight experienced teachers, four suggested that teacher rela-tionships were significantly improved over the previous year:

The rapport has been much, much better. Much deeper. Perhaps individual differences thatwere noted at the workshop have improved a lot. I know that even with the other [name ofcourse] teacher our rapport has been much better. (Respondent 1)7


Definitely – a change from last year. As I mentioned to you before, I think there are peoplewho had conversations that hadn’t spoken to each other in years. There were deep rifts andwhile they may still be strained – the bridge is there. (Respondent 9)

Another three of the returning eight teachers suggested that relationshipswere slightly better than in previous years, with the remaining returningteacher suggesting that the mathematics teachers no longer shared thesame wing and were scattered throughout the building. In previous years,the mathematics teachers in this school had classrooms located along thesame hallway, which they referred to as the “math wing”. However, thefall after the workshop, teachers’ classrooms were reassigned to allow forthe creation of ninth grade teams. Thus, not all the mathematics teacherswere on the same wing. This teacher felt that, in spite of the department’spositive experiences in the summer workshop, the lack of proximity to oneanother was a significant barrier to improving departmental relationships.

While the fact that seven of the eight experienced teachers reportedsome improvement in department relationships as a positive result, theconsideration of the pre-existing conditions in the department makes itmore impressive. While nearly half of the experienced teachers (four ofthe nine original ones) had reported tension among faculty members inthe pre-workshop interviews, not one of the faculty, new or experienced,suggested that level of tension still existed. In general, teachers’ responsesto the questions about their relationships with their colleagues was muchmore positive in post-workshop interviews than in pre-workshop inter-views. None of the teachers mentioned the existence of the “clique” ofsix teachers, as they had in the pre-workshop interviews. From teacherdescriptions, it appeared that teachers as a group were more integrated andgenerally felt more comfortable with all of their colleagues – not just aselect few.

As might be expected, the four new teachers to the departmentsuggested that they benefited significantly from spending time in theworkshop with their new peers. Here are two representative comments:

The first [positive] thing I point to [about the workshop] is meeting everybody. Forget whatthe content of the workshop was – the relationships have been invaluable. (Respondent 12)

I was very comfortable coming into this school because I already knew where everythingwas. I knew all the math people. I was comfortable with them. I felt comfortable enoughto say anything to them or ask anything of them. I can’t imagine what it would have beenlike for me if I hadn’t had that. (Respondent 3)

Despite having to deal with problems such as large class sizes andadjusting to a new school (and for one of these teachers, a new career),all four of the new teachers reported having positive interactions with the


other mathematics faculty members. They felt their experienced peers weresupportive, approachable and helpful.

What Led to This Reaction?

Teachers’ overwhelmingly positive reaction to the workshop and theirbelief that the workshop helped make department relationships strongeractually surprised us as researchers. Given the pre-existing conditions, wedid not expect such a universally positive response to our intervention.This reaction leads us to ask the question: What was it about this experi-ence that helped these teachers grow closer as a group? Based largely oninterviews with teachers and researcher field notes, three factors appearedto contribute to the growth of teacher community in this setting. Thesefactors were determined through the coding system applied in the analysisof the data.

In analyzing the transcripts of the post-workshop interviews and of thevideotapes of workshop interactions, we labeled each response or interac-tion with a different code. For example, when a teacher compared theirexperiences in other professional development workshops with his/herexperience in our workshop, we labeled that response “experience withother workshops”. Because we were using a computer program to label ourdata, at the end of the coding and labeling process, we were able to sort bylabels. When we sorted the data concerning teachers’ positive reaction toour workshop, we found that a substantial portion of the data fit into threebroad categories. Each of the categories is discussed below.

INTERACTING WITH COLLEAGUES OVER MATHEMATICS:THE REVELATION OF DIFFERENT PERSPECTIVES

When asked to explain why they thought the workshop had been sobeneficial to teacher relationships in their department, one of the mostfrequent responses given by teachers concerned their interaction in solvingmathematics problems. Throughout the workshop, teachers were askedto work on problems in pairs or small groups. Teachers were instructedto switch partners or groups frequently in order to work with a largernumber of people. For some teachers, this small group work provided themwith a mathematical focus outside of themselves about which they had tocommunicate.

Just the act of problem solving together and putting everything aside and turning to acolleague and saying: “I don’t know how to do this or I don’t quite understand the wayyou’re approaching this. Could you explain it to me?” . . . was important. (Respondent 9)


The way you all had us shift partners – even though my first thought was: “Oh God, no.I just got comfortable with this one and now I’ve got to go and get to know another one.”But it ended up being really good because when you’re both thinking about a problem,you’re not thinking about each other. And the relationship just sort of happens without youknowing it. (Respondent 11)

The act of working on mathematics together gave teachers a sense thatthey were interacting over shared territory. This was not a cocktail partywhere the individuals comfortable with small talk would excel. This wasexploration of mathematics – something with which they all had experi-ence and in which they had all invested. As Westheimer (1998) points out,community is not built in and of itself – but is built over a set of substantiveissues important to all participants. In this case, the mathematical contentgave these participants a reason and purpose for interacting. The result wasthat participants developed relationships by way of their problem solvingactivities and their discussions of mathematics.

However, even more common than the belief that the problem solvinggave teachers a substantive focus, was the feeling among these teachersthat the problem solving activities allowed them to see how each indi-vidual thought. During the workshop, after the small groups had a chanceto work together on a problem, each group was asked to report its find-ings to the large group. In these reports, teachers saw and appreciated thevarious approaches different groups took in solving problems. At times,the variation across solutions was significant.

For example, in a problem where teachers were asked to come up witha model for the growth of the chambers of a nautilus shell (commonlymodeled with a geometric sequence), teacher investigations had extremedifferences. Each group took measurements of successive chambers in areal nautilus shell, and then plotted this data on a graph or made tables ofthe data in an effort to find a mathematical model that described their data.One group of teachers used graphing tools on the computer in an effortto find a curve that approximated the plot of their data. Another groupattempted to linearize the data and used a graphing calculator to draw aline of best fit. Still another group manipulated the data through calcula-tions. They took the logs of their data in an effort to derive a series ofrational functions that would best fit the data. In listening to presentationsof these types of non-traditional approaches, teachers were challenged intheir attempts to understand the methods used by others.

In post-workshop interviews, teachers were asked to “Describe oneevent or interaction during the workshop that you feel was important orhad an impact on your relationship with the other faculty members inyour department.” Eleven of the twelve remaining Tree teachers suggestedthat hearing other people’s methods for solving problems had a significant


impact on their relationships with their colleagues. There seemed to betwo reasons why teachers felt the understanding of each other’s problemsolving approaches helped improve their relationships. The first was thatsimply by hearing the way others approached a problem, teachers felt thatthey started to understand the way others thought and thus got to knowthem better.

We were very actively involved and participated in what was going on so that it allowedus to just share. I still think of Susan as having a certain approach – from her background– to problems. And I see Carol having a unique approach. And Sam of course [was] thegenius of the group. And I had a chance to work with them but also to learn a little ofwho they are and how they think and just have fun solving problems together at that level.(Respondent 6)

A second reason why hearing other people’s perspectives seemed to besignificant for workshop participants was simply being able to see thatmathematics can be explored in a variety of ways. The discussion ofdifferent approaches to problems seemed to allow these teachers an oppor-tunity to air, acknowledge and accept the diverse perspectives from whichthey approached mathematics. Doing this in the somewhat neutral contextof a problem solving activity, as opposed to a more formal departmentmeeting, seemed to allow teachers to step back and simply appreciateothers and their perspectives.

Respondent: [The workshop] allowed for opinions to be expressed. And people listened toeach other’s opinions for the first time.Interviewer: Do you think it was listening or do you think it was airing opinions in the firstplace?Respondent: Yes, airing the opinions and then exposure to perhaps this is a different wayto approach it. This is a new idea. . . .

Interviewer: And that had never happened before?Respondent: Right . . . if it had happened, nobody was willing to listen to someoneelse’s viewpoint. And without recognizing each other’s professional value. You know, justbecause you don’t do it my way – that doesn’t mean that you’re stupid or that you don’tknow what you are doing as a teacher. (Respondent 1)

The benefit of hearing others’ perspectives on issues of mathematics cameup repeatedly in the interviews. It was almost as if the existence of thesedifferences among their faculty was a revelation for this group of teachers.Understanding the perspective from which a peer was coming seemedto bring with it a deeper appreciation and respect for the various posi-tions of that individual. Such understandings also brought with them asense of acceptance of the different individuals and their unique view-points.

I enjoyed some of the freewheeling discussions where . . . you got to know how peoplethought a lot more – when people were fairly free in asking questions or challenging what


was happening . . . just watching and listening and thinking – “Oh, OK, I didn’t know heor she thought this way or came from that point of view.” (Respondent 7)

The fact that teachers came away with a better appreciation of each other’sperspectives on mathematical ideas is not a total surprise. One of our goalsin the workshop was to promote the development of multiple perspec-tives and representations of mathematical content. The workshop itselfcontained many structures to allow different viewpoints and approachesto emerge. First and foremost was the creation of an environment whereteachers were expected to collaborate in problem solving. Group workwas the norm, and we hoped small and large group activities would allowteachers to see that not all of us think alike. In learning to appreciate thediversity among themselves, we hoped teachers would begin to understandthe diversity that exists in student thinking. Perhaps the most importantstructural feature which led to the airing and appreciation of different view-points was the work of the lead facilitator. There were five team membersimplementing this workshop. Three team members were graduate students,one was a staff member, and one was the principal investigator of the grantsupporting this workshop and the leader of the research team. The researchteam leader, who is also second author on this paper, acted as the leadfacilitator for the workshop. The three graduate students (one of whomis first author of this paper) and the staff member supported her workas facilitator by preparing materials, assisting teachers as they workedon the problems and with the computers, leading certain activities, anddealing with various logistical concerns. The three graduate students alsovideotaped all the workshop sessions and took field notes of their observa-tions of the workshop activities. It was clear from these observations that,although the teachers had friendly and positive relationships with all fivemembers of the workshop team, the lead facilitator was the person whomthe teachers held in highest regard.

From the beginning of the workshop, the facilitator made clear thatmultiple interpretations and solutions in mathematical problem solvingwere not only acceptable but also expected. As mentioned above, theNautilus problem elicited a variety of solutions from teachers. The facili-tator went around to each group, offering support and posing questions forfurther explorations. In organizing the group presentations, the facilitatortried to organise the order of the group presentations so that each methodshown complemented the next and further broadened the problem. Themethods went from being straightforward to being fairly sophisticated.

The last group to present chose a fairly complicated approach tomodeling their data. The leader of this group presented her method to therest of the teachers, who were rather awe-struck.


Penny: We did a graph of the points. . . . Graph of area versus chamber number – I think itlooks parabolic. . . . Well, anyway, I didn’t start trying to guess what the power was. WhatI did was take the log base 10 of the area.Diedre: Why did you even know to do that?Penny: Because I’ve done it before. And it’s the nature of logs. Log x4 is equal to 4 log x.So plot log base 10 of y versus log base 10 of x, then the slope of that line will give me thepower relationship. So I don’t have to guess the power – I can find it.Facilitator: Let’s check in – does everyone follow?Penny: Do you see because of the nature of logs, that slope will give me the power relation-ship between y and x? And it doesn’t have to be a whole number power. It can be y = x5.Facilitator: The trick is that log x4 = 4 log x. Because if you are plotting log x versus logx4, then the claim is you’re plotting log x versus 4 log x. So the 4 is functioning as m in theline mx. Basically, you’re giving the same function to both variables, but it’s bringing thepower down in front.Penny: And that gives me my power and once I have my power that gives me the relation-ship between y and x.Facilitator: That’s a way to find out the relationship if you think it’s a power function [like]x2, x3, etc. . . . some kind of polynomial. It’s a way to model something.Penny: But it doesn’t have to be a polynomial.Facilitator: Yeah, you can do rational functions too – but I’m worried that they’re not thatclean.Penny: And this also is wonderful because if I plot log x versus log y and get a disjointedline – that tells me I have two power relationships. I have two functions at work on differentparts of the data. That happens all the time like with magnetic fields – strength of themagnetic field drops off at different rates depending on distance away from the field. Thisis what we do in our physics labs. (Some teachers whistle like this is impressive or wayover their heads.) (Day 5 – Tape 2).8

The teachers in the audience found this approach challenging. Several ofthem asked clarifying questions and a few of them made it clear theyfound it difficult to understand. “I thought I knew logs until you starteddoing this,” said one. But despite this very sophisticated method and thediscomfort it caused many of the teachers, the facilitator made clear theimportance of bringing it into this discussion.

As a teacher I would say – I came around and saw what they were doing and gulped –I said: “We’re going to have to understand and present this method.” And I told my staffnot to disrupt this – I could have guided them to use geometric sequences and gotten you[referring to the small group] to stop. But I think you offered the group a really importantinsight – which got brought into the group. (Day 5, Tape 2).

By stressing the importance of appreciating multiple approaches to prob-lems, the facilitator not only modeled what we believed teachers shouldbe doing with students in their classrooms, but she also helped teachersappreciate the diversity and strength of perspectives existing among theirfaculty.


Certainly, the teachers recognized that the workshop was structuredto allow for varying perspectives – and appreciated the freedom andopportunity to explore mathematical content in this way.

We were never told: You have to do it this way. We were allowed all these different ideas. Imean I did it one way, Juanita did it another way, Penny did it another way, Timothy anotherway. But we were all doing the same thing and we were allowed to do that. Nobody toldus how to work anything. I like that. I like that. (Respondent 2)

Thus, the freedom to have a unique interpretation on a mathematicalproblem gave these teachers the opportunity to share their individualitywith their colleagues. These mathematical interactions became the catalystfor their sharing of themselves and listening to the perspectives of theirpeers.

NAVIGATING NEW TECHNOLOGY: WE’RE ALL IN THESAME BOAT

Giving participants the time and opportunity to interact over mathematicalcontent may also have helped these mathematics teachers face a significantbarrier to improved department relations: the strength of each individual’smathematical content knowledge. The whole issue of content knowledgeis a touchy one for teachers – especially in the state of Texas where teachercompetency testing has an interesting and not very teacher-friendly history(Shepard & Kreitzer, 1987). For a mathematics teacher, to work with one’speers on mathematics problems is to make oneself vulnerable for there isalways the chance of making mistakes and looking foolish.

At points, the teachers in this workshop did make comments thatreferred to their discomfort with some aspect of content. When the facili-tator was discussing the derivation of the formula for the sum of ageometric sequence, several of the teachers mentioned that this derivationwas new to them.

Diedre: Uh-uh. I’ve never seen this before. I mean I’ve seen the formula and I’ve taught theformula – but I didn’t really know where it came from. And I guess I never had the time tosit down and play with it and see it.Allison: No one ever explained it this way and I understand it now – why the formulais the way it is. That’s why I wanted to be real sure about what you were saying. Nobodyexplained it to me this way and I was confused about these formulas. But now I understand.Facilitator: You’ve got to fight that thing that you’re stupid because you just didn’t gettaught it. And if you didn’t get taught a tool for thinking something through . . . then whatcan you do? (Day 4, Tape 1).

Given some of the teachers’ insecurities regarding mathematical contentknowledge, the work on mathematical problem solving throughout the


workshop could easily have been a divisive force among these teachers.However, teachers were exploring mathematics content with technologythat was unfamiliar to all of them. While some teachers had reportedhaving previous experience with computers, none of the teachers had usedthe main tool of the workshop, a program called Function Probe (Confrey,1997), and none of them had seen mathematics problems presented in amultimedia format.

Adding technology to the problem-solving mix seemed to maketeachers feel as if they were covering mathematics content that was newto all participants. For many teachers, the technology seemed to level the“content knowledge playing field” and led to more relaxed and supportiveinteractions between participants. The idea that everyone was on “equalfooting” came up several times when teachers were asked what aspect ofthe workshop helped improve teacher relationships.

It wasn’t just one thing, it was the same thing that just kept happening over and over whenwe were working with the problems and working with the computer with software that wasnew to everybody. So even if you were good with another computer we all sort of startedon equal footing there – so you didn’t have to feel like a total dummy. (Respondent 11)

Equality – starting out on an equal footing – that none of us had used Function Probe andwe were all exploring it together. Putting us all on the same plane there . . . in that maybesome people came with computer literacy and many of us didn’t. And being able to get tothe same level with that. (Respondent 1)

The belief that everyone was “in the same boat” in relation to thenew technology also led to the growth of helping relationships amongthe teachers. They were eager to understand both the technology andthe content and generally were not shy about asking for help. Teacherswillingly assisted each other with both technological and content issues.When teachers made mistakes or got stumped in their presentations, otherteachers came to their rescue.

Tree teachers continued these helping relationships well after the work-shop. At the end of the workshop, teachers took home a computer in orderto continue working on workshop-related activities. In dealing with prob-lems with setting up and using these new computers, teachers were quickto call each other for assistance. In many cases, a more “expert” teacherwould make a house call to a peer who was having computer problems.Similarly, in finishing the problem solving activities and written reflectionpieces that made up the teacher portfolios for the course, several teachersworked together to complete these assignments.

Among the remaining twelve teachers at Tree High School, eightof them reported getting or giving assistance with workshop-relatedtechnology or activities on their own personal time. While some skeptics


may suggest that such helping relationships are common in professionaldevelopment workshops, one of these teachers recounted a similar type ofworkshop with participants from all over the country where other groupmembers were not very helpful.

You would be in the computer lab and you would ask some of these teachers that knewmore than you did and they felt bothered because you were slower and you were holdingthem back from finishing their work. And in this [workshop], people were more helpful. Ifyou didn’t know how to do it, somebody else came to your rescue. So it was much, muchdifferent. (Respondent 5)

At the start of school, the helping relationships that teachers had startedto build in the workshop were evident. The beginning of school year 1997–1998 was chaotic for many of Tree’s mathematics teachers. Because ofsome changes in the school’s structure, almost all of the mathematicsteachers at Tree had to change classrooms. Teachers helped each otherwith moving materials and books from one room to another. In addition,many of Tree’s mathematics teachers were teaching classes they had nottaught for many years, if at all. More experienced teachers of these coursesoffered to share lesson plans or activities with their peers. Finally, evenwith large class sizes and hectic schedules, teachers managed to find timeto check in on each other. Several reported stopping by each other’s roomsafter school to see how they had survived the day. The helping relationshipsstarted in the workshop were weathering the challenges of the beginningof the school year.

Starting school we had a great advantage in already having a working relationship witheach other. People felt like they could ask things of other people – whether it was helpwith doing something, resources, where to go – I mean we were already beyond whateverbarriers there usually are . . . I think the fact that we sat down and we were learners together[meant] that nobody had anything to hide anymore. I mean everyone knew who was whoand what was what. And I think that made for a very comfortable working relationship andI don’t think anyone felt isolated in starting – which frequently happens. (Respondent 9)

Thus, working with new technology, which was not comfortable for any ofthese teachers, led them to bond. In helping each other face the commonchallenge of developing these new technological skills, these teachers alsodeveloped stronger collegial relationships.

HANDS-ON LEARNING: BEING ENCOURAGED TO TRYTHINGS FOR OURSELVES

The majority of teachers from Tree stated that they had never been to aworkshop where they had been so active.


Most of the workshops I’ve gone to we just sit there and listen to somebody. In this work-shop, we did the work and then we exchanged ideas. I haven’t been to that many workshopswhen we’ve done that . . . Actually sitting down and doing it. Seeing what you’re doing. Iwill not learn if you tell me this is how it’s done. I have to do it. (Respondent 2)

Over the nearly 40 years I’ve been teaching school, the only workshops that I ever attendedthat I came back with more than being numb on both ends, were the ones where somebodywas down to earth enough to tell you something and then let you do it. [This workshop]was a hands-on situation. It was a combination of having a computer everyone could gettheir hands on, having the different pieces of hardware here like the motion detectors andthe overhead projection screen. . . . I was really fascinated with everything. (Respondent11)

Most of these teachers suggested that the hands-on nature of the coursemade it more enjoyable and allowed everyone to have a more positiveexperience. However, few of them attributed the hands-on nature of thecourse to improvement of faculty relationships.

Nevertheless, a strong case can be made that the hands-on learningsupported the growth of relationships in this workshop. Some of this argu-ment is intuitive. Allowing teachers to be more hands-on gave them theopportunity to have a variety of different kinds of interactions with eachother. They not only were discussing mathematics or debating the validityof a certain method – but they were struggling to find accurate measuresof the volume of a nautilus shells chambers, or they were trying to figureout how to arrange themselves to get the motion detector to produce a stepgraph. Very frequently, they were attempting to figure out how to get that“blasted” computer to perform certain tasks. In each of these contexts, thenature of their learning was different – so the nature of their interactionsand resulting relationships would be broader and perhaps deeper.

Another reason hands-on learning can be cited as a factor in helpingthese teachers improve their collegial relationships comes from work ofWestheimer and Kahne (1993). One of their five means for fostering andsustaining community is through breaking norms as a way for creatingopportunities for new relationships. In essence, this involves giving peopleexperiences different in nature or context from the course of their “normal”interactions. Such nontraditional experiences give people the chance to“recast” their relationships with each other. For teachers, workshop activ-ities such as rolling a basketball up and down a ramp in front of a motiondetector or sending goofy email messages to each other while sitting inthe same room could be classified as norm-breaking activities. The levelof humor and interaction during these activities was high, and it seemsclear that such activities gave teachers a chance to see each other in adifferent light. Given the positive effect that teachers claim the workshophad on their relationships with each other, it seems likely that the chance


to work together on hands-on activities contributed to the growth of theserelationships.

GROWING PAINS

Despite the fact that in post-workshop interviews, nearly all of Tree’smathematics teachers reported positive growth in the collegiality of theirdepartment, just as many of them mentioned school factors that may act asbarriers to further departmental growth. First and foremost was the lack ofproximity that these mathematics teachers have to each other. While manyof these teachers are just down the hall from each other, several of theteachers are in separate parts of the building. With the limited amount oftime teachers have to interact with each other during the school day, severalteachers see having classrooms in close proximity to other departmentmembers as vital to maintaining positive teacher relationships.

Some of the distance between department members is due to theschool’s creation of ninth grade teams. These teams present another poten-tial barrier to the continued growth of the mathematics department’sbudding professional community. A ninth grade team consists of a groupof teachers from each of the main subject areas who teach the same set ofninth grade students. All of the teachers on a given team have classroomsnear one another to facilitate communication about student progress andcurricular issues. Mathematics teachers on these teams would thus haveclassrooms away from the majority of their department.

While certainly positive in some respects, the creation of teamsaccording to grade level may undermine the power of content-specificdepartments. In the case of Tree’s mathematics teachers, those teacherson ninth grade teams have responsibilities to both their department andtheir team. For several of them, this means they are in either department-sponsored or team-sponsored meetings for four of their five planningperiods per week. A few complained about being “burned-out” on meet-ings and being separated from their non-team peers in the mathematicsdepartment.

Another barrier to community for Tree mathematics teachers is staffturnover. In the year after our workshop, the mathematics departmentat Tree lost five faculty members, four of whom had participated in theworkshop. Because replacement faculty members will not have had theworkshop experience, the mathematics department will need to find someway to integrate these newcomers into the larger group. Failure to do thiswill certainly limit the improvement of collegial relationships within thisdepartment.


Several teachers mentioned lack of support from administrators asbeing a barrier to a stronger department. A few teachers thought theywere on administrators’ “bad list” because they had been given undesirableteaching schedules. Some other teachers felt that school administratorsdid not take the time to find out what was really going on in mathe-matics classrooms. No administrators ever attended department meetingsand some members of the department felt they were being neglected. Inter-estingly enough, interviews with administrators revealed that they werequite pleased with the progress that the mathematics faculty had made overthe past year. They noted better communication among faculty membersand increased use of technology and innovative teaching practices inmathematics classes.

If Tree’s mathematics department is going to continue to grow, the riftthat many teachers described between the department and school adminis-trators needs to be repaired. A considerable amount of energy seems tobe expended in the department complaining about administration andbureaucratic demands – energy that could be used better to promoteimproved classroom practice. In addition, just as faculty members needpeer support to undertake classroom reform, they also need the support ofschool administration. Such support allows teachers to remain connectedto and make use of the larger structure of which they are a part.

The final barrier to continued growth of the relationships in this mathe-matics department is time. As almost any teacher anywhere will report,there simply is not enough time to do all the things teachers need or want todo. In the case of Tree’s mathematics department, almost all of the teacherspointed to time constraints as being a barrier to continued improvementof their faculty relationships. Many of the teachers are dealing with largeclasses of 30 or more students. Several of these teachers do extra curricularwork such as coaching or tutoring of students. The current schedule at Treedoes not permit all of the mathematics faculty to share the same planningperiod or the same lunch period. This makes group interaction and plan-ning difficult. If a mathematics faculty community is to be fostered andmaintained in this setting, the time will have to be found to allow teachersto continue to build it.

EPILOGUE

In the three years following our workshop, our research group continuedto work with the Tree mathematics teachers on a number of initiatives.The year following the workshop (’97–’98) the researchers and teachersworked together on an assessment measuring students’ understanding of


simultaneous equations. In the summer of 1998, teachers participated infive more days of workshop activities with researchers, focusing on othercontent topics and constructing a replacement unit for Algebra I courses(Castro-Filho, 2000). During school year 1998–1999, the mathematicsdepartment at Tree, with the help of researchers, piloted two experimentalcourses with students: one integrating math and science and a second using“mathematics as modeling” as an organizing theme (Castro-Filho, 2000;Confrey, Castro-Filho & Wilhelm, 2000). In the summer of 1999, teachersand researchers reviewed the student results of the piloted courses andmade revisions to the courses as needed (Confrey, Bell & Carrejo, 2001).

Through our observations of teacher relationships in these subsequentprojects, we watched as the Tree mathematics teachers continued to worktowards maintaining the strong professional relationships they developedin our initial workshop. As mentioned previously, one major issue for theseteachers was the constant influx of new faculty members into their depart-ment. By 1999, only 7 of the 13 original workshop participants still taughtmathematics at Tree High School (Confrey, Bell, & Carrejo, 2001).

Our continued work with the Tree mathematics teachers provided onemeans to integrate the new faculty members into the department andfamiliarize them with the issues that had been discussed in our initialworkshop. In addition, the Tree mathematics teachers continued to meeton a regular basis to discuss issues related to curriculum and studentprogress. They also instituted a monthly potluck dinner to welcome newfaculty members and to continue to improve and develop their faculty’srelationships (Wilhelm, personal communication).

CONCLUSIONS

It is clear from the data presented in this article that teacher relationshipsdid improve and intensify in the period following the workshop. However,can it be said that a professional community was developed through thisintervention? The answer to this question probably rests on how onedefines community. Westheimer (1998) suggests a “community” will havethe following five characteristics: a large degree of interaction and partici-pation, interdependence among its members, shared interests and beliefs,concern for individual and minority views, and concern for personal rela-tionships. Our data suggests that while the relationships among Tree’smathematics faculty have some of these characteristics, they do not haveall of them. Thus, by this definition, a community was not developed.

However, other studies (McLaughlin & Yee, 1988; Little, 1982;Rosenholtz, 1989; McLaughlin & Talbert, 1993) define professional com-


munity much more loosely. Professional communities in these studiesare characterized by high levels of teacher interaction and collegiality.Using this criterion and the fact that Tree’s mathematics teachers reportedthey were interacting much more frequently and substantially than everbefore, it would seem the Tree teachers’ relationships were exhibiting thehallmarks of a professional community.

Probably more important than whether or not the effect of the work-shop was to produce a professional community (by any definition) isdetermining what we have learned about professional development andprofessional communities from this experience. Knowing that faculty rela-tionships and department interactions often have a long history, we, asresearchers, did not expect that our initial two-week intervention wouldresult in a full-blown teacher community. However, we did hope that theprocess of interacting over substantive mathematical content and curricularissues would help this faculty begin the process of improving teacher rela-tionships. By studying the early formation of a potentially budding teachercommunity, we hoped to contribute to the growing understanding of howschools and teachers can begin to move toward a sense of community andthe subsequent realization of reform goals.

So what have we learned? When the teachers are back in school, anumber of constraints become evident which impede continued relation-ship building. The multitude of demands on teachers’ time and thedistance between their classrooms create barriers of space and time whichprevent teachers from communicating regularly. This causes us to ques-tion the extent to which the workshop enabled participants to continuerelationships in the school setting.

However, by combining teacher “teaming” in a professional develop-ment setting with teachers’ exploration of mathematical content, webelieve we have developed a powerful vehicle for achieving two goalsin pursuit of improving mathematics instruction. First, given the role thatteacher community and interaction has been shown to play in supportinginstructional change (Little, 1982; McLaughlin & Yee, 1988; Rosenholtz,1989), structuring professional development activities such that they alsowork as community development may increase the chances that suchactivities will support subsequent changes in curriculum and proposedteaching innovations.

In addition, given that one of the major concerns in reforming mathe-matics instruction is the depth of teacher content knowledge (Ball, 1991,1994), shared mathematical inquiries in a professional development settingnot only give teachers an opportunity to work with one another andbuild community, but they also give teachers a means to develop deeper


mathematical understandings. These two objectives, building both contentknowledge and community through professional development, are thusintegrated and mutually reinforcing.

As an illustrating example, the two teachers at Tree who consistentlyteach Algebra II had a somewhat strained relationship before the work-shop. After their experiences in the workshop, they decided to try collabo-rating more frequently. When we arrived at the school to do post-workshopinterviews, both teachers, at separate times, excitedly shared the tale ofhow they had recently debated the “real” meaning of the function concept– and enjoyed the debate. For these two teachers, the content is at the centerand focus of their relationship, which continues to grow.

Westheimer (1993) has argued that teacher community does notdevelop in and of itself. In fact, as a consultant, Westheimer is oftenapproached by school district administrators who want him to come totheir schools and “show” teachers how to build a professional community(personal communication). In response, Westheimer always asks theinquirers: What are the pressing, important issues that your teachers arefacing or are concerned about? He then advises district personnel tohave teachers start working on and facing those issues. According toWestheimer, it is only in wrestling with these issues that teacher com-munities can begin to develop (Westheimer, personal communication).In our case, the issue over which the Tree teachers engaged was themathematics, which is central to their teaching and the success of theirstudents.

There is very little in the literature discussing the development orexistence of teacher communities that addresses the notion of using mathe-matical content (or other subject content) as the “issue” around whichteachers can interact and professional communities can develop. There arestudies which discuss the growth of teacher communities around the devel-opment of a new school-wide curriculum (Sergiovanni, 1994), recognitionof community-shared values (multiculturalism, equity, etc.) (Westheimer,1998) or implementation of new instructional practices (Moses, Kamii,Swap & Howard, 1989). In addition, several studies look at the correla-tion between school success and teacher community (Little, 1982) or thedifferent levels of teachers’ interactions that promote reform (Little, 1990).However, no study seems to look at field-specific content explorations (likemathematics) as the basis for building teacher community.

Thus, this study is important in that we learned that teachers’ jointmathematical investigations CAN act as a means to build teacher com-munity. Our experience clearly illustrates that by allowing teachers toexplore rich, open-ended mathematical problems with a variety of tools,


they can learn to appreciate both the mathematics and a given individual’sperspective on the mathematics. Diversity of thought and personalityamong department members, a potentially divisive force, is thus givenroom to flourish through the exploration of these problems. This lessonhas been so powerful for our research group, that professional developmentinterventions that simultaneously develop teacher community and contentknowledge are now an integral part of our proposed model for imple-menting standards-based curricula and technology innovations (Confrey,Bell & Carrejo, 2001).

“What we want for our children, we should also want for their teachers”(Hargreaves, 1995, p. 27). In the end, we found that what supports thedevelopment of community among secondary mathematics teachers issimilar to what we would define as a “good” mathematics program forstudents: the pursuit of interesting mathematical investigations, the use oftechnological tools, and the stimulating discussion of mathematics. Allthese structures promote an environment that can “grow” professionalcommunities while at the same time develop teacher content knowledge.We believe that together, these two components – content and community –provide a strong foundation for teachers to undertake instructional reformin their mathematics classes.

ACKNOWLEDGEMENTS

This research was supported by a grant from the National Science Founda-tion (RED 9453876). All opinions and findings are those of the authorsand not necessarily those of the Foundation.

APPENDIX A: WORKSHOP EVALUATION ITEMS

Below are a sample of the post-workshop written evaluation items.

Part I: Likert scale items: The response scale is as follows: 1 = Strongly Disagree,2 = Disagree, 3 = Neutral, 4 = Agree, 5 = Strongly Agree.

1. This course changed the way I think about algebra.2. I feel more comfortable with computers since taking this course.3. This course did NOT help to improve my relationship with the other members

of my department.4. I was unimpressed with the technology used in this course.5. Viewing and analyzing videos of students working on problems increased my

understanding of how students think about mathematics.


6. Using technological tools allowed me to gain more insight into algebracontent.

7. I liked exploring the multimedia problems presented in Netscape.8. For me, this workshop has raised some concerns about the directions in which

reform in mathematics education is headed.9. This experience has helped me realize how important hands-on materials are

to student learning.10. I plan to use more technological tools in my teaching this year because of this

course.11. This course made me realize how difficult it can be to really understand what

students are thinking.12. In general, I feel this course was a positive experience for our entire math

faculty.

Part II: Open-ended questions.

1. Please describe the BEST aspects of this workshop for you.2. Please describe the WORST aspects of the workshop for you.3. In your opinion, how has this workshop affected your content knowledge?4. In what ways, if any, do you believe this workshop will impact your

relationships with the other mathematics faculty in your school?5. How do you think your experiences in this workshop will impact your

teaching in the next year?6. Would you be willing to take a similar course to this one focusing on

trigonometric and exponential functions? Please give reasons for your answer.

NOTES

1 To find successful schools, Little examined aggregate standardized achievement testdata for a three year period. She also considered nominations for successful schools fromdistrict administrators.2 All proper names relating to participants in this study are pseudonyms.3 In the United States, the traditional sequence of mathematics courses in high schooltends to be as follows: an introductory course in algebra in the first year of high school, anintroductory course in geometry in the second year of high school, and a more advancedcourse in algebra in the third year of high school. In the fourth and last year of high school,students can choose from a variety of courses including trigonometry or pre-calculus.4 A “credit hour” is the unit of measure used in universities in the United States todetermine the amount of time a given course of study is worth. Generally, one credit houris equivalent to approximately 16 hours of contact/class time with a university instructorover the course of a 16 week semester period. For each hour of contact/class time, a studentis expected to spend approximately three hours preparing for class or doing out of classassignments. In the United States, university classes are generally structured to be worththree credit hours. Thus, this workshop was structured as a three credit course. Teacherparticipants were given the option of taking the workshop for university credit.


5 In this large, urban district, students from the middle schools in the city have some voiceas to which high school they will attend. The city has several “specialized” high schools(a performing arts school and a science and technology school) to which middle schoolstudents could apply. Thus, Elm Middle School was only one of the several middle schoolswhose students went to Tree. Despite efforts to get math teachers from the other middleschools which feed into Tree, only two teachers from Elm Middle School were able toattend the workshop. Elm Middle School is only one mile from Tree High School andserves a very similar student population to Tree High School.6 Although the intent was that the two participants from Elm Middle School would alsobe included in the development of the Tree math teachers’ professional community, thisdid not happen. Both Tree and Elm teachers felt that the biggest barriers to the inclusionof middle school teachers in this community was time and distance. Since the two schoolswere on separate campuses, and the teachers’ schedules were so demanding, it was difficultfor the Tree and Elm math teachers to collaborate once the workshop was over.7 In this study, each participant was assigned a number by which data related to that indi-vidual would be identified. To preserve confidentiality, only the researchers know whichnumbers match to which participant. Quotes from post workshop interviews in this text arelabeled with the appropriate respondent’s number. This allows readers to see that a varietyof respondents’ views are represented throughout the text.8 As previously mentioned, all workshop sessions were videotaped. The videotapes werelabeled by the workshop day on which they were used and the order in which they wererecorded. On most days, three videotapes were needed to record the days events. All quotesfrom videotapes are labeled by the videotape from which they came.

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ANDREA LACHANCE

Cortland Education DepartmentState University of New YorkP.O. Box 2000Cortland, New York 13045-0900USAE-mail: [email protected]

JERE CONFREY

Department of Curriculum and InstructionUniversity of TexasCampus Mail Code D5700Austin, Texas 78712USAE-mail: [email protected]

FRAN ARBAUGH

STUDY GROUPS AS A FORM OF PROFESSIONALDEVELOPMENT FOR SECONDARY MATHEMATICS TEACHERS


ABSTRACT. In this paper I examine the value(s) that seven secondary geometry teachersplaced on their participation in a school-based study group. I also examine the organiza-tional structure of the study group, and the impact of that structure on teachers’ continuedparticipation in the study group. This study draws upon both written and oral interviewdata as well as transcripts of study group meetings. Results of analysis show that theteachers were supported in four areas: building community and relationships, makingconnections across theory and practice, curriculum reform, and developing a sense ofprofessionalism. Further, analysis also indicated that participation in the study grouphad an impact on teachers’ self-efficacy. Building on the study group literature (contain-ing predominately non-mathematics education research), this study has implications forsecondary teachers’ professional development, particularly those types of professionaldevelopment experiences that seek to be ongoing, school-based, and teacher-centered.

KEY WORDS: in-service teacher education, professional development, secondarymathematics teachers, study groups, teacher education

INTRODUCTION

A true profession of teaching will emerge as teachers find ways and are given the opportu-nities to improve teaching. By improving teaching, we mean a relentless process in whichteachers do not just improve their own skills but also contribute to the improvementof Teaching with a capital T. Only when teachers are allowed to see themselves asmembers of a group, collectively and directly improving their professional practice byimproving pedagogy and curricula and by improving students’ opportunities to learn, willwe be on the road to developing a true profession of teaching. (Stigler & Hiebert, 1997,p. 21)

Like Stigler and Hiebert (1997, 1999) many members of the educationcommunity call for professional development opportunities that rely onteacher collaboration (e.g., Little, 1987, 1990; Loucks-Horsley, Hewson,Love & Stiles, 1998). Responding to the call for research on professionaldevelopment involving teacher collaboration is a small but growing bodyof literature on what is generally termed “teacher study groups” (e.g.,Birchak et al., 1998; Jones, 1997; Meyer et al., 1998; Short, Glorgis &Pritchard, 1993).


140 FRAN ARBAUGH

Within the U.S. mathematics education community, the call for teachereducators to move away from the one-day in-service model of teacherdevelopment to a more collaborative and comprehensive model of profes-sional development is also strong (e.g., Brown, Smith & Stein, 1996; Cobb,Wood & Yackel, 1990; Fennema, et al., 1996). Sprinthall, Reiman & Thies-Sprinthall (1996) describe the in-service model as “Knowledge How-ToModels” of professional development. Although these models differ invarious ways, an underlying assumption of these types of professionaldevelopment experiences is that in-service teachers lack knowledge aboutcertain aspects of their profession, and they need to be exposed to the infor-mation to “fill in the gaps.” More collaborative teacher-centered models ofprofessional development, which Sprinthall et al. call “Interactive Models”now exist, and many mathematics educators are involved in understandingthose models and their effects on in-service teacher learning and classroompractices (e.g., Britt, Irwin & Ritchie, 2001). Despite the fact that researchconducted outside of the mathematics education community suggests thatstudy groups are a useful means to support teacher learning and profes-sional growth (Birchak et al., 1998; Lewison, 1995; Meyer et al., 1998),little empirical research on the use of study groups as a form of profes-sional development for U.S. teachers has been undertaken by investigatorsin mathematics education. (One example of this type of research can befound in Hammerman, 1995, 1997.) If the use of study groups is indeeda productive professional development mechanism, then the mathematicseducation community benefits from research on the use of study groups asa form of professional development for practicing teachers. Such researchwill help the community better to understand the kind of support studygroups provide to in-service teachers.

The research reported in this article is part of a larger research project inwhich I sought to understand the ways in which participation in a partic-ular study group influenced in-service secondary mathematics teachers’knowledge, thinking, and practice regarding the teaching of mathematics(Arbaugh, 2000). Other papers resulting from this research present anexamination of the extent of teacher participants’ learning and professionalgrowth (see, for example, Arbaugh & Brown, 2002). This paper focuseson understanding the context in which the teacher learning occurred – thestudy group itself. To that end, I address the following questions:

1. What value did teachers place on their participation in an ongoingstudy group?

2. What organizational aspects of the study group were important forcontinued participation?

STUDY GROUPS FOR SECONDARY MATHEMATICS TEACHERS 141

Teacher motivation to learn, much like student motivation to learn, is inter-twined with the context of the educational setting. Research similar to thatreported in this paper has been conducted in subjects other than mathe-matics and with study group participants other than secondary teachers(e.g., Klassen & Short, 1992). As Klassen and Short state: “Few researchstudies look at how beliefs and practices are developed, shifted, andchanged and what contexts and conditions exist to promote or inhibit thosechanges” (p. 343). Understanding the value that secondary mathematicsteachers place on their participation in a study group, as well as whatorganizational aspects of the group meetings (release time for meeting,how often to meet, how long to meet, the number of members in the group)are important for continued participation, helps us to provide authenticexperiences and contexts for teacher growth. Listening to what teachers tellus about their motivation for learning is vital for understanding in-serviceteacher education issues.

In studying two school-based study groups whose members wereelementary teachers, Birchak et al. (1998) found that the teachers’comments regarding what they valued about participating in the studygroup fell into four categories: (1) building community and relation-ships, (2) making connections across theory and practice, (3) supportingcurriculum reform, and (4) developing a sense of professionalism. Wouldsecondary teachers value similar aspects of the study group experience?Would other valuable aspects of study group participation emerge? Usingthese four categories as an analytical frame, I investigated the value thathigh school mathematics teachers place on their participation in a studygroup.

BACKGROUND

What Is a Teacher Study Group?

The term “study group” as used in education is but one of the termsattributed to a group of educators who come together on a regular basisto explore different aspects of education. Other terms that carry a similarconnotation are evident in the literature on professional development inthe United States and include “teacher inquiry groups” (Hammerman,1997), “professional study groups” (Mitchell, 1989), “teacher supportgroups” (Rich, 1992), and “teacher professional groups” (Avalos, 1998).For this research, “study group” is defined to be a group of educatorswho come together on a regular basis to support each other as they workcollaboratively to both develop professionally and to change their practice.

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The “Toyota Time Study Group”

In spring of 1999, the geometry teachers at Ericson Valley High School(EVHS) were awarded a Toyota Time Grant (sponsored by the NationalCouncil of Teachers of Mathematics) to support a long-term goal todevelop (or find) and implement a new geometry curriculum that wouldbe more student-centered and inquiry-based than the curriculum they werecurrently using. They knew that using a more inquiry-based curriculumwould also require different pedagogical practices. Recognizing thatchange is often a difficult process to undertake and implement, thegeometry teachers at EVHS welcomed the opportunity to participate ina professional development experience that would help them focus on thetasks that they use in their geometry classes.

Brian, the math department chair, contacted me for assistance indesigning and implementing the professional development experience – acollaborative, teacher-centered environment in which these teachers couldbe provided support as they worked towards reaching their goals. Given myinterest in study groups, coupled with the need at EVHS for professionaldevelopment, Brian and I decided to begin a study group that would focuson the teaching of geometry. Nine geometry teachers and I began meetingin October 1999. Two of the teachers did not complete data collectionactivities for this research (due to circumstances not related to the studygroup itself), leaving seven teachers on whom this research reports.

We met, as a group, ten times from October 1999 through March 2000(approximately once every two weeks). The length of meetings varied;they lasted 1.5 hours (after school; meetings 2 and 8), 2.5 hours (duringschool hours; meetings 1, 3, 4, 5, 6, 9), or 6.5 hours (during school hours;meeting 7). Meeting 10 was also held after school hours and was 2.5 hoursin length. Teachers were supported with release time (substitutes wereprovided for meetings held within their contract hours) and with stipendpay for time that exceeded their contract hours.

We engaged in many activities as a group over the course of the 10 studygroup meetings. The Mathematical Tasks Framework formed the umbrellafor many of our activities. (See Stein, Smith, Henningsen and Silver (2000)for an extended discussion of this framework.) This framework focuseson the levels of cognitive demand required by mathematical tasks and thevarious phases tasks pass through in their instructional use. We used thisframework over the course of our work together to consider many aspectsof the teachers’ practice.

Study group meetings were typically organized around two to four maintopics of discussion, as indicated in Table I. The teachers initiated many


study group activities; Brian and I initiated other activities. At the end ofeach study group meeting, the agenda for the next meeting was negotiated.

While this table provides an abbreviated look at the activities of thestudy group, it does not provide any indication of the spirit of the discus-sions. Discussions were often lively and animated, with the teachers, Brian,and me participating by telling stories, asking questions, and initiatingtopics. A full account of the study group discussions is beyond the scopeof this article, but can be found in Arbaugh (2000).

THE STUDY

Study Participants

Seven high school mathematics teachers, Annie, Brian, Carl, Craig, Ed,Megan, and Pamela,1 all of whom taught at least one section of geometryduring the year in which this research occurred, participated in this study.At the time of the study, all seven of the teachers taught at EricsonValley High School (school population of approximately 1750 studentswith 16 mathematics teachers during the year of the study). Five of theseven teachers (Annie, Brian, Craig, Ed, and Pamela) taught the samelevel geometry course, and used the same textbook. Carl taught a “low-level” geometry course, and Megan taught a geometry course designedfor students in an alternative program within the high school. These twoteachers used the same textbook, but a different book from other studygroup members, in their classes.

The seven teachers had varied teaching experience. Two of the teacherswere in their first year of teaching (Annie and Pamela), having recentlycompleted bachelor’s degrees, which included becoming state certified toteach secondary mathematics. The other five teachers had the followingnumber of years of experience in teaching mathematics: 3 (Craig), 8(Ed), 15 (Megan), 24 (Brian), and 32 (Carl). Although Craig and Ed hadprevious teaching experience, they, like Annie and Pamela, were new toEVHS. Craig was new to the area, having taught high school math in aneighboring state. Ed had recently transferred to EVHS from a middleschool in the same district. Brian had also taught at the same middle school,having transferred to EVHS three years prior. Megan and Carl could beconsidered the “old-timers” of the group, each having taught at EVHS forthe majority of their careers. Additionally, during the year of this study,Brian was in his first year as the mathematics department chair.

My own role was one of group participant/facilitator and researcher,and I played an active role in study group meetings. I was a participant

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TABLE I

Study group activities

Study Topics/activitiesgroup #

1 • Brainstorm “What does student reasoning mean?”• Reading/Discussion about the Mathematical Tasks Framework (Stein &

Smith, 1998)• Task-sorting activity (Stein, Smith, Arbaugh, Brown & Mossgrove, in press)• Negotiating next meeting agenda

2 • Continuation of task-sorting activity discussion• Sharing methods used to introduce triangle congruency∗• Negotiating next meeting agenda

3 • Discussion about student motivation∗• Brainstorming topics for inclusion in study groups• Gathering high-level tasks∗• Negotiating next meeting agenda

4 • Sharing about activity done in teachers’ classrooms (The EnvelopeGame)∗

• Assembly of tasks collected at last study group meeting• Engaging in MIRA activities∗• Negotiating next meeting agenda

5 • Sharing about MIRA activity done in teachers’ classrooms∗• Analysis of semester exam∗• Negotiating next meeting agenda

6 • Engaging in activities based on ideas of Competency-Based Instruction(CBI)∗

• Negotiating next meeting agenda

7 • Engaging in The Geometer’s Sketchpad activities∗• Writing reflections on classroom observations• Sharing of activities done during classroom observations• Researcher feedback on classroom observations• Negotiating next meeting agenda

8 • Sharing of activities done during classroom observations• Negotiating next meeting agenda

9 • Writing reflections on classroom observations• Sharing activities done during classroom observations• Reading/Discussing four articles on classroom discourse∗• Negotiating next meeting agenda

10 • Teacher writing time• “Where do we go from here?” discussion• Study group wrap-up dinner∗

∗Teacher initiated activity.


in all discussions – asking probing questions and challenging the teachersto reflect verbally on their knowledge and teaching. I found articles forus to read based on the requests from the teachers. After receiving feed-back from the teachers on what they wanted to accomplish at the nextmeeting, Brian and I, working together, finalized agendas for the meetings.At my request, the teachers completed written reflections. As researcher, Icollected all data necessary to complete the study.

Data Collection and Analysis

For the larger research project, multiple sources of data were collectedin order to support data analysis and subsequent research conclusions, aswell as to strengthen analysis through triangulation (Ernest, 1997). Threeof those data sources are relevant to this paper: the final interview writtenpreparation, the final interviews, and the audiotaped accounts of studygroup meetings. These data sources and their analyses are explained below.

During the last study group meeting, I asked the teachers to writeanswers to a set of questions (I call this the “final interview written prepara-tion”). These questions, and the teachers’ written answers, formed the basisfor much of the final interview.

For the “final interview written preparation,” I asked teachers to respondto the following questions:

• What aspects of your participation in this study group have had animpact on your knowledge about issues in mathematics education?

• What aspects of your participation in this study group have supportedyou in planning for instruction in your geometry class(es)?

• What aspects of your participation in this study group have supportedyou in implementing instruction in your geometry class(es)?

• What else about your participation in this study group has had animpact on any aspect of your classroom practice?

• What did you like most about participating in this study group?• What did you like least?• How important was it that we were able to meet, most of the time,

during school hours? Why important/not important?• Is there anything else you’d like to tell me about your participation in

this study group?

Along with predetermined interview questions (used to inform anotherpart of the large research project), I used an individual’s written responsesto the questions above to prompt discussions in the final interview. Iconducted all interviews, and every interview was audiotaped and thentranscribed. Individual interviews lasted approximately 45 minutes, were

146 FRAN ARBAUGH

conducted at school during the teacher’s planning period, and took placewithin a month following the last study group meeting.

All study group meetings were audiotaped and transcribed. For thisstudy, I coded all transcripts using a coding scheme recommended byCoffey and Atkinson (1996). In describing what they mean by coding,Coffey and Atkinson write:

In practice, coding usually is a mixture of data reduction and data complication. Codinggenerally is used to break up and segment the data into simpler, general categories and isused to expand and tease out the data in order to formulate new questions and levels ofinterpretation. (p. 30, authors’ emphasis)

Following their advice, I first coded the interview transcripts to break upthe data, reducing it to chunks that would inform different parts of the largeresearch project. For this study, I then used Birchak et al.’s (1998) fourcategories (building community and relationships, making connectionsacross theory and practice, supporting curriculum reform, and developinga sense of professionalism) as codes to reduce the data further. I also codedcomments not falling in these categories and looked for emergent themes.Using the qualitative research software NUD∗IST4, I “chunked” the datafor further interpretation. Transcripts of study group meetings were thenanalyzed to find supporting evidence for the results of analysis of theinterview transcripts.

RESULTS: TEACHER PARTICIPATION

Teacher participation in this particular study group can be characterized asstrong, based on attendance at meetings. Of the seven teachers who partici-pated in this study three teachers attended all 10 study group meetings, twoteachers missed one meeting each, one teacher missed two meetings, andone teacher missed a total of three meetings.

Given the strong attendance rate, I was somewhat surprised that intheir final interviews more than one teacher indicated skepticism at somepoint about participating in this group. Carl, for example, told me abouta time when he thought about dropping out of the group. He talked aboutfeeling obligated to his students and that he “was feeling like maybe I wasneglecting them.” I asked him why he did not leave the study group at thattime and he said:

One of the things I’ve done over the years is say, ‘Well I don’t often have the opportunityto improve my skills. If I don’t take them when I have them, then there is no one to blamebut oneself.’


Pamela, a first-year teacher who attended all of the meetings, told me aboutthe hesitation she felt about being in the group in the first place:

I think I was hesitant at the first of the year, because all through my schooling I heard ‘don’tburn yourself out the first year. Don’t do too much. Don’t try to add in a million activities.Just go straight through it and learn from [it] and get better each year.’

Based on study group attendance and evidence presented elsewhere, thesetwo teachers and others felt that the overall experience was worthwhile(Arbaugh, 2000), yet as Carl and Pamela indicated in their statementsabove, some were hesitant about participating. The seeming inconsistencybetween their hesitancy at the beginning of the study and their willing-ness to stick with it and continue to participate throughout the studyraised certain questions in my mind. Was their willingness to participatebased only in the acquisition of new pedagogical knowledge, which inturn helped them to begin to change their practice? Or is there somethingelse about our use of a study group format that can explain the teachers’involvement? To help address those questions, the questions addressed inthis study must be answered.

Results of analysis indicated that the teachers placed a high valueon their participation in this study group (Arbaugh, 2000). The teachersreported that certain aspects of participation were most useful to theirprofessional growth and also indicated that certain organizational aspectsof the study group were important for their continued participation. Theresults of analyses are discussed below.

Building Community and Relationships

Four of the seven study group participants reported that they valued thestudy group because it provided the opportunity to build community andrelationships with the other teachers in the group. Annie wrote and talkedabout the amount of interaction she had with the other members of thestudy group, even outside of study group meeting time. She commented onthe lack of interaction with mathematics teachers who were not in the studygroup: “It’s not happening with the other teachers . . . we don’t interactquite as much,” even though they taught common courses.

Annie also felt that she had received a good deal of moral support fromthe other study group members, particularly the teachers who had a numberof years of teaching experience. “Part of what I think I benefited from ishaving the more experienced teachers [in the group] and then interactingwith them more than I would have otherwise.” On a number of occasions,Annie came to study group feeling frustrated with what was happening inher classroom. For example, during the 7th study group meeting, Anniesaid:

148 FRAN ARBAUGH

I know that we [my classes] don’t do well with whole group discussion, that we don’treally have a discussion. I don’t know how to teach the kids to have a whole-group discus-sion, because I don’t think they know how. They don’t understand that that’s what we’resupposed to be doing.

The response from the other group members, including the “old-timers,”was that they perceived that they were not very good at orchestratingwhole-class discussions either, and that Annie should not feel “bad” aboutit. The group decided that reading articles about orchestrating whole-groupdiscussions would help them all to improve on that aspect of teaching. Wedid just that at a subsequent study group meeting.

Similar to Annie, Pamela spoke about how she did not talk to otherteachers in the mathematics department in the same way that she talked tothe teachers who were in the study group:

Occasionally I talk with one [other teacher], an algebra teacher, just to find out wherethey are. . . . In pre-calculus, occasionally we get together and talk about different labs[activities] to do, and where we are, and etc. But it was nice to have all this time to talk tothem [other members of the study group] about what they are doing.

Craig also commented that the thing he liked best about the study groupwas “the interaction and discussion of ideas” that occurred during meet-ings. Megan was a little more expansive in her comments about the studygroup environment: “I like the atmosphere that we had. I felt like I got toknow the other teachers much better. Being able to spend that time withthem made me more comfortable.”

Brian, the department chair, spoke about the frustration he feltregarding one aspect of the community that was being created. In hiswritten responses, he stated, “People need to come [to study group] willingto share openly but not dominate.” When I asked him to elaborate duringhis interview, he added:

I think the bigger problem with [our study group] was not that people were dying todominate. I don’t think we necessarily had people that that was their goal, to dominate. But,we had people, and very likely because they are new to the building or new to teaching, butnot that willing to share.

Brian perceived the community somewhat differently than did Annie andPamela, who were first year teachers, Craig, who was new to EVHS, orMegan, who was the lone mathematics teacher in the alternative programat EVHS and did not teach in the mainstream education program. All fourof these teachers had a need to build community and relationships, andit appears that the study group afforded them a unique opportunity to doso. Brian, in his role as department chair, saw the community buildingin a different light from the other members of the group. His look at the


community occurred through a different lens – that of a supervisor ormentor.

Making Connections across Theory and Practice

Four of the seven teachers reported that participating in the study grouphelped them “think through connections between their beliefs and theirpractices” (Birchak et al., 1998, p. 24). For example, Brian spoke abouthow the discussions in the study group helped him think about questioninghis students during class:

What really affected me on a daily basis was thinking about the questioning. I find myselfin class – I know it’s not the same thing as video-taping – but almost like I’m watchingmyself on a video tape. Analyzing what you’re doing while you’re doing it . . . in a muchmore intense way than I would have.

Furthermore, he said:

I’ll tell you what would happen without the study group. Yes, I would kind of think aboutthose things and I would kind of know that it’s better to probe and get a kid to keep thinking. . . but I wouldn’t know why. And I think not only does it help me to know why, but it helpsme in my conversations with other teachers.

Brian came into the study group with a strong belief that studentsshould be challenged on a daily basis with problems that really makethem think. He appreciated the work we did with the Mathematical TaskFramework and Levels of Cognitive Demand (Stein et al., 2000). In hiswritten responses, he stated: “I have thought a great deal about cognitivelevels of students’ tasks in the past. But this experience has given me someknowledge base and shown me the importance of having some knowledgebase.”

In a similar manner, Carl commented that the study group had influ-enced how he thought about the mathematical tasks he used with hisstudents:

It’s just making me think differently. Every time I give them problems, I [think] ‘Am Igiving this assignment because I gave this last year or this assignment because I think thisis what’s going to help them?’ I think differently about those things and how I spend timein class. I definitely have changed that.

Carl also commented that he liked the way that the articles we readin the study group informed his classroom practice. In the study groupwe often talked about how to get students working in math class – how,as the teacher, to stop doing the thinking for the students. Carl com-mented that the articles helped him conceptualize what he called “just thewhole process,” meaning a student-centered, reform-oriented mathematicsclassroom.

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The same conversations and articles also had an influence on Megan,helping her examine connections between her beliefs and her practice.She talked about this aspect of the study group on two different occasionsduring her interview. The first time she addressed this subject was duringa conversation about supporting student reasoning:

It [the study group] has helped me a lot to think about getting my students to reason better.We were talking about ways to question the kids and to get them to come up with theanswers on their own. I think that’s just a huge step . . . me not constantly guiding them.I wasn’t guiding them, I was fully answering for them. And it’s something that I thoughtabout before, but I think about much more now.

Later in her interview, Megan returned to the topic of questioning students.In response to my question, “If you had to pick just one aspect of yourteaching that has been influenced because of your participation in the studygroup, what would it be?” Megan again brought up questioning students,and attributed reading articles and then discussing them as supporting herin thinking more deeply about this aspect of her classroom practice.

Pamela also talked about thinking more deeply about her practice,and how her participation in the study group had helped her redefine herexpectations in the classroom:

I think my goal at the start of the year was that I wanted my students to ‘try.’ And I thinkI’ve refined my definition of ‘try.’ Not just write down a proportion, because you see somenumbers there. That’s not trying. But actually tell me how you came up with that proportionand tell me how you reasoned through that. So I try to ask them questions and give themactivities so that they have to tell me.

Brian, as indicated earlier in this section, found that connecting theoryand practice not only helped him as a teacher, but also in his role as depart-ment chair. Later in his interview, our conversation turned to providingprofessional development and how hard it is sometimes to “convince”teachers to change their practice. He said that he thought this studygroup helped him with this aspect of his job by providing him with moreknowledge of the theoretical basis for his beliefs.

Each of these teachers found that participating in the study group helpedthem deepen connections between theory, their beliefs, and their practice.Brian, as indicated above, also found that the study group also helped himconnect theory and practice in his role as department chair.

I believe that these teachers were well supported in making connec-tions between theory and practice because of the conscious effort I madeto ground many of our discussions in theory. When designing this studygroup, Birchak et al. (1998) provided invaluable guidance regarding thefacilitation of study groups. They advise that study groups, among other


things, (1) challenge teachers thinking, and (2) connect research and prac-tice. Our use of the Mathematical Tasks Framework (Stein et al., 2000) as aframing concept for this study group allowed the teachers to examine theirpractice within a conceptual framework. As reported elsewhere (Arbaugh,2000), the Mathematical Tasks Framework served a vital role in influenc-ing these teachers’ thinking about mathematical tasks and provided theteachers with support for thinking about instructional decisions and/orimplementation issues.

Supporting Curriculum Reform

One of the main foci of the study group was to develop (or find) and imple-ment a new geometry curriculum that would be more inquiry-based thanthe curriculum the teachers were currently using. We spent a considerableamount of time in the study group meetings discussing task selection fortheir classrooms, and then reflecting on implementation of those tasks. Itis not surprising that all seven teachers wrote and/or spoke about the studygroup helping them by supporting curriculum reform.

The teachers who were new to teaching Geometry (Annie, Craig, Ed,and Pamela) each commented that they liked the sharing of “good” mathe-matical tasks that occurred among study group members. They felt theneed to supplement their adopted geometry text with student-centered,inquiry-based activities that require a high-level of student thinking. Ed’scomments during his interview represent what these four teachers saidabout how the study group supported a change in curriculum:

I also like to see the problems that [the other teachers] really liked . . . or problems thatthey did that I really liked to give me some ideas of where to go from here. Because thisbeing my first year of teaching this course, I’m coming in with nothing . . . I have no files,I have nothing besides what I scrounge around and find or what we’ve talked about or theone day when we put the binder together.

The study group had other influences on curriculum reform for theseteachers. For example, Pamela reported, “I tend to bring in more outsidematerial than I might have if I’d not been in this study group.” Evidenceexists (Arbaugh, 2000) that Pamela did increase her implementation ofhigh-level, supplementary tasks over the length of the school year.

Carl, an “old-timer,” spoke about how the study group influenced theamount of material from the text that he felt he needed to cover: “I don’tfeel like I ‘covered’ the vast amount of material that I did before. But theend result, [is that the students] end up knowing as much or more than justcovering material.” Brian talked about how the study group afforded himthe time and impetus to explore other geometry curricula:

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Because of the study group and this project, I’ve looked a lot at [a commercially-availableinquiry-based curriculum] and they have a different approach to many things that wehave done in the past. Not necessarily that radical of content change, but a very differentapproach.

