journal of mathematics teacher education_7

BARBARA JAWORSKI

EDITORIAL

Contributing to Knowledge↗Research: ↘Enhancing Practice

It seems clear that research in mathematics teacher education has theresponsibility to contribute to knowledge in the field and also to theenhancement of practice. Of course, what we perceive as “knowledge”might be a source of debate. What knowledge do we value as teachers andeducators? In what ways do we use knowledge in our practice, or promoteknowledge through our practice?

Research that is reported in JMTE often springs from practice andoffers perspectives on practice. Reports of research offer analyses of datacollected from practical settings. Theoretical perspectives and contextualfactors influence analyses and are fundamental to findings. We might seethe reported research contributing to a body of knowledge in the field ofmathematics teacher education. How can we characterise such knowledgeand to what extent is it theoretically or contextually situated? What do welearn from papers in JMTE?

The three papers in this issue are different in a number of ways:reflecting on these differences raises interesting questions about howresearch contributes to knowledge and practice in mathematics teachereducation. Two papers discuss the learning of pre-service teachers withrespect to teacher education programmes; one paper looks at the learningof teacher educators through their in-service activity with practisingteachers. Here we can ask questions about teacher learning and the asso-ciated development of teaching; for example, in what ways processesand practices differ according to the knowledge and experience of theteachers involved. In what respects, perhaps, pre-service teacher educationcan gain insights from issues in the learning of experienced teachers, orotherwise. Research approaches and methods also differ. One paper offersdetailed case studies of just two teachers. One uses a questionnaire surveyto access teachers’ viewpoints. A third uses a complex combination ofresearch methods and theoretical models to track and characterise teachers’

Journal of Mathematics Teacher Education 7: 1–4, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

2 BARBARA JAWORSKI

learning. To what extent is what we learn from these papers, about teacherlearning and teaching development, related to the research methods thatare used and the theoretical perspectives that underpin these methods?

Orit Zaslavsky and Roza Leikin discuss research which has exploredthe professional development of mathematics teacher educators throughtheir engagement in practice. The practice referred to here is thatof working with junior and senior high school mathematics teachersin an in-service professional development programme (14 mathematicsteacher educators working, over 5 years, with about 120 teachersgrouped according to the age of pupils taught and year of beginning theprogramme). The researchers developed their own conceptual frameworkbased on models in the published literature and applied their frameworkto emerging patterns and issues from a grounded theory analysis of datafrom video recordings of workshops, interviews with teacher educators andsummaries of staff meetings. Analyses resulted in the generation of storiesrelating to the learning, or growth, of teacher educators within the project.These stories were theorised in relation to the conceptual framework ofthe project. Thus, while the stories are highly personalised, their analysisvia the framework allows perceptions of generality that can be extended toother practices.

From this research we gain insights into the professional growth ofteacher educators through their practice of working with teachers in in-service courses over months or years. We see the development and useof theoretical models which contribute to a characterisation of teachereducators’ learning. We see further examples of the use of methods ofdata collection and analysis, some common (e.g., interviews) some lesscommon (e.g., stories, as a medium for capturing complexities of practice).

Amy Roth McDuffie discusses a study of the reflective thinking ofelementary pre-service teachers during their student teaching of mathe-matics. The study draws on theoretical notions such as Schön’s conceptsof reflection on and in practice and Shulman’s concept of PedagogicalContent Knowledge, using these as a basis for identifying and discussingdilemmas in student teachers’ practice. The study provides both theoreticaland practical insights: the former through relating layers of reflection togrowth of mathematical and pedagogical knowledge, and the latter throughthe nature of particular dilemmas that arise and their contribution to thelearning of particular student teachers. This paper addresses the learningof two teachers through in-depth analysis of one lesson for each teacher,which raises questions about representativity and generalisability. It isimportant to recognise nature and source of issues and dilemmas arisingfrom student teachers’ practice, rather than particular teachers or lessons or

EDITORIAL 3

issues or dilemmas. Growth of knowledge can be observed through waysin which dilemmas provoke reflection.

Here we see in depth into student teachers’ practices and the, sometimespainful, recognition that limited knowledge constrains practice. Yet, thepotential and potency of such recognition shows power in the reflectiveprocess. Although the activity of the teacher educator (the author of thispaper) is not analysed per se, one conjecture is that her (research) interven-tions have provided a frame for the student teachers’ conceptions relatingto their practice.

Margaret Walshaw’s study also concerns the learning of studentteachers with its particular focus on identity, or subjectivity, within a post-structuralist theoretical frame based on the work of Michel Foucault. Herpaper reports an exploration of the formation of identity with particularattention to one cohort of students in their second year of a teacher educa-tion programme. Data were collected on one particular day when studentspresent (n = 72) responded to a questionnaire about their recent teachingpractice experience. Students’ written responses from the questionnaireswere analysed according to Foucauldian themes to capture aspects ofknowledge contributing to teachers’ subjectivities regarding their teaching.While at one level the paper is about what it means for the student teachersto engage in pedagogical work in primary schools, and the particularinsights the Foucauldian analysis affords, at yet another level we perceivethat alternative epistemological frameworks can afford or constrain incertain ways, challenging us to question ways in which knowledge isformulated.

Our insights into the thinking of student teachers in this study is limitedto the kinds of responses a questionnaire affords. The poststructuralistanalysis directs our attention to certain characteristics of this data relatedto the subjectivities of the teachers whose words are analysed. We mightsee the importance of such research less in the particularities of students’thinking at this stage of their course, but rather in opportunities to perceivesuch thinking through a particular frame of reference.

I suggested in an earlier editorial (JMTE 6.1) that taking any threepapers it is always possible to find some theme in their content. I thusexpected that I should discern such a theme here on which to reflect.However, the insights I gained from my (re)reading of these papers wentbeyond the identification of a theme, to recognition of alternative percep-tions of knowledge, and their relatedness to theoretical formulations andpositions in both substance and methodology. I am left with an aware-ness of profound differences in the ways these papers formulate theirperspectives on knowledge and practice, of the significance of alterna-

4 BARBARA JAWORSKI

tive perspectives for practices in teacher education and of the potentialafforded to individuals and communities of teachers and educators to(re)conceptualise practice.

We would welcome papers in which authors reflect on their “reading”of JMTE papers either in relation to a piece of research which the paperreports, or in a contribution to our sections on Mathematics TeacherEducation around the World, or in a short Reader Commentary paper.

ORIT ZASLAVSKY and ROZA LEIKIN

PROFESSIONAL DEVELOPMENT OF MATHEMATICS TEACHEREDUCATORS: GROWTH THROUGH PRACTICE1

ABSTRACT. In this paper we present a study conducted within the framework of an in-service professional development program for junior and senior high school mathematicsteachers. The focus of the study is the analysis of processes encountered by the staffmembers, as members of a community of practice, which contributed to their growth asteacher educators. We offer a three-layer model of growth through practice as a conceptualframework to think about becoming a mathematics teacher educator, and illustrate how oursuggested model can be adapted to the complexities and commonalities of the underlyingprocesses of professional development of mathematics teacher educators.

KEY WORDS: community of practice, growth-through-practice, mathematics teachereducators, professional development, sorting tasks

ABBREVIATIONS: MT – Mathematics Teacher; MTE – Mathematics Teacher Educator;MTEE – Mathematics Teacher Educators’ Educator

INTRODUCTION

In the past decade there have been several calls for reform in mathematicseducation that are based on the assumption that well prepared mathe-matics teacher educators are available to furnish opportunities for teachersto develop in ways that will enable them to enhance the recommendedchanges. However, there are relatively few formal programs that provideadequate training for potential mathematics teacher educators, let aloneresearch on becoming a mathematics teacher educator. Our study examinesthe process of becoming a mathematics teacher educator within a profes-sional development program for mathematics teachers. More specifically,we analyze the growth of mathematics teacher educators through theirpractice.

In our work, the community of mathematics educators is viewed asa community of practice. Theories of communities of practices provideus with tools for analyzing the professional growth and various kinds oflearning of teacher educators (Rogoff, 1990; Roth, 1998; Lave & Wenger,1991). These theories consider teachers’ knowledge as developing sociallywithin communities of practice. We take these theories a step further tolook at teacher educators’ professional knowledge.


6 ORIT ZASLAVSKY AND ROZA LEIKIN

An integral characteristic of the community of practice of mathematicseducators is associated with the notion of reflective practice (Dewey, 1933;Schön, 1983; Jaworski, 1994, 1998; Krainer, 2001). Following Dewey’semphasis on the reflective activity of both the teacher and the student,as a means for advancing their thinking, there has been transition froma theoretical perspective of the constructs of reflection and action to amore practical position. Consequently, the notions of reflection on-actionand reflection in-action emerged and have been recognized as an effectivecomponent contributing to the growth of teachers’ knowledge about theirpractice. In our study, reflection is a key issue for the development ofteacher educators.

The work described in this paper was conducted within the frameworkof a five-year reform-oriented in-service professional development project(“Tomorrow 98” in the Upper Galilee2) for junior and senior high schoolmathematics teachers. The reform of mathematics teaching requires thatteachers play an active role in their own professional development (NCTM,2000). In order to help teachers see new possibilities for their own practicethey must be offered opportunities to (a) learn challenging mathematicsin ways that they are expected to teach and (b) engage in alternativemodels of teaching (e.g., Brown & Borko, 1992, Cooney & Krainer, 1996).Accordingly, the main task of the project staff members was to offer suchopportunities for the participating teachers.

Our paper focuses on the professional development of the project’sstaff members as teacher educators within the framework of the project.We analyze some processes in which the project members engaged asthey became more proficient, and the conditions that contributed to theirtraining and professional growth within the community of mathematicseducators.

CONCEPTUAL FRAMEWORK

As mentioned earlier, the design of both the program of the project andthe research that accompanied it was driven by constructivist views oflearning and teaching. According to this perspective, learning is regardedas an ongoing process of an individual or a group trying to make senseand to construct meaning based on their personal experiences and interac-tions with the environment in which they are engaged. It follows that thecommunity of mathematics educators (i.e., teachers and teacher educators)can be seen as learners who reflect continuously on their work and makesense of their histories, their practices, and other experiences.

TEACHER EDUCATORS’ GROWTH-THROUGH-PRACTICE 7

In order better to understand the learning processes involved in thiscommunity, we adapt and expand two explanatory models of schoolmathematics teaching – Jaworski’s (1992, 1994) teaching triad and Stein-bring’s (1998) model of teaching and learning mathematics as autonomoussystems. We combine these two models into a three-layer model thatoffers a lens through which to examine the interplay between the learningprocesses of the different members of the community. More specifically,our model strives at reflecting how “Knowledgeability comes from partic-ipating in a community’s ongoing practices. Through this participation,newcomers come to share community’s conventions, behaviors, view-points, and so forth; and sharing comes through participation” (Roth, 1998,p. 12).

EXTENSION OF THE TEACHING TRIAD

Jaworski (1992, 1994) offers a teaching triad, which is consistent withconstructivist perspectives of learning and teaching. The triad synthes-izes three elements that are involved in the creation of opportunities forstudents to learn mathematics: the management of learning, sensitivity tostudents, and the mathematical challenge. Although quite distinct, theseelements are often inseparable. According to Jaworski “this triad forms apowerful tool for making sense of the practice of teaching mathematics”(1992, p. 8). We borrow the idea of this teaching triad for describing andanalyzing the practice of the leading members of our community, namely,the mathematics teacher educators. Analogous to the way mathematicsserves as a challenging content for students, the teaching triad serves asa challenging content for mathematics teacher educators. Accordingly, weconsider the teaching triad of a mathematics teacher educator to consist ofthe challenging content for mathematics teachers (i.e., Jaworski’s teachingtriad), sensitivity to mathematics teachers and management of mathematicsteachers’ learning (see Figure 1).

In the current study we distinguish between three different groups ofmathematics educators: The mathematics teachers (MTs) who participatedin the program, the project staff members who served as mathematicsteacher educators (MTEs), and the project director/leading researcher (thefirst author of this paper), who can be seen as a teacher educators’ educator(MTEE). Zaslavsky and Leikin (1999) describe in their table (p. 146, ibid.)some similarities and differences between the teaching triad relevant toeach group of the community. It should be noted that within the frameworkof the project, each group of mathematics educators took different rolesalternately, depending on the occasion, either as facilitators of learning or


Figure 1. The teaching triad of mathematics teacher educators.

as learners. These dynamic movements between the roles of a facilitatorand of a learner fostered many learning-through-teaching processes of theparticipants, as discussed by Zaslavsky and Leikin (ibid).

A MODEL OF GROWTH THROUGH PRACTICE OFMATHEMATICS TEACHER EDUCATORS

A further description of the learning-through-teaching process may beseen in Steinbring’s (1998) model of teaching and learning mathematics.According to this model, the teacher offers a learning environment for hisor her students in which the students operate and construct knowledge ofschool mathematics in a rather autonomous way. This occurs by subjectiveinterpretations of the tasks in which they engage and by ongoing reflectionon their work. The teacher, by observing the students’ work and reflectingon their learning processes constructs an understanding, which enables himor her to vary the learning environment in ways that are more appropriatefor the students. Although both the students’ learning processes and theinteractive teaching process are autonomous, these two systems are inter-dependent. This interdependence can explain how teachers learn throughtheir teaching.

Stemming from our reflections on our work within the project, weadapt Steinbring’s model and use Jaworski’s terminology to help us thinkabout and offer explanations to some ways in which mathematics teachers


(MTs), teacher educators (MTEs) and mathematics teacher educators’educators (MTEEs) may learn from their practice. Parallel to the extensionof Jaworski’s teaching triad, suggested earlier (Figure 1), we present inFigure 2 a model of teaching and learning as autonomous systems for thedifferent groups of learning facilitators, which is an extension of Stein-bring’s model (1998). Our three-layer model consists of three interrelatedfacilitator-learner configurations (dotted, light shaded, and dark shaded),each of which includes two autonomous systems; one system describesthe main actions in which the facilitator of learning engages (depicted bycircular-like arrows), while the other system describes the main actions inwhich the learner engages (depicted by rectangular-like arrows).

According to the model depicted in Figure 2, any member of thecommunity may be part of two different configurations at different pointsof time. For example, a mathematics teacher (MT) may switch roles froma facilitator of students’ learning to a member of a group of learners whoselearning is facilitated by a mathematics teacher educator (MTE). This isexpressed by the two different colors in the rectangle representing MT’sknowledge as well as in the two kinds of arrows that come out of thisrectangle. The rectangular-like arrow indicates how MTs, as learners, workon their learning tasks (including solving problems that are mathema-tical, pedagogical or both), make sense of them and construct meaningin subjective ways. By reflecting on their actions and thoughts and bycommunicating them, MTs develop their knowledge of the teaching triadthrough learning. The other kind of arrow, the circular one surroundingthe inner facilitator-learner configuration of our model (i.e., the dottedone), indicates the process underlying MTs development of their knowl-edge through teaching. This inner configuration, which also relates tostudents’ learning, is borrowed from Steinbring (1998). Similarly, MTEsdevelop their knowledge of the MTE Teaching Triad (Figure 1) in twoways: through learning, as facilitated by a MTEE, or through teaching,when they facilitate MTs learning.

The horizontal bi-directional arrows that come out of the three rect-angles representing the knowledge of the different teaching triads indicatetwo additional kinds of exchange: one exchange has to do with sharing,consulting and exchange of ideas through direct interactions between thedifferent facilitators and learners. The second kind of exchange has to dowith a switch from the role of a certain type of facilitator (e.g., MT) to arole of another facilitator (e.g., MTE). The latter exchange may occur asspecial experiences of a number of MTs on their way to becoming a MTE.In the same way similar switches and exchanges occur between MTEs andMTEEs.


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Krainer (1999) points to the unique complexity of the project that is thefocus of this paper, particularly in relation to the challenge of interweavingthe professional growth of mathematics teachers (MTs) with the growth ofmathematics teacher educators (MTEs). The purpose of our paper is tocharacterize the nature of this complex symbiotic growth and to illustratehow our model applies to practice. This is done using a story through whichwe gain insight into the underlying processes of the MTEs professionalgrowth. We begin by describing in more detail the overall setting of theproject and some characteristics of the team members, followed by thestory.

THE GOALS OF THE STUDY

Our three-layer model presented above is actually one outcome of a largerstudy, in which we attempted to understand and make sense of the waysin which the members of the community developed through their prac-tice. Our goal in this paper was twofold: first, to analyze and understandbetter the growth through practice of teacher educators as members of thecommunity of mathematics educators. Second, we aimed at testing thedescriptive and explanatory power of our theoretical model by applyingit to this specific context.

The Overall SettingThe goals and design of the project were very much in line with whatCooney and Krainer (1996) consider as essential components for teachereducation programs. More specifically, the project goals included thefollowing:

• Facilitating teachers’ knowledge (both mathematical and pedagogi-cal) in ways that support a constructivist perspective to teaching, by:

– Offering teachers opportunities to experience alternative ways oflearning (challenging) mathematics;

– Preparing teachers for innovative and reform oriented approachesto management of learning mathematics (particularly, the kindswith which they have had very limited experience);

– Fostering teachers’ sensitivity to students and their ability to assessstudents’ mathematical understanding.

• Promoting teachers’ ability to reflect on their learning and teachingexperiences as well as on their personal and social development.

• Enhancing teachers’ and teacher educators’ socialization and devel-oping a supportive professional community to which they belong.


ParticipantsRoth (1998) points to the connection between the length of time for whicha community is designed and its goals. Accordingly, our community wasdesigned for a period of five years, with newcomers gradually joiningduring the first three years.