These teachers came to the study group with an agenda of reformingtheir geometry curriculum. This study group, which occurred in the 1999–2000 school year, afforded them an opportunity to begin to think deeplyabout their curricular choices and begin to change the way in whichthey implemented their curriculum in their geometry classes. Accordingto Brian (personal communication, July 2002), the geometry teachers atEVHS adopted a reform-minded, inquiry-based curriculum for use in theirgeometry classes beginning in the fall of 2002.

Developing a Sense of Professionalism

Birchak et al. (1998) write that the teachers in their study groups “didn’tcome to the group because they felt they needed to make changes in theirteaching, but as part of their professional lives as learners” (p. 26). In asimilar manner, all of the teachers in this study group came to appreciatehow participating allowed them the opportunity to engage in the learningprocess through building their knowledge base as well as being affordedthe time to reflect on their learning and practice.

All seven teachers commented that they learned from hearing otherteachers in the study group discuss implementing specific tasks in theirclassrooms. Annie, for example, described this type of collaboration:

First of all, going over tasks together. Choosing some things to try or some people saying,‘I want to try [teaching it] this way’. And then several people trying it and then discussingwhat they tried. You learn a lot from what somebody else did, [and you think] ‘gosh wellnext year I can try it that way’ – ‘I didn’t think of doing it that way.’

Carl expressed a similar thought when he talked about how hearing otherteachers’ strategies for supporting student thinking made him really thinkabout his own practice, and who was actually “doing the math” in hisclassroom. I asked Carl if he had subsequently used these specific tech-niques in his own teaching. He replied, “I think that it was the big picturefor me,” and went on to explain that he had begun to think about teaching ina different manner, and that these specifics helped him formulate a visionof what that “big picture” should include. Similarly, Craig, in his finalinterview written preparation, wrote, “Hearing [other teachers] tell whatwent well and what didn’t helped my lessons be better.”

Evidence presented elsewhere (Arbaugh & Brown, 2002) indicates thatthe teachers also built their knowledge base in other ways. The teachers notonly increased their knowledge base about teaching through participation


in the study group, they also found that participating in the study groupencouraged them to reflect on their teaching more often than they wouldhave without the study group. For example, Craig, in his written responses,stated:

I’d not spent a lot of reflection time before. When I did, I mainly thought of how the kidscould learn better. Now I look at things I could do to help them learn better. I look more athow I can create an opportunity for them to learn.

Similarly, Megan and Annie both talked about how they had begun toreflect on their teaching differently from the way they had before. Megansaid that her reflections had become focused on two questions: “Am Igetting my kids to reason?” and “Am I getting them to do the math oram I doing it for them?” Annie, a first-year teacher, made a more generalcomment regarding the opportunity to reflect on her teaching: “It [the studygroup] has forced me to reflect more. I wouldn’t do this as much if not forthe group. I would be surviving this year with little reflection.” With thislast statement, Annie indicated that without the study group, she wouldhave done little reflection over the course of the past school year.

Brian again looked at the opportunity for teacher learning that occurredthrough participation in the study group through the lens of a departmentchair:

One thing the study group does is get around the fact that people don’t naturally spend a lotof time thinking about their teaching and thinking about their lessons outside of the schoolday. Well, we put them in a situation where inside the school day they can do that. To methat’s one of the big pluses of study group.

An Emerging Category: Teacher Self-Efficacy2

The influence of study group participation on some of the members’ self-efficacy was an unexpected result, and one that emerged during analysis.Missing from the literature on study groups is a discussion of increasedself-efficacy as a result of study group participation. Three teachers, Craig,Pamela, and Ed, talked about an increased confidence in teaching, and theircomments indicate that as the teachers worked together to improve theirteaching, confidence about their own teaching practices grew.

For example, Craig indicated several distinct times that participatingin this study group had given him more confidence in his own teaching.During the final interview, Craig said the following: “Sometimes it givesyou the confidence if someone else did the same thing you did and theyhad success with it also. [It] builds up your confidence that you’re doing itthe right way – a successful way.” Craig also indicated an increase in self-efficacy in his response to the question, “What else about your participation

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in this study group has impacted you as a math teacher?” He wrote: “Ithink I have gained confidence as a teacher. It helped to know that I wasdoing some things that others were.” When I asked him to elaborate onthat response during his interview, he began by saying, “[It helped] hearingother people [say] ‘That’s a good idea. That’s a nice way to go about it.’ ”Craig went on to say that hearing that kind of praise from Brian and othermore experienced members of the study group really influenced how hefelt about himself as a teacher. He appreciated receiving feedback from hispeers that indicated that he was doing a good job.

Craig appears to have gained self-confidence as a result of study groupmembers talking about what happens behind the closed doors of theirclassrooms. Teachers rarely have the chance to make their teaching publicto colleagues and receive critical feedback. Our focus on classroom imple-mentation of high-level tasks provided the opportunity for these teachers tomake their teaching more public. His own sharing influenced Craig, as didsubsequent feedback from peers, listening to other teachers and comparingtheir teaching stories with his own.

Provisional Summary

The study group in which these teachers participated provided an oppor-tunity and support for them to build community and relationships, makeconnections across theory and practice, engage in curriculum reform,and increase their sense of professionalism. Further, this study indicatesthat some teachers perceived an increase in self-efficacy as a result ofparticipation in this study group.

RESULTS: ORGANIZATION

Were there organizational aspects of our study group that particularlysupported these teachers’ efforts? In the next section, I address my secondresearch question “What organizational aspects of the study group wereimportant for continued participation?”

Organizational Aspects of Study Group Meetings

During their final interview, I asked each teacher to comment specificallyabout the ways in which we organized the study group meetings. In thefollowing sub-sections, I examine what they told me about four aspectsof our study group’s organization: release time, requirements outside ofstudy group meeting time, frequency and length of study group meetings,and number of study group members.


Release time. Overall, the teachers characterized release time as being animportant organizational characteristic of our study group. Three of theseven teachers indicated that release time was “very important.” In addi-tion, Annie described the release time as being “helpful” and Carl thoughtthe release time was a “mixed bag” but ultimately decided it was “thebest of the options.” He thought that having to prepare for a substituteadded work to his already-full plate and, at times, he felt as though he wasneglecting his students by being absent. In conclusion he said:

But I think we would have been more frustrated if it had been when we were already tiredat the end of the day, or on Saturdays and we were giving up time with our families and soon. . . . So I think it was the best of the options, but there were still things there that madeit tough to deal with.

Pamela, Megan, and Brian were willing to consider after-school meetingtimes on a continual basis, although each expressed “problems” withmeeting so late in the day.

After-school meetings during this project were a mixed bag. The first ofsuch meetings was our second meeting, and most of the teachers were ableto attend (although Megan had to leave after 45 minutes to attend to familymatters). Our second after-school meeting did not fare nearly as well.Craig could not attend because of basketball coaching responsibilities,Carl because of union responsibilities, Ed because of family responsibil-ities. Megan, Annie, Brian, Pamela and I met until Megan had to leave 45minutes into the meeting to attend to family matters, when the rest of usdecided that the meeting was an exercise in futility.

Release time was important to these teachers because they leadvaried and busy lives. Beyond the time outside of contract hours thatthe teachers need for grading and planning, high school teachers areoften expected to contribute to the school on a larger basis by sponsoringclubs, coaching sports, and participating in policy-making capacities.Most of these activities occur in the hours right after school. This wasthe case with some members of our group. Teachers also often havefamily responsibilities that make it difficult for them to spend an extendedamount of time at school after contract hours, which was the case withother members of our group. If release time had not been provided fora majority of our meetings, Craig, Ed, and Carl would have had seriousreservations about participating.

Requirements outside of study group time. In their final interviews, fourout of the seven teachers said they preferred having time to complete theirreflections and read articles during study group meetings as opposed tobeing required to do those sorts of things outside of study group time. Of

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the four, Carl provided perhaps the strongest statement regarding any timerequirements outside of study group time when I asked him to commenton that issue:

I think that deals with the issue of how much time do you devote to this, and where it isgoing to be spent. And I personally would have been more inclined to [leave] the studygroup, if we would have had to spend more time outside.

Conversely, Megan and Pamela both indicated that the reading andwriting that I asked them to do could have been done outside of the studygroup meetings. Brian, caught in a conundrum between the “ideal” andactual practice, said “in the ideal it [reading and writing] really ought to bedone outside of study group.” But in reality, Brian himself had difficultycompleting early outside-of-study-group writing.

Overall, while some of the teachers said that they could (or should)read and write outside of study group time, the voices of the other teachersspoke loudly regarding this issue. Not only did the teachers indicate theirpreference verbally, they also showed their preference by not completingoutside-of-study group assignments early in the study. Further, in theirfinal interviews, two of the teachers (Carl and Craig) indicated that theirparticipation in the study group would have been affected had they beenrequired to do these types of activities outside of study group time.

Why was this aspect of study group organization so important for Carland Craig? I contend that the reasons are similar to those attributed tothe importance of release time. The teachers have “full plates,” as Carlindicated. Any reading or written reflections for the study group endedup far, far down on a list of priorities for these teachers. Brian spoke ofhaving to make choices on a daily basis about what to attend to that day.Study group “homework” was not at the top of his priority list, oftenbecause of other school responsibilities.

Frequency and length of study group meetings. Most of the teachersagreed that the frequency of our study group meetings was “just right” (wemet about once every two weeks). Meeting once a week seemed to bringup a concern regarding being out of the classroom too much. The teachersalso indicated that meeting less often would lessen the connections fromone study group to another.

Number of study group members. As a group, these teachers expressed theopinion that the number of members of our study group worked in our ownsituation, but many of them speculated that a study group could also havefewer members. Many felt that having too few members (less than about4) would mean that there was a lack of diverse experiences and ideas, and


that having too many members (more than about 10) would lead to a groupwhere individual contributions would be lessened.

DISCUSSION, CONCLUDING REMARKS ANDUNANSWERED QUESTIONS

High school mathematics teachers often work in isolated conditions,rarely collaborating with their colleagues about the mathematics theyteach (Stigler & Hiebert, 1999). The contact they do have with otherteachers in their departments is often in the form of department meetings,where the focus of discussion is most likely administrative concerns. Highschool mathematics teachers may also share a common “planning” space,a department workroom or teacher lunchroom – spaces where teachersgather during the school day. It is in these spaces that teachers share phys-ical space while they individually grade papers, write quizzes and tests,and plan for upcoming lessons.

Grossman, Wineburg, and Woolworth (2001) write, “The simple factis that the structures for ongoing community do not exist in the Americanhigh school” (p. 947). The Toyota Time Study Group provided a struc-ture in which teachers could begin to build community and relationships.Release time for meetings was an important part of that structure. Wouldthe group have been as “successful” if funds were not available for substi-tutes? Evidence provided in this study suggests that the answer to thatquestion is a resounding “NO.” Other organizational aspects of our studygroup do not appear to be as much of a “deal breaker” as release time andoutside requirements. The recommendations these teachers make aboutlength and frequency of meetings and number of study group memberscan assist other educators who are considering using a study group as aform of professional development for secondary mathematics teachers.

In their work with high school English and history teachers, Grossmanet al. (2001) begin to theorize about the development of community at thehigh school level and also discuss obstacles to community. They arguethat “community is difficult to attain and even harder to sustain” (p. 993,authors’ emphasis). If community is indeed a difficult undertaking, theyask, “Why bother?” (p. 993). In response to that question, Grossman andher colleagues assert that several reasons exist that make the effort ofsupporting teachers to build community worthwhile.

The first reason Grossman and her colleagues give for the importance ofteacher community is that “we cannot expect teachers to create a vigorouscommunity of learners among students if they have no parallel communityto nourish themselves” (p. 993). They also argue that this type of intellec-

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tual community can help in the retention of teachers. Further, they contend,“teachers will always need to find ways to stay abreast of developments inthe subjects they teach” (p. 994).

As indicated earlier in this article, the teachers in the Toyota Time StudyGroup found that our work together and the time we spent in the studygroup meetings allowed them to study teaching and learning in a waythat would not have been probable without the study group. It providedthem with a venue in which they could make connections between theoryand practice, and strengthen their knowledge base about reform-orientedmathematics teaching and learning. They also found that the study groupwas a setting that encouraged reflection. Unfortunately, contrary to whatGrossman and her colleagues contend about teacher retention, two of theteachers in the Toyota Time Study Group left the teaching profession thesummer following our work together. Annie and Pamela both decided thatthe teaching profession was not the right ‘fit’ for them at that point in theirlives, and both left to seek opportunities outside of education.

Concluding Remarks

This study provided an opportunity for these seven teachers to “speak”about their experiences in this study group. As indicated by study groupmeeting attendance, and then supported by the teachers’ final interviews,these teachers were actively involved in study group activities and foundseveral aspects of those activities useful. These teachers, though fewin number, speak loudly about the benefits and usefulness of providingopportunities for teachers to collaborate on a regular basis over an extendedperiod of time. It is interesting to note that although this study wasconducted during the 1999–2000 school year, the teachers at EVHScontinued to find ways to support sustained collaboration. In the 2000–2001 school year, some members of the “Toyota Time Study Group” begananother study group focused on the teaching of algebra. Some EVHSmathematics teachers who were not members of the original study grouphave subsequently joined the algebra group.

My goal in this study was to further the understanding of the use ofstudy groups as a form of professional development for U.S. high schoolmathematics teachers. I had hypothesized that a study group atmospherewould allow these teachers to begin to develop a more collegial relation-ship (Little, 1990). What became clear to me during this study is that,for these teachers, having the opportunity to collaborate with their peerson a regular basis and in a meaningful manner was one of the mostuseful aspects of study group participation. Repeatedly, these teachersreiterated that working with their peers helped them to reflect on their


own teaching and build new pedagogical knowledge.3 Brian, in his finalinterview, summed up the experience as follows:

Without this study group, would those people have gotten together and had some of thediscussions we’ve had this year? No, they would have talked about geometry. They wouldhave talked about ‘what lesson are you on? When are you giving the chapter 8 test?’ But itwouldn’t have been the kinds of discussions that we had in study group. And long term Ithink those will affect peoples’ teaching . . . they affected our teaching some already.

This research supports the literature on the use of study groups as aviable form of professional development for in-service teachers (Birchak etal., 1998; Cramer, Hurst, & Wilson, 1996; Short et al., 1993). It extends ourunderstanding of using study groups as a context for professional develop-ment for mathematics teachers, and begins to fill a gap in knowledge aboutusing study groups with high school mathematics teachers. It highlightsimportant outcomes for high school mathematics teachers who participatein a study group – particularly with regard to the extent to which they valuecollaboration with their peers.

Study groups are only one approach for designing collaborative andteacher-centered professional development. Other examples of profes-sional development opportunities that allow teachers to work collabora-tively exist in the U.S. as well as in other countries, and many of theseapproaches have been studied and found to be “successful.” The resultsof this research study should not be interpreted as suggesting that studygroups are the only venue for this type of collaboration. Indeed, the“success” of any professional development opportunity often is dependentupon the context of the experience and the people who are involved. Thisresearch provides an opportunity for the U.S. educational community tohear the voices of teachers who were involved in a professional experiencethat they found valuable.

To that end, these teachers overwhelmingly recommended, no matterthe organizational details as discussed in this paper, that high schoolmathematics teachers be provided with opportunities to collaborate, ona regular and sustained basis, with their peers to improve mathematicsteaching and learning. Grossman et al. (2001) suggest that this type ofcollaboration can support teachers as they build community.

Unanswered questions for future research. When I was conceptualizingthis research, I struggled with the recommendations from the liter-ature on study groups pertaining to the use of teacher-initiated contentfor group study juxtaposed with my own beliefs about what consti-tutes “productive” professional development for in-service mathematicsteachers. Richardson (1992) describes this struggle as “the agenda-setting

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dilemma in a constructivist staff development process” in an article of thesame name:

The dilemma of agenda-setting relates to the dual and sometimes conflicting goals of intro-ducing participants to a particular content, and creating an empowering and emancipatoryenvironment that requires that the participants own the content and process. The achieve-ment of both goals simultaneously seems to be particularly difficult when the processis undertaken in a context in which teachers expect a staff development program to betop-down, with an “expert” leading the process. Who, then, owns the content? Who is(are) the expert(s)? The second goal suggests that power must be shared equally by theparticipants, and that each participant contributes expertise. However, with a preconceivedand negotiated formal content that is seen to reside in the staff developers, the goal ofshared power may be difficult to reach. (p. 288)

Particularly in this era of reform in the United States, when so muchof what we want teachers to investigate and understand is pedagogicallyand epistemologically new to them, further research on the use of studygroups with mathematics teachers is imperative. Can and will mathematicsteachers be self-directive in their professional development? What wasmy role as facilitator in encouraging an environment where the teachers“dug deeper?” Was the study group a success because of the structure,or because of my influence on topics and direction of teacher talk duringstudy group meetings? Results of a study regarding the topics that a groupof middle school mathematics teachers talked about during common plan-ning time (Brown, Arbaugh, Allen & Koc, 2000) suggest that the presenceof an “expert other” or resource partner was a determining factor in thequality of teacher talk. How important is the “expert other” in a study groupto the teachers involved?

Whether or not an “expert other” is on hand, will mathematics teachersbe self-directive in setting the agenda in such a study group, and will thetopics chosen be conducive to professional growth? I suggest that teachers’perceptions of what “professional development” entails have probablybeen colored by the workshops they attended in the past, where they wereengaged in activities that they could use with their students. If teachers donot (or are not able to) see outside of the experiences they have traditionallyhad in professional development situations, can they take ownership ofsetting a professional development agenda that will lead to professionalgrowth? Do we need to think about supporting teachers in a more directiveway as they begin to grow professionally and then slowly “loosen the apronstrings?”

Other questions remain about the way in which this particular studygroup was organized. Was part of the “success” of this study group exper-ience based on the fact that all the teachers taught at the same school?Further research could also be conducted that examines the advantages/


disadvantages of school-based study groups in comparison with district-wide study groups, or even study groups that utilize the distance educationtechnology available via WebCT or satellite hook-ups.

In the end, studies such as the one conducted for this paper contribute tothe mathematics education community’s understanding of supporting highschool mathematics teachers as they seek to grow professionally. Still, wehave much to learn.

ACKNOWLEDGEMENT

I would like to acknowledge Mitzi Lewison and Sandra Abell for theirthoughtful feedback on early drafts ot this article.

NOTES

1 All teachers’ names, as well as the name of the high school, in which they taught, arepseudonyms.2 Self-efficacy is defined as “an ability construct . . . that refers to individuals’ beliefsabout their capabilities to perform well” (Graham & Weiner, 1996).3 Please see Arbaugh (2000) and Arbaugh and Brown (2002) for research resultsregarding teachers’ growth in knowledge.

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University of Missouri-Columbia303 Townsend HallColumbia, MO 65211USAE-mail: [email protected]

TEACHER EDUCATION AROUND THE WORLD

RENUKA VITHAL

TEACHERS AND ‘STREET CHILDREN’: ON BECOMING ATEACHER OF MATHEMATICS �


ABSTRACT. The phenomenon of ‘street children’ where young people, for variousreasons, live on the streets of towns and cities is found all over the world in varyingdegrees and forms. In South Africa, one approach to take care about the plight of thesechildren has been to set up and run what are referred to as ‘street shelters’. One such streetshelter, the only one exclusively for girls in the city of Durban, is Tennyson House. In thispaper I describe an innovative outreach programme integrated with a university curriculumin which a group of pre-service teachers taking mathematics education as a major wereinvolved in teaching mathematics to girls at Tennyson House. From the vantage point ofa mathematics teacher educator in the programme, I describe and reflect on what wasexperienced and learned from the intervention in terms of three aspects: learning aboutlearners; learning about teaching (mathematics) and learning about relationships.

KEY WORDS: care and compassion in mathematics classrooms, preservice mathematicsteacher education, teaching diverse mathematics learners

INTRODUCTION

HIS NAME IS TODAY1

We are guilty of many errors and many faults.But our worst crime is abandoning the children.Neglecting the fountain of life.Many things we need, can wait.The child cannot.Right now is the time his bones are being formed.His blood is being made andHis senses are being developed.To him we cannot answer “Tomorrow”

His name is Today.Gabriella Mistral

� A shorter version of this paper appears in L. Bazzini & C.W. Inchley (Eds.), Proceed-ings of CIEAEM 53: Mathematical Literacy in the Digital Era, 21–27 July, Verbania (pp.301–306). Milano: Ghisetti E Corvi Editori.


166 RENUKA VITHAL

What does it mean to provide a teacher education that develops knowl-edge, skills, attitudes and values not only for working with the diversityof learners found in the vast majority of societies today but also one thatreaches its margins – those learners we do not usually have in our mindswhen we think and talk about mathematics education – such as so-called‘street children’? This was not the question I started out with, but I pose itnow to all of us mathematics teacher educators.

This paper takes the form of a retrospective reflexive account of anoutreach programme, the Tennyson House Project, seeking to supportthe mathematics learning of so-called ‘street children’ in relation toformal school mathematics. Its focus is on student teachers who mediatedlearning for such children, and it draws lessons for teacher education. Thestrong integration of the programme with a mathematics teacher educa-tion curriculum helps us not only to understand better and work withsuch learners but also points to gaps and silences in theories, practices,and research in mathematics education; and in the education and trainingof teachers of mathematics. The description of the programme unfoldsthrough the paper as it evolved over a two-year period. I share my reflec-tions and learning from this experience as teacher educator and researcher,and write not from analysis of research data, but as analysis of praxis –“the dialectical tension, the interactive, reciprocal shaping of theory andpractice” (Lather, 1986, p. 258). How do we produce teachers who demon-strate not only competence in the content and pedagogy of mathematics,but also, build the courage and capacity to speak back to the theoriesand practices we advocate, arguably, without sufficient attention to thediversity within which they are expected to be given meaning? Perhapsmost importantly, how do we develop an ethic of caring, compassion andkindness that notices and seriously considers those at the periphery usuallylabelled as failures in the mathematics classroom, and that reclaims theminto mathematics education.

I tell the story of the Tennyson House Project, as it came to be called,through my journal, lecture notes, the portfolios of two of the studentteachers – Amisha Jugoo (AJ) and Shaleen Jugoo (SJ) – and memorieswhich no doubt are tempered through the passage of time. I draw also ona paper a group of student teachers and I presented jointly as a plenaryat a regional mathematics teacher conference and that later appeared ina journal (Vithal et al., 2001). My focus now is on student teachers andmathematics teacher education, hence the children’s voices are mediatedby data made available by the student teachers where it was directly quotedor referred to in their portfolios and the co-authored paper. In particularI explore: firstly, how we come to learn about learners and teachers of

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mathematics; secondly, learning to teach mathematics at the boundaries,which opens up challenges to current theories and practices; and thirdly,learning about relationships, which hold the first and the second aspectstogether in any mathematics education endeavour.

BACKGROUND AND CONTEXT

This project was a collaboration between Tennyson House and the Facultyof Education at the University of Durban-Westville where I am a mathe-matics teacher educator. Referred to as a historically disadvantaged univer-sity which was created by apartheid South Africa for those it chose toclassify “Indian”, in post-apartheid South Africa, the university has rapidlychanged to a student body that is predominantly “Indian” and “African” butwith a staff complement that remains still largely “Indian” and “White”.2

I was working with undergraduate student teachers, in the third year of theBachelor of Paedagogics, specialising in mathematics education to becomeprimary school teachers. I was looking for an opportunity to give them theexperience of working in small groups with mathematics learners beforethey embarked on teaching practice in their fourth year of study where theywere likely to be thrown into the deep end by being placed in large under-resourced multilingual classes. In particular the intention was to work withpupils who were ‘failing’ mathematics to understand better, interpret andact on the theories and practices they were learning in the mathematicseducation coursework.

This opportunity was also important for another reason. When studentteachers are asked to write about their experiences in mathematicsclassrooms, I have found quite consistently, over several years, that thevast majority reflect on these as cold and indifferent places, even violent,where life is often lived in fear of the subject and/or of the teacher. Toenculturate prospective teachers into a different way of being and actingin their mathematics classroom has been difficult, as years of socialisationtakes over when they return to the school setting. However, as I observedand interacted with them during their teaching practice, I found somethingprofoundly different about the group of student teachers who worked atTennyson House, that created an imperative to share the outcomes of thisprogramme. It seemed possible to challenge and change their deeply heldvalues, beliefs and attitudes in ways that also impacted on their knowledgeand skills for teaching mathematics.

The idea for this project arose from a conversation with Cheryl Smith,an educational psychologist in the faculty at that time, who had justcompleted her doctorate working with the staff and students at Tennyson

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House. Her experiences suggested that student teachers could providevaluable input for the girls’ education in mathematics and I recognised thisas potentially the opportunity I was looking for. Our first meeting to get theproject off the ground was with the director of Tennyson House, RobynHommens and it sought to ensure reciprocity of the goals and purposes forboth the girls and the student teachers. The main intention was to supportthe girls who were brought to the shelter by care/social workers or thepolice, assessed, counselled, and then placed into the formal public schoolsystem. Our task was to provide support for the girls in their mathematicslearning, to improve their success in school, which, it was reasoned, mightcontribute to reducing the possibilities for them to return to the street;and it was also to support the counselling they received to contribute totheir being reunited eventually with their families and the community fromwhich they came. The invitation to the student teachers to participate in theprogramme required them to take on the responsibility for the mathema-tical growth and development of a young learner for an entire year – theperiod that Tennyson House accommodated the girls until their placementback with their families. Being given choice to participate recognised thefragility and sensitive nature of the work, setting and learners. Ten outof eleven student teachers agreed to take up this opportunity and the oneremaining student, who could not accommodate the programme due totimetabling difficulties, took up the alternative option of working with a‘failing’ mathematics learner from his neighbourhood.

Although there are many programmes that attempt to provide educa-tional opportunities for ‘street children’, most of these are located withinthe shelters or some other special arrangement but are relatively inde-pendent of the formal education institutions or system (e.g. le Roux, 1994;Amin, 2001). The innovation of this initiative was therefore the strongintegration of an outreach educational effort with the formal programmeof a university course and the link with public schools.This contributed tothe programme being taken seriously (not seen as a voluntary work) andalso viewed as being essential to the preparation of students in becomingteachers of mathematics.

LEARNING ABOUT LEARNERS

The learners in this project were, on the one hand the children at TennysonHouse to whom the student teachers were teaching mathematics, and onthe other hand, the student teachers themselves who were working withme to learn about teaching mathematics.


The issue of labelling and characterising children as street children,runaways, homeless, strollers, youth-at-risk, and so on, and its conse-quences, is widely debated in the literature by those who study the livesof these children (Amin, 2001; Swart-Kruger & Donald, 1994; Richter,1989). A distinction often made in the literature is between ‘children of thestreet’ and ‘children on the street’ (Rooyen & Hartell, 2002; Amin, 2001).In the South African context ‘street children’ is typically used to refer tothe former category of children, who for various reasons are abandoned orleave home (Chetty, 1997) to live on the streets of cities with almost nocontact with their families (Rooyen & Hartell, 2002).3 As a result of theravages of apartheid, these children are mainly “African”. Although themajority of street children in South Africa are boys, this project focussedon a shelter that served girls only. Tennyson House could accommodateat most 10-12 girls at any one time (although the number there at anytime fluctuated enormously) and attempted to do so in a warm home-likeenvironment.

The student teachers’ learning began with a presentation from Cherylabout the characteristics and life experiences of street children in generaland of the girls at Tennyson House in particular. One student teacherreflected:

Cheryl’s chat with us before our trip was a critical moment in our project, She explained tous that we would have to earn the right to visit the girls’ rooms, be careful with our choiceof words i.e. do not refer to them as “street children”, earn the right to be their friends, etc.(AJ)

Students and teachers are part of society and therefore carry into educa-tional settings knowledge and attitudes acquired from everyday lives.Often these children are viewed in very negative ways. The need to helpstudent teachers to deconstruct their perceptions of these children, tocounteract any negativity from their past experiences, has been shown tobe important in helping student teachers to work with such learners (Amin,2001). This was borne out in one of the student teacher’s reflections afterthe first meeting with the girls.

My attitude towards street children was always a negative one. I personally felt that thesechildren were a nuisance on the street. However my narrow view has now been broadened.I would do my best to ensure that the girls are given help to the fullest of my ability. I feelthat Cheryl’s input on the girls has been a great help to me. I am able to look at the girls aspeople who are part of our community. I now feel its time we cope and accept the facts ofstreet children. (AJ)

The intention therefore in providing information about these girls was toengender genuine respect and caring and to shape their relationship with

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the girls in positive ways. But it was important also not to create a patronis-ing or victim ideology. Cheryl made available to us the research and workdone in this area to make us aware that many of these young peoplecould be shown to have demonstrated considerable strength and couragein seeking to better their lives when many others in similar situations ofabuse, violence, poverty and neglect remained in similar circumstances. Itwas crucial that the programme, as far as possible, should not exacerbatein any way the difficult life they had already experienced. This was furtherreinforced by Robyn on their first visit to Tennyson House; she made veryclear to the student teachers, especially the males, how they were expectedto handle themselves given that many of the girls were sexually active– several were sex workers and she reminded us that they were still justteenagers.

Another reason for giving student teachers as much information aspossible about the learners was to help them plan their lessons. Many ofthe girls had a disrupted schooling. Student teachers were informed aboutthe level of mathematical development as gauged from various psycho-logical assessments done for each girl who entered Tennyson House andavailable documents such as school report cards, which guided the girls’placements into various grades in school. Both the girls and the studentteachers participated by choice and required commitment to teaching andlearning. This presented an important challenge for the student teacher whohad to strive much harder to be really innovative to find ways to make theirlessons interesting and relevant for young girls. To do this student teachersacquired as much knowledge as possible about their learners and this drewsharp attention to the importance and need to create meaningful curriculumexperiences for learners.

It raised more general questions of where and how do teachers acquireauthentic knowledge about their students if they are to make real connec-tions to them in ways that respected and valued their lives? To this endstudent teachers developed ‘mathematical life histories’ for each of thelearners. These biographies were constructed by interviewing the girlsabout their experiences of teaching and learning mathematics and their pastperformance; how they went about learning for tests, the textbooks avail-able and how they were used; mathematical topics they liked or disliked, orthat they found easy or difficult; what things the teacher did that helped orhindered their understanding of mathematics, such as forms of assessmenttheir teacher engaged and so on. They reviewed the girls’ mathematicsworkbooks, test books, jotters and report cards. They gained additionalinformation from Cheryl, and the care workers or other staff at TennysonHouse about particular learners assigned to them. Some student teachers


asked the girls to write about themselves. Mpu wrote “about her school,her friends and mathematics” as asked by SJ who reported her words asfollows4

My school’s name is SM Primary School. I like my school so much. I am in Std 4C. Miss Gis my class teacher, in our class we are 45 of us – we are 20 girls and 25 boys. I sit at theback in my classroom. My favourite subjects are Maths and Geography. I like them verymuch but I do not understand them properly.

I have two friends. I like them a lot, they are from my class. They help me in my work.Their names are Z and N. In my school they used to help me in mathematics becausethey understand mathematics. Some mathematics is difficult for me, like long division andsubtraction. But I will try my best to work hard because I know that education is veryimportant to us. My teacher used to tell us how learning will help us when we grow up,how much education is important to us, that is about my school and my school work.

Based on all this information and what the learners were currentlydoing in mathematics classes, the student teachers negotiated areas, topicsand developed activities for lesson planning. To address both the past andpresent learning needs, a variety of different aspects had to be dealt with,for example AJ recorded as objectives of one lesson:

Revise work done on 20/5; Use of counter to aid Nca; Allow Nca to generate her ownexamples; Multiplication of two-digit numbers; Rectangle – concept, shape, formation:Problem solving using the Constructivist Approach to teaching; Notion of place holder;Homework to be given: find things/objects that are circular, triangular and square in shape.

Being involved in the Tennyson House project as a teacher educatorprovided opportunities for me to learn about and interact with studentteachers beyond and outside the formal coursework programme and thismade available much more detailed information about the student teachers.The diversity and limitations of their own experiences in learning mathe-matics had to be engaged. Their strengths and weaknesses in teachingmathematics became more visible and were discussed in this extendedtime. The effect was that the student teachers themselves became awareof and had an opportunity to express and act on their own learning needsas a teacher. Their knowledge, skills, attitudes, values, fears and levels ofconfidence in becoming teachers of mathematics were much more openlydiscernible through their interactions with the girls, each other, the teachereducators and their written work. This made it possible for me to tailor theteacher education programme to respond more directly and explicitly tothe diversity of needs and concerns of the student teachers in their devel-opment as mathematics teachers. To illustrate, AJ reflected in writing onthe case presentations which were explicitly built into the lecture periods:

Shaun mentioned the issue of language preference by learners. This debate concerninglanguage is a vexed one and continues to remain an area of investigation and controversy. I

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would appreciate it if we are enlightened on some approaches that we could use to bridgethe language barriers.

I responded in her portfolio “This (topic on language and mathematicseducation) will be done in the second semester – I could give you aresource in the meantime, please check with me”. In this way the projectopened up possibilities for learning about a special group of childrenbut also about what prospective teachers of mathematics were concernedto know, and that needed to be included or focused on more in thecoursework.

LEARNING ABOUT TEACHING (TO TEACH MATHEMATICSDIFFERENTLY)

The ten student teachers who agreed to participate in the programme,where possible, were grouped in pairs to work with one learner to ensurecontinuity and security for the girls if any student teacher was absent. Thiswas an important decision.

I did not go to Tennyson House as I was unwell. However Amisha filled me in. Mpu feltthat I had abandoned her and was all lost. I did a reading on this and found that thesechildren had been let down by close family members and they need some stability andassurance in their lives . . . However I sent Mpu the hundreds board and place value table.Shivani did enjoy working with Mpu. (SJ)

Furthermore, this arrangement facilitated collegiality and reflective prac-tice as the student teachers jointly prepared lessons and evaluated theoutcomes of activities. This was also important because of the constantlychanging arrival of new girls or absence of one of the girls.

The tutorial lessons took place each Tuesday afternoon, after school, atTennyson House for approximately one hour. Both Cheryl and I supervisedthe tutorial programme, observing, participating in teaching, and makingnotes for following up on issues. Detailed review of the case for eachlearner took place during the coursework on campus where time was setaside for the student teachers to make presentations. Some student teachersreferred to this as “group therapy sessions”. However, what they valuedwas the opportunity for reflexivity in multiple domains and the resourcesavailable to them, including themselves. AJ reflects:

Collaborative learning and sharing should be the focus of our project. I found that we learnta lot from each other by sharing our ideas and experiences. I observed that each of us hadunique things/ideas to share. Our math ed 3 group seem to be very enthusiastic about theproject from what I can gather. They seem genuinely concerned about the girls and theireducational needs. With regards to our lessons and their content we need to specify ouroutcomes. After the lesson we need to reflect on whether our outcomes are met or not.


Both Cheryl and Robyn attended these sessions sometimes to give feed-back and additional information about individual learners or about theprogramme. Each student teacher collected all such information as wellas lesson plans, learner’s work, post lesson reflections and evaluations in aportfolio which I reviewed periodically and which counted toward the finalassessment of the course.

Debates and discussions about the connections between field workand coursework are long standing and have recently seen an increase inresearch (see e.g. Ebby, 2000) but much of this work locates opportu-nities for teaching practice mainly in schools and typically in wholeclass situations. The Tennyson House tutorial programme made availablea site for student teachers to interrogate and build on the ideas beingpresented to them in their course work. Working with learners on themargin in an intensive micro setting allowed student teachers to learnmuch more about the girls as well as to develop their teaching knowl-edge and skills; and to challenge the theoretical ideas as they interpretedthem and made them their own for practice. They were constantly beingforced to reflect on the ideas and their actions from the perspective of thelearners. For example they questioned current curriculum ideas they werelearning in their coursework such as ethnomathematics and critical mathe-matics education; and also, in the new South African curriculum, reformsthat emphasized connecting mathematics with learners realities, about theextent to which teachers could draw on learners’ backgrounds to makemathematics education relevant and meaningful when those backgroundsare painful associated with poverty, abuse, life as a sex worker or begging.Student teachers were given some of this sensitive information about thegirls and had to adjust their lessons, their pacing and their expectationsabout learning. But there were boundaries that could not be transgressedby a teacher of mathematics. As they worked for instance, with girls whohad suffered neurological damage from substance abuse and short memoryand attention spans which required greater patience and perseverance inteaching, they had to be careful in probing learners’ previous learning toestablish the gaps, levels and quality of mathematical understanding. Inthis they touched the outer limits of social, cultural, political perspectivesand practices expounded in the current mathematics education literature.

The work with the girls provided a resource with which student teacherscould engage their own learning in becoming teachers. The link to theirteaching and learning was explicit, direct and sustained as all opportunitiesand spaces for reflexivity were exploited. Even as we drove back to campusafter the sessions we talked about what had happened in the lessons thus

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serving as a kind of debriefing session. For example in the previous weekI had observed a lesson which SJ described as follows:

In Mpu’s book, there existed three triangles, isosceles, scalene, and equilateral [which areall drawn looking the same but are marked with respective equal sides]. She explained thatshe is having difficulty with triangles as she does not know how to distinguish them as toher they are all the same. The teacher had just told them to copy them from the board withno explanations. So I started with a deductive way of reasoning. I drew many shapes andasked her whether she could tell me which of those shapes are triangles. She was able todistinguish, therefore I knew that she knew what a triangle was.

It was clear that Mpu was still struggling with the concept of differenttriangles. In the subsequent lessons SJ experimented with different kindsof worksheets including an inductive approach learnt in their first year.In the many informal spaces created by the project setting, I shared myobservation that several of them were teaching various aspects of geometrybut still did not use manipulatives or any concrete materials nor drawon work done in their mathematics education courses. So in the nextsession I brought along some of the materials that we had discussedand developed in the previous years. My intervention was experienceddifferently in this environment from its impact during course work as thepower of learning about teaching by working with actual learners becameimmediately evident:

Mpu was very excited as this [geostrips] was something new and that she could physicallywork with it (concrete) . . . Without telling her anything she automatically started joiningthem. She made an equilateral triangle. Every time she made a triangle, I asked her why shemade the triangle in that way. She responded quite well. The only problem she is having isspelling the words and pronunciation. She asked me what were geostrips, I explained thatwe use them in order to understand some mathematical concepts. I asked her whether ourfirst experience was better than our present one and she said it was working with geostripsthat she enjoyed and understood. (SJ)

At first their teaching was very traditional but the failure of learnersto understand and do the tasks they gave them soon forced the studentteachers to consider more progressive pedagogy that was more learner-centred and activity-based. But there were other consequences: “this wasconfusing for the learners because the teachers in school only used tradi-tional algorithms” (Vithal et al., 2001, p. 10). This was a constant anddifficult tension that had to be dealt with. After having spent consider-able time helping Mpu to work with the different operations there wasmuch disappointment when she wrote her next test and got 9 out of 40. SJanalysed the difficulty.

She got her subtraction and addition sums correct, however multiplication and divisionwrong. We used the example from her test [327 × 23 = 301]. I guess she got confused as her


teacher showed her the basic algorithm without showing/telling them that multiplication isrepeated addition . . . She explained to us that she just added throughout and even then shegot the addition wrong. (SJ)

As work with Mpu continued, SJ wrote ‘she did grasp the concept but whatis taking up time is her times tables’. Several student teachers reflected onhow learners often lost track of the main problem if everything had to beworked from first principles. They discussed the problem of rote learningbut agreed that there was a need to learn and know basic number factsand made charts for the girls. From their observations of teachers in theschool and the class work books it appeared that the rise of valuing child-generated strategies in mathematics classrooms inspired by constructivisttheories of learning was being interpreted by teachers in de-emphasizingthe learning of basic numbers facts. They questioned the absence of discus-sion on the role and place of memory in progressive theories of learningand the difficulties for solving problems efficiently and quickly to succeedin school mathematics. Given that the dominant forms of assessmentremain time limited paper and pencil tests, this facility was rather crucialfor improved mathematics performance.

From the outset it was agreed that a key indicator of the success ofthis intervention would be improvement of performance in school mathe-matics. Indeed all the girls improved their results in mathematics and insome cases by more than 100 percent. To this end, the student teachersneeded to build on what the teachers were doing, as AJ noted:

What was evident in our discussions is that we need to speak to the girls’ teachers and findout their perspective on the progress of the girls. The source of information we are currentlyreceiving is only from the learner’s perspective. We now need multiple perspectives to havean overall perspective on the learner’s ability.

A whole day visit was organised to the public school for the studentteachers to strengthen support for the girls by developing a working rela-tionship with the teachers in whose class the girls were placed. The studentteachers held discussions with the mathematics teachers, observed themteaching and the girls’ work in class, and spent time meeting the girls’friends. This school, a former “Indian” school, with all “Indian” teachers,now accepted large numbers of “African” students. For some studentteachers this was a troubling experience. SJ reflected: “We were greetedat the gate by a teacher who had a stick in her hand”; and on meeting withanother teacher Mr. H who “has been teaching for 34 years, he commentedthat Grade 6c is the worst class and that was Mpu’s class”. She was deeplyconcerned at the way in which he used words such as “stupid” and “lazy” inclass. Several student teachers also took up their concerns with the teachersabout incorrectly marked work in the tests and books. They pointed out that

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besides their difficulties with mathematics, the girls also faced the stigmaof being labelled “street children”. Nevertheless some teachers were kindand considerate and some of the girls had made close friends with theseteachers.

As they stayed with the girls in the class and playgrounds, the powerof the student teachers’ learning lay in experiencing the brutal reality ofschool life from the perspective of those learners who are on the marginof their teachers’ concern. Even though they realised that they had tosuccumb to teaching the traditional algorithms as the schoolteachers wereexpecting, the student teachers learnt that it was crucial that in developingand restoring any basic knowledge and skills, meaning always needed tobe preserved. In preparing themselves for later teaching practice, not onlydid the project provide opportunities to explore relations between theoryand practice, it pointed to an important silence in the literature. They posedquestions about how practices arising from current theories of mathematicslearning, which are often themselves the outcomes of analysis involvingsmaller numbers of learners, can be given meaning in the real life ofschools and large whole class settings in ways that account for all learnersin the classroom.

The student teachers learned to work from were the learner was,listening and carefully drawing on their main concerns and interests inplanning their teaching. AJ describes how she facilitated this:

Although I went with prepared activities for the whole session, this session was going tobe Nca’s choice in work. She enthusiastically showed me all her work covered during theweek . . . Nca is at the moment learning about time in mathematics. The teacher has givenNca worksheets on time. Each worksheet requires the learner to fill in the time in wordsas illustrated by the hands of the clock . . . Nca has asked me to do activities dealing withtime. I welcome and appreciate Nca’s input as this enables me to meet the needs of mylearner. Nca suggested that she would like to show me how she does her homework. Thisprovided me with an opportunity to observe how Nca handles mathematics.

The student teachers became thoroughly acquainted with the officialschool mathematics curriculum and the expectations of different learningoutcomes for the different grade levels and were able to identify the gapsin the learners’ knowledge as they realised that the girls were performingwell below the grade levels at which they were placed. For instance, theycite what one particular learner was able to do and not do as follows:

counting to hundred by ones, twos, fives and tens, but cannot count in threes, fours, sixes,etc. . . . telling time in a digital watch but cannot read international time . . . measures feetand inches, but is unfamiliar with even the vocabulary of the metric system. (Vithal et al.,2001, p. 9)

The immense work this required is evident as in any one session studentteachers often focussed on many different aspects of mathematics because


they were providing tuition in work already done as well as the workcurrently being engaged. In addition they confronted head on the issue ofteaching English second language learners who had varying competencein English. Some mathematics lessons were in effect language lessons butthe student teachers found novel ways in which to integrate language withmathematics. In teaching Nca about time AJ for example produced anactivity in which “She had to match the letters of the alphabet with theappropriate time on the clock” and used this topic to teach how to spellnumbers, improve her vocabulary, spelling, punctuation and pronunciation.Invariably, the girls also asked for help in other subject areas hence studentteachers found themselves supervising general homework. The benefitof this was reciprocal because the student teachers brought content andmethodologies from other disciplines into their mathematics teaching. Astudent teacher who assisted learners with their science learning throughthe use of puppets and story telling, then also used this approach toteaching time.

Through the constant reflections in which the student teachers engaged,about the learners and mathematics teaching and learning, which charac-terised the programme, they were being inducted into a kind of teacherprofessionalism. During case discussions they presented achievements anddifficulties, how they had overcome some of these, and what was still to bedone. They asked for advice from the teacher educators but also listenedcarefully to their colleagues who were grappling with similar or relatedissues and sought out relevant literature. SJ notes one student teacher’sdiscussion

Pum is progressing well. Wanted ways on how to approach measurement as she is havingdifficulty conceptualising the rule and how it works. She can answer questions but cannotexplain herself. We suggested the use of Fuys & Tischler [activities from a reader] to helpwith measurement, what we did in Math Ed II.

The problems were real and there were real consequences to being able todeal with these effectively. They could see quite visibly, feel and experi-ence the power of their own actions in real time as the learners in their careresponded. The collective brought pressure to bear if any student teacherstayed away or did not do the required work because it was made open andtransparent and had direct consequences for others who had to cover forthose who did not deliver for whatever reason.

LEARNING ABOUT RELATIONSHIPS

The student teachers’ commitment to the girls was considerable and strongbonds of care, compassion and friendship emerged. The student teachers

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collected and displayed readings, newspaper articles, etc., about streetchildren and created a special notice board in the lecture room on whichinformation could be shared. AJ reflects on how she found these useful:

I have been doing some reading on street children and have used some techniques inestablishing a relationship with Nca. I must honestly say that these techniques proved tobe a great help to me on this visit. I gave full attention to Nca, sitting facing her and havingeye-to eye contact with her. I observed her facial expression, how she related to Natashaand I, her readiness to learn, etc. But most of all, I listened to her paying attention to contentas well as her expression of feeling. I listened more than I spoke.

The student teachers facilitated a resource material collection drivein the university to provide a small library for Tennyson House andfound shelves for the books. They developed charts to help the girls withtables, bonds, and language development and filled the tutorial room at theshelter. Worksheets and activities were personalised with the names andbeautiful drawings. A warm relationship ensued throughout as birthdayswere remembered and gifts, cards or letters exchanged. Nca expressed herappreciation to AJ in a letter:

Dear Natasha and Amisha

I would like to thank you for all you have done in teaching me math and English.I really appreciated it and the patience you had.I pray that you pass your exam and that all goes well with your studiesI hope you like my note.

God bless youLove

Nca

I really like the present you gave me.I like it very much

The first time each student teacher was invited to their learner’s roomin the house was valued and appreciated. Not surprisingly, the femalestudent teachers bonded with the girls differently from the males andthey struggled to deal with the relationships as they forced reflections andcomparisons with their own lives while holding onto and integrating afocus on mathematics. AJ wrote:

Everything about me is planned and organised in a particular way to deal with situationsin life. But this time round no amount of planning could prepare me for what it was Ncawas about to say. I was going to be dictated by my heart and interest in her. Nca has a sonwho is a year old. She loves him dearly. She spoke of her mum and said that her mumhas no care or interest in her well being. She does not know her siblings because they arealso disregarded in a similar manner. Nca began to question us on our families. She askedwhether we had both our parents. She further questioned us about our life with them andespecially if they cared about us. There were tears in her eyes but she tried to cover up


the situation by mentioning her love for her grandmother. I felt sad about what she hadsaid about her mum and her mum’s attitude towards her. It took me a while before I couldprovide some assurance to Nca and this was not easy for me to handle. I have decided toprepare some work on geometry for next week to gain insight on her concept of shapes. Iwant to educate Nca to become creative in mathematics and enjoy life to the fullest.

As the programme proceeded, the idea for the girls to visit the univer-sity emerged. “This was done with the primary aim of getting the girls to bemore responsive, to see the educational possibilities beyond their currentlife situation” (Vithal et al., 2001 p. 11). The girls were taken on a tour ofthe university – from the library to the café, the lecture rooms, laboratoriesand the temple and mosque, they were introduced to their different friendson campus, they had tea with the dean of the faculty and finally they hada party with gifts for each of the girls. This visit led to some parts ofthe tutorial programme being moved to the university so that some of theresources available could be used in their teaching and learning plans.

In this way the relationship between the girls and the student teachersdeepened and developed with evidence of much care, trust and friend-ship. Perhaps one of the most important and unexpected outcomes of theprogramme for post-apartheid South Africa was that this compassion forthe learners served to deracialise relationships between the learners andtheir teachers but also between the student teachers themselves. The fivemale and five female student teachers were encouraged to pair themselvesacross gender and race differences but chose to group themselves as twopairs of racially mixed males (“Indian and African”), two pairs of “Indian”females and one gender mixed. The girls at Tennyson House were all“African”. For most of the “Indian” student teachers it was their firsttime of working with “African” learners. The realities of South Africanlife penetrated into the programme as SJ openly confronted her fears andprejudices:

This project as whole has been an eye-opener for me. I have become a caring, tolerantperson. After the terrible incident of being held up at gunpoint by two gunmen, I grewanti-black with fear and disgust for them. These girls have shown me that they are innocentindividuals looking for some support, love and caring from me. Not all are alike and theyhelped me to open up from my fears by working so close to them. Whilst working withMpu, I see no colour just a girl wanting my help.

AJ also reflected that “My interaction with Nca has definitely changed myattitude toward African children . . . The impact Nca and her friends arehaving on me is a positive one.” The male students mentioned that havingto work closely together across the colour lines resulted for the first timein shared jokes and sometimes getting together during their free time oncampus, an important outcome for a university and a society still fraughtwith racial conflicts.

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In building up these strong relationships with learners there was greatercommitment to the tasks. But there was also much disappointment whensome of the girls chose to leave the shelter and return to the street or didnot return from having gone home for the vacations. AJ wrote:

22 July was dominated by a sense of disappointment and sadness by me. I was informedthat Nca had not returned to Tennyson House after the holidays. The reason for this wasnot known. I can’t understand why she had not returned. I have grown so attached to herand felt so unhappy when she did not appear to give me a big hug as she usually does . . .

Words are not enough to express what I am feeling.

These losses created some delay and difficulty as the student teachersresolved their feelings of despondence and began work with anotherlearner.

What is evidenced here is the deep caring relationship that developedbetween the student teachers and the learners they had been made respon-sible for. The extent to which this capacity had indeed developed onlybecame clear much later when the student teachers were placed in schoolsfor their teaching practice. They cared and paid attention to those whowere in various ways at the margins of their teaching, those who wereusually made invisible and ignored because they fail to learn, fail torespond according to expectations. This notion of caring, as an integralpart of mathematics educational practice, is rather under-developed thoughit has been theorised, for example, in feminist educational literature.Nel Noddings (1984, 1991), who also writes about a politicised mathe-matics classroom (1993) in bringing an ethic of caring to interpersonalreasoning, discusses several features which can also be identified in thestudent teachers relations to each other and to the learners. We saw anattitude of caring based on a recognition of differential power relations;attentiveness and receptivity to the needs and goals of each person in therelation; flexibility in both the means and the ends of the engagement; andthe considerable effort necessary to cultivate and sustain the relationship(Noddings, 1991). To develop this caring and reasoning, extended teacher-student contact is needed so that relations of trust and interpersonal skillscan be cultivated and nurtured; so that a genuine participative relationshipbetween the parties can come to exist; and so that “students can begin thesensitive work of learning to be carers as they see caregiving modelled”(Noddings, 1991, p. 167).

My relationship with the student teachers was also an open one. Theyknew that we were on new ground not knowing how each of our experi-ences was going to turn out and impact on those around us. This servedsignificantly to equalise the inescapable hierarchical relations of poweras we reflected on what happened after each session and planned for the


next step. All three women in the project, Cheryl, Robyn and myself, alsomodelled the care and commitment to both the learners and the studentteachers in the way we spoke to each other and the way we valued thework of each individual toward the greater goal. There is no doubt that the‘realness’ of the situation of shaping a young life contributed to the careand sensitivity given to the words and actions. By creating an opportunityto express caring, caring develops and comes to be integrated in the waywe think about learners and how we plan our teaching and learning. Butin its wake there lies also all the pain and hurt of separation when caringrelationships come to an end. The end of year party at Tennyson Housewas as sad as it was joyous. Some of the girls were apprehensive aboutreturning to the life they had escaped, not sure if they would continueschooling in the poverty stricken rural areas and sad at leaving their youngteachers.

CONCLUSION

One of the most important issues this experience demonstrates is howworking at the margin can raise important issues of theory and practice inmathematics teaching and learning that are equally valid and present, if notmore so, at the centre where they may be difficult to render visible (Vithal& Valero, 2003). Moreover, it showed how it may be possible for teachereducators to provide experiences and opportunities in the programmes theyoffer student teachers, enabling their journey to becoming mathematicsteachers to be one that is simultaneously productive of caring, kindnessand compassion integrated with their pedagogical and content knowledgefor teaching and learning mathematics.

As a teacher education intervention this programme was considered tohave been a success. The year following its implementation the facultysecured funding and expanded it to other places that catered for younglearners, which included also detention centres. A community serviceprogramme was built into the teacher education curriculum and madea requirement for all prospective teachers (Keogh & Paras, 2001). Thisexpansion posed new challenges and raises many other issues.

NOTES

1 The student teachers began a presentation to the mathematics educators with this poem(Vithal et al., 2001). Working with the children, learning to teach mathematics in the projecthad somehow touched somewhere much deeper.

182 RENUKA VITHAL

2 As repugnant as these racial descriptors are, they are used here because they still havemeaning in the South African context to address the ravages of apartheid; and also becausethey have relevance for the reflections made in the paper.3 I should mention that the well known street mathematics studies in Brazil (e.g. Nuneset al., 1993), which were discussed (among others) during the coursework of the teachereducation programme, focus on the mathematics of children who work on the street frompoor backgrounds but who may or may not be living with their families. As a descriptionand analysis of an educational programme rather than a research project, this paper focuseson a specific group of ‘street children’ and their mathematics education with the specificgoal of integrating it with school mathematics.4 In some cases I have changed the English slightly to present grammatical text which isintended to be clearer for the reader without changing the sense of what was written.

REFERENCES

Amin, N. (2001). An invitational education approach: Student teachers interactionswith at-risk youth. University of Durban-Westville, South Africa. Unpublished M. Eddissertation.

Chetty, V.R. (1997). Street children in Durban: An exploratory investigation. HumanSciences Research Council. Pretoria: HSRC Publishers.

Ebby, C.B. (2000). Learning to teach mathematics differently: The interaction betweencoursework and fieldwork for pre-service teachers. Journal of Mathematics TeacherEducation, 3(1), 69–97.

Keogh, M. & Paras, J. (2001). Teacher education through community service: A casestudy at a university in transition. African Forum for Children’s Literacy in Science andTechnology. South Africa: University of Durban-Westville.

Lather, P. (1986). Research as praxis. Harvard Educational Review, 56(3), 257–277.Le Roux, J. (1994). Street-wise: Towards a relevant education system for street children in

South Africa. Education and Society, 12(2), 63–68.Noddings, N. (1984). Caring: A feminine approach to ethics and moral education.

Berkeley: University of California Press.Noddings, N. (1991). Stories in dialogue: Caring and interpersonal reasoning. In C. Wither-

ing & N. Noddings (Eds.), Stories lives tell: Narrative and dialogue in education. NewYork: Teachers College, Columbia University.

Noddings, N. (1993). Politicizing the mathematics classroom. In S. Restivo, J.P.V.Bendegem & R. Fischer (Eds.), Math worlds: Philosophical and social studies of mathe-matics and mathematics education (pp. 150–161). Albany: State University of New YorkPress.

Nunes, T., Schliemann, A.D. & Carraher, D. (1993). Street mathematics and schoolmathematics. Cambridge: Cambridge University Press.

Richter, L.M. (1989). South African ‘street children’: Comparisons with Anglo-Americanrunaways. Paper presented at the Second Regional Conference of the InternationalAssociation for Cross-Cultural Psychology, 27 June–1 July. Amsterdam.

Rooyen, L. & Hartell, C.cG. (2002). Health of the street child: The relation betweenlife-style, immunity and HIV/AIDS – a synergy of research. South African Journal ofEducation, 22(3), 188–195.


Swart-Kruger, J. & Donald, D. (1994). Children of the South African Streets. In A. Dawes& D. Donald (Eds.), Childhood and adversity: Psychological perspectives from SouthAfrican research. Cape Town: David Philips.

Vithal, R. & Valero, P. (2003). Researching mathematics education in situations of socialand political conflict. In A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick & F.K.S.Leung (Eds.), Second international handbook of mathematics education. Dordrecht:Kluwer Academic Publishers.

Vithal, R, Paparam, A., Jugoo, S., Majola, W., Jugoo, A. & Mpalami, N. (2001) Learning toteach mathematics at Tennyson House. KwaZulu-Natal Mathematics Journal, 6(1), 3–14(Also presented as a plenary paper at the Durban-Central Conference of the Associationof Mathematics Education of South Africa, Durban Girls College, 10 Oct, 1998.)

School of Educational StudiesUniversity of Durban-WestvillePrivate Bag X54001Durban, 4000South AfricaE-mail: [email protected]

BOOK REVIEW

Wenger, E. (1998). Communities of practice: Learning, meaning andidentity. Cambridge, UK: Cambridge University Press. ISBN 0521430178 hbk; 0521 66363 6 pbk

Wenger’s book is stimulating, insightful, and challenging. In it, hedevelops substantially some of the themes from his earlier work with JeanLave (Lave & Wenger, 1991) which itself was a move on for many of thekey ideas of situated cognition in Lave’s (1988) book. Many researchersin education generally (Kirshner & Whitson, 1997) and in mathematicseducation in particular (for example, Stein & Brown, 1997; Lerman, 2001;Graven, 2002), have found that psychological cognitivist paradigms werelimited in exploring learning as part of a socially constructed world. Asituated cognition perspective is appealing since it seems to provide abridge between cognitivist perspectives and sociological perspectives.Lave and Wenger (1991) explain:

The notion of situated learning now appears to be a transitory concept, a bridge, betweena view according to which cognitive processes (and thus learning) are primary and a viewaccording to which social practice is the primary, generative phenomenon, and learning isone of its characteristics (p. 34).

The work of Lave and Wenger (1991) is increasingly being drawnon to describe and explain student and teacher learning in the field ofmathematics (see Adler, 1996, 1998, 2001; Boaler, 1997, 1999; Boaler& Greeno, 2001; Lerman, 1998; Santos & Matos, 1998; Stein & Brown,1997; Watson, 1998; Winbourne & Watson, 1998). Furthermore, math-ematics educators are increasingly noting the importance of Lave andWenger’s (1991) work for analysing mathematics teacher education. Theirperspective on learning has some political motive in the sense that it movesaway from theories that reduce learning to individual mental capacitysince these often “blame marginalized people for being marginal” (Lave,1996, p. 149). They emphasise the importance of “shifting the analyticfocus from the individual as learner to learning as participation in thesocial world” (Lave & Wenger, 1991). The injustice, and indeed theoret-ical inadequacy, of ‘blaming individuals’ will resonate with many readers’experiences, in that research in the context of pre-service and in-service


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projects often tends to blame individual teachers for the lack of take-up ofnew ideas.

In the Introduction Wenger explains the aims and achievements ofhis earlier work with Lave but notes that the concepts of identity andcommunity of practice, while central to their work, “were not given thespotlight and were left largely unanalysed” (p. 12). In this work Wengermoves away from a focus on legitimate peripheral participation (it ismentioned only twice in his book) to give a greater focus on the conceptsof communities of practice and identity. Referring to his 1998 work, hewrites: “In this book I have given these concepts centre stage, exploredthem in detail, and used them as the main entry points into a social theoryof learning” (p. 12).

Wenger explains that communities of practice are everywhere andbecause they are so informal and pervasive they are rarely focused on.Focusing on them allows us to deepen, to expand and to rethink our intu-itions. He relates communities of practice to the learning components ofmeaning, practice, community and identity as follows:

On the one hand, a community of practice is a living context that can give newcomersaccess to competence and also can invite a personal experience of engagement by whichto incorporate that competence into an identity of participation. On the other hand, a wellfunctioning community of practice is a good context to explore radically new insightswithout becoming fools or stuck in some dead end. A history of mutual engagement arounda joint enterprise is an ideal context for this kind of leading-edge learning, which requiresa strong bond of communal competence along with a deep respect for the particularity ofexperience. When these conditions are in place, communities of practice are a privilegedlocus for the creation of knowledge (Wenger, 1998, p. 214).

As we have seen above, the work of Lave and Wenger (1991) movedaway from psychological and cognitive explanations of learning to a moresocial and situated view of learning and a shift from a focus on the indi-vidual as learner to learning as participation in the social world. So, too,the work of Wenger (1998) is situated within this broader field. He notesthat his work is a social theory of learning that does not aim to replaceother theories of learning but does have its own set of assumptions andits own focus. His work can be considered a theory in that it constitutes acoherent level of analysis and yields a conceptual framework from whichto derive general principles for understanding and enabling learning.

In the Introduction, Wenger also goes to great lengths to explain the‘intellectual context’ (p. 11) of his social theory of learning by placing it atthe intersection of two ‘axes’ of intellectual traditions. The vertical axis hasthe two ends labelled ‘theories of social structure’ and ‘theories of situatedexperience’. The former emphasises institutions, norms, cultural systems,discourses and history while the latter emphasises agency and intentions.In this sense ‘learning as participation’ is caught in the middle. He explains

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It [learning] takes place through our engagement in actions and interactions, but it embedsthis engagement in culture and history. Through these local actions and interactions,learning reproduces and transforms the social structure in which it takes place (p. 13).

However Wenger points out that the horizontal axis (the ends of which arelabelled ‘theories of social practice’ and ‘theories of identity’) is the axiswith which his work is mostly concerned but adds that this is “set againstthe backdrop of the vertical one” (p. 13). At one end of the horizontalaxis, theories of social practice focus on the production and reproductionof ways of engaging with the world while emphasising social systems ofshared resources. At the other end, theories of identity focus on the socialformation of the person, the creation of membership and the formationof social categories. Wenger explains that on this horizontal axis learningis again caught in the middle since it “is the vehicle for the evolutionof practices and the inclusion of newcomers while also (and through thesame process) being the vehicle for the development and transformation ofidentities” (p. 13).

Wenger clarifies his intentions as follows:

The purpose of this book is not to propose a grandiose synthesis of these intellectual tradi-tions or a resolution of the debates they reflect; my goal is much more modest. Nonetheless,that each of these traditions has something crucial to contribute to what I call a socialtheory of learning is in itself interesting. It shows that developing such a theory comesclose to developing a learning-based theory of the social order. In other words, learning isso fundamental to the social order we live by that theorizing about one is tantamount totheorizing about another (p. 15).

He provides a theory of learning in which the primary unit of analysisis neither the individual nor social institutions but ‘communities of prac-tice’. The theory explores systematically the intersection of the learningcomponents: community, practice, meaning and identity and these providea conceptual framework for analysing learning as social participation.

Wenger’s (1998, p. 4) work is based on four premises:

1. A central aspect of learning is that people are social beings;2. Knowledge is about competence with respect to ‘valued enterprises’;3. Knowing is about active engagement in the world;4. Meaning is ultimately what learning produces.

Furthermore, he emphasises that learning is inevitable since failing tolearn something involves learning something else. However, he adds thatreflection on learning, a key motive of his book, despite its inevitability, isimportant because:

We wish to cause learning, to take charge of it, direct it, accelerate it . . . Therefore ourperspectives on learning matter . . . It is our conception of learning that needs urgentattention when we choose to meddle with it on the scale which we do today (p. 9).

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Thus he appeals to teacher educators (and others in education) since wetoo are compelled to reflect more systematically on learning assumptionsand are directly involved in ‘meddling’ and ‘taking charge of’ the learningof teachers.

Wenger (1998) identifies four components of learning namely:meaning, practice, community and identity. These components of learningare defined as follows:

1. Meaning is a way of talking about our ability to experience the worldas meaningful;

2. Practice is a way of talking about shared historical and socialresources, frameworks and perspectives that sustain mutual engage-ment in action;

3. Community is a way of talking about the social configurations inwhich our enterprise is defined and our participation is recognisableas competence;

4. Identity is a way of talking about how learning changes who we are.

These four components together provide a structuring framework for asocial theory of learning. Wenger (1998, p. 5) summarises this frameworkin the following diagram:

Figure 1. Components of a social theory of learning: An initial invientory.

Wenger notes that the elements are “deeply interconnected and mutu-ally defining” (p. 5) and points out that one could “switch any of the fourperipheral components with learning, place it in the centre as the primaryfocus, and the figure would still make sense” (p. 5).

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After the introduction, Wenger tells the story of Ariel in two vignettes,a story of her participation in a workplace setting of claims processing ina medical insurance office. The story is well written and ideally serves hispurpose, enabling these key ideas to emerge. The remainder of the bookelaborates the key terms we have defined, and he does this in two sections,Practice and Identity. The final section focuses on design, prefaced bythe remark that “Learning cannot be designed” (p. 225)! Nevertheless,after defining design as “systematic, planned, and reflexive colonization oftime and space in the service of an undertaking” (p. 228), he makes someproposals for both organisations and education, emphasising the priority ofaddressing identities and modes of becoming and only secondarily skillsand information.

TEACHING AS A COMMUNITY OF PRACTICE?

Many acknowledge that Lave and Wenger’s (1991) perspective has not yetbeen developed into a complete theory of learning and that there are manydifficulties that arise when applying such perspectives to learning math-ematics or learning to teach mathematics (Adler, 1998; Watson, 1998).Furthermore, there are few studies that focus on how learning is enabledfrom such a perspective. What are the mechanisms that enable learning totake place from a perspective of ‘learning as becoming’? Thus while math-ematics teacher education researchers are creating contexts that enableteacher learning and describe what teachers learn in social terms, little hasbeen done to explain how those contexts enable learning (Wilson & Berne,1999).

Wenger’s vignettes, of course, present a story of participation in anactivity that is quite far from teaching. Further, Wenger continues thetheme from Lave and Wenger (1991, pp. 40/1) of separating teachingfrom learning and focusing very firmly on the latter and they imply that inaddition to teaching not being necessary for learning, teaching is not partic-ularly useful for learning. In this sense Lave and Wenger have reconstitutedlearning but they have not fully reconstituted teaching. Their disregard forteaching in relation to learning, although understandable in apprenticeshipcontexts where teaching is more incidental than deliberate, is problematicfor us in the field of mathematics teacher education research. While weagree that much learning takes place without intentional teaching and thatmuch teaching does not lead to intended forms of learning, in some caseseven the most didactic forms of teaching have led to successful learning interms of certain desired outcomes. It is likely that much of the learning ofreaders of this journal occurred through such forms of teaching.

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Lave tries to address the implications of their work for teaching in her1996 paper:

Teaching, by this analysis, is a cross-context, facilitative effort to make high quality educa-tional resources truly available for communities of learners. Great teaching in schools isa process of facilitating the circulation of school knowledgeable skill into the changingidentities of students (p. 158).

While Lave (1996) addressed the need for the reconceptualisation ofteaching in relation to Lave and Wenger’s (1991) perspective on learning,this is not taken up by Wenger (1998). He continues to undermine the valueof teaching to the point that he asks: “How can we minimise teaching soas to maximise learning?” (p. 267). Wenger’s avoidance of the concept of‘teaching’ per se stems from the apprenticeship context from which hisand Jean Lave’s work developed. In this context there are no ‘teachers’only ‘masters’ and much of the learning is tacit. However, Wenger doesnot use the term ‘master’ in his 1998 work, nor does he develop a thor-ough discussion of the central role of such a person in a community ofpractice or more specifically in a learning community. Perhaps this isbecause the development of his perspective is based on the vignette ofAriel’s involvement in the practice of claims processing rather than on aperson undergoing apprenticeship training or formal learning. The resultis that much work needs to be done in order to translate Wenger’s (1998)perspective on learning (based in the context of learning on the job) tolearning in more formal education contexts where teachers (or facilitators,co-ordinators etc.) have a central role in ensuring that successful learningoccurs and are furthermore held accountable for such learning. That is, thesuccess of a teacher’s vocation depends on successful learning.

Thus we argue that, since the corollary of ‘teaching is not a preconditionfor learning’ is not ‘teaching does not result in learning’, it is important toask the following: Where is teaching in learning? What conceptualisationof teaching is needed to help maximise learning? What does it mean to bea teacher when it is argued that the practice of teaching should be minim-ised? We would argue that teaching in most pre- and in-service teachereducation settings and in most schools occurs within a community of prac-tice. What sense can we then make of a community of practice where theknowledge, skills and identities to be developed are to be minimised withinits sphere of activity?

One attempt to answer these questions has been by Graven (2002). Inher study Graven extended Wenger’s model of inter-related componentsof learning to describe and explain teacher learning that occurs within amathematics in-service programme in South Africa that was stimulatedby radical curriculum change. The study used qualitative ethnography in

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which the researcher performed the dual role of both coordinator andresearcher of the in-service practice. As a relatively longitudinal studythe phenomenon of confidence, an independent and additional componentof learning, emerged in teachers’ descriptions and explanations of theirlearning. The extension of Wenger’s theory to include the overarchingand interacting component of confidence was embedded in and derivedfrom data analysis of the learning of ten teachers, over a two-year period.Graven (2002) identified seven different categories of confidence that theten inset teachers repeatedly referred to when describing and explainingtheir learning at the end of their inset programme. These categories relatedclosely to Wenger’s four components of learning and included confidencein relation to: classroom practice; access to knowledge resources; access tosupport resources; increased status and recognition bestowed on them byothers; increased participation in broader educational activities; affectivefactors and understanding one’s own limitations. While the first fivecategories relate clearly to the components of practice, meaning, identityand community, the latter two components could not be subsumed withinthese components. This challenged Wenger’s four-component model asbeing sufficient to explain learning in all contexts.

The research of Graven (2002, pp. 303–304) shows that “many teachers(in the inset) changed their understanding of what it meant to be acompetent professional mathematics teacher and began to see learningas an integral part of being a professional, irrespective of one’s level offormal education”. It is highlighted that this “can be especially difficult forteachers since they are usually constituted as ‘all knowing’. Teachers aslearners in an INSET context differ from other learners in other contextssuch as schools or apprenticeship contexts. . . . Teachers expressed confid-ence in the acceptance that indeed one cannot know everything but onecan become a life-long learner within the profession of mathematicsteaching. This new approach to learning was both a result of confidence,and provided teachers with increased confidence.”

SOME FURTHER CHALLENGES IN APPLYING WENGER’SFRAMEWORK

An interesting paradox occurs in relation to the co-ordinator of an INSETproject being equated to a ‘master’ in an apprenticeship practice. In thecontext of in-service teacher education that is co-ordinated from outsideof schools (for example, from universities) the ‘masters’ are not in thesame vocation as the teachers. That is, the ‘master’ is not a schoolteacherbut rather a teacher educator. Thus, while teachers are learning about the

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profession of teaching through their participation in the INSET, they arenot being ‘apprenticed’ into teaching. The ‘apprenticeship’, instead, if theyso choose, is into the practice of being a teacher educator. A key differencefrom other apprenticeship contexts examined by Lave and Wenger (1991)is that this form of apprenticeship is seldom the intention of INSET andtherefore is only taken up by some teachers, depending on their trajectoriesand career goals.

Just as Wenger (1998) avoids the notion of master, he does not engagewith the notion of ‘mastery’. ‘Mastery’ of the profession of mathe-matics teaching is clearly much broader than mastering the practice ofteaching learners mathematics, or in Wenger’s terms, successfully organ-ising a community of practice in which mathematics learning takes place.Graven’s study has demonstrated that mastery, in relation to becominga professional mathematics teacher, involves becoming confident in rela-tion to: one’s professional knowledge and experiences, one’s participationin professional activities, one’s membership in a range of professionallyrelated communities and one’s identity as a professional mathematicsteacher.

An additional point of clarification is needed in relation to wherecommunities of practice fit in relation to Wenger’s figure ‘Componentsof a social theory of learning’ above. According to Wenger’s definition,communities of practice clearly involve all four components of learning.Wenger explains that his use of the concept of communities of practicewas as a point of entry into a broader conceptual framework of which it isa constitutive element, and that the analytical power of the concept is that itintegrates all four components. In this way, ‘communities of practice’ is theprimary unit of analysis in relation to his theory of learning. For teacherlearning this allows the primary unit of analysis to be not ‘the teacher’,nor the ‘learning community’ but the teacher-in-the-learning-community-in-the-teacher (see also Lerman, 2000). In this respect, the communityof practice of a teacher education programme is primary and permeatesthe analysis of teacher learning in relation to each of the components oflearning.

CONCLUDING REMARKS

This book is recommended reading for all researchers interested inlocating learning beyond individual cognitive development. To relate thesepowerful ideas to learning to become a teacher of mathematics remainsa challenge but a worthwhile one as the concepts Wenger presents and

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develops, those of meaning, practice, community and identity, provide richand functional dimensions for research and development.

REFERENCES

Adler, J. (1996). Secondary school teachers’ knowledge of the dynamics of teachingand learning mathematics in multilingual classrooms. Doctoral Dissertation, Faculty ofEducation, University of the Witwatersrand, South Africa.

Adler, J. (1998). Resources as a verb: Reconceptualising resources in and for schoolmathematics. Proceedings of the 22nd conference, psychology of mathematics education,Vol. 1 (pp. 1–18). South Africa: University of Stellenbosch.

Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht: KluwerAcademic Publishers.

Boaler, J. (1997). Experiencing school mathematics. Teaching styles, sex and setting.London: Open University Press.

Boaler, J. (1999). Participation, knowledge and beliefs: A community perspective onmathematics learning. Educational Studies in Mathematics, 40, 259–281.

Boaler, J. & Greeno, J. (2001). Identity, agency, and knowing in mathematics worlds. InJ. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171–200). Westport CT: Ablex Publishing.

Graven, M. (2002) Mathematics teacher learning, communities of practice and thecentrality of confidence. Doctoral Dissertation, Faculty of Science, University of theWitwatersrand, South Africa.

Kirshner, D. & Whitson, J. (Eds.) (1997). Situated cognition: Social, semiotic, andpsychological perspectives. London, UK: Lawrence Erlbaum Associates.

Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life.New York, USA: Cambridge University Press.

Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture, and Activity, 3(3), 149–164.

Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. NewYork: Cambridge University Press.

Lerman, S. (1998). Learning as social practice: An appreciative critique. In A. Watson(Ed.), Situated cognition and the learning of mathematics (pp. 33–45). Oxford: Centrefor Mathematics Education Research, University of Oxford Department of EducationalStudies.

Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.),Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport CT:Ablex Publishing.

Lerman, S. (2001). A review of research perspectives on mathematics teacher education. InF.L. Lin & T. Cooney (Eds.). Making sense of mathematics teacher education (pp. 33–52). Netherlands: Kluwer Publishers.

Santos, M. & Matos, J. (1998). School mathematics learning: Participation through appro-priation of mathematical artifacts. In A. Watson (Ed.), Situated cognition and thelearning of mathematics (pp. 105–125). Oxford: Centre for Mathematics EducationResearch, University of Oxford Department of Educational Studies.

Stein, M.K. & Brown, C. (1997). Teacher learning in a social context: Integrating collab-orative and institutional processes with the study of teacher change. In E. Fennema &

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B.S. Nelson (Eds.), Mathematics Teachers In Transition (pp. 155–192). New Jersey:Lawrence Erlbaum Associates, Mahwah.

Watson, A. (1998). Situated cognition and the learning of mathematics. Centre for Math-ematics Education Research. Oxford: University of Oxford Department of EducationalStudies.

Wenger, E. (1998). Communities of practice: learning, meaning, and identity. New York:Cambridge University Press.

Wilson, S.M. & Berne, J. (1999). Teacher Learning and the Acquisition of ProfessionalKnowledge: An Examination of Research on Contemporary Professional Development.Review of Research in Education, 24, 173–209.

Winbourne, P. & Watson, A. (1998). Learning mathematics in local communities of prac-tice. In A. Olivier & K. Newstead (Eds.), Proceedings of the Twenty-Second AnnualMeeting of the International Group for the Psychology of Mathematics Education, Vol. 4(pp. 177–184). Stellenbosch, South Africa.

Mellony Graven and Stephen LermanUniversity of the Witwatersrand and Southbank University

TERRY WOOD and BETSY BERRY

EDITORIALWHAT DOES “DESIGN RESEARCH” OFFER

MATHEMATICS TEACHER EDUCATION?

In a previous editorial for JMTE (5.3) I argued for the importance ofgenerating and sharing knowledge about the complexity in mathematicsteaching. One reason I gave for the need to create a shared knowledge baseof mathematics teaching was that this would allow us to create approachesto teacher development that would “insure that teachers can accomplishthe substantial changes to meet the demands of teaching” (p. 202). For thisissue, Betsy Berry and I consider the landscape of research in educationwith respect to teacher education approaches, programs, or models in lightof ‘design research’.

The research in general teacher education and mathematics teachereducation is voluminous and provides extensive results and evidence aboutapproaches to teacher development. And yet we still are unable to identifyapproaches to teacher education to insure that teachers meet the demandto develop relative to the complexity in mathematics teaching. In thisEditorial, we argue that “design research” and the accompanying “designexperiments” offer a possible way to tackle this situation.

Design research as a type of research is currently receiving consider-able attention in the United States as evidenced by the recent special issueof the Educational Researcher (Kelly, 2003) published by the AmericanEducational Research Association (AERA). Design research is commonlyviewed as an approach that makes use of existing traditional research find-ings to develop some type of ‘product’, in the case of teacher education,programs for professional development.

Design research can be characterized in the following ways: First, aphysical or theoretical artifact or product is created. For the researcher/teacher educator the product being developed and tested is the profes-sional development model itself. For the teacher, the product that theydesign and study is specific to their students and might be an assess-ment tool or strategy or implementation guideline for a particular mathe-matics lesson, and so forth. Second, the product is tested, implemented,reflected upon and revised through cycles of iterations. The model isdynamic and emergent as the process progresses. Third, multiple models


196 TERRY WOOD AND BETSY BERRY

and theories are called upon in the design and revision of the products.Fourth, design research of this nature is situated soundly in the contextualsetting of the mathematics teachers’ day-to-day environment, but resultsshould be shareable and generalizeable across a broader scope. Fifth, theteacher educator/researcher is an interventionist rather than a participantobserver in a collaborative, reflective relationship with the teacher(s) asthe professional development model evolves and is tested and revised.

Consider for example, a professional development design that com-bines and weaves together a collaborative learning experience for teachersthat includes selected elements from action research (Miller & Hunt,1994), Japanese lesson study (Lewis, 2000) or reflective practice groupsand communities of practice (Stein et al., 1998). All of these modelshave supporting studies that document success for teacher developmentdescribed in different ways. Some describe growth in mathematics contentknowledge or changes in teacher beliefs or teaching efficacy or perhapschange in classroom practice and increased questioning skills. All ofthese concerns are inherent in the professional development of both pre-service and practicing mathematics teachers. While it remains impossibleto address all things at all times, design research allows us to consider thecomplexities of mathematics teaching and of teacher development in ourinitial design and in our reflections and revisions of the model.

In our proposed design, participating teams of teachers might designclassroom environments and teaching tools for the implementation ofcomplex problem solving activities with their students as they collaboratewith the researchers/professional developers. Researchers and professionaldevelopers might combine elements of reflective practice to promoteteachers learning through reflection. They might also bring teacherstogether to create collaborative groups in which the joint work centers oncrafting of lessons that comprises lesson study.

A research design of this nature would provide parallels and integra-tion of the work of the teacher educators with the work of the teacherswith the work of their students. At all levels there would be an environ-ment of dynamic cyclical change. In this research model teachers mustpass through cycles as they consider the “problem” of facilitating the“perfect” environment for their students learning and teacher educatorswould pass through cycles as they design the “perfect” environmentfor teacher learning. The design of this professional development modelrequires that teacher educators and teachers collaborate and share the jointwork of all aspects of the process including the designing, reflecting on andrevising of the model throughout the cycles but in different ways and fordifferent purposes. It has been said of education, and mathematics educa-

EDITORIAL 197

tion in particular, that we’re trying to “fly the airplane and fix it at thesame time”. Taking a design research approach allows us to acknowledgethat we (teachers, teacher educators, researchers, policy makers) are andcan be pilots, maintenance crew, designers and test pilots for the airplanethrough out the process. Our roles and responsibilities may change, evolveand emerge, but throughout the design research process, we will be able tolook at “old” models in new ways and combine them and re-combine themto enhance mathematics education.

Taking a design research approach allows teacher educators the oppor-tunity to develop approaches to teacher education that are situated in theprofessional lives of the teachers they work with. This creates a dynamicprocess that allows for immediate changes through a process of cyclesof iteration. But most importantly, the success of using a design researchapproach lies in the creativity of the teacher educator and her or his capa-bility to combine and recombine elements drawn from research on teachereducation to ‘design’ an approach that is useful and effective within thespecific context they are working.

This brings us to the article by James Hiebert, Anne Moss, and BradGlass in which they describe an ‘experiment’ model for teaching andteacher preparation in mathematics education. On the surface, this appearsto be similar to design research as described above, but in actuality theirapproach is different in that their thinking is grounded in two goals formathematics teaching and the importance of creating a shared knowledgebase of teaching. For them, the difference lies in a focus on the processof developing teaching rather than the creation of a specific product toenhance teacher development. Nevertheless we see this as a valuable exten-sion of the idea of a ‘product’ – the processes involved can become theproduct that is sought. We publish this article, despite its offering themodel only in theory, to emphasize the process-as-product possibility, butrecognize that the model needs the iterations of practical implementation toreach the deeper issues of manifesting a theoretical model in the realitiesof practice. We shall be interested in reports of research that study theimplementation of this or other process-into-product models.

Jeremy Kahan, Duane Cooper and Kimberly Bethea offer a frame-work to guide research on the relationship between mathematics teachers’knowledge of mathematical content and their teaching of mathematics.In this framework they relate elements of teaching and the processes ofteaching in which knowledge of content is of consequence and illustratetheir use of the framework through vignettes from their own work aseducators with pre-service secondary mathematics teachers. They high-light the complexity of investigations into the relationships involved. The

198 TERRY WOOD AND BETSY BERRY

paper from Jennifer Szydlik, Stephen Szydlik and Steven Benson exploreschanges in beliefs of prospective elementary teachers about the nature ofmathematics during a pre-service mathematical content course designedto provide participants with authentic mathematical experiences and tofoster autonomous mathematical behavior. Beliefs about the nature ofmathematical behavior were studied both at the commencement and atthe completion of the course. Students’ changes in beliefs were attributedto work on ‘big’ problems with underlying structures, a broadening inthe acceptable methods of solving problems, a focus on explanation andargument, and the opportunity to generate mathematics as a classroomcommunity. Both of these papers take up issues of mathematics teachingpreparation linked to knowledge of mathematics as raised in the paperfrom Hiebert et al. Each one offers insights into practices and theirrelation to theoretical perspectives, and addresses issues relating to thecomplexity of teaching and teacher education. These papers together offerimportant perspectives to educators seeking to address ‘The Teaching Gap’in mathematics teaching development (Stigler & Hiebert, 1999).

In his Reader Commentary, John Mason responds to Alan Schoenfeld’sreview (in JMTE 6.1) of his book Researching your own practice: Thediscipline of noticing (Mason, 2002). The book, review and commentarytake up issues of how educators can work with teachers to encourageteacher reflection on and inquiry into teaching practices, aiming to enhanceteaching for effective mathematical learning. Thus, together they offer aperspective apposite to the theme of this editorial and the papers in thisissue of JMTE.

The editors would like to draw attention to the Mason paper here as thefirst example of Reader Commentary – a new style of paper welcomedby this journal (see the inside front pages of this issue for submissiondetails). Here we welcome shorter and less formal papers than would beexpected in the category of Research Papers; for example, papers thatmake a response to a published JMTE paper or which offer an idea ortheoretical perspective for discussion or further development. Such paperswill be reviewed as appropriate at the discretion of the editors. We shouldlike to use this category to encourage debate in the field of mathematicsteaching development and teacher education.

REFERENCES

Kelly, A. (Ed.) (2003). Theme issue: The role of design in educational research. Educa-tional Researcher, 32.

EDITORIAL 199

Lesh, R. (2002). Research design in mathematics education: Focusing on design experi-ments. In L. English (Ed.), Handbook of international research in mathematics education(pp. 27–49). Mahwah, NJ: Lawrence Erlbaum Associates.

Lewis, C. (2000, April 2000). Lesson study: The core of Japanese professional devel-opment. Paper presented at the annual meeting of the American Educational ResearchAssociation, New Orleans.

Mason, J. (2002). Researching your own practice: The discipline of noticing. London:Routledge/Falmer.

Miller, L. D. & Hunt, N. P. (1994). Professional development through action research. In D.B. Aichele (Ed.), Professional development for teachers of mathematics (pp. 296–303).Reston, VA: National Council of Teachers of Mathematics.

Stein, M. K., Silver, E. A. & Smith, M. S. (1998). Mathematics reform and teacher devel-opment: A community of practice perspective. In S. Goldman (Ed.), Thinking practicesin mathematics and science learning (pp. 17–52). Mahwah, NJ: Lawrence ErlbaumAssociates.

Stigler, J. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers forimproving education in the classroom. New York: Free Press.

Wood, T. (2002). Demand for complexity and sophistication: Generating and sharingknowledge about teaching. Journal of Mathematics Teacher Education, 5(3), 201–203.

JAMES HIEBERT, ANNE K. MORRIS and BRAD GLASS

LEARNING TO LEARN TO TEACH: AN “EXPERIMENT” MODELFOR TEACHING AND TEACHER PREPARATION IN

MATHEMATICS

ABSTRACT. This paper describes a model for generating and accumulating knowledgefor both teaching and teacher education. The model is applied first to prepare prospectiveteachers to learn to teach mathematics when they enter the classroom. The concept oftreating lessons as experiments is used to explicate the intentional, rigorous, and systematicprocess of learning to teach through studying one’s own practice. The concept of planningteaching experiences so that others can learn from one’s experience is used to put intopractice the notion of contributing to a shared professional knowledge base for teachingmathematics. The same model is then applied to the work of improving teacher prepara-tion programs in mathematics. Parallels are drawn between the concepts emphasized forprospective teachers and those that are employed by instructors who study and improveteacher preparation experiences. In this way, parallels also are seen in the processes usedto generate an accumulating knowledge base for teaching and for teacher education.

KEY WORDS: knowledge for mathematics teaching, knowledge for mathematics teachereducation, learning to teach, lesson study

An enduring problem in mathematics education is how to design prepara-tion programs that influence the nature and quality of teachers’ practice(Borko et al., 1992; Cooney, 1985, 1994; Ebby, 2000; Lortie, 1975).The absence of strong effects resulting from such programs is notedprimarily when prospective mathematics teachers are asked to developteaching practices different from those they have experienced. This canbe explained, in part, by the observation that teaching is a cultural practice(Gallimore, 1996) and changing cultural practices is notoriously difficult.People learn to teach, in part, by growing up in a culture – by servingas passive apprentices for 12 years or more when they themselves werestudents. When they face the real challenges of the classroom, they oftenabandon new practices and revert to the teaching methods their teachersused.

The absence of strong effects of preparation programs also can beexplained, in part, by the lack of a widely shared knowledge base for bothteaching and teacher education (Grimmett & MacKinnon, 1992; Hiebertet al., 2002; Holmes Group, 1986; Huberman, 1985; Raths & McAninch,1999; Yinger, 1999). Prospective teachers studying to enter the profes-


202 JAMES HIEBERT ET AL.

sion cannot consult a common source of knowledge that allows them tobegin where their predecessors left off. They often start anew, developingteaching methods that work for them. In a parallel way, teacher educatorslack a shared knowledge base for building more effective teacher prepar-ation programs. Teacher colleges and universities might learn from eachother about program features and requirements, but little shared informa-tion is at the instructional level and even less is supported by research oneffectiveness. Like schoolteachers, teacher educators mostly start anew,learning how to teach preparation courses more effectively.

If mathematics teaching showed signs of continuing improvement andif students were learning mathematics well, the concern about the effect-iveness of teacher preparation programs would be less urgent. But theaverage classroom in the United States reveals the same methods ofteaching mathematics today as in the past (Fey, 1979; Stigler & Hiebert,1999; Welch, 1978). U.S. students continue to learn disappointingly littlemathematics (Gonzales et al., 2000; Silver & Kenney, 2000) and are espe-cially deficient in the competencies required to understand mathematicsdeeply and use it effectively (National Research Council [NRC], 2001).

Given these facts, is it possible to nurture, during a preparation pro-gram, the knowledge, competencies, and dispositions that teachers willneed to become expert mathematics teachers when they enter the class-room? Probably not. The model we propose claims that it is both morerealistic and more powerful to help prospective teachers learn how tolearn to teach mathematics effectively when they begin teaching. In otherwords, preparation programs can be more effective by focusing on helpingstudents acquire the tools they will need to learn to teach rather than thefinished competencies of effective teaching.

The model for teacher preparation described in this article is built ontwo primary and over-arching learning goals. We believe that achievingthese goals will provide prospective teachers with the tools they needto become increasingly effective mathematics teachers as they enter theclassroom. The goals are:

• Become “mathematically proficient” (NRC, 2001).• Develop the knowledge, competencies, and dispositions to learn to

teach, with increasing effectiveness over time, in ways that help one’sown students become mathematically proficient.

In this paper, we elaborate the two primary goals and then describemore fully how we interpret the concept of learning to teach – the conceptthat underlies the proposed model of teacher preparation. We then describethe kinds of environments that prospective teachers must create in order tosustain their own learning and that of the profession. Finally, we step back

LEARNING TO LEARN TO TEACH 203

to describe the larger educational program in which preparation programsoften are embedded and apply the same concepts of learning from experi-ence to the task of building a knowledge base for teacher preparation andimproving the effectiveness of preparation programs.

TWO PRIMARY LEARNING GOALS FOR PROSPECTIVETEACHERS

Goals are expressions of values. The two goals described below providecomplete statements of the values built into the proposed model. Alldecisions about teacher preparation programs that are aligned with themodel, both in substance and in the processes used to develop them, aredriven by the desire to help prospective teachers achieve the two goals.Likewise, the knowledge needed to improve the effectiveness of prepara-tion programs is the knowledge of how to help prospective teachers achievethese goals. For the reader to understand the proposed model, it is essentialto understand the nature of the goals and why they were selected.

Goal 1: Become Mathematically Proficient

The mathematics education community in the United States is in the midstof a debate about the future of mathematics education. The core of thedebate is about what mathematical outcomes are of most value for schoolstudents (Hiebert, 1999; Kilpatrick, 1997). In other words, what learninggoals should be set for students? In a deliberate attempt to address these“math wars” in the United States, the National Research Council issued areport that offers recommendations on appropriate mathematics learninggoals for students in the 21st century (NRC, 2001). The recommendationsare based on widely solicited expert advice and on a synthesis of researchon mathematics teaching and learning.

The mathematics learning goal for school students proposed in the NRCreport is to become “mathematically proficient”. In brief, mathematicalproficiency is the simultaneous and integrated acquisition of five kinds ofmathematical competencies, or “strands:

• conceptual understanding – comprehension of mathematical con-cepts, operations, and relations

• procedural fluency – skill in carrying out procedures flexibly, accur-ately, efficiently, and appropriately

• strategic competence – ability to formulate, represent, and solvemathematical problems


• adaptive reasoning – capacity for logical thought, reflection, expla-nation, and justification

• productive disposition – habitual inclination to see mathematics assensible, useful, and worthwhile, coupled with a belief in diligenceand one’s own efficacy” (NRC, 2001, p. 116).

We endorse the goal of increasing mathematical proficiency for schoolstudents. This means, in turn, that we endorse the goal of increasingmathematical proficiency for prospective teachers.

The concept of mathematical proficiency carries with it the notion thatsuccess in mathematics is achieved by making progress along each of thefive strands rather than by completely mastering any one individual strand.Furthermore, there is evidence that progress is made more easily alongeach strand when all five strands are interwoven and treated simultaneouslythan when one strand is singled out for prolonged attention (NRC, 2001).But the common teaching methods in the United States often have sepa-rated these strands and emphasized some at the expense of others (Fey,1979; Stigler & Hiebert, 1999; Stodolsky, 1988). Indeed, this is one way ofdescribing the impoverished nature of U.S. school mathematics teaching.

Because it is unrealistic to expect prospective teachers to learn to teachfor mathematical proficiency without becoming proficient themselves, theproposed model focuses on the integrated development of the five strandsof mathematical proficiency. The mathematical topics of school curricula,along with related mathematical ideas, must be studied in ways thatencourage attention to all five of the strands. In addition, the study ofschool students’ thinking and the ways in which it can reveal mathematicalproficiency (or its absence), and how such proficiency develops, must beregular features of teacher education courses.

Goal 2: Prepare to Learn to Teach for Mathematical Proficiency

The second goal can be elaborated in two parts – preparing to learnto teach, and preparing to learn to teach for mathematical proficiency.Preparing to learn to teach is a relatively uncommon way of conceivingthe goal of a teacher preparation program and requires some justification.

Learning to learn is not an easy task (Bereiter & Scardamalia, 1989)and brings its own set of challenges. Learning to learn to teach cannotbe an easy goal to achieve. Why do we believe it is a more appropriategoal for prospective teachers than the more conventional goal of masteringsome aspects of preferred teaching practices by graduation day? First,the complexity of teaching and the difficulty of mastering all aspects ofeffective teaching, especially as defined by the new and ambitious learninggoal of mathematical proficiency, nearly ensure that prospective teachers


cannot become experts, or even accomplished novices, during a relativelybrief program. Even if the current knowledge base identified the completeset of skills and dispositions for effective teachers, it is unlikely thatprospective teachers could acquire these competencies in a relatively briefpreparation program. Without such a knowledge base, it becomes essen-tial for beginning teachers to know how to learn to teach with increasingeffectiveness over time, taking advantage of new knowledge generated bythemselves and others.

A second reason for targeting the goal of learning to teach is thatthe richest environments for learning to teach effectively are schoolclassrooms (Ball & Cohen, 1999; Clark, 2001; Jaworski, 1998; Schön,1991). Teachers who are equipped with tools for learning from their exper-iences are in a strong position to learn more effective methods over the fullcourse of their careers.

A third reason for focusing on tools for learning to teach is thatteacher preparation programs are suited better for developing the knowl-edge, dispositions, and competencies prospective teachers need to takeadvantage of their experiences when they become teachers than for simu-lating the daily experience of teaching. Formal study of teaching mathe-matics, in structured courses and field experiences, allows novices to slowdown the classroom and examine its apparent chaos. Course experiences,such as studying cases or interviewing students, which pose problematicteaching situations in meaningful but digestible chunks permit prospectiveteachers to consider the various elements of classrooms – subject matter,students’ thinking, teacher-student interactions – and to develop tools formonitoring and examining these elements as they enter the classroom andbegin experiencing life as a teacher (Masingila & Doerr, 2002; Moyer &Milewicz, 2002).

A final reason for selecting learning to teach as a goal for preparationprograms is that many schools in the United States today do not haveorganizational structures that provide novice teachers with the supportthey need in order to take advantage of the rich potential of classroomsas learning sites (Darling-Hammond, 1997). Learning to teach, in plannedand systematic ways, is not a process into which beginning teachers willbe inducted as they enter the average school. It is a process that they willneed to create. This makes it even more essential that beginning teachersare equipped with the tools and are encouraged to develop the dispositionsthat will enable them to learn from their experience.

Preparing to learn to teach for mathematical proficiency requires, inaddition to one’s own proficiency and in addition to knowing how tolearn to teach, the appreciation of the importance of setting mathematical


proficiency as the learning goal for one’s own students. This appreciationcomes from two sources: the recognition of the long-term benefits foryoung students of becoming mathematically proficient, and of the impor-tance of measuring teaching effectiveness against a well-defined goal, inthis case mathematical proficiency. To know whether changes are im-provements in practice, or merely changes, one needs to measure theireffects against a clear, consistent standard. The second learning goal forprospective teachers includes appreciating the importance of using mathe-matical proficiency as the standard for measuring one’s own teachingeffectiveness.

LEARNING TO TEACH BY TREATING LESSONS ASEXPERIMENTS

We believe that a teacher preparation program aligned with the proposedmodel – a program self-consciously committed to helping students achievethe two goals identified above – would be designed quite differentlyfrom conventional teacher preparation programs. Because the concept ofpreparing to learn to teach is a distinguishing feature of the model, weconsider the concept further and describe how it might look in practice.

Preparing to learn to teach means knowing how to learn from classroomteaching experiences. It means planning these experiences in a way thataffords learning and then reflecting on the outcomes in order to maximizethe benefits that can be gained from the experiences (Artzt, 1999). AsC. Roland Christensen phrased it, “Every good teaching plan has an exper-iment in it” (J. Simon, 1995). In our re-phrasing: Prospective teachersshould be inclined and able to treat the lessons they teach as experiments.Treating lessons as experiments is, in our view, precisely what is needed tolearn to teach. Phrased in this way, the goal has a distinct and clear focus.

The notion of treating lessons as experiments carries the recognitionthat experience, by itself, does not ensure better knowledge or improvedperformance (Sullivan, 2002). In order to take advantage of their exper-ience, teachers need to design lessons with clear goals in mind, monitortheir implementation, collect feedback, and interpret the feedback in orderto revise and improve future practice.

An additional explanation is needed for our use of the word “experi-ment”. We can clarify our intended meaning by comparing our use withboth the increasingly popular phrase “design experiment” in educationand the social sciences and the more orthodox use of “experiment” in thenatural sciences. First, our use of experiment shares many features withthose highlighted in descriptions of design experiments (Brown, 1992;


Cobb, Confrey, diSessa, Lehrer & Schauble, 2003; Design-Based ResearchCollective, 2003; Kelly & Lesh, 2000). When teachers treat lessons asexperiments, they engage in many of the practices critical for conductingdesign-based research or design experiments. For example, the goalsinclude both the actual improvement of classroom environments and thegeneration of shareable knowledge about such environments. The processplays out through continuing cycles of planning, enactment, analysis, andrevision, and hypotheses about connections between teaching and learningare used to drive each cycle of the process. However, the model we proposedoes not treat the process of experimenting with lessons as a particularresearch method, applied with researchers’ expertise and resources anddesigned to collect data for a defined period of time and to generate aspecific product. Rather, we use the treating of lessons as experiments asa way of making some aspects of teachers’ routine, natural activity moresystematic and intensive. Teachers routinely plan lessons and then wonderabout their effectiveness. Treating lessons as experiments provides a moresystematic way to engage in these activities by focusing attention on, andmaking more explicit, the process of forming and testing hypotheses, aprocess that is contained in most definitions of “experiment”.

We also note some similarities and differences with the way in whichwe use the word experiment and its more orthodox connotations. Theprimary difference is that, by experiment, we do not mean the traditionalform that involves a controlled study with random assignment of parti-cipants. The methods we have in mind are not randomized assignmentof participants to comparison groups but rather replications and observa-tions of individual classroom experiments over multiple trials (Hiebert etal., 2002). But our use of the word does share the traditional connotationof intentional learning from carefully planned experiences. Experiment isused to emphasize the systematic and rigorous way in which teachers canstudy and reason about their practice in order to improve their teaching.Experiment also is used to emphasize the open and public process neededto grow a shared knowledge base for teaching.

The term experiment is useful, in addition, because it helps to identifythe knowledge, dispositions, and competencies that prospective teachersmust develop in order to design, implement, and learn from instructionalexperiences. In fact, one way to identify the requirements is to ask what isneeded to conduct more traditional forms of experiments.1

Clarifying the Research Question

When teaching lessons, clarifying the research question means articulatingtwo related but distinct statements about the lesson: the learning goalsfor the lesson, and the hypotheses that link planned instructional activ-


ities with expected learning outcomes. Both are needed in order to makedecisions about the lesson design. Learning goals set parameters on thepool of potential learning activities and establish the criteria for judgingthe lesson’s effectiveness. Hypotheses about how the learning activitieswill support students’ achievement of the goals direct the selection andsequencing of activities. Explicit statements of learning goals and instruc-tional hypotheses allow the lesson to be a learning opportunity for both thestudents and the teacher.

In what form should learning goals be expressed? This is a nontrivialquestion because, just as in other scientific experiments, the research ques-tion (learning goal) shapes everything that follows. First, the learning goalfor a lesson should fit within the over-arching goal of developing mathe-matical proficiency. This means that making progress along the multiplestrands for mathematical proficiency should be part of the goal. Second,the goal should be precise enough to guide lesson design decisions. Thismeans, among other things, that the goal should be measurable. To learnfrom teaching the lesson, a teacher needs to know whether the lesson waseffective in helping students reach the goal. But goals do not need to takethe form of performance objectives, popular in the past (Gagné, 1985).Students’ progress toward the goal can be measured in ways other thanwritten performance on narrowly defined tasks. We believe that it is moreuseful to define learning goals for lessons in terms of students’ thinking(Wittrock, 1986; M. Simon et al., 1999): How does the teacher expectstudents to be thinking about a concept or procedure before the lessonbegins and how is their thinking expected to change during the lesson?Expressing learning goals in terms of students’ thinking has the advantageof providing a rich source of information that the teacher can use to assessprogress along multiple strands of mathematical proficiency.

To set appropriate learning goals for lessons, prospective teachers needto be making progress toward the first of the twin goals – becoming math-ematically proficient. Part of mathematical proficiency is the constructionof a mental map of the curriculum, with key concepts, skills, reasoningforms, and dispositions as landmarks, along with possible routes fortraversing the territory. Knowing how to select and express appropriategoals for individual lessons means knowing how the goal for a singlelesson fits within the larger sequence of learning goals.

Formulating hypotheses for a lesson that predicts changes in students’thinking due to instructional activities moves the lesson from a plannedlearning experience solely for the school students to a planned learningexperience for the teacher as well. It transforms the lesson into an exper-iment that yields an empirical test of a teacher’s local theory of howstudents learn and how instruction facilitates learning. It allows the teacher


to generate knowledge of teaching that can be recorded, preserved, andapplied in order to improve teaching in the future. It is, in a very real sense,the heart of the process of treating lessons as experiments and improvingteaching in a gradual but steady and continuing way.

Designing the Experiment

As the lesson is designed, decisions are guided by the learning goalsand the hypotheses about which instructional activities will help studentsachieve them. The hypotheses might describe why particular activities willhelp students change their thinking in particular ways, or how studentswill be thinking at a particular point in the lesson and why a particularinstructional task will trigger a desired change in thinking. In this way, theplanned lesson becomes a series of researchable questions about students’thinking and instructional moves. This kind of cause-effect analysis is atthe heart of lesson design. The more explicit teachers can be about whythey selected particular activities, and what they expect will happen atspecific points of the lesson and why, the more they can learn from thefeedback they receive from the students.

Treating lessons as experiments means that lesson designs are morethan sequences of activities to keep students occupied during a 45-minuteperiod. Lessons-as-experiments require, on one hand, constructing localtheories regarding the relationships between teaching and learning and, onthe other hand, the tying of the theories to this learning goal in this context.The knowledge is concrete but the local theories search for patterns andfind connections between this knowledge and more general principles ofteaching and learning.

When lessons are treated as experiments, the usual emphasis on makingappropriate spontaneous decisions while implementing a lesson shifts tomaking appropriate predictions and decisions while planning a lesson.Greater emphasis is placed on clarifying the learning goal(s), on specifyingthe activities used to help students achieve them, on providing an accom-panying rationale for each activity in the form of teaching/learning hypoth-eses, and on justifying every facet of the lesson before it is implementedin real time. To plan with this level of explicitness and detail, the teachermust consider the content, the students, the information collected fromprior implementations, and so on.

Designing the mathematics lesson as an experiment requires consid-erable mathematical proficiency. Guided by the learning goal for thelesson, prospective teachers must be able to create or select mathematicsproblems that afford their students the opportunity to engage the variousstrands of proficiency.2 They must predict how students are likely to solvethe problems in order to build on students’ thinking and plan discus-


sions about solution methods that help students improve their thinking.They must think about how students’ contributions to the lesson can bemaximized and expanded. They must anticipate what ideas students canconstruct while working independently and what information students willneed to move forward in their thinking. They must also be able to eval-uate students’ mathematical proficiency and the validity of their solutionmethods.

Gathering Data

The success of conventional scientific experiments can be judged by thequality of the data collected. Do the data help to answer the researchquestion, do they help the researcher understand the phenomena beingstudied, and do they help the researcher formulate a follow-up experimentthat might be even more useful? In a parallel way, does the information theteacher collects during the lesson indicate that students are moving towardthe learning goal, does it help the teacher understand why and how thelesson worked well or not, and does it help the teacher plan an even moreeffective lesson?

Explicitly stated learning goals and hypotheses about how students’thinking will change during the lesson suggest what kinds of informationthe teacher needs to collect and when to collect it. With this approach, theevaluation of student learning and thinking is not something that is tackedonto the end of the lesson; instead, it becomes an essential and ongoingpart of the lesson. Student data can be collected systematically during thelesson to evaluate students’ learning, to help the teacher think about thelesson from the perspective of the learners who took part in it, to help theteacher interpret the results of the experiment, and to inform the lessonrevision process.

The kind of information the teacher collects during the lesson is crucialto the success of the lesson. As noted earlier, students’ thinking provides arich source of information. But there is more to this than is at first apparent.The lesson must be designed so that students’ thinking is revealed acrossmultiple strands of mathematical proficiency. A natural way in which thiscan occur is by centering the lesson on solving significant mathematicalproblems and then listening to students discuss methods that can be usedto solve the problems (Chazan, 2000; Lampert, 2001). As noted earlier,this approach fits well the goal of helping students acquire mathematicalproficiency. Now one can see that it also provides an environment in whichteachers can assess changes in students’ thinking.3

Gathering useful data depends both on a lesson designed to revealstudents’ thinking and on a teacher competent and disposed to solicitand hear key ideas. The second of these requirements makes additional


demands on the mathematical proficiency of prospective teachers. Itprovides another reason for the widely held belief that teachers themselvesmust be mathematically proficient. To hear emerging expressions of profi-ciency and to know how to support and improve them, teachers must befamiliar with the domain.

Interpreting Data and Drawing Conclusions

Experiments are considered to be useful to the extent that something islearned about the research questions. Of course, there might be unintendedlearning as well. But the planned learning occurs as the researcher reflectson the data and thinks about how the data address the research questions.So, too, it is with teaching. Teachers learn whether and how well the lessonsupported the learning goals for students as they reflect on the informationthey collected.

The cause-effect analysis used to construct the lesson comes underspecial scrutiny. Are the hypotheses – that changes in students’ thinkingwill be prompted by particular lesson activities – supported by theevidence? Did the lesson activities have their intended consequences?Why or why not? Did the lesson facilitate students’ achievement of thelearning goal(s)? To the extent that the evidence collected can address thesequestions, the particular lesson can be revised. But, more important, theanswers to these questions also produce more refined hypotheses aboutteaching and learning that can be tested further in future lessons. Theexperimentation is used as the basis for making more informed decisionslater. This refinement of hypotheses and of local theories creates the kindof knowledge base for teaching that sustains continuing improvements(Stigler & Hiebert, 1999).

CREATING ENVIRONMENTS FOR GENERATINGKNOWLEDGE FOR TEACHING

Acquiring the Identity and Dispositions of a Professional Teacher

Up to this point, we have treated prospective teachers as individuals,working to prepare themselves to become effective teachers. The previousdiscussion of treating lessons as experiments allows for an individualteacher to engage in this process alone, gradually improving his/her prac-tice. Preparing to learn to teach, the second goal for prospective teachers,means more than this. It means becoming part of a profession (see alsoDarling-Hammond & Sykes, 1999; Shulman, 2000).

Becoming a professional teacher, in our view, means drawing from, andcontributing to, a shared knowledge base for teaching. It means shifting the


focus from improving as a teacher to improving teaching. This requiresmoving outside the individual classroom, surmounting the insularity of theusual school environment, and working with colleagues with the intent ofimproving the professional standard for daily practice. This also requiresredirecting attention from the teacher to the methods of teaching. It is notthe personality or style of the teacher that is being examined but rather theelements of classroom practice.

Shifting from a vision of effective teachers to effective teachingrequires a major shift in mindset. It requires a change in the culture identi-fied in the opening paragraphs of this article. When teachers work togetherto experiment with lessons, with the intent of sharing what they learn withtheir professional colleagues, they are engaged in something more thanbecoming a better teacher; they are contributing directly to the knowledgebase upon which a true profession is built. They are doing what members ofmany other professions do, but what teachers seldom have had the chanceto do.

Shifting from a vision of effective teachers to effective teaching alsorequires a new set of obligations. Rather than considering only what oneis learning from one’s own experience, teachers must ensure that otherscan learn from their experience, and that they are disposed to learn fromothers’ experiences. Planning teaching so that others can learn is differentfrom planning teaching so that you can learn.

Considering what others can learn from your experience requirescollaboration with other teachers who share the same learning goals forstudents. Such collaborations are characterized by a number of featuresthat increase the chances of generating knowledge of teaching that is usefulfor the profession (Darling-Hammond & Sykes, 1999; Hiebert et al., 2002;Loucks-Horsley et al., 1998). Working with colleagues ensures that thelearning goals, lesson designs, and data interpretation become explicit andpublic so they are accessible to others. Collaboration allows teachers toassist each other in collecting the kinds of data that can inform efforts toimprove teaching and learning. Making these elements public also meansthat they can be examined, critiqued, and replicated in other contexts. This,in turn, yields further information about effective lessons, and about thehypotheses that shaped the lessons.

Mechanisms already have been developed through which this kindof collaborative teacher experimentation can occur. One form of such amechanism – often called lesson study – has a 50-year history in someAsian countries and currently is being adapted and developed in local sitesaround the United States (Fernandez et al., 2001; Lewis, 2002; Lewis &Tsuchida, 1998; NRC, 2002). Lesson study is an especially promisingmechanism for our model because it fits well the learning goals proposed


for prospective teachers and, as will be seen shortly, it fits well theknowledge generation and continuing improvement processes proposedfor teacher preparation.

Acquiring the Dispositions and Skills Needed to Create LearningEnvironments in Schools

Learning to teach effectively is a long-term enterprise. If prospectiveteachers are going to use the knowledge, dispositions, and competenciesthey develop for learning to teach, they need to work in schools that allowthem to engage in this work. As noted earlier, many schools in the UnitedStates today do not offer such environments (Darling-Hammond, 1997).Often, beginning teachers are expected to know how to teach, even to bequite skillful. They are expected to fit into the on-going culture, workingindependently and projecting confidence in their performance as a teacherfrom the very beginning. Not surprisingly, in these cultures there are fewprovisions for learning to teach. For example, there is little time in theworkweek for teachers to collaborate on designing and improving lessons.

Consequently, to achieve the goal of preparing to learn to teach formathematics proficiency, prospective teachers need to acquire skills tocreate environments for themselves that support their learning as teachers.They must prepare to be agents for change. In our view, this does not meanthat they enter the school armed with the answers for teaching effectively,but rather that they recognize the value of working with colleagues toimprove their practice and that they possess the skills needed to create suchenvironments. Of course, they cannot be expected to transform the cultureof every school they enter, but they need to know how to connect with otherteachers and form collaborative groups aimed toward improving teaching(Clark, 2001; Britt et al., 2001). They also need to learn to value this kindof work enough to invest the time and energy in order to create the environ-ments that afford it. This can be achieved, in part, by providing supportedpractice in teacher-directed study groups during the teacher preparationprogram.

CREATING ENVIRONMENTS FOR GENERATINGKNOWLEDGE FOR TEACHER EDUCATION

Goals for Teacher Educators

In the model we propose, the principles and processes that are proposedfor the generation of the knowledge needed to improve teaching are thesame as those that are proposed for the generation of the knowledgeneeded to improve teacher preparation. Assuming that mathematics


teacher educators are proficient mathematically, we identify one majorlearning goal for teacher educators that parallels the learning-to-teach goalfor prospective teachers: to learn to teach prospective teachers in waysthat support the achievement of their learning goals and to do this inways that generate a shared knowledge base for teacher education. Inour model, the same process of experimenting with lessons is used as themeans to achieve this goal. Treating lessons (for prospective teachers) asexperiments becomes the routine, on-going activity for course instructors.The requirements to do this well are the same ones elaborated earlierwhen describing the components of treating lessons as (scientific) exper-iments. Prospective teachers now are the students and university facultyand doctoral students now are the teachers.

A Sample Learning Environment for Improving Teacher PreparationPrograms

The best way to expand on the proposed model in the context of teacherpreparation programs is to describe an example of how the process oftreating lessons as experiments can be put into operation at this level. Akey feature of this example is that the process is designed intentionally sothat it will be sustainable over generations of teacher educators with theproducts always viewed as unfinished.

Imagine a teacher preparation program at a typical university in whichmultiple sections of courses are offered in mathematics and/or methods ofteaching mathematics for prospective teachers. Now imagine the group ofinstructors for each course (e.g., doctoral students and faculty) meeting, atleast weekly, to jointly plan the course and study its effectiveness. In lessonstudy fashion, the group sets clear goals for the course (specific sub-goalsof the two primary goals identified earlier), identifies particular lessons thatare key sites for helping students achieve these goals, plans these lessonstogether, implements them with careful monitoring of students’ thinking,and revises the lessons for use the following term. Each term, the groupof instructors for a particular course inherits a set of lesson plans (detailedin a special way that emphasizes the cause-effect analysis of the lesson)that provides the current knowledge base for effective instruction in thatcourse. Each semester, the group of instructors takes up the challenge ofincreasing the knowledge base by improving the effectiveness of a targeted(perhaps different) small subset of lessons.

The lesson plans for each session of the course are the repositoryfor the collective knowledge for teaching the course effectively. In addi-tion to detailing the sequence of activities for the lesson and the role forthe teacher in presenting tasks and leading discussions, the plans predict


the responses of the students (based, increasingly, on past experience),suggest how to use these responses to further the goal of the lesson,provide rationales for the instructional decisions specified in the plans andhypothesize how and why particular instructional activities will facilitateparticular learning. In fact, the plans are local theories of teaching andlearning with the planned lesson serving as an example. The theories offertargeted, micro-hypotheses about the way in which teaching promoteslearning. The implemented lesson serves as a test of the hypothesesproposed in the lesson plan, and the feedback received from the students isused, not only to revise the plan, but to revise the hypotheses and theoriesas well.

As noted earlier, local theories, with examples, are a useful form inwhich to package the knowledge generated about teaching a particularcourse effectively. Knowledge expressed in this form retains its connectionto the context so it remains immediately useful for future instructors of thecourse, but it also rises above the details that vary unpredictably from classsection to class section and from term to term and thereby moves toward amore principled knowledge base for teaching and teacher education.

Because teaching and learning are too complex and variable to presumethat one could learn all there is to know from a single implementation of acourse, the proposed model allows for continued authorship of the coursesover time. A final version of a course is not expected. Rather, it is expectedthat instructors will learn to reason about teaching in an increasingly usefulway and to accumulate knowledge for effective teacher preparation. Eachterm, an implemented course can be thought of as the accumulated wisdomof the previous instructors/authors. The process produces a single text,but the reading of the text and the accumulated knowledge always leadsto remolding, reworking, rewriting, reevaluating, and reinterpreting overtime. The text is both a repository of the wisdom and knowledge of thetime and the locus of change as knowledge increases.

A Learning Environment for Program Improvement Provides an Imagefor Prospective Teachers

Central to the proposed model is the goal of treating lessons as experi-ments. As described earlier, this includes the ability to set clear learninggoals, design lessons that support students’ achievement of the goals (witha rationale for why the lesson might do so), collect data to evaluate thelesson’s effectiveness, and interpret the data to revise the lesson accord-ingly. These are ambitious goals. One might wonder how prospectiveteachers can achieve them in a relatively short preparation program.


Two course-design strategies can help prospective teachers learn to treatlessons as experiments. The first is for instructors to include some of thecomponent skills needed to treat lessons as experiments in each course.For example, treating lessons as experiments depends on listening care-fully to students’ thinking and assessing its apparent mathematics potential(validity of reasoning, connection to later ideas, etc.). Listening to (andreally understanding) students can be set, early in the program, as a goalfor courses. Techniques for supporting these goals include, for example,using videos of interviews with school students to generate the mathe-matical ideas that then are explored more deeply. Skills such as creatingtasks that elicit students’ thinking, inferring changes in students’ thinkingfrom changes in their solution methods, and expressing lesson goals interms of changes in students’ thinking, can be addressed in later courses asprospective teachers proceed through the program.

A second strategy that can help prospective teachers treat lessons asexperiments is based on the fact that the knowledge, dispositions, andcompetencies that enable prospective teachers to treat lessons as experi-ments parallel, quite closely, the knowledge, dispositions, and competen-cies that instructors must develop collaboratively as the courses themselvesare improved. The process of course improvement in which the instructorsare engaged can be made transparent for prospective teachers so that theycan see how the courses they are taking are being planned, evaluated, andrevised. This provides an image of how the process can work to generateknowledge for, and improve, teaching.

One way in which the process of course improvement can be revealedto the prospective teachers in the program is through a gradually intensi-fied experience, first as observers and informants, then as apprentices, andfinally as full participants. In other words, prospective teachers first couldbe observers and informants in studying the improvement of courses theyare taking. As observers, they might view a videotape of their instructorsplanning the lesson they just completed. As informants, they might viewa videotape of a lesson in which they just participated as a student andprovide feedback about critical learning moments during the lesson andabout their own thinking during these points. As teachers, they mightdesign lessons for school students which they treat as planned learningexperiences – for the students, for themselves, and for others in their class.By playing different roles within the system of teaching improvement, theopportunities increase for prospective teachers to internalize the notions oflearning to teach and, over time, improve their own teaching and enableothers to learn from their experiences.


WHAT ABOUT CURRENT THEORIES OF LEARNING ANDTEACHING?

Noticeably absent from the entire previous discussion of learning to teachis an endorsement, or at least an analysis, of grand theories of learning(e.g., behaviorism, constructivism, social-constructivism) and their corol-lary theories of teaching. Our model for teacher preparation is silent aboutthese theories because we believe no a priori endorsement of particulartheories is necessary. What is necessary, in our view, is that learning goalsfor students are precisely and explicitly articulated, and that hypotheses areformulated and tested for how the instructional activities will help studentsachieve the learning goals.

The role for current theories of learning and teaching is to provideresources that can help predict what kinds of instructional activities willbest support students’ efforts to achieve the learning goals. In this sense,they provide shortcuts for what otherwise would be a rather lengthy andchaotic process of trial-and-error. They suggest instructional approachesthat can be translated into lesson designs and then tested for effectiveness.But no particular learning or teaching theories are privileged at the outset.Only the two learning goals of mathematical proficiency and preparingto learn to teach are privileged. How these can best be accomplished isthe continuing task for those engaged in building the knowledge base formathematics teaching and teacher education.

LESSON AS THE UNIT OF ANALYSIS AND IMPROVEMENT

A danger of building a model for teaching on the planning and analysisof lessons is the possible (mis)perception that individual lessons containall of the information needed to construct a knowledge base for teaching.School learning occurs over sequences of lessons, and an adequate knowl-edge base will include information on students’ learning trajectories andhow these can be supported over time (M. Simon, 1995). We expect thatextensions of the model proposed here, which devote explicit attentionto students’ learning over time, can and should be made. But this earlyversion of the model focuses on individual lessons in order to anchor itto a nearly universal unit of teaching (the daily lesson) and because thereare some good reasons to focus on a lesson, at least initially, in order tostudy and improve teaching. The individual lesson is a big enough unit ofteaching to contain all of the complex classroom interactions that influ-ence the nature of learning opportunities for students. At the same time,the individual lesson is the smallest natural unit for teachers that retains


such interactions. The benefit of defining small units is that they allow thedetailed analyses of teaching/learning relationships that make up the coreof a knowledge base for teaching.

CONCLUSION

Fractals are curious geometric objects in which each piece of the object isidentical to the larger object. A distant view of the whole object or a close-up view of a particular piece reveals the same basic design. The modelwe propose has some fractal characteristics. From a distance, the modelas a whole looks like a learning system with an emphasis on continualstudy and incremental improvement. Zooming in reveals teacher educatorsengaged in learning how to study their teaching of prospective teachers.They are valuing incremental improvement and contributing to the profes-sional knowledge base for teacher education. Zooming in further revealsthe prospective teachers themselves engaged in learning how to study theirown teaching, to value incremental improvement, and to contribute to theprofessional knowledge base for teaching.

We believe the model is promising, in part, because of the similarityof intellectual activity at all levels and sites. Research, teaching, andlearning are tightly intertwined and are actively engaged, albeit in some-what different ways, at all levels. As with fractals, it is possible to seesimilar structures and mechanisms at work in each level, and in the systemas a whole.

We also believe the model is promising because it addresses the twoproblems identified at the outset – the culture of teaching which passesalong, in a relatively unexamined way, the teaching methods of the past,and the absence of a knowledge base for teaching and for teacher prepar-ation. The model outlines a system designed to achieve particular learninggoals, for school students and for prospective teachers, using a processof continuing improvement through learning from planned instructionalexperiences. Such a process involves changing the culture of teaching inways that afford building a professional knowledge base.

Is the model realistic? Can preparation programs be designed to helpprospective teachers accomplish the twin goals of mathematics proficiencyand learning to learn to teach for mathematics proficiency? Not in thenear future. Among other obstacles, a sufficient knowledge base for math-ematics teaching or for preparing mathematics teachers does not yet exist.The model we propose,however, is intended to guide long-term growthof knowledge in a gradual and incremental yet steady and lasting way. We


believe that this goal can be achieved but achievement depends on buildinglocal cultures that value this kind of gradual and continuing progress.

ACKNOWLEDGEMENTS

Preparation of this article was supported, in part, by the National ScienceFoundation (Grant #0083429 to the Mid-Atlantic Center for Teaching andLearning Mathematics). The opinions expressed in the article are thoseof the authors and not necessarily those of the Foundation. Thanks toDaniel Chazan, Christopher Clark, Ronald Gallimore, James Raths, MartinSimon, James Stigler, Terry Wood, and an anonymous reviewer for theircomments on earlier drafts of the paper.

Correspondence concerning this article should be addressed to JamesHiebert, School of Education, University of Delaware, Newark, DE 19716([email protected]).

NOTES

1 Thanks to Stephen Hwang for suggesting this parallel.2 Theoretical and empirical work suggest that instructional methods that ask students tosolve challenging mathematical problems help students integrate, rather than separate, thefive strands of mathematical proficiency (Hiebert et al., 1996; NRC, 2001; Schoenfeld,1985; Silver, 1985). If the problems are appropriate, students will call on most or all ofthese strands while constructing and examining solution methods, thereby integrating themand becoming increasingly proficient.3 A single lesson will not provide a precise measure of every student’s progress. Thismeans that, in order to chart students’ progress toward mathematical proficiency, manylessons will need to be planned and implemented in ways that make students’ thinkingtransparent. This kind of a lesson cannot be a one-time event. It also means that anindividual student’s progress will be measured over time; any single snapshot will beincomplete. In this sense, students’ progress is better conceived as a movie than a snapshot.