In total, about 120 teachers participated in the program. The teacherswere grouped according to the grade levels they taught (junior/senior highgrades) and the year in which they enrolled in the program. Altogether,there were six groups of teachers each consisting of about 20 teachers.During the first three years of the project, two new groups of teachersjoined the program each year – one junior high and the other senior highschool level.

The teachers who participated in the full program took part for fourconsecutive years in weekly professional development meetings for sixhours per week, throughout each school year. The meetings consisted ofa wide range of activities led mainly by the project team. Some of theteachers gradually became more involved in the program and towardstheir third year assumed responsibility for many of these activities. As theprogram progressed, the location of the activities shifted from a centralregional location into the schools in the region.

The project was designed to enhance the development of the projectmembers hand in hand with the development of the in-service teacherswho participated in the program. The design of both the staff enhance-ment component of the project and the research that focused on the staffmembers’ professional growth stemmed from the project’s goals, and wasbased on two main assumptions: (a) Similar to the ways in which teacherslearn through their own (teaching) practice (Brown & Borko, 1992; Mason,1998; Steinbring, 1998; Leikin, Berman & Zaslavsky, 2000), teachereducators learn through their practice; (b) There are learning aspects thatare fundamentally inherent to the structure and nature of the community ofpractice, which the project team constitutes (Lave, 1996; Roth, 1998).

The project team consisted mainly of experienced and highly reput-able secondary mathematics teachers. Although there were altogether over20 team members, only 14 were involved in the project from its earlystages until its completion. The team members varied with respect totheir expertise and experience, one of the characteristics that Roth (1998)considers essential to a community. None of them had any formal training(such as the Manor Program reported by Even, 1999). Some did not haveany previous experience in mentoring or teaching other teachers. In thefirst three years of the project, the tasks of the staff members were mostlydirected toward designing and carrying out in-service workshops. Some


were in charge of the in-service activities with the teachers, and othersfacilitated the activities by participating in the weekly in-service meetingsor by assisting in the preparation of resources that were required for thein-service meetings.

It should be noted that, at the beginning of the project, there weremany staff members who were not very confident of their qualifications asteacher educators, and who expressed a need for guidance by the projectdirector or other more experienced members. Thus, the project directorassumed the role of teacher educators’ educator (MTEE) in various ways.The staff members expected to gain expertise as teacher educators withinthe framework of the project, in order to become more competent in theirwork. It was only towards the end of the project that most of the staffmembers considered themselves proficient teacher educators.

METHODOLOGY

The methodology employed in the study followed a qualitative researchparadigm in which the researcher is part of the community underinvestigation. It borrows from Glaser and Strauss’s (1967) GroundedTheory, according to which the researcher’s perspective crystallizes as theevidence, documents, and pieces of information accumulate in an inductiveprocess from which a theory emerges. The methods, data collection andanalysis grew continuously throughout the progressing study as integralparts of the professional development project. The researcher acts as areflective practitioner (Schön, 1983) whose ongoing reflectiveness andinterpretativeness are essential components (Erickson, 1986). In our case,the researchers were members of the community of practice which theyinvestigated.

Data Collection

Within the grounded theory paradigm, data collection served two inter-related roles. First, it served to investigate the processes involved in thedevelopment and growth of mathematics teachers engaged in the project.Second, it served to influence iteratively the design of the project, as themain task of the staff members was to plan and carry out weekly workshopsand related activities with the in-service teachers, with no readily availablecurriculum. Multiple data were collected during the 5 years of the project.

Staff members were asked to provide written and oral reactions totheir colleagues. In order to create situations in which less experiencedmembers would learn from more experienced ones in an apprenticeship


like manner (Rogoff, 1990) staff members were required, as part of theirworking load, to observe their colleagues’ workshops (including takingpart in the workshops’ activities when they felt comfortable to do so) andto provide written and oral reactions to their colleagues. These writtenreactions served three main purposes: to provide feedback to the learningfacilitator, to enhance the capacity of the team members to reflect and totrace the development of reflection skills among the team members.

Many of the workshops were videotaped, first in order to extend thepossibilities of team members to observe each other at their conveniencealong the previous line and second, to allow us to analyze the profes-sional development of team members, the changes in their teaching styles,their sensitivity to the teachers, the nature of the challenging mathematicaltasks, the structure of the workshops and the management of learning thatthey facilitated.

In order to foster reflection and self-analysis of the team membersthey were also required to give written accounts of the workshops forwhich they were in charge (Borasi, 1999, reports the significance ofwriting for enhancing reflection). From the research perspective, thesewritten accounts combined with the videotapes helped us follow changesin teachers’ reflective abilities. Semi-structured interviews were conductedwith the team members (1–2 with each one) fostering reflection ontheir personal professional growth and the ways in which they relate itto the project’s goals and to the various activities in which they wereinvolved. Their reasoning about their involvement in the project providedinformation about how they perceived their own development within theframework of the project, and helped us identify significant sites in theprocess of MTE professional development.

Detailed written summaries of staff meetings were available for allstaff members. Staff meetings were conducted on a regular and frequentbasis. In these meetings, staff members could reflect on their work, sharetheir experiences, consult with their colleagues, and negotiate meaningwith respect to the goals and actions of the project. The summaries of themeetings helped us realize commonalities as well as differences in MTEs’positions regarding various issues that were meaningful to the overallprogress of the project. They also led us to understand how these positionswere modified as a result of interactions between the staff members.

The underlying assumption of the project was that ownership andresponsibility, which are indicators of professionalism (Noddings, 1992),would contribute to MTEs’ positions in their community of practice. Thus,staff members were continuously encouraged to initiate ideas and suggestnew directions and actions within the project. This aspect was also mani-


fested in a series of resource material for mathematics teachers and teachereducators that evolved as a result of identifying and reflecting on mutualinterests and successful experiences of the team members. From a researchperspective, we were able, using these materials, to identify changes inMTEs’ conceptions of the goals and activities of the project, in general, andwhat constitutes challenging and worthwhile tasks for MTs in particular.

In addition, the project director kept a personal journal includingdetailed notes of all events and interactions with and among the teammembers in which she was present. Through these notes, together withother sources, many stories emerged.

To summarize this section, it seems worthwhile to note that the abovecomponents that were designed to contribute to MTEs’ professionalgrowth address all four dimensions, which Krainer (1998, 1999) considersas describing MTs’ professional practice: Action, Reflection, Autonomy,and Networking. In our work, these dimensions refer to MTEs’ profes-sional development. From a research perspective, the multiple sourcesallowed triangulation and supported the validity of our findings.

Data Analysis

As described earlier, the project was documented in numerous ways. Thedata for this paper were analyzed by the two authors through an inductiveand iterative process. Our analysis focused mainly on the MTEs, i.e.,the team members, through: their performance (by analyzing videotapedworkshops), their utterances that contributed to our understanding of theirexperiences (by analyzing written self reports of MTEs, protocols of theirindividual interviews, protocols of the staff meetings) and specific eventsthat seemed meaningful or explanatory to us.

Data analysis focused on two main groups of MTEs who differed withrespect to their starting points in the project. One group included memberswho had some previous experience as MTEs. The other group consisted ofvery experienced and highly reputable secondary school MTs who did nothave any previous experience as MTEs.

The findings, drawn independently, were discussed in order to share theinterpretations and meanings that we elicited. In the course of our discus-sions new ideas came up and new questions were raised. As a result a setof “big ideas” emerged and were formulated – describing the professionaldevelopment encountered by the above two groups of MTEs in the project.Subsequently, the protocols and videotapes were analyzed again in orderto crystallize our conception and understanding of these processes.

Finally we chose to present these big ideas by using two staff members(Tami and Rachel), each representative of a different group of MTEs. Tami


had some previous experience as MTE while Rachel had no such previousexperience. In addition, their storylines intersected at a particular point inthe life of the project, which served as a fruitful context for professionaldevelopment of MTEs. This kind of intersection was also typical in thiscontext. The participatory nature of the research called for special cautionwith regard to our personal roles in the project as MTEE and MTE respec-tively. Our involvements in the episodes chosen for analysis strengthenedthe reflective elements of our analysis. However, we tried to be aware ofthe influence of our personal biographies as mathematics educators on ourtheoretical sensitivity to the research process.

Data Presentation

As mentioned earlier, most of the documentation served both as meansfor enhancing the professional development of the team members and forthe research purposes. Thus, it was used to support the emerging personaland collective stories that portrayed the nature of the processes of growththrough practice and participation encountered by the team members.Personal stories have been acknowledged as means of presenting mean-ingful processes of teaching and learning to teach (Schifter, 1996; Chazan,2000; Krainer, 2001; Lampert, 2001; Tzur, 2001). Krainer (2001) points tothree learning levels that may be connected by stories:

Firstly, stories provide us with authentic evidence and holistic pictures of exemplary devel-opments in the practice of teacher education. Secondly, through stories we can extendour theoretical knowledge about the complex processes of teacher education. Thirdly, andpossibly most important of all, stories are starting points for our own reflection and promoteinsights into ourselves and our challenges, hopefully with consequences for our actions andbeliefs in teacher education (p. 271, ibid.).

One such story that evolved over time had to do with the theme of “cooper-ative learning and learning to cooperate” (Zaslavsky & Leikin, 1999).In the next section we present another story that is related to the abovethree learning levels with respect to the growth through practice of MTEs.Indeed, this story, as well as other similar stories that emerged from ourstudy, served as starting points for our reflections and insights, which led usto the development of the theoretical model presented earlier in this paper.By communicating our story here, we try to provide authentic evidenceof an exemplary development in the growth through practice of teachereducators.

There are five characters in the story, representing the diversity of themembers of the community of mathematics educators in the project: Tami,Alex, Rachel, Hanna, and Keren. Tami and Alex represent experiencedMTEs, while Rachel and Hanna represent experienced MTs in transition


to becoming MTEs; Keren, represents the MTEE, who often assumed therole of a MTE. As mentioned earlier, we chose Tami and Rachel as themain two characters, through which we convey the interplay and trans-ition from MTE to MTEE (by Tami’s storyline) hand in hand with theinterplay and transition from MT to MTE (by Rachel’s storyline). Thus,our story focuses on the professional development of these two MTEs(Tami and Rachel), while the other characters in the story provide thenecessary background and serve to shed light on MTEs’ interactions withother members of the community and on their growth through practice. Itshould be noted that Tami’s experience encompassed components of MTs,MTEs, and MTEEs roles, while Rachel’s included just MTs and MTEsencounters. In addition, Tami was actively involved in the transitions ofRachel and Hanna from MTs to MTEs. Therefore, Tami’s storyline is thedominant part of the story.

THE STORY: HOW TO SORT IT? OPENING THE TASK AS ATRIGGER FOR OPENING NEW HORIZONS

The story described in this paper took place during the second year ofthe project. At that time Tami,3 who was a team member from the begin-ning of the project, worked on the preparation of a workshop for seniorhigh school teachers that focused on the role of the domain and range ofa function in solving conditional statements (i.e., equations and inequal-ities). Another staff member, Alex, who joined the team at the end ofthe first year, worked independently on a workshop with a similar mathe-matical context. Both of them had similar mathematical background and,based on their personal experiences and professional knowledge, theyboth considered this topic a problematic one for MTs. At a certain point,Keren (the project director who assumed the role as MTEE) realized thatthey were both interested in working on the same mathematical topic andsuggested that they collaborate and prepare a workshop together. Tamiand Alex had different teaching styles; Tami, who had special expertisein developing and implementing cooperative learning approaches in math-ematics, suggested managing the workshop in a cooperative learningsetting; contrary to Tami’s suggestion, Alex was inclined to organize theworkshop in a more teacher educator centered fashion, where he wouldlead the teachers towards the consideration of the use of the domain andrange of a function in solving equations and inequalities.

The first stage of collaboration included discussion of and an agreementon the specific mathematical tasks on which they would base the learningoffers for the teachers. Thus, they each composed a collection of mathe-


Figure 3. The group assignment – the task and the setting of the workshop.

matical objects – equations and inequalities. The combined set includeda total of 40 objects of which they negotiated the selection of a sub-set.Consequently, they ended up with the selection of 22 items (see the finalset in Figure 3). The final set of objects varied with respect to two maincategories: (1) the kind of family of functions that constituted the equations


Figure 4. A sorting sheet for the group assignment (for details see Figure 3).

and inequalities and (2) the kind of role the domain and range of functionsplayed in the solution processes.

At the second stage, Tami and Alex discussed the management of theteachers’ learning. Tami, who felt very strongly that the use of a cooper-ative learning setting would be significant, convinced Alex, who had littleprior experience in using cooperative learning methods, to go along withher. Thus, Tami and Alex began to design a structured cooperative learningworkshop according to the exchange of knowledge method that Tami hadpreviously developed (Leikin & Zaslavsky, 1999). For this purpose sheproposed to group the 22 equations and inequalities according to the


families of functions (category 1 above), while Alex insisted on groupingthem according to the role of the domain and range of the functions insolving the conditional statements (category 2 above). Tami argued that aninitial grouping of the 22 conditional statements according to the familiesof functions would allow the teachers to infer the role of the domain (orrange) when solving the equations and inequalities. On the other hand,Alex wanted to make sure that the different roles of the domain and rangewere made explicit. At this point, Tami and Alex decided to present thetwo alternative suggestions to Keren.

This meeting yielded another, more open, approach based on Keren’sexpertise and professional experience: instead of grouping the equationsand inequalities in advance, the decision was made to ask the teachers tosort the 22 equations and inequalities openly, in as many ways with whichthey could come up (e.g., Silver, 1979; Cooney, 1994). This change in themathematical task implied a change in the management of learning froma structured method to a more open setting in which the cooperation wasfostered by the nature of the challenging mathematical task rather than bystructured role assignments (see Figures 3 & 4). After some adaptation ofKeren’s suggestion to their personal styles, Tami and Alex felt able to carryout the revised plan.

They conducted two consecutive workshops, the first one with 19 seniorhigh school MTs who were in their second year of enrollment in theproject, and the second with 22 senior high school MTs who were in theirfirst year in the project. In both workshops a number of staff members werepresent (among them were Rachel and Hanna, who had just joined theMTE team). Rachel and Hanna participated in the first workshop and thenobserved some of the group activities in the second workshop. Altogetherthere were 9 small groups who worked according to the group assignmentsdescribed in Figure 3: in the first workshop the work was organized in 4small groups; in the second one there were 5 small groups.

Generally, the teachers in both workshops began sorting by what weterm surface features (i.e., features that can be observed without solvingthe statements), and only after a while turned to more structural features(that may be identified only by solving the statements). Tami and Alex hadanticipated that the teachers would suggest only two ways of sorting thegiven conditional statements: according to the families of functions thatappear in the statements (a surface feature) and according to the roles ofthe domain and range of the functions in solving the equations and inequal-ities (a structural feature). Much to their surprise, overall the 9 groups ofteachers proposed 11 different criteria for sorting the statements, as shownin Figures 5 & 6. Note that the shaded areas in Figure 5 (criteria ii, ix &x) indicate sorting according to the criteria initially planned by Tami and


Figure 5. Teachers’ shared classification of the statements – a summary of the first twoworkshops.

Alex, whereas, criterion ix was originally included by Tami and Alex as asub-category of criterion x.

The number of different sorting criteria employed by the small groupsvaried as follows: three criteria (by 5 groups); four criteria (by 2 groups);six criteria (by one group); and seven criteria (by one group). There wasonly one sorting criterion which all 9 groups raised (criterion ii, which


Figure 6. Distribution of teachers’ classification of the statements.

was one of the two originally planned). Another rather popular (surface)criterion was suggested by 6 groups (criterion i). The rest of the criteriawere raised by at most 4 groups of MTs. Interestingly, the main categorythat Tami and Alex saw as the target concept of the workshop (criteria ix &x) was addressed only by 3 groups, none of which fully addressed all thesub-categories connected to the role of the domain and range in solvingconditional statements. In addition, for the criteria that were suggested bymore than one group, there were differences in many cases in the sub-categories specified by the MTs.

At the end of the first workshop, Keren took part as facilitator ofthe reflective whole group discussion, while Tami followed this role ina similar manner in the second workshop. These concluding discussionsfocused on the mathematical challenge of the task, the management ofthis cooperative learning setting, and the implication of the MTs’ learningexperiences to their knowledge of learners – themselves as well as theirstudents. With respect to the mathematical challenge, the teachers claimedthat at first the task appeared to be rather simple and straightforward.However, they realized as they continued working on the task that it offeredmultiple levels of difficulty and complexity with which they had to deal.There was a consensus that although they were not implicitly required tosolve all the conditional statements, they were driven to do so as theyproceeded in the task, moving from surface features to structural ones.Even those who had a particularly sound mathematical background foundthe task challenging. There was an agreement that the small groups’ sharedideas and findings led each group to the recognition of additional criteriaand sub-categories. Special attention was given to the specifics of the set


of statements, which were regarded by the MTs as rich, profound, andinstrumental in enlightening the role of the domain and range in solvingvarious conditional statements. Several MTs expressed disappointment inthe fact that no small group, on their own, reached the complete sortingscheme that was generated by the entire group (as shown in Figures 5 & 6).