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School of EducationUniversity of DelawareNewark, DE 19716USAE-mail: [email protected]

JEREMY A. KAHAN, DUANE A. COOPER and KIMBERLY A. BETHEA

THE ROLE OF MATHEMATICS TEACHERS’ CONTENTKNOWLEDGE IN THEIR TEACHING: A FRAMEWORK FOR

RESEARCH APPLIED TO A STUDY OF STUDENT TEACHERS

ABSTRACT. The authors develop and explain a framework to guide research on therelationship between mathematics teachers’ knowledge of content and their teaching. Theframework is two-dimensional. The dimensions are (a) the elements of teaching and (b) theprocesses of teaching in which knowledge of content is of consequence. The interplaybetween the elements of teaching and the processes of teaching is discussed theoreti-cally, considering connections to existing literature about the role of teachers’ subjectmatter knowledge. Three vignettes from the authors’ work with pre-service secondarymathematics teachers investigate further the relationship between content knowledge andeffective mathematics teaching. The vignettes also serve to illustrate the complexity ofinvestigations into this relationship. Direction is offered for use of the framework in futureresearch.

KEY WORDS: mathematical content knowledge, mathematics teaching, pedagogicalcontent knowledge, subject matter knowledge

INTRODUCTION

When policy makers and professional mathematicians consider the prob-lems of school mathematics teaching and learning, they frequentlyconclude that students would learn more mathematics if their teachersknew more mathematics. In his landmark review of 30 studies of teachers’knowledge of mathematics, Begle (1979) concluded: “It seems to betaken for granted that it is important for the teacher to have a thor-ough understanding of the subject matter being taught” (p. 28). TheGlenn commission (U.S. Department of Education, 2000) agrees, “High-quality teaching requires that teachers have a deep knowledge of subjectmatter. For this there is no substitute” (p. 22). Similarly, the Confer-ence Board of the Mathematical Sciences (2000) advocates a series ofmathematics courses to help teachers develop a deep understanding ofmathematics. These common-sense conclusions and related recommenda-tions for teacher preparation rely in part on anecdotal evidence – reports ofincidents where prospective or in-service teachers have shown limited or


224 JEREMY A. KAHAN ET AL.

flawed understanding of key mathematical ideas or facts during instruction,a lack, in turn, assumed to hinder student learning.

In reviewing related research, Darling-Hammond (2000) finds rigorousresearch ambiguous in linking a teacher’s content knowledge and herteaching. In his research, Begle (1979) reviewed 17 studies and foundthat “once a teacher reaches a certain level of understanding of the subjectmatter, then further understanding contributes nothing to student achieve-ment” (p. 51). Monk (1994) determined a positive relationship betweensecondary students’ (who scored similarly on a pre-test) performance onmathematics post-tests derived from the National Assessment of Educa-tional Progress and their teachers’ number of courses (typically semesters)of collegiate mathematics. He found that this relationship weakens afterfive semesters of mathematics classes for the teacher. Indeed, Monkconcluded, “Courses in undergraduate mathematics pedagogy contributemore to pupil performance gains than do courses in undergraduate mathe-matics” (p. 130). Ball & McDiarmid (1989), Ball & Wilson (1990), andMa (1999), make the case effectively that counting the number of coursesprovides too blunt a measure of content knowledge for teaching. Chineseteachers with the equivalent of a 9th grade mathematics education and 2 or3 more years of teacher education outperform college-trained United Statesteachers when asked to respond to four mathematical teaching scenarios(Ma, 1999). Mathematics majors performed no better than non-majorswhen asked to explain why one cannot divide 7 by 0 and worse than non-majors when asked to develop a story to model 13

4 ÷ 12 (Ball & Wilson,

1990). Similarly, Askew (1999) found that for British elementary teachers“being highly effective was not positively associated with higher levelsof qualification in mathematics. The amount of continuing professionaldevelopment in mathematics education that teachers had undertaken wasa better predictor of their effectiveness than the level to which they hadformally studied mathematics” (p. 96). As measured in most past studies,content knowledge alone does not ensure effective teaching performance,and may not be the best investment of teacher development time.

Nonetheless, interest in the connection between teacher knowledgeof mathematics and student learning persists. The implication for themathematical preparation of teachers is real, especially given that teacherpreparation programs tend to work in a zero-sum game environment whereadditional preparation in mathematics will result in decreased preparationin some other area (such as pedagogy). To facilitate further research in thisdomain, we have developed a framework (see Figure 1) that guides ourresearch and will perhaps stimulate others who explore the relationshipof a teacher’s mathematical content knowledge (MCK) and her teaching.

ROLE OF MATHEMATICS TEACHERS’ CONTENT KNOWLEDGE IN TEACHING 225

The framework provides an organized way in which to examine this multi-faceted relationship. This article introduces the framework, explains it intheory and in relation to prior research, and then illustrates its applicationin our research about pre-service teachers. In our research, the frame-work provides structure for the analysis of MCK in teaching. This role istwofold. First, the framework helps us to focus on the aspects of teachingwe need to include in our study. Second, the framework helps us toorganize the findings of such study.

CONTENT KNOWLEDGE

In this article, the authors focus on MCK of teachers and its role intheir teaching. As an example of mathematics to be taught, consider thefollowing quick method for testing if a given number, for example 15,234,is divisible by 9. To do so, one can sum the digits to obtain 15, see that 15is not divisible by 9, and conclude that neither is 15,234. Ability to executethis divisibility test is an element of MCK a teacher would need to knowbefore teaching the test to students. In using the term “MCK” the authorsare specializing Shulman’s (1987) use of “content knowledge” (p. 8) as thefirst element of the knowledge base for teaching.

There are several elements of MCK worth describing. Bransford,Brown, and Cocking (2000) note that competence in an area requires threefeatures: (a) “a deep foundation of factual knowledge”, (b) understandingof the “facts and ideas in the context of a conceptual framework”, and(c) organization of the knowledge “in ways that facilitate retrieval andapplication” (p. 16). Although our work in this domain began before 2000,Bransford et al.’s definition accords with our conceptualization of MCK,and our use of the term comprises all three elements. We value factualknowledge, but we value it most when it is coordinated with deeper under-standing and is ready for application. We now articulate some key featureswe believe are implicit in this definition.

We concur with Grossman, Wilson & Shulman’s (1989) enumera-tion of syntactic knowledge, “knowledge of how to conduct inquiry inthe discipline” (p. 29), as a dimension of subject matter knowledge forteaching, and we believe that is included by “application” in Bransfordet al.’s (c) “application”. This reading of Bransford et al. is supportedby their subsequent contention that knowing how to learn in new situa-tions should be an integrated part of what teachers teach students. We arealso influenced by the work of Hiebert & Lefevre (1986), who stress thatknowledge has both procedural and conceptual components. The proce-dural component roughly corresponds to the facts and ideas mentioned


by Bransford et al., and these ideas may include mathematical methodsand algorithms. Teachers’ conceptual understanding of the mathematicsroughly parallels features (b) and (c) in Bransford et al., including whetherteachers know why the methods and algorithms work. In the divisibility by9 example, we believe teachers ought to know not only the procedure, butalso why it works and how it generalizes. Our use of “MCK” in this papershould be taken to include all the features described in this paragraph andthe preceding one.

Content knowledge in the subject area does not suffice for goodteaching. Grossman (1990) compared beginning teachers of English whohad deep content knowledge (e.g., being well versed in Shakespeare),but different extents of preparation as teachers. She found that teacherswith preparation as teachers tended to think of how to relate the material(e.g., Hamlet) to student experiences whereas those without preparationas teachers tried to teach secondary students as the teachers themselveshad learned in college seminars, introducing them early on to jargon-intense literary theory. The latter approach was predictably less successfulat engaging student interest, which suggests that content knowledge is notthe only part of the knowledge base for teaching. Later in this article,several examples of practice by mathematics teachers will illustrate thissame point, but we now turn to elaborating the other components of theknowledge base for teaching.

In his inventory of the knowledge base for teaching, Shulman(1987) includes content knowledge, general pedagogical knowledge (e.g.,classroom management strategies), and pedagogical content knowledge(PCK), which he defines as a “special amalgam of content and pedagogythat is uniquely the province of teachers, their own special form of profes-sional understanding” (p. 8). In the example of testing for divisibility by 9by exploring the sum of the digits, being able to anticipate the question ofwhether the divisibility by 9 test will always work or why it works consti-tutes PCK. Knowing some possible responses to the question of why thetest works (e.g., letting students experiment using a variety of appropriatemanipulatives incorporating base ten) is also an example of PCK. Thisknowledge is not simple, because a generic teacher approach of tellingstudents to use manipulatives might lead to the use of teddy bear coun-ters, which do not reinforce place value and accordingly would frustratestudents and not lead them in a direction likely to be productive. Noticehow PCK is content-specific and at the same time goes beyond simpleknowledge of mathematics because a mathematician may not possess it.

We grant that PCK matters, but in this article, MCK is in the fore-ground. For the divisibility by 9 test, the focus in this article would be,


what mathematics does a teacher needs to know to address the topic well?The teacher needs some understanding of place value and modulo (clock)arithmetic to see that the test depends on the fact that ten is one morethan one times nine. It would help to know bases other than ten to explorewhether the divisibility by 9 test works in base 3. Such knowledge wouldalso let the teacher vary the problem while preserving the mathematicalstructure by asking students to devise a test for divisibility by 2 in base 3.The exploration of this example from an MCK perspective illustrates thisarticle’s focus.

FRAMEWORK FOR RESEARCH ON TEACHER KNOWLEDGEAND TEACHING

Our framework for the analysis of content knowledge and its relation-ship to teaching appears in Figure 1. This framework resembles a figureappearing in Grossman (1990) and its elements, as detailed below, reflectthe authors’ research-informed choice of key elements of teaching. Theprocesses are similar to Anders’ (1995) dimension of planning, predicting,interpreting, and reflecting and to Shulman’s (1987) seven or Artzt &Armour-Thomas’ (1999) three stages of teaching. Like Artzt and ArmourThomas’ model, our framework is a two-dimensional array that illustratesthe interaction of elements of teaching with four teaching processes. Itguides our research agenda and thinking on the role of teacher MCK inmathematics teaching and is intended to inspire like-minded researchers.Finally, it provides a way of examining and categorizing current researchin this field, with an eye toward articulating what is known and determiningwhere there are gaps to be addressed.

Figure 1. Framework for research on teacher knowledge and teaching.


DISCUSSION OF FRAMEWORK

Processes and Elements

The columns in the framework refer to four teaching processes: prepara-tion, instruction, assessment, and reflection. We differ here from Artzt &Armour-Thomas (1999), who examine mathematics instruction throughtemporal stages of “pre-active (planning), interactive (monitoring andregulating), and post-active (evaluating and revising)” (p. 213). We preferto identify teaching processes rather than phases to indicate that theprocesses may be ongoing and overlapping. For example, assessment andreflection happen not only after instruction, but also during it as well,as when teachers use questions to assess what the class understands orcirculate to supervise student work.

The rows of the framework contain the following elements of teaching:a) goals and objectives, b) selection of tasks and representations, c) motiva-tion of content, d) development: connectivity and sequencing, e) allocationof time, points, and emphasis, and f) discourse. These elements representthe authors’ research-informed judgment of facets of teaching in whichcontent knowledge will matter most. The framework is extensible bydesign, and readers may wish to divide processes or elements more finelyor add others, but the authors attempted to keep the number of processesand elements small, lest the framework become unwieldy.

In many instances, key elements pertaining to mathematics content willmanifest themselves differently during the different processes of teaching.We postulate that researchers investigating the manifestations of mathe-matics teachers’ MCK will find some cells (e.g., 6B features prominentlyin the report of our research below) of the framework array better sourcesof evidence than others do. Each cell represents an intersection of teachingprocesses and elements, and research can explore how MCK plays out inteaching behavior manifest in each cell. More generally, one can use thecells to classify data or research by the subject to which it pertains, orto focus on specific features of teaching. We now elaborate on each cell,ordered by element and within that by process.

Cells

A teacher’s knowledge of mathematics, or lack thereof, may affect thegoals and objectives of a single lesson or of a unit of instruction (in interac-tion with external sources such as textbooks and government guidelines).Teachers’ goals and objectives will manifest themselves in their prepara-tion in lesson and unit plans, where objectives are often explicitly stated,and so could be studied in the context of cell 1A. Instruction can be studied


to determine whether or not questions posed of the students are consistentwith the teacher’s expressed goals; such a study would be included incell 1B. In the same way, evidence of the teacher’s objectives is observ-able in the teacher’s assessments of students (cell 1C). Discussion andperhaps revision of goals occur during the teacher’s reflection on a lessonor unit just completed (cell 1D). Thus one can proceed across a row of theframework, articulating how the element plays out in the various processes(columns), and what the role of MCK is in each of these.

The tasks on which students work in and out of class are importantfeatures of a lesson or an assignment as are the representations used by theteacher to guide investigations and explain or clarify ideas. The primaryevidence of task and representation selection researched to date has beenin class preparation (cell 2A), but the framework leads us to consider taskselection during instruction (cell 2B) (as when selecting a task to catch anunplanned teachable moment), and assessment (cell 2C), or reflection onthe lesson (cell 2D), (when the teacher considers modifying or replacingtasks or representations).

Element 3, motivation of content, indicates the teacher’s ability toaddress or preempt the perennial student questions of “Why is thisimportant to know” and “When are we ever going to use this?” Evenwith common tasks or a common text, teachers will experience differentlevels of success in motivating students to study a topic or learn a lesson.Research could explore the conjecture that a stronger mathematics back-ground leads to a better understanding of why certain mathematics topicsare in a curriculum and thus a better ability to convey that value to students.For example, in justifying the introduction of graph theory in the secondarycurriculum, teachers who can plan lessons (cell 3A) that result in sharingwith students (cell 3B) how the theory plays out in efficient design of layerson microchips or routing of long-distance phone calls will be more motiva-ting. In assessment (cell 3C), motivation of content matters contributesrelevance and purpose to the work as demanded by the question “How doesthe assessment engage students in relevant, purposeful work on worthwhilemathematical activities?” (NCTM, 1995, p. 14). At testing time, studentsare likely and entitled to wonder why what they are being asked to do isimportant, and greater mathematical knowledge expands a teacher’s abilityto write tests that address that question. Finally, teachers reflect on whetherthe examples they thought would motivate student interest in this contentactually did motivate this group of students (cell 3D). In this paragraph,we have focused on motivating the mathematics through applications, butMCK may also enhance a teacher’s sense of the structure, power and


beauty of mathematics and their students may come to share intrinsicmotivation for the mathematics as a whole.

The mathematical development of a lesson or unit is important foreffective teaching. The content should not appear to be a collection ofdisjointed, isolated topics, and it should be sequenced so that topicsare studied in a sensible order with prerequisite content being taught orreinforced as needed. The present evidence of the role of mathematicalknowledge in connectivity and sequencing has been in a teacher’s prepara-tion (cell 4A) and instruction (cell 4B). Exploring whether a teacher’sassessments ask students to demonstrate how the ideas in a unit connect(cell 4C) or how teachers reflect on how the lesson or unit might have beenreorganized to have maximum effect (cell 4D) may also be worthwhile.

Related to the aforementioned choices a teacher must make about whatto teach and ask are the matters of how long to spend on a task or howmuch value to give a test item or class topic. Here, knowledge of the mathe-matics under instruction allows a teacher to determine if a topic deserves asubstantial time allotment. A teacher’s MCK will be reflected in allocatingtime in a lesson plan (cell 5A) and reallocating it during instruction (cell5B). Points or criteria in a rubric are allocated on the teacher’s formalassessments (cell 5C). Finally, in reflecting on instruction and assessment,the teacher often reallocates time for these students or for the ones whowill be in the same course next time (cell 5D). Thus each cell of row 5 ispopulated, although research has focused on cell 5A. MCK is a factor influ-encing a teacher’s decisions about allocation of time, points, and emphasis,but we recognize there are often strong internal (e.g., teacher beliefs aboutwhat mathematical proficiency is) and external factors (e.g., curriculumguidelines or standardized tests) in such decisions.

In the 1990’s, much attention was given to the quality of discourse(element 6) in mathematics classrooms. For example, three of thesix standards for teaching mathematics in the Professional Standardsfor Teaching Mathematics (NCTM, 1991) explicitly mention discourse,including the teacher’s role in discourse. We use discourse in the sense ofthat document:

The discourse of a classroom – the ways of representing, thinking, talking, agreeing anddisagreeing – is central to what students learn about mathematics as a domain of humaninquiry with characteristic ways of knowing. Discourse is both the way ideas are exchangedand what the ideas entail: Who talks? About what? In what ways? What do people write,what do they record and why? What questions are important? How do ideas change? Whoseideas and ways of thinking are valued? Who determines when to end a discussion? Thediscourse is shaped by the tasks in which students engage and the nature of the learningenvironment; it also influences them (NCTM, 1991, p. 34).


Discourse cuts across all processes of teaching. In preparation, theteacher plans the likely classroom discourse, scripting questions andplanning responses to anticipated student comments (cell 6A). Duringinstruction, the mathematical communication occurs (cell 6B). We agreewith Dawson (1999b) that such communication is two-way, and that weignore what students bring to it at our peril, but here our focus is on thepossibilities created by what the teacher brings to the communication.Teacher weaknesses may become manifest in inaccurate mathematicalstatements, careless and otherwise. Research suggests that a mathemati-cally strong teacher is flexible enough to ask impromptu questions and toaddress unexpected statements or conjectures that arise in the classroom(Ball, 1997; Haller, 1997). Similarly, “mathematical skills of the teachers. . . are essential for recognizing the differing mathematical potential ofresponses from the class” (Breen, 1999, p. 117). Finally, we will providean example below in which mathematical knowledge creates the possi-bility of seizing unforeseen opportunities to relate the content to other bigideas. The discourse of written tests and examinations is asynchronous,but present, and discourse is a significant part of ongoing assessment asthe teacher asks and answers questions of individuals, groups of students,or the entire class (cell 6C). Finally, teachers reflect on the discourse thattranspired, and evaluate it (cell 6D).

For simplicity, we have presented the framework as two-dimensional.That choice accorded with our focus on the role of MCK. The more generalmodel is 3-dimensional, with the third dimension representing the facet ofthe teacher being explored. These might be elements of the knowledgebase for teaching such as pedagogical knowledge or PCK, but they mightalso be other facets such as the teacher’s affect, or the teacher’s beliefsand conceptions about mathematics and mathematics teaching (Boulton-Lewis, Grossman, Wilson & Shulman, 1989; Smith, McCrindle, Burnett& Campbell, 2001; Lerman, 1990; Ponte, 1999; Thompson, 1984). We arewell aware that teacher factors other than MCK are at work in teachingin practice, as are conditions such as time, material, and pupils (Brown &McIntyre, 1993). Our focus on the role of MCK is not intended to excludeother factors.

Models are necessarily simplifications, and so the authors acknowl-edge that our framework misses some nuances; nonetheless, we believeit can provide a useful structure for researchers to examine the role ofmathematics teachers’ MCK with considerable specificity. We now turnto a study of pre-service teachers’ MCK and how it related to severalcells of the framework. The framework helps us observe several trends andfacilitates our analysis of the role of content knowledge in teaching. The


study raises several issues in the role of MCK in teaching, and also gives apreliminary “proof-of-concept” for using the framework to guide research.

METHODOLOGY

Participants

At the beginning of the 1999-2000 school year, students in the secondarymathematics methods course were asked to participate in a research studyabout pre-service teachers’ MCK. All sixteen students who were enrolledin the course consented. These participants were all pre-service secondarymathematics teachers in their last stage of formal pre-service education ata large research university in the eastern United States. Data were collectedduring the students’ mathematics methods course and during their studentteaching experiences. The participants had completed all of the requiredmathematics courses including Advanced Calculus, Linear Algebra, andAbstract Algebra. All the teachers had some previous field experiencethrough their participation in a departmental tutoring program. Most ofthe sixteen participants were in their senior (fourth) year of undergraduatestudy at the time they took the methods course (although some were juniors– 3rd years); all were seniors at the time of student teaching.

Procedures, Instrument, and Data Collection

For these participants, several types of data were collected. An MCKtest for teachers was compiled mostly from tasks generated by theresearchers and colleagues and to some extent from related researchstudies (e.g., Even, 1993; Markovits, Eylon & Bruckheimer, 1988). Thetest assessed the prospective teachers in three major strands of thesecondary school mathematics curriculum – number, algebra/functions,and geometry/measurement. The items were chosen to represent some keyideas and problems in those content strands and to test students’ factualknowledge, conceptual understanding, and ability to apply that knowl-edge (consistent with Bransford et al., 2000). Sample questions from thealgebra/functions and geometry/measurement strands of the assessmentare provided in Figure 2. Note how the geometry items demand factualknowledge of an area formula (item 2.1(a)), a conceptual framework thatrelates it to other area formulae (item 2.1(b)), and the ability to retrieveand apply geometric properties to a new situation in order to evaluate anew theory (item 7.2). The design of the algebra items (e.g., 5.1 and 5.2) issimilar. The MCK test was given to students during the first week of theirmethods course.


Figure 2. Sample questions from the MCK test

Transcripts for these students were also evaluated and ranked forthe number and level of mathematical courses taken. The students wereassigned ordinal rather than cardinal test scores on each test, a consensusranking from 1 (best) through 16 from the individual results of threescorers. Moreover, the 16 collegiate transcripts were examined by thescoring team and ranked comparatively from 1 to 16 based on two compo-nents taken together: students’ grades in mathematics courses and breadth


of advanced mathematical coursework. The students had completed at leasttwo semesters of calculus and one semester of linear algebra, and they hadcompleted between zero and seven advanced undergraduate mathematicscourses. While our MCK test was a rough measure, its results were posi-tively correlated (Spearman’s rho = 0.618) with the results of the transcriptanalysis; furthermore, 12 of the 16 participants had ranks within 4 placeson the two orderings (see Table I). That the MCK test rank correlated withanother measure of mathematics achievement lends it some validity.

For these students, lesson plan assignments from the methods coursewere (after being blinded) reviewed by the third author for depth of mathe-matical content as demonstrated through selection of worthwhile tasks,motivation of content, development, and discourse. (Since students weregiven the objective and since this was just one lesson, neither the goalnor the allocation of time was rated.) On this basis, the lesson plans wereassigned holistic rating of strong (1), good (2), or fair (3).

TABLE I

MCK test rank, transcript rank, and lesson plan rating

MCK 1 2 3 4 5 6.5 6.5 8 9 10 11 12 13 14 15 16

Transcript 1 3 2 7 9 6 4 15 10 13 14 16 5 12 8 11

Rank

Lesson Plan 2 1 ∗ 3 1 1 3 3 2 3 2 2 2 3 3 3

1 = strong; 2 = good; 3 = fair; ∗ = missing.

In addition, the first author, who was at the time unaware of studentMCK or transcript ranks, observed the student teachers during their studentteaching experiences in their last semester of school. Student teacherswere observed on at least two different occasions and post-observationconferences were held to discuss the lessons. The observer took field notesduring the observations and the conferences, and these were consulted infurther analysis. At the time of the observations and their initial analysis,the observer was unaware of the framework, so, unlike the lesson planassignment, whose ratings are based directly on framework elements, itis only in subsequent re-analysis and discussion that these observationswere linked to the framework. Thus, the framework’s connection to theanalysis of the lesson plans is tighter. The next sections of this paper usethe framework in analyzing lessons these teachers planned (but did notteach) and other lessons they taught.

It is worth noting a limitation in this set of data: it is removed frompractice. Lesson plans that contain high-level mathematical tasks, in the


course of teaching, often slip to lower-level mathematical activity or nomathematical activity at all (Henningsen & Stein, 1997). Furthermore, datafrom pre-service teachers, even actual student teaching, only indicate howstudents are reacting to their preparation. Ensor (2001) found that whenAustralian pre-service teachers become in-service teachers, they use theterms they learned in teacher preparation to describe their activity, but maynot be faithful to what they learned and espoused during teacher prepara-tion. Factors in the new job environment are paramount in the transitionto teaching. Nonetheless, lesson plans are windows on teacher thinkingabout teaching, as Grossman (1990) shows in analyzing teachers’ plansfor teaching a particular poem and reflecting on the strengths and weak-ness these reveal about them as teachers. Similarly, observation of studentteaching is at least a window on teaching potential.

RESULTS OF CORRELATIONAL ANALYSIS OFLESSON PLANS

Generally speaking, the top scorers for the MCK and transcript rank alsoproduced strong lesson plans. Likewise, those teachers who scored near thebottom on the MCK and transcript rank produced weaker lesson plans (seeTable I). The lesson plan rating was somewhat more strongly correlatedwith the MCK rank than with the transcript rank (Spearman’s rho was0.436 for the former, 0.301 for the latter), so the MCK test is plausiblya rough guide, relevant to the analysis of the relationship between MCKand planning (column A). The data suggest that the teacher’s MCK, as wehave measured it, does play a role in the quality of the planning process inparticular and perhaps in other teaching processes as well. If so, our MCKtest may serve as a less blunt measure than grade point average or coursecounting. We now explore more deeply the relationship between MCK thelesson plans.

Preparing for Periodicity

The sixteen pre-service teachers were asked to submit a lesson plan onperiodicity with the following two objectives: (a) the students will under-stand (as demonstrated by use in exercises and by verbal explanations) theeffect of a parameter ß on the functions y = sin ßx and y = cos ßx; and(b) given the desired period, students will be able to determine a rule ofa function having that period. By the end of the lesson, students will beexpected to have the general idea about periods of sinusoidal functions,not necessarily expertise because homework and subsequent class content


will reinforce the idea. Teachers were asked to provide sufficient detailfor others to teach the lesson, to consider the motivation for the lesson, topose appropriate questions to guide thinking and to determine if studentsunderstood the concept, and to identify specific examples to use during thelesson. Teachers chose the length of the lesson. In preparing the lesson,teachers could assume that students were drawn from a 10th–12th gradeclass, were comfortable using, graphing, and determining values of the sineand cosine functions and that they understood why the period, in radians,of each function is 2π .

In the following examples, our interpretation is that there is a rangeof quality of planning (column A), manifest in all its elements: goals andobjectives; selection of tasks and representations; motivation of content;development; allocation of time, points, and emphasis; and discourse.Rather than organizing discussion of these lesson plans by order of appear-ance of framework rows, we present the most salient aspects first, andtouch on others only briefly. We also note that this range aligns roughlywith teacher MCK, although other factors probably played a role, too.

In planning motivation for the lesson, cell 3A, teachers showed a rangeof performance, and one that aligned with MCK measures. The plan ofMs. Wong began with warm-up exercises that asked students to graph thefollowing functions over one period:

1. y = sin x2. y = cos x3. y = sin 2πx4. y = cos 2πx

The planned discussion that followed posed the following questions:

(1) How did you approach warm-up #3 & #4? (Did you use yourcalculator?)

(2) What was different about the graph of #3 & #4? What made itdifferent?

(3) If the period of sine and cosine is 2π , what is the period of the functionin #3 & #4?

(4) At what x value did the function complete the sine or cosine graph?

In our judgment, Ms. Wong’s plan provided limited motivation for thetopic of periodicity. We agree with Prestage and Perks (2001) that the useof the graphing calculator as a resource can be motivating, but only insofaras the teacher uses it to “open up the possibilities within the particularquestions” (p. 82), discover possibilities, and use these as the basis ofquestions and activities. We do not see these elements in Ms. Wong’slesson. The lesson went on to have students graph (by hand) several sine


and cosine functions and to determine for what values of x each graphwas 1, 0, and –1. The lesson also asked students to be prepared to discussthe period of each function. It is unclear from the plan if students wouldmerely discuss the value of the period or if they would discuss the meaningof the period. There was never any discussion in the lesson that would leadstudents to be able to do the latter. It appears that if students could obtainthe period procedurally, they would have succeeded in mastering thislesson. To account for this limited (proposed) communication of periodi-city, one might suppose that Ms. Wong has a relatively weak mathematicsbackground (among many other possibilities). Interestingly enough, Ms.Wong ranked fourteenth out of sixteen on the MCK test and twelfth out ofsixteen on the transcript rank.

By contrast, we were impressed with Mr. Wilson’s motivational activityfor periodicity (cell 3A). Mr. Wilson began the (proposed) lesson with astory about a sea-faring community and used the water level in the lock ofa dam as a function of time to exemplify periodicity.

“A boat goes into a chamber that is then sealed and filled with water. The boat reaches thetop and exits. The water is then drained and the process can start over. This process is donewithout ceasing, as there are too many boats for the lock to take breaks.” The followingquestions are then asked about the process:

(1) Where did the water [level] start?(2) Where did the water [level] end?(3) Did anything happen in the middle? What?(4) Could this [cycle] happen again?(5) Would the boat always have to get to the top in the same amount of time?

Mr. Wilson’s planned lesson went on to review the function y = sin xand its period. Then Mr. Wilson asked, “If we were going twice as fast,would the period be the same?” Mr. Wilson connected this concept back tothe dam by telling students to imagine filling and draining the lock twiceas fast. How does the rate at which water is added or drained affect theproblem? The lesson continues with more questions that lead students todiscovering how the period affects the graph and why. In our interpretationMr. Wilson’s proposed lesson motivates periodicity more appropriatelythan Ms. Wong’s, and based on this lesson plan one could conjecturethat Mr. Wilson has a strong mathematics background (supported by adeveloping understanding of teaching periodicity, although an expert couldrefine his questions and consider students’ interactions with the ques-tions and desired learning trajectory in greater depth). This conjectureis supported by Mr. Wilson’s rank results, which were fifth on the MCKand ninth on the transcript rank. We further note that although MCK heresupported ways to use applications to motivate the topic, we believe there


is a range of ways to motivate it (chapter 9 in Ollerton & Watson, 2001,describes five approaches and twenty-seven related strategies), and that inthese too, MCK enhances the possibilities for the teacher.

It is important for students to see how a new concept connects to previ-ously learned concepts (row 4). Part of MCK, as construed broadly abovedrawing on Bransford et al. (2000) and Hiebert & Lefevre (1986), is tosee “the big picture” so mathematics does not appear to be a collection oftopics that make no sense. Teachers with poor mathematical backgroundsmay not understand how some topics relate with each other and so itmay be impossible for them to help students appreciate such relationships.Additionally, teachers with poor mathematics backgrounds may have diffi-culty in selecting appropriate tasks (row 2) to help students make theseconnections.

To address connectedness (cell 4A) and selection of worthwhile tasks(cell 2A) in preparing a good lesson, reconsider the previously mentionedlesson about the lock on the dam. Through questions, Mr. Wilson leadsstudents to discover how a sine function may be a reasonable model of thelevels of water in a dam, and how the period of the function relates to themodel. For example, the following questions are posed:

(1) What does it mean that the water goes in and out twice as fast?(2) If it takes half the time, what can be said of the number of times the

water procedure can occur in the same time?(3) If two boats now complete the cycle in the same time that one

previously could, what is the new period?(4) What might the function look like?

At this point, more time (cell 5A) is spent on the function itself andwhat to do to it to make the period change. This lesson suggests that Mr.Wilson understands the “big picture” (cell 1A) and knows how to selectappropriate tasks to help students achieve this level of understanding.

Interestingly enough, some of the pre-service teachers providedadequate lesson plans and planned some appropriate questions of students(cell 6A), such as Ms. Wong’s series (1)–(4), while not fully demonstratingthorough knowledge of the concept. One such teacher, Ms. Feller, ledstudents to discover how ß affects the period by presenting various sineand cosine functions for students to graph and having students comparethe graphs. However, Ms. Feller never planned to discuss with the classwhat period means. It is not clear from the lesson plan that Ms. Fellerunderstands. Students in this class could leave with a computational under-standing of period, that is, they could know that the period of y = sin 2x isπ , but they might have no idea of what this means. Ms. Feller ranked sixand a half on the MCK and fourth on the transcript rank.


Other teachers appear to have a better understanding of periodicity,but they do not develop this understanding for their students (cell 4A) toour satisfaction. Ms. Adler is one such teacher. Ms. Adler mentions whathappens to a sine function when the absolute value of the parameter ß insin(ßx) is greater than 1, when the absolute value of ß is between 0 and 1,and when the absolute value of ß is equal to 1. Ms. Adler uses mathematicalterms correctly like contraction, expansion and reflection; however, in ourjudgment, the lesson does not provide students with enough opportunity tolearn what Ms. Adler knows. Ms. Adler ranked eleventh on the MCK andfourteenth on the transcript rank.

The framework guided our analysis of the lesson plans, and our inter-pretation of the plans written by the pre-service teachers in this classsuggests that MCK played a role in preparing appropriate lessons. Sothe framework itself appears useful, at least for those who share its valueand find our interpretations of the lesson plans plausible. The elements ofteaching, as defined in the framework, play roles in the planning process,and the quality of teachers’ plans with respect to these elements, bothbased on a holistic rating and closer examination, do appear correlated withteachers’ MCK. Just as we expect MCK to be a factor in lesson prepara-tion (column A), we also expect it to have a prominent role in classroominstruction (column B) in mathematics. The next vignette illustrates how,to some extent, that expectation is disappointed and provides a caution thatstrong MCK does not ensure strong instruction.

Eratosthenes Revisited – An Episode from Student Teaching

VignetteMs. Geary was the highest mathematical achiever in her cohort of studentteachers, attaining the top ranks on the aforementioned MCK test andtranscript analysis. She was also fastidious in planning her lessons. In hergrade 10 and 11 class on Principles of Geometry and Algebra, Ms. Gearybegan her lesson by telling students that today they would learn about dirt.She then asked the students what a sieve was. After some discussion, Ms.Geary produced a sieve and modeled how to use it to separate “big dirt”and “little dirt”.

Ms. Geary then produced an overhead transparency illustrating theinformation contained in Table II, as well as giving printed copies tostudents.

Ms. Geary then circled 1 and wrote that it was not prime. Mick, astudent, objected, offering, “A prime number is something that can bedivided only by itself”. Ms. Geary corrected him noting that a prime “hastwo factors, one and itself”.


TABLE II

A hundred chart

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Ms. Geary asked what multiples of 2 would be prime? After the classcalled out some answers, Ms. Geary answered the question saying that nomultiples of 2 would be prime, “2, 4, 6, 8, and so on.” The class correctedher, pointing out that 2 is prime. Ms. Geary proceeded to cross off 4, 6,8, . . ., 100 on the chart and explained that the sieve was at work. Thecomposites sifted out were the big dirt, Ms. G. asserted. Some discussionensued about whether crossing off the composites was like sifting out bigdirt or small dirt.

Ms. Geary, working with the class, proceeded to cross off multiples of3, 5, and 7. At this point, the chart looked like Figure 3.

Figure 3. The sieve of Eratosthenes.

Ms. Geary proceeded to cross off the multiples of 11. After Patricia,a student, commented that they were all on a diagonal and Ms. Gearypraised the observation, the class became impatient, claiming that therewas nothing more to cross out. Ms. Geary responded, “By this point mostshould be out”. She then proceeded to cross out multiples of 13. Again


the class protested that these were all already crossed out. Ms. Geary said:“You should have got most of them. Before proceeding to 17, a forlornMs. Geary asked the class observer whether she could stop. The observerconsidered saying “I think you should discuss that with the class”, butinstead told her she could stop.

At Ms. Geary’s request, Lindsay, another student, read the list of primesless than 100 aloud. Ms. Geary assigned homework, and encouragedstudents to play with the sieve in the remaining time. She closed by sayingthat as they looked back on their mathematical experiences this year, thestudents might not remember all the details, but they would remember bigdirt and little dirt. She dismissed them, telling them to “Have a wonderfulday”.

In the post-observation conference, the observer probed Ms. Geary onwhy 1 is not prime She said, “Because some theorems won’t work.” Askedif she thought her students could relate to this explanation, she said no. Inthe conference, Ms. Geary and the observer also discussed that she couldhave stopped crossing off multiples of primes when she reached 11 sinceshe had surpassed 10, the square root of 100. Ms. Geary then recalled thatthe square root came into play and said she had once known that. Whenpushed to say why, she was able to reason that any composite number lessthan 100 which had a prime factor greater than 10 would also have anotherone less than 10.

Analysis

Our analysis of this student teaching observation begins with a caution,from Brown & McIntyre (1993). The teacher and students have a moreextended perspective on how the teacher teaches: “And teacher educatorsare exposed to only a narrow window in the teacher’s life at a time whenthe demands, novelty and artificiality in the situation may well producebehaviours which are uncharacteristic of the individual in the longer term”(p. 23). So it would be both arrogant and foolhardy to use these lessonsto identify what the teachers do not do but should. We submit, though,that they are worth investigating providing that we do not over generalizefrom our interpretations of these observations and take care to focus on thelessons rather than the teachers.

The task in this lesson was taken from materials prepared by the county,but Ms. Geary enhanced it with the physical model and the big dirt andlittle dirt idea, clarifying the sieve metaphor for students. In this respect, webelieve she showed pedagogical wisdom in her preparation for discourse(cell 6A). This lesson had the potential to be memorable because Ms.


Geary gave it a clear beginning, middle, and end, and she connected aphysical and mathematical model.

In reviewing the instruction process of Ms. Geary’s discourse (cell6B), we have several concerns. She omitted an explanation of why 1 isnot prime, so students had to just accept it as a fact that, if they wereperceptive enough, they could see followed from a definition; they werenot privy to Ms. Geary’s mathematical knowledge that that definition wasa deliberate choice. She was inaccurate in crossing off 2 as composite.She was inaccurate in saying most (not all) composites were gone andpersisting in searching with 11 and 13. Finally, her teaching was inflexiblein not listening to the class when they said she had gone far enough andin not letting them discuss the issue as a class. In Ms. Geary’s reflectionson that class’ discourse (cell 6D), one does see evidence of her MCK.Her statement that one chooses not to make 1 prime so that theorems willhold is perceptive. Rather than saying that primes have 2 distinct factors bydefinition, she recognizes that the definition is chosen that way so one neednot say 1 is prime. She is correct that some theorems about primes will nothold for 1. When she reflects on why she could stop at 10 in checkingfactors, she is able to construct an explanation.

This case raises some issues about the relationship between MCK anddiscourse, row 6 in the framework. Is it reasonable to expect that a studentteacher with superior MCK would be able to explain why 1 is not prime,even to a high-school class? Could she be expected to know when shecould stop the search for composites, to heed the class when they told herso, or to let the question be discussed? The first author thought so, and wassurprised to learn that Ms. Geary had outperformed her peers in measuresof MCK.

One possible explanation of Ms. Geary’s difficulties in discoursedespite apparently strong MCK, is that her MCK was built lookingforward, rather than backward. Henry Pollak (personal communication,4/1999), past-president of the Mathematical Association of America,claims that prospective teachers ought to have courses in real analysisor abstract algebra separate from mathematics majors so that they couldlearn how the material relates to the high-school curriculum, not to thenext college course. Similarly, Ma (1999) holds that MCK will not makebetter teachers unless it is explicitly used to view the earlier content froman advanced perspective. Perhaps a simpler theory is that college mathe-matics courses need to (but usually do not) give teachers opportunities tounpack their understanding (Ball & Wilson, 1990). Had Ms. Geary under-taken such a course or perspective, perhaps she would have produced theprime factorization theorem as an example of a theorem that is undermined


by letting 1 be prime (would 21 factor to 7 × 3, 7 × 3 × 1, 7 × 3× 1 × 1, . . .?). Given such preparation, perhaps Ms. Geary would alsohave applied her knowledge about primes and factors to determine howfar to check in the sieve. Recalling that the definition we use of contentknowledge includes organizing knowledge in ways that facilitate applica-tion, Ms. Geary’s gap between her formal knowledge and that applied inher teaching is a limitation in her content knowledge. This interpretationsuggests that further research on content knowledge needs to take seriouslyZeichner & Schulte’s (2001) contention that research needs to attend tothe character and quality of the preparation, not just to which courses wereincluded. This interpretation also suggests that mathematical preparationof teachers should bridge the gap Ms. Geary exhibits and that assessmentsof MCK should check for it.

Another possible explanation of Ms. Geary’s case relates to affectiveconsiderations. Ms. Geary’s inexperience teaching and her anxiety aboutbeing observed may have constrained her teaching. She simply may nothave been comfortable enough in her teaching role to let the class discussan issue of which she herself was unsure or which she had not anticipated.Ms. Geary certainly knew that 2 is prime, but the error in crossing it offcould have been the result of being nervous about being observed.

Dodes (1953) suggests that lessons given by beginning teachers oftenhave extremely strong motivation and beginnings, but “when he has finallycreated an edifice of motivation, he attempts to stuff the content of thelesson into its attic” (p. 307). Moreover, a hallmark of experienced teachersis that they anticipate areas of difficulty in their planning. If so, it is alsopossible that the flaws in Ms. Geary’s lesson resulted from her inexperienceas a teacher.

Ms. Geary’s case and our analysis of it are similar to a case study of aprospective primary teacher by Rowland et al. (2000). Despite a strongmathematics background as measured by an audit they developed, thisprospective teacher initially struggled with teaching. This student becamean effective teacher because of strong MCK together with her willing-ness to explore the foundations of mathematical concepts that would betaught, her diagnosis of what students already know, her reflectivity andher increasing confidence in her ability to do and teach mathematics.Ms. Geary’s MCK may yet grow to include relating it to the secondarycurriculum, and she may grow in the other areas as well.

As noted in our introductory caution after Brown & McIntyre (1993),this episode represents a thin and potentially unrepresentative slice of Ms.Geary’s teaching. Brown and McIntyre contend further that many condi-tions play a role in teaching in practice. Maybe teacher factors were at


work (Ms. Geary may have had a cold – and as the observer recalls, shedid – or otherwise been having a bad day). Or perhaps time constraintsor the materials provided by the county were limiting. So our suggestionsabove to explain Ms. Geary’s teaching in this lesson should be taken assuch. The case of Ms. Geary suggests that MCK alone may not ensureexemplary teaching. Our analysis of the next case suggests that a lack ofMCK narrows the scope of what is possible for teaching.

Introducing Circles and Chords – Another Episode from Student Teaching

VignetteMs. Lehava was intermediate in mathematical achievement in her cohort ofstudent teachers, ranking eighth of 16 students on the test of their MCK. Inthe observer’s opinion, her observed lessons challenged students to thinkabout the mathematically big ideas. For example, in a geometry lesson toaccelerated 8th graders, Ms. Lehava began by swinging a stone attachedto a rope, and then asking, “What do you think we’re going to do?” Fromthis basis, Ms. Lehava was able to elicit from students the term radius andwhat role it played in construction of a circle. Students observed that radiiof equal length give congruent circles, which they later used when Ms.Lehava challenged them to use compasses to copy a given circle. She alsochallenged students to extend their thinking by spinning the rock throughspace instead of constraining it to a plane. They were able to guess thislocus described a sphere. Only then did she proceed to identify and namechords, arcs, radii, and other terms related to circles. In introducing chords,Ms. Lehava asked the class how many chords there were in a circle, andthey responded “infinity”.

Subsequently the class discussed an exercise asking them to name achord on the diagram reproduced below.

Figure 4. Name a chord.


Ms. Lehava was able to use this question to get many students involved.After about 10 chords had been named, the class agreed that there weremore, and Ms. Lehava proceeded with the lesson.

One question that arose in the post-observation conference was whetherMs. Lehava had considered asking the class how many distinct chordscould be named using the points labeled on the circle. She replied thatshe had not. When the observer asked if that might not be a good previewof 8C2, the number of combinations of 8 things taken 2 at a time, Ms.Lehava was unfamiliar with the notation, nor could she see at a glancehow to calculate the number of distinct chords. Although her mathe-matics preparation was relatively strong, in transitions between acceleratedprograms in several countries and schools, Ms. Lehava had missed thistopic.

Analysis

In Ms. Lehava’s lesson, we see several strengths. She used a physicalmodel to engage the students and to extend beyond 2 dimensions. Themodel also underscored the centrality of radius (as opposed to diameter)in circle constructions. Ms. Lehava also pushed students to contemplatethe infinite. As a lesson in student teaching, we see this one as relativelystrong, owing in part to Ms. Lehava’s enthusiasm for mathematics and herattending to and modeling the big ideas, not just the skills. We commendher prepared (column A) and implemented (column B) selection of tasksand representations (element 2).

A less positive aspect of this lesson was that Ms. Lehava did notperceive what we see as a teachable moment. The class was naming chordafter chord with no end in sight. How could they know if they had themall? If they continued, how long might they go on (if Ms. Lehava letthem)? These questions were waiting for Ms. Lehava to raise them (sinceno student did). Why did they not occur to her?

The reader may object that the idea occurred to the observer and not toMs. Lehava by pure luck, but as Pasteur noted “Luck favors the preparedmind”. Just as most of us do not dream of atoms dancing in benzene rings,the (smaller) inspiration to ask about the number of distinct chords onecould name for this circle did not occur to Ms. Lehava. The followingparagraphs explore suggestions of why not and their implications for MCKin column B, instruction, intersecting several teaching elements.

This case suggests an hypothesis about MCK and discourse, namelythat MCK is a factor in recognizing and seizing teachable moments. Haller(1997) used differences in teacher MCK to account for which teacherscapitalized on opportunities to connect student work on probability with


issues of representing fractions, decimals, and percentages. Here, too, hadMs. Lehava had some knowledge of combinatorics, the question of howmany chords would have been familiar in its structure, so the connec-tion might have occurred to her as a teachable one (row 4: Development:connectivity and sequencing), and become part of the classroom discourse(row 6). And had she known that questions of counting are a big idea inmathematics, once the question arose she might have made the judgment toallocate time to it (row 5) and had the confidence to select the task (row 2).

Another point that bears consideration in light of Ms. Lehava’s case isthat content knowledge relative to teaching may be specific to a particularcontent area (Ponte et al., 1999, make the same point about teacher beliefs).Ms. Lehava’s MCK was not poor, and in areas where it was strong, such asunderstanding circles, her lesson reflected extensions into multiple dimen-sions and models that helped students to focus on what was mathematicallycentral. Yet a serious gap in Ms. Lehava’s combinatorics background (astrand the MCK assessment did not probe) inhibited her recognition of ateachable moment. Perhaps a teacher with general mathematical experi-ence and confidence and teaching experience would have thought of thequestion anyway, and felt comfortable pursuing it with the class evenwithout knowing how to find the answer; such a decision would alsodepend on the teacher’s syntactic knowledge, whether she knew how topursue such a mathematical inquiry. But, especially for a beginning teacherlike Ms. Lehava, the specific knowledge was critical for the opportunity tobe realized. Recall too that content knowledge in the sense of Bransford etal. (2000) includes not only the fact (8C2) but also a conceptual framework(how it fits into combinatorics) and application (recognizing and solvingcombinatorics problems or knowing how to inquire into the mathemati-cally unknown), and all these aspects of MCK play a role in perceivingand seizing the teachable moment.

In this case, too, MCK alone might not be sufficient for seeing orpursuing the question on counting the chords. Several other factors favoredthe observer over Ms. Lehava. First, as with Ms. Geary the affective factorsof being watched and the pressure for time might inhibit creative flashes.Also, as one without teaching experience, Ms. Lehava may not have beenused to looking for unplanned teachable moments. Even if the question hadoccurred to her, a lack of curricular knowledge that such counting prob-lems would be important to these students or the student teacher’s value ofand security in staying with the plan (as described by Brown & McIntyre,1993) might have led Ms. Lehava to decide not to pursue this explora-tion. In fact, there would be cases where an exemplary teacher might seethis teachable moment, and deem it inappropriate (e.g., because it takes


time away from geometry) to pursue at that time. Following Dawson’s(1999b) cautionary words against teacher educators attempting to fix whatis wrong with teachers, our point here is not to judge that Ms. Lehavamade an incorrect decision in not enacting the possibility of this teach-able moment, but that she didn’t make a decision at all; her lack of MCKnarrowed the scope of what was possible. Taken together, the cases of Ms.Geary and Ms. Lehava support Monk’s (1994) claim that “a good grasp ofone’s subject area is a necessary but not sufficient condition for effectiveteaching” (p. 142) and Dawson’s (1999a) that “improved mathematicalknowledge is a necessary but far from sufficient condition to foster changein the teaching and learning of mathematics” (p. 8).

RESEARCH DIRECTIONS

A case studied by Ball (1991) serves to illustrate a vexing facet ofresearch on the role of MCK in teaching – the “implications issue”. Balldescribes an elementary teacher who in response to an interview ques-tion explains her approach to teaching multi-digit multiplication using thespecific example 123 × 645. An intermediate product appearing in thestandard algorithm is 123 × 4 = 492, which, procedurally speaking, will beshifted over one place. In teaching, the teacher does not explain to students,though, in terms of the place value concept, that the shifting occurs becausethe multiplication by 4 really represents multiplication by the 40 (or fourtens) in 645 with a result of 4,920 (or 492 tens).

A researcher, with or without the benefit of our framework, whowitnesses this teacher’s presentation (situated at 6B in the framework)would be unable to make a conclusion about the teacher’s MCK withoutother supporting evidence. Perhaps the teacher only knows the multipli-cation algorithm procedurally and does not understand conceptually whyit works. If that were true, she certainly would not be able to explain theconcept to her students. However, the teacher described by Ball does knowwhy the procedure works, as she explains in her full response to Ball’sinterview question, but she purposely does not delve into discussion aboutit because “her concern is to get her students to be able to perform thecomputation” (Ball, 1991, p. 22).

This scenario is representative of the difficulty in finding necessary andsufficient conditions between evidence and conclusions about a teacher’sstrength or weakness in mathematics. For instance, if a teacher makessignificant errors, beyond “careless” errors, in the processes of teaching,a researcher (and an aware student) can conclude that the teacher is notstrong mathematically, at least in the content area being taught. However,


an error-free lesson plan or examination is not by itself sufficient toconclude that the teacher has a strong knowledge base of mathematics.Here the framework can help the researcher, by guiding her to explore thisconjecture with data from other cells in the framework array. Perhaps it islikely that a teacher who exhibits fear of mathematics during the instruc-tion or reflection processes is not very strong mathematically. However,knowledge that a teacher has a solid mathematical foundation does notenable a researcher to conclude that the teacher will exhibit enthusiasmor strong motivation during instruction, nor can the researcher make theconverse assumption. Lester & Wiliam (2000) discuss this very diffi-culty on the nature of making justifiable conclusions from evidence inmathematics education research. Indeed, they assert that “establishingthe strength of the link between evidence and a claim often is based onprobability” (p. 133).

The research pursued here suggests a need for studies that attend tothe dimension of the framework not explicitly in our figure. Although weintended to focus on MCK, our analyses repeatedly invoked PCK andaffective considerations to explain surprising results. Studies that gathersuch data in an attempt to characterize teaching in terms of all facets ofthe teacher are desirable. Such studies could provide a better explanationof how what a teacher brings to teaching affects what occurs. Such studiesmay take the form of developing instruments to assess not only MCK, butalso coordinating these with measures of PCK and inventories of teacherbeliefs, then proceeding by statistical analysis of their relationship withteaching performance. Or such studies could be more qualitative in nature,with several teachers studied in depth with respect to all these aspects andseeing what patterns of teaching emerge.

We submit that such research has important implications for teachereducation. There is much feeling about what teachers ought to know andwhere their preparation should focus. Working with our proposed frame-work, researchers could gather evidence that bears on such questions.Especially if such studies attend to the nature of the preparation, whatopportunities for learning are in the courses rather than just counts ofcourses, then we may learn more about what knowledge and beliefs variousprograms promote in their teachers and in turn how those teachers teach.We do not expect some grand positivist vision of finding the best teacherpreparation program, but we do think the identification of models thatprepare teachers well can become more based in data.

The framework itself may be a useful tool for teacher education, too.Programs may wish to see if the experiences they provide give studentsopportunities to learn and reflect on every cell; this motivation, secondary


for us, was the driving factor in Artzt & Armour Thomas’ (1999) devel-opment of a similar framework. Teacher education programs may alsowish to see if they assess students on the contents of every cell. Outsideevaluators of teacher education programs might use the framework in asimilar way. Teacher education programs can also use the framework as aprogram design tool or a means to articulate their design. We also acknowl-edge that some programs may forego addressing every cell in a deliberatedecision to focus in depth on the ones most valued by the program. Thusthe framework can serve as a way to review and improve teacher educationprograms.

The framework presented here can guide research in two ways. First,the framework is an organizer for thought about the many facets of thequestion of how content knowledge plays out in teaching. Second, thespecific cells of the framework array are delimiters that offer useful bound-aries for the scope of investigations. Within this framework, researcherscan engage in focused inquiry within a cell, or broader study, across a rowor down a column. Research may reveal that the role of content knowledgeis more pronounced in some cells than in others. Or a review of researchmay show that some cells, rows, or columns, have been studied in greaterdepth than others, indicating gaps research needs to target. In these ways,the framework guides research that engages the challenging and importantquestion of the relationship of what mathematics teachers know to howthey teach.

ACKNOWLEDGEMENTS

We thank James T. Fey and Anna O. Graeber for their contributions to thedevelopment of the ideas herein and Martha H. Bigelow, Joan E. Hughes,and Julie S. Kalnin for their painstaking editing of several drafts. We alsothank the editor and reviewers of JMTE for their detailed suggestions.

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JEREMY A. KAHAN

University of MinnesotaTwin Cities(Author for correspondence:Department of Curriculum and Instruction230·C Peik HallUniversity of MinnesotaMinneapolis MN 55455E-mail: [email protected])

DUANE A. COOPER

Morehouse College

KIMBERLY A. BETHEA

University of MarylandCollege Park

JENNIFER E. SZYDLIK, STEPHEN D. SZYDLIK and STEVEN R. BENSON

EXPLORING CHANGES IN PRE-SERVICE ELEMENTARYTEACHERS’ MATHEMATICAL BELIEFS

ABSTRACT. Research literature (e.g., Thompson, 1992) suggests that teachers’ beliefsabout the nature of mathematics provide a strong indicator of their future teachingpractices. Moreover, current reform efforts (e.g., NCTM, 2000) ask teachers to lead mathe-matical explorations that allow their own students to construct mathematics. Understandingprospective teachers’ mathematical beliefs and the circumstances under which these beliefsmight be changed is therefore critical to teacher educators. In this paper we describethe culture of a mathematics content course for prospective elementary teachers that isdesigned to provide participants with authentic mathematical experiences and to fosterautonomous mathematical behaviors. Using both survey and interview data, we exploredparticipants’ beliefs about the nature of mathematical behavior both at the commencementand at the completion of the course. We argue that the participants’ beliefs became moresupportive of autonomous behaviors during the course. We report that students attributedchanges in beliefs to specific classroom social norms and sociomathematical norms thatincluded facets of work on “big” problems with underlying structures, a broadening in theacceptable methods of solving problems, a focus on explanation and argument, and theopportunity to generate mathematics as a classroom community.

KEY WORDS: autonomy, elementary education, mathematical beliefs, problem solving,sociomathematical norms

INTRODUCTION AND BACKGROUND

Hersh (1986) writes, “One’s conception of what mathematics is affects one’s conceptionof how it should be presented. One’s manner of presenting is an indication of what onebelieves is most essential in it . . . The issue, then, is not, What is the best way to teach?but, What is mathematics really all about?” (p. 13).

Mathematics is a human activity; problems are encountered, interpretedand refined, assumptions are made and clarified, and arguments arecreated. Halmos (1985) explains, “When you try to prove a theorem, youdon’t just list hypotheses and start to reason. What you do is trial anderror, experimentation, guesswork” (p. 321). On the other hand, whileinsight comes from exploration, conviction in mathematics arises throughdeduction and consistency. In its rigorous form, mathematics is basedon fundamental assumptions and validity arises from those assumptionsthrough proof. Tall (1992) writes, “Advanced mathematical thinking –


254 JENNIFER E. SZYDLIK ET AL.

as evidenced by publications in research journals – is characterized bytwo important components; precise mathematical definitions (includingthe statement of axioms in axiomatic theories) and logical deductionsbased upon them” (p. 495). Reason, logic, and internal consistency aresources of conviction for the mathematical community. Moreover, the actof producing new mathematics is inherently creative; problem refinement,exploration, and arguments come from within the community, not froman external authority. Thus, arguably, an authentic mathematical experi-ence consists of exploration, sense making, and creating arguments usingdefinitions and logical deduction.

The literature suggests that prospective elementary teachers see mathe-matics as an authoritarian discipline, and that they believe that doingmathematics means applying memorized formulas and procedures to text-book exercises (Carpenter, Linquist, Mattews & Silver, 1983; Ball, 1990;Shuck, 1996). In fact, traditional mathematics classroom norms appearto foster this belief. They do not provide authentic mathematical experi-ences or promote autonomy; instead, they encourage students to rely onan external authority (their instructors or textbooks) for the determinationof mathematical validity (Schoenfeld, 1989; Frid & Olson, 1993). Green(1971) asserts that a classroom emphasis on rules and memorized proce-dures would be better called indoctrination, and he distinguishes teachingas the process of helping an individual to know based on rational evidence.Cooney, Shealy & Arvold (1998) write, “An indoctrinated view of mathe-matics minimizes the impact of rationality in favor of memorization. Thisview constitutes the antithesis of considering mathematics as a humanendeavor” (p. 311).

That elementary education students hold beliefs inconsistent withautonomous behavior is particularly worrisome given that teachers’ beliefsabout the nature of mathematical behavior influence the way they conveymaterial to their students (for a synthesis of this literature see Thompson,1992). For example, Thompson (1984) found that a teacher who viewsmathematics as a collection of facts and rules to be memorized andapplied is more likely to teach in a prescriptive manner, emphasizing rulesand procedures conveyed by the teacher. On the other hand, a teacherwho holds a problem-solving view of mathematics is more likely toemploy activities that allow students to construct mathematical ideas forthemselves. If we are to educate future teachers who are able to leadmathematical explorations and to allow their own students to constructmathematics, that is, if we are to help them teach in a way consistentwith current calls for reform (NCTM, 2000), we must help them seemathematics as a discipline that makes sense.

MATHEMATICAL BELIEFS 255

Unfortunately, pre-existing beliefs about teaching, learning, and subjectmatter tend to be tenacious and resistant to change (Lerman, 1987; Brown,Cooney & Jones, 1990; Pajares, 1992; Foss & Kleinsasser, 1996). Kagan(1992) observed that pre-service teachers tend to leave their universityprograms holding primarily the same beliefs with which they arrived, withmany of their initial biases having in fact grown stronger throughout theirprogram. She asserted “If a program is to promote growth among novices,it must require them to make their pre-existing personal beliefs explicit;it must challenge the adequacy of those beliefs; and it must give novicesextended opportunities to examine, elaborate, and integrate new informa-tion into their existing belief systems. In short, pre-service teachers needopportunities to make knowledge their own” (p. 77).

Yackel & Cobb (1996) have suggested that cultural and social processesare integral to mathematical activity and have posited that the culture ofthe mathematics classroom is central to the development of mathematicaldisposition among students and to bringing about change in mathematicalbeliefs. This perspective of the classroom as a culture is described byBauersfeld (1993):

“[T]he understanding of learning and teaching mathematics . . . support[s] a model ofparticipating in a culture rather than a model of transmitting knowledge. Participating in theprocesses of a mathematics classroom is participating in a culture of using mathematics,or better: a culture of mathematizing as a practice. The many skills, which an observer canidentify and will take as the main performance of a culture, form the procedural surfaceonly. These are the bricks of the building, but the design for the house of mathematizingis processed on another level. As it is with cultures, the core of what is learned throughparticipation is when to do what and how to do it” (p. 4).

Yackel and Cobb propose that within this culture there is a reflexiverelationship between beliefs and classroom norms: the beliefs that studentsbring to the classroom will interact with the agenda brought by the teacher,and together the students and teacher will negotiate norms and taken-as-shared meanings. The student beliefs will influence the classroom normsand those norms, in turn, will influence the beliefs of students.

Sociomathematical norms are those that provide access to mathematicaldistinctions. For example, the expectation that students respond to eachother’s ideas is a social norm. What counts as a different mathematicalsolution or argument, a relevant mathematical comment, and an elegantsolution are sociomathematical norms (Yackel & Cobb, 1996). Similarly,the level of rigor expected in justifying one’s claim and the acceptedform of an argument are sociomathematical norms. These norms informstudents when it is appropriate to contribute to the classroom discourse(Is their idea different? Is it relevant? Is it more elegant or efficient than


what has been previously shared?), and how to make it (Is the contributionvalid? Rigorous? Convincing? In an accepted form?).

In this work, we describe a classroom for elementary education studentsin which participation in the classroom culture meant engaging in authenticmathematical behaviors (in the sense we described previously), and weexplore participants’ beliefs about the nature of mathematical behavior,both at the commencement and at the completion of the course, and reporthow students attribute changes in beliefs to specific classroom social normsand sociomathematical norms. By autonomous mathematical behavior, werefer to that behavior that involves sense-making rather than memorizationor appeals to authority. Within the class community, we focus on sociallynegotiated meaning and on promoting community autonomy (as distinctfrom reliance on an outside authority) rather than on the autonomy of indi-viduals. In fact we assume “. . . the development of individuals’ reasoningand sense-making processes cannot be separated from their participationin the interactive constitution of mathematical meanings” (Yackel & Cobb,1996).

The classroom culture we attempt to nurture was inspired in part byclassrooms described by Lampert (1988); Wilcox, Schram, Lappen &Lanier (1991); Civil (1993). These educators were also concerned withcreating classroom communities in which the mathematical behaviors ofexploring, conjecturing, and reasoning are the norm. All attempted tochallenge authoritarian assumptions held by the students about the natureof mathematical behavior and to convince them that mathematics madesense. As Lampert (1988) writes, “I assumed that changing students’ideas about what it means to know and do mathematics was a matterof immersing them in a social situation that worked according to rulesdifferent from those that ordinarily pertain to classrooms, and then respect-fully challenging their assumptions about what knowing mathematicsentails” (p. 470). Like Lampert, we hoped that the culture of such a classwould nurture student beliefs and would make them powerful practitionersof mathematics. Furthermore, we hoped to attribute changes in beliefs tospecific norms.