With respect to the cooperative learning setting that was managed in theworkshop, the groups reported that they encountered genuine cooperativework that was fostered by the nature of the assignment (see Figure 3). Oneof the most common ways in which they collaborated was by dividing theset of statements to be solved among the members of the small group.This, in their opinion, increased the efficiency of the group’s work, andallowed them to focus on wider and deeper aspects of the statementscompared with those they could have reached individually. The need topresent their results to the whole group was an incentive to their progress.There were a number of MTs who admitted that, at first, they had resentedthe requirement to work cooperatively in small groups, since they alwayspreferred to work individually and did not believe in cooperative learning.However, as the workshop progressed, they became involved in the groupinteractions and felt that they had contributed to the group and that, atthe same time, they had gained insights through these interactions. Withrespect to the knowledge of learners the teachers pointed to what theyconsidered an effective learner-centered environment, which was enhancedby the task and setting. For them it was a meaningful manifestation thatsuch an environment can be implemented and be influential to them aslearners. Some of them expressed their willingness to try out a similaractivity with their students. Others argued that it would take them toolong to design such a learning environment for their students, although,in reflecting on their own learning claimed that they would have liked toprovide their students with similar experiences.

At the end of the workshops, when Tami and Alex reflected on theevents it turned out that they both felt they had learned a lot. Theyconsidered the most critical in the entire process of designing and carryingout the workshop the exchange of ideas with Keren and her suggestion tomodify their original task to an open-ended sorting task. What seemedto them at first as a rather minor change, proved to make a cardinaldifference. They realized the potential of this kind of task in enhancingthe professional growth of MTs with respect to the MTs’ teaching triad.They attributed this to the cognitive nature of the task as well as to theopenness of the setting. Tami claimed that she had gained appreciation ofthe power of open-ended mathematical tasks in enhancing MTs’ abilityto construct shared mathematical and pedagogical knowledge, with verylittle interference of the MTEs. In addition, Tami explicitly mentioned how


her observation of the way Keren facilitated the concluding whole groupdiscussion at the first workshop (as well as observing Keren in other similarsituations) had influenced the way she conducted the discussion at the endof the second workshop. Alex expressed a similar view of the above eventsand their impact on his attitude towards his role of MTE.

As mentioned earlier, Rachel and Hanna were two staff members(MTEs) who took part in these two workshops, first as learners and thenas observers. Their participation in the second workshop was initiated byRachel, who offered to take part in it as an observer and to provide Tamiwith feedback about her management of MTs’ learning. Rachel’s reflectivediscussions with Tami addressed several perspectives which included: themathematics in which the MTs engaged, the impact of the openness of thetask on the mathematical discussions that took place, the interactions andcooperation between the MTs, Tami’s flexibility throughout the workshop,and Tami’s way of conducting the whole-group discussion.

Together with the MTs, Rachel and Hanna developed an appreciation ofthe potential of sorting tasks for enhancing mathematical understanding aswell as for creating an open learning environment. Consequently, Racheland Hanna joined Tami and became involved in further endeavors toenhance teachers’ implementation of sorting tasks in their classrooms. Thiscollaboration began by extensive guidance that Tami provided for Racheland Hanna in preparation for conducting a similar workshop on their own.Tami observed this workshop and met with them after it was over to reflecttogether on the MTs engagement in the sorting tasks and on Rachel’s andHanna’s collaborative roles as MTEs. Before conducting more such work-shops, Rachel turned to Tami for further advice and suggestions on how tolead an improved workshop.

These iterative experiences led to the design of a series of workshops inwhich the MTs collaboratively developed sorting tasks in various schoolmathematics topics for their own students. Following these workshops,there were MTs who began applying these tasks in their classes, andshared their implementation experiences with their colleagues and withstaff members at the weekly meetings. A year later, Rachel offered, onher own initiative, to conduct her adaptation of the series of workshops onsorting tasks with a new group of MTs.

ANALYSIS AND DISCUSSION:THE MODEL AND THE STORY

The purpose of this section is to demonstrate the explanatory power ofour three-layer model of growth through practice. For this purpose, we


Figure 7. Adaptation of the model of MTEs’ growth-through-practice to Tami’s storyline.

offer an analysis of our story in light of the model. This analysis conveysthe complexities and commonalities of the underlying processes of profes-sional development of MTEs within the framework of the project. Theparticipants of the story are seen as members of a diverse communityof practice in which its various participants shift roles and continuouslyengage in learning and facilitating activities.

We turn to an analysis of the professional growth through practiceof Tami and Rachel – the key characters of our story – who representtwo typical storylines of MTEs’ development. Our analysis indicates theconnections between their professional development and the interactivenature and structure of the community of practice to which they belonged.We attribute these connections to the different interactions between thevarious kinds of members of the community of practice – MTs, MTEs,and MTEE – at the three layers of our model. Figures 7 and 8 depict themain benchmarks of Tami’s and Rachel’s storylines respectively along ourmodel.


Figure 8. Adaptation of the model of MTEs’ growth-through-practice to Rachel’sstoryline.

The first part of the story exhibits a representative type of interactionbetween two collaborating MTEs with a similar expertise, i.e., MTE-MTE interactions. Tami’s first benchmark (see a in Figure 7) involvesher collaborative interactions with Alex – another MTE with a similarbackground – in the course of the planning stage. Tami’s initial design ofthe target workshop originated from Tami’s MTE’s knowledge of MTE’steaching triad that consisted of both a strong mathematical background andher expertise in cooperative learning methods. The collaborative processof preparing this workshop on the selected mathematical topic – the roleof domain and range of function in solving equations and inequalities –with another MTE (Alex) presented Tami with a doubtful situation. Thiscase was similar to many other situations in which MTEs, who worked ona workshop collaboratively, realized that there were differences betweentheir own and their colleagues’ positions. We found these differences tobe complementary, competing, or contradictory. These doubtful situationsdrove MTEs to try to reach, by discussion of their different positions, ashared position for preparing the workshop of common interest. This, inreturn, facilitated their learning.


The next part of the story represents MTEs’ learning through consultingwith MTEE, i.e., through MTE-MTEE interactions. Tami’s decision toconsult with Keren at the initial planning stage (see b in Figure 7) wasmotivated by her disagreement with Alex. Such doubtful situations ledother staff members in various situations in the project to consult with moreexperienced members. The exchange of ideas between Tami and Alex andtheir brain-storming with Keren enhanced Tami’s knowledge of the MTE’steaching triad iteratively by adding sorting tasks to her repertoire of poten-tial learning tasks for teachers. Consequently this led to an improved planof the learning offers that Tami facilitated for the MTs.

The third part of the story, that includes the two workshops thatTami conducted, represents MTEs’ learning through implementation ofthe workshop with MTs, i.e., through MTE-MT interactions. This partinvolved Tami’s learning from teachers’ ideas through her interactions withthem in the course of providing the intended learning offers for MTs andobserving their work (see c and d in Figure 7). During the first workshopTami was surprised to realize the numerous approaches of the MTs tothe sorting task. By observing the teachers’ work on the problems andreflecting in and on action (see d in Figure 7), Tami became more sensitiveto MTs’ ways of thinking and became more aware of what may be expectedof MTs. She also became convinced of the potential of sorting tasks as avehicle for professional development. This learning of Tami led her to varythe learning offers in the second workshop (see e in Figure 7). Note thatTami’s learning through observing included her observation of the wholegroup discussion that was led by Keren (the MTEE) in the first workshop.This part can be seen as involving MTEE-MTE-MT interactions.

The fourth part of the story that deals with Tami’s collaborative workwith Rachel exhibits another type of MTE-MTE interactions, i.e., inter-actions between two collaborating MTEs with differing expertise. Interac-tions of this type occurred when a more experienced MTE mentored a lessexperienced MTE. In many cases an experienced MTE, after designing andimplementing a workshop and encountering the learning entailed in thisprocess, began preparing other MTEs to conduct this workshop in futureoccasions. This can be considered as a transition stage in which an exper-ienced MTE began taking the role of a MTEE. We analyze the learningthat occurred in the course of such interactions from two perspectives:The learning of the more experienced MTE (in this case – Tami), and thelearning of the less experienced MTE (in this case – Rachel).

After gaining expertise in planning and running the above workshops,Tami was prepared to mentor other less experienced MTEs (e.g., Racheland Hanna). In the beginning, she provided extensive guidance to Rachel


and Hanna in conducting the workshop. After they had gained their firstexperience in its implementation, Tami offered them some additionaladvice towards their second round of implementation. By this process Tamideveloped her MTEE’s knowledge of MTEs learning (see f in Figure 7 andE in Figure 8). Overall, the model illustrates how Tami’s MTE’s knowl-edge developed and transformed into a MTEE knowledge (see the specialarrow above b in Figure 7).

Rachel’s development as a MTE started by participating in Tami’s firstworkshop as a MT (see A in Figure 8), and reflecting on her own as wellas on her colleagues’ work during that workshop (see B in Figure 8). Thisinteraction with Tami led first to the growth in Rachel’s MT’s knowledge.Following this experience, Rachel offered to participate in the secondworkshop as an observer and to provide Tami with feedback about themanagement of the workshop and the learning processes the MTs experi-enced. Through observation of and reflection on the second workshop (seeC in Figure 8), her knowledge evolved further. In her subsequent inter-actions with Tami surrounding the second workshop, they both reflectedon and analyzed the workshop from several perspectives (see earlier, inthe story). Through the exchange with Tami, Rachel developed her MTE’sknowledge and felt ready to take the role of a MTE (see C in Figure 8).

Rachel’s growth and transition from MT to MTE was similar to thatof Tami – from MTE to MTEE. Thus, Rachel’s planning stage of a newworkshop involving other sorting tasks (which was done in collaborationwith Hanna who underwent the same process) was similar to Tami’s initialplanning stage (see D in Figure 8 in comparison to a in Figure 7). Theneed Rachel felt to consult with Tami regarding the learning offers shehad planned, was somewhat like the need Tami felt to consult with Kerenbefore the first workshop (see E in Figure 8 in comparison to b in Figure 7).Again, the exchange of ideas and consultation led to an improved plan ofthe learning offers that Rachel facilitated for the MTs (see F in Figure 8in comparison to c in Figure 7). Altogether, similar to the way Tami’sMTE’s knowledge grew through her practice and evolved into a MTEE’sknowledge, Rachel’s MT’s knowledge developed into a MTE’s knowledge.

Our analysis conveys the iterative nature of the growth through prac-tice of the different members of the community of mathematics educatorsand through their different interactions, continuously switching roles fromlearners to facilitators. The story that we chose to present is one of manysimilar stories that emerged throughout this five-year project.


CONCLUDING REMARKS

In this paper we introduced a three-layer model of development ofmembers of a community of mathematics educators (Figure 2). The modelis used to analyze the growth through practice of teacher educators. Byapplying it to one particular professional development program, this modelis shown to have a descriptive and explanatory power.

Throughout the paper we examine the mathematics educators involvedin the project as members of a special community of practice that evolvedover time in the context of the project. Our analysis focused on the differentopportunities that were offered within this context and the ways in whichthese opportunities led to the development of the various members of thiscommunity, with an emphasis on two different members. In particular, wedescribed interactions between experienced members of the communityand new comers who were less experienced. Our analysis is a detailedaccount of how the “knowledgeability comes from participating in acommunity’s ongoing practices” (Roth, 1998, p. 12). Moreover, it points tothe ways in which these interactions contributed not only to the newcomers(e.g., Rachel) but also to the more senior and experienced members of thecommunity (e.g., Tami).

We speculate that the three-layer model that we offer and use in thispaper may be useful in similar ways in shedding light on and describingsuch processes within other professional development frameworks. Wealso are rather confident that awareness of the complexities involved inMTEs work, as portrayed through the model (see Figure 2), may drawattention to and shed light on the necessary ingredients of this evolvingprofession. We suggest further that this model may be integrated as part ofMTEs education. Thus, fostering MTEs reflection by explicitly referring tothe above model and its relevance to their profession may enhance MTEsgrowth-through-practice.

To conclude, we point to the role that mathematical challenge and thespecific mathematical tasks played in this context. A major concern of themathematics education community has to do with teachers’ mathematicalknowledge. There is an agreement that teachers must have a deep andbroad understanding of school mathematics in order to be able to offerstudents challenging mathematics. Unfortunately, formal higher educa-tion in mathematics does not always address this need. In our work, wehave developed numerous activities aimed at this goal (e.g., Zaslavsky,1995, in press; Leikin, 2003). Our challenge was to identify mathematicalsubtleties in the junior and senior high school curriculum and to designlearning activities that, under certain modifications, are equally applicable


to students as well as to MTs. The sorting task, which constituted the corecontent of the story we presented earlier, is one example of this approach.While such learning activities are the main goal for student mathematicallearning, for MTs they served as a vehicle for professional growth beyondthe mathematical knowledge.

As indicated in our story, the sorting assignment proved challenging forMTs as well as for MTEs in terms of the mathematics involved. However, italso provided a rich opportunity for teachers to experience a different kindof learning (compared with their previous experiences as learners), whichrequired activeness and cooperation. Reflection on their work as learnerswas a key factor in the transition from dealing with the mathematics itselfto elevating their experience to encompass the MTs teaching triad. Ourwork may be viewed as an attempt to shift from looking at Jaworski’steaching triad mainly as three distinct interrelated components to treatingit as an entity in itself that forms the challenging content of MTEs (SeeFigure 1).

This process, in itself, of designing such mathematical activities forMTs proved instrumental in the professional growth of MTEs (Zaslavsky,Chapman & Leikin, 2003, refer to this process as an example of indirectlearning). The design of such activities required drawing on sound mathe-matical knowledge, sensitivity to MTs as learners, and implementation ofinnovative approaches to management of MTs learning. In addition, it alsoinvolved a reflective state of mind and a collaborative disposition.

NOTES

1 This paper shares the same setting and method described in Zaslavsky and Leikin(1999), though the conceptual framework has been refined and extended with referenceto a different story.2 The mathematics component of the “Tomorrow 98” project in the Upper Galilee wasdirected by the Technion – Israel Institute of Technology, and funded by the Israeli Ministryof Education.3 All the names in the paper are pseudonyms.

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Department of Education in Technology and Science Orit ZaslavskyTechnion – Israel Institute of TechnologyHaifa, 32000IsraelE-mail: [email protected]

University of Haifa Roza LeikinHaifa, Israel

AMY ROTH MCDUFFIE

MATHEMATICS TEACHING AS A DELIBERATE PRACTICE: ANINVESTIGATION OF ELEMENTARY PRE-SERVICE TEACHERS’

REFLECTIVE THINKING DURING STUDENT TEACHING

ABSTRACT. In this case study I examine the reflective practices of two elementary pre-service teachers during their student teaching internship. I extend current views of reflectivepractice to create a framework for a ‘deliberate practitioner’. With this framework, I inves-tigate the pre-service teachers’ thinking with regard to reflective processes and how theyuse their pedagogical content knowledge in their practices. My findings indicate that thepre-service teachers use their pedagogical content knowledge in anticipating problematicevents, and in reflecting on problematic events in instruction. However, limits in pedago-gical content knowledge and lack of confidence impede the pre-service teachers’ reflectionwhile in the act of teaching. They were more likely to reflect on their practices outsideof the act of teaching. Implications for teacher educators and pre-service teachers arediscussed.

KEY WORDS: mathematics education, pedagogical content knowledge, reflectivepractice, student teaching, teacher education

With the emergence of recent reforms in education in the United States(e.g., National Council of Teachers of Mathematics [NCTM], 1989, 1991,2000), researchers and educators have re-examined teaching by movingaway from a technical model of teaching by prescribed methods to onethat regards it as a, complex, demanding practice. Two separate butcompatible perspectives have made substantial contributions as to how weview teaching, and correspondingly, to how we approach teacher educa-tion. First, viewing teachers as reflective practitioners has underscoredthe problem solving nature of teaching (McIntyre, Byrd & Foxx, 1996;Russell & Munby, 1991; Schön, 1983, 1987; Valli, 1992; Zeichner, 1993).Consequently, the focus of many teacher education programs is on thedevelopment of reflective practitioners (Christensen, 1996). This focusis consistent with a constructivist perspective for teaching and learningthat is the basis of many teacher education programs (e.g., McIntyre,Byrd & Foxx, 1996). Second, the conceptualizing of pedagogical contentknowledge (Grossman, 1990; Shulman, 1986, 1987) as a unique typeof knowledge for teaching has helped researchers, teachers, and teachereducators gain an understanding of the knowledge base that teachers needfor successful practice.


34 AMY ROTH MCDUFFIE

In studying the knowledge of mathematics teachers, Ball, Lubienskiand Mewborn (2001) call for research on “how teachers are able to usemathematical knowledge in the course of their work” (p. 450) and “what[teachers] are able to mobilize mathematically in the course of teaching”(p. 451). One approach to understanding the use of knowledge is to investi-gate how teachers think about their practice. The purpose of my study wasto intersect the constructs of reflective practice and pedagogical contentknowledge (PCK) in order to examine how pre-service teachers use theirmathematical PCK in thinking about their practice, both in planningand classroom teaching. I investigated the reflective practices in mathe-matics of two pre-service elementary teachers during their student teachinginternship.

THEORETICAL FRAMEWORK

I shall provide a summary of reflective thinking and pedagogical contentknowledge as I applied and merged these two concepts in my work withthe pre-service elementary teachers.

Reflective Thinking

While a range of interpretations exists for what is considered to bereflective practice, I referred primarily to the ideas of Dewey (1910) andSchön (1983, 1987). Reflection is a practice that has gained considerableattention in the past two decades, yet Dewey began this discussion earlyin the last century. Dewey argued that reflective thinking begins whenteachers experience a difficulty or troubling event (i.e., a problem). A keyaspect of the reflective process is that teachers act on their reflections:

Reflection involves not simply a sequence of ideas, but a consequence – a consecutiveordering in such a way that each determines the next as its proper outcome, while eachoutcome in turn leans back on, or refers to, its predecessors . . . Each phase is a step fromsomething to something . . . There are in any reflective thought definite units that are linkedtogether so that there is a sustained movement to a common end. (pp. 2–3)

Thus, acting on reflections distinguishes reflective practice from justthinking back and may be an important aspect in the development ofteaching.

Schön (1983, 1987) developed these ideas further and separatedreflection into two forms, reflection-in-action and reflection-on-action.Reflection-on-action is a deliberate process of looking back at problematicevents and actions, analyzing them, and making decisions. Russell andMunby (1991) explained that reflection-on-action “refers to the ordered,

MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 35

deliberate, and systematic application of logic to a problem in orderto resolve it; the process is very much within our control” (p. 165).Reflection-in-action is a more immediate consideration and resolution ofan identified problem in the act of teaching and learning (Schön, 1987).Both types of reflection and the triggering (problematic) events wereconsidered in this study.

Mewborn (1999) studied the reflective practices of pre-service teachersduring field experience as part of their mathematics methods course.Mewborn examined the elements of mathematics teaching and learningthat were problematic for the teachers and the thinking they engaged into resolve those problems. She found that pre-service teachers did engagein reflective thinking. However, this reflective thinking was not evidentuntil they internalized their own authority to generate, reason about, andtest hypotheses in order to examine children’s mathematical thinking.Mewborn argued that to facilitate reflection pre-service teachers need anon-evaluative atmosphere and relationship with cooperating teachers anduniversity faculty so that they are encouraged to generate hypotheses andto arrive at resolutions to problematic events without fear of judgment.

Inherent to all discussions of reflection is a problematic or puzzlingevent triggering reflection and, thus, I sought to investigate reflection thatarose from problematic situations. While it could be argued that reflec-tion might occur during or after a successful lesson (a non-problematicsituation), I chose to adopt the constructs for reflective thinking ofDewey (1910) and Schön (1987) by focusing on pre-service teachers’thinking that was inspired by identified problems. I synthesized the variousviews and definitions of reflection considered for this study as a cyclein teaching practice. In this cycle, the problematic event initiates theprocess of reflection. Following this problematizing, either during instruc-tion (for reflection-in-action) or after instruction (for reflection-on-action),the teacher analyzes the problem and the options and/or approaches forresolving the problem. Next the teacher decides on a resolution or planfor action. Finally, the plan is implemented in practice and the resolutionis tested. At this point the process either ends for this event or results ina subsequent problematic event (an unresolved issue), and the reflectivecycle continues. As Dewey (1910) contended, the cycle does not neces-sarily move directly from one phase to the next. One could, for example,re-define the problem while in the process of planning an action.

Using Pedagogical Content Knowledge to Guide Reflective Thinking

Along with a focus on teachers developing reflective thinking practices,researchers have been concerned about teachers developing a sufficient


knowledge base to guide their thinking about children’s ways of under-standing mathematical concepts and process (e.g., Ball, Lubienski &Mewborn, 2001; Borko & Putnam, 1996). Indeed, pedagogical contentknowledge is an important resource for teachers to use as they reflect inpractice. Shulman (1986, 1987) first defined pedagogical content knowl-edge as,

The blending of content and pedagogy into an understanding of how particular topics,problems, or issues are organized, represented, and adapted to the diverse interests andabilities of learners, and presented for instruction. Pedagogical content knowledge is thecategory most likely to distinguish the understanding of the content specialist from that ofthe pedagogue. (1987, p. 8)

Building on Shulman’s work, Grossman (1990) delineated four centralcomponents of pedagogical content knowledge. These components are:conceptions of purposes for teaching subject matter (i.e., forming goals);knowledge of students’ understanding, conceptions, and misconceptionsof particular topics in a subject matter; curricular knowledge; and knowl-edge of instructional strategies and representations for teaching particulartopics. Grossman acknowledges that “these components are less distinctin practice than in theory” (p. 9), but this general framework was usefulin thinking about the ways in which the pre-service teachers in this studyused their pedagogical content knowledge.

Researchers have found that novice teachers tend to have inadequate orunderdeveloped mathematical pedagogical content knowledge for use inpractice (e.g., Borko & Putnam, 1996; Borko, Eisenhart, Brown, Under-hill, Jones & Agard, 1992). In elementary mathematics education, severalprojects have as a goal the development of teachers’ understanding aboutchildren’s learning such as Cognitively Guided Instruction (e.g., Carpenter,Fennema, Franke, Levi & Empson, 1999) and SummerMath (e.g., Schifter& Fosnot, 1993; Schifter, Bastable & Russell, 1999). These projects haveproduced materials (e.g., case studies for exploring teaching and learn-ing) that have the potential to build pedagogical content knowledge forpre-service teachers before entering the classroom. However, researchis needed to investigate how these teachers use this knowledge in theirpractice.

My study investigated the role of reflection in practice by examiningthe reflections and experiences of pre-service teachers during studentteaching. Although Mewborn (1999) focused on pre-service teachers, herresearch was conducted earlier in the teachers’ educational preparation.In my study, I focused on the form of reflective thinking pre-serviceteachers’ exhibited and how they employed pedagogical content knowl-edge in mathematics in their thinking. The specific research questions


for pre-service teachers’ mathematics instruction were: (a) When the pre-service teachers demonstrate reflective thinking, what forms does it take(i.e., reflection-in- or reflection-on-action)? (b) During reflective thinking,how do these teachers use their pedagogical content knowledge?

METHODOLOGY

I studied two pre-service elementary teachers, pseudonymous Gerri andDenise, during their semester-long student teaching internship. I conductedthis study from a perspective that combined ideas of interactionism andconstructivism in viewing the process of becoming a teacher. According tothe perspective of interactionism, people construct and sustain meaningsthrough interactions and patterns of conduct (Alasuutari, 1995; Blumer,1969). This position is in accordance with the constructivist perspective oflearning in that individuals develop understandings based on their exper-iences and knowledge as it is socially constructed (Cobb & Bauersfeld,1995). This framework supported this study in that the pre-service teachersreflected and constructed meanings based on their participation in, andobservation of interactions and patterns of conduct with their students andcolleagues. Because I was interested in describing and interpreting thethinking and experiences of pre-service teachers during student teaching, Iselected a qualitative case study as the most promising mode of inquiry(LeCompte, Millroy & Preissle, 1992; Stake, 1995). The cases werebounded by the semester-long student teaching experience and focusedon reflection in regard to their mathematics instruction using incidents ofreflecting-in and reflecting-on practice as units of analyses.

Data Collection

Context and ParticipantsTeaching program. Gerri and Denise were enrolled in a Master in Teachingprogram at a state university. This two-year master degree programserved pre-service teachers who already held a baccalaureate degree ina field other than education but desired to become teachers. Two primaryobjectives of this program were:

1. To educate teachers to become effective practitioners who areinformed scholars with the leadership and problem solving skills tohelp schools and communities meet the needs of the 21st century andto enlighten thought and practice by bringing the inquiry method of aresearch university to bear on the entire educational process.


2. To empower teachers as reflective practitioners by helping themdevelop the multiple and critical decision making skills essential fortoday’s classrooms. (University program description document)

This research-based approach to developing reflective practitioners wasevident in the design of the student teaching internship. Requirementsof the internship included: 12 weeks in a K–8 school placement; writingweekly in reflective journals; setting goals and reflecting on meeting thesegoals weekly; writing lesson and unit plans; observing and reportingon other teachers’ instruction; and completing a classroom-based actionresearch project on their own teaching. Gerri and Denise completed studentteaching during the spring semester.

For the action research project, the pre-service teachers designed theirstudies during the previous semester as part of a course titled “ClassroomFocused Research”. Using two texts (Hubbard & Power, 1993; McNiff,Lomax & Whitehead, 1996) as a framework for study, the pre-serviceteachers studied methods of designing and conducting action research, andplanned original classroom-based research projects. The action researchproject focused on a specific teaching strategy or approach. Each teacherworked with a faculty committee (a supervisor, with expertise in theselected area for research, and two additional faculty members). The pre-service teachers wrote literature reviews in their areas of study as part ofa full study proposal. These proposals were submitted to their supervisorsfor feedback and review before submitting a final version at the end of thesemester. Then they implemented their studies during student teaching. Inthe month following their student teaching internship, Gerri and Deniseanalyzed their data, wrote, and presented reports of their studies to afaculty committee.

Given that pedagogical content knowledge is a focus of the study, I shalldescribe the mathematics methods class that served as Gerri’s and Denise’sprimary source of this knowledge prior to student teaching. In the firstsemester of the program, Gerri and Denise completed Elementary Mathe-matics Methods for which I was the instructor. The primary goals of thiscourse aligned with Grossman’s (1990) four components of pedagogicalcontent knowledge. This course emphasized developing understandingsfor: reform-based visions of teaching and learning mathematics (goalsin teaching and learning); how children think and learn about mathe-matics, including common misconceptions in elementary mathematics(knowledge of students); the range of resources and curriculum materialsavailable for mathematics instruction (curricular knowledge); and devel-opmentally appropriate strategies for teaching and learning (knowledge ofinstructional strategies).


Van de Walle’s (1998) methods text was used along with supple-mental research-based readings. Instructional approaches included in-classexplorations with textbook activities and extensive use of theoreticalframeworks and case studies from Cognitively Guided Instruction (e.g.,Carpenter, Fennema, Franke, Levi & Empson, 1999) and DevelopingMathematical Ideas (e.g., Schifter, Bastable & Russell, 1999). Courseassignments included: interviewing children and analyzing their thinking,understandings, and dispositions for mathematics; writing lesson plans thatincluded an analysis of possible children’s approaches and misconcep-tions; and collecting and critiquing problem-centered tasks from reform-based publications (e.g., Teaching Children Mathematics and Mathematicsin the Teaching in the Middle School).

Gerri. Gerri held a Bachelor’s degree in engineering and consequentlyhad substantial college-level coursework in mathematics. She entered themaster’s degree program to begin a career in teaching after staying homewith her children for several years. While Gerri had volunteered exten-sively in her children’s schools, she did not have any formal teachingexperience prior to entering the program. Gerri was regarded by Univer-sity faculty and her field specialist as having strong content background,especially in mathematics and science.

Denise. Denise was a recent graduate and held a Bachelor’s degree inFrench with mathematics coursework through first-semester calculus. Sheentered the master’s degree program two years after completing her under-graduate program. During those two years, she had worked as an educa-tional assistant at an elementary school. During this time, she was exposedto hands-on, student-centered approaches to teaching. Denise explainedthat, as an educational assistant, she had recognized the value and bene-fits to student learning in using these approaches. However, Denise feltthat she lacked the theoretical foundations and framework needed to planeffectively and to manage student-centered instruction because she did nothave an academic background in education prior to entering the program.

Researcher. I served as a participant observer in that I researched thepre-service teachers’ practice while acting as their university supervisor.I also had an established relationship with Gerri and Denise prior to thestudy as their mathematics methods course instructor. Additionally, as theirMaster’s degree Committee Chair, I advised Gerri and Denise on theiraction research projects during all phases of their research.

Prior to the study and heeding the advice of Mewborn (1999) regardingthe creation of a non-evaluative environment to promote reflection, I


discussed with Gerri and Denise my role as a researcher and as theirsupervisor. I explained that the purpose of my study was to understandhow they reflected on their practice. I also explained that, as their super-visor, I perceived my primary role to be a resource to them and to providesupport during their internship. During my observations and conferenceswith Gerri and Denise, I followed constructivist frameworks for studentteaching supervision that emphasized the developing of self-directed,reflective practitioners (cf., Sullivan & Glanz, 2000). Correspondingly, myfocus was on facilitating the pre-service teachers’ skills in analyzing theirpractice. For example, every conference began with my asking the pre-service teachers for their perceptions of: “What went well?” and “Whatwould you like to change?” I encouraged them to identify their strengthsand problem areas; helped them to clarify their thoughts and plans; andasked about their progress with their goals in follow-up meetings.

As is the case with most pre-service teachers in this program, Gerriand Denise were confident they would pass student teaching (only passor fail grades were assigned). Therefore, they were more concerned abouttheir professional growth than official evaluations, and Gerri and Denisestated that they also viewed me primarily as a resource. Indeed, neitherGerri nor Denise was ever at risk of failing student teaching. Throughoutthe semester, both Gerri and Denise commented that they viewed me asa support and resource rather than an evaluator. However, ultimately, asthe university supervisor, I was responsible for evaluating their internship,and thus it would be naïve to think that we had an entirely non-evaluativerelationship.

Data SourcesThe primary data sources were: audio-taped interviews and conferenceswith the participants; my observations of classroom teaching; reflectivejournal entries and weekly goal statements; lesson and unit plans; andparticipants’ data collected as part of their action research projects andtheir final research reports. Nine observations and semi-structured inter-views and/or conferences, occurring approximately every week to twoweeks, were conducted with each participant throughout the semester. Irecorded field notes for the observations, and each observation lasted aboutone hour. Each interview lasted about 30 minutes and was transcribed. Theparticipants wrote at least one journal entry weekly, and wrote lesson plansdaily.

Data Analysis

To address the first question, I analyzed the data by analytic induction: Isearched for patterns of similarities and differences for when reflection did


and did not occur and also for the forms of reflection (Bogdan & Biklen,1992; LeCompte, Millroy & Preissle, 1992). I classified events as reflectivethinking if evidence existed that the pre-service teachers completed allcomponents of the reflective cycle. To find evidence, I first observed forproblematic events that initiated reflective thinking, and/or I read the pre-service teachers’ journals for their reporting of surprising or puzzlingevents. During observations, the pre-service teachers usually demonstratedthat something puzzling occurred through facial gestures, pauses in theirspeech, talking aloud, and/or a lesson that departed from the previouslywritten lesson plan. As part of journal writing, the pre-service teachersfocused on problematic events and wrote about them directly.

Next, I investigated whether the pre-service teachers analyzed anddeveloped a possible resolution to the problem. I obtained this evidencefrom questioning and from their journal writing about considerationsin making an instructional decision. In analyzing the journals, to helpsort out whether their reflections were in-action or on-action, the pre-service teachers placed an asterisk next to any thoughts or ideas thatoccurred to them after teaching rather than just writing about their thoughtsduring the lesson. Finally, I examined the data for a resulting action orimplementation of a plan in instruction.

It should be noted that the findings were often supported, at least inpart, from the participants’ self-reporting of reflective experiences. While Iendeavored to validate these reports through triangulation of data sources,the nature of this research did rely on participants’ reports of reflection.Consequently, I made efforts to ensure that the participants had operation-alized reflective experiences in a manner consistent with my understand-ings of reflection-in-action and reflection-on-action as presented in thismanuscript. We discussed these notions thoroughly in the initial interview(with examples and non-examples of reflection), and then summarizedthese ideas again throughout the semester.

For the initial coding of data, I used codes of reflection-in-actionand reflection-on-action to identify and track the forms of reflection.Through the data analysis process, I found a need to distinguish furtherforms of reflection into subcategories (immediate reflection-in-action,delayed reflection-in-action, short-term reflection-on-action and long-termreflection-on-action), and a new category emerged: deliberate practice.Each of these forms of reflection is represented in Figure 1 and describedbriefly below as they were used in coding and analyzing data. They areexemplified further in the presentations of the case findings.

Immediate Reflection-In-Action (referred to as immediate reflectionhereafter) represents the thinking when pre-service teachers made imme-diate decisions while completely in the act of teaching. This form of


Figure 1. Processes of deliberate practice.

thinking was difficult to identify, and thus I relied heavily on focusedobservations and follow-up interview questions to determine if thepre-service teachers identified a problem during instruction, and thencompleted the reflective cycle. Moreover, I asked the participants directlyabout occurrences of and their ability to analyze and make decisionabout problems identified during teaching. Delayed Reflection-in-Action(referred to as delayed reflection hereafter) represents the thinking thepre-service teachers exhibited when a pause or break occurred in theact of teaching (e.g., students completing individual work or recess).Similar to immediate reflection, delayed reflection resulted in analysis, adecision, and an action for the lesson in progress or for the plans for theday. However, a break in activity distinguished delayed reflection fromimmediate reflection. Immediate reflection and delayed reflection bothcorrespond to Schön’s (1987) description of reflection-in-action, but theydiffer in the level of instructional activity and demands occurring duringreflection. In Figure 1, the double arrow between Reflection-In-Action andthe Teaching and Learning Episode represents how instruction triggersreflection and consequently, reflection-in-action influence teaching whileinstruction is taking place.

With regard to forms of Reflection-On-Action, Short-Term Reflection-On-Action (referred to as short-term reflection hereafter) was exhibited


when the pre-service teachers thought back over a short period of time aftera lesson or day was over (e.g., reflecting on a lesson as they drive homeor on the week’s instruction over the weekend). Short-term reflection isdifferent from delayed reflection in that the pre-service teachers were notunder pressure to reflect, resolve, and implement the action to address aproblem immediately during the problematic lesson or day. This reflec-tion was often about the success of a lesson in contributing to learninggoals in order to guide planning for the next lesson or unit. Long-TermReflection-on-Action (referred to as long-term reflection hereafter) wasexhibited when the teachers systematically analyzed and examined theirpractice over an extended period of time for the purposes of understandingand improving practice more globally. They looked for emerging patternsand developed personal theories about teaching and learning. Most often,the reflective cycle for long-term reflection took place over several months.