Our effort focused on establishing two primary mathematical beliefsthat support autonomous behavior: 1) mathematics is a logical andconsistent discipline as opposed to a collection of facts, and therefore2) mathematics is something that can be figured out as opposed to some-thing that must be handed down by an authority. We attempt to documentand make sense of changes in student beliefs about these and relatednotions in the context of the social and sociomathematical norms thatmight influence (and be influenced by) these beliefs. We use the term


norms broadly to encompass the expected ways of participating in theclassroom culture.

The remainder of the paper is organized in five sections. In the first,we provide a description of the culture of the mathematics classrooms inwhich our intervention took place. As part of this description, we attemptto detail explicitly the social and sociomathematical norms of that culturevia quotes from transcribed videotape. In the second section, we describethe methodology of this study. In the third section, we describe partici-pant responses, at the start of the course, to a questionnaire and structuredinterviews about the nature of mathematical behavior and autonomy. Inthe fourth section, we explore changes in students’ beliefs based on thefinal interview data, and we report students’ attributions of changes intheir beliefs, focusing on both social and sociomathematical norms ofthe classroom. In the final section, we reflect on the study and provideconcluding remarks.

THE NUMBER SYSTEMS CLASSROOM

Overview

Number Systems is the first of three required mathematics courses forelementary education majors at our university (a public, comprehensive,undergraduate institution in the United States). The content of the courseis typical of many first mathematics courses for prospective elementaryteachers. It focuses on sets, whole number and integer operations, modelsfor operations, algorithms, rational numbers and their operations, logicand number theory. The class meets for an hour three times each weekfor 14 consecutive weeks. The participants in the study were 93 studentsenrolled in three sections of the Number Systems course. Each of thethree sections consisted of approximately 30 students. Typically, thestudents had completed three years of high school mathematics preparationincluding two years of algebra and a year of geometry. The first author ofthis article was the instructor for the three sections. She is a mathematicianand a mathematics educator.

The course was generated almost entirely by the conversations thatarose out of community work on a set of demanding problems, each ofwhich had an underlying mathematical structure that formed a part of thecourse content. Furthermore, the problems generally allowed for a varietyof problem solving strategies. As a typical example, consider the (perhapsfamiliar) Locker Problem:


The lockers at Martin Luther King Middle School are numbered from 1 to 100. Onemorning a teacher opens all of the lockers. Then another teacher closes every second locker(that is, those numbered 2, 4, 6, . . .). Then the first teacher changes every third locker(numbers 3, 6, 9, . . .): if it is open, she closes it and if it is closed, she opens it. Then thesecond teacher changes every fourth locker, and so on. At the last stage, one of the teacherschanges every 100th locker. Which lockers are open at the end of all this and why?

The open lockers are those showing perfect square numbers. Why?Each locker is changed once for each factor its number has, and the perfectsquares are exactly those numbers with an odd number of factors (thesquare root factor will have no “partner”). Hence they will be the onlyopen lockers. This mathematical structure can be discovered by collectingdata, solving a smaller version of the problem, considering several specificcases, or by logical considerations.

During a typical class meeting, students worked for 20 to 30 minutes ona problem, like the one above, in small groups of three or four students. Theclass then convened in a large semicircle for a discussion of their findings,strategies, solutions and arguments. In these discussions, the instructoremphasized the necessity of mathematical justification; complete solu-tions required logical arguments. However, the class was designated as themathematical authority. The instructor declined to give the final word onthe correctness or the completeness of any solution and there was no text.

In both small and large group discussions, the instructor saw herprimary roles as that of a motivator, scribe, challenger and guide. Inthese roles, she often engaged consciously in the following behaviors:intent listening, feigned (and sometimes real) confusion, skepticism andsilence. The goal was for the essential mathematics and underlying struc-ture of each problem to be revealed, but by the students, not the instructor.We caution that this did not mean that the instructor contributed littleto the discourse or withheld information needed to solve a problem. AsCobb, Yackel & Wood (1992) argue “[t]he conclusion that teachers shouldnot attempt to influence students’ constructive efforts seems indefensible,given our contention that mathematics can be viewed as social practiceor a community project” (pp. 27–28). When a student made a conjecturethat the instructor could not immediately evaluate, she acted as one of thecommunity, joining in the attempt to argue or create counter-examples. Asstudents described mathematical ideas, the instructor gave those ideas thestandard names and assigned notation. There were a few occasions whenthe instructor suggested a known (to her) approach or organized the dataon the board in such a way as to reveal structure or suggest an argument,and, on three occasions, the instructor presented some mathematics (e.g.,a proof that the square root of two is irrational). However the spirit of theclass was consistently one of community inquiry.


Though, generally, the class had discovered a solution to a courseproblem by the end of class, this was not always the case, and frequentlythere remained further mathematics to be uncovered. On six occasionsthroughout the semester, students were asked to produce written reports ontheir problem-solving efforts. These assignments provided opportunitiesfor further reflection and discussion. In the reports, students were requiredto focus their mathematical thinking by describing the problem, discussingthe strategies they used to work on the problem (including those that leddirectly to a solution and those that did not), providing a solution, and,finally, arguing that their solution was complete and valid. In some casesthese reports were produced as group projects.

Classroom Norms

We structured the course with the intent of establishing social and socio-mathematical norms that might foster autonomy. The norms identified(based on a priori design of the course, the video tape, and reflections ofthe instructor) within the classroom culture included the following:

Social Norms

1. The environment of the classroom is informal and respectful. Studentsare not afraid to contribute their ideas.

2. Doing mathematics is a community effort.3. In a whole class discussion, students share ideas with each other as

opposed to just the instructor. Students respond directly to each other’sideas.

4. Solutions and arguments come primarily from the students and not theinstructor.

5. The role of the instructor is to guide the discussion and to provideencouragement.

Sociomathematical Norms

6. The content of the course is generated by big problems that haveunderlying structures. Doing mathematics means working to uncoverand describe the structure.

7. A variety of solution strategies (including trial and error, physicalmodeling, data collection, pattern seeking, and consideration of aparticular example), and arguments (including algebra, logic, exhaus-tion, and revealing diagram) are valued. An argument or solution isdifferent if it uses a different strategy, reveals the structure in a differentway, or reveals a different aspect of the structure.


8. Complete (if informal) deductive argument is required for all mathe-matical claims, and all claims are subject to scrutiny.

9. The inquiry does not end until the given problem is solved andunderstood. To understand a problem is to see the underlying structure.

10. Elegant solutions are those that most directly reveal the underlyingstructure thus providing a powerful explanation for an observed patternor solution.

11. Students are expected to reflect on the entire problem-solving process,using precise and careful language, often in writing.

Students may have experienced some of these norms, such as the socialnorm of an informal but respectful classroom, in other courses. Many ofthe norms, however, were foreign to the students, and negotiating them inthe classroom culture initially caused them discomfort. Establishing andmaintaining the norms required a conscious effort and consistent rein-forcement on the part of the instructor. In fact, though not every classactivity provided an opportunity to reinforce every identified norm, thevery structure of the course continually supported these norms.

We illustrate the classroom culture with two episodes from a sectionof a Number Systems class. Although the classes were not observedon a regular basis, this typical session was videotaped about halfwaythrough the semester. In this particular period, the “big problem” wasto find the number less than 1000 that has the most factors. Underlyingmathematics included prime factorization, the fundamental theorem ofarithmetic, theorems about numbers of factors, and rules of exponents. Theepisode highlights the classroom culture and the ways the instructor andstudents reinforced the norms.

Episode 1: [All the students have formed a large semi-circle opening toward a black board.The instructor is in front of the board with chalk in hand. Amy, Sarah, and Cari are workingtogether in an attempt to argue that (xn)m = xnm. They have asked the instructor to write(xn)m = (x x x x . . . x) (x x x x . . . x) . . . (x x x x . . . x). Underneath each group of ‘x’sthey have had her write “n of these” and under the entire right hand expression they haverequested that she write “m of these ‘n’s.” John and others expected to see n ‘x’s followedby m ‘x’s.]

Instructor: “Is this clear?”John: “Nope.”Instructor: “What’s not clear John?”John: “What’s not clear to me is that you’re taking the n . . . The first part, like see thatlittle part right there, that’s clear. The second part is not clear because it’s not n. That’sm.”Amy [and others]: “That’s not m.”John: “It should be.”Tara and Dawn [agreeing with John]: “Yes. Yes”Sarah: “It doesn’t matter what it is.”


Cari: “It’s representing the same thing.”John: “No. If they’re different numbers, then they are going to be different . . . youcan’t call that just anything. I don’t think you can call it n.”Sarah: “Another way of writing ‘to the n power’ is whatever number of ‘x’s.”Amy: “Would you rather see it like using numbers?”John: “Yeah, I would. You can’t call a 3 a 2.”Amy, Sarah, Cari: “No. No. No.”Sarah: “Those ‘x’s from the brackets, its representing n ‘x’s.Kaila: “But aren’t you saying n and m are the same thing here?”[Several students are talking at once. The instructor interrupts the confusion to call onTina who has her hand up.]Tina [siding with John]: “I think that we had a miscommunication in our group becauseI don’t agree with that [indicating the board]. What I was trying to say is that if youmultiply those two exponents, the second group would be ‘m of these.’ ”[Others agree][Cari wants to propose a different notation. She asks that the instructor write (xn)m =(xn)(xn) . . . (xn) with “m of these” under the right hand expression.]Sarah: “That’s the same thing. Those ‘x to the n’ are repeating x, n times.”John: “Write (23)2 equals (2 × 2 × 2) × (2 × 2). So what they’re saying is 2 × 2 is 2. . .”Several: “No!”Amy: “That’s not it. What it should be is (2 × 2 × 2) × (2 × 2 × 2). It is 2 to the third,twice.”Instructor: “Which one is right?”Cari: “The top one [referring to Amy’s suggestion].”John: “Oops. I guess, yeah. Yeah, that is right!” [Laughs][General Laughter]John: “Say it’s not squared. If it’s 2 cubed to the third, would there be three groups ofthem then?”Several: “Yes!”

This episode illustrates the following social norms: First, the class wasfocused as a community on understanding an argument; second, it was anenvironment where students were not afraid to contribute their ideas; andthird, ideas came from the students and not the instructor. In fact, in thisepisode the instructor contributed no mathematical ideas. Fourth, studentsresponded directly to each other; note how little the instructor spoke duringthe exchange. Furthermore, we see that students were focused on providinga deductive argument and that mathematical claims were subject to criti-cism and scrutiny. Finally, we see that the inquiry continued until Johnunderstood.

Episode 2: The students have returned to the problem of finding the number less than 1000that has the most factors. 720 = 24325 is the best candidate so far, with 30 factors. Studentsare in the large group discussion semi-circle. The instructor is at the board.

Instructor: “Did anyone do better [than 30 factors]?” [5-second pause] “No? No onegot better. Can you argue why this will be the one with the most factors?” [15-secondpause] “Big silence. No?”


Cari: “I think it’s all about exponents.”Instructor: “Ok, what do you mean by that?”Cari: “When you use that formula, you use the number of exponents. When you havehigh exponents, you are going to get a bigger number of factors.”Sarah: “Because you want to have more exponents that you’re multiplying.”[Instructor clarifies these comments and suggests using a fourth prime in the product.Several students say that this will not work.]Cari: “No, it will be over 1000.”Sarah: “Well, the next prime will have to be 7.”Instructor [to another student who said “no”]: “Why do you say no, Dayna?”Dayna: “Because I tried it, like I took the same exponents and made . . . [5 secondpause] . . . Can I think for a minute?”Instructor: “Yeah, you can. Cari, did you want to say something?”Cari: “If you put a seven, I don’t think it’s going to work because you’ll have to belowering the exponents on the other ones.”Instructor [noting that time is running out for this class period]: “How many peoplethink we can do better than 30?” [several students nod or raise their hands] “How manyare pretty sure 30 is the best we can do? I see several ‘pretty sure’s.’ I’m going to leavethis (until tomorrow) . . .” Several: “NO!” [we hear groaning and the sound of a handslamming on a desk]

In this episode, we see that classroom discourse was focused on thestructure that underlies this “big problem”. It is a sociomathematical normthat doing mathematics means searching for the structure and making argu-ments based on it. Note that the sociomathematical norm that mathematicalclaims are subject to scrutiny extended to the instructor; her suggestion (touse a fourth prime) was not automatically accepted as a correct way toproceed. Note also that there was an expectation among the students thatthe inquiry should not end until the problem is understood, and severalpeople were disappointed to end the class period in confusion.

METHODOLOGY

On the first day of class, the 93 participants completed a ten-item, ten-minute questionnaire (see Table I for the list of items) with Likert Scaleresponse options. The questionnaire was adapted from work by Szydlik(2000) and Frid & Olson (1993) on describing students’ sources of convic-tion in mathematics. Its design was based on two fundamental constructs:1) Mathematics is logical and consistent, and therefore 2) Mathematicsis something that can be figured out. We assume that these beliefs arecentral to the fostering of autonomous behaviors. A diagram showing our(partial) view of the relationships among the questionnaire constructs isshown in Figure 1. Note that while all of the constructs are expressednegatively in the diagram, some are framed positively on the questionnaire.


Figure 1.

Arrows point to beliefs that are logical consequences of another belief.The instrument was used to provide a rough assessment of the nature ofstudents’ mathematical beliefs at both the start and end of the NumberSystems course, to provide a protocol for interviews, and to identify thosestudents with beliefs consistent, and those with beliefs inconsistent, withautonomous behaviors.

Each student was awarded a score on each item of the question-naire. Positive scores indicated beliefs about mathematics supportingautonomy and negative scores indicated beliefs that suggest dependenceon an external authority. For example, students were given the optionto “Strongly Agree”, “Agree”, be “Not Sure”, “Disagree”, or “StronglyDisagree” with the item: “In mathematics I need to memorize how todo most things”. Responses were coded –2, –1, 0, 1, and 2 respectively.Using the same response options, Item 2: “In mathematics everythinggoes together in a logical consistent way” was scored 2, 1, 0, –1, and –2respectively.

Each student was assigned a cumulative score based on the sum of thescores on the ten individual items. In the first two weeks of the semester,


eight students chosen at random from each of the upper quartile, middlehalf, and lower quartile of the cumulative scores were invited to participatein structured interviews. (The middle half of the students was containedin a sufficiently small range that, rather than interview eight from eachquartile, we chose to interview a total of eight from that half.) In thatinterview, students were given the opportunity to clarify and expand oneach questionnaire response. For example:

Interviewer: Item [6] says, [“I am interested in knowing how mathematical formulasare derived or where they come from.”] And you said that you [disagree]. Why did youanswer like that?Student: [response]Typical Probes: What do you mean when you say [whatever they said]? How did youinterpret that question? Can you tell me more about what you were thinking about[whatever is confusing]? Anything else?

The typical interview lasted approximately fifteen minutes. All inter-views were audio-taped.

The remainder of the beliefs data was collected in the last two weeksof the semester. All the students again completed the questionnaire andthe same 24 students were invited to complete a final interview. Twenty-two students accepted the invitation and two declined (one of them hadwithdrawn from the class). In the final interview, students were again askedto clarify and expand on their questionnaire responses. At the end of thefinal interview, students were asked two additional questions: 1) Is thereanything about this class that has changed your view about mathematics inany way; and 2) What is it about the way the class was run or structuredthat allowed you to see [whatever is was they said had changed].

All interviews were transcribed and each transcript was analyzed bytwo researchers and was coded based on several a priori themes. Theseincluded the role of an authority in mathematics (Questionnaire Items7, 8, and 9), the role of memorization in mathematics (QuestionnaireItems 2 and 9), distinctions and expectations about the nature of mathe-matical arguments and truth (Questionnaire Items 1, 3, 4, 5, 6 and 10),student expectations of classroom culture, sociomathematical norms, andthe process of changing beliefs. Based on the initial interview, we classifieda student’s beliefs as either supportive of autonomous behavior, mixed, orsupportive of non-autonomous behavior. The final interviews (this timeincluding the student responses to the two additional open-ended ques-tions) were analyzed using the same methodology. Students were againsorted into the three categories. In the two cases where there was disagree-ment between the researchers as to which category a student belongedbased on the final interview, we placed the student in the lesser autonomouscategory.


STUDENTS’ MATHEMATICAL BELIEFS AT THECOMMENCEMENT OF THE COURSE

Initial Questionnaire Responses

In this section we will give a brief report of the questionnaire data collectedfrom the large sample (N = 93) and discuss some of apparent contradic-tions among student responses in light of student explanations of theirquestionnaire responses in the initial interview. In the following discus-sion, survey data is reported for the large sample, whereas student explana-tions are always based on statements made by the 24 interview participants.Students’ cumulative scores on the initial questionnaire ranged from –14to 10 out of a possible range of –20 to 20 points. The median score waszero, and half of the students scored between –3 and 3.

The questionnaire data revealed that students generally agreed thatmathematics is logical and consistent and also agreed that, in mathematics,they need to memorize how to do most things. The initial interview datasuggest at least two reasons for the apparent inconsistency. First, severalstudents interpreted logical to mean “precise” or “rigid”. For example,Stephanie explained that mathematics is logical and consistent because,“In math there is a certain way you have to do everything”, and Cyndisaid “. . . there are different formulas used for every problem but there isonly one precise, exact answer”. Second, for some students the responseto the first item was hypothetical; interview participants said mathematicsis logical and consistent although they do not personally understand it.For example, Jenni explained, “I’m sure it does [go together in a logicaland consistent way], if you know what you are doing . . .”. The responseto the second item was practical. Students said that they must memorizeformulas, procedures, or template problems in order to work new prob-lems. As Tanya explained, “I have to have a couple of example problemsin order for me to even attempt to try the next, or another example, withoutgiving up”.

Students were divided on whether mathematics is something theycan usually figure out for themselves; 44% expressed agreement, 38%expressed disagreement, and 18% said they were not sure. The interviewparticipants indicated that this division is not as strong as it appears.Because these students may have interpreted this item in the context oftheir prior school mathematics culture, “figure things out” likely meantthey could do a problem on their own after they have been shown howto do a similar one or when they are given a formula. As Ann explained,“They show you how to do the problem and give a couple examples withthe numbers in them, you know, actually do them. That helps. So if I


have examples, I can figure things out”. Terrence agreed, “. . . as longas it’s pretty black and white on the chalkboard or in the textbook, it’sjust a matter of following what’s been told to you”. If the interviewedstudents are representative of the larger sample, then the vast majority ofstudents actually disagreed that they could “figure out” mathematics at thecommencement of the course.

Prior school norms also appeared to influence student responses toItem 7 (“If I’m given a problem that is quite a bit different from theexamples in the book, I can usually figure it out myself”) and Item 8 (“Ihave to rely on the teacher or textbook to tell me how to do the problems”)in similar ways. Several of the students who were interviewed indicatedthat they could not imagine being asked to do a problem significantlydifferent from those in the text or having a teacher who did not first showthem how to do similar problems. For example, although Shannon agreedwith Item 7, she explained her answer this way: “Usually the books willhave like a step to step process. They show you how to do it, and so if Ido the problem exactly how they do it in the book, then I can usually getthe answer right”. A summary of student responses to the initial question-naire can be found in Table I. However, in light of the above discussion,the reader is cautioned that this table provides only a rough measure ofstudents’ beliefs.

Classification of Interview Participants Based on their MathematicalBeliefs

Our reservations about the validity of the questionnaire convinced usto focus primarily on the interviews rather than questionnaire scores asa measure of students’ beliefs. While the questionnaire score was anaccurate measure of beliefs for some students, it was not for others, and,as described above, some participants interpreted items in ways that wedid not intend or they simply were able to express themselves more fullyduring the interview. Based on the initial interview, we classified students’beliefs as either: non-autonomous (11 students), mixed (10 students), orautonomous (3 students). To give the reader a sense of student beliefs atthe onset of the course, we will describe a proto-typical example of eachtype of interviewed student.

Case 1, Mandy. (cumulative score of –8 on the initial questionnaire)Mandy exhibited primarily non-autonomous beliefs. Overall, she appearedto interpret the questionnaire items as intended and was consistent inher responses. She explained that mathematics must make sense to somepeople, but it does not to her personally: “Maybe to some people whounderstand it more clearly, everything would make sense and there would


TABLE I

Initial survey responses: (N = 93)

Initial Survey Responses Mean σ

SA A NS D SD

1. In mathematics everything 8 54 10 20 1 0.52 0.96

goes together in a logical 8.6% 58.1% 10.8% 21.5% 1.1%

and consistent way.

2. In mathematics I need 14 57 6 16 0 –0.74 0.92

to memorize how to 15.1% 61.3% 6.5% 17.2% 0.0%

do most things.

3. In mathematics I can 1 40 17 32 3 0.04 0.98

usually figure things 1.1% 43.0% 18.3% 34.4% 3.2%

out for myself.

4. Mathematics has never 7 25 5 46 10 0.29 1.19

made too much sense 7.5% 26.9% 5.4% 49.5% 10.8%

to me even though I

can often get the

right answer.

5. Drawing pictures or 16 53 14 9 1 0.80 0.88

imagining real physical 17.2% 57.0% 15.1% 9.7% 1.1%

situations is a main

thing that helps me

do mathematics.

6. I am interested in knowing 11 33 26 20 3 0.31 1.04

how mathematical formulas 11.8% 35.5% 28.0% 21.5% 3.2%

are derived or where they

come from.

7. If I’m given a problem 0 28 32 28 5 –0.11 0.90

that is quite a bit different 0.0% 30.1% 34.4% 30.1% 5.4%

from the examples in the

book, I can usually figure

it out myself.

8. I have to rely on the 5 46 18 22 2 –0.32 0.97

teacher or the textbook 5.4% 49.5% 19.4% 23.7% 2.2%

to tell me how to do

the problems.


TABLE I

Continued

Initial Survey Responses Mean σ

SA A NS D SD

9. I learn mathematics best 31 47 7 8 0 –1.09 0.87

when someone shows me 33.3% 50.5% 7.5% 8.6% 0.0%

exactly how to do the

problem and I can

practice the technique.

10. In mathematics if I know 2 35 37 17 2 0.19 0.84

a few concepts I can 2.2% 37.6% 39.8% 18.3% 2.2%

figure out the rest.

always be a right answer, but I sometimes don’t understand why certainanswers are considered right”. Because she does not make sense of mathe-matics, she feels forced to memorize how to solve problems. She admitted:“if I don’t understand it, the only way to get it right is to memorize it . . .”.She says she relies on examples to use as templates for working problems.

As with the vast majority of the surveyed students, Mandy agrees thatshe learns mathematics best when someone shows her exactly how to dothe problems and she can practice the technique. “If somebody shows mehow to do it, then I know I’m doing it right and I won’t question why I’mdoing it”. For Mandy, mathematics is a collection of rules and proceduresthat come from an outside authority. It is this authority that decides whethereach procedure is valid. She cannot personally decide and indeed has littleinterest in doing so. When asked if she was interested in how the formulaswere derived, she responded that, while perhaps people who were good atmath might want to know this, she personally does not have a “strong urgeto know”. Ten other students who were interviewed (eight women and twomen) had interviews similar to that of Mandy.

Case 2, Tara. (cumulative score of 4 on the initial questionnaire) Taraexhibited a mix of beliefs. She agrees that mathematics is logical andconsistent and she claims that she has personally made sense of mathe-matics on at least some occasions. She explains, “. . . when you figureout a problem . . . there’s a reason why – why it is going together”. Inparticular, she recognizes that knowing where formulas come from givesinsight in mathematics, and therefore it is important to her to see the under-lying logic and derivations. However, she does not think she could create


derivations on her own and expects to be shown by a teacher or textbook.In other ways she is much like Mandy, agreeing that the “easiest” way tolearn mathematics is to be shown a technique and then practice. She lacksconfidence in her ability to figure things out on her own; she is “not sure”whether she could do a problem that is quite a bit different from examplesin the book, but thinks she could do a problem if shown how to solve asimilar one first. She explains, “. . . if it’s a problem you’ve already had inthe past you know how to do it and you don’t need a textbook or a teacherto say how to do it”. Nine other of the students who were interviewed (fivewomen and four men) provided responses to questionnaire items similarto those of Tara.

Case 3, Donald. (cumulative score of 11 on the initial questionnaire)Donald exhibited primarily autonomous beliefs. He strongly agreed thatmathematics is logical and consistent and he disagreed that he needs tomemorize things: “There is way too much to memorize, you’ve just got toknow little bits and pieces and how to string them together”. Althoughhe said he is not interested in how mathematical formulas are derived,his theme of making connections is prevalent throughout his interview.Because mathematics reflects reality, drawing pictures helps him work outproblems. “Most of the time it is usually about something that exists in thephysical world, drawing a picture helps you visualize it. That just makesit a lot easier to actually figure it out”. Donald is confident in his ability todo mathematics and likes to figure out problems on his own. “If somebodyshows me how to do it and I just practice the technique, I won’t rememberhow to do it the next time. But if I figure it out myself it will be mine, and Iwill know it for sure”. Two other of the students interviewed (both women)exhibited beliefs similar to that of Donald.

CHANGES IN STUDENT BELIEFS

As mentioned earlier, because student questionnaire responses were some-times inconsistent with their verbal explanations, we elected to focus ourdiscussion of changes in beliefs primarily on the qualitative data. We donote, however, that the quantitative data is consistent with qualitative data,showing a statistically significant shift (3 points) in the cumulative averageon the questionnaire in the direction of more autonomous beliefs (seeFigure 2) for the large sample (N = 84). A summary of the quantitativedata for the final questionnaire can be found in Table II.


Figure 2. Distribution of cumulative student initial and final survey scores.

TABLE II

Final survey responses: (N = 84)

Final Survey Responses

SA A NS D SD Mean σ

1. In mathematics everything 14 53 4 12 1 0.80 0.93

goes together in a logical 16.7% 63.1% 4.8% 14.3% 1.2%

and consistent way.

2. In mathematics I need 6 34 6 35 3 –0.06 1.12

to memorize how to 7.1% 40.5% 7.1% 41.7% 3.6%

do most things.

3. In mathematics I can 4 42 17 19 2 0.32 0.96

usually figure things 4.8% 50.0% 20.2% 22.6% 2.4%

out for myself.

4. Mathematics has never 3 17 3 49 12 0.60 1.08

made too much sense 3.6% 20.2% 3.6% 58.3% 14.3%

to me even though I

can often get the

right answer.


TABLE II

Continued

Final Survey Responses

SA A NS D SD Mean σ

5. Drawing pictures or 28 49 5 2 0 1.23 0.66

imagining real physical 33.3% 58.3% 6.0% 2.4% 0.0%

situations is a main

thing that helps me

do mathematics.

6. I am interested in knowing 4 27.5 18.5 33 1 0.01 0.98

how mathematical formulas 4.8% 32.7% 22.0% 39.3% 1.2%

are derived or where they

come from.

7. If I’m given a problem 3 39 21 18 3 0.25 0.95

that is quite a bit different 3.6% 46.4% 25.0% 21.4% 3.6%

from the examples in the

book, I can usually figure

it out myself.

8. I have to rely on the 2 27 14 39.5 1.5 0.14 0.97

teacher or the textbook 2.4% 32.1% 16.7% 47.0% 1.8%

to tell me how to do

the problems.

9. I learn mathematics best 21 38 11 14 0 –0.79 1.01

when someone shows me 25.0% 45.2% 13.1% 16.7% 0.0%

exactly how to do the

problem and I can

practice the technique.

10. In mathematics if I know 5 53 19 6 1 0.65 0.75

a few concepts I can 6.0% 63.1% 22.6% 7.1% 1.2%

figure out the rest.

Changes Expressed by Interview Participants

Based on their final interview explanations, we classified six of the 22students as holding non-autonomous beliefs, ten students as having mixedbeliefs, and six students as showing primarily autonomous beliefs. (Note:Two students we had classified as having primarily non-autonomousbeliefs at the initial interview did not participate in the final survey orinterview.) A total of five students changed categories from the initial tothe final interview, all in the direction of more autonomous beliefs.


As a group, the students who were interviewed described three specificways in which their beliefs changed. First and foremost, students said thatthe course had allowed them to see mathematics as a more sensible discip-line (As Gwen said, “It’s a lot more logical than I thought it was beforeeven”); now they know there is an underlying reason why formulas andprocedures work the way they do. Tanya explained, “I feel like I know alot . . . I feel like I know, since I know why [things] happen, and I dig muchdeeper on how they’re discovered . . . and I have a better understanding ofmath as whole. It made me think so hard some days . . . I never thought Icould think that hard”. Sam said, “Before mathematics was just numbersand . . . all I cared about was getting the right answer. I never really thoughtabout it and I never really did that well in math. I guess [now I] really wantto know why, why things work and why they came out the way they are. . . now I’m doing a lot better because I know why it’s happening . . .”.Nancy agreed, “When I was in high school, I would just get an answer andthen I just figured that was the answer. But this math class taught me why. . . like, what is the reason behind the answer . . . and why things workout like they do . . .” Sarah explained, “Usually through high school andelementary you’re just [using] numbers and spitting numbers out . . . butthis, you actually see the theories and stuff . . . I didn’t realize any of thesetheories. I appreciate math a lot more because it’s not so dry anymore,there’s actually some meaning behind it”. Thirteen of the 22 interviewedstudents volunteered a comment like those above when asked how theirview of mathematics had changed.

Students indicated that they are now aware that mathematics is a humancreation and they can be a part of making mathematics themselves. Annexplained it this way: “I think I’m not as intimidated by [mathematics].By the way you gave us the problems and had us figure it out and told usthat ‘Okay, even if you think the way you’re doing it is stupid, if it works,it’s just as good as any other solution’. I realize I can figure things outfor myself. I don’t have to be told by the teacher”. Ann’s interpretationof the instructor’s encouragement needs to be placed in the context of aclassroom where it was the norm for students to solve problems in waysthat made sense to them rather than in ways they perceived were expectedby the teacher. While a mathematician might argue that not all solutionstrategies are equivalent in terms of elegance, efficiency, and their abilityto be generalized, it is important to appreciate Ann’s realization that astrategy that makes sense to her is better that an elegant solution that shedoes not understand.

Terrence said that he has seen for the first time that there are manydifferent approaches to problems. He explained:


“. . . you [the instructor] would say, ‘Okay, here’s a question, now everybody come up withan answer’. And when I sat down to think it out, I would think, ‘this is the most logicalway to do it’ – and all of a sudden, there’s about at least a dozen different ways of doing it.And as they’re all going up [on the blackboard], I’m thinking, ‘Wow, this is a way I wouldhave never thought of – but it’s so easy’. It’s just amazing that there’s all these differentpeople in the class thinking a totally different way – probably thinking that this is the waythat it is done . . . [The class] has changed my whole attitude as far as school, as far as theway I will teach”.

Six of the interviewed students made similar comments to those made byAnn or Terrence.

Finally, several students expressed more confidence in their own abili-ties to solve problems and more interest in mathematics in general. Samsaid, “I don’t think of [mathematics] so much as the enemy anymore”.Barb explained, “I’m more interested in [mathematics] because I know alot of why things work the way they do”. Six students made commentssimilar to those made by Sam and Barb.

Interviewed Students’ Attributions for Changes in Beliefs

The structure of the Number Systems course was a conscious attemptto establish social and sociomathematical norms in the classroom thatencourage more autonomous student beliefs about mathematics. Indescribing the culture of the classroom, we identified eleven such norms(see p. 259). In the last question of each final interview, the student wasasked if there was anything about the way the class was structured thathelped change his or her view of mathematics (if indeed that view hadchanged). A typical probe by the interviewer on this question took thefollowing form:“What is it about the way the class was run or structured that allowed you to see mathe-matics in this different way? Was it the problems used? Was it the group work? Was it thediscussion? Is there some specific aspect that you could point to?”

We were able to match all of the students’ attributions with “goal norms”of the class. Table III provides a summary of the student attributions.

Not surprisingly, the norm most frequently cited by students was thesocial norm that mathematics was done as a community. They said thatworking together helped give them confidence and allowed them to seethe ideas of peers. Ann explained, “You didn’t have to figure it out all byyourself. Everybody would contribute what they thought”. Jane said, “Noteveryone knows as much as everyone else, so you get to show each otherways of thinking” and Nancy explained that she liked “having differentpeople in class give different views of it . . . because then it’s not likethere’s only one reason this works, there’s more than one reason and there’sdifferent ways to find it”.


TABLE III

Student attributions for changes in beliefs about mathematics

Social/Sociomathematical Norm # of studentattributions

1. Respectful classroom environment. 5

2. Students work as a community. 14

3. Students interact with each other in large group discussion. 3

4. Solutions come from students. 7

5. Instructor acts as a guide. 2

6. Content generated by “big” problems. 11

7. Variety of strategies and arguments valued. 7

8. Complete deductive argument required. 1

9. All problems must be solved and understood. 1

10. Elegant solutions reveal structure. 0

11. Students reflect on the problem-solving process. 4

Eleven students stated that the sociomathematical norm of having thecourse content generated by big problems with an underlying structure(Goal Norm #6) helped them to see the sense in mathematics. Studentssaid that the problems were deep and challenging enough to allow for realthinking. Ben liked this aspect: “I guess it was because you didn’t hand usa book and say, ‘Go and do problems 1 through 15 for Wednesday’. Yougave us a big problem, like a story problem, and it was more challengingand interesting because it kind of had other stories behind it”.

Seven students cited the sociomathematical norm that a variety ofstrategies and arguments were valued as helping them to see that peoplethink about mathematics in many ways (Goal Norm #7). As Nancyexplained, “Having different people in the class give different views ofit, different examples from yours so that it was easier to understand. Causethen it’s not just like there’s only one reason this works, there’s more thanone reason and there’s different ways to find it”.

Seven students said the social norm that they (and not the instructor)were the ones who decided whether a strategy worked or a solution wasvalid (Goal Norm #4) changed their beliefs about mathematics. Gwennoted, “You never ever, ever gave us the answer . . . and so that I reallyliked a lot” and Bob explained, “there’s really no book to rely on, or youknow, you don’t go around and give the answers right away. You let us pickat [the problems] for awhile, and that’s helped to give me confidence”.

Students also appreciated the informal yet respectful nature of theclassroom (Goal Norm #1). Five students identified this norm as an


important aspect of the course. As Bob stated in his interview, “I thoughtthe discussions were very, very open. You know, the floor was open andyou could say anything you wanted to even if it was wrong . . . You didn’tmake anybody feel like their answer was dumb even if it was wrong oranything like that”. Ann expressed similar feelings: “It was good the waythat you really tried to convince us that any way of solving the problemcame up with was ok even if it was kind of dumb . . . just the way youmade sure everyone knew nothing was too stupid to say and raise yourhand”.

Several students remarked on the effect of reflecting on the problem-solving process (Goal Norm #11). Steve noted, “I liked the way that wehad one sheet, usually the one problem a day. We worked in groups andthen we had the discussion and that helped. With some of the problems, weactually had to say, ‘Ok, what are the steps for doing this?’ ” Jane was evenmore explicit in her attribution: “. . . and it was the papers that we had towrite, those assignments where you had to explain, you know, what’s thequestion about, why did you get the answer, why is that the only answer.It was really good that way. At first it seemed like, oh my gosh, I can’tdo this. But it was really helpful. I liked that”. Four students attributedchanges to this particular sociomathematical norm.

Not all of the classroom goal norms were widely identified by thestudents (in particular, Goal Norms #8 and #9 were only mentioned byone student each, while Goal Norm #10 was not mentioned at all). Thisis not surprising, however, since the student attributions for changes intheir mathematical views were free responses. The interviewer did notprovide a complete list of the goal norms; rather, the students were invitedto contribute their own thoughts. It is reasonable to expect that studentswould tend to attribute clearly observable norms more often than moresubtle norms such as emphasis on complete deductive reasoning.

While the students responded overwhelmingly that the course had beenan experience that made them more powerful in doing mathematics, thiswas not the case for one student. Sarah expressed a loss of confidence inher ability to solve problems, and this seemed to be a direct consequenceof her experience of the course: “. . . being in class, there’s a lot of stuffthat I thought I would be quicker on. But a lot of these problems just didn’tcome to me right away – there were very few of them actually. I had toreally work at them harder. I had to listen to you, or my classmates gotthem before I did. So . . . this class wasn’t really logical and I had a reallyhard time figuring them out for myself unless I spent a lot of time”. Sarahbegan the course with beliefs supportive of non-autonomous behavior andended it with even less confidence.


CONCLUDING REMARKS

The study lends additional support for the adoption of the NationalCouncil of Teachers of Mathematics (1989, 2000) process recommenda-tions regarding problem solving, reasoning and proof, and communication.Our analysis suggests that a classroom focusing on problem solving usinga variety of strategies, reflection on the process of problem solving,and engagement in the process of exploration, conjecture, and argumentcan help students develop mathematical beliefs that are consistent withautonomous behavior. Specifically, a classroom culture in which studentssolve challenging problems without external assistance and where mathe-matical conviction is determined based on logic and consistency can alterstudent beliefs about mathematics, at least in the context of the new culture.Many students in the Number Systems course expressed mathematicalbeliefs that were more consistent with autonomous behavior at the endof the course than they did at its commencement, and many were able tocommunicate changes in their beliefs in an insightful manner. Studentssaid they saw – many for the first time – the meaning in mathematicsand gained confidence that they can construct that meaning themselves.This result is particularly important for prospective teachers who will beexpected to help their future students make, refine, and explore conjecturesbased on evidence and make arguments to support or refute those claims(NCTM, 2000).

Students in our study attributed changes in their beliefs to three primaryaspects of this classroom culture. First, the course was built around care-fully constructed and very challenging problems – each with an underlyingmathematical structure or lesson. Students began to have faith that eachproblem would yield this structure or story (why it worked) and weremotivated to uncover it. Second, the course instructor provided almost noassistance in the problem solving aspect of the course and no answers wereprovided for problems. The only way for the class to understand a problemwas to figure it out. The only way to know they were correct was to find aconvincing argument. Third, the community work on the problems madethe process less frustrating for students, allowed them to see the ways inwhich their peers did mathematics, and showed them that problems couldbe solved in more than one way.

Did the Number Systems experience fulfill Kagan’s (1992) condi-tions for changing beliefs (making personal beliefs explicit, challengingthe adequacy of those beliefs, and providing extended opportunities toexamine, elaborate, and integrate new information)? We assert that it did.In part, it was the methodology of our study that helped students makebeliefs about mathematics explicit. All participants, but particularly those


who participated in the interviews, had the opportunity to reflect on theirmathematical beliefs through their work on the questionnaire items, as wellas through course assignments. The classroom culture showed studentswhat authentic mathematical experiences look like (often revealing theirmathematical beliefs to be inadequate), and provided opportunities forreflection on the process of doing mathematics through group work andwritten problem reflections. In the future, the course may be improved bymaking these processes even more transparent to the students.

There must be two qualifications to this work. First, one of theresearchers was the instructor for the course. It is certainly possible thatstudents (either unconsciously or consciously) made statements in the finalinterviews in order to please their teacher. Even if this is true to someextent, however, we assert that the students could not have made (and didnot make) such insightful statements regarding the nature of mathema-tical behavior at the commencement of the course; they simply did notknow enough about what mathematics was supposed to be about. This issomething that they learned from the class, and that alone is an achieve-ment. Second, students’ initial interpretations of terms and phrases suchas “understand”, “figure it out”, and “problem solve” were made in thecontext of prior mathematical experiences. This means that in the initialsurvey “understand” possibly meant “recall the procedure to use”, “figureit out” possibly meant “get the right answer”, and “problem solving” mayhave referred to doing more problems just like a given example. Statementsprovided by the interviewed students suggested that the meaning of theseterms changed as their beliefs about these ideas changed. This makes itmore difficult to compare initial survey and interview responses with finalsurvey and interview responses.

Our data supports the assertions of others (Schram et al., 1988; Wilcoxet al., 1991; Civil, 1993; Yackel & Cobb, 1996) who advocate focusingon classroom culture as a means of affecting beliefs and adds weight toviewing this avenue as a powerful agent of change. This is an area that weintend to investigate further. We are now interested, for example, in how“non-traditional” classroom norms are established. Specifically, what arethe interactions related to specific themes or content (Bauersfeld, 1994)that form the mathematical habits of the students? What is the teacher’sway of being in the classroom that might help create a new culture? Infuture research, we will attempt to document how the Number Systemscourse culture is established and nurtured, and we will attempt to measureautonomous behavior directly, both by interviewing participants and byobserving ways in which the students work problems.


ACKNOWLEDGEMENTS

This work was supported in part by a Faculty Development Grant from theUniversity of Wisconsin Oshkosh. We would like to thank the reviewersand editors of JMTE for their thoughtful advice on revisions of this paper.

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JENNIFER E. SZYDLIK

Mathematics DepartmentUniversity of Wisconsin Oshkosh800 Algoma Blvd.Oshkosh, WI 54901, USAE-mail: [email protected]

STEPHEN D. SZYDLIK

University of Wisconsin Oshkosh

STEVEN R. BENSON

Education Development Center

READER COMMENTARY

JOHN MASON

SEEING WORTHWHILE THINGS

Response to Alan Schoenfeld’s Review of Researching Your OwnPractice in JMTE 6.1

In his, to my reading, extremely generous and supportive review ofResearching Your Own Practice: the discipline of noticing, Alan Schoen-feld agreed, I think, that “noticing” lies at the heart of both learningand teaching, and hence also of professional development. Furthermore,systematic, disciplined noticing makes learning, teaching and professionaldevelopment more effective and more efficient. He then pointed to one ofhis own projects and raised the important question of how teachers mightselect what they choose to notice, or at least to look out for and questionin and around their teaching. More pointedly, how might things worthy ofnoticing come to the attention of practitioners? He referred to some of hisown work with teachers in which “big questions” are raised in order tostimulate discussion and hence, presumably, in order to promote change inpractices.

OUTLINE

The discipline of noticing provides a method of working on issues todo with practice, at every level and in any context. Indeed, the basicmechanism is the only way to develop practice intentionally that I havebeen able to locate. It does not, however, claim or aim to decide whatis worth noticing. What I consider worth noticing depends on a valuesystem; what I notice reveals something of that system. Although thereis much in people’s own practices worthy of attention, outside stimulusis also valuable in focusing attention on issues which might otherwisebe overlooked. However, where some person or some agency wants tobring about change, there are practical and ethical difficulties: practicalbecause of the global failure (even where there is local success) of mostreform movements; ethical because there are parallels to the “constructivistdilemma” in teaching addressed by Paul Cobb (see later), concerning how


282 JOHN MASON

to get teachers to engage in personal and professional development and toadopt practices that are deemed valuable or even necessary.

VALUES

There is an obvious difficulty, for who decides whether changes in prac-tices are to be deemed valuable or necessary? An obvious criterion isthat they enhance learning. But who specifies the criteria for decidingwhether learning is being or has been enhanced? Asking learners for theiropinions is unsatisfactory at best, for they are caught up in their ownconcerns as learners and are not generally aware of choices being madeby teachers or of how things might be otherwise. Accepting someoneelse’s criteria (performance on standard tests) abrogates responsibility formaking choices, bows to social pressure to conform, and stifles creativityand participation in choices. Being satisfied by your own criteria can all tooreadily become solipsistic, even if or perhaps especially when, there is alocal-community-of-practice (Winbourne & Watson, 1998; Wenger, 1998).Something more is required, which the Discipline of Noticing formu-lates as constantly seeking resonance in an ever-expanding community ofsceptics.

EFFECTING CHANGE

The book concentrates on issues which arise for teachers in their teaching(including planning and reflecting), for several reasons. First, every practi-tioner has issues about their own practices, even if these are not explicitlyacknowledged. Consequently own-practice provides a suitable domain ofexperience and concern with which to try to resonate in a text which isdescribing ways of working. Second, people are much more likely to wantto work at changing something if they feel a need for change, so the mosteffective fulcra for leveraged change are issues which teachers recogniseand feel are within their reach, within their power to change. On the whole,this means their own actions, for the actions of others are very hard to influ-ence. Third, even if stimuli come from an outside source initially, changesin practice eventually depend on individuals, supported by colleagues.A group of colleagues can develop into a local-community-of-practicethrough trading accounts of incidents. Over time they can develop a sharedvocabulary, can find their noticing sharpened and extended, and therebyfeel supported in trying out possible alternative or fresh actions adapted totheir situations.

SEEING WORTHWHILE THINGS 283

However, whether the stimulus comes from general discussions, fromforces in the institutional hierarchy, from individuals, or from some localor even global community of practice, ultimately, personal change is apersonal matter. Only I can make changes to my practice, wittingly or not.It is an individual matter within a social context. The social context maymake it difficult to effect change, but literature resounds with accountsof people resisting social forces. What matters most is resonance: afeeling that you are not alone. A local-community-of-practice provides asounding board within which to develop confidence to counteract forcesfrom outside.

I know that colleagues committed to strong forms of social constructi-vism may wish to stress rather more strongly the influence and role of alocal-community-of-practice, but I myself find that this only goes so far,as support. Some people may follow in the wake of activists; others maynot detect possibilities or even desirability of change. But in every case,something is needed from the individual which is more than adoption ofcurrent practices. For in education, practices are not enough. No matterhow precisely I specify a lesson for someone else to carry out, they haveto bring something of themselves, they have to bring their “mathematicalbeing” to the teaching. Indeed, the more precisely someone tries to followa lesson plan, the more likely it is that real teaching will not take place(Mason, 1998).

I respect the care which has gone into preparing the questions whichAlan describes offering to teachers in order to stimulate movement outof their immediate concerns and practices. Just as John Wesley circulatedissues for local groups to discuss each week, it is valuable to have outsidestimulus that jars people out of their immediate concerns and provokesreflection, discussion, and awareness, stimulating them to look at theirpractices “from a distance” as well as “from the inside”. But provoca-tion that generates reaction is not nearly as effective as stimulus to evokeconsidered response.

It is also vital that ways of working develop so that individuals areminded and re-minded to question their behaviour as part of their practice,and this is what the Discipline of Noticing may contribute. But it is easyfor individuals and groups to become dependent on outside stimuli, on anexternal “fix” around which discussion and investigations revolve. Notethat there is a subtle but important difference between welcoming stimuli,and becoming dependent on an outside force to take initiative. However,dependency is dependency, no matter how good the intentions. How thenare teachers to be supported in turning general discussion into effective

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changes in practice without being forced into specified practices and yetwithout becoming dependent on outside stimuli?

One of my guiding principles is that desiring other people to changesays a great deal more about me and my desires than it does about them.What am I hiding in myself, or from myself, by focusing on others? Is it awishful or blaming sentiment of

“if only they would . . .”,

by which I try to pass responsibility onto others? If so, then the real task isto work on myself. Is it an evangelistic sentiment of

“I really enjoy, got benefit from . . . so others could (should?) do likewise”,

trying to urge my experience upon others? What is this teacherly lust (MaryBoole in Tahta, 1972, 1991), this desire for others to have experiencessimilar to mine? If this is the force, then the real task is to invite andinveigle rather than force others to become interested in what interests me.I am reminded of Aesop’s tale of the competition between the sun and thewind to get a person to remove their coat: gentle invitation always worksbest. In either case, trying to force other people to change in specified waysis ultimately unproductive, and even ethically dubious.

This is a difficult point, for missionary zeal is easy to produce, at leastat first, though it is rather more difficult to sustain. I, and I think Alan,and many others like us, would like learners to experience and develop theuse of their powers to think mathematically. But my desire becomes anobstacle, an impediment, as soon as I desire specific practices, specificchanges, specific pleasures. I consider it to be ethically sound to wishfor others that they have continued opportunities to choose to participatein activities through which they may experience something fresh, someexpansion of their current awareness. But I am adamant that at everymoment I respect their choice to opt out. I do not, however, respect a choiceto revert to (even worse, to subside into) mechanical and unreflectivebehaviour. I consider that to be unprofessional.

I consider it to be the professional responsibility of people leadingprofessional development to develop stimuli which will engage teacherswith issues that are considered important in professional practice. Forexample, if it is deemed important for teachers to think about the “bigideas” in mathematics, I would look for mathematical tasks which had agood chance of bringing some of these “big ideas” to the surface. Then Icould draw attention to them, and after reflection, invite further examples.But the principal interest for practicing teachers would be how they mightgo about exposing these themes or “big ideas” to their learners. Drawing onthe Discipline of Noticing (because what the teacher aims to achieve is to


increase the sensitivity of their learners to notice mathematical themes and“big ideas” when they meet them again) it would make sense to seek taskswhich fit within standard curriculum topics and to offer teachers imme-diate experience of such tasks themselves, together with the opportunity toreflect upon them, discuss them with colleagues, and imagine themselvesin their own situation using some adaptation or modification. In this way Iam “preaching my practice” rather than endeavouring to “practice what Ipreach”.

TRUST

My “position of restraint” in which I respect others’ choices whileexpecting them to be awake and alive to their teaching is tenable preciselybecause there are ways of working which are effective in attracting othersto want to experiment, to reflect, to challenge their habitual practices.Specifically, there are ways of working on and with mathematics whichenrich the experience of learners. But these ways of working are not simplybehavioural practices which can be performed mechanically. They arisethrough the very processes I am talking about: reflecting, experimenting,noticing, preparing myself to notice in the future, catching myself justbefore a habit kicks in, seeking resonance in others through describingincidents and seeking resonance within my own past and future experience,and so on. These all go to make up the Discipline of Noticing.

MECHANISM AND FREEDOM

In my view, using cause-and-effect as the underlying deterministic mecha-nism in professional development or in teaching is at least unhelpful, andat worst destructive. Treating people as mechanisms produces mecha-nical behaviour. The Discipline of Noticing comes from, and is aimedat, replacing mechanical behaviour by sensitive intentional and consciousresponse, leading to an experience of personal freedom when fully partici-pating in choices in the moment. I am convinced that the same holds forany community-of-practice as well as for individuals. If some practices atleast are not in flux, are not being challenged and investigated, then thepractice itself is likely to become mechanical and sterile.

Always there is the inescapable circularity in my position, for thecriteria for improved learner experience colours and blinkers what isnoticed and taken as evidence, as well as what opportunities are noticedin which to act freshly and responsively, and what “acting freshly and

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responsively” is taken to mean. This circularity can be seen as recursiverather than circular, constantly acting upon itself to alter, if ever so slightly,current conditions.

CONDITIONS FOR CHANGE

What I am talking about is not simply exchanging some practices forothers. I am talking about being sensitive and awake to possibilities toact moment-by-moment in the unfolding of a lesson. People develop theirpractices when they detect a disparity, a disturbance to their habitual func-tioning. Two things are required in order to arrange conditions so thatpeople work on themselves and their practice effectively. The first is tobe in the presence of personal and professional development (consistentwith Vygotsky’s notion (1934) that higher psychological functioning isinternalized through being in the presence of that functioning in others).Hence the theme of the book is that “I cannot change others; I can work atchanging myself”. The second is to have access to practices which fosterand support personal and professional development, which is what thediscipline of noticing offers. A necessary consequence is to have access toalternative practices whether through the literature, through attending otherpeoples’ classes, or through listening to and working on other peoples’accounts of classroom incidents.

Although there may be other ways to work on professional develop-ment, my enquiries and experience suggest that every approach that hassome influence boils down in the end to individuals becoming sensitisedto notice more and different details, so that it is possible to surmounthabits and to act freshly and responsively. Individuals here means teachers,teacher educators, professional development support agencies, anyoneinvolved. And to what end? To the end that learners themselves are sensi-tized to notice in new ways as a result of their experiences. This requiresboth alternative actions to take when deemed appropriate, and catching themoment before a habit kicks in, in order to participate in choosing to actdifferently. The problem of habits, as William James (1890) pointed out, isthat they kick in before you have a chance to choose to act differently. Thediscipline of noticing, distilled from multiple ancient and modern sources,provides the only means I have been able to find for arranging to participatein choice rather than be slave to habit. Additional and alternative practicesmay improve how it works, but working to become aware in the momentso as to participate in choice is what is required.


MAKING CHOICES

I conjecture that one of the features of modern classrooms which blockslearners full participation is being unable to participate in choices andunable to influence what happens. The same applies at all levels of institu-tionalised education. Teachers who feel they cannot exercise choice them-selves are not going to arrange for learners to exercise choice. Teacherswho are not trusted and respected are unlikely to trust and respect learners.But these sentiments reveal something of what I value. The Discipline ofNoticing is formulated to reflect those values, but does not depend uponthem.

Regarding “choice”, we often say “to make a choice”, but close exami-nation reveals that choices are made a very long time before the moment,and automatised as predisposition or habit. What we think are “choicemoments” are usually a fleeting instant before a habit kicks in. My desire isto “participate in a moment of choice” rather than “be subject to choice”.For example, if you have ever said to yourself “I won’t do that again”,and then found yourself doing it again, you may recognise that even ifyou are aware that you ought not to, or that part of you doesn’t want todo something, another part does, and did. I am thinking here of dietarychoices, resolving not to say something, resolving not to shout at kids,resolving to ask more and tell less, and so on.

WHAT DRIVES BEHAVIOUR?

In his review, Alan subtly points to an issue which has exercised manyof the contributors to this journal, namely the gap between beliefs andbehaviour. As Paul Ernest (1989) put it so succinctly, there is a sometimesstartling difference between espoused and enacted practice. Every attemptat educational reform has foundered on the rock of teachers learning to“gargle the discourse” without significant changes in observable beha-viour (Cohen, 1988). Examples include constructivism, zone-of-proximal-development, scaffolding, investigative teaching, and practical maths, toname just a few. Yet this is an entirely natural response to forces fromoutside trying to make “me” change. Interestingly and perhaps unex-pectedly, there are also examples of teachers changing their practices(including reverting to older practices) while bending current establisheddiscourse to describe it (Houssart, 1999). Espoused beliefs certainly donot drive behaviour. Whether there are deep-seated beliefs which do drivebehaviour, or whether “beliefs” are the narratives we use to account forbehaviour and to express our ideals is another matter.

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If I think that beliefs drive behaviour, then I need to work on peoples’beliefs in order to affect their behaviour. Inviting them to discuss “bigquestions” would be one approach. Inviting them to articulate, refine andquestion their beliefs would be a parallel action. Presumably it is mostuseful if beliefs and behaviour are worked on together so that the twocan develop in tandem. The plethora of research studies suggests to methat teacher demeanour and behaviour certainly can influence learners,though not in deterministic ways which are easy to capture as evidence.Sometimes the response will be consonant and sometimes dissonant, asadolescents either follow or reject individuals as role models.

An alternative is that behaviour is driven by habits, and that espousedbeliefs are attempts at narrative to account for the behaviour of which weare aware, drawing on the socially prevalent forms of discourse (thoughagain, some people may deliberately refuse to use a current discourse). Forexample, when you are confronted with behaviour that you do not recog-nise as typical, it is tempting to “explain it away”. This view is consistentwith personal observation (using the discipline of noticing!) as well as withthe view that human beings are narrative animals.

It is not clear to me that beliefs exist, or that people actually even hold“beliefs”. Rather, beliefs could be how people account for their behaviour.For example, when asked about mathematical thinking, I tend to revert tomy habitual patterns of speech; with effort I can enter the question freshly,but what I say is an attempt to speak from experience, not an espousalof beliefs which drive my behaviour. Beliefs may be what interviewersand observers construct from observed behaviour and from the responsespeople make when they are asked questions about it (Ruffell et al., 1998).And as it emerges near the end of the book, perhaps what I learn from theseconstructions is as much about the interpreter’s sensitivities to notice, asabout the person being observed and interviewed.

From an enactivist perspective, the distinction between belief and beha-viour is artificial: acting is knowing and knowing is acting (Maturana &Varela, 1988). Your observed behaviour is what you know to do, and whatyou know to do is enacted. I haven’t yet resolved for myself how thissits with my own experience of an inner life often in conflict with outermanifestations: I intend not to do something, but find myself doing it; I setmyself to do something yet I forget or shrink from doing it.

INFLUENCE THROUGH BEING

Earlier I mentioned “being mathematical”. In order to stimulate teachers towant to work on their own mathematics as a key to appreciating learners’


experiences, I have often said to teachers that perhaps what matters mostin the mathematics classroom is the “mathematical being” of the teacher.I draw from this the consequence that it is important for teachers to “bemathematical with and in front of their learners”, for it is their “being”which communicates so much to learners, and is likely to be a majorinfluence. But “being” here involves much more than superficial practices,put on for show like a cloak to impress inspectors and to “get through theday”. Heidegger used the notation to remind readers that he meant moreby the word than the simple “being” of “human being”. Something more ispossible than experiencing “higher psychological functioning” when in thepresence of a “real teacher” (Mason, 1998). “Being” is what is not institu-tionalised in habit; it is how I contact and transform creative energy. Suchenergy flows through and is channeled by attention and awareness whichare somewhat like a force field. From being in the presence of an awareprofessional it is possible to pick up much more than surface behaviour,just as apprentices pick up more than practices from being in the presenceof experts. It is of course, not inevitable. Some tuning is required, someletting go, some subordination to “other”.

A PROFESSIONAL DEVELOPMENT DILEMMA

Paul Cobb (1988 unpublished interview, Open University) drew attentionto a constructivist dilemma in teaching: the teacher knows that learners willconstruct their own meaning, so how does a teacher arrange that learnersconstruct an appropriate meaning that conforms with current mathematicalpractice? His search for a resolution led him to stress the social dimen-sion over the cognitive, locating meaning in social practices rather than inindividuals (Cobb, 1994). My resolution would stress the evident appealof mathematical structures to human neural structures, and consequentconsonance between mathematical structures and brain functioning, whilestressing the role of individual will and associated will-power, and altera-tions of the structure of attention, all supported (to lesser or greater extent)by the socio-professional milieu (Brousseau, 1997).

There is a similar dilemma within professional development. How canteachers be led to (re)construct for themselves the sorts of practices whichresearchers have shown to be effective? A social constructivist answer ispresumably to develop a discourse and to expose teachers to experts whomodel behaviours appropriate to that discourse. I agree. Where I mightdiffer is in whether it is intended that specific changes be made. My answeris that those who support professional development of others need to trustindividuals and their local communities of practice as long as they are

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evidently working on issues. They need to stimulate them to question theirhabits, but also, importantly, to respect teachers’ conclusions.

SOLIPSISM

There is no question in my mind that it is easy to become wrapped up andabsorbed in your own solipsistic and idiosyncratic practice. That is whyan essential part of the Discipline of Noticing concerns validation, even ifits form runs against the zeitgeist of Popperian falsifiability (Popper, 1972)and deterministic rationality. As part of the discipline of noticing, or indeedany method, it is vital to test sensitivities to notice against the experienceof new colleagues. It is vital to continue to refine and hone task-exerciseswhich are intended to promote specific noticing, by trying them with newcolleagues, not because you want colleagues to change, but in order toguard against your own solipsism.

PROMOTING ALGEBRAIC THINKING

As a specific example, Alan mentions the ongoing issue of algebra,that watershed in mathematics for so many people. Starting with somecurriculum-based tasks which invite teachers to experience and expressgeneralities for themselves, it is possible to offer the thought that

“a lesson without the opportunity for learners to generalise, is not a mathematics lesson”

and for this to stimulate them to be on the lookout for such opportuni-ties for themselves. In Mason (1998a) I called such statements protases,because they are like the first (general) statement of a syllogism whichwhen juxtaposed with a particular (arising from individuals’ experience)leads to some “action” in the form of discussion and decision to “trysomething out at the next opportunity”. Like a grain of sand in an oyster,a statement acts like a protasis when it niggles away at the back of yourmind, with the effect of sensitising you to notice more than you noticedbefore. A protasis is an offering, with no expectations: if it is taken up,it can be worked on, but if it is not taken up, then it disappears again.Cause-and-effect are replaced by trust in organisms and selves dealing withwhat they can and ignoring what they cannot. True, respect for the personoffering a protasis may urge them to work on it with more intensity thanif it is encountered in some other context. But that is the nature of socialforces and influence.


RESONANCE

Alan reports that not all of the tasks in the book resonated directly withhis experience, and that there is so much to be noticed that it is hardto know where to start. But as the Chinese adage has it, “a journey ofa thousand miles begins with a single step”. If I pick something that isniggling me, and enlist the help of colleagues to work away at it in areasonably, perhaps increasingly systematic manner, I find I am muchbetter prepared to engage with big and little questions that others wantme to think about. Steps are made, at least locally, towards elements ofa “science of education”, something that Caleb Gattegno (1990) was sodedicated to encouraging. He recommended starting with enquiring intothe “universe of babies” (Gattegno, 1973) because this yields evidencefor the fundamental powers everyone is born with and from which all theacademic disciplines arise. But it is enough, in my experience, to start withyour own personal experience and the experience of colleagues, informedand directed by “big themes”. What matters is to develop some practiceswhich enable intentional work on sensitising yourself to notice and tochoose to act against habits.

REFERENCES

Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactiques desmathématiques, 1970–1990. In N. Balacheff, M. Cooper, R. Sutherland & V. Warfield(trans.). Dordrecht: Kluwer.

Cobb, P. (1994). Where is the mind? Constructivism and sociocultural perspectives onmathematical development. Educational Researcher, 23(7), 13–20.

Cohen, D. (1988). Teaching practice: Plus ça change . . .. In P. Jackson (Ed.), Contribu-ting to educational change: Perspectives on research in practice (pp. 27–84). Berkeley:McCutchan.

Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.),Mathematics teaching: The state of the art (pp. 249–254). London: Falmer Press.

Gattegno, C. (1973). In the beginning there were no words: The universe of babies. NewYork: Educational Solutions.