Deliberate Planning was exhibited when pre-service teachers purpose-fully used existing knowledge, theories, and reasoning about teaching andlearning to design plans for particular students’ learning. While reflectionson past experiences may be part of an existing knowledge or theory base,this form is different from reflection. Reflective thinking was initiated bya problematic event. Conversely, deliberate planning involved analysis ofinstructional options prior to teaching, often preparing to avoid anticipatedproblematic events. All five forms of thinking compose a framework forthe deliberate practitioner.

After establishing this framework for deliberate practice, I used codesof immediate reflection, delayed reflection, short-term reflection, long-term reflection, and deliberate planning to classify the forms of thinking,and correspondingly, revised my research questions to consider thinking inthese five forms (versus Schön’s [1987] two forms). Additionally, for moreprecise coding, I used qualitative data analysis software with a combina-tion of open coding (Strauss & Corbin, 1990) and indexing of text (Miles& Huberman, 1994). Once I identified and classified reflective and/ordeliberate events (the first research question), I analyzed these events forhow pedagogical content knowledge was used (the second research ques-tion). That is, I examined how these teachers applied pedagogical contentknowledge (as described in Grossman’s [1990] four component model) inplanning for or reflective thinking about a problematic event. Throughoutthis process, the multiple data sources were compared to generate acomplete picture of each event and to confirm or refute emerging patterns.


FINDINGS

For each teacher case study, I describe an instructional episode. Followingthe description of each episode, I present my analysis and interpretationsby discussing the forms of deliberate practice and how pedagogical contentknowledge was used in reflection and deliberate planning. I chose theseepisodes because they represent Gerri’s and Denise’s thinking and prac-tices, and they provide examples of how the various processes of deliberatepractice connect to each other and to PCK.

Gerri

Gerri’s Philosophy and Field PlacementGerri’s philosophy of teaching seemed to be focused on ensuring thateach student was learning and was forming a conceptual understandingof the subject areas. In an interview prior to beginning student teaching,Gerri made it clear that her foremost interest was in student learning. Shewas distressed that current educational environments present barriers tofocusing on students’ understanding of concepts. In referring to a fieldexperience prior to the student teaching semester, Gerri stated,

I have seen that there is a real expectation . . . to get a lot of information to the kids . . . .In the practicum that I just completed, I saw the kids pushing forward in the curriculumwithout really having a good basis of what we were trying to teach . . . . Kids don’t reallyhave the concepts (Interview, December 30).

Gerri’s concern for students gaining strong conceptual understandingwas evident in her choice for her classroom-based action research project.She investigated the efficacy of her teaching practices in developing herstudents’ conceptual understanding of multiplication and division.

Gerri completed her internship in a third-grade classroom. Whileher field specialist, Mrs. Baker, was well respected in the school andthe community, she was not perceived as a reform-based teacher. Mrs.Baker had a predominately teacher-centered style and focused primarilyon skill mastery. Both Gerri and I observed this more traditional style.Gerri described Mrs. Baker’s teaching as, “Based on my observations,[Mrs. Baker has] a pretty traditional classroom. They [the students] areencouraged not to visit with their neighbors, to stay on task” (Interview,December 30). With regard to teaching mathematics, and just before takingover the class, Gerri said, “The students are learning their multiplicationfacts . . . . [Mrs. Baker] is a traditional teacher, relying on the text1 to teachmath. Her students have not really worked in small groups or with manipu-latives, so I can expect some problems when I first start teaching math”(Journal, January 3).


Consequently, Gerri understood that Mrs. Baker was not likely tobe able to support her in her efforts to implement more reform-basedapproaches to teaching mathematics. Additionally, Gerri’s journals and myobservations indicated that Mrs. Baker tended to be a reactive mentor,offering suggestions to solve existing problems or dilemmas, but notconsulting with Gerri prior to teaching a lesson.

Making Connections: Skip Counting Patterns and MultiplicationThe instructional episode. Early in Gerri’s student teaching experience,she taught a lesson on multiples. While the students had not necessarilyacquired all of their multiplication facts, Mrs. Baker had provided instruc-tion on all of the basic facts for multiplication. Gerri’s goals for this lessonwere: “Students will find the multiples of one digit numbers using theirknowledge of 0–9 multiplication facts. Students will understand what ismeant by the term multiple in mathematics” (Lesson plan, January 21).

The lesson began with Gerri writing, “0 2 4 6 8 10 12” on the board,and asked, “What is the pattern for these numbers?” A student responded,“Multiples of 2”. After another example with fives, Gerri wrote, “0 36 9 12 15” on the board, and again asked, “What’s the pattern here?”Another student responded, “Goes up by three”. After showing a fewmore examples with similar questioning, Gerri handed out a worksheetwith similar exercises on it (e.g., “Find the first 5 multiples of 6”). Upto this point, I observed that Gerri had followed her lesson plan exactly.Gerri then went back to her desk, where four students promptly lined upfor individual help (a practice established by Mrs. Baker during individualseatwork time). The students did not know where to start on the questionswhen a pattern had not been presented (as the example listed above). AsGerri worked with individual students, she asked the students how theyfound the next number in a pattern. When the responses relied on addingto the last number, she realized that while the students could complete thepattern, they did not relate this pattern to multiplication.

Indeed, from my observation many of the students seemed to bring onlya recursive interpretation to the pattern, observing that the next numberincreased by a fixed amount from the last number. When one studentsaid, “Multiples of 2”, it was not at all clear that the student linked theword “multiples” to multiplication. Perhaps she just used “multiples” asa word that preceded the quantity of increase in the pattern. Gerri did notemphasize this connection, and accepted responses such as “goes up bythree”, without asking for further explanation or exploration. Later duringseatwork, Gerri recognized this lack of connection in the concepts, andidentified a problem in meeting her stated goal of developing an under-


standing for the term multiple. She explained in an interview immediatelyfollowing the lesson,

We started with the patterns, and they caught right on. They were skip counting. But then,I didn’t make the connection very strongly that, besides skip counting, you could also dothe multiplication facts. I felt like I kind of missed that. And when they came back [to mydesk] and I was working one-on-one [with students], I realized that I felt like I could havemade that [connection] stronger. (Interview, January 21)

After realizing the problem, Gerri stopped the students during theirindividual seatwork and began a whole class discussion at the board usingproblems on the worksheet to build the connection she recognized wasmissing.

Forms of thinking. This episode illustrates aspects of deliberate plan-ning, delayed reflection during the lesson, and long-term reflection. Whilepresenting the patterns, Gerri deliberately planned for the students tounderstand the term multiple through patterns. However, she did notconsider that simply being able to provide a correct response to the ques-tion as posed (i.e., “What’s the pattern?”) was not sufficient to determineif her goal of understanding the term multiple had been met. Once sheworked with students individually, she began to probe their thinking furtherby asking, “How did you find the pattern?” and through this deeperquestioning, she was able to assess that they, at best, had a limited under-standing of the term multiple. In not planning to question the students onhow they arrived at their solutions (showing limited deliberate planning),an approach emphasized in reform-based approaches to teaching (e.g.,NCTM, 2000), she missed an opportunity of on-going assessment and achance to encourage the students to build their understandings. However,once a break in the lesson occurred, and Gerri worked with students indi-vidually (without the demands of facilitating a whole class discussion),she had time to consider and assess individual students’ approaches andidentify a problem in instruction: the students were not understanding“multiple” as referring to multiplication of numbers. She analyzed thesituation and implemented a different approach in instruction, emphasizingthe connection between the pattern of products and each product’s corre-sponding multiplication facts. Given that this all transpired during a lesson,it represented delayed reflection. Because the problem identification andreflective thinking did not occur until Gerri had a break in action, I did notclassify it as immediate reflection.

Long-term reflection also was evidenced in Gerri’s conclusions aboutquestioning and seeking explanations from students. This lesson repre-sented one of a series of efforts for Gerri to examine the efficacy of


teaching multiplication and division for understanding as part of heraction research project. Based on earlier field experience prior to studentteaching, Gerri identified a problem of students having only procedural orfact-based knowledge of these operations. Reflective thinking was exhib-ited as Gerri planned her study on a global level, and then planned eachlesson as part of implementing her study on a daily level. As Gerri imple-mented her plans to facilitate her students’ conceptual development ofmultiplication and division, she completed the action of the reflectivethinking cycle.

Use of pedagogical content knowledge. In planning the lesson Gerri knewthat it is important for students to understand the meaning of multipleand to look at patterns in different ways (multiples as a pattern and asproducts), consistent with reform goals (e.g., NCTM, 2000) and repre-senting Grossman’s (1990) notion of PCK for goals in teaching andlearning. Indeed, once Gerri realized that her goal of understanding had notbeen met, she deemed that this idea was important enough that she shouldadjust the lesson, and return to a whole class discussion to emphasize theconnections. Although use of PCK was evident in planning the lesson,Gerri had not yet developed the capacity to apply this knowledge whenprobing the students’ thinking during a whole class discussion. Yet, duringindividual instruction (while other students were working at their seats),Gerri demonstrated that she did have PCK for questioning strategies andfor understanding students’ thinking (Grossman, 1990) by recognizing theproblem (students’ limited understanding of the word multiple). Thesefinding are consistent with Corwin (1996) that emphasized the complex-ities involved in listening and questioning students to support mathematicslearning in whole class instruction. In sum, Gerri more effectively appliedher PCK when she was not orchestrating the lesson: during planning,during a break in teaching, and after the lesson.

Denise

Denise’s Philosophy and Field PlacementLike Gerri, Denise’s fundamental philosophy for teaching and learningwas to focus on students’ understanding in learning. This focus was evidentin that her planning and self-analysis consistently relied on students’conceptual learning as a referent over concerns for covering textbook orexternal curricular guidelines, as will be illustrated in the episode below.Moreover, Denise’s action research project was developed around herconcern for students’ attitudes and anxiety as factors in learning math.Denise theorized that by using approaches from multiple intelligences in


teaching math, her students would learn to utilize their individual strengthsand feel more successful in learning math. Her plan was to implementa multiple intelligences approach (e.g., Gardner, 1993) in mathematicsinstruction in order to facilitate her students’ learning. Denise was awarethat Gardner’s work has been questioned for its scientific merit in theresearch community. In choosing her approach to action research, sheacknowledged limitations of the theoretical basis, but still wanted to seethe effects of these approaches in practice.

Denise was placed in a fourth- and fifth-grade combination classthat was team-taught. Both team members, Mrs. Knight and Mrs. Earl,served as mentors to Denise, although Mrs. Knight was her official fieldspecialist. Mrs. Knight and Mrs. Earl used predominately student-centered,reform-based approaches and materials. They focused on concept devel-opment and active, hands-on learning, and they frequently used cooper-ative learning strategies. Mrs. Knight was a teacher leader in the region.Mrs. Knight tended to be both a proactive and reactive mentor. Mrs.Knight discussed, on a daily basis, possible options and considerationswith Denise prior to Denise’s teaching lessons and reviewed events anddecisions after lessons.

Developing Meanings for Decimals: Representing Decimals on GridsThe instructional episode. The lesson was part of a unit that served as anintroduction to decimals. Prior to this lesson, the students had discussedeveryday uses of decimals, decimal equivalents for common fractions (e.g.,1/2 and 1/4), approximate values of decimals (e.g., 0.49 is almost equal to0.5 which is the same as 1/2), and why we need both decimal and fractionalforms of numbers (Lesson plans for March 10 and March 13). Denisedeveloped these lessons in collaboration with Mrs. Knight. Denise reliedprimarily on Mrs. Knight’s advice and outside resources (e.g., materialsfrom Investigations [TERC, 1998]) and, indeed, I did not ever see Deniseusing or referring to the district textbook, Addison-Wesley Mathematics(Eicholz, 1991).

For this lesson, Denise’s goal was for students to gain an understandingof “representing decimals on grids; reading, writing, and ordering decimals[using grids]; and adding decimals on a grid” (Lesson plan, March 14).Immediately prior to teaching the lesson, Denise and I had a pre-lessonconference. Denise explained her goals to me and said that she thought thedecimal grids (see Figure 2) were an effective representation for studentsto gain an understanding of the value of various decimal numbers. Denisehad been exposed to decimal grids both in her mathematics methods courseand as an educational assistant.


Figure 2. Decimal grids Denise used in March 14th lesson.

Denise discussed her lesson plan with me and said that she plannedto have a whole-group discussion on representing various decimals onthe grids. She planned to ask them to color one tenth (written as “0.1”on the board) on each of the four grids and then to choose the best gridto show one tenth. Next, she wanted students to express the decimal ofone tenth in fractional forms that correspond to each grid (e.g., 10/100 forthe hundredths grid, 100/1000 for the thousandths grid, etc.). Denise thenplanned to “Write on the board 0.1 = 0.1000. Ask the students to figureout how they know these decimals are equal. Have the students share theiranswers with their partners” (Lesson plan, March 14). Denise planned to


end the lesson with students showing various other numbers on the grids(e.g., 0.75, 0.125) and discuss fractional equivalencies for each.

After Denise shared her plans, I asked what challenges students mighthave using the grids. She replied: “They have to show one tenth, so it’s easyon this one [showing one column on the tenths grid]. But I’m afraid they’renot going to know it’s the same shape on here [the hundredths grid]. I’mwondering if they’ll color in one little box [representing one hundredth]”(Pre-lesson conference, March 14). At this point Denise revealed that thislesson was the first time her students had seen or used decimal grids. Uponhearing this information, I suggested that she may be incorporating toomany ideas in a first experience with grids, and that perhaps she shouldhave the students just work with the tenths grid and become comfortablerepresenting various numbers on this grid before moving on to comparingthe grids and the multiple representations for decimals. As we ended theconversation, Denise decided that she was not comfortable changing thelesson at this point, but she would consider slowing down the discussion ifthe students seemed to be having difficulties.

Denise began the lesson as she had planned it and asked a student tocome to the overhead and color in one tenth on the tenths grid. The studentcorrectly colored one column. Next, another student (Maria) colored in onetenth on the hundredths grid by coloring a row of ten of the one hundredthssquares, and then labeled it as “0.01”. Denise then explained that thissecond representation was “still one tenth because it was ten hundredths”,but she did not address Maria’s incorrect labeling of one tenth as “0.01”.

Pointing at the thousandths grid, Denise said, “100 times 100 is a thou-sand okay?” Indeed, Denise was describing the ten-thousandths grid whilepointing to the thousandths grid. She paused, looked back at her notes, andsaid, “Well this is still one tenth”, as she pointed to a column representingone tenth on the thousandths grid. She paused again, and corrected herself,“Wait, 100 times 100 is 10,000, right? But this is still one tenth”, againpointing to one column on the thousandths grid. Somewhat rushed in herspeech, Denise said, “So all of these are one tenth right?” as she pointedto each of the tenths, hundredths, and thousandths grids. Uncharacteristicof her typical approach to teaching, Denise did not call on students andthe students were silent throughout this part of the lesson. Denise, seemingflustered, omitted what she had planned for developing ideas of which gridis most appropriate for which numbers and fractional equivalents of thedecimal numbers (as described in her plans above) and quickly began thelast part of her lesson plan.

Denise asked a student to represent 0.75 on a grid. The student coloredseven columns on the tenths grid. Denise questioned further and asked


the student, “Now what are you going to do with the rest?” The studentlooked again and shaded half of another column, and he explained, “five[pointing to the five in the hundredths place] is half of ten”. Next, Denisehad another student show 0.75 on the hundredths grid. This student coloredthe same shape as the previous student but on the hundredths grid, coloringin 75 squares. Denise asked the class, “Do you agree?” Without muchresponse, Denise asked the student to explain her work. Without shadinganything on the thousandths grid, Denise asked, “What do you think it willlook like on the thousandths grid?” Another student responded, “The samething”. Denise asked this student to show this, and asked again if everyoneagreed (again little response from the class). Next, she turned to the ten-thousandths grid, and asked a student (Carl) to show 0.75 on this grid. Carlshaded seven and a half hundredths squares across the top row (showing0.075). Again, Denise asked who agreed. With no response, Denise askedanother student (Mark) to show how he thought this should be represented.Mark correctly colored the same pattern that had been shaded on the othergrids. Denise stated that they were both correct (but did not explain howthese two representations could be considered correct), and began a finaldiscussion reminding students how to pronounce the numbers correctly(e.g., seventy-five hundredths).

From my observations, it was not at all clear that most students under-stood how to use the various grids. While a few students were able toprovide correct responses at the overhead, based on facial expressionsand lack of response from other students (not typical for this class), thestudents working at the overhead seemed to be the only students whowere confident in their understandings. Moreover, all of the students whocontributed volunteered to do so. Maria’s and Carl’s work indicated thatnot all students understood these representations and the values of thedecimal numbers. For this second part of the lesson (starting with repre-senting 0.75), Denise’s teaching style seemed to be more representative ofwhat I had observed of her teaching in that she asked students to showand explain their work, rather than Denise doing most of the talking (asin the first part of the lesson). Yet, unlike her usual practice, Denise didnot delve into deeper explanations for their thinking and solutions, andskimmed over the idea that both Carl’s and Mark’s representations “werecorrect”.

Immediately following the lesson, Denise and I discussed the lesson.As soon as we sat down Denise said, “I wasn’t sure how to use that tenthousandths grid! You can tell! . . . I still don’t know if I was completelycorrect”. I asked her to explain what she was unsure about, and she said,“The representation of the examples”. She went on to say that she became


confused when the two boys represented 0.75 on the ten-thousandths grid.I reminded her, “You said that they were both correct”. She replied, “Theyweren’t!” I went on to explain how they could both be considered correctif the first boy defined the unit one to be a row (what had been one tenth inother representations), but that if we were to assume that the unit consistedof the large square (as was the case, but was never directly stated in thelesson), he was not correct. We continued to discuss this idea of decidingand declaring the unit one as a first step in using grids, and Denise indicatedthat she had remembered this kind of discussion in the methods class.Indeed, it seemed as if Denise’s vague recollection of the methods classdiscussion led her to the idea that they both could be correct, but hermemory and understanding was not sufficient to recall how both studentscould be correct.

Next we discussed her confusion with her statement that 100 times 100is 1,000. She immediately said, “It’s not one thousand; it’s ten thousand!It gets confusing!” After more discussion Denise said, “This makes senseto me now . . . . But you think this is [only] decimals, why should it beso hard! . . . There’s a lot of stuff here, and I didn’t know that until I gotup there I guess”. Denise continued to think about these issues, and in asubsequent lesson on March 15 and 16, revisited the topics in this lesson,incorporating the ideas from our discussion in facilitating her students’understanding of and representations of decimals.

Forms of thinking. This episode illustrated deliberate planning, the diffi-culty of immediate reflection, and short-term reflection. In planning thelesson, Denise had clear goals for students to understand the value ofdecimals (number sense with decimals) and for using grids as a represen-tation to facilitate this learning. She was aware of the possible difficultiesthat students might encounter in using the grids (e.g., seeing that a numberwould cover the same area on any of the grids), and had specific ques-tions planned to draw out connections among the decimal number, thegrid representation, and the fractional equivalent. These plans and ideaswere consistent with recommendations from NCTM (2000) and from hermethods course and text (Van de Walle, 1998).

While teaching the lesson, Denise encountered problems in her ownunderstanding of what the grids represented (i.e., her confusion with thethousandths grid representing 100 times 100) and how to use the gridsto represent decimal values (i.e., her confusion with Carl’s and Mark’sdifferent representations for 0.75). While she identified a problem, shedid not resolve her confusion until after the lesson was completed. Sheadmitted in the post-lesson conference that indeed she did not realize the


limits of her understanding until she was teaching in front of the class.In her confusion, she began a short teacher-led explanation of how torepresent one-tenth on each grid, without following her plans for devel-oping these ideas with the class. In doing so, Denise essentially movedaway from her goals and reform-based approaches of facilitating under-standing, towards more traditional approaches of presenting material.Later in the lesson, Denise’s inadequate understanding of the representa-tion again appeared in trying to sort through Carl’s and Mark’s two ways torepresent 0.75. Because Denise had not developed the idea of identifyingthe unit (either in her mind or in her students’ minds), while she had somesense that both representations could be considered valid (from a vaguerecollection from the methods course), she was unable to make sense ofthis in the act of teaching. So, she avoided a conceptual discussion andmoved on to other parts of the lesson.

These moves were consistent with the findings of Borko et al.’s (1992)case of Ms. Daniels. When Ms. Daniels was confronted with her ownconfusion over understanding division with fractions while teaching, sheattempted to use a representation she had learned in her math methodscourse. Misapplying this representation, she realized that she had madean error, but while teaching, Ms. Daniels did not fully understand theproblem. With no resolution to the problem, Ms. Daniels resorted to aprocedural, rule-based explanation. Laying my framework for deliberatepractice over Borko et al.’s findings, I would interpret this episode asthat, when Ms. Daniels was faced with a gap or weakness in PCK, herimmediate reflection on the problem led her to follow the pull of tradition.

Similarly, Denise, when challenged to analyze and make sense ofher confusions regarding the grid while teaching, evinced inexperi-enced teachers’ difficulties in accessing instructional options to resolve adilemma. Without a repertoire of options to consider during immediatereflection, Denise opted for more familiar traditional methods. Deniseresorted to telling her students, “that they all show one tenth”, and indeedomitted a significant portion of her plans to develop and unpack the ideasbehind how they each show one tenth. That is, when confronting an instruc-tional dilemma, Denise demonstrated immediate reflection and resolvedthe problem by moving toward the tradition of telling and abandoningplans to implement more innovative approaches.

To resolve further this dilemma, Denise later exhibited short-termreflection. Starting in our post-lesson conference and continuing in herplanning for the two lessons, Denise made sense of the representations inher own mind, planned ways to facilitate her students’ understanding, andimplemented these plans. Thus, given time to analyze the problem, Denise


reflected further and returned to her goals for developing the students’understandings.

Use of pedagogical content knowledge. Denise demonstrated PCK ineach of Grossman’s (1990) four areas; however, her underdeveloped PCKcaused the problems identified above and her difficulties with immediatereflection in resolving those problems while teaching. First, in estab-lishing her goals for the lesson, Denise demonstrated that she understoodpurposes and strategies for learning decimals, consistent with reform-based goals (e.g., NCTM, 2000). In planning for the lesson, Denisewas aware of and chose materials effectively (i.e., drawing from Investi-gations and using the decimal grids) for facilitating understanding ofdecimal values, thus demonstrating curricular knowledge and knowledgeof instructional strategies. Moreover, in our pre-lesson conference, Deniseindicated her understanding of students when she explained her concernfor how they might misuse the grids and the importance of connectingdecimals, fractions, and the grids in learning.

Denise’s PCK needed further development in understanding fully whateach grid represented (e.g., her error in confusing the thousandths and theten-thousandths grids), and in understanding the role of identifying the unitas part of understanding the multiple representations. Her limited PCK wasnot apparent to Denise until she began to teach, despite her careful plan-ning. Moreover, these limitations impeded Denise in her understanding ofthe complexity of teaching and learning decimals (c.f., Ball et al., 2001),and correspondingly resulted in a lesson plan that had too many conceptspacked into an introductory lesson, without sufficient time to explore anddevelop each of these issues. In sum, these limitations in Denise’s PCK ledto some confusion and a dilemma during instruction, and upon immediatereflection, she decided to depart from her plans for developing students’understandings of how to use the grid representation in favor of moretraditional approaches of telling what the grid represented.

DISCUSSION AND IMPLICATIONS

The cases of Gerri and Denise illustrated forms of thinking in deliberatepractice and how they used each of Grossman’s (1990) four components ofPCK during deliberate practice. Below, I discuss and suggest implicationsfor the various forms of reflective thinking as they were exhibited, and howthe pre-service teachers’ use of their PCK played a role in this thinking.


Supporting Pre-service Teachers through Collaborative DeliberatePlanning

Using their PCK Gerri and Denise deliberately planned for many ofthe possible problems of teaching and learning. It is important to notethat the examples presented here in which Gerri and Denise demon-strated thoughtful, research-based practice were just a sample of the manyinstances I found. It seemed that the use of research-based resources(e.g., Carpenter et al., 1999; Van de Walle, 1998) in their teacher educa-tion program prepared Gerri and Denise to approach instruction as aproblem solving endeavor, focusing on facilitating students’ understandingand anticipating problems in teaching and learning. However, they stillencountered problems in teaching that most probably would have beenavoided by more experienced teachers. For example, in Denise’s case,while her plans indicated strong PCK for decimals, it was not until sheactually taught the lesson that gaps were revealed in her PCK. Despitethe PCK Gerri and Denise had gained through their professional course-work (as evidenced in their pre- and post-conferencing and planning), theirunderstandings were limited without having used it in practice. This isconsistent with Borko and Putnam’s (1996) argument for inadequate PCKin pre-service teachers. In considering implications from this study, thequestion then became, “Do we just accept these gaps as part of noviceteaching? Or, can teacher educators and field specialists better utilize anddevelop pre-service teachers’ existing PCK and, correspondingly, enhancetheir skills in deliberate planning?”

As I considered Denise’s episode in particular, I realized that if I hadhad the pre-lesson conference on decimals with Denise prior to the day ofthe lesson, I could have better supported her in understanding the complex-ities and issues in teaching this lesson. As it was, while I suggested someideas for her to consider, she was not in the position to act on them just 15minutes ahead of teaching. Mrs. Knight, as an expert mathematics teacher,did serve as this kind of resource at times. Indeed, Denise’s decision touse the decimal grids was motivated by a planning discussion with Mrs.Knight and access to Mrs. Knight’s extensive reform-based resources.However, even this discussion did not include in-depth, lesson-specificPCK that Denise might need to bring to this lesson. Moreover, it is notclear that it is reasonable for mentor teachers to serve in this role. Mentorteachers’ primary focus needs to be on supporting pre-service teachers’ intheir learning to do the daily work of a teacher, and this may mean thatdeveloping pedagogical content knowledge is left in the background.

Conversely, university supervisors could, more aptly, support pre-service teachers’ development of pedagogical content knowledge by usingthe pre-service teachers’ experiences as the context for learning. To


support effectively our pre-service teachers’ knowledge development andcorrespondingly their skills in deliberately planning for instruction, univer-sity supervisors need opportunities to discuss in depth the teaching andlearning issues for specific lessons to pose various problematic scenarios,both after teaching episodes and well in advance of lessons. Throughthese discussions we might be able to provide opportunities for pre-serviceteachers to consider possible mathematical teaching and learning prob-lems that might not occur to pre-service teachers due to their lack ofexperience. The model used for supervision in this study that empha-sized reflective practice in general (Sullivan & Glanz, 2000), using apre- and post- conference on the day of the lesson, did not allow suffi-ciently for subject-area specific support prior to a lesson to build onthe pre-service teachers’ deliberate planning. The pre-service teachers’experiences did serve as rich contexts for reflection and making sense ofmathematics for teaching and learning after the lessons were over, andthese discussions were an important part of the pre-service teachers’ devel-opment. However, it seems that opportunities for teachers’ learning couldbe expanded by conducting in-depth conferences to plan lessons well inadvance of a lesson. The purpose of these conferences could be to examineand explore the mathematics and appropriate pedagogical approachesrelated to a lesson. Correspondingly, in order to enact this recommend-ation, pre-service teachers need subject-area specific supervisors and/orfield specialists that could structure and prompt these conversations.

Using Pedagogical Content Knowledge during ImmediateReflection-in-Action is Difficult

In the instructional episodes, I found substantial evidence for the difficultyof immediate reflection. Both Gerri and Denise stated that they wanted toteach lessons as they had planned them, and did not have the confidence orflexibility to change these plans in the middle of the lesson. Correspond-ingly, they found immediate reflection to be difficult and it rarely occurred.Indeed, I only found one instance of immediate reflection that resulted ina continued effort to develop students understanding (i.e., continue withplans to implement reform-based goals). Instead, for all other observations,when immediate reflection occurred this reflection resulted in a decisionto abandon innovative approaches in favor of more familiar traditionalapproaches (as illustrated in Denise’s case). Delayed reflection as a specificform of reflection-in-action recognized the pre-service teachers’ need fora brief break in activity to provide the opportunity for reflection duringinstruction, and short-term reflection allowed the pre-service teachers toexamine their practice outside of the act of teaching. Gerri’s perspective onher reflective practice illustrated how it was easier for her to find opportuni-


ties for delayed reflection or short-term reflection than it was for immediatereflection:

You almost need a quiet time [to reflect] . . . A big time for me was a half an hour beforelunch, we’d have prep time. That was a big time to think about, “How did the morning go,and what am I going to do in the afternoon? And is there something in the morning that’sgoing to affect what I was planning to do in the afternoon? . . . Then, maybe I had to makea quick shift there. It’s hard to do on the fly because that crunch of time. You don’t getmuch of a chance to say, “Did that work okay?” . . . You do have to have that quiet time.(Interview, April 26)

Gerri’s and Denise’s reflective practices were consistent with Borkoet al.’s (1992) report of the experience of Ms. Daniels and with Ball etal.’s (2001) notions regarding the complexities of analyzing a problem andits representations during instruction and the need for confidence in one’sown mathematical understanding enough to adjust instruction “on the fly”(p. 438).

It is important for teacher educators, pre-service teachers, and their fieldspecialists to consider how pre-service teachers find it difficult to reflect inthe form of immediate reflection but have the capacity to reflect throughdelayed reflection and short-term reflection. While this study examinedonly two cases, it seems that reflecting while in the act of teaching mightbe difficult for many novice teachers, especially given the limits of PCKfor novice teachers (c.f. Borko & Putnam, 1996). Teacher educators, pre-service teachers, and field specialists might benefit from recognizing thatthis form of thinking could be difficult for novice teachers by: devel-oping strategies for fostering this form of thinking through pre-teachingconferences (as described above with regard to deliberate planning); and/orrecognizing that immediate reflection might be beyond the skills of somenovice teachers who are wrestling with heavy cognitive demands duringinitial teaching experiences. In those cases where we recognize that imme-diate reflection might not be possible, we might instead focus on delayedreflection, and help pre-service teachers to structure their instructionalplans so that slight breaks in activity are included to allow time forreflection and adjustments to teaching and learning. Additionally, we canfacilitate short-term reflection through post-lesson conferences and journalwriting, as illustrated in Gerri’s and Denise’s cases.

Encouraging Long-term Reflection for Sustaining Growth

The long-term reflection exhibited by the pre-service teachers seemedto be an important part of their reflective practice for future teaching.Through long-term reflection, the pre-service teachers put together thepieces of individual reflective episodes to realize a pattern that informedtheir practices. These patterns over time may be even more important


in sustaining professional growth than individual incidents of reflection.Certainly, while Gerri learned from the multiples lesson that she should notassume that students’ correct use of terms indicates that they understandunderlying concepts, it was more important that she learned that deeperquestioning and students’ explanations are needed for learning. Gerri’saction research project helped her to make these connections in that herproject compelled her to look at her practice and her students’ learning(for teaching multiplication and division) over time, and also, to considerapproaches from her research-based reading and her university coursesin planning. Thus, action research should be considered as a means ofencouraging pre-service teachers to: experiment with innovative teachingand learning strategies, consistent with those explored in their univer-sity courses; focus on students’ understandings and learning process; gobeyond teaching approaches modeled by their field specialist; and thereby,stimulate using PCK for long-term reflection on action.

In summary, by extending Schön’s (1983, 1987) ideas about reflectivepractice, I presented a framework for deliberate practice that encouragesteachers and teacher educators to focus on preparing for instructionaldilemmas as well as reacting to them through use of PCK. Through aware-ness of this framework and the complexity of thinking involved in teachingas a problem-solving and a problem-anticipating endeavor, pre-serviceteachers can be made aware of their thinking and strive to move awayfrom the technician model of teaching that they might have experiencedas students. More specifically, by thinking about teaching as a deliberatepractice, they might focus better on using PCK to anticipate problems inlesson planning, and not simply deliver a curriculum as it is presentedfrom an external source. Consequently, pre-service teachers may be morepredisposed to enter the field feeling empowered as decision makers, ratherthan feeling obliged to follow external guidelines without critical analysis.

Additionally, teacher educators can help pre-service teachers becomeaware of the complexities of thinking and problem solving in the practiceof teaching. Teacher educators can support pre-service teachers by helpingthem to recognize patterns in their forms of thinking, when and how theydraw upon their pedagogical content knowledge, and the consequencesfor teaching and learning from their thinking and decisions. Indeed, ifDenise were not being observed on the day she used the grid represen-tation, the consequence of reverting to direct teaching my have becomea pattern in her practice. Instead, through support and discussion with ateacher educator, she became aware of the need for more careful planningwhen using representations with students and reform-based pedagogicalapproaches. Teacher educators should focus on more than developingreflective practitioners who react to problematic situations, but on devel-


oping deliberate practitioners who use theory, research, and experience toanticipate problematic situations.

Further research is needed to examine the applicability and useful-ness of this emerging framework for teachers and pre-service teachers. Inparticular, investigations are needed with more experienced teachers thatpotentially have a stronger knowledge base in place. Additionally, researchis needed to determine whether experienced teachers exhibit immediatereflection more as a form of reflection and the corresponding implicationsfor fostering immediate reflection in novice teachers.

NOTE

1 Eicholz, 1991, Addison Wesley Mathematics.

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College of EducationWashington State University Tri-Cities2710 University DriveRichland, WA 99352USAE-mail: [email protected]

MARGARET WALSHAW

PRE-SERVICE MATHEMATICS TEACHING IN THE CONTEXT OFSCHOOLS: AN EXPLORATION INTO THE CONSTITUTION OF

IDENTITY

ABSTRACT. This paper engages with poststructural ideas for a discussion on what itmeans to engage in pedagogical work in the context of elementary/primary school math-ematics classrooms. Central to the analysis are the pre-service student and the part thatthe teaching practicum plays in the ‘making’ of a teacher. Drawing on insights from thework of Foucault on power and subjectivity, instances of teaching knowledge in produc-tion, as interpreted by pre-service teachers, are examined. The view is towards developingtheory from readings of specific regulatory strategies that impact powerfully on pre-serviceteachers’ constructions of themselves as mathematics teachers.

KEY WORDS: Foucauldian analysis, identity, normalization, pre-service teaching, power,surveillance

One of the most immediate goals in teacher education is the develop-ment of professional expertise for effective practice. Teachers’ identitiesare threatened. Today, these have been affected by policy shifts andprofessional development initiatives developed over the past decade (e.g.,Australian Education Council, 1990; Ministry of Education, 1992; NCTM,1991, 2000; OFSTED, 1994). In such identifications issues of classroompractice and affect are linked precisely to the effective teacher. We witnessa blurring of the emotional/logical and a melée of methods, which willprovide access to those identities. In particular, research on mathematicsteachers’ beliefs and/or knowledge (e.g., Cooney, 2001; Sowder, Philipp,Armstrong & Schappelle, 1998) sketches out the rationale for mappingwhat teachers believe and know, often before and after a process ofprofessional development or curriculum change. The body of researchon reflective practice, emphasizes principled thinking, reason and criticaljudgment (e.g., King & Kitchener, 1994; Lerman, 1994). Other researcherswork long-term with individual teachers (Ball & Bass, 2000; Simon, Tzur,Heinz, Kinzel & Smith, 2001). Through their commitment to teacherempowerment, they have given teachers a personal voice and enabled themto confront, analyse, and develop their public practice. These approacheshave combined the personal and the public in imaginative ways.