Gattegno, C. (1990). The science of education. New York: Educational Solutions.Houssart, J. (1999). Teacher’s perceptions of good tasks in primary schools. In E. Bills

(Ed.), Proceedings of BSRLM, February (pp. 31–36). Warwick University.James, W. (1890) (reprinted 1950). Principles of psychology, Vol 1. New York: Dover.Mason J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and

structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267.Mason, J. (1998a). Protasis: A technique for promoting professional development. In

C. Kanes, M. Goos & E. Warren (Eds.), Teaching mathematics in new times: Proceed-ings of MERGA 21, Vol 1 (pp. 334–341). Mathematics Education Research Group ofAustralasia.

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Maturana, H. & Varela, F. (1988). The tree of knowledge: The biological roots of humanunderstanding. Boston: Shambala.

Popper, K. (1972). Objective knowledge: An evolutionary approach. Oxford: OxfordUniversity Press.

Ruffell, M., Mason, J. & Allen, B. (1998). Studying attitude to mathematics. EducationalStudies in Mathematics, 35, 1–18.

Tahta, D. (1972). A boolean anthology: Selected writings of Mary Boole on mathematicseducation. Derby: Association of Teachers of Mathematics.

Tahta, D. (1991). Understanding and desire. In D. Pimm & E. Love (Eds.), Teaching andlearning school mathematics (pp. 220–246). London: Hodder & Stoughton.

Vygotsky, L. (1934). E. Hanfmann & G. Vakar (trans.) 1962, Thought and language. M.I.T.Press.

Winbourne, P. & Watson, A. (1998). Participation in learning mathematics through sharedlocal practices. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd interna-tional group for the psychology of mathematics education, Vol. 4 (pp. 177–184). SA:Stellenbosch.

Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge:Cambridge University Press.

Centre for Mathematics EducationMathematics FacultyOpen UniversityMilton Keynes MK7 6AAUKE-mail: [email protected]

PETER SULLIVAN

EDITORIALINCORPORATING KNOWLEDGE OF, AND BELIEFS ABOUT,

MATHEMATICS INTO TEACHER EDUCATION

One often hears discussions among science teacher educators that theprospective teachers with whom they work have limited knowledge ofthe breadth of science and, more importantly, have thought little aboutthe nature of scientific thinking. The first, no doubt, is a product of thediversity of science courses offered as part of undergraduate studies thatsatisfy the requirements for secondary teaching. It is suspected that thesecond is a product of the way that university science courses are taughtand assessed. Both issues – discipline knowledge and beliefs about thenature of the discipline – suggest challenges for all teacher educators.

It is interesting to consider how these teacher educators should respondto possible gaps in the knowledge of prospective teachers of the specificdomains of science. At secondary level, it is possible that there may besome general science teachers who have not studied physics, for example,but who would expect to teach selected physics topics in the juniorsecondary years and learn the necessary physics along the way. At theprimary level, we do not expect teachers to be familiar with the full rangeof aspects of scientific knowledge that are required for their teaching.It is important that they know how to learn the science they need. Thesame is true for mathematics. Even at secondary level, we anticipatethat prospective teachers would not have studied all of the mathematicaldomains that might be required in their teaching. For myself, I did notstudy any statistics when at university, but it was not difficult to learn thenecessary statistics required for teaching. Likewise I have learned to usespreadsheets and dynamic algebraic and geometric software as teachingand investigative tools, changing the nature of learning in those domains.We expect that primary teachers will be familiar with the mathematicalconcepts required for teaching in the early years, and even if they are uncer-tain about some topics in the upper grades, we hope that they will learnthose concepts as required. In other words, so long as teachers have theorientation to learn any necessary mathematics, and the appropriate foun-dations to do this, then prior knowledge of particular aspects of contentmay not be critical. Rather than ensuring complete coverage of all possible


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topics, teacher educators could perhaps define the appropriate foundations,and the elements of the desired orientation to learning, and find ways tosupport the prospective teachers in achieving them.

The second issue, that of the nature of the discipline, may be morecritical. It seems obvious that beliefs about the nature of science willbe a key influence on the way that teachers teach science. The same,no doubt, is true for mathematics. I can recall attending a professionaldevelopment session during my first year of teaching, after completinga degree majoring in mathematics and a Graduate Diploma course withsubstantial mathematics education studies, and being asked to define thenature of mathematics. I was surprised at the diversity of replies from theother participants, but more astonished that I had reflected so little on thequestion. It highlights that at the time, at least for me, the nature of knowl-edge was not considered problematic, and certainly was not promoted inmy undergraduate mathematics studies.

These two issues, knowledge of, and beliefs about, mathematics area unifying theme in the three articles that make up this volume. Theconsistent message is of using reflection on experiences as a key tool inthis teacher learning. The argument is that reflection by the prospective andin-service teachers on their prior mathematical experiences has consider-able potential to reveal insights into the nature of mathematical knowledgeand its learning. This volume makes an important contribution to ourunderstanding of the way that such reflection may contribute to teacherlearning.

It is worth noting that reflection on the experiences that the prospectiveteachers have had prior to their teacher education studies may not be neces-sary for all aspects of mathematics teacher education, especially thosethat are considered seriously for the first time in their teacher educa-tion program. These include the ways to teach particular topics (suchas decimal place value), the difficulties that some students might experi-ence, the structure of the mathematics curriculum, awareness of availableresources and approaches, as well as more general issues such as an orien-tation to learning about teaching from the study of practice, and ways ofaddressing diversity and equity. While reflection on their prior experiencesmay be of some use, these are issues that are theorised for the first timeduring initial teacher education, and so reflection on them is less revealing.

In contrast, the issues of mathematical content knowledge and beliefsabout the nature of mathematics are formed by experiences prior to theteacher education program. This has two dimensions. First, it would beuseful for mathematics teachers and undergraduate university mathematicslecturers to consider the appropriate formative experiences that will foster

EDITORIAL 295

the capacity for ongoing mathematical learning and reflection on thenature of mathematics, especially for prospective teachers. To give just oneexample, in Victoria, Australia, we have an interesting challenge related tographing calculators. Such tools are more or less essential in the study ofsenior secondary mathematics, and indeed are used effectively in muchjunior secondary teaching (the larger screen makes them suitable for iter-ative number investigations, for example). These tools have an impact,inter alia, on the way functions are represented, the ways that represen-tations are linked, the consideration of the impact of variables on therepresentation, and the ways of considering particular feature of represen-tations, such as turning points. This changes the nature of the mathematicsbeing studied. Yet graphing calculators are not incorporated in all univer-sity level mathematics courses, for either teaching or assessment. So wehave a situation where senior secondary students use graphing calculators,with the resultant impact this has on the methods and the nature of themathematics studied, they move to undergraduate studies in which thetools are limited in their use, and then they move to teacher education andteaching where the graphing calculators become important once more. Inother words, a range of educators contribute to the formation of beliefsabout the nature of mathematics. Clearly the contribution to teachers’orientation to learning, and their beliefs about the nature of mathematicsare not solely the responsibility of teacher educators.

In an article that gives prominence to the nature of mathematicalknowledge, Leikin reports on a study of teachers’ preferences for solvingmathematics problems, for explaining their solutions, for liking theirsolutions and for teaching the approach to students. Working with in-service teachers, Leikin offered participants an unfamiliar approach tofamiliar problems. She posed some problems which are commonly solvedanalytically using introductory calculus, but which are also amenableto geometrical solutions based on symmetry. Her basic premise is thatteachers need to be fluent problem solvers, and in particular to be able tosolve problems in more than one way. This highlights the need for teachereducators to go beyond focusing on knowledge of mathematics to beliefsabout the nature of mathematics and mathematical learning. The teachersin the project chose to solve problems in ways they considered easier, theypreferred to explain solutions in the way that was easier, even though theyliked the more elegant solutions. Leikin suggested that the curriculum andassessment regimes acted as a constraining influence on the approachesteachers might choose to use.

Working in a similar way, with primary level teachers, Farmer,Gerretson and Lassak also examine the nature of mathematics through

296 PETER SULLIVAN

reports of three case studies from a professional development programwhich focused on mathematical learning. Through an examination ofparticular mathematical activities, they emphasize an orientation toproblem solving, student thinking, and sense making. The professionaldevelopment program described by Farmer et al. intended to build onthe experience and mathematical learning of the teachers, as well as theirfundamental disposition and beliefs. They sought to engage teachers inauthentic mathematical learning experiences through the posing of sixproblems, some of which were open-ended, and by fostering small andlarge group discussions. Farmer et al. noted three levels of approach fromthe teachers, from focusing on activities, to focusing on understanding,attitudes and beliefs, and ultimately to teaching as inquiry. They notedthat factors contributing to effectiveness of the professional developmentwere the inclusion of appropriate and adaptable student-centred learningactivities, supported opportunities for discussion, and an inquiry stancetaken by the facilitators.

Focusing also on the nature of mathematical learning, as well as theexperiences that prospective teachers have had in undergraduate programs,Goulding, Hatch and Rodd argue that consideration of the previous mathe-matical experiences of prospective teachers can be a fruitful experience.They surveyed prospective teachers in a one-year graduate course abouttheir experience of undergraduate mathematics at university. Many of theirprospective teachers felt the transition from school to university was diffi-cult, some found the demand to struggle challenging and did not valuethat experience, and some resented the unresponsive style of teachingand assessing in their undergraduate mathematics. Goulding et al. arguethat teacher educators can use the process of reflection on undergraduatemathematics by the prospective teachers as a prompt for discussion onhow their own learning experiences might impact on their future beliefsabout teaching. They noted seven key issues to be considered in teachereducation including transition, understanding, the nature of independenceand support, the task of teaching, assessment, the experience of learningmathematics, and the value and nature of mathematics as part of thisprocess.

Taken together, the articles raise the critical issues of knowledge of,and beliefs about, mathematics, and pose creative and interesting waysthat teacher educators can build on the prior experience of the prospectiveteachers to enhance their mathematics teacher education.

ROZA LEIKIN

PROBLEM-SOLVING PREFERENCES OF MATHEMATICSTEACHERS: FOCUSING ON SYMMETRY

ABSTRACT. The aim of the study presented in this paper was to explore factors thatinfluence teachers’ problem-solving preferences in the process of (a) solving a problem,(b) explaining it to a peer, (c) liking it, and (d) teaching it. About 170 mathematics teacherstook part in the different stages of the study. A special mathematical activity was designedto examine factors that influence teachers’ problem-solving preferences and to developteachers’ preferences concerning whether to use symmetry when solving the problems.It was implemented and explored in an in-service program for professional developmentof high-school mathematics teachers. As a result, three interrelated factors that influ-ence teachers’ problem-solving preferences were identified: (i) Two patterns in teachers’problem-solving behavior, i.e., teachers’ tendency to apply a stereotypical solution to aproblem and teachers’ tendency to act according to problem-solving beliefs, (ii) the wayin which teachers characterize a problem-solving strategy, (iii) teachers’ familiarity witha particular problem-solving strategy and a mathematical topic to which the problembelongs. Findings were related to teachers’ developing thinking in solving problems andusing them with their students. The activity examined in this paper may serve as a model forprofessional development of mathematic teachers and be useful for different professionaldevelopment programs.

KEY WORDS: mathematics teachers, problem solving, professional development,symmetry

PROLOGUE

PART I: I Can SOLVE the Problem That Way but This Way I CanUNDERSTAND it

My friend’s son (Ron) asked me to help him solve the followingproblem:

Problem 1: Of all the triangles with a given side s and given area A,which one has the minimal perimeter? (See Figure 1)

This problem from a high-level Israeli matriculation examination issupposed, according to the curriculum, to be solved using calculus tools.Ron was one of the best pupils in his class, yet he had an unusual diffi-culty when solving this problem on his own. During our meeting, wediscussed which elements of the triangle could be chosen as a variable


298 ROZA LEIKIN

Figure 1. Problem 1.

for an appropriate function and what is the most convenient choice forsolution. Eventually Ron solved this problem easily by performing a well-known algorithm (i.e., writing a function, finding its derivative, etc.). Atthis stage, I suggested solving this problem using symmetry. He had notstudied symmetry in school but solved the problem based on symmetryconsiderations with my guidance (see Appendix, Card 1, Solution 2).

Although he said he “enjoyed this solution and it was not difficult atall”, Ron’s reaction was unusual. He became sad, and then after someminutes of silence he said:

Why don’t they teach us to do it this way? I can solve the problem that way [using calculus],but this way I can understand the solution. I can see it, I can feel it, and the result makessense.

My decision to conduct this study was motivated by Ron’s “Why” and bymy conviction that using symmetry in problem solving could help teachersteach mathematics for understanding.

PART II: Teachers’ RELUCTANCE TO USE SYMMETRY when SolvingProblems

In this study, symmetry in its general sense was considered as an inter-disciplinary concept within mathematics. I first conducted a pilot study inwhich about 40 high school and junior high school mathematics teacherswith different teaching experiences took part. The purpose of this pilotinvestigation was to get a first impression about whether mathematicsteachers use symmetry when solving problems. “Thinking symmetry” waspresented to the teachers as a way of thought and as a problem-solvingstrategy that was an alternative to those based on different traditionalschool mathematics tools and that belonged to different mathematicaltopics. The teachers were asked to solve Problem 2 (see below), which isa well-known example of the use of symmetry in mathematical problem-solving. This problem is similar to Problem 1, which may be considereda special case of Problem 2. Eight high school teachers were interviewedindividually, six junior high school teachers were observed while working

TEACHERS’ PROBLEM-SOLVING PREFERENCES 299


in pairs and twenty-two teachers (working in either senior high schoolor junior high school) solved the problem in small groups. Additionallyfour preservice teachers were interviewed before teaching Problem 2 andwere observed when presenting pupils with this problem and managing itssolution in microteaching settings.

Problem 2: Let l be a straight line and A and B two points on the sameside of the line. Find a point C on line l, such that the sumof its distances from point A and from point B is minimal.(See Figure 2)

Problem 2 (as in the case of Problem 1) is usually included in Israelisecondary mathematics textbooks in the topics of Calculus and AnalyticGeometry without any reference to the concept of symmetry. Thus, notsurprisingly, most of the teachers (38 out of 40) did not use symmetrywhen solving the problem. Moreover, when the teachers were presentedwith a “symmetry solution” they expressed uncertainty as to whether thissolution was admissible. I found that all the teachers demonstrated twopatterns of problem-solving behavior: (a) applying a stereotypical solutionto a problem, and (b) acting according to problem-solving beliefs.

Applying a stereotypical solution: Such teachers’ problem-solvingbehavior is based on an automatic connection between problems of aparticular type and a particular problem-solving strategy. Application ofa stereotypical solution relies mainly on teachers’ previous mathematicalexperience. For most of high school teachers, using the derivative of anappropriate function served as a stereotypical solution when finding aminimal distance. Secondary school teachers made a connection betweenconcepts of “minimal distance” and “perpendicular to a line”.

Acting according to problem-solving beliefs: When teachers approached aproblem based on their feeling of “what school mathematics is” or “what

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is good for their pupils” their mathematical behavior was attributed tothe teachers’ problem-solving beliefs. Usually this kind of behavior wasobserved despite the facts that the teachers knew how to solve a problemusing symmetry and that their pupils seemed comfortable with solutionsbased on symmetry. In these cases the teachers often made comments like“it [the solution based on symmetry] is beautiful but doesn’t prove” or that“this is not a solution, it is only an illustration”. Teachers’ beliefs regardingtheir pupils were expressed when they found solutions based on symmetry“good for a teacher but not good for a pupil”. In many cases when actingaccording to their problem-solving beliefs, the teachers tended to apply astereotypical solution.

In summary, the pilot investigation revealed a conflict between pupils’preferences for using symmetry when solving problems and teachers’competence, or willingness, to satisfy these preferences. Consequently themain study had two principle purposes (a) to examine factors that influ-ence teachers’ preferences of a particular way of solving a problem and(b) to explore the possibility of developing teachers’ preferences for usingsymmetry when solving problems.

THEORETICAL BACKGROUND

Research on problem-solving in general and teaching problem-solving inparticular has been the focus of mathematics educators for many years.One of the directions that the US National Academy of Education (1999)emphasizes as being of great importance is how teachers can promotestudents’ understanding of various problem-solving methods. The currentstudy stems from an assumption that, if teachers are to be able to promotemeaningful problem-solving activity, they have themselves to be fluent inmathematical problem-solving. This problem-solving expertise is seen tobe strongly connected to teachers’ content knowledge (e.g., Polya, 1963;Silver & Marshall, 1990; Yerushalmy, Chazan & Gordon, 1990). Oneof the valuable components of this knowledge, which is connected toproblem-solving expertise, is solving problems in different ways.

Research on teaching mathematics has found that to teach for under-standing the teachers need deep and connected knowledge of the relatedsubject matter (see, for example, Chazan, 2000; Askew, 2001). Accordingto Askew (2001), a “connectionist orientation” distinguishes some highlyeffective teachers from others. Thus, solving problems in different waysis especially important as it may develop connectedness of one’s mathe-matical knowledge (NCTM, 2000). Ma (1999), in her study on teachers’mathematical understanding, demonstrated that for Chinese elementary


teachers who have deep conceptual understanding “solving one problemin several ways [yiti duojie] . . . seemed to be an important indicator ofability to do mathematics” (Ma, 1999, p. 140). Moreover, these teachersconsidered it one of the ways in which they could “improve them-selves”. Dhombres (1993) suggested that providing two different proofsfor one particular theorem opens different routes for solvers in theirmathematical knowledge, each of which may be available when appro-priate. Schoenfeld (1985) points out that awareness of opportunity to solveproblems in different ways helps students not to give up when solvingproblems.

However, teachers usually do not solve problems in different waysthemselves or in their classrooms (Schoenfeld, 1988). Moreover, some-times teachers do not accept students’ solutions that are different fromthose that they had taught in the mathematical lessons. This leads todevelopment of students’ beliefs that, for any mathematical problem, thereis only one acceptable solution, to be achieved by applying one particularproblem-solving strategy. Such teachers’ instructional behavior is rootedin their content knowledge and in their beliefs (Cooney, 2001; Calderhead,1996; Schoenfeld 2000; Sullivan & Mousley, 2001; Thompson, 1992).Thomson (1992) contends that teachers’ beliefs, knowledge, and prefer-ences concerning the discipline of mathematics constitute their conceptionof the nature of mathematics. Scheffler (1965) distinguishes betweenknowing and believing by claiming that knowing has “prepositional andprocedural nature” whereas believing is “construable as solely proposi-tional” (p. 15, ibid.). Cooney (2001) adds an additional condition toScheffler’s definition of beliefs as “abstract things, in the nature of ahabit or readiness” that express “a disposition to act in a certain wayunder certain circumstances” (Scheffler, 1965, p. 76). Cooney claims thatverbal proclamations of beliefs must be followed by actions, which areconsistent with these beliefs and provide evidence of the beliefs. In thisstudy, teachers’ statements regarding a particular problem-solving strategythat were substantiated either by applying this strategy or by reluctance touse or accept it were considered as expression of teachers’ beliefs.

Considering problem-solving-in-different-ways as an important pieceof teachers’ instructional behavior, this study examines the possibilityof influencing teachers’ preferences for varying their problem-solvingstrategies by presenting them with different solutions of mathematicalproblems. One of the presented strategies was always symmetry-based.Symmetry was chosen as a mathematical context for this study forseveral reasons. One the one hand, many mathematicians and mathematicseducators stress the important role that symmetry has in problem-solving

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in various branches of mathematics (e.g., Dreyfus & Eisenberg, 1990;Polya, 1973; Polya, 1981; Schoenfeld, 1985; Yaglom, 1962; Weyl, 1952).Solutions of mathematical problems using symmetry are often elegant andreveal the essence of the problem conditions. On the other hand, in theIsraeli secondary school mathematics curriculum, symmetry is mentionedjust once, in connection with the quadratic function. Dreyfus and Eisen-berg (1990) and my pilot investigation demonstrated that teachers inIsrael do not usually use symmetry when solving mathematical problems.Consequently one of the aims of the study was finding a way to advanceteachers’ use of symmetry in problem solving.

In the framework of this study a special mathematical activity “Solvingproblems in two different ways” was designed and implemented within anin-service professional development project for senior high school mathe-matics teachers (for details see Zaslavsky & Leikin, 1999). The goals anddesign of the activity were aligned with what Cooney (1994), Comiti andBall (1996), and Ball (1997) suggest as essential: providing constructiveand reflective opportunities to deepen teachers’ mathematical knowledgein general and their problem-solving expertise in particular. This studyembodied a reflective component in which teachers explicitly consideredthe possibility of the implications of their own learning experiences fortheir teaching (Cooney & Krainer, 1996, p. 1162). Teachers’ reflection-in-action (Schön, 1983) and teachers’ reflection-on-action (Jaworski, 1994)were fundamental parts of the mathematical activity that was designedfor the purposes of this study (as depicted in Figure 3 in the Methodo-logy section). Teachers’ reflective activities comprised various aspects,such as pedagogical considerations, implications for students’ learning,and concerns regarding teachers’ own mathematical practice. In this way,these reflective activities served both research and learning purposes.Furthermore, all the research instruments in this study were constructed toenhance reflection by the participating teachers, in the belief that reflectionis a key issue in teachers’ learning.

OBJECTIVES AND RESEARCH QUESTIONS

The aim of the study presented in this paper was to explore factors thatinfluence teachers’ preferences when choosing a particular way to solvea mathematical problem. The possibility of developing teachers’ prefer-ences to solve problems using symmetry was examined and the changesin teachers’ problem-solving preferences were analyzed. Following theobjectives of the study, two main research questions were considered:


1. What factors influence teachers’ problem-solving preferences?2. How does the designed activity change teachers’ problem-solving

preferences?

METHODOLOGY

The study was carried out in two stages. In the first stage of the study, themain characteristics of different problem-solving strategies, as perceivedby the teachers, were identified and research instruments were constructedand examined. During the second stage of the study the research instru-ments were used to seek data to answer the two research questions.

POPULATION

One hundred experienced (at least 5 years of teaching) high school mathe-matics teachers participated on a one-time basis in the first stage of thestudy. Each of these teachers took part in a two-hour learning-researchsession in a group of 20 teachers; thus five sessions were organized at thisstage of the study. Teachers worked in pairs according to the Exchange-of-Knowledge method (as presented in the next section) and were askedto complete a Response Questionnaire at the end of the session. None ofthese teachers participated in the second stage of the study.

Thirty-three experienced (at least 5 years of teaching) high schoolmathematics teachers participated in the second stage of the study. Theteachers were experienced in teaching such topics as Calculus, AnalyticGeometry, Advanced Algebra (i.e., exponential and logarithmic equa-tions, complex numbers, number sequences, mathematics induction, andcombinatorics). Some of these teachers also had experience in teaching3D-Geometry, and Vectors. The teachers took part in four two-hour work-shops managed according to the Exchange-of-Knowledge method once aweek for a month. These workshops were incorporated at the end of thesecond year of a five-year professional development program, in whichthese teachers participated voluntarily (for details of the program seeZaslavsky & Leikin, 1999). These teachers completed Characterizationand Preferences questionnaires as described below.

Note here, that the teachers who took part in the second stage ofstudy continued their participation in the professional-development projectduring two years following the study. Thus, as can be seen in the ResearchInstruments section, these teachers provided supplementary data for thestudy.

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THE ACTIVITY

The teachers who participated in the study took part in a special mathe-matical activity “Solving problems in two different ways”. The activityexplicitly emphasized importance of two elements of Jaworski’s (1992)teaching triad: (i) mathematical challenge that involves stimulating mathe-matical thought and inquiry, and (ii) the management of learning thatconcerns the creation of a learning environment and establishes classroomvalues and expectations (Jaworski, 1992; Zaslavsky & Leikin, 1999).Therefore, solving-problem-in-different-ways was presented as an explicitmathematical challenge for the teachers and the teachers worked insmall-group learning settings that incorporated worked-out examples.Teachers were informed that symmetry was chosen for these activities toenable solving problems in different ways in different mathematical topicsbecause of its interdisciplinary nature within mathematics.

Firstly, the activity was based on the learning method Exchange-of-Knowledge in Pairs (Leikin, 1997; Leikin & Zaslavsky, 1999). The methodcombined the advantages of cooperative small-group settings with thoseof learning from worked-out examples. Based on research findings thatsome small-group learning methods have been shown to increase learners’communications (Good et al., 1992; Leikin & Zaslavsky, 1997; Webb,1991), small-group learning settings for the high school mathematicsteachers were used as a research tool to create and observe teachers’ inter-actions in the course of problem-solving. Noddings’ (1985) analysis ofsmall groups as a setting for research on mathematical problem-solvingand use of “pair problem-solving” in other studies in the past (e.g., Schoen-feld, 1985) showed “pair problem-solving” to be an effective tool to evokericher problem-solving procedures. As these studies suggested, learnersworking in pairs are far more verbal than learners working alone, and theirsolutions are often more elaborate and advanced. The learning setting usedin the study allowed teachers to choose between different ways to solutions– one of which was based on symmetry (see Appendix 1) – in order tosolve problems individually and in order to explain problems to the otherteacher.

Secondly, research in mathematics education demonstrated that the useof worked-out examples facilitates the acquisition of knowledge requiredfor problem-solving in mathematics and science (Ward & Sweller, 1990;Zhu & Simon, 1987). Consequently, learning materials built for the studyincluded worked-out examples (see Appendix 1) to alleviate the pressurethat may be present when teachers participate in educational research.The same learning setting (as presented below) was used in the first


and the second stages of the study accompanied by different types ofquestionnaires.

Working cardsFor the purpose of the study, eight special working cards were designedby the researcher, each consisting of three parts (for two examples, seeAppendix 1).

Part I presents a worked-out example of a problem solved in two ways,one of which is based on the use of symmetry.

Part II includes a similar problem that can be solved in either of thesetwo ways.

Part III includes a different problem that can be solved by means ofsymmetry.

Different cards included problems from different topics from thesecondary mathematics curriculum: geometry, algebra, combinatorics,calculus, and complex numbers. The alternative solutions presented onthe cards were based on different types of symmetry: (i) geometric, i.e.,connected to geometric transformations of geometric figures: (ii) alge-braic, i.e., connected to permutations of algebraic symbols; and (iii)logical, i.e., connected to symmetry in proofs (for details see Leikin,Berman & Zaslavsky, 1998). Thus overall the activity presented theteachers with different ways of solving problems.

The settingThe Exchange-of-knowledge method consists of two main stages:(i) preparation of “card experts”, and (ii) pairs of exchange-of-knowledge.

Preparation of “card experts”: Teachers worked in pairs. If there was anodd number of participants, then three teachers worked together, and therest worked in pairs. Equal numbers of pairs (including the group of threeif there was one) of teachers received two different working cards chosenfor a session. The teachers in a pair received the same working card andwere asked to read and understand the two methods of solution presentedin Part I of the card. After that the teachers were asked to solve a problemfrom Part II of the card in one of the two ways presented in Part I. Theteachers could do this individually or together with their partner, as theydesired. The process of deciding how to solve a new problem may beviewed as reflection-in-action. Finally, at this stage of work, the teacherswere requested to discuss their solutions of the problem from Part II in apair.

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Problems in Part III of the cards served as a reserve for the teachers whofinished two parts of the cards earlier than others. Usually the problems inthis part were both different from, and more difficult than, the two previousproblems.

Pairs of exchange-of-knowledge: After completion of the work with thiscard, the teachers were called “card experts” (experts in the field of thecard) and changed their peers for one who had a different card and workedin a new pair. In these new pairs the “card expert” teachers were requiredto choose one of the ways to solve the problem presented in Part I of theircard in order to explain the problem to the other teacher. The need to makea choice brought teachers into reflective analysis of the solutions. Afterexplaining solutions, teachers exchanged their cards and were asked tosolve problems from Part II of the cards that were new to them. Teacherscould communicate when solving the problems, ask questions and helpone another.

The whole group discussion: Each workshop ended with a whole groupdiscussion that took from 15 to 20 minutes. The teachers were asked toreflect on their participation in the activity. Usually during these discus-sions the teachers expressed their difficulties, suggestions, and concernsregarding mathematical and didactical issues of the sessions.

THE DATA COLLECTION AND ITS ANALYSIS

The research data were collected from several sources: research ques-tionnaires, videotaped pairs work and group discussion, and some othersupplementary data. This section describes data collection and the waysthe data were analyzed.

Questionnaires

Three different questionnaires were constructed for this study: at the firststage of the study – a response questionnaire, at the second stage of thestudy – a preference questionnaire and a characterization questionnaire.

The Response Questionnaire: The questionnaire was administered on aone-time basis to 100 high-school mathematics teachers at the first stageof the study. One of the purposes of the study was to explore factorsthat influence teachers’ preferences when choosing particular ways ofsolving a mathematical problem. Teachers’ views regarding the advan-tages and disadvantages of the different solution strategies were observed


in the course of the pilot study. To verify this observation and makeit more precise, the teachers were asked to list advantages and disad-vantages of the different solutions that were presented in the workingcards (see Appendix 2). Advantages and disadvantages listed by theteachers were categorized according to their similarity and the identifiedcategories were called “problem-solving-characteristics”. Each one of thecategories was defined as a characteristic if it was found in not fewerthan 30 to 100 response questionnaires. Reliability of the categorizationwas examined by inter-coder agreement between four experts. The expertswere asked to classify teachers’ responses according to the categoriesdefined by the researcher or to suggest new categories. A category wasaccepted if at least three of the four experts agreed about the classifi-cation. These problem-solving characteristics were used for constructionof the characterization questionnaire that was used in the second stageof the study. The identified characteristics represented teachers’ opin-ions on how particular ways to solution were (i) difficult (when teaching,solving, explaining, and understanding), (ii) interesting, (iii) conventional(i.e., regular, standard), (iv) convincing, (v) inspiring (i.e., developingcreative mathematical thinking), (vi) challenging (i.e., demanding creativemathematical thinking), (vii) beautiful.

The Characterization Questionnaire (see Appendix 3): Together with thepreference questionnaire (see Appendix 4), the characterization question-naire aimed to analyze the relationship between the ways teachers charac-terize different problem-solving strategies and their preferences regardingthe use of these strategies. The questionnaire consisted of propositionseach with a particular problem-solving characteristic that had been definedon the basis of teachers’ answers to the response questionnaire. In theresponse questionnaire, there was no consistency among the teachers asto which characteristics were associated with regular problem-solvingstrategies and which were associated with symmetry-based solutions.Thus, for each characteristic the teachers were asked to answer which ofthe two ways of solution fits the proposition better. Each of the problem-solving characteristics appeared in the response questionnaire as bothadvantages and disadvantages of the solutions. Thus, the teachers wereasked to report whether they consider a proposition an advantage or adisadvantage of a particular problem solving strategy. All the proposi-tions were formulated as positive statements to make the characterizationquestionnaire easy to use. When classified, all the responses in whichteachers claimed that a solution based on symmetry was more difficult,were considered equivalent to those that claimed that a solution based

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on calculus tools was easier. Similarly, teachers’ responses in which theystated that conventionality of a solution was an advantage were consideredidentical to those in which the teachers claimed that unconventionality ofa solution was its disadvantage.

The Preference Questionnaire (see Appendix 4): In this study teachers’problem-solving preferences are considered for the four differentprocesses in which teachers of mathematics are often involved: solvinga problem, explaining a problem’s solution, liking a solution, teaching tosolve a problem. The preference questionnaire was used for two purposes.Firstly, combined with the characterization questionnaire it aimed to showrelationships between the ways in which teachers characterize differentproblem-solving strategies and teachers’ preferences regarding the use ofthese strategies. Secondly, it aimed to follow changes in teachers’ prefer-ences regarding solution strategies in each of the processes listed above.The questionnaire was administered at the end of each of the four work-shops. Thus, each teacher answered these questionnaires four times. Onehundred and eleven questionnaires of each kind were received.

Overall, the research procedure integrated a mathematical activity, inwhich the teachers were involved, with reflection on the activity in parti-cular and on their teaching practice in general. The need to choose differentways to solutions for different purposes required reflection-in-action andthe need to answer questionnaires required reflection-on-action. Solvingproblems in different ways was embedded in small-group learning settingsthat allowed the teachers to choose problem-solving strategies freely, andallowed the researcher to keep track of these choices (See Figure 3).

Videotapes

To analyze teachers’ problem solving strategies and factors that influenceteachers’ preferences for using a particular strategy when solving the prob-lems, four teachers were videotaped during each session: first as two pairsof “card experts” and then as two pairs of “exchange-of-knowledge”. Eachpair of “card experts” was given a different working card; the original pairswere then split at the “exchange-of-knowledge” stage so that members ofthe new pair were experts in different cards. In addition to the pair work,all the group discussions that concluded each of the meetings at the secondstage of the study were videotaped. The videotaped data were used mainlyto support numerical data from the questionnaires and qualitative data fromthe pilot study.


Figure 3. The setting.

Supplementary data

During the second stage of the study, two staff members observed interac-tions (which were not videotaped) between the teachers in two pairs andconducted written protocols of these observations to collect additional dataabout teachers’ problem-solving strategies and their preferences.

As mentioned earlier, most of the teachers who took part in thesecond stage of the study continued their participation in the professional-development project during two years following the study. The researchercontinued her work on the project as a staff member. Thus, some supple-mentary data were collected during these two following years from severalsources. First, the data were collected from reports of other staff memberswho conducted the workshops for these teachers and observed interestingepisodes connected with the use of symmetry in their workshops. Second,

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all the workshops in the project were videotaped and different episodesrelevant to the study could be observed from the videotapes. Third, theresearcher continued to communicate with the teachers during meetings,through e-mail messages and telephone calls, and the teachers shared withher their mathematical and pedagogical experiences related to the use ofsymmetry. Note that no systematic investigation took place in teachers’classrooms.

RESULTS OF THE STUDY

The prologue in this paper described briefly the main findings of the pilotstudy. This section presents results of the main study that included twostages: the first stage aimed at the construction of the research tools andthe second stage aimed at answering the research questions. The findings inthis section are from the second stage of the main study and are organizedin accordance with the research questions.

What factors influence teachers’ problem-solving preferences?

Three main factors that influence teachers’ problem-solving preferenceswere identified in this study:

i. The ways in which teachers characterize different problem-solvingstrategies;

ii. Patterns of teachers’ problem-solving behavior, i.e., teachers’ tend-ency to apply a stereotypical solution to a problem and teachers’tendency to act according to their problem-solving beliefs; and

iii. Teachers’ familiarity with the type of symmetry and with the mathe-matical topic to which the problem belongs.

All of these factors are interrelated and are connected to teachers’ mathe-matical and teaching experiences.

The ways in which teachers characterize different problem-solvingstrategies and their problem-solving preferences

Identification of problem-solving characteristicsAnalysis of the response questionnaire indicated that different problem-solving characteristics were attributed to different solution strategies,i.e., symmetry-based strategies and those that do not use symmetry. Forexample, sixty-two teachers (of one hundred) referred to difficulty of thesolution. Among them, thirty-two teachers thought that a solution based


on calculus tools was easier for a solver whereas thirty teachers founda solution based on symmetry easier. In addition, some of the problem-solving characteristics were considered by different teachers to be eitheradvantages or disadvantages. For example, 95 teachers referred to theconventionality of a solution. Sixty-one teachers found the calculus solu-tion more conventional. Fifty of these teachers claimed that this wasan advantage of the solution and 11 teachers thought that this was itsdisadvantage. Thirty-four teachers found a solution using symmetry lessconventional, 22 of them thinking that this was the advantage of thesolution and 12 teachers considering it its disadvantage.

Relationship between the ways in which teachers characterize differentproblem-solving strategies and teachers’ preferences towards thesestrategiesCorrelations between teachers’ responses on the preferences questionnaire(Appendix 4) and their responses on the characterization questionnaire(Appendix 3) were examined. As a result, the following relationshipsbetween teachers’ problem-solving preferences and the ways in which theteachers characterize the problems were found:1

• Teachers prefer to solve a problem using the way they consider easierto solve, easier to explain, easier to understand, or more convincing

• Teachers prefer to explain a problem using the way they considereasier to explain, more interesting, more convincing, more beautiful,or more challenging.

• Teachers like the way they consider more beautiful, easier to solve,easier to understand, or easier to explain.

• Teachers prefer to teach a problem using the way they consider moreconvincing.

Some of these relationships between teachers’ problem-solving prefer-ences and the ways in which the teachers characterize the problems (e.g.,preferring to solve a problem using the way they consider easier to solve)seem to be very natural. Some other results may be considered as unex-pected, like for example, teachers’ preferences regarding teaching. It couldbe desired that the teachers would choose for teaching not only solu-tions that they considered to be more convincing but also solutions theyconsidered to be challenging and beautiful. Note that no correlation wasfound between teachers’ preferences of different types and their perceptionof a solution as more likely to develop mathematical thinking.

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The two patterns of teachers’ problem-solving behavior and theirproblem-solving preferencesTeachers’ mathematical and teaching experiences are mainly expressedin the two patterns of teachers’ problem-solving behavior, i.e., teachers’tendency to apply a stereotypical solution to a problem and teachers’ tend-ency to act according to their problem-solving beliefs. These patterns wereidentified first in the course of the pilot study, as described in the prologue.

Both in the pilot study and during the second stage of the main study,some teachers, despite knowing how to solve a problem using symmetry,were reluctant to use symmetry and even to accept these solutions,expressing their feelings of the mathematical or didactical insufficiencyof a solution. The teachers could explain that “The solution is not inthe curriculum”, “The solution will not be accepted by the examinationcommittee”, “The solution is only demonstration but not a proof”, and“The solution it is not so good for most of the students”. These and otherexpressions of teachers’ feelings about mathematical nature of solutionsand about its didactical value were followed by the teachers’ actions inthe form of rejection of a solution and by attempts to apply a stereotypicalsolution. In other cases, even if the teachers were not competent to solveproblems using symmetry, they persisted in approaching the problemsusing symmetry because they found this way of solution “being moreelegant”, “being challenging”, “developing mathematical thinking”, and“explaining the result of solutions”. In the cases when the teachers aban-doned a particular solution, despite the fact that they were competent tofind the solution, or when the teachers kept on unsuccessfully solving aproblem in a particular way, their actions were considered as providing anevidence of their beliefs (Cooney, 2001). Such mathematical behavior wasconsidered to be based on teachers’ problem-solving beliefs.

Most of the videotapes of the pair work that were recorded during thesecond stage of the main study include episodes that demonstrate the twopatterns of the teachers’ problem-solving behavior. For example, duringthe first workshop two teachers (Naomi and Shelly) who worked as apair of “card experts” with card 1 solved the problem from part II of thecard (see Appendix 1) first using symmetry and then using a derivative“to check the solution”. They found the symmetry solution “elegant andbeautiful”. Nevertheless, they were not certain regarding the propriety ofusing this solution in the classroom. At the end of the solution, Shelly said:“Do you think you may solve [the problem] this way with the students?Maybe it is not strong enough? What about exams?” In the same work-shop, the teachers in the second videotaped pair of “card experts” also


attempted to solve the problem from part two of Card 2 (see Appendix 1)by using symmetry. They were able to connect algebraic symmetry of theexpression and geometric symmetry of its graph. However, the teachersfelt uncertainty regarding the correctness of the solution and decided tosolve the problem in the regular way. The following excerpt from the tran-script relates to teachers’ solution of the problem from Part II of Card 2(Appendix 1).

Rasan: She stated [the researcher in the solution presented in Part I of thecard] that the graph is symmetrical with respect to y = x becauseyou may exchange x and y. You may change x with y and y withx and you will get the same value. Do you understand?

Simon: Yes.Rasan: O.K. There is no difference between x and y [in the first problem].

Here there is no difference between x and (–y).Simon: Yes, it is symmetrical according to x = –y.Rasan: Thus it is perpendicular to it.Simon: Are you sure? I do not know. . . . Is this correct? I think we should

go to the first way, a regular way.Rasan: Let’s solve it [using the derivative].

When these teachers changed their peers and worked in pairs of“exchange-of-knowledge”, Naomi worked with Rasan who named thesymmetry solution “philosophical”. In the following excerpt, Rasan andNaomi discuss the ways in which they solved Problem 2 on their cards:

Naomi: Which way did you do it?Rasan: I did it the first [standard] way. This way [symmetry-based] is a

little philosophical . . .

Naomi: This is the standard way. I chose this method [symmetry-based]and I did it both ways . . .

Rasan: Ah. You chose a problematic one . . . Sophistication.

Thus these two pairs of the teachers, from the beginning, tried to usesymmetry when solving the problems. Similar to the findings in the pilotstudy, at the second stage of the study some teachers, such as Naomi,sometimes used both solutions “to check themselves” or “to feel secure”.Other teachers, such as Rasan, who considered a symmetry solution “notmathematical enough” or “too philosophical”, rejected symmetry-basedsolutions and preferred to solve problems in the regular way.

During the third workshop of the second stage of the main study Rasanand Simon worked together again. Rasan solved a problem (Problem 3: seeFigure 4) using logical symmetry and presented it to Simon who persisted

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Figure 5. The triangle must be isosceles.

in finding a stereotypical solution to the problem. Despite Rasan’s enthu-siasm, Simon was reluctant to accept his solution. Simon’s reluctanceresulted in Rasan’s feeling that symmetry is “not mathematical enough”.Consequently, he expressed his belief that the symmetry solution is just“blah-blah” and joined Simon in his attempts to find a stereotypicalsolution as presented in the following episode.

Problem 3: Among all the triangles inscribed in a circle which one hasthe maximal area, assuming that such a triangle exists?(See Figure 4)

Rasan: Ah. Let’s do it this way. Let’s take one side. [Points to side AB,Figure 5]

Simon: Why? Why should we go this way? The angle . . .

Rasan: Let’s do it this way. Suppose that this is constant [points to thechosen side AB, Figure 5]. To have maximal area for the triangleit must be . . . this vertex [vertex C] must be here [Rasan drawsan isosceles triangle having the chosen side AB as a base. AC =BC].

Simon: O.K. it is. We have said this.Rasan: This is all. We have finished. Let’s go to another side.


Simon: This is the same idea [here Simon made a connection betweenRasan’s solution and other problems (see for example Problem 1in Part III of Card 1, Appendix 1) that were solved using logicalsymmetry in proofs].

Rasan: This is the same idea; that is why this [the triangle] is a symmet-rical from all the directions, in other words it is equilateral.[While saying this he pointed to the isosceles triangle shown inFigure 5. He did not draw a new triangle.]

Simon: Yes . . . [continues solving in this way, trying to find a function.]Rasan: Do you understand my idea?Simon: Yes [saying it automatically – does not listen, continues to

struggle with the stereotypical solution]. SilenceRasan: But this is “Blah-Blah”, not mathematics [joins Simon in his

solution]

This excerpt shows that Rasan used geometric symmetry when statingthat the triangle was isosceles and logical symmetry in proofs to explainwhy it must be an equilateral triangle. Nevertheless, Simon, who claimedhe understood Rasan’s approach, did not accept the solution. Moreover,he persisted in writing a function of two variables with which he didnot know how to proceed while, based on the fact that the triangle isisosceles as stated by Rasan, he could have found a function in one variableand approached the solution more straightforwardly. As noted before theexcerpt, Simon’s reluctance to use symmetry caused Rasan to doubt themathematical nature of his own solution. The transcript demonstrates, thatafter some minutes of silence he concluded [to himself]: “But this is blah-blah, this is not mathematics” and joined Simon in his attempt to solve theproblem using derivatives.

Familiarity with the type of symmetry and of the mathematics of theproblemThe familiarity with the type of symmetry and the mathematical topicof the problem were found to influence teachers’ problem-solving pref-erences. Use of geometric symmetry was the most familiar to the teachers,while algebraic symmetry was less familiar. During the first meeting ofthe second stage, most of the teachers reported that they never heard aboutthe concept of algebraic symmetry. Teachers differed in their familiaritywith geometric and algebraic symmetry and this affected their preferenceas to which type of symmetry they used. Thus, during the first session ofthe second stage of the main study when solving a problem that could besolved using geometric symmetry (see Appendix 1, Card 1, Part II), 60%

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of the teachers preferred to solve the problem using geometric symmetry.At the same workshop, only 14% of the teachers who may have solvedthe problem using algebraic symmetry (see Appendix 1, Card 2, Part II)preferred the algebraic symmetry solution. Over the course of the mainstudy, the teachers were involved in the discussion of a unifying approachto the concept of symmetry (Leikin et al., 2000), as they consideredapplications of algebraic symmetry in different mathematical topics. Inother words, the concept of algebraic symmetry became more familiarto these teachers. Thus at the last workshop almost an equal number ofteachers preferred using different kinds of symmetry: 64% of teachers whosolved systems of equations preferred solving the problems using algebraicsymmetry: 67% of the teachers who where asked to find the value of anintegral preferred doing this using geometric symmetry.

The mathematics topic to which a problem belongs also influencedteachers’ preferences for using symmetry when solving a problem. Forexample, during the third meeting, 45.5% of teachers preferred to usegeometric symmetry for the “section of a cube” problem whereas at thesame workshop 75% of teachers preferred to solve the combinatorialproblem using geometric symmetry.2

Changes in teachers’ problem-solving preferences

To follow the changes in teachers’ problem-solving preferences, thepercentage of teachers (i) who preferred a solution without the use ofsymmetry, (ii) who preferred two ways of solution, and (iii) who preferreda solution based on symmetry were calculated. This calculation was basedon the preference questionnaire, taking each type of preference separatelyfor each session. Table I presents the percentage of teachers who reportedthat they preferred symmetry-based solutions only and who preferred usingtwo different ways of solution at the first and at the last workshops. It alsopresents the total percentage of the teachers who used symmetry at the firstand at the last workshops.

Overall, the activity influenced teachers’ preferences for different typesof problem-solving with regard to the use of symmetry. Note that at thebeginning on the second stage of the main study the teachers did notfeel confident enough with using symmetry. Thus, about 36% of teacherswho reported that they preferred to solve a problem using symmetry werecurious to compare the two ways of solutions and the other teachersdecided to “check correctness of the symmetry-based solution using thetraditional one”. In this way the teachers became involved in solving theproblems in different ways and could experience the advantages of this


TABLE I

Percentage of teachers who preferred a symmetry-based solution

Preferences For solving For explaining In liking For teaching

a problem a problem to a peer

First Last First Last First Last First Last

meeting meeting meeting meeting meeting meeting meeting meeting

Symmetry-based

solution only 24.1% 65.4% 27.6% 46.2% 44.8% 80.8% 17.2% 38.5%

Two different

ways one of

which is based

on symmetry 13.8% 7.6% 17.2% 25.9% 3.4% 7.6% 34.5% 39.5%

Total 37.9% 73.0% 44.8% 72.1% 48.2% 88.4% 51.7% 77.0%

process. In the course of the intervention, teachers’ explanations for whythey prefer to use two ways of solving a problem changed. During the thirdand the fourth meetings they claimed that they “found it interesting to solvea problem in two different ways and to compare them”, and that now they“can feel why it is important to let students know different problem-solvingstrategies”. Note that at the end of the intervention there were teachers(about 30%) who still preferred standard solutions. These were mostlymathematics teachers who had long teaching experience. They expressedtheir unwillingness to change “clear and easier” ways of solution for thenew ones, which are not included in the secondary school curriculum. Thismay be attributed to inertia in the teachers’ tendency to use stereotypicalsolutions and in the teachers’ problem-solving beliefs. Additionally wemay conjecture here that the period of investigation was relatively short(four weeks) to develop these teachers’ readiness for change.

Interestingly, different types of problem-solving preferences changedin different manners. The percentage of teachers who liked symmetryincreased quickly (from 44.8% at the first session to 63.6% at the secondsession). The percentage of teachers who preferred to solve problems usingsymmetry only changed slowly at the beginning and then increased quickly(from 24.1% at the first session to 27% at the second session to 61% at thethird session). The manner in which teachers’ preferences to use symmetrywhen explaining a problem changed was similar to that in which teachers’preferences to use symmetry when teaching increased and was differentfrom those when using symmetry in solving problems.

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Videotaped whole group discussions support the numerical data thatwere obtained from questionnaires. For example, at the end of the thirdworkshop Alina and Dorit felt that symmetry had become an essential partof their mathematics.

Alina: At the beginning it was difficult [to use symmetry]. Today I seethat it is easier to work with symmetry than it was at the previousmeetings. We learned to think in this way. Now I feel this is apart of my mathematics. Today I am convinced that symmetry isa strong problem-solving tool.

Dorit: It [symmetry] is easy to understand. It is connected to manytopics. It can be used instead of algebraic manipulations, insteadof derivative. You can use it instead of congruence and trigono-metry. You just need to see symmetry.

The written questionnaires, the observers’ protocols and the videotapesshow that during the study more than 70% of teachers started to solve prob-lems using symmetry, as in the example of Rasan in the excerpt presentedearlier.

SUMMARY AND DISCUSSION

The study presented in this paper aimed to explore factors that influ-ence mathematics teachers’ problem-solving preferences when solving aproblem, when explaining it to a peer, in liking it and when teaching it.For the purpose of the study a special professional-development mathe-matical activity “Solving Problems in Two Different Ways” was designed,focusing on the concept of symmetry. Using different strategies to solve aproblem and then reflecting on the mathematics being used was an integralpart of teachers’ learning (Romberg & Collins, 2000). In sum, the proposedactivity provided an effective research and professional-development back-ground that was based on emphasizing a reflective component of anin-service program and explicit consideration of the implications of theteachers’ own learning experiences for their teaching practice. Three mainfactors that influence teachers’ problem-solving preferences were identi-fied in the study: (i) Two patterns of teachers’ problem-solving behavior,i.e., teachers’ tendency to apply a stereotypical solution to a problem, andteachers’ tendency to act according to their problem-solving beliefs; (ii)The way in which the teachers characterize a problem-solving strategy;(iii) Teachers’ familiarity with the type of symmetry and with the mathe-matical topic to which the problem belongs. Each of these factors is based


on the teachers’ mathematical and teaching experience and all of thefactors are interrelated.

Teachers’ tendency to apply a stereotypical solution was considered inthis study to relate mainly to teachers’ mathematical knowledge. Basedon their personal mathematical experience and on their teaching experi-ence, the teachers connected a particular mathematical topic to a particularproblem-solving strategy. For example, most of the teachers who partici-pated in this study approached maxim-minima problems either with aderivative or by constructing a perpendicular. Then this strategy seemedmore “standard”, “conventional” and “acceptable”. When explaining whythey chose not to use symmetry when solving the problems, the teachersreferred to students’ need for a conventional problem-solving procedure.As Ball (1992) point out, “typically, students experience mathematics as aseries of rules to be memorized and followed. Speed and accuracy are whatcount; justification and reasonableness play little role” (p. 84). So, ourteachers, when discussing different ways of solving problems in generaland of using symmetry in particular, often asked: “Will they accept thissolution if a student uses it at the matriculation examination?” Our argu-ment that any mathematically correct solution must be accepted was notconvincing. The use of a stereotypical procedure, at least at the beginningof the intervention, seemed more acceptable.

In this study, teachers’ mathematical behavior that was based ontheir feeling of “what school mathematics is” or “what is good for thestudents” was attributed to teachers’ problem-solving beliefs. Interviewswith teachers in the course of the pilot study and videotapes of the secondpart of the main study demonstrated that teachers’ problem-solving beliefsstrongly influenced their mathematical performance and their preferencesfor using different problem-solving tools. As was demonstrated in thepaper, some teachers did not use symmetry because they did not believethat it was mathematical enough, whereas others did not use symmetrybecause they believed that “the use of symmetry could be good for teachersbut not good for their students”.

The other important factor that influenced teachers’ problem-solvingpreferences was the way the teachers characterize a problem-solvingstrategy. Figure 6 depicts correlations between different types of problem-solving strategies and different problem-solving characteristics. Severalproblem-solving characteristics that were identified at the first stage of themain study as frequently mentioned, i.e., conventionality of the solution,difficulty in teaching, and inspiration, were not found to be significantlyrelated to any type of problem-solving preference. Some of the problem-

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Figure 6. Correlations between the way the teachers characterize a particularproblem-solving solution and teachers’ preferences regarding this solution.

solving characteristics seem to have a strong relationship to the teachers’preferences, the others seem to have little influence. For example, difficultyof the explanation of the solution was found to be related to the three typesof preferences with high levels of significance (see Figure 6). Teachers’opinion that a solution is more convincing also related to three types ofproblem-solving preferences. At the same time, teachers’ opinion that asolution is interesting or challenging related only to teachers’ preferenceswhen explaining a solution to a peer.

Modes in which problem-solving characteristics are related to teachers’problem-solving preferences sometimes disclosed very natural connec-tions. For example, teachers chose to solve a problem in a way that theyconsidered “easier to solve”, to explain a problem in a way that was“easier to explain”, and liked a strategy that was “more beautiful” intheir opinion. Understandably, teachers chose to solve a problem usinga strategy that seemed to be easier to explain and to understand. Teacherschose to explain a solution to their peer in a way that they found to bemore interesting, more beautiful, more challenging and more convincing.However, the finding that there was no significant correlation betweenteachers’ problem-solving preferences regarding a particular strategy andtheir opinion that the strategy is more inspiring (i.e., may better developstudents’ mathematical reasoning) was one of the disappointing aspectsof this study. The pressure of the curriculum and the preparations for thematriculation examinations in the upper grades created an emphasis on thetechnical procedures. Thus the teachers liked solutions that they thought


were easier when solving, explaining, and understanding. Note also thatteachers preferred to teach a problem in a particular way if they consideredit more convincing.

All the factors that were found to influence teachers’ problem-solvingpreferences may be seen as grounded in the rules of mathematicalclassroom discourse and its socio-mathematical norms and practices estab-lished in the classroom community (Cobb, 2000; Sfard, 2000). As Sfardpoints out, roles of the participants in a mathematical discourse affect whatthey see as a purpose of the activity, what they count as a convincing argu-ment, and what they see as a required form of argument. On the one handteachers’ teaching and learning experiences were reflected in the findingsof this study. On the other hand, norms that were founded in the mathema-tical activities of the study influenced problem-solving preferences of mostof the teachers. In terms of Cobb (2000), the teachers when participating inthe study were involved as learners in contributing to the establishment ofthe social norms. Consequently they “reorganized their individual beliefsabout their own role, others’ roles, and the general nature of mathematicalactivity” (Cobb, 2000, p. 322).

The activity described in this paper was found to furnish an effectiveprofessional-development environment. The supplementary data in thisstudy show that the teachers acquired a shared conviction that they hadto create a variety of challenging learning conditions for their students.The teachers found both symmetry and solving problems in different waysto be attractive not only for themselves but for their students as well, andtherefore some teachers started implementing this in their own lessons.

The activity presented in this paper served as a model for profes-sional development of mathematics teachers and may be recommended forimplementation in different professional development programs focusingon other big ideas or different learning environments. Implementation ofthis activity in a professional development program developed teachers’awareness of the importance of such problem-solving characteristics asthe elegance of a solution and the mathematical challenge and inspirationof a solution in teaching mathematics.

One of the limitations of this study is its limited application toclassroom practice. Even though the teachers’ self-reports allowed usto see that some teachers implemented their learning experience in theclassroom, no systematic investigation of this issue took place. It mightbe interesting to examine systematically how often and in which way theteachers who took part in this study use symmetry in their classroom. Dothese teachers indeed teach their students to solve problems in different

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ways in other mathematical topics? If they do, how do their students learnin this way? How do teachers’ preferences for using different problem-solving strategies influence and depend on their students’ mathematicalperformance and problem-solving preferences?

EPILOGUE

I would like to think that if Ron asked me now “Why don’t they teach usto solve problems using symmetry?” I could answer him: “Some teachersdo”. The teachers who took part in the second stage of the study continuedtheir participation in the professional-development project during twoyears following the study. The teachers voluntarily informed me abouttheir different experiences related to using symmetry in the classroomand in other situations. They told me their stories when we met at theworkshops, they sent me e-mails and called me to share their experiencesof using symmetry. From this supplementary data it appeared that someteachers started to use symmetry on their own and to search for the casesin which symmetry might be applied. The teachers often were surprisedthemselves that they used symmetry even when they did not plan it.Their problem-solving behavior had changed. Most of the teachers beganto believe in symmetry as a powerful mathematical tool. The followingexcerpt from the interview with the teachers who participated in the projecttwo years after the end of the intervention exemplifies such a change:

Activities related to symmetry were very important. These activities left me with manyopen questions that I could not answer quickly. Several times it took me more than a weekto find a solution for these questions. These activities influenced my approach to teachingproblem solving and my definition of a good student. Before these activities I thought thatany good student could solve any problem immediately. Now when my students fail tosolve a problem and feel unsatisfied because of this failure, I ask students to talk aboutthings they do not understand and to solve the problem again. I think that ability to thinkabut difficulties, to talk about mathematics and to try solving a problem several times candefine a good student. This is the main change in my approach to problem solving.

ACKNOWLEDGMENTS

The paper is based on the D.Sc. thesis of the author submitted to theTechnion – Israel Institute of Technology, Haifa, under the supervisionof Abraham Berman and Orit Zaslavsky and was partly supported bythe Miriam and Aaron Gutwirth Memorial Fellowship. The study was


conducted within the framework of “Tomorrow 98” project in the UpperGalilee – A Model for Improving Mathematics High School Education,Orit Zaslavsky – Project Director.

I would like to thank Sara Hershkowiths, Barbara Jaworski, BebaShternberg, Anna Sfard and Michal Yerushalmy for their helpfulcomments on the earlier versions of the paper.

NOTES

1 Note that for all the reported relationships between the ways in which teachers char-acterize different problem-solving strategies and teachers’ preferences regarding the usethese strategies significant corrections were found. However these correlations were ofdifferent strengths. Space does not allow details to be included in this paper.2 For reasons of length, it has not been possible to include details of all the problems. Ifreaders are interested in these problems they can contact the author directly.

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APPENDIX 1


APPENDIX 2

Response questionnaire

Please, complete the following table: list advantages and disadvantages of the

different solution strategies that were presented in your card.

Way 1: Regular Solution Way 2: Based on Symmetry

Advantages

Disadvantages

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APPENDIX 3

Characterization questionnaire

Please compare the two ways of solving the problem presented in Part I of your original card.Complete the table.

Way 1 Way 2 Is this an advantage or a(mark X if (mark X if disadvantage of the solution?

appropriate) appropriate) Advantage Disadvantage Difficult todetermine

Mark X where appropriate

1. The way is easier when solvingthe problem.

2. The way is more interesting.

3. The way is more challenging(needs more creative thinking).

4. The way is easier to understand.

5. The way is more conventional.

6. The way is more convincing.

7. The way develops thinking more.

8. The way is easier to explain.

9. The way is more beautiful.

10. The way is easier to teach.


APPENDIX 4

Preference Questionnaire

Please complete the table regarding different ways of solving of the problempresented in your card.

Way 1 Way 2 Another way Why?(Mark X if (Mark X if (What is it?)

appropriate) appropriate)

In which way did you solvethe problem from Part II of

the first card?

In which way did you explainthe problem from Part 1 of the

first card to your peer?

Which way do you like better?

Which way would you use inyour classroom if you were to

teach problems of this kind?

Faculty of EducationUniversity of HaifaMount CarmelHaifa, 31905IsraelE-mail: [email protected]

JEFF D. FARMER, HELEN GERRETSON and MARSHALL LASSAK

WHAT TEACHERS TAKE FROM PROFESSIONAL DEVELOPMENT:CASES AND IMPLICATIONS

ABSTRACT. In this article, we report on an 18-month long mathematics professionaldevelopment project with elementary school teachers. Using a model we developed, threeparticipant case studies were analyzed with respect to not only the professional develop-ment milieu, but also how these teachers interacted with the professional developmentexperience. In particular we found that having teachers reflect on new, authentic reform-oriented mathematics learning experiences leads some teachers to take an inquiry stanceconcerning their own teaching, resulting in self-sustaining changes in their mathematicsinstructional practices. This implies that professional development for elementary mathe-matics teachers should include challenging mathematics learning experiences completewith opportunities to reflect on personal and professional implications.

INTRODUCTION

Entering the third decade after the publication of An Agenda for Action(National Council of Teachers of Mathematics [NCTM], 1980) and thenational conversation that it spawned, visions of reform have been articu-lated by many organizations (American Association for the Advancementof Science [AAAS], 1993; NCTM, 1991; NCTM, 2000). With the recentpublication of Principles and Standards for School Mathematics {PSSM}(NCTM, 2000), the question of how to support practicing teachers inimplementing the reforms it envisions looms ever larger. One of the twocore premises from the Glenn Report (US Dept. of Education, 2000)is that better teaching is the lever for change and effective professionaldevelopment is the indispensable foundation for high-quality teaching.

There are many kinds of mathematics professional developmentprojects possible (Loucks-Horsley et al., 1998; Sparks & Loucks-Horsley,1989). They vary in scale, purpose, audience, length of intervention,content, and structure. The question of which are most effective inimproving instruction is not trivial. Ball (1995) encourages mathematicsprofessional developers (and mathematics education researchers) to takean inquiry stance toward this question, experimenting to discover what can“work”. Moreover, there is the question of what it means for professionaldevelopment to work: What do teachers actually take from it?


332 JEFF D. FARMER ET AL.

Teachers may participate in professional development to gather specificactivities for classroom use or in order to learn how to implement particularcurricula, improve their instructional practices, obtain college credit, andfor other reasons. It may not be that teachers embark upon a professionaldevelopment activity in order to change their attitudes or beliefs, althoughthis is a common hoped-for outcome on the part of providers (Shifter& Simon, 1992). If we take Ball’s inquiry stance, we must uncover therelationships among professional development design, individual teachercharacteristics and actual outcomes.

To deal with these issues, we describe some of the general complexi-ties of designing professional development, and relate how our designemerged from negotiations in our design team. To discuss the resultingresearch project, we first describe our theoretical perspective and the emer-gence of our own model and research questions from our interactions withteachers and early formative feedback. Second, we present descriptions ofour methodology and three specific case studies that we chose to examinein greater depth, using the model. We conclude with implications for thedesign and implementation of mathematics professional development.