64 MARGARET WALSHAW

Those accounts that attempt to map out the process of learning to teachoften describe the process “as a personal journey” (Cooney, 2001, p. 2).However, the description of that journey often subscribes to a theoreticalposition (e.g., Cooney, 1985) that is unable to account for forms of regu-lation and ‘political tactics’ which govern the individual teacher withinthe university and school contexts. Lerman (2001) has argued that “theconcepts, beliefs, and actions in one context and those in another . . .

are qualitatively different by virtue of those contexts” (p. 36). Lerman’sargument is furthered in studies in which content knowledge is givenless attention in pre-service teachers’ planning and teaching than are theirobservations of supervising teachers (Calderhead, cited in Poulson, 2001,p. 46). Poulson summarises Furlong and Maynard’s (1995) findings: “Evenwhen students had sound knowledge they did not draw upon it in theirplanning and teaching, but preferred to copy and adapt ideas suggestedby their supervising teacher/mentor” (p. 46). Lave’s work, like that of theactivity theorists (e.g., Davydov & Radzikhovskii, 1985), might be drawnupon to help us understand pre-service teachers’ work in schools. In aseries of well-known classic studies (Lave, 1988; Lave & Wenger, 1991),Lave developed a model of learning as participation in a community ofpractice. In brief, the idea is that “[d]eveloping an identity as a memberof a community and becoming knowledgeably skilful are part of the sameprocess, with the former motivating, shaping and giving meanings to thelatter, which is subsumed” (Lave, 1988, p. 65). Adler (1998) is one of afew researchers who has seized upon the pedagogic potential of Lave’stheories to “capture the complexity and tensions in the teaching-learningprocess” (1998, p. 163). Ensor (2001), on the other hand, is drawn to thenotion of re-contextualizing (Bernstein, 1996; Dowling, 1998) to producean analysis and a theoretical account of the transformation of discoursefrom pre-service to beginning secondary teaching.

I come to the issue of engaging pre-service teachers’ identities some-what differently. Goldsmith and Schifter (1997) press for forms of signi-fication that offer teachers new narratives for conceptualizing their work.They ask that researchers give due consideration to “qualitative reorgan-izations of understanding” (p. 21), and, in responding to this request,my point of departure arises from the problem of taking teacher’s iden-tity for granted. In my mind, it is not enough to approach the issueof identity in some a priori way, as an outcome of belief change, ameasure of content knowledge acquisition, or an aftermath of being therein the classroom community of practice (Britzman, 1991). Such taken-as-successful approaches (see Grant, Hiebert & Wearne, 1998) vary widely inappeal, in remedy, and in theoretical stance, but they all share some under-

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lying thematic celebration of first-hand experience. Centering experience,in this way, as the organizer of perception, however inadvertently, scriptsidentity as synonymous with the teacher’s role and function and tends tooverlook what I see as the more important aspects of learning to teach. Ibelieve there is more at stake for mathematics teaching than assessmentsof teachers’ knowledge, beliefs and learner-empowering efforts.

My starting point is with de-centered identities and with politicallydeliberate things – like the expectation that individual teachers are mastersof their thoughts and actions. Focusing on that period of time when pre-service teachers work in schools, the suggestion is that the classroomcommunity of practice is anything but neutral, however anaesthetized itmay appear to be. Once we theorize that the school in which the pre-serviceteacher is placed is constituted of, and by, material and embodied relationsof discourse and practice, we can begin to see that identity might not beas much assigned as consented to through constant social negotiation. Weare drawn into a context of inevitable power relations. What becomes para-mount is our attending to the interplay between discursive regulation andthe subjective investment in and reworking of, and even resistance to, thosenormative practices. Learning to teach could then be viewed as inextricablybound up with a critique of relations of power surrounding the pre-serviceteacher’s participation and negotiation in a community of practice. Thismove away from the autonomous individual is, in my mind at least, aproductive means of talking about and investigating the constitution of pre-service teachers’ identity in school situations since, first, it opens up otherpossibilities for teacher education. Second, it offers a more complex andlayered notion of the teacher. Third, such a move allows us to engage thehistorically specific relationship between pre-service pedagogical practiceand forms of social control and possibility.

In the first section, I engage with poststructural theory in order toconstruct a conceptual model for the identity formation of pre-servicemathematics teachers. I introduce elements from the exacting scholarshipof Michel Foucault to explore their significance for mathematics teachereducation. This approach is not new to teacher education (see Hamilton& McWilliam, 2001), and this is not the first time connections betweenFoucault’s work and mathematics teacher education have been made (e.g.,Walkerdine, 1988, 1994; Klein, 2002; and Brown, 2001). Foucault’s workis part of a wider intervention initiated from poststructuralists and fromfeminists, concerning debates in which issues of agency, reflexivity andidentity formation are paramount. Foucault (1984) asks explicitly that wegive up thinking about the self as a centre of coherence. He asks insteadthat we explore identity as a ‘technology of the self’, directing our attention

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to the political and institutional processes central to its constitution. Forhim, identity is historical and situationally produced; it exceeds singulardefinition precisely because it is always contingent and precarious. Butler(1997) means just this when she claims that there is no ‘first’ momentestablishing pure identity that can be rationally unpacked.

Thinking about the constitution of teacher identification in this wayshifts our preoccupations as mathematics educators with teaching otherpeople to teach, towards a consideration of the conditions and circum-stances for the making of a teacher. At the same time, it demands anacknowledgment of the tentative and shifting balance between pedago-gical theory and classroom experience. In the second section, I give abrief description of the methods I used to gather and analyze the researchdata. Instead of asking questions about the defining, natural and universalstate of pre-service teachers, I ask one general question concerning theengagement of student teachers with classroom practice: What does thedevelopment of teaching identity look like? To answer that question Ipose another specific problem: How are pre-service mathematics teachersworking in schools constituted as teachers?

In the third section, I explore the ‘making’ of pre-service teachers in thecontext of work in New Zealand primary schools. It is acknowledged thatthis exploration takes place in a very specific location, but the questionsthat it raises will have relevance for others. I draw upon the empirical datafrom pre-service teachers’ observations and documentation of ordinarydaily classroom practice to reveal the discursive practices at play in theprocess of becoming a mathematics teacher. In this account, I attempt toexplain how certain versions of ‘good’ teaching, and not others, cometo be intelligible to pre-service teachers. From the analysis, I argue thatassessing how identity is shaped and lived in the mathematics classroomcan provide some insight into the politics surrounding pre-service teachers’work in contemporary mathematics classrooms.

A FRAMEWORK FOR THE FORMATION OF TEACHERIDENTITY

What seems necessary is a language of justification for pedagogical iden-tification. Foucault’s language is particularly helpful because it providesus with a theoretical expression of the politics of identity. Foucault doesnot engage the common quest of uncovering one’s true identity or self.His notion of identity is far more fragile than the cohesive and sover-eign individual, and places into question the very idea of authenticity. InDiscipline and Punish (1977) he develops the theme of the governmentality


of individuals in which processes of identification are explored as theyare lived by individuals in relation to both structural processes and livedexperiences. To capture this more nuanced understanding of an on-goingchanging relationship between the individual and that to which he or sheidentifies, the word ‘identity’ is often replaced by the term ‘subjectivity’.

For Foucault, politics enter into any discussion of subjectivity. Socialinstitutions such as schools have particular modes of operating, partic-ular forms of knowledge, and particular positionings. Particularities thatrelate to the school, the classroom, the associate/supervising teacher, theuniversity course, previous classroom experiences, personal biography,and so forth, all have their place in constituting the pre-service teacher as‘teacher’. Foucault (1977) is not concerned with the essentially negativeaspects of power as much as with its enabling and constitutive aspects.Drawing upon his reformulation of power, we can argue that practices of‘normalization’ and ‘surveillance’ which construct pedagogical relationsare productive and cannot be dissociated from knowledge. Foucault’s argu-ment is that power and knowledge directly imply, but are not coextensivewith, one another: that is, that there can be no power relation without thecorrelative constitution of knowledge, nor any knowledge that does notsimultaneously, presuppose and constitute a power relation.

In Foucault’s understanding, systems of power both produce and sustainthe meanings that people make of themselves and it is through thesesystems that identities and subjectivities are strategically fashioned andcontested in the dynamics of everyday life. Thus, integral to the construc-tion of subjectivity and identity formation, is an a priori set of rules offormation governing beliefs and practices in such a way as to produce acertain network of material and embodied relations: they “do not merelyreflect or represent social entities and relations: they actively constitutethem” (Walshaw, 2001, pp. 480–481). For the pre-service teacher, thismeans that he or she is the production of the practices through which he orshe becomes subjected. Britzman (1991) and Davies (2000) speak of thisas the discursive constitution of pedagogical subjectivity.

This idea is important because it suggests practices of discipliningand regulation that are, simultaneously, practices for the formation ofa classroom teacher. He or she needs interaction with others and theirdiscourse in order to form a self-concept. Indeed, the very possibility offorming and articulating concepts of one’s self (e.g., as a teacher in theclassroom) is ultimately dependent on language and the meanings of otherpeople (Blake, Smeyers, Smith & Standish, 1998). They define what isnormal, and state that what is not normal creates the need for normal-ization, through surveillance procedures that are made both explicit and

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implicit. Institutionalized practices exercise control over the meaning ofteaching by normalizing and providing surveillance practices to keep suchmeanings in check. Full engagement by a pre-service teacher in thosepractices would reveal that he or she has learned to perform and enactnot only the genres that constitute the knowledge, modes of operating andtheories and practices of the classroom, but also the particular position-ings and embodied practices that construct mathematics in the associate’sclassroom. Individual pre-service teachers become ‘successful’ when theirengagement is self-actualized and no longer requires regulation.

In this study, I explore the making of the teacher through processesof normalization and surveillance. It is through an examination of thoseprocesses that we begin to see the political and strategic nature of modesof operating, knowledge, and positionings that are ordinarily considered tobe either relatively independent of power, or linked only in a vague or inad-equate way to organizational or institutional power. I offer a Foucauldianexplanation through an examination of the “simultaneous articulations of adispersed and localized shifting nexus of social power” (Haywood & Macan Ghaill, 1997, p. 268), surrounding the world of mathematics teachingin schools.

METHOD

Collection of Data

I invited all second-year internal pre-service teachers to participate in thestudy. Students in this course are undergraduates, ranging in age from 19to 53 and most are women. Only a small proportion of them is over the ageof 40 and these are, in the main, change-of-career students.

All 72 students who were in attendance on the day when the data werecollected chose to participate. I asked them to respond to a questionnaire(see Appendix) about their recent teaching practice experience. During thethree weeks in which they were out in schools in the third of four schoolterms, they worked closely with one classroom teacher (the associate),endeavoring to build a professional partnership with this teacher withinthe supportive environment formed by links with the university and theschool. A few days previously, at the university, an opportunity had beenprovided for pre-service teachers to discuss their teaching practice in classdiscussion. That discussion had created an opportunity for the individualpre-service teacher to name similar and different experiences in relation toand in reaction to the class.

The questionnaire developed from my involvement with the courseduring the two previous years. In those previous class discussions, as


well as in my conversations with individual pre-service teachers, apparenttensions had surfaced when pre-service teachers spoke about their prac-ticum experience. I developed the questionnaire to allow the pre-serviceteachers currently enrolled in the course to give expression to their exper-iences. In addition, I asked for more quantitative information about theschool and the class. The intention was that each pre-service teacher mightbe given the opportunity to construct a sense of how teaching identity iscreated in relation to people and practices: how it is socially structured andhistorically inflected. More importantly, by making those processes visiblethrough the questionnaire, it was hoped that pre-service teachers might seehow similar processes could be acted upon in future practicums. BecauseI wanted to understand how pre-service teachers constituted themselvesas teachers in mathematics and how they constituted themselves as moralsubjects responsible for their own actions, the observations and reports ofassociate teachers were extraneous to the analysis. I use quotes from whatpre-service teachers write to provide “a profuse and diverse specificity . . .

where voices are juxtaposed and counterposed so as to generate somethingbeyond themselves” (Lather, 1991, p. 134). It was for epistemologicalreasons, to do with the objectives of the study, that a relatively traditionalmethod of data collection was used. Without question, the questionnairecould only provide a partial and incomplete view of the development ofteaching identity. However, any research that examines the ways in whichpre-service teachers construct truths about themselves within the schoolcontext will be limited in insight. This is the case even for that researchwhich employs extended and in-depth interviewing. There are many well-articulated reasons for these limitations, not the least of which, in theFoucauldian explanation, is the understanding that identity is constantlyon the move. In any event, the questionnaire provided information that ledto new knowledge about identification held by pre-service teachers.

Analysis of Data

The method used to analyze the data examines written text, systematic-ally, as moments of inter-subjectivity in very localized sites. This methodbegins in Foucault’s theory of language and social power and takes seri-ously the discursive constitution of subjectivity. The method is not to beconfused with discourse analytic methods in mathematics education thatdeveloped from the application of interactional sociolinguistics, ethno-graphy of communication and ethnomethodology to the study of classroomtalk (e.g., Cobb & Bauersfeld, 1995; Simon & Blume, 1996; Yackel,2001). These approaches, that ordinarily focus on the study of languageuse per se, tend to analyze language as a way of explaining the psycholog-ical intents, motivations, skills, and competences of individuals. In these

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approaches, speech acts are taken as ‘transparent’. In contrast, I am lookingat the analytically prior question of how pre-service teachers’ accountsof their teaching experiences are crafted. I am interested in pursuinghow discourse constitutes versions of the social and natural worlds and,specifically, how discourse creates different material effects with regard tothe constitution of pre-service teachers’ identification as teachers.

The analysis of the discursive production of teaching practice willcentre on three Foucauldian themes. I look first at how the teaching prac-ticum takes place in a transitory space. Then I explore how practices ofnormalization intrude into the ways in the ‘making’ of the pre-serviceteacher. These practices are investigated through two distinct forms: firstI explore how organizational patterns in schools impose rhythms andtemporal conditions that induce teachers into a particular programmableorder. I look then at how pre-service teachers draw upon the pedagogiesnormalized in the work during the course to frame their view of the routinepractices in the classroom. In the final section of the analysis, I investigatehow the practicum functions as part of the technology of surveillance anddiscipline. Each of the sections is grounded in words generated through theinterview responses from pre-service teachers with varying backgrounds(mathematical and otherwise) in the mathematics teaching course. Inputting their quotations together, I try to make underlying structures ofpower visible.

THE DISCURSIVE PRODUCTION OF TEACHING PRACTICE

Transitory Positionings

In poststructuralist thought, identity is transitory and mobile. At the begin-ning of the practicum, pre-service teachers position themselves in relationto the available discourses. During the practicum, other forms of discoursecome to play that in some ways confirm and in other ways contradict thoseearlier ones. I explore ongoing transitional positions by posing the questionas to how pre-service teachers position themselves at the beginning andat the end of the practicum. I use words taken from the questionnaire inresponse to this question:

Beginning: I was looking forward to teaching. I felt confident and enthusiastic, thoughsometimes I am unsure about how to explain some maths ideas.

End: I was more confident and able to see students developing understanding.

Beginning: Prior to posting I felt inadequate and concerned in algebra and problemsolving.

End: I realise now that teaching any curriculum required me to learn the subjectfirst. Teaching maths made me learn maths.


Beginning: I felt a bit uncomfortable – wanted more in-depth knowledge of how to teachit. How to get it across. How to start and introduce a unit.

End: More confident. Timing and pacing is now more sorted out. I learned whichresources, activities, and strategies worked for students.

Beginning: I was lacking confidence.End: I knew the material well and I was comfortable with maths. However I still

lacked management strategies.

Beginning: I felt that I was familiar with the resources and activities but I had no ideahow to run the maths lessons.

End: I was still very in the dark with what she did with each group individually.That’s because I was always helping the very new group.

Beginning: Scared of teaching maths.End: Feeling a lot more relaxed. Children really seemed to enjoy the activities,

especially using money.

Beginning: I was a bit shaky and nervous about teaching maths as it is not my favouritesubject. I feel it is necessary to give children a sound base of understanding inmathematics and mathematics skills so I was therefore skeptical of my ownability.

End: I appreciated being able to teach at different levels and gain strategies forteaching each level. I feel a lot more confident in teaching maths. HoweverI am going to further my maths education by taking maths papers next year[electives].

The identity of pre-service teachers during the practicum is mapped ontoa complex grid of formal and informal educational discourse and practice.Steeped initially in the as-yet-still-developing self-constructions producedto some extent by the university course and by their own mathematicalbackground, many are aware that they enter a new range of discoursesand identities which will constitute them as a teacher of mathematics.On the threshold of something new, slightly over fifty percent of the pre-service teachers worried about the development of their teaching voice andhow they might constitute the teaching of mathematics within their ownsubjective experience of mathematics. Britzman (1991) has reported onthe difficulties of attempting the delicate work of educating others whilestill being educated oneself.