TENSIONS IN MATHEMATICS PROFESSIONALDEVELOPMENT

There are many things that can be meant by mathematics education reform.In addition, many reform documents articulate a rather coherent vision,but fall far short of explicating clear paths of implementation (Ball, 1995).The importance of problem-solving, attending to students’ thinking andencouraging sense-making were obvious to the authors (as mathematicseducation researchers), while the daily choices of problems, tasks, ques-tions, and processes are constrained by multitudinous factors which differamong grade levels, schools and individual teachers. This tension betweenthe vision of reform and the practicalities of teaching was the first tensionconfronted.

Another important tension revolves around ownership; namely, thatfor teachers actually to implement changes in instruction, they must beinvolved in creating and redesigning it. Yet, the very essence of changeis that something novel must happen; without the clear articulation of adifferent vision, what is designed is destined to resemble closely what hascome before. This conundrum is difficult to address and often overlooked(Ball, 1995).

There is also a conflict between time and content. We must recognizethat many current elementary teachers’ mathematical understanding is far

WHAT TEACHERS TAKE FROM PROFESSIONAL DEVELOPMENT 333

from ideal (Ma, 1999). Certainly there is not enough time in a one-weekworkshop to support teachers in the learning of mathematics to the depththat is considered normative in the PSSM (NCTM, 2000). Moreover, asingle workshop, without periods of gestation and sustained support, doesnot afford sufficient time for teachers to develop a deep understanding ofthe mathematics they teach; such understandings develop, if at all, overlonger periods (Ma, 1999).

An additional dimension to the time vs. content dilemma, madeapparent by PSSM (NCTM, 2000), concerns teachers’ knowledge of cogni-tion and research on how children learn mathematics. Given the numberand kinds of topics in most elementary curricula, familiarizing teacherswith even a representative portion of this knowledge base seems daunting.Yet we know, from experience with such projects as Cognitively GuidedInstruction [CGI] (Carpenter, Fennema & Franke, 1996) that such knowl-edge can be profitably and effectively used by teachers to improve theirinstruction.

ASSUMPTIONS AND PERSPECTIVES

Given these tensions and complexities, designing effective mathematicsprofessional development is no simple task. Among the designers of ourprogram (the first two authors of the article, a master elementary mathe-matics teacher and two district curriculum specialists), two fundamentalareas of agreement emerged which guided us.

First, while a professional development project provides “raw material”for change, teachers themselves ultimately determine what the impact willbe, and they bring a wide variety of experiences and needs to any project.We viewed mathematics content and effective reform-oriented pedago-gical practices as important needs, while teachers might view themselvesas having other needs (professional support, specific classroom activitiesthey could use, methods for teaching particular skills, etc.). Our designintended to take into account teachers’ expressed needs, and to supportchanges they might decide to make in their instructional practices, ratherthan prescribing our own paradigm.

Second, teachers’ mathematics learning must have a central positionin the project; such learning can serve as an entry point for addressingother goals. For example, by modeling constructivist pedagogies in ourmathematical activities, we entered into discussions of their usefulnessin elementary mathematics instruction. Furthermore, by using activitiesthat were adaptable to the elementary level and sufficiently challenging to


the teachers, we could address the desire of some teachers to add to theirrepertoire while still meeting a need to learn more mathematics.

We selected mathematical topics for their maximum relevance to ourteachers’ perceptions of their current needs, given our state’s mandatoryhigh-stakes testing environment. We also were forced to cull ruthlessly thepedagogical topics and focus on nurturing oral and written communication,understanding constructivist principles and using mathematical problem-solving, with some attention to a few additional principles from the PSSM(NCTM, 2000), such as equity and assessment.

Next, and perhaps more importantly, we realized that the only way tohave sustained impact would be to address explicitly teachers’ fundamentaldispositions and beliefs about the teaching and learning of mathematics.We focused on supporting teachers to become life-long mathematicslearners and inquiring, reflective practitioners, given the project’s timeframe. This development in our thinking parallels that of Rhine (1998):

I propose that our human, bounded rationality dictates that the value of educationalresearch to the teaching community is not the acquiring of research-based knowledgeof student understanding, but the process of teachers engaging with that knowledge andconsidering implications for their instruction. (p. 27)

Instead of focusing mostly on research-based knowledge of learning,we decided to engage our teachers in authentic mathematics learningexperiences. We use the word “authentic” here to mean being relevantto teachers’ professional needs and intellectually challenging, giventheir current mathematical understanding. By focusing the workshops onsupporting teachers both in learning mathematics and in reflecting on howthey learned it, we had the potential of influencing teachers’ (a) mathema-tical knowledge, (b) view of mathematics learning and teaching, (c) atti-tudes toward mathematics and mathematics learning, and (d) beliefsabout the nature of mathematics, mathematics learning and mathematicsteaching. These are important components of instructional change (Ernest,1989).

THE EMES PROJECT DESIGN

The objectives of the resulting Enhancing Mathematics in the ElementarySchool (EMES) project were to:

1. Increase participants’ knowledge of mathematical content relevant toelementary instruction in ways that model appropriate standards-basedinstructional practices.


2. Increase participants’ familiarity with national and state teaching andlearning standards for mathematics, the state mathematics assess-ment plan and increase their ability to implement standards-basedinstruction and assessment.

3. Increase participants’ awareness on issues of diversity and equityrelated to mathematics education and enable participants to changetheir instructional practices to be more supportive of all students’mathematics learning.

4. Enhance participants’ skills in problem-solving, critical thinking andmathematical communication in ways that can be directly applied toinstructional practices.

5. Support participants in integrating and applying knowledge gainedfrom project activities into their curriculum and instructional practices.

6. Support participants in professional collaboration and networking.

In general, we agreed that the teachers needed positive, reform-orientedmathematics learning experiences (such as solving interesting problems insmall groups, discussing ambiguities and hidden assumptions in problems,considering multiple representations of mathematical ideas, engaging inmathematical writing, etc.), discussion of pedagogy that would challengetraditional views of mathematics teaching and learning, and opportunitiesfor ongoing support to put new ideas into classroom practice. All of thiswas presented in two one-week summer institutes with regular Saturdaysessions held during the academic year. The “Probability and StatisticsInstitute” was offered the first and second summers, while the “Arithmetic,Algebra, and Geometry Institute” was also offered the second summer.

Although we were clear about what mathematical content to examineand how to facilitate learning of mathematics and reflection on pedagogy,we were less clear about how teachers’ implementation of reform prin-ciples would (or should) actually look. This was compounded by thediversity of assignments: among the approximately eighty participantswere teachers of kindergarten through grade six. We decided to encourageand support implementation, but to leave its actual direction and focusto the teachers. To assure their work was well-directed and focused, wemonitored their choices and engaged in some suggestion and negotia-tion. Our hope was that the changes they chose to make would be moreauthentic, effective and permanent than anything we could create andimpose. In short, we applied the principles of constructivist epistemologyto think about what and how our participants would learn (Loucks-Horsleyet al., 1998).

Virtually no one involved in the project (neither facilitators nor parti-cipants) experienced the kind of elementary school mathematics instruc-


tion envisioned in reform documents: rich in problem-solving, orientedtowards conceptual understanding and sense-making, supportive of allstudents’ mathematical thinking and focused on students’ activity andcognition rather than on textbook material. Thus, we designed everyproject session to include each of the following:

• Mathematical communication• Discussion of mathematical concepts and principles• Analysis of the learning process• Reflection on pedagogical principles (particularly those which chal-

lenge traditional views of teaching and learning)• Discussion of implications for classroom instruction and/or planning/

debriefing implementation activities

We describe one illustrative activity that took place in the proba-bility and statistics institute both summers. It began with the entire groupreviewing the definitions of the three measures of central tendency (mean,median and mode) and range. This was accomplished by questioning parti-cipants and discussing the definitions they provided; the group settledfairly quickly on the usual definitions.

Participants, working in small groups, were then given six problems.Each described attributes of a data set; participants were to construct adata set with the given properties. For example: “The mean of the dataset is 8, the median 10, the range 16; construct a possible data set”.Several problems were easy but some required more thought and onewas impossible (or barely possible, depending on rounding conventions).Groups that became stuck while working were encouraged to clarify theirthinking and explain their difficulties. After all the groups solved at leasta majority of the problems, solutions and problem solving methods werediscussed in the whole group.

Each group shared their data set for each problem. The participantslooked for similarities in these data sets, which provided clues to the rela-tionships among the measures of central tendency. When the impossibleproblem was discussed, other issues arose: What kinds of numbers arepermissible in a data set, can rounding be used, exactly how do thestated constraints combine to make the problem impossible, etc. Discus-sions often led into interesting mathematical byways. Participants reportedgaining a deeper understanding of the measures of central tendency.

This activity models the kind of question that Sullivan and Clarke(1991) call a “good question” in their book, Communication in theclassroom: The importance of good questioning. This book was providedto participants in the project. Some properties of such questions are thatthey:


• have more than one correct mathematical answer;• require more than recall of a fact or reproduction of a skill;• are designed so that all students can make a start;• assist students to learn in the process of solving the problem;• support teachers in learning about students’ understanding of mathe-

matics from observing/reading solutions.

Answering these questions provides opportunities for teachers to exper-ience firsthand deeper levels of conceptual knowledge construction, andshows them how small groups can work together to solve such complextasks.

As part of this activity, we discussed explicitly with participants howwe used Sullivan and Clarke’s (1991) procedure to create this task. Inanother session we asked teachers to modify traditional recall tasks fromtheir own curriculum to create similar good questions. The mathematicaland pedagogical processing of this activity takes over two hours. At theend of an activity (but before the full discussion of pedagogy), teachersmay write in their mathematics journal about the content learned. After thepedagogical discussion, teachers may be asked to reflect either in writingor orally in pairs or groups.

An important aspect of our facilitation involves the stance we takein leading the discussions and responding to questions. While bringingto light mathematical connections that were relevant to material beingdiscussed, we also made clear that we learned from the discussions. Wepointed out proposed solution methods that were novel to us, and tried tobe explicit about the expectation that we would all learn from the prob-lems, solutions, and discussions. The teacher-leader who co-facilitated theworkshops with us was also explicit about how she gains new insights inthe process of teaching mathematics.

In addition to small and large group discussions, participants engagedin a structured activity called a dyad (Weissglass, 1996) in which twopeople take turns talking, without interruption, comment or evaluation, forabout two minutes concerning the question at hand. Dyads were sometimesused during the discussion of mathematics problems, to help participantsfocus on prior knowledge at the beginning of an activity or to facilitatereflection and planning. Each institute day included significant time forteachers to plan implementation of their projects based on their reflectionand learning. Participants wrote in mathematical journals, gave writtenfeedback at the end of each session and wrote expository papers describingtheir instructional innovations; those participating in one institute wereasked to email the facilitators periodically with reports on their classroomimplementation.


THEORETICAL PERSPECTIVE AND EMERGENCE OF THERESEARCH PROJECT

The research project emerged for two reasons: a desire to know what wasbeing accomplished in EMES, and what our participants were getting fromit. We were hoping to see teachers change their disposition toward mathe-matics (viewing mathematics as being constituted by numerical and logicalrelations rather than in terms of skills), toward themselves (as mathe-matics learners) and toward mathematics teaching (viewing mathematicsteaching as supporting learning by facilitating conceptual understanding).A preliminary answer was provided by an external evaluator who usedpre- and post-tests, written comments from teachers and observations ofworkshop sessions to show that the project was operating consistentlywith the original proposal and that the project goals (stated earlier) weresubstantively met (Shaw, 2000). The research team (authors of this paper),however, became interested in looking deeper into how individual teachersinteracted with the project, and particularly in describing and analyzing itsimpact on them.

Our theoretical perspective can be described as holistic constructivism(Noddings, 1990). We accept that knowledge is constructed individu-ally by an organism through interaction with its environment, whileacknowledging that for humans, our environment is largely social. Weexplicitly reject the tendency to view either the psychological or the socialperspective of mathematical learning as primary, but instead maintain thatboth have a vital role to play in a rich description of the reality that emergesin a mathematics classroom.

This position is not taken for naively pragmatic reasons, but withthe principled assumption that reality is multilayered and complex, bothidiosyncratic and socially mediated. To “locate” cognition solely eitherwithin the individual brain or within the social milieu is fundamentallyto “give up” on understanding some important aspect of it. We locateknowledge construction (meaning-making) in the interactions of an indi-vidual (complete with prior knowledge schemas, experiences and socialidentities) with her or his (primarily social) environment.

This position is reflected in the design of the workshops through ouremphasis on individual reflection and conceptual understanding along withcreating group norms, social interactions and a supportive professionallearning community. It is also reflected in our stance as researchers; weare engaged in an inquiry process, constructing for ourselves and ourcommunity new understandings of the mathematics professional develop-ment processes. We worked, through cycles of design, implementation,


data collection, analysis, and description, to develop our individual under-standings of what we were studying and to negotiate them with each other,with our participants and finally with the broader mathematics educationcommunity.

As the project progressed, it became clear that various participantshad rather different ways of interacting with it, and hence, seemed toexperience different effects from their participation. Some appeared to bemostly interested in obtaining specific activities for use in their classroom,or in receiving credit for their participation. Others were interested inenhancing their professional skills, and their understanding of the subjectmaterial. Still others seemed to be “turned on” to a different way ofthinking about and doing mathematics, and eager to uncover implicationsfor their students and classrooms. We began to create a model describinghow teachers were interacting with the project, and tried to learn in greaterdepth what they were taking from it.

A REFLECTIVE MODEL OF MATHEMATICS PROFESSIONALDEVELOPMENT

In order to capture the different ways we observed teachers interacting withthe EMES project, we developed a reflective model of our professionaldevelopment milieu and its relationship to the participants’ classrooms.This model, though designed to capture some of the specific aspectsof the EMES project, is adaptable to other settings. In particular, itallows for analysis of connections between project activities and teachers’professional practice when professional development is designed to modelreform-oriented mathematics instruction.

A model with a similar purpose can be found in Fennema et al.(1996). That model is also an attempt to characterize teachers’ interactionswith a professional development project. One difference is that it focusesentirely on beliefs and actions regarding CGI, while our model attempts tocharacterize a broader collection of interactions.

A fundamental agreement in the design process was that we wouldmodel for participants, in ways that were appropriate to their learning asadult professionals, activities consistent with the PSSM (NCTM, 2000).To describe the effects of this on teachers, we developed a model of levelsof appropriation that represents elements of both the professional develop-ment milieu and the classroom milieu.. The model is displayed in Figure 1.The elements of the professional development milieu parallel those of theclassroom milieu, displayed on the right.


Professional Development Milieu Elementary Classroom Milieu

P1: Teacher as (mathematics)learner

C1: Student as (mathematics)learner

P2: Project mathematicsactivities and content

C2: Classroom mathematicscontent

P3: Professional developer asfacilitator/instructor

C3: Teacher as teacher

P4: Facilitator’s knowledge(of participants, theirknowledge, mathematics,mathematics learningin adults and children,facilitation skills, etc.)

C4: Teacher’s knowledge(of students, theirmathematical knowledge,how children learnmathematics, mathematics,teaching techniques)

P5: Professional developer asinquirer into professionaldevelopment

C5: Teacher as inquirer/scholarwithin the classroom and itsprocesses

Figure 1. Model of the interaction between the professional development milieu and theelementary classroom milieu.

The first three entries of each milieu can be thought of as the tradi-tional part of the model. These describe the participants and their rolesin a mathematics problem-solving activity. These elements appear in thetraditional instructional triangle joining teacher, student and mathematicscontent. There are two additional elements in each milieu. One is thefacilitator’s knowledge of professional development content and pedagogyparalleling the teacher’s knowledge. The second represents the facilitatorin the inquiry role, as in Ball (1995), of learning what works in the profes-sional development milieu; the parallel element on the right is the teacheras learner in the process of teaching. These two elements are sometimesconsidered in a “larger” instructional triangle (Mumme et al., 2003). Weinclude them because we consider mathematics, not instruction, to be ourprimary content, although mathematical activity serves as an entry point toprofessional conversations.

Levels of Appropriation within the Model

In order to characterize how mathematics professional development inter-acts with the teachers’ professional selves, we identified three levels ofappropriation:


Level one: Concrete activity and content. In level one, teachers parti-cipating in a mathematics professional development activity appropriatecontent such as specific mathematical skills or concepts that they willactually teach or specific pedagogical techniques to implement in theirclassrooms. Furthermore, they look for specific mathematical problems,tasks or games to use with their students.

Participants who appropriate at this level are focusing on the specificparallel between the mathematics activity in the seminar and mathematicsactivities in their own classrooms. They are likely to report that they didmany similar activities with their students that we did in the professionaldevelopment. The teacher may report that it worked well, their studentsliked it or difficulties were experienced, but without much analysis. Inthe model above, these appropriations could be represented by parallelhorizontal lines from P1 to C1, P2 to C2 and P3 to C3.

Level one is the most basic level of appropriation. Russell et al. (1995)point out that some teachers characterize mathematics as “an ad hocaccumulation of facts, definitions and computational routines” (p. 18).Likewise, some teachers may characterize teaching as an accumulation ofvarious kinds of skills and knowledge of practical routines, uninformedby general principles. Such teachers might only be able to make level oneappropriations.

In our experience, teachers often evaluate single professional develop-ment sessions based on whether they were able to appropriate material atthis level. The expectation of being able to make concrete appropriationsmay create dissonance when teachers participate in sessions not primarilydesigned for this purpose. In the EMES project, teachers were generallyhappy from the start, finding that they were almost always able to appro-priate a concrete activity or some directly relevant mathematics content, asmany of our activities were easily adaptable.

Level two: Professional principles and understandings; attitudes andbeliefs. In these kinds of appropriations, participants view themselves asprofessionals who are gaining additional knowledge from the session.In content, they look for and construct mathematical ideas that willallow them to integrate, connect, and explain the mathematical conceptsthat they will teach. In pedagogy, they may attempt to gain an under-standing of strategies that can be useful in mathematics instruction, suchas cooperative learning, journal writing for mathematical understanding,constructing challenging mathematical tasks. Level two appropriationsarise when participants view themselves simultaneously as learners (P1)and as teaching professionals (C3), and appropriate skills and knowledgeto improve as professionals. They may build their knowledge (C4) from


reflecting on any of the elements in P1 through P4, either individually orwith other participants.

Level two appropriations can look quite similar to those of level onein some cases. The difference lies in whether teachers are taking indi-vidual activities or “bits” of content which they can use in practice, orwhether they are organizing them into general categories and developingprinciples that they can flexibly use as professionals. Repeated appropria-tion of concrete elements can lead to changes in attitude or belief, or to theconstruction of general principles. These level two appropriations can beinferred when teachers create new classroom activities that are consistentwith reform principles, in addition to adapting and extending activitiesgained from a session. Teachers who appropriate mathematical under-standings at level two are interested not only in knowing the mathematicsthat they teach, but also in understanding connections among variousmathematical ideas, and in understanding more background and depth incontent, to better teach their students. They are interested in principles ofmathematics learning and teaching.

Examples of level two appropriations have been described in othermodels by placing the traditional instructional triangle (joining C1, C2 andC3) as the content inside a larger instructional triangle involving teachersas participant-learners and facilitators as teacher-instructors (Mummeet al., 2003). Such a model assumes that teachers in a professionaldevelopment setting are learning mostly about teaching. Our model ismore comprehensive in that it allows description of teachers’ mathe-matics learning and their learning about learning (through self reflectionand group discussions with peers about current mathematics learningexperiences) as well as their learning about pedagogy.

Level three: Teaching as inquiry. Teachers who are constructing knowl-edge at this level have a different perspective. In addition to being ableto use and adapt concrete elements, learning mathematical ideas, andapplying general principles for mathematics teaching, these teachers alsosee themselves as learning from (or, perhaps more appropriately, in)the process of teaching. They view themselves as mathematical learnersalongside their students, acknowledging that they can never know enoughmathematics to support and teach each student perfectly as they strugglethrough a school year. They also view themselves as learners about theirstudents’ cognition, striving to understand how their students are thinkingand why, and how to pose interesting worthwhile tasks.

Teachers at this level appropriate the same elements that are found atlevels one and two; the main difference is in how the elements are viewed.Here they become tools of inquiry for the teacher/learner who is facilitating


mathematical learning in the classroom. Teachers who are constructingknowledge in this way view themselves as learners all the time (not justin the professional development session), and particularly as persons wholearn from their students.

This stance probably requires a significant amount of professional andpersonal maturity. It certainly requires both the opportunity and the will-ingness to reflect critically on one’s own (and others’) practice. Because ofthis, we say teachers who appropriate elements from professional develop-ment largely at level one are taking a practitioner’s stance toward theproject, those who also appropriate and integrate knowledge (level two)are taking a professional stance and those who, in addition, begin to takeon the role of a learner in their own teaching process (level three) are takingan inquiry stance.

In constructing our model, we wanted to capture not only the reflectivenature of the project, but also the ways teachers were interacting withit. Relating the professional development milieu to the classroom milieuallows us to describe not only how teachers view themselves as learnersand professionals in the professional development setting, but also theknowledge they construct and its relationship to practice.

There is an interesting parallel in this model with one developed byRussell et al. (1995) to describe beliefs about learning and teaching mathe-matics that guide teachers in making instructional decisions. The nature ofthe levels in the two models is strikingly similar: they describe a develop-ment ranging from seeing mathematics as an ad hoc collection of factsand procedures (their level one) to systematic inquiry organized around“big” mathematical ideas (their level four). The difference is that in ourmodel, the levels revolve around teachers’ professional learning about bothmathematics and mathematics teaching, encompassing a larger range ofinteractions.

The two milieus of our model also roughly parallel the first twofoci of the model developed by Tzur (2001) to describe the multi-layered reflective processes involved in developing as a mentor of teachereducators. The elements of his model (like ours) allow for a rich descrip-tion of the reflective processes involved in learning from the practice ofteaching (or facilitating professional development).

DEVELOPMENT OF THE RESEARCH QUESTIONS

Our model represented an attempt to create a tool to aid in the descriptionand analysis of what we were observing. To study the interactions between


professional development design and teacher change, we then developedthree questions:

1. What does an inquiry stance toward mathematics teaching look like?2. How does this stance develop?3. What was the role of the EMES project in its development?

We focused on these questions because they have significance for theoverall understanding of what makes for successful mathematics profes-sional development.

While it is clear that there may always be a need for professionaldevelopment activities that teachers can learn from as practitioners andprofessionals, the development of an inquiry stance represents a self-sustaining level of autonomy on the part of the teacher. This is not tosay that such teachers no longer need support; however, by learning withand from their own students, they become the directors of their ownprofessional development.

If real reforms are to be sustained, this type of learning from profes-sional practice must become a reality (that such continual learning fromreflection on teaching is possible, see again, Ma [1999]). Thus the inquirystance embodies the target for effectiveness; analyzing cases may help uscreate opportunities for more teachers to develop this stance. In addition,because of the autonomy the teachers enjoyed in implementation, the casemethod seemed likely to reveal teachers’ varied experiences.

METHODOLOGY AND CASE STUDIES

We chose three teachers to study based on several factors. One, we wantedto look at teachers who appeared to be learning from the project. Weselected teachers who (a) had been involved from the beginning or nearthe beginning of the project, (b) appeared to be interested and enthusi-astic in some way, (c) appeared to be learning from the project, and (d)were willing to allow us to interview them and observe their classes. Thethree teachers chosen were at different places in their careers and teach atdifferent schools. They and their students vary in ethnicity. Our selectionof participants did not include any who embraced a practitioner’s stance,since analysis of such cases would be unlikely to uncover any new insightsrelated to teacher change and the development of an inquiry stance.

For each case, at least two interviews and two classroom observationswere completed. Emails teachers sent and final implementation reportswere collected. We also had access to teachers’ daily reflections on workthey did during both the summer institutes and the Saturday seminars.


The data were coded, codes were refined and each case was analyzed forimportant themes. These were then related to the model, and a descrip-tion of the kinds of appropriation was generated for each case. Each casedescription was written by one person from the research team, with theother researchers providing feedback. Individual participants read theircase descriptions and were given the opportunity for comment regardingsuggestions for corrections, clarity of descriptions and their consistencywith that person’s experience. The researchers are still in contact withall three teachers in the case studies, and have interacted with themoccasionally since the completion of the initial data collection.

Note: In the case studies, all names used are pseudonyms. Directexcerpts from data are indicated in the following way: (I) interviewexcerpt, (D) daily reflections during a workshop, (E) e-mail report froma teacher participant, and (R) participants’ final written reports on theirwork.

Case One: Donna McBride

Donna has been teaching elementary school for three years. During hersecond year of teaching, she attended EMES Saturday sessions. Shefollowed this by participating in the “Arithmetic, Algebra, and Geometry”EMES summer institute. For her pedagogy project, Donna focused onpromoting and improving written communication in her classroom. Thecontent project Donna chose focused on number sense. Specifically, shewanted her students to gain a better understanding of decimals. In additionto her projects, Donna indicated that she used many ideas and specificactivities from the EMES workshops during her third year of teaching.

In an early interview, Donna said she had been trained in the use ofmanipulatives and cooperative grouping. However, she indicated that shedid not know how to implement the use of those techniques effectivelyin her own classroom. The EMES project gave Donna the knowledge shedesired:

And so it’s . . . been a great benefit of the project because it’s heightened my refinement ofa lot of the methods I was taught in college. And so, it’s made me a much better teacher . . .

as a result of that. (I)

However, Donna also expressed a desire to better understand the “whys”behind mathematical concepts and believes the project provided this: “Ifeel more grounded in the concepts. I mean, . . . I’ve known the conceptsand I can . . . I can do math, but . . . (it) . . . has never really felt a part of meas much as it does now” (I).

Donna’s pedagogy project involved using more written communicationin her classroom. But during the course of her third year of teaching,


Donna observed that written and oral communications are closely linkedand realized that communication promotes conceptual understanding:

I really wanted the kids to take time and reflect – why is this answer correct, what was yourthinking, how did you get that, where are you going with this. Which takes a long time andeffort. And it gets very painfully laborious but . . . but their thinking is just heightened andlight bulbs start going off when you slow down and say. ‘Why?’ And they really have tothink through it and when . . . they don’t have the ‘Why’ right there, they slow down andstart thinking and start talking aloud and all of a sudden the concept I wanted them to getin the first place finally is there. (E)

One area of communication Donna focused on was her students’tendency to describe mathematical processes themselves instead of theirrationale. This was illustrated by a writing assignment on division wherestudents explained how division works. Her students wrote descriptionsof the process one goes through in order to divide and not the reasoningneeded to understand the process. In this case, when her students’ perfor-mance fell short, Donna did not seem to contemplate why; instead shefocused on improving their performance.

Donna’s work on communication resulted from her experiences atEMES sessions where group work and dyads were used to discuss mathe-matical ideas, along with good questioning and reflective writing. Donnatook the information acquired and adapted it for use in her own classroom.

Donna’s mathematics content project revealed another area of change.To promote the learning of decimals, Donna used real world problemscenarios from the Summer Olympics integrating technology, good ques-tioning, manipulatives, communication, and grouping. Her students wroteabout how decimals are used in the Olympic contests; they watched someof the Olympic games and studied the statistics from them.

She gave her students activities to encourage flexible thinking aboutplace value as well. One of these games involved taking three numbers andmaking as many decimal numbers as possible and developing strategiesto systematically check that the list was complete. Donna said that herstudents were more successful than when following the district-adoptedtextbook’s approach to these topics.

Extensions and connections to decimals were made throughout the restof the semester and were not limited to mathematics lessons. One follow-up lesson dealt with money. Donna tried a lesson from the textbook aboutmoney and felt that it was entirely unsuccessful because her studentsfound it uninteresting. So, she had the students look at foreign currency,research the countries in which the currency originated and then, comparethe currency values. Donna wrote about how:

In science and reading, we had been learning about earthquakes. We studied the Richterscale and learned what range of magnitudes the scale represents. We constructed a model of


the earth with shoeboxes and they replicated an earthquake. If they chose a magnitude of,say, 7.0, I then asked them to show me what a 7.8 might look like to see if they remembered. . . decimals. They remembered, all right! They loved creating massive destruction withtheir models! (E)

As her methods of instruction changed, Donna felt she began to growas a teacher. She realized she had previously been bound by the text-book stating, “Textbooks simply do not offer this type of teaching thatthe seminar does” (I). Her students were empowered by what they werelearning and were now able to connect their new knowledge to othersubject areas.

During the academic year that followed the EMES summer institute(her third year of teaching), Donna began to move away from the text-book and use explorations with other standards-based teaching techniques.She observed how her students benefited from inquiry-driven experiences.Furthermore, despite anxiety about the annual state-mandated assessment,she knew this was better than teaching to the test. Donna found that, “WhenI let go of my dependency on the text and began moving towards theseother approaches, I finally felt comfortable and excited to teach math” (I).Now she is able to integrate the exam material into her daily teaching. Butshe still must resist going back and teaching the “old way” because of heranxiety.

In the Reflective Model, Donna appears to be taking a professionalstance toward the professional development activity. Given her back-ground, Donna was already having experiences and taking actions thatappear consistent with this stance. Moreover, Donna did not come to theEMES sessions with the sole purpose of gathering new teaching tech-niques: she wanted to learn not only effective methods of teaching, butalso the “why’s” behind mathematical concepts.

Donna did begin her interactions with the project by appropriatingmany specific elements (level one). This emphasis is consistent withwhat is known about beginning teachers, namely that one of their maindevelopmental tasks is to “develop standard procedural routines that inte-grate classroom management and instruction” (Kagan, 1992, p. 129). Butshe also developed her views about what constitutes good mathematicsteaching, adapted activities and processes to her own situation and creatednew activities that were consistent with reform principles (level two). Instating that “I hope I am never where I think I want to be because thenmy effectiveness will begin to diminish” (I), she seems to embrace profes-sional learning as a lifelong process. This indicates her commitment to herown professional development. This interest in continued learning signalsthat she may develop an inquiry stance as she matures professionally.


Case Two: Eva Pantoja

Eva Pantoja has been teaching elementary school for 13 years, 10 of themas a fourth and fifth grade teacher. She started the project the first summer,completing the institute on “Probability and Statistics” and returning to herfifth grade classroom to implement her project. During the second summershe participated in the “Arithmetic, Algebra and Geometry” institute, andcame to two of the three Saturday seminars held in the fall.

Eva made significant changes in her classroom practice as a result of herparticipation in the project. During the first year, the changes were ratherclosely tied to either the summer institute or the Saturday seminars, andrequired some courage to attempt:

I tried a lot of things last year that were completely new to me because I had learned themin the class. And as I was going to the Saturday seminars and I kept finding out moreand trying more things and sometimes I felt like I was just stepping off of a cliff and notknowing what I was doing and not knowing where I was going. I just knew that it wasbetter than what I was doing before. (I)

Two events helped Eva solidify her commitment to change the first year.One involved students in her fifth grade class who had been in her fourthgrade class the previous year:

. . . the year before last I had some kids who, when I was teaching the traditional way, werevery low, but last year when I started trying new things, those were the kids who werehaving the light bulbs go on left and right. (I)

The other significant event involved a change in her students. The fifthgrade teachers, at the end of the year, ability-grouped their classes and putone group with each teacher. Eva was assigned the high-ability group:

. . . I had the high group but I had to get through so much material in a short amount oftime . . . and so I went back to my traditional teaching thinking these are the high kids andthey’ll catch on fast and we’ll just go right through it . . . they learned very little . . . theywere bored, they didn’t like it, the whole atmosphere changed. (I)

Eva came to the institute the second summer highly committed toimproving her instructional practices. Eva decided on place value asher content emphasis, and on mathematical journaling as her pedagogyemphasis.

For her place value unit, Eva created her own instructional design. Shetold her students she had been abducted and taken to Pluto, and whilethere, learned about their strange monetary system (base three). After thestudents worked on the Plutonian system for a while, she modified it tobase 10. Eventually she moved her students into working with base-10blocks and creating strategies for addition and subtraction of multi-digitnumbers with chip-trading. All of this took quite a bit of instructional time,


and her students ended up working on place value and associated conceptsand operations for a number of weeks.

There are many possible questions about this instructional design. Wasit effective? How were students assessed? Was the trajectory appropriate?Efficient? Did all students find it within their grasp? Was the motivationalstory engaging? From the point of view of our research questions, theseissues do not interest us directly. What is significant, however, is that thesequestions were of great importance to Eva. While some participants tendedto emphasize positive aspects of their implementation, Eva reports mostlyabout what she learned and what she would change:

In my ‘action research questions’, one question I did not pose was, ‘What will I learn aboutplace value when I teach this unit?’ I thought I understood our place value system becauseI knew that there was an exponential relationship between each place. I had hoped to guidemy students to some understanding about how to ‘move around’ in our place value system.I had hoped that they would become more mentally flexible when using our number system(to add, subtract, solve problems, etc.) if they could understand how our system works. Ithink I achieved this goal to some extent, but what I learned is that our place value system isjust a system of symbols designed to simplify the representation of numbers. Other systemsexist which do the same. I did not make this clear to my students and I realize now that Ishould have. (R)

Eva’s report is replete with things that she found surprising which ledher to revise the unit. In one case, she realized that students find it difficultto deal with large numbers in base ten, even after developing a concreteunderstanding of the system by using base-10 blocks. She shared this withother teachers at a Saturday seminar and found not only that some hadsimilar experiences, but that one teacher had an activity designed to dealwith the situation.

This report shows that Eva is open to learning new mathematical ideasas well as new methods of instruction. As she says:

One issue I was constantly reminded of while tackling this unit is that I myself amconstantly learning. I am not the best teacher of place value, but because I was braveenough to try this unit and reflect on it, I know I am better now than I was at the beginningof this school year, and next year I will be even more effective. (R)

In choosing to use mathematics journals with her fourth grade students,Eva joined a large portion (about two-fifths) of the teachers in the insti-tute who chose to implement some aspect of mathematical writing. Thismay have been because writing was an important (if small) part of theprocessing of almost every mathematics activity during the summer insti-tute; teachers kept journal entries describing mathematical ideas they hadlearned and, on occasion, these were collected and read by the facilita-tors. On the other hand, the state-mandated mathematics assessments havenumerous open-response items that require students to write solutions;


implementation of these assessments occurred at the same time as theEMES project activities. Discussions with teachers indicated that both ofthese factors likely contributed to the choice of so many to implementwriting.

During mathematics time, Eva gave the students a problem, had themwork on it and discuss it, either as a whole class or (less often) in pairs orgroups, and at the end of the period, write about it. She initially character-ized their writing as very disorganized, but she says it has improved. Evasought out information by reading Writing in Math Class (by elementaryeducator Marilyn Burns) but still created her own methods. She has modi-fied the prompts she gives (making them more specific to the lesson). Atfirst she picked up every student’s notebook, but has learned to collect afew each day. She uses them as an informal assessment tool, and some-times uses a student question from a journal as a starting point for classdiscussion.

Other changes Eva identifies are tied to the idea of mathematicalcommunication. She has worked consistently to implement the idea ofcreating good questions, and spends considerable time in class askingstudents to explain their responses. She and her students call this “back-wards math”. She listens carefully to student explanations, at times askingthem to clarify if she or other students do not understand. Eva believesthat it is important for students to have time to work out solutions withoutinterruption from the teacher, and to see multiple solutions. She now givessome explicit attention to problem-solving strategies. This contrasts withher previous emphasis on having students practice saying numbers and onthe basic operations of arithmetic.

During the first year of the project, Eva took a professional stancetoward the professional development experience, appropriating bothconcrete activities and content and principles and ideas. She used specificactivities that were modeled or developed in either the summer instituteor the Saturday seminars, modifying them as appropriate for the level ofher students. She also significantly altered her approach towards classroomcommunication, spending much more time on problem-solving and payingattention to student explanations. A key factor in creating many of thesechanges involved a shift in her understanding of mathematics learning:

And I realized that there is a whole different way to approach things kind of like the whole-language versus skills-based reading. I didn’t realize that math was constructive, you know,I thought it was sequential and so I guess that was a big catalyst. (I)

Later, she clarified these ideas:

Giving the students experiences giving them opportunities . . . to construct their under-standing of math. You know that is the main thing that I learned in this program is that. . . I know that kids construct literacy. But I had no idea that they construct their math


understanding too. (Laughter) You know, I thought I just fed it to them. (Laughter). So Ithink . . . that good questioning is very important. (I)

By the middle of the second year, after two summer institutes and half adozen Saturday seminars, Eva developed a strong inquiry stance towardsher mathematics teaching. Eva took the suggestion (made in the secondsummer institute) of keeping a journal of her mathematics teaching exper-iences quite seriously. She uses the journal to reflect on her own practice,which she says “helps me think of what to do next, it is a great opportunityfor me to problem-solve and trouble-shoot my own math teaching” (R).Her final report on the progress of her plan is deeply reflective, containingsignificant analysis of the practices in which she engaged, as well as alengthy discussion of what lies ahead: what she will try next and why,and what outcomes she is looking for. Her stance is also indicated byher careful attention to students’ thinking, and her orientation towardsdifficulties; if things do not go well, she says, “at least I’m learning” (I).

It is clear that, at least with regard to mathematics teaching, there wasno real sense of inquiry on Eva’s part before participating in the project.She entered the project with a significantly felt need for improving hermathematics teaching, combined with a willingness to try new things, bothof which may have contributed to the development of her inquiry stance.In addition, it is worth noting that the engagement and reactions of herstudents to the way that she teaches mathematics is her touchstone forassessing and modifying her practices. This is clear from both classroomobservations and the way Eva talks about what has been effective for herstudents. Modifications can occur in as short a time frame as a single lessonor in the overall planning for her mathematics curriculum, and are based onstudent engagement with tasks and communication about what they havelearned. This finding is consistent with Fennema et al. (1993) who foundthat increased learning on the part of students was a highly significantmotivating factor in one teacher’s change process.

Another factor in Eva’s development appears to be the reflectionthat participation in the project engendered. While not every teacherwho is encouraged to reflect on practice engages in the same level ofanalysis, Eva’s reflections are thoughtful, self-critical and highly focusedon improving instruction for her students. Some of the action researchquestions included in her plan were: “Will I ask questions that lead studentsalong at a good pace? Will base ten seem harder than base three? Whatquestions will my students have and what problems will ‘hang them up’?What discoveries will they make?” (R). The importance of a teacher’s own(written) reflection on her instruction as a catalyst for change was alsonoted by Edward and Hensien (1999).


A real sense of empowerment, as a teacher and a learner, developedthrough this process. This is evidenced by the fact that, during the time Evaimplemented her place value lessons and focused her students on writing,she was receiving pressure from the district level to “cover everything”(E). Her decisions to implement her own ideas and give her students suffi-cient time to work on the activities were viewed (by her) as not exactlyin line with current policies. This sense of empowerment as an inquiringprofessional also appears in her characterization of her own progress:

I’m finally really teaching math. That’s wonderful! You know, I know when my kids arelearning and I know when they are not. I’ve really been looking for (something) . . . allthe years I’ve been teaching . . . knowing I haven’t been doing a very good job at teachingmath, so I’m getting better at it now. And that’s wonderful. I love feeling that I know whatto do. You know, I have a mathalosophy. (Laughter) I have a math teaching philosophy.Whereas before I just said ‘I’ll do it from the book’. So that’s good. (I)

It is this idea of getting better at it that clearly characterizes Eva’s currentpractice; it contrasts with her previous image of herself as a somewhatinadequate mathematics teacher.

Case Three: Vera Holloway

Vera Holloway is a fourth-grade teacher who entered the EMES projectwith 25 years of experience. She joined the project with a high level ofself-awareness, reflectivity and interest. She had been through mathematicsprofessional development workshops the previous two summers and feltthat she had learned a lot; however, she wanted more.

Vera’s very first reflective writing piece at the end of the first day ofsummer work indicated that her expectations exceeded those of a practi-tioner; “I hope we don’t come away with a ‘cute’ unit like we do in so manyeducation classes. I would like you to know that I want to be challenged,to refine my teaching and to have unanswered questions” (D).

Three major themes emerge from Vera’s reflections and interviewsduring the course of the project. The first involves Vera’s own empower-ment as a mathematics learner, her changed views of the nature ofmathematics and her deeper understanding of mathematical concepts. Thesecond theme follows from the first: Vera (immediately) articulated adesire to empower her students mathematically. For her, this followeddirectly from her own mathematics learning experiences in the project.The third theme relates to the method that she chose to accomplish hergoals for students: posing appropriate problems, asking good questions,and paying close attention to their thinking.

Vera immersed herself into the project’s mathematics activities withinterest and enthusiasm. She found herself revisiting concepts that shehad learned many years before, or those that were related to content she


teaches, and gaining new understanding and insight. She was very ener-gized by these “ahas” and by the support she received in thinking creativelyin a mathematical context. She appreciated being encouraged to think ofherself as a mathematician/mathematics learner, and readily took on thisidentity for herself.

Vera is a professional writer as well as a teacher; empowering herstudents in literacy has been a strength for her because of this identifi-cation:

Three years ago, four years ago, five years ago in my classroom . . . I would have told mychildren . . . ‘you are a writer, just like I’m a writer’. I never did that with mathematicsbecause I was on the outside looking in. I wasn’t a mathematician. And so I never said,‘Oh, mathematics! I’m a mathematical thinker’. (I)

From the beginning, Vera was interested in giving her students similarexperiences to those that she had in the EMES class. The similarity,however, did not relate to the specific nature of the activities, nor did itonly encompass the kinds of interactions that occur between teachers andstudents; mostly, for Vera, the point was for the students to have similarinternal or personal experiences (the “aha” experiences that she had) asthey created their own understanding of mathematical content.

Early on in the project she stated, “I want to excite, challenge andincrease the math competence of my students, to teach them to persevereand to teach them to be mathematically confident” (E). Vera’s desire toempower her students and her constructivist teaching philosophy led to herinterest in the concept of finding ways to ask good questions of students.She latched on to this concept and constantly expressed the desire to applyit to her entire mathematics curriculum.

For her pedagogy emphasis, Vera chose to work on creatingchallenging, non-standard problems to send home with students. Shecreated problems (sometimes in the form of games that pupils wereinstructed to play with family members) that were closely related toconcepts being covered in the project, often using the results of thehomework in her instruction.

Observations of Vera’s Teaching Clarify the Picture

At the beginning of one period, Vera asked students to write in their note-books the rules of the function game, which they had learned the previousday, with Vera choosing the rule and the students guessing. After a fewminutes, Vera asked for volunteers to read their description to the studentswho had not been present the day before.

To play the game today, students were asked to volunteer a rule andrun the game. After listening to the first volunteer explain his rule, Veradecided he did not yet understand the game and asked him to sit down and


TABLE I

Student “rule” data from Vera’s class

7 56

9 90

8 72

5 30

1 2

0 0

3 12

watch for one turn. The next student had a rule much more complicatedthan the ones previously used. So it took a long time for other students toguess it. Table I shows how the chalkboard read.

The students were a little frustrated but eagerly engaged. At one point,Vera asked the students to discuss at their tables what might be going on(students sit in groups of 4 or 5), giving them a hint by suggesting that therule might be doing “more than one thing”. After some further discussion,students at one table were able to characterize the rule as multiplying bythe same number and then adding it on.

Fairly early in the process, one student who felt he had the rule begansaying “7 times 8”, “9 times 10” and so on, specifying different rules foreach set of numbers. Vera pointed out that it had to be the same rule eachtime. After someone came up with the rule that worked for every case, theclass checked it collaboratively (again working at tables) to see if it reallyworked and if they understood it. Vera showed the students the algebraicnotation for the rule [(n × n) + n] on the board, explaining it as involvinga variable (a term she had discussed with them earlier), and they checkedthat this representation worked in each case. Then all students took about10 minutes to write in their notebooks what the table was and what the rulewas.

This episode illustrates how Vera has worked to create activities toempower her students as mathematicians. Her instruction contains richmathematical discourse, both among students and between student andteacher. Students’ explanations of their thinking are listened to and valued,and this atmosphere is reflected in the interactions of the students.

The description also illustrates how Vera changed her homeworkassignments. Now, instead of practice problems, homework is either aproblem to solve or some sort of game. These assignments integrate withher classroom instructional activities.

In terms of the reflective model, what did Vera appropriate from theproject? She did take some activities directly to use with her students


and also improved her professional knowledge in a variety of ways,using processes (such as the idea of math notebooks) or principles (usingmultiple representations of a mathematical concept) that were used in theproject. Mostly, however she took a sense of empowerment, inquiry, andnew modes of interacting with students around mathematics:

I can truly say now, ‘If you’re good at language you can be good at math. Math is alanguage . . . it’s more natural than I ever thought it was. I thought it was, I truly believedthat mathematics was for a limited number of people who had mathematical minds. I reallydid believe that. And I’d read many places that this simply isn’t true. But I didn’t believeit, in my heart. And I believe it in my heart now – that’s changed in three years. (I)

When asked what it was in the project that helped her get to this point, shemade clear that it was not just that we were saying the same things that shehad read before, rather:

It was your doing with me . . . what I need to do with children. Modeling with me . . .

letting me live through the experience . . . of being . . . a constructor of a mathematicalworld. Which I’d never been allowed to be before, I just had to find the right answer andspit it out enough times in the right context. (Laughter) And . . . pray a lot, you know. Butnow (laughter) . . . I get to . . . I get to play. (I)

It is this sense of play that connects Vera’s interactions in the project asa mathematics learner and interactions with her students as a teacher. Sheclearly receives great enjoyment from creating opportunities to investigatemathematics with her students and from watching them reflectively as theylearn. It is clear that Vera has adopted an inquiry stance in her mathematicsinstruction.

DISCUSSION

These cases show three different teachers, at different stages in theircareers, working to improve mathematics instruction through professionaldevelopment. Vera, the most experienced teacher, came into the projectwell-prepared (almost primed) to act as an inquirer, explicitly uninterestedin level one appropriations and looking for something more, which shefound in her own learning experiences. Her reactions to, and reflections on,her own mathematical learning were immediately and explicitly applied toher thinking about her students and her instruction. Her experiences increating and reflectively improving student-centered pedagogy in literacyallowed her to create such pedagogy in her mathematics instruction, onceshe reflected on her personal experiences.

Eva, in the middle of her career, moved more slowly into the inquirystance, beginning with level one and two appropriations and graduallycoming to see herself as a mathematics teacher-learner, driven at least


partly by her intense interest in her students’ reactions to her pedagogyand their mathematical thinking. Donna, a new teacher, began the projectas a practitioner, interested primarily in learning how to do the things thatshe had been taught that she should do while in her teacher educationprogram. As the EMES project progressed, she became more interestedin the principles behind the activities and worked to adapt and create newinstructional elements with them in mind, moving her into a professionalstance.

Just as mathematics learning can look very different from one studentto another, so these cases show significant variation among teachers forwhom this particular project “worked”. For both Eva and Vera, theirown mathematical learning and their attention to their students’ thinkingbecame important influences for instructional change. This dynamic hasbeen seen among pre-service teachers as well (e.g., Ball, 1988) and, there-fore, is not entirely attributable to the teachers’ classroom and careerexperience.

Both Vera and Eva remarked that the project allowed them to bringto their mathematics instruction student-centered elements that had onlybeen previously present in their literacy instruction. A common elementfor them was the catalytic nature of the mathematical learning experiencesin the project. In attempting to explain the success of professional develop-ment programs (such as CGI) which focus on providing teachers access toresearch-based knowledge of learning, Rhine (1998) speculates:

. . . the power of these projects may not be primarily due to developing teachers knowledgebase of specific strategies students employ. Perhaps the major impact of these projects onadvancing educational reform is, instead, due to teachers’ engagement with research thatserves as a catalyst for their new orientation toward inquiry into students thinking andvaluing students’ knowledge and thinking processes. With this focus, teachers then inte-grate their assessment of students’ understanding into their instructional decision-makingprocess. (p. 28)

In our project, a different element (teachers’ own mathematics learningexperiences) served a similar purpose. Level two and three appropriationsin content and pedagogy are the goal of many professional developmentprojects. Schifter (1998) explores the need for examination of disciplinarycontent and examination of student thinking and, in addition, discoversthat powerful synergy can arise between the two. We found that teacherswho take an inquiry stance toward professional development utilized thissynergy for their own learning.

It is interesting to speculate about the role of literacy instruction in thetwo cases of Eva and Vera. How important for these teachers was the factthat they already had a well-developed constructivist theory of learningrelated to literacy? Did the fact that they had this schema allow them to


develop more quickly such a theory for their mathematics instruction andact on it?

For Donna, the new teacher, the dynamic was quite different, thoughalso influenced by her experiences. She appeared not to have any conflictwith the pedagogical aspects of the project; these were consistent withher preservice teacher education program and she had not built up anyrepertoire of teaching experience that might have conflicted with it.

Contrary to this, as a new teacher she viewed herself to be in need ofspecific ideas for implementing student-centered pedagogy (Kagan, 1992).As she continued making these kinds of appropriations during the courseof the project, her classroom experiences led her to construct professionalknowledge for this purpose. It seems that for Donna, this met her currentneeds; we may speculate that as she gains confidence as a practitioner andprofessional, she may develop an inquiry stance toward her own teachingand future professional development opportunities.

IMPLICATIONS FOR PROFESSIONAL DEVELOPMENTDESIGN

We return to Ball’s (1995) suggestion and examine the implications of thisstudy for the design of mathematics professional development. In the caseswe studied here, the following elements were effective:

• Inclusion of authentic and readily adaptable student-centered mathe-matics learning activities

• Rich opportunities for discussion and reflection• An open, learner-centered implementation component• An inquiry stance taken by the facilitators

Including authentic mathematics learning experiences that are alsoeasily adaptable to the classroom milieu allowed teachers to adapt theactivities directly and also to reflect on mathematical concepts or pedago-gical principles (either in the workshop or as a result implementation). Themodeling of student-centered instruction also allowed teachers to adoptinstructional techniques directly, integrating them into instructional prac-tices and activities that were obtained from other sources. Using suchmathematics as the “entry point” into professional discussions allowsteachers to interact with content in ways that are specifically useful forteaching (Shulman, 1986).

Furthermore, opportunities for discussion, journaling and reflectivewriting, centered on mathematical ideas and issues of pedagogy, allowedteachers to construct mathematical and professional meanings for them-selves from the project activities. This is true, even though the construc-


tions clearly varied considerably among our research cases and – based onour impressions – even more among other project participants.

In addition, the implementation component of the project was individu-ally designed by each teacher to meet her or his own perceived needs asa classroom teacher. Our input usually involved suggestions or questionsdesigned to clarify issues. This not only created a sense of ownership onthe part of the teachers for the instructional changes, it also supportedteachers as professionals in making a wide variety of appropriations ofproject elements, encompassing all three levels identified in the reflectivemodel.

Finally, all three of the above components were supported by theinquiry stance of the EMES project facilitators. We entered both theprofessional development project and later the research project with adeep awareness of how little we knew. We know that the teaching weare trying to help teachers learn is not completely defined. It is embeddedin a vision of reform, rather than a collection of procedures. Likewise,our understanding of professional development that can support teachers’learning is a mix of myth, belief, and conjecture (Ball, 1995). We made noattempts to hide from our participants the fact that we were learning: facili-tators worked with teachers as co-learners of mathematics and as inquirersinto their classroom milieus. Facilitators read participant feedback aftereach session and not only used it to modify the next day’s activities, butoften shared this feedback and our modifications explicitly with the group.Questions from teachers about how they should implement what they werelearning were usually met with comments such as “That’s up to you” and“We don’t know how this can be adapted to your grade level – that’ssomething we expect to learn from you”. Such explicit attempt by thefacilitators to model an inquiry stance in discussions with participants maybe an important factor in stimulating teachers’ thinking about themselvesas inquirers (through the teaching process). Those who design professionaldevelopment and wish to support teachers in their adoption of an emergingmodel of the profession of teaching may reap significant benefits fromtransparency regarding their own inquiry processes.

ADDENDUM

We thank the editors for pointing out to us the similarity of our workwith that of Cochran-Smith and Lytle (1999). Although there are someinteresting parallels that could be drawn between our levels of appropria-tion and the three views of knowledge that they identify, the parallelsare not exact. There is a close match, however, between what we foundand called the “inquiry stance” and their view of “inquiry as stance”;


some differences are that (a) they seem to explicitly locate this stancein communities of teachers, whereas we identified it in several individualcases (who were however indeed part of a learning community) and (b)they include a critical stance toward the current educational system as apart of their description whereas our cases (while quite possibly possessingsuch critical views) were not analyzed from that perspective.

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JEFF D. FARMER & HELEN GERRETSON

University of Northern ColoradoMathematical SciencesGreeley, CO 80639E-mail: [email protected]@unco.edu

MARSHALL LASSAK

Eastern Illinois UniversityDept. of Mathematics and Computer ScienceCharleston, IL 61920E-mail: [email protected]

MARIA GOULDING, GILLIAN HATCH and MELISSA RODD

UNDERGRADUATE MATHEMATICS EXPERIENCE: ITSSIGNIFICANCE IN SECONDARY MATHEMATICS TEACHER

PREPARATION

ABSTRACT. This paper reports views of student teachers on a one year secondary teacherpreparation course about their undergraduate experiences of learning mathematics. Writtenresponse data were collected from 173 student teachers (trainees) from several differentinstitutions and their views were collated and thematised. The principal issues that arise arethose of discontinuity of experience from school to university, the lack of preparedness for“struggle” in the face of challenging mathematics at university and an unresponsive styleof teaching and assessing. The significance of these views to the students as prospectiveteachers and the ways in which they could be used by teacher educators on training coursesis discussed.

KEY WORDS: experience of undergraduate mathematics, mathematics teacher educa-tion, qualitative data, student-teacher reflections, teaching and learning undergraduatemathematics

INTRODUCTION

How do prospective mathematics teachers deal with their undergraduatedegree mathematics? What strikes them as important about their experi-ence after they have graduated and while they are on their teacher trainingcourse? How do these views feed in to their development as mathematicsteachers? This paper presents firstly a synthesis of views of pre-serviceteachers with regard to gaining a suitable degree in order to teach mathe-matics in secondary schools in the UK, and then addresses how suchpre-service teachers’ views of their undergraduate university experiencemight be woven into their teacher training programme. The questionwe address particularly is: “How can a one-year secondary mathematicsteacher preparation programme incorporate intelligently the undergraduateexperiences of students?”

In the UK, the vast majority of those who train as secondary mathe-matics teachers become qualified teachers by completing successfullya one-year Post Graduate Certificate of Education (PGCE) course aftercompleting an undergraduate degree. (The PGCE is one of several possible“initial teacher training” (ITT) courses available to prospective teachers.)


362 MARIA GOULDING ET AL.

To be accepted on to a PGCE course a candidate is required to haveat least 50% mathematics or content “strongly related” to mathematicswithin their degree.1 We wanted to investigate questions specific to mathe-matics education- exemplified in the paragraph above, about the nature ofa prospective teacher’s personal educational “journey”. Thus we aimed tofind out what was important for PGCE students to say about their under-graduate mathematical experience as they began teacher training; our dataon this were obtained from students’ brief, written responses to a promptsheet, as explained below.

BACKGROUND TO THE STUDY

Our interest in pre-service secondary mathematics teachers’ views of theirundergraduate mathematics education is concerned with these students’beliefs about and attitudes to the teaching and learning of mathematics.Work in mathematics education on beliefs and attitudes has been activeinternationally since the early 1980s. Thompson’s (1992) review of theliterature on “Teachers” Beliefs and Conceptions’ traces the rise of interestin this cognitive dimension of teaching, away from the dominance ofbehaviourism, with Shulman’s general work on teacher cognition (e.g.,Shulman, 1987) and the concept of reflective practice (Schön, 1983). Thesetheoretical constructs led us to the view that the undergraduate mathe-matics experience contributed to pre-service teachers’ attitudes and beliefsand that reflection on these experiences could play a significant role insecondary mathematics teacher preparation.

Recent research on the role of beliefs in teacher preparation (e.g.,Smith, 2001; Artz, 1999) rests frequently on the aforementioned previouswork about the importance of (serving) teachers’ attitudes and conceptionsto their classroom practice. While Smith’s research shows that “there isnot a deterministic relationship between beliefs about mathematics and thestudent teacher’s classroom teaching style” (p. 136), Artz gives evidencethat her structured reflection programme can make a difference to a studentteacher’s approach to instruction. These recent findings, from case studyresearch, suggested to us that the experiences which shape beliefs needto be made explicit and worked upon if they are to influence the studentteachers’ approach to teaching.

Our research looked for patterns across a large number of individualresponses. What was noticed in the data, of course, was a function of ourbackgrounds and cultures. We recognise that people are generally situ-ated within a social context and community (Wenger, 1998) from whichreification of concepts emerges. Specifically, each of us, the co-authors of

UNDERGRADUATE MATHEMATICS EXPERIENCE 363

this paper, has been involved in teacher education in ten different highereducation institutions over a considerable period of time. Consequentlywe are sensitised to ways in which our community uses terms like“understanding”, “ability” or “abstraction”. Also we have a professionalobligation to respond to government policy on the preparation of teachers.We have all been members of a working group concerned with theeducation of undergraduate mathematicians, the Teaching and LearningUndergraduate Mathematics group (TaLUM2) for several years. Further-more, our value systems have a shared domain that includes a delight inmathematics, a despair of assessment-driven curriculum “delivery” anda commitment to teacher education as a professional enterprise. Withinthe notion of teacher education, we concur with the value of reflectionas essential for professional development (Jaworski, 1994). Specificallywith regard to mathematics teacher-education, Mason’s (1998) develop-ment of Gattegno’s notion of awareness into principles for focussingattention is consonant with our common view that preparing mathematicsteachers goes beyond “training” and requires development of sensitivityto mathematical structures as well as being able to see things from apupil’s perspective. Although we are working within the context of teachereducation in England and Wales, the value placed on reflection is commoninternationally, as is the reality of working within the constraints ofnational schools and university policies. The insights obtained thereforehave relevance beyond England and Wales.

In the United Kingdom, Northern Ireland and Scotland have their owneducation systems and policies on assessment and are different from thosein England and Wales. In England and Wales, policy since the EducationReform Act (DES, 1988) has placed particular emphasis on assessmentas a central feature of school children’s experience. This is related to thedesire for accountability: standards must be seen to be going up and aschool’s performance is judged by league tables of results published inthe newspapers. Government control and demand for accountability hasextended to teacher education, resulting in the “initial teacher training”(ITT) requirements (DfEE, 1998; DfES, 2002). The three of us work inEngland where award of “Qualified Teacher Status” (QTS) depends onsuccessful completion of “a course of ITT” (DfEE, 1998) in which atleast two thirds of the time on any secondary PGCE course is spent inschool, with day-to-day “training” undertaken by school based mentors.The words “training” and “trainee” are part of the official rhetoric inthe PGCE national curriculum and have thus, inevitably, entered the UKdiscourse of teacher education. The assessment on PGCE courses includesattaining the standards for initial teacher training set out under categories


of “professional values and practice”, “knowledge and understanding”, and“teaching” (DfES, 2002, pp. 6–12). Since these standards are imposedupon teacher educators, university teacher education institutions have toreconcile this reality with the need to provide courses that acknowledgethe importance of research and deliberation:

These requirements, however, do not in themselves constitute the teacher trainingcurriculum. They are only one of the many, if sometimes competing, components thatmake up the broad spectrum of a teacher’s professional knowledge that underpin initialteacher education courses. (Bourdillon & Hutchinson, 2002, p. xi)

There are clear tensions here, not least since the quest for accountabilityhas distorted what it ostensibly was there to protect. As the philosopherO’Neill says about trust and accountability:

We are imposing ever more stringent forms of control. We are requiring those in the publicsector and the professions to account in excessive and sometimes irrelevant detail to regu-lators and inspectors, auditors and examiners. The very demands of accountability oftenmake it harder for them to serve the public sector . . . we are galloping towards central plan-ning by performance indicators, reinforced by obsessions with blame and compensation.(O’Neill, 2002, p. 2)

This cultural picture is relevant to our “trainees” both as products ofthe higher education system and as beginner teachers within the schoolsystem: assessment is now a central feature of school life and, in thisview, learning is measurable and measuring instruments are mathematicsassessments. In many institutions over the past decade new assessmentprocedures have been developed. For example, the work of Williams(2000) documents techniques for widening modes of assessing mathe-matics. The dominance of the timed exam, however, continues becauseof the increased numbers of students to higher education in UK in thelast decade. Furthermore, assessment procedures have to be “accountable”,as O’Neill notes, and the mathematical community in Higher Educationretains its confidence in “traditional” exams (Lawson, 2000, pp. 19–20).

Consequently, it is predominantly through passing a traditional timedexam that students make transitions from one phase of their education toanother. Transitions can involve discontinuities of experience and indeedour data indicate that respondents have felt both exhilaration and terrorin bridging the gap between school or college and undergraduate study.Almost all of our respondents had taken GCE (General Certificate ofEducation) Advanced Levels (“A-levels”) which are two-years coursesdesigned for students of 16–18 years in their last two non-compulsoryyears of schooling. The standard entry qualifications to UK universitiesfor students from England, Wales or Northern Ireland is three A-levels and


from Scotland four “Highers”, a qualification usually taken by Scottishpupils at 17.

Recent UK writing about transition from school to university includes:concerns with the students’ mathematical capabilities (e.g., LMS, 1995)including their conceptions of mathematical proof (Jones, 2000), reportson the change in Advanced level (A level) syllabuses (e.g., Hirst, 1991),and research on the number of students taking A levels (e.g., Kitchen,1996). In contrast, this article discusses the experiences of trainee teachersthat were reported at least three years after the transition took place. Thestrength of feeling expressed in their comments after such a gap in timeindicates that for many this transition was memorable.

METHODOLOGY

Data Collection

In coming to a decision about what method of data collection would allowus to collect responses from a relatively large and representative sample ofstudent teachers, three main factors were taken into account.

(i) We would be relying on the goodwill of PGCE tutors in other insti-tutions for the actual collection of the data, so the instrument had tobe simple to administer.

(ii) The time available on a PGCE course for anything but essentialsis very limited because of the high proportion of the course thatis spent in school. The instrument therefore had not to be tootime-consuming. A full questionnaire would not have been feasible.

(iii) We wanted to tap into reflective responses rather than rapid superfi-cial ones.

We therefore used a prompt sheet of comments on a single sheet of A4paper (Appendix A). This response sheet was developed via a pilot surveywhich used comments extracted from students’ assignments from a moduleon mathematics education taken as part of an undergraduate course. Thesix comments used in the pilot survey were selected from those available toshow a balance between positive and negative responses, to cover differentissues, to be reflective, concise and dense in meaning. It was important thatthe comments used indicated that a non-simplistic standard of responsewas expected.

In responding, students were free to choose either to develop thethemes suggested by the prompts or to introduce their own ideas about thenature of their university experience. A few also chose to tick or highlight


comments on the prompt sheet, although this response was not suggestedto them. However, almost all expressed their ideas in their own words. Itwas also hoped that, having chosen this form of data collection involvinga request for short comments, we would be tapping into their dominantresponse to their degree courses. The results from the pilot survey havealready been described (Anderson et al., 2000).

For the main survey, we decided that it would be preferable to useresponses made by other PGCE students as prompts. We carried out thesame process of comment selection using the comments generated by thepilot survey. In fact, there was little difference in the issues raised betweenthe two surveys. Nevertheless, in this report, we have confined ourselvesto responses from the main survey executed in the academic year 2000–2001. The sheet in the main survey also asked for the title of the student’sinitial degree, the university at which it was studied and the date of gradu-ation. The students were invited to put their names on the sheets, althoughthis was optional. Nevertheless, most students did identify themselves andtherefore we were aware of the gender of the respondent in the majority ofcases.

The one-page response sheet was administered early in the courseby the PGCE students’ mathematics education tutors. The students wereasked to read all their comments before adding their own comments onthe space at the bottom. In general, these were a few sentences about theirown experience of mathematics at university; few turned over the A4 sheet.The data therefore consist of short summaries of what first came into theirminds, although the use of prompts meant that this was not completelyspontaneous. We acknowledge that another research instrument could haveelicited a different range of responses. This does not detract from theauthenticity of this data, merely emphasises the fact that each student wrotewhat came immediately to mind after reading the prompts.

The Sample

In all, 173 student-teachers from 10 PGCE courses, well distributed bothgeographically and in terms of “old” and “new”3 universities, completedthe prompt sheets. It was of interest that they had attended 65 differentfirst universities, whose locations include all the countries in the UnitedKingdom and several from abroad. The sample therefore gives us descrip-tions of the experience of doing mathematics, or a related degree, at aconsiderable variety of institutions. It will be clear from the relatively smallnumber of PGCE courses compared with the number of universities atwhich the students had studied, that in the UK many students choose to


train to teach at a university different from that at which they did their firstdegree.

Table I below shows the distribution of types of degree, Table II thegender distribution of respondents and Table III the profile of years ofgraduation. In the UK a single honours degree is one in which mathematicsis the major subject studied. A joint honours degree involves the study oftwo major subjects in approximately equal proportions. The number ofmales and females identified were roughly the same. The majority of thestudents did not come straight from a university course to train to teach,though a sizeable proportion did.

TABLE I

Distribution of degree type

Single Joint Engineering Accounting Physics Other

Honours Honours with

Mathematics Mathematics

73 (42%) 38 (22%) 26 (15%) 7 (0.4%) 6 (0.3%) 23 (13%)

e.g., Maths/ Some on e.g., Economics

French 2 year Psychology

teacher

training

courses

TABLE II

Gender

Male Female Not declared

69 72 32

TABLE III

Year of graduation

Year(s) of 2000 1995 to 1990 to 1980 to Pre 1980 Not

graduation 1999 1994 1989 declared

No. of students 74 52 15 20 8 4


Data Analysis

The initial classification of the data was undertaken by a single member ofthe team, using a grounded theory type approach in which, after repeatedreading of the data, headings were created and comments were classifiedin relation to these (Glaser & Strauss, 1967). The process was continueduntil all the issues raised by a significant number of respondents had beenembodied in a heading. These analysed data were then read by a secondmember of the team who reviewed both the completeness and the robust-ness of the classification. As a result, a few new headings were addedand the descriptors were refined. The analysed data were then subject todiscussion by the whole team during which the classification was furtherrefined before the themes on which we report in this article were selected.The themes were a re-association of a group of related headings from theanalysis.

MAIN FINDINGS

Our survey results fell readily into two major categories: transition touniversity and university experiences of learning mathematics. The first ofthese categories of response constituted a “theme” in itself and responses inthe second category were discerned in terms of “themes”: transition fromschool to university, understanding: cognitive demands; independenceand support; teaching at university; assessment; emotional experiences inlearning mathematics; the value and nature of mathematics: proof; pureand applied mathematics; mathematics as a connected discipline.

While not surprising, it is of interest that students’ responses wererelevant to their future as mathematics teachers, and hence to those ofus involved in mathematics teacher education. There is also much ofrelevance and interest for lecturers of undergraduate mathematics courses,although such discussion is outside the scope of this paper. This sectiongives the frequency with which certain experiences and feelings werereported, using illustrative quotations. For each category representativeexample(s) are given. Although we looked for associations betweenpatterns and other variables (e.g., gender, previous degree) we could notfind any noteworthy links.

1) Transition from School to University

This transition was described vividly, even by nine students who hadgraduated over five years before joining the PGCE course, and was feltlargely to have been a leap rather than a smooth process. Table IV shows


TABLE IV

Transition experience

Number commenting Jump/Leap/ Repetition A natural Complex

difficult development

58/173 i.e., 33% 34 9 3 12

the different categories of experience and the number of responses in eachcategory.

Representative quotations illustrate the four categories.

Jump: The jump from A-level was great – not only in terms of leveland pace of the material but also in presentation i.e., veryintensive lectures consisting of lots of new material. (Mastersin Mathematics, graduated 2000)

For the first time I found myself out of my depth –literally in the case of fluid dynamics. (Single Honours,graduated 1990)

Repetition: I found my first year, especially statistics, a lot of my ‘A’level course repeated. (Statistics and Operational Research,graduated 1998)

Natural: I enjoyed the development of my A Levels into a mainly puremaths degree. (Joint Honours with Management, graduated1993)

Complex: The first year Maths was strange – in some instances it testedme and in others it was way below my level. (Joint Honourswith Economics, graduated 1998)

While the issue of transition from school to university was offered byremarkably many students, the majority of comments were about experi-ences as an undergraduate student studying mathematics. Some studentswere able to stand back and give an overview of their experience, forexample:

University maths was more interesting than much of school maths because you wereable to go into much more depth starting from basic axioms and developing the theoryfrom there. It was good to see behind the various things you learnt in school. (Masters inMathematics, graduated 1999)

I found the whole experience of learning mathematics at X – the combination ofself-study, lectures and tutorials – challenging, rewarding and very rigorous. The idea ofbeing ‘able to do well without doing anything original or creative’ was given no time in X,


as it is false. Being pushed to think is a major attraction of mathematics, and the universitycourse. I always felt the presence of the over-arching philosophy of the course and thepursuit of mathematical knowledge. (Masters in Mathematics, graduated 2000)

Many students, however, expressed strong views about more specificaspects of their experience. These have been grouped in themes (2)–(7)(below). For each theme, comments were classified under headings withsome respondents including a comment under more than one of theseheadings. This applies to a number of the subsequent tables.

2) Understanding: Cognitive Demands

The majority of those who mentioned understanding had achieved it overthe course of their study, often after a struggle, and with a resultant feelingof satisfaction, but almost as many had been unsuccessful; see Table V.

TABLE V

Comments on “understanding”

Number Achieved Little or no Understanding Other

commenting understanding understanding was necessary

39/173 i.e., 23% 16 12 5 6

Three quotations illustrate these categories.

Achieved: It was a struggle at the beginning of each year, but satisfyingwhen I finally understood the concepts. (Single Honours,graduated 2000)

Little: It was a struggle to maintain enjoyment in the subject, when Inever felt completely satisfied that I understood the conceptsin front of me. (Single Honours, graduated 2000)

Necessary: I liked pure maths because it was more important tounderstand than just to remember proofs of the theorems.I remember being quite proud of myself when I finallymanaged to understand and see logic and beauty of someproofs. (Physics, graduated 1989)

3) Independence and Support

Students commenting in this theme realised that they were expected towork independently and while some rose to this challenge and were offeredenough support, slightly more felt abandoned. Peer support was greatly


valued but was instigated by students, rather than being built into courses;see Table VI.

TABLE VI

Comments on independence and support

Number Need to learn Lack of support Enough support Peer

commenting independently from tutors from tutors support

54/173 i.e., 31% 27 10 7 13

The categories are illustrated by the following quotations.

Independence: The increased independence was challenging yetrewarding. As a result more time was spent investigatingideas for myself which I feel had a very positive effect.(Joint Honours with Physical Education, graduated 1999)

No support: Most lecturers and my tutor were completely unapproach-able – you couldn’t ask questions during lectures and feltvery alone. (Single Honours, graduated 2000).

Support: At times I found working by myself hard – but luckilyI could discuss things with my tutors and others at mycollege. (Single Honours, graduated 2000)

Peers: Students attending the Mathematics classes were closeknit and often tended to study as a group outside theclassroom. This . . . was extremely effective as everyonefelt involved and supported even at the most difficulttimes. (Business Studies, graduated 2000)

4) Teaching at University

Of those students who chose to comment about the university teaching theyhad experienced, most were very critical of the teachers and the teachingmethods, particularly lectures, as indicated in Table VII.

TABLE VII

Negative and positive views on university teaching

Number commenting Negative Positive It depends on the lecturer

40/173 i.e., 23% 30 4 6


Quotations reflect the tenor of most comments.

Negative: I stopped attending lectures at the start of my second year andnever went back. To sit taking meaningless notes all day wasjust too much . . . I enjoyed the tutorials and feel that lecturescould have been replaced with handed out notes and some realteaching allowed to take place . . . most lecturers could notteach to save their lives. (Single Honours, graduated 1976)

Positive: At uni for the first time I was taught in a logical way and couldsee the patterns in it. (Physics, graduated 1999)

Dependent: The quality of the lecturer’s presentation impacted onlearning and understanding of the subject/topic. For example,a foreign lecturer with poor writing on blackboard ledto a large number of students lacking understanding, andcomplaining to Dept. Lectures given by enthusiastic, clearteachers were welcomed and students performed better inexams. (Joint Honours with Management, graduated 1995)

5) Assessment

Almost all the students who mentioned assessment referred to rote learningfor examinations and were critical of examinations as a measure of abilityor understanding; see Table VIII.

TABLE VIII

Comments on Assessment

Number commenting Exams – negative Exams – positive Exercises Coursework

25/173 i.e., 15% 23 2 2 1

The following comments are representative of what respondents wrote.

Negative: It seemed you just had to learn proofs and regurgitate themin order to pass an exam, we didn’t even have to understandit, just remember it. (Engineering, graduated 2000)

Positive: It was only when it got to the Summer Term and exami-nations that I got involved with the subject through inde-pendent study. Then I would see those links which areso exciting and reach a level of understanding that wassatisfactory if not good. (Science, graduated 1999)

Exercises: It was difficult to keep on top of the exercise sheets as therewere no official deadlines for the work. I usually ended up


leaving all the work until very close to the exam time. (Jointhonours with French, graduated 1999)

Coursework: I found that the final exams were the worst way of testingmy ability and the marks I received did not reflect my abilitywell at all. I did one ‘coursework’ paper which was my bestpaper by far and I feel that more options like these wouldhave enabled me to do much better in my degree. (Singlehonours, graduated 2000)

6) The Emotional Experience of Learning Mathematics

Feelings of struggle and frustration were common amongst those reportingthe emotional experiences of learning mathematics at university. This waslargely a negative or lonely experience but sometimes rewarding, withabout a quarter of these students experiencing moments of pure joy wheninsight was obtained; see Table IX.

TABLE IX

Comments on emotions experienced

Number commenting Struggle and frustration That Eureka feeling Loneliness

41/173 i.e., 24% 30 9 10

The following comments illustrate responses.

Struggle: I hated it most of the time . . . It became difficult to enjoy someof the maths modules if you were struggling and couldn’t getany help (Joint Honours with Education, graduated 2000)

Eureka: I sometimes feel there is no light at the end of the tunneluntil one day, it just all clicks into place. (Single Honours,graduated 1989)

Loneliness: However I did feel isolated and this became more apparentin the final year because the difficulty level increased. (Jointhonours with Statistics, graduated 2000)

7) The Value and Nature of Mathematics

Most of the comments about the value and nature of the mathematics beingstudied were about proof or pure and applied mathematics, together withsome about mathematics being connected.


ProofAlthough an identifiable group of students enjoyed the rigour of proof,others felt it was emphasised at the expense of application, and was sonew and difficult that they resorted to memorising set proofs for exams;see Table X.

TABLE X

Comments on proof

Number Proofs Over – Needed Enjoyable Proofs Positive: Negative:

commenting were emphasised to be were rigour for exams

difficult memorised new or logic only

52/173 i.e., 30% 7 7 3 6 6 11 12

Categories are illustrated by the following comments.

Hard: I found pure [mathematics] very hard – thinking up myown proofs seemed impossible. (No details supplied)

Over-emphasised: I felt too much emphasis was placed purely on learningdifferent proofs. (Joint Honours with Finance gradu-ated 1997)

To be memorised: [Proofs] could be memorised in the short term, repro-duced in the exam and then almost immediatelyforgotten never to be needed again. (Joint Honours withFinance graduated 1997)

Enjoyable: I love the idea of formal proof by opposition to thescientific proof (Single Honours, France, graduated1999)

New: It was lecture & book & try to understand – this isdifficult when theorems & proofs are flying about &you’ve never really had any experience of them. (Jointhonours with European Studies, graduated 1995)

Logic & Rigour: I remember being quite proud of myself when I finallymanaged to understand and see logic and beauty ofsome proofs. (Physics, graduated 1989)

For exams only: Much emphasis was placed on recalling proofs forexams rather than application. (Single Honours, gradu-ated 1994)


Pure and applied mathematicsThe majority of those commenting about the balance of their courses interms of pure and applied mathematics had been able either to see wheretheory was applied or actually to apply it themselves in problem solvingsituations; see Table XI.

TABLE XI

Enjoyment of pure or applied mathematics

Number Enjoyed Found the Enjoyed Liked the

commenting applied mathematics pure maths balance

mathematics too theoretical

and irrelevant

39/173 i.e., 23% 20 7 11 1

The following comments represent responses given.

Liked applied mathematics: It was rewarding to use calculus to solveproblems relating to the real world e.g., pres-sures exerted by water in pipes. (Geography,graduated 2000)

Too theoretical/irrelevant Little attempt was made at making the workrelevant . . . This was true of both depart-ments [Pure and Applied]. (Economics,graduated 2000)

Liked pure mathematics: Learning real pure maths was a revelation,school mathematics suddenly seemed verysmall. (PhD Maths, graduated 1992)

Liked the balance: The emphasis on practical application ofmaths proved very valuable but at the sametime, for me, the potential for exploring puremaths more deeply was exciting and enjoy-able. (Business Studies, graduated 1982)

Mathematics as a “connected” subjectFinally, although only 13 students (8%) commented on mathematics as a“connected” subject, for them the connections between different aspects ofmathematics was both an aspect of progression and of their appreciationof coherence within the discipline itself, as the following quotation shows.


Could get an overview of the ‘whole’ subject for the first time and see that the waysof thinking needed in one area were the same as in an apparently totally different one.(Physics, graduated 1993)

INTERPRETATION AND DISCUSSION OF RESULTS FORTEACHER PREPARATION

We now return to our initial prompt from the introduction:

How can a one-year secondary mathematics teacher preparation programme incorporateintelligently the undergraduate experiences of students?

In order to respond to this question, which is essentially to interpret ourresults for other contexts as well as our own, we revisit each of the sevenmajor themes which have been theorised from analysis of the data –transition, understanding, independence, teaching, assessment, emotionalaspects, nature of mathematics – reported in the previous section, and foreach of these themes, we:

(a) delve further into issues in the individuals’ responses;(b) highlight the relevance of these issues for their preparation to be

teachers;(c) interpret the significance for mathematics teacher educators.

By widening the lens from a) consideration of individuals’ reports, tob) developing individuals to teach mathematics, to c) planning for teacherpreparation, we show how student teachers’ undergraduate experiences areresources relevant to mathematics teacher preparation.

1) Transition

a) Issues for student teachersMany of the student teachers in our sample experienced a curriculumdiscontinuity because university staff assumed that incoming undergradu-ates knew more than was actually the case, although others foundthemselves repeating work they had grasped at school.

It was a huge shock. Not only was the rate at which new work was covered rapid (!), but Ifelt that I had very little help when things weren’t going well (Single Honours, graduated2000)

They also had to deal with being, relatively, less successful.

It was very disheartening to go from getting very high marks at school to getting 20% in1st year Analysis, simply because it was such a jump from A-level & we’d done nothinglike it before. (Single Honours, graduated 2000)


This “shock” indicates a problem with transition from school to univer-sity which concerns students and threatens their sense of security. Schoolchildren also undergo transitions when they move school or opt into moreadvanced mathematics classes. In this “transition” section we focus onthe issues of (i) curriculum discontinuity and (ii) change in perceivedcompetence relative to a new class of students.

b) Relevance for their education as teachers(i) Curriculum discontinuityThese prospective teachers have experienced several transitions in theireducation from, for example, primary school to secondary school andfrom the end of compulsory schooling (at 16 years in the UK) to tertiaryeducation (from 16–19 years). The transition from school to universityhas emotional, social as well as cognitive aspects. Reflection on their owndiscomfort or exhilaration on this transition may help student teachers toappreciate school pupils’ transitions and prepare their pupils appropriately.

In the significant transition that occurs for those sixteen year olds inEngland who choose to study mathematics at A level in post-compulsoryeducation, the pupils can be almost immediately over-taken by gaps andlimitations in their grasp of mathematics. For example, from Bruce’s(Bruce, 1994) study of pupils who started an A-level course, after onemonth, James,

felt frustrated in that he would start questions and they would all go wrong and he wouldbe unable to complete them . . . This was putting him off, especially when some of the classwere enjoying the work without stumbling in this way. (p. 112)

Curriculum discontinuity can involve repetition as well as curriculum gaps.Repetition, which some experienced at the start of their university courses,can be compared with starting secondary school and learning nothing newin mathematics in the first year (Cockcroft, 1982).

(ii) Change in perceived competenceMany university mathematics undergraduates will have only rarelystruggled with school mathematics so it is not surprising it is difficult forthem to adjust to struggling at university. If these mathematically ablepupils had been challenged at school would they have dealt with thestruggle and challenge better? Pupils’ mathematical awareness will notbe deepened if everything is too easy. “We need to educate our pupilsto know that out of difficulties encountered and overcome, emerges real‘know-how’ ” (Hatch, 1999, p. 108).


c) Significance for teacher educatorsTrainee teachers need to be prepared to support those pupils who maymake the specific transition from school to university mathematics. Theneeds of mathematically able pupils have been brought to the fore recentlyin the UK by government actions: teacher educators need to stress theimportance of thinking about and providing for the needs of those whowill later study mathematics at the university. This can be done by linkingthe student teachers’ experiences with the Standard requiring trainees toplan tasks “which challenge pupils and ensure high levels of pupil interest”(DfEE, 1998).

In terms of transitions in general, it is important to note that someof the causes of low attainment in mathematics have been attributed tocurriculum discontinuities (Haylock, 1991). Consequently, the NationalCurriculum (DfEE, 1998) and the recent National Numeracy Strategy(DfEE, 1999) have been designed to include remedies for this problem.In particular, the National Numeracy Strategy provides bridging units,designed to be started in the last term of primary education and completedin the first term of secondary education. Trainees can be encouraged to usethese bridging units as a device to diminish curriculum discontinuity.

The awareness that transitions are points where difficulties can arise isan important one for all prospective teachers. Teacher education require-ments in England demand that new teachers are cognisant of the progres-sion between “Key Stages”. To this end, the one-year PGCE course can usethe deeply felt transitional experiences from university to help prospectiveteachers understand the problems which pupils face. This experienceincludes the emotional shock of being in a very different situation, amongothers who often seem dauntingly better prepared or more able. Bothaspects of the experience can be discussed as relevant to school life, withthe common facets of all transitions being brought out and recognised asthe powerful forces that they are.

2) Understanding

a) Issues for student teachersIn describing aspects of the university experience of learning mathematicsbeyond the transitional phase, some prospective teachers claimed thatunderstanding could be sacrificed:

As part of the undergrad degree it was possible to rote learn without ever fully under-standing. Maths assignments could be completed from text books and the underlyingprinciples completely missed. Slightly abstract teaching often not conducive to improvingunderstanding and leading to extra work having to be completed in own time to facilitatethis. (No details provided)


whereas others came to value it for its own sake and were sceptical aboutbeing able to succeed in exams without some understanding.

However, once I had come to terms with the fact that getting correct answers wasn’t theonly way to feel that I had achieved and that just being able to use the work I did understandto explore new concepts was good in itself, I began to enjoy the challenge instead of fearingit. (Joint Honours, graduated 2000)

These differences are of importance when considering the teaching andlearning of mathematics in school.

b) Relevance for their education as teachersWhat are the problems and possibilities in teaching for understanding?There seem to be useful parallels between the compromises that sometimeshad to be made when doing a mathematics degree and those which haveto be made by the classroom teacher if her pupils are to get the qualifica-tion they need. One part of the compromise involves aiming for a balancebetween long-term and short-term goals, and knowing at which point in acourse to stress one or the other. It may be tempting for teachers to aim fora high performance in the examinations at the expense of understandingand without looking to where the pupil is going next – on to the next phaseof education, often in a different institution. It is only too easy to choosethe short-term goal if the next stage is taught by someone else at a differentestablishment.

c) Significance for teacher educatorsAs part of their teacher-education, student teachers can be asked toreflect on occasions when they have not understood some mathematics or“crammed” for an exam. Questions such as: “Did not understanding feelsatisfying?”, “Was “cramming” adequate preparation for the next stageof learning?”, can be used to raise student-teachers” awareness of theacademic and affective consequence of failing to teach for understanding.

It is difficult, however, to imagine a teacher education course in mathe-matics that does not stress the importance of teaching for understanding,or does not introduce Skemp’s (1976) distinction between relationaland instrumental understanding. The arguments for developing relationalunderstanding are compelling, yet clearly it is difficult for teachers toachieve the aim. Teacher educators, working with trainees who havestruggled with the tension themselves as learners, have material to workwith here. They can explore Skemp’s arguments in the light of these experi-ences, but they can also examine why teaching for understanding is sodifficult.


Teacher educators can also unpick the student teachers’ automaticallyretrieved knowledge of procedures and make sure that the meanings andconnections grounding these procedures are exposed and discussed. Whenintroducing models for teaching, the idea of returning to meanings andunderstandings cyclically (e.g., Booth, 1984) could be advocated. This isin contrast to models that move from meaningful explanations at the begin-ning of a teaching sequence to reinforcement and applications, withoutever returning to the original underpinnings. For instance, when planningwork on proof with older pupils, trainees could return to the justification ofcompressed rules such as “minus times minus is a plus” which may havebeen justified when the pupils were younger with analogies and patternbut which can be re-examined using an axiomatic approach in the contextof teaching proof. For those trainees who have not experienced curriculawith an emphasis on proof, this may be the first time they have consideredusing this kind of work with older pupils or indeed encountered such ajustification themselves.

The implications of failing to teach for understanding have repercus-sions not only on learners’ mathematical understandings but on their imageof the subject itself, as a meaningless collection of arbitrary rules to beremembered for examinations and then forgotten. In teacher education,these issues are also significant in work on pupils’ attitudes and on thejustification for the amount of time pupils spend on mathematics in theiryears of compulsory schooling.

3) Independence and Support

a) Issues for student teachersOne of the most striking contrasts between the school and universityexperience is the reduction in contact time with teachers or tutors and theonus on students to organise their own working days.

Later on when things began to require a lot more detail I found I struggled and only as aresult of fantastic support from both my tutors & fellow students, was I able to enjoy itand predominantly conquer it. (Joint Honours (biology), graduated 2000)

There was a lot of emphasis on doing things independently. The lectures weremore a guideline than anything. (Masters in mathematics, graduated 1999)

b) Relevance for their education as teachersTrainees may believe that pupils in school have to be spoon-fed so thatthey get the grades they need and that only those who have successfullyachieved the right qualifications for entry to further study of mathematicsare capable of independent learning. Those who have felt lonely and


unsupported in learning mathematics at university may over-compensatewhen they become teachers: perhaps “breaking the work into small steps,repeating explanations and giving rules (short cuts)” (Scott Hodgetts,1995, p. 50). This kind of teacher behaviour tends to create pupils whowould say that “the teacher’s job was to convey their knowledge and thattheir own role was to absorb it” (p. 50). Wigley (1991) argues cogentlyagainst what he calls “the path smoothing model” (Wigley, 1991, p. 4) asdescribed above and advocates instead “the challenging model” (Wigley,1991, p. 6) which he believes offers independence and insight for pupils ofall abilities.

Each individual has a different level of tolerance of challenge anddifferent capacities to push themselves in the face of challenge. Somepeople may require more stability than others and the amount of stabilityneeded may vary over time and in different contexts. Hence, it is notsurprising that for some students, university mathematics was seen asan exciting challenge, while for others, the lack of adequate support in“finding the way” had a serious negative effect. Mathematics teachingattracts both those who find security in routines and methods and alsothose who want to leap chasms into abstract territories using the preci-sion of mathematics. The need to feel that there is help available whenthe struggle becomes unbearable is described keenly. However, that kindof support was not experienced by everyone at university, although thosewho did were appreciative of it. When these students begin to teach,their experience should sensitise them to the crucial and critical natureof support.

c) Significance for teacher educatorsHow can we help student teachers to include challenge and struggle in theirteaching of pupils of all ages in order to develop independent thinking?One approach is to make an analogy with their own lack of preparationfor university work. From this comparison, independence can be seenas a vital long-term goal for mathematics teaching at school level. Astrainees, they could share their own preferences for risk and security,recognise similarities and differences and consider the role of context indetermining these orientations. They could take the initiative by settingup specific opportunities to help individuals and small groups of pupilsas well as teaching whole classes, and to investigate pupils’ preferencesfor learning styles. Learning to plan appropriately for pupils with differentlearning preferences, and to incorporate gradual withdrawal of support aspupils develop competence, develops the aim of promoting pupil agencyin learning. This promotion of pupil agency can be further developed by


setting up and investigating the effectiveness of peer group or cross-agetutoring (FitzGibbon, 1988).

When planning sequences of lessons, there are ways of incorporatingchallenge and independent thinking within a supportive environment.Bell’s (1993) research showed that sequences of gently graded tasks do notresult in long-term learning. Trainees could be introduced to his principlesfor the design of teaching which include:

[exposing] the cognitive conflict and [helping] the learner to achieve a resolution. This isone type of intervention . . . another [intervention] is adjusting the degree of challenge . . .

(Bell, 1993, p. 9)

The use of these principles can be illustrated with reference to theCognitive Acceleration in Mathematics Education (CAME) project(Adhami, Johnson & Shayer, 1998) for pupils aged between 11 and 13.The trainees can be shown how the CAME activities have been designed topromote cognitive conflict which is to be resolved in peer group discussionwith teacher mediation, followed by reflection and review. In this project,although they are being well supported, pupils are expected to struggletowards conceptual understanding.

4) Teaching

a) Issues for student teachersMathematics teaching had no structure or direction at university. (Joint Honours statistics/economics, graduated 1986)

At times lecturers were poor teachers and textbooks were not clear or provedbaffling. (Masters in mathematics, graduated 2000)

These students generally succeeded at school, whether or not the mathe-matics was “delivered” via a rather formal exposition; they were operatingwithin their domain of competence. However, at university, some foundthemselves near the limits of their competence and then the formal exposi-tion characteristic of the university lecturing style was not an effectivemethod to communicate new mathematical material. Many pupils at schoolfind mathematics difficult, just as many of these student teachers had founduniversity mathematics difficult. From the many years of mathematicsteaching experience we have between us, we would speculate that nearthese limits of competence the quality of the teaching becomes very muchmore significant at any level.

b) Relevance for their education as teachersTrainees’ experience of the ineffectiveness of formal lecturing to help themlearn when the mathematics is difficult could help them to become aware


of the inappropriate nature of such a “lecturing style” for their pupils. Byconsidering the relationship between a teacher’s personal understandingand a learner’s current knowledge, student teachers’ awareness of the needfor knowledge for teaching can be motivated. Furthermore, their experi-ence indicates that the way in which a dialogue is set up between teacherand pupil, both academically and personally, may affect the quality oflearning profoundly. Our shared experience suggests that, at interview,many applicants for teacher training seem aware that making the workinteresting to the pupils is important. They have reason to feel that theirjob is to ensure that their pupils are, in the words of a respondent, “reallycaptivated and inspired” (Joint Honours (finance), graduated 1997).

c) Significance for teacher educatorsThe challenge for teacher educators, working in partnership with teachersin schools, is to help develop teachers with the appreciations, knowledgeand skills to teach mathematics so that it is meaningful, connected andinteresting to all pupils. UK government requirements require trainees tomeet specified standards before they qualify, including:

3.3.3. They teach clearly structured lessons or sequences of work which interest andmotivate pupils and which:

• make learning objectives clear to pupils;• employ interactive teaching methods;• promote active and independent learning that enables pupils to think for themselves,

and to plan and manage their own learning. (DfES, 2002, p. 12)

These standards apply to all subject areas and they rely on a certain amountof interpretation by tutors and by mentors in schools. It may be that tutorsin different subjects stress different aspects of the standards and that, inthe case of mathematics, where there is a shortage of suitably qualifiedteachers, the requirement to interest and motivate pupils may not be judgedas stringently as in other subjects. At the time of writing, a governmentinquiry on the teaching of post-14 mathematics has been set up to “tacklethe widespread unease at the state of maths education” (The Times Educa-tional Supplement, 2002, p. 8). The chair, Professor Adrian Smith, claimsthat “Nobody is telling us how exciting maths is” (p. 8). The problemrevealed by our research for teacher educators is that many of our traineessee their school experience as a golden time compared with their univer-sity experience. There is clearly a danger then that poor school practicewill be recycled, and that the one-year PGCE course will have difficultyin interrupting this process. The personal experiences of trainees providerich opportunities to investigate the attractiveness and the limitations of


didactic approaches, and the need to present the subject in a stimulatingway which stresses its fascination and relevance.

5) Assessment

a) Issues for student teachersI found a lot of people saying that anybody can do a maths exam as all you need to do islearn formulas. However, I believe that a person can only be a successful mathematician ifthey understand the concept of the subject. It is impossible to learn everything by memory.(Single Honours, graduated 2000)

Although many of the students in our survey, the majority of whom wererecent graduates, will have had some experience of different forms ofassessment within their university course, within our data, there are fewcomments about any sort of coursework or formative assessment. This isunsurprising: the vast majority of assessments the students have experi-enced were formal, timed examinations, and because of the degree ofstress that such an examination stimulates, they are more memorable thanother forms of assessments. There are frequent references to memoryand “regurgitation”. A smaller number aims to work from understandingthe concepts. This needs to be considered together with the data aboutunderstanding, where many indicated that they had given up any idea ofattempting to understand, so that examinations were perforce reduced tomore or less meaningless rehearsals of memorised information.

b) Relevance for their education as teachersAs teachers these students will be faced with the dilemma of how bestto prepare their pupils for the almost annual public testing which is thelot of English school children today. Pupils themselves and indeed theirparents, are showing an almost obsessive concern with the grades whichthey get at GCSE or at A level. Rises and falls in the overall pass rates,together with queries about the fairness of the grading system, are taken upeach year in the political arena. The pressure on teachers to groom pupilsfor examinations rather than to teach for understanding can be enormous.Their experience of university mathematics offers many of the traineeteachers access to the feeling of pointlessness caused by the reductionof mathematics, under the pressure of examinations, to endless repetitionof meaningless procedures. This could help them, as teachers resist thepressure just to tell their pupils “how to do it”.

It is difficult to predict how these students’ limited experiences ofassessment styles will influence their attitudes to assessment as teachers.In their case study of mentor-student teacher interaction, Butterfield,Williams & Marr (1999) found that serving teachers tend to view assess-


ment in terms of examinations, tests and class control rather than in termsof pupil learning. However, these trainee teachers will need to be flexibleenough to respond to new assessment instruments and new methods ofstudent assessment. For example, the importance of formative models ofassessment is now well known (Black & Wiliam, 1998), yet the students’experience of being assessed in mathematics seems to have been almostentirely summative. The crucial message from the Black and Wiliamresearch is we now have evidence that good formative assessment isassociated with improved examination performance.

c) Significance for teacher educatorsTeacher educators need to help the student teachers to appreciate thatthe formal teaching of lectures assessed by examinations caused many ofthem to relinquish the ambition to understand. Since this is presented asan implicit contrast to previous experience we can conjecture that, beingthe more able pupils in mathematics, they did understand what they didat school. They can be led to explore whether, if mathematics teachingis presented in a formal way at school and assessed by summative tests,pupils too may be led into rote learning in response to work which they feelmeaningless. It is very common in secondary school in the UK for a topicto be taught and a formal test set at the end of the teaching, which mightbe at the end of only a fortnight’s work. Such practices can be questioned.

There is a great contrast between the student’s university experiencesof assessment and the requirements of the standards for qualified teacherstatus that the trainees have to satisfy in relation to assessment of pupils:

3.2.1 They make appropriate use of a range of monitoring and assessment strategies toevaluate pupils’ progress towards planned learning objectives, and use this information toimprove their own planning and teaching.

3.2.2 They monitor and assess as they teach, giving immediate and constructivefeedback to support pupils as they learn. They involve pupils in reflecting on, evaluatingand improving their own performance. (DfES, 2002, p. 10)

Teacher educators have to help trainees to appreciate the broad nature ofassessment described in these standards and the implications of them forthe way they should teach and weave informal assessment opportunitiesinto their day-to-day teaching. The last sentence of 3.2.2 describes a jointprocess of formative assessment undertaken by teacher and pupil, in whichthe pupil takes some responsibility for knowing what she/he can or cannotdo. Reflection on what the university experience did not provide in this areacan be used in sessions on the PGCE in order to help the student teachersto reformulate their understanding of how teaching and assessment areinterlinked.


6) The Emotional Experience of Learning Mathematics

a) Issues for student teachersI found a maths degree very rewarding though and you can’t beat the feeling of satisfactionand achievement when a problem you’ve been struggling on suddenly is solved! (SingleHonours, graduated 2000)

I hated it most of the time . . . It became difficult to enjoy some of the mathsmodules if you were struggling and couldn’t get any help. (Joint Honours with Education,graduated 2000)

Such contrasts in descriptions of emotional experience point towards theimportance of emotion in mathematical learning.

b) Relevance for their education as teachersThese trainees’ curriculum requires them to “understand how pupils’learning in the subject is affected by their physical, intellectual, emotionaland social development” (DfEE, 1998, p. 11). Reflection on their ownfeelings about learning mathematics at a level where they were emotion-ally engaged – sometimes threatened, sometimes elated – can be usedto sensitise trainee teachers to pupils’ feelings and make them receptiveto research evidence on mathematics anxiety (e.g., Evans, 2000; Buxton,1981; Hoyles, 1982). Some of the adults in Buxton’s study had no expecta-tion of being able to plan solutions to problems, were unable to plan them,or were unable to plan them in time. In Hoyles’ (1982, p. 369) study ofschool children, failure in mathematics was associated with “feelings ofinadequacy and anxiety”. Such pupils may crave security and step-by-stepexplanations, resulting in highly dependent learners. To avoid this, Buxtonargues that a high level of success needs to be coupled with a level offailure that can be tolerated.

c) Significance for teacher educatorsThere were some students in our survey who expressed positive emotions,as did some of the school pupils identified by Hoyles. Questions for thetraining period, then, involve exploring the different emotional responsesevoked in typical classroom settings, and investigating constructivestrategies which might alleviate mathematics anxiety without making thework boring and unstimulating. The teacher educator’s job is to drawattention to the emotional qualities in classroom scenarios. This can bedone with the class of student teachers themselves. Role playing anxiety-provoking situations (like not being allowed “enough time” to complete amathematical task, or being shouted at) and, in contrast, experiencing calmand focussed classroom climates (achieved, for example, by setting taskswhere individuals can contribute within unthreatening groups or giving


individual meditative tasks, such as pattern work), the student teachers’awareness of the emotional dimension of learning mathematics can beenhanced.

7) The Value and Nature of Mathematics

a) Issues for student teachersBy the end of my four year course I was forced to specialise. Choosing pure maths throughinterest I found a lot of the higher algebra and group theory contrived and irrelevant.(Masters in mathematics, graduated 2000)

What do we understand words like ‘contrived’ and ‘irrelevant’ to mean,where mathematics is concerned, and how is this important? Othercomments link such perceptions to the learning of ‘proof’ (see Table xabove, et sequ.).

b) Relevance for their education as teachersWhile some students’ responses communicated a positive purpose tolearning mathematics, some of these university undergraduates commu-nicated an alienation consonant with that experienced by some pupilsat school. Boaler’s (1997) longitudinal case study of school classroomsindicates that different teaching methodologies give rise to different pupilviews of the value and nature of mathematics. At the undergraduatelevel, Crawford and colleagues’ (e.g., Crawford, Gordon, Nicholas &Prosser, 1998) research revealed two conceptions of mathematics, the“fragmented” and the “cohesive”.

More specifically, students who were pupils in English schools in the1990s are unlikely to have met the idea of a mathematical proof beforestudying at university, yet the most recent English mathematics curriculum(DfEE, 1999) re-emphasises its importance. Hence within their year ofteacher preparation, many of these trainees need to learn school-relevantproof techniques and how to provide appropriate classroom experiences,as well as to see themselves as teachers of mathematical reasoning andproof (Rodd & Monaghan, 2001). The current emphasis on proof withinthe secondary curriculum involves pupils explaining and justifying theirconclusions in an atmosphere of collective enquiry rather than writingdown some formal reasoning, stereotypical of a university mathematicsproof. Thus there is a big challenge for new teachers in this area of thecurriculum.

The re-emergence of proof within the school mathematics curriculumis an example of national policy affecting the nature of the school mathe-matics discipline. The extent to which ICT is to be incorporated into schoolmathematics is another area where the nature of school mathematics is


affected by policy. For these student teachers’ development, it behovesthem to be aware that the nature of school mathematics is not immutableand as teachers they will be expected to adapt to new revised curricula.

c) Significance for teacher educatorsIn different eras and societies different knowledge, skills and under-standing in mathematics are valued by that society. In our own culture,mathematics is central within the school curriculum. We have, therefore,an opportunity to discuss with our student teachers why this should be andwhether, given mathematics teacher shortages (in England), mathematicswill retain its importance in the education of new generations. For example,we could discuss whether a value of mathematics is its link to rationality.At a school level at least, this idea is expressed in the quotation chosenfor the introduction to the National Curriculum for Mathematics by DrColin Sparrow: “[mathematics] lies at the core of scientific thinking, andof rational and logical argument” (DfEE, 1999, p. 57). Since rationality,as characterised by logical deduction and exemplified by mathematics, isa potential tool of thought for future citizens and is accorded huge impor-tance (even if the practice is wanting) in Western, and now global, thought,it could be said that school children have an entitlement to be inculcatedinto being able to operate in that mode.

CONCLUSION

In standard graduate teacher training in the UK, new PGCE studentsarrive in the autumn of an academic year and, if successful, start theirteaching career in the autumn of the following year. Success consists oftheir attaining the four “Standards for Initial Teacher Training’ (DfEE,1998; DfES, 2002) which address “knowledge and understanding”, “plan-ning and preparation”, “classroom teaching” and “professional issues”.Any one-year course, such as the PGCE, uses the students” mathematicalknowledge acquired at university (as well as at school) to enable themto plan and deliver mathematics lessons as mathematics teachers them-selves. Our claim is that – with respect to teacher training – the fruitsof the undergraduate degree are not just mathematics, but also includethe products of the experience represented by the seven themes we havediscussed. In other words, while we have always in these one-year coursesincorporated the students’ mathematical knowledge, other aspects of theundergraduate experience (specifically: transition, understanding, inde-pendence, teaching, assessment, emotion and nature of mathematics) are


available and suitable for generalised use in the preparation of secondarymathematics teachers.

Although clearly the experiences of an adult are not identical to thoseof school pupils, research with in-service teachers (Hatch & Shiu, 1999;Hatch, 2000) indicates that teachers can gain valuable classroom insightsfrom consideration of their own learning. Those who gave up their desireto understand mathematics and just memorised calculations need to be ableto examine this painful awareness and draw out its implications for theirown teaching. Those who valued struggle and subsequent triumph need tobe helped to attain the pedagogic confidence to build this expectation intotheir classrooms. Those who found examinations an unhelpful experiencemay come to realise that the common technique of “teach a topic, test atopic and move on” may be a questionable way of building understandingof mathematics. Those who appreciated the value of peer tuition may beable to link this to small group discussion and its value in the classroom.

Some of the products of experience represented by the themes identifiedin this research have already been incorporated into the statutory teachertraining curriculum for England and Wales (DfES, 2002) as well as intothe courses of individual institutions. For example, trainees are requiredto understand issues related to transition and, in many courses, studentteachers are asked to reflect on their own learning journeys (Goulding,1997) in order to increase their empathy with their future pupils’ struggles(emotion) as well as raise their awareness about understanding or teaching.Let us exemplify further the claim that these aspects of undergraduateexperience of learning mathematics can be used to enhance a one yearteacher preparation course: We consider the research finding that traineeteachers tend to teach the way they were taught at school (Powell, 1992),which was predominantly in a transmissive mode. As teacher trainerscharged with preparing these student teachers to teach the full range ofsecondary children, we are mindful that these student teachers were rela-tively successful at school mathematics and we are required to preparethem to teach pupils who may lack their facility, their resources or theirdetermination. Representatives of our student teacher class will have hadrelative success at school mathematics delivered transmissively but experi-enced significant struggle at university mathematics delivered in a similarway. These student teachers are in a position to illuminate why teachingas telling can work for students positioned advantageously, but fails forstudents who are struggling; instead of taking refuge in comfortable schoolmemories, the student-teachers can make use of the university experienceof being at or beyond the limits of their competence to learn about teachingmathematics.


In designing learning experiences for prospective teachers on one-yeartraining courses, teacher educators have a magnificent resource available:the student teachers’ own undergraduate experiences of studying mathe-matics. We have presented significant themes within this experience asreported by student teachers and related each of these themes to a studentteacher’s own development and to issues for secondary mathematicsteacher education. By acknowledging difficulties and successes in theirown experience of learning mathematics at university, student teachersare well placed to grasp that learning inevitably involves a student’s feel-ings and their life in general. By thus positioning our trainees holisticallyas feeling people, learners of mathematics and of the teaching/learningprocess, we address not only our statutory “Standards” but also, we hope,keep alive the desire to teach mathematics.

APPENDIX A: THE PROMPT SHEET

Name: Title of degree:

University attended (PGCE): University attended (Degree):

Date of degree:

Here are some statements about university mathematics written by PGCE mathematicsstudents. Please write a few sentences about your own experience of mathematics atuniversity.∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

• It was the ultimate challenge because for the first time I found myself strugglingespecially in the first year. Maths at university was initially a very big jump from A level.

• I enjoyed learning mathematics at a higher level. My favourite subject was puremaths – understanding theorems and how they worked. There was much moreemphasis on proof and rigour.

• The more complicated structures at university are not taught butrather thrown at you and you are expected to hear something onceand remember and understand it straight away. Most of the time itwas self study working out by yourself how to answer problems andlearning it for the exams.

• It was frustrating sometimes to spend two hours puzzling over aconcept that could feasibly have taken 2 minutes to thrash outwith the lecturer in person. I felt very much more alone. This didhowever increase the personal satisfaction of solving a problem.

• I felt that some understanding was necessary in order to rote learn for exams . . . Itwas possible to do extremely well without doing anything original or creative. Afterthe exam you don’t need to know half the stuff. It’s a case of learning the procedurefor the exam then forget it all.


• In the commercial workplace I can say that the rigour and tests oflogical reasoning I learned have to be of real advantage − I canthink through ideas for myself.

NOTES

1 This is a requirement although students have a variety of degrees and the institutions whoaccept them on to PGCE Mathematics courses will have to exercise their own judgementabout what counts as strongly related mathematics content in a non- mathematics degree.2 The TaLUM group was started by the Mathematical Association in the UK and has beensupported by a wide range of other associations concerned with mathematics. The sub-group that worked on this study has been supported largely by the Association of Teachersof Mathematics.3 In the British context a “new” university is an institution which was classed as a poly-technic before the 1990s. An “old” university is any other university. There were 7 “old”universities and 3 “new” universities in the sample of institutions where the students weredoing their PGCE courses.

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MARIA GOULDING

Department of Educational StudiesUniversity of YorkYork YO10 5DD

GILLIAN HATCH

Centre for Mathematics Education, Institute of EducationManchester Metropolitan UniversityDidsbury, Manchester M20 2RR

MELISSA RODD

Centre for Studies in Science and Mathematics EducationSchool of EducationUniversity of LeedsLeeds LS2 9JT

ACKNOWLEDGEMENT

The Editors of JMTE acknowledge most gratefully the collaboration of thefollowing colleagues in reviewing papers for JMTE. The quality of JMTEis dependent on sympathetic and cogent reviews of the articles submitted.We thank our reviewers most sincerely.

Adler, JillAinley, JanetAn, ShuhuaAnderson, JudithAnthony, GlendaArbaugh, FrancesBall, DeborahBattista, M. P.Becker, JerryBecker, JoanneBell, AlanBlanton, MariaBlume, G.Boaler, JoBobis, J.Borasi, RafaellaBowers, JanetBreen ChrisBurton, LeoneBush, W. S.Campos Lins, RomuloCarrillo, JoseCestari, Maria LuizaChapman, OliveChazan, DanChick, HelenChronaki, AnnaCivil, Marta

Cockburn, AnneD’Ambrosio, BeatriceDa Ponte, JoaoDahl, HenrikDawson, SandyDoerr, HelenDreyfus, TommyEmpson, SusanEven, RuhamaFranke, MegaGates, PeterGellert, UweGoffree, FredGoos, MerrilynGoulding, MariaGraven, MellonyGrevholm, BarbroGroves, SusieGrouws, DougGutierrez, AngelHaggarty, LindaHalai, AnjumHanna, GilaHarel, G.Heaton, RuthHerbst, PatricioHoch, JodyHorne, Marge

Hoyles, CeliaJahnke, Hans NielsJones, KeithJones, TonyKoyama, MasatakaKahan, JeremyKieren, TomKlosterman, P.Knuth, EricKoleza, E.Laborde, ColetteLachance, AndreaLassak, MarshallLeder, GilaLerman, StephenLin, Fou LaiLloyd, GwendolinMason, JohnMcLeod, DougMewborn, DeniseMok, IdaMulligan, JoanneNardi, ElenaNemirovsky, RicardoNeubrand, JohannaNickerson, SusanPerks, PatPeter Koop, Andrea

Journal of Mathematics Teacher Education 6: 395–396, 2003.

396 ACKNOWLEDGEMENT

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Skott, JeppeSteinbring, HeinzSzendrai, JulianaSztajn, PaolaThangata, FionaTirosh, DinaTsamir, PessiaUtairat, SuwattanaVan den Heuvel

Panhuizen, Marja

Vincent, JillWarfield, JanetWatanabe, TadWatson, AnneWilson, MelvinZaslavsky, OritZawojewski, J.Zevenbergen, Robin

Journal of Mathematics Teacher Education

INSTRUCTIONS FOR AUTHORS

EDITOR-IN-CHIEF

Professor Barbara Jaworski Editors: Konrad KrainerHøgskolen i Agder University of Klagenfurt, Austria(Agder University Collete) Peter SullivanInstitutt for Matematiske Fag La Trobe University, AustraliaServiceboks 422 Terry Wood4604 Kristiansand Purdue University, U.S.A.Norway

AIMS AND SCOPEThe Journal of Mathematics Teacher Education (JMTE) is devoted toresearch into the education of mathematics teachers and developmentof teaching that promotes students’ successful learning of mathematics.JMTE focuses on all stages of professional development of mathematicsteachers and teacher-educators and serves as a forum for considering insti-tutional, societal and cultural influences that impact on teachers’ learning,and ultimately that of their students. Critical analyses of particularprogrammes, development initiatives, technology, assessment, teachingdiverse populations and policy matters, as these topics relate to the mainfocuses of the journal, are welcome. All papers are rigorously peerrefereed.

Papers may be submitted to one of three sections of JMTE as follows:

1. Research papers: these papers should reflect the main focuses of thejournal identified above and should be of more than local or nationalinterest.


398 INSTRUCTIONS FOR AUTHORS

2. Mathematics Teaching Education Around the World: these papersfocus on programmes and issues of national significance that couldbe of wider interest or influence.

3. Reader Commentary: these are short contributions; for example,offering a response to a paper published in JMTE or developing atheoretical idea.

Authors should state clearly the section to which they are submittinga paper. As general guidance, papers should not normally exceed thefollowing word lengths: (1) 10,000 words; (2) 5,000 words; (3) 3,000words.

Critiques of reports or books that relate to the main focuses of JMTEappear as appropriate.

Manuscript Submission

Online Manuscript Submission

Journal of Mathematics Teacher Education has a fully web-enabled manu-script submission and review system. This system offers authors the optionof tracking in real time the review process of their manuscripts. The onlinemanuscript and review system offers easy and straightforward log-in andsubmission procedures. It supports a wide range of submission file formats,influding Word, WordPerfect, RTF, TXT and LaTeX for article text andTIFF, EPS, PS, GIF, JPEG and PPT for figures. PDF is not a recommendedformat.

Manuscripts should be submitted to:

http//JMTE.edmgr.com

Authors are requested to download the Consent to Publish and Transferfor Copyright form from this system. Please send a completed and signedform either by mail or fax to the Journal of Mathematics Teacher EducationOffice.

NOTE: By using the online manuscript submission and review system, itis NOT necessary to submit the manuscript also as printout + disk. In caseyou encounter any difficulties while submitting your manuscript online,please get in touch with the responsible Editorial Assistant by clicking on“CONTACT US” from the toolbar.

INSTRUCTIONS FOR AUTHORS 399

Electronic figures

Electronic versions of your figures must be supplied. For vector graphics,EPS is the preferred format. For bitmapped graphics, TIFF is the preferredformat. The following resolutions are optimal: lime figures – 600–1200dpi; photographs – 300 dpi; screen dumps – leave as is. Colour figurescan be submitted in the RGB colour system. Font-related problems canbe avoided by using standard fonts such as Times Roman, Courier andHelvetica.

Figures

All photographs, graphs and diagrams should be referred to as a ‘Figure’and they should be numbered consecutively (1, 2, etc.). Multi-part figuresought to be labelled with lower case letters (a, b, etc.). Please insert keysand scale bars directly in the figures. Relatively small text and great vari-ation in text sizes within figures should be avoided as figures are oftenreduced in size. Figures may be sized to fit approximately within thecolumn(s) of the journal. Provide a detailed legend (without abbreviations)to each figure, refer to the figure in the text and note its approximate loca-tion in the margin. Please place the legends in the manuscript after thereferences.

Colour figures

Colour figures may be printed at the author’s expense. Please indicateat submission which figures should be printed in colour, the number ofcolour pages you prefer and to which address we can send the invoice. Inaddition, please specify if figures are to appear together on a colour page.Our standard price are: for one page C= 795, for two pages C= 1250, for threepages C= 1480 and for each subsequent page an additional C= 230.

Tables

Each table should be numbered consecutively (1, 2, etc.). In tables, foot-notes are preferable to long explanatory material in either the heading orbody of the table. Such explanatory footnotes, identified by superscriptletters, should be placed immediately below the table. Please provide acaption (without abbreviations) to each table, refer to the table in the textand note its approximate location in the margin. Finally, please place thetables after the figure legends in the manuscript.


Language

We appreciate any efforts that you make to ensure that the language iscorrected before submission. This will greatly improve the legibility ofyour paper if English is not your first language.

Reviewing ProcedureJOURNAL’S NAME follows a double-blind reviewing procedure. Authorsare therefore requested to omit their name and affiliation from the manu-script. Self-identifying citations and references in the article text shouldalso be avoided or left blank when manuscripts are first submitted. Authorsare responsible for reinserting their name, affiliation, self-identifying cita-tions and references when manuscripts are prepared for final submission.

AbstractPlease provide a short abstract of 100 to 250 words. The abstract shouldnot contain any undefined abbreviations or unspecified references.

Key WordsPlease provide 5 to 10 key words or short phrases in alphabetical order.

AbbreviationsIf abbreviations are used in the text, please provide a list of abbreviationsand their explanations.

Section HeadingsFirst-, second-, third-, and fourth-order headings should be clearly distin-guishable. Headings should follow APA guidelines.

AppendicesSupplementary material should be collected in an Appendix and placedbefore the Notes and Reference sections.

NotesPlease use endnotes rather than footnotes. Notes should be indicated byconsecutive superscript numbers in the text and listed at the end of thearticle before the References. A source reference note should be indicatedby means of an asterisk after the title. This note should be placed at thebottom of the first page.

Cross-ReferencingPlease make optimal use of the cross-referencing features of your softwarepackage. Do not cross-reference page numbers. Cross-references should


refer to, for example, section numbers, equation numbers, figure and tablenumbers.

In the text, a reference identified by means of an author’s name should befollowed by the date of the reference in parentheses and page number(s)where appropriate. When there are more than two authors, only the firstauthor’s name should be mentioned, followed by ‘et al.’. In the event thatan author cited has had two or more works published during the same year,the reference, both in the text and in the reference list, should be identifiedby a lower case letter like ‘a’ and ‘b’ after the date to distinguish the works.

Examples:Winograd (1986, p. 204)(Winograd, 1986a; Winograd, 1986b)(Flores et al., 1988; Winograd, 1986)(Bullen & Bennett, 1990)

AcknowledgementsAcknowledgements of people, grants, funds, etc. should be placed in aseparate section before the References.

ReferencesReferences to books, journal articles, articles in collections and confer-ence or workshop proceedings, and technical reports should be listed atthe end of the article in alphabetical order following the APA style (seeexamples below). Articles in preparation or articles submitted for publica-tion, unpublished observations, personal communications, etc. should notbe included in the reference list but should only be mentioned in the articletext (e.g., T. Moore, personal communication).

References to books should include the author’s name; year of publication;title; page numbers where appropriate; publisher; place of publication, inthe order given in the example below.

Mason, R. (1995). Using communications in open and flexible learning.London: Kogan Page.

References to articles in an edited collection should include the author’sname; year of publication; article title; editor’s name; title of collection;first and last page numbers; publisher; place of publication, in the ordergiven in the example below.


McIntyre, D.J., Byrd, D.M. & Foxx, S.M. (1996). Field and laboratoryexperiences. In J. Sikula, T.J. Buttery & E. Guyton (Eds.), Handbook ofresearch on teacher education (3rd. Ed.; pp. 171–193) New York: Simon& Schuster.

References to articles in conference proceedings should include theauthor’s name; year of publication; article title; editor’s name (if any);title of proceedings; first and last page numbers; place and date ofconference; publisher and/or organization from which the proceedings canbe obtained; place of publication, in the order given in the examples below.

Yan, W., Anderson, A. & Nelson, J. (1994). Facilitating reflective thinkingin student teachers through electronic mail. In J. Willis (Ed.), Technologyand Teacher Education Annual, 1994: Proceedings of the Fifth AnnualConference of the Society of technology and teacher Education. VA:Association for the advancement of Computing in Education.

Blanton, M.L., Westbrook, S.L. & Carter, G. (2001). Using Valsiner’sZone Theory to Interpret a Pre-service Mathematics Teacher’s Zoneof Proximal Development. In M. Van den Heuvel-Panhuizen (Ed.),Proceedings of the 25th Conference of the International Group for thePsychology of Mathematics Education. The Netherlands: FreudenthalInstitute, Utrecht University.

References to articles in periodicals should include the author’s name;year of publication; article title; full or abbreviated title of periodical;volume number (issue number where appropriate); first and last pagenumbers, in the order given in the example below.

Thomas, L., Clift, R.T. & Sugimoto, T. (1996). Telecommunication,student teaching and methods instruction: An exploratory investigation.Journal of Teacher Education, 46, 165–174.

References to technical reports or doctoral dissertations should includethe author’s name; year of publication; title of report or dissertation;institution; location of institution, in the order given in the example below.

Hughes, M. (1975). Egocentrism in pre-school children, EdinburghUniversity, Edinburgh, unpublished PhD thesis.


PROOFSProofs will be sent to the corresponding author by e-mail (if no e-mailaddress is available or appears to be out of order, proofs will be sent byregular mail).

Your response, with or without corrections, should be sent within 72 hours.Please do not make any changes to the PDF file. Minor corrections (+/–10) should be sent as an e-mail attachment to: [email protected]/Always quote the four-letter journal code and article number and the PIPSNo. from your proof in the subject field of your e-mail. Extensive correc-tions must be clearly marked on a printout of the PDF file and should besent by first-class mail (airmail overseas).

OFFPRINTSTwenty-five offprints of each article will be provided free of charge.Additional offprints can be ordered by means of an offprint order formsupplied with the proofs.

PAGE CHARGES AND COLOUR FIGURESNo page charges are levied on authors or their institutions. Colour figuresare published at the author’s expense only.

COPYRIGHTAuthors will be asked, upon acceptance of an article, to transfer copyrightof the article to the Publisher. This will ensure the widest possibledissemination of information under copyright laws.

PERMISSIONSIt is the responsibility of the author to obtain written permission for aquotation from unpublished material, or for all quotations in excess of 250words in one extract or 500 words in total from any work still in copyright,and for the reprinting of figures, tables or poems from unpublished orcopyrighted material.

ADDITIONAL INFORMATIONAdditional information can be obtained from:

Journal of Mathematics Teacher Education, Kluwer AcademicPublishers, Attn Publishing Editor. Marie Sheldon, P.O. Box 17, 3300AA Dordrecht, The Netherlands, tel.: +31-78-6576183; fax: +31-78-6576254. E-mail: [email protected]

CONTENTS OF VOLUME 6

Volume 6 No. 1 2003

EDITORIAL / JMTE and Professional Development 1–3

VICTORIA SÁNCHEZ and SALVADOR LLINARES /Four Student Teachers’ Pedagogical Reasoning onFunctions 5–25

WIM VAN DOOREN, LIEVEN VERSCHAFFEL andPATRICK ONGHENA / Pre-service Teachers’Preferred Strategies for Solving Arithmetic andAlgebra Word Problems 27–52

PAOLA SZTAJN / Adapting Reform Ideas in Different Mathe-matics Classrooms: Beliefs Beyond Mathematics 53–75

Book Review

Mason, John (2002). Researching your own practice: Thediscipline of noticing (ALAN H. SCHOENFELD) 77–91

Volume 6 No. 2 2003

EDITORIAL / KONRAD KRAINER / Teams, Communities &Networks 93–105

ANDREA LACHANCE and JERE CONFREY / Intercon-necting Content and Community: A QualitativeStudy of Secondary Mathematics Teachers 107–137

FRAN ARBAUGH / Study Groups as a Form of ProfessionalDevelopment for Secondary Mathematics Teachers 139–163

Teacher Education around the World

RENUKA VITHAL / Student teachers and ‘street children’: Onbecoming a teacher of mathematics 165–183


406 CONTENTS OF VOLUME 6

Book Review

Wenger, E. (1998). Communities of practice: Learning,meaning and identity (MELLONY GRAVEN andSTEPHEN LERMAN) 185–194

Volume 6 No. 3 2003

EDITORIAL / TERRY WOOD and BETSY BERRY /What Does “Design Research” Offer MathematicsTeacher Education? 195–199

JAMES HIEBERT, ANNE K. MORRIS and BRAD GLASS /Learning to Learn to Teach: An “Experiment” Modelfor Teaching and Teacher Preparation in Mathematics 201–222

JEREMY A. KAHAN, DUANE A. COOPER and KIMBERLYA. BETHEA / The Role of Mathematics Teachers’Content Knowledge in their Teaching: A Frameworkfor Research applied to a Study of Student Teachers 223–252

JENNIFER E. SZYDLIK, STEPHEN D. SZYDLIK andSTEVEN R. BENSON / Exploring Changes in Pre-service Elementary Teachers’ Mathematical Beliefs 253–279

Reader Commentary

JOHN MASON / Seeing Worthwhile Things: Response toAlan Schoenfeld’s Review of Researching Your OwnPractice in JMTE 6.1 281–292

Volume 6 No. 4 2003

PETER SULLIVAN / Editorial: Incorporating KnowledgeOf, and Beliefs About, Mathematics into TeacherEducation 293–296

ROZA LEIKIN / Problem-Solving Preferences of MathematicsTeachers: Focusing on Symmetry 297–329

JEFF D. FARMER, HELEN GERRETSON and MARSHALLLASSAK / What Teachers Take from ProfessionalDevelopment: Cases and Implications 331–360

CONTENTS OF VOLUME 6 407

MARIA GOULDING, GILLIAN HATCH and MELISSARODD / Undergraduate Mathematics Experience:Its Significance in Secondary Mathematics TeacherPreparation 361–393

Acknowledgement 395–396

Instructions for Authors 397–403

Contents of Volume 6 405–407

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