Pre-service teachers in this study identify and name categories ofunknown knowledge: the teacher’s mathematical explanations; algebra andproblem solving; starting and introducing a unit; how to get it across;how to teach it; how to run the mathematics lessons; general mathema-tical knowledge. By naming these categories, pre-service teachers haveestablished their own personal classifying grid for the development of ateaching identification for mathematics. At the end of the practicum, pre-service teachers register a growing confidence. The categories they identify

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become less general and more specifically tied to the construction of actualpedagogical practice within the classroom. Variously named as timing,pacing, resources, activities, workable strategies, management strategiesor teaching to groups or teaching different levels or intention to enrol in anelective mathematics paper, they all become the parameters of what willbe identified as successful pedagogical work. Yet such identifications donot have the full measure of pre-service teachers’ subjectivity preciselybecause subjectivity is an interactive weaving together of many complexselves in relation to the available discourse and to the complex selves ofothers. The associate and the students in the classroom are implicated inthe identification to which pre-service teachers assign themselves, to theextent that those identities are constantly refashioned, as investments arelived and rearranged within the classroom. Because pre-service teachersare deeply invested in the essentially human subject, such theorizing maynot make much sense to them. They see themselves as individuals witha real and essential core, whose outer layers are a series of roles andfunctions that can be cast off to reveal the true and real self. Arguably,they would consider that a teaching identity comes about not by theories,but by actualized practice, and, like many pre-service teachers, they wouldview the practicum as an opportunity in which the true teaching self mightmanifest itself. Given that, it is difficult to say whether they would holdonto an understanding of themselves as teachers caught up in the avail-able discourses and in the working of power. However, in the Foucauldiananalysis, teaching identities are continuously being transformed, displaced,and extended and it is that understanding of identity formation that is, inthe main, unaccounted for in the normative story of mathematics teacheridentity.

Normalization Practices

Pre-service teachers are not only redefining their teaching identities in rela-tion to the available discourse in the classroom and to the complex selves ofothers, they are also learning what is defined as ‘normal’ practice throughthe school’s organizational procedures. Arguably, institutional practicesare measures and techniques that produce identities implicitly rather thanby repressive force, yet the rationalities underpinning their specific waysof doing and knowing have the same purpose of securing the conformingindividual. The school, construed as a regime of power, constructs specificregulatory practices for the normalization of and ultimately the produc-tion of the self-governing individual teacher. To address the question ofhow principles of organization, as developed by schools, are implicated inthe production of governable individual pre-service teaching, I asked pre-


service teachers to state the ‘Usual time of day for mathematics lessons’;‘Usual duration of lessons’; and ‘Total number of mathematics lessonsoccurring during teaching practice’.

The student teachers in this study stated that mathematics was routinelytaught in the morning (89%). For fifty-six percent, mathematics was sched-uled for early morning, and for a third, mathematics took place betweenthe morning break and lunchtime. Timetabling arrangements that differ-entiate curriculum areas impose temporal conditions through which thepre-service teacher is constructed to perform a designated teaching taskand discouraged from teaching any other. The widespread practice ofmorning mathematics, taken together with commonsense understandingsof positive effects of morning learning, would suggest that the socialsignificance of mathematics is not lost on schools.

A third of the pre-service teachers saw mathematics taught on a dailybasis, and only a relatively small number (15%) reported that mathematicstook place less than four times per week. Students noted that the sched-uled length of time was consistent from one day to the next. However,the expected duration varied considerably from one classroom to another.Whilst the median time spent on mathematics during the school weekwas three hours twelve minutes, one student came to expect five hoursregularly each week. Such practices of school administration provide thepre-service teacher with a particular cyclic order for the teaching of math-ematics. Because these institutional practices fix limits, controlling the‘time’ around which pedagogical reality might take place, they contributeto what Foucault (1977) calls ‘normalization’.

Within the classroom, the associate teacher invests in particular dis-cursive codes of mathematics pedagogy. Those codes of practice, in turn,place particular processes and practices in the foreground: they shapeplanning as well as the enactment of those plans in the classroom. Course-work at the university also imposes specific ways of doing. I askedpre-service teachers to describe a typical lesson in order that I mightunravel the part that the associate’s ways of doing things played in pre-service classroom work. Here I wanted to make explicit how the work ofclassroom teachers is often assessed against the normative understandingsof teaching provided through course work. In doing this, I wanted to under-stand the ways in which the course work functions as part of the technologyof normalization.

As in many educational practices, the typical lesson structures pedago-gical arrangements for school work, establishing a set of institutional andsocial relations for the teacher and learner in the classroom. Accordingto pre-service teachers’ observations, in most classrooms the teacher first

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maintains prior knowledge (e.g., “Quick 10”; “problem solving game”;“Strike III”; “Around the World”; “basic facts drill”; “times tables”; “warmup game”; “10 fast facts”; “recapping”; “maintenance”; “10 mental”;“10 teasers”; “family of facts game”; “hundreds board”). The teacherthen introduces new concepts for the day, sometimes making links withprior knowledge, and provides explanations (e.g., “teacher was explainingdifferences between parallel and perpendicular. The class went outside tofind these two on the concrete”).

The teacher models (e.g., “teacher guides one group at a time”;“teacher gives examples and reinforces ideas”; “models on the board”;“teacher leads class discussion”; “teacher was modeling with equip-ment and encouraging children”; “teacher modeled an example from thebook”). Questions are posed for the children and checks made on theirunderstanding (e.g., “teacher was questioning and explaining and demon-strating”; “asking questions about what they understood or saw beingdemonstrated”; “observing and assisting when needed”; “roving class andhelping”; “talking to the children”; “scaffolding”; “whole class discussionand questions generated”).

It is the teacher who supplies work and activities to enable practice ofmathematical skills and ideas (e.g., “children do activities”; “the 2 groupsnot working with the teacher work from maths box”; “teacher set task frombooks or task sheets”; “dispatches children to activity boxes”; “handed outworksheets”; “pupils’ activities including use of equipment”; “then theystarted to work through the pages”; “gave children a chance to manipulateequipment and work out problems for themselves”).

Finally, the class reflects on the work (e.g., “going over everything”;“question time”; “mark work and stick worksheet into maths book”;“review of lesson”; “wrap up and clean up”). In this logic the teachermoves reflexively from talk, to writing on the board, to observing, totalk and questioning, all the while grounding understanding through theprocess of children’s activity and written work.

However, each classroom produces its own truths about teaching prac-tice and what is taken as ‘true’ in one classroom is not to be consideredas universal nor indeed even necessary in others. Teaching practice in anyone classroom becomes intelligible through its reliance on certain tech-niques that are accepted, sanctioned, and made to function as true. Power,knowledge and truth become coordinates that constitute relationships inthe classroom.

Each society has its regime of truth, its ‘general politics’ of truth: that is, the types ofdiscourse which it accepts and makes function as true; the mechanisms and instances whichenable one to distinguish true and false statements, the means by which each is sanctioned;the technique and procedures accorded value in the acquisition of truth; the status of thosewho are charged with saying what counts as true. (Foucault, 1984, p. 73)


Each classroom has its particular regime of truth that legitimizes and sanc-tions a discursive space for certain practices and social arrangements.Pre-service teachers observed, in most classes, teacher talk and expos-ition, and, in most classes, children engaged in whole class discussionand debate and worked with hands-on equipment. Children often workedon worksheets. Pre-service teachers estimated that group and cooperativeactivities occupied half of the class time. A quarter of the pre-serviceteachers observed peer assessment and a third of the children marked andcorrected their own work. Such theoretical decisions about learning haveimportant implications for the ways in which pedagogical relations canbe conceptualized and enacted. In creating particular modes of activity,ways of being and interpersonal relationships, decision-making makespossible both what can be said and what can be done within the mathe-matics classroom. Knowledge, then, including practitioners’ knowledge,is implicated not only in the practices of administration and normalization,but also in the production of forms of sociality.

In order to explore the ‘take up’ by pre-service teachers of theseobserved practices I asked them to comment on the way their expectationswere met regarding the way in which mathematics was taught. Eighteenpercent chose not to answer this question. Forty-seven percent claimedthat their expectations were not met. Some of those responses follow:

Worksheets weren’t great all the time. I expected to see more hands-on work as well.No use of equipment.Disappointed that only maths text books and worksheets were used.I felt that by not having group work that some children were slipping through the gaps.I didn’t like children working through a textbook – this was not my idea of teaching.I think the Associate sees herself as a facilitator to pass out worksheets.I almost felt as if I knew more about teaching maths and portraying it effectively.

However, in other classrooms, thirty-five percent of the pre-serviceteachers stated that the way their mathematics experiences in the schoolhad met their expectations:

HANDS ON! Children did enjoy the practical activities.Group work went well as children are closer in ability.[I was interested to see] that concepts were put into real, relevant contexts and that childrenwere able to experience these.The teacher integrated maths into the morning roll call, as children counted how manychildren were at school, how many boys/girls, the difference between number of boys/girls,etc.Teacher always asked ‘how did you work that out?’ and got children to explain theirworking out.Math was very much made relevant and hands on for the children who experienced a lot ofdifferent activities, e.g., popcorn (mass/weight), cooking recipes, different food containers.Of interest was the way many children in my class supported each other in their work, orwere willing to tutor each other.

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Mathematics teacher education requires an object of study and placesthat object in the foreground through particular knowledge and particularpedagogic modes of operating. Through the knowledge and modes of oper-ating that it advocates and promotes, the course establishes a benchmarkfor what will count as ‘doing mathematics’. Through explicit engagementswith the official curriculum statement and its theoretical representationsof development, cognition, pedagogy, assessment, and the learner, pre-service teachers had learned what counts as evidence and the kinds ofquestions central to the field. They knew what particular pedagogic modesare legitimated and the types of classroom arrangements deemed central toknowledge facilitation.

Ensor (2001) refers to the symbolic and material resources mappedout in the university course as a “privileged teaching repertoire” (p. 299).In this study, the teaching repertoire privileged in course work promotesthe use of apparatus and technology, recognizes difference, and validatesproblem solving, group activities, integrated learning, and collaboration.The teacher’s role is to create a supportive learning environment, facilit-ating and empowering rather than posturing as the authoritative validatorof thinking. Thus the teacher is expected to “provide the setting, posethe challenges, and offer the support that will encourage mathematicalconstruction” (Davis, Maher & Noddings, 1990, p. 3), knowing when tointervene and when not to interfere. Whole class discussion and smallgroup collaborations are encouraged (Wood, Cobb & Yackel, 1991) inwhich learners’ explanations and justifications are given opportunity forexpression.

By producing the terms of school mathematics and thus the para-meters of school mathematical practice – the possibility and effectivenessof classroom teaching – the course had powerfully created an identifi-cation for the mathematics teacher. Pre-service teachers read classroompractice through the terms and parameters made available by the course.Often operating below conscious awareness, those terms establish norm-ative judgements about what can pass as teaching mathematics. As Bordo,cited in Lather (1991) says: “We always ‘see’ from points of view thatare invested with our social, political and personal interests” (p. 139). Increating knowledge and operating modes for teachers in school mathe-matics, the course works as a powerful cultural institution, positioning,defining, enabling and regulating prospective teachers. In Foucauldianunderstanding, by normalizing particular pedagogical practices, the coursewas part of the technology of normalization.

[power] . . . applies itself to immediate everyday life which categorizes the individual,marks him by his own individuality, attaches him to his own identity, imposes a law oftruth on him, which he must recognize and which others have to recognize in him. It is a


form of power which makes individuals subjects. (Foucault, in Dreyfus & Rabinow, 1982,p. 212)

Technologies of Surveillance and Discipline

Like practitioners in other fields of professional practice, prospectiveteachers are confronted with learning the discursive codes of practice. Asthey move from the university course into the school, they enter a differentnetwork of political and social discursive practices. The identity posi-tions and politics that such discourse offers provide pre-service teacherswith access to a differential engagement and positioning in relation to theregime of ‘knowledgeable’ practice operating in schools. Yet neither thesediscursive codes, based on theoretical decisions, nor how those codes aretaken up, are always made explicit to the novice: “Maths just seems tohappen in this classroom. It just arrives along like all the other curriculumareas” [respondent]. To the associate teacher, however, teaching constitutesa closely scripted strategy of how teacher’s work is to be enacted in theclassroom.

Lacking the full credentials to live in the world of teaching, pre-serviceteachers try to carve out a teaching voice in a setting already createdfrom others’ ideas and intentions. They work hard at embodying thosepractices that will elevate their subordinate and less influential position.Gaining better access to this ‘knowledgeable’ practice demands atten-tion to those strategic practices and orientations in schools which, takentogether, signify the subject position of the teacher. But attention to detailis not enough: that attention must be monitored, and the pre-serviceteachers’ practice placed under the panoptical gaze (Foucault, 1977);assessed against the associate’s standards. It is those very practices ofsurveillance that provide continuities within the pedagogical site. In thissection, I investigate how particular surveillance procedures, exercised inrelation to the associate’s classroom practices, regulate and sanction thework of the pre-service teacher at the classroom level.

My associate was very well organized and supportive. She shared all her plans andresources with me. She provided quality feedback with positive ideas for me to improveon.She [the associate] gave me a lot of freedom to use my ideas. She supported me and askedif I needed anything and shared resources.Full of ideas, very supportive of new approaches. Happy to share information. Associatehappy to learn herself.

Not only are pre-service teachers working with the technicalities ofmathematics practice they are also, among other things, exploring theposition of their mathematical teaching in someone else’s classroom. Inthe associate/pre-service teacher relation, the pre-service teacher is one of

78 MARGARET WALSHAW

the primary subjects of disciplinary institutional power, the most pervasivedisciplinary practice being, to borrow Foucault’s term, ‘the gaze’. The‘gaze’ is delicate and seemingly intangible, yet its networks can determinethe very texture of teaching and its possibilities. For these pre-serviceteachers, subtleties within the networks of power were shaping a love ofand passion for teaching mathematics. Such alliances, however, were notalways apparent:

[At the end] I was disillusioned at the lack of encouragement I received from my A. T.[Associate Teacher]. Nice enough person but I think that I made her feel I was taking herclass away as they were so responsive to all the new ideas that I brought into the classroom.I could have used some help in developing new ways for the children to think and try thingsbut the A. T. had tried and true methods of working and that was the way it was.After slowing down I got the hang of taking a maths lesson. I tried doing more excitingactivities with the class which they enjoyed, but I found after a couple of days it was bestfor them to go back to the structured routine of book work . . . The lessons should not bestructured so much so that children can’t handle change.I was enthusiastic and ready to put ideas to practice about how to be an effective teacher,but the topic based maths programme didn’t allow for it.I wanted to introduce new ideas but did not have enough confidence. I just followed myAssociate’s plans. I felt I could not try new things as my Associate was set in the waythings were done.At first I was very enthusiastic and full of ideas but found that due to the teaching styleof my associate it was difficult to implement my plans. [At the end] I had adjusted mypersonal style to fit the class culture. It is difficult to force your way. You really just haveto fit the class as it already is.I was forced to follow her methods of teaching in maths as that is what she had plannedand wanted maintained. I am confident in maths but was given little opportunity to expressmy confidence. Could not go outside the square.

Like the pre-service teachers in Britzman’s study, the transfer from theuniversity course into the school brought differential institutional practicessharply into focus. For them this new space was fraught with ambiguousand sometimes painful negotiations to produce individual subjectivity. Thepre-service teacher who invests in the discursive practices of the associatesignals an engagement with the technologies and practices through whichmathematics teaching is managed in a particular classroom. It is an engage-ment made with a glance towards the pre-service teacher’s desire for thepedagogical – a glance oriented prospectively to continued and futureplacement within this classroom. Pre-service teaching, then, depends asmuch on embodied relations of power between people in the practicumas it does on choosing which material resources or any underpinningeducational philosophy.

Teaching practice is a strategic and interested activity. Preciselybecause the pedagogical relation between pre-service teacher/associate isfused with networks of power, it is impossible for pre-service teachers


to practise disinterestedly, since their practice in schools always worksthrough vested interests, both their own and others’ rhetoric of opinionsand arguments. It is not the intention here to suggest any straightforwardcausal link between the determining structures (associate’s practice) andthe action of individuals (pre-service teaching practice). Foucault criticizesthis tradition on two counts: the mode of the essential human subject whichit employs, and the practices which it projects and regulates about theindividual. Although we cannot claim linearity between associate practiceand pre-service teaching development, this is not to suggest that there iscomplete lack of unity between the levels of action. The pre-service teacherengages with, negotiates, and contests the cultural logics of the associate’spractice. Whilst classic studies on mathematics teacher education mightbe able to talk about pre-service teaching in social and cultural terms, theyare unable to take into account the way in which the beginning teacher isinscribed within and, at the same time, refashions classroom existence inrelation to others.

Unanalyzed elements exist in many authoritative arguments about the“gap between the ‘espoused’ and ‘enacted’ ” (Sfard & Kieran, 2001,p. 186). In the wider education literature, Zeichner and Tabachnik (1981)tried to explain the lack of perfect fit between practices advocated bycourse work and actual teaching practice as a problem of the school setting.Others have set some score by the idea of pre-service teachers’ variedengagements with course work (e.g., Lacey, 1977). These explanationsderive from and sustain conceptions of instrumental rationality: that peopleand circumstances can be matched. Within this conventional paradigm,there is no place to consider the pre-service teacher in any terms otherthan in a model of normality/pathology. For Foucault, a perfect fit betweenself and society, and between social relations and psychic reality, is animpossibility. Perfect fits are impossible between the associate and thenovice teacher, between the ideal or imagined teacher and the real teacher,and between course work and classroom practice. Practices always placeboth parties within circulating and competing relations of knowledge,desire, and power.

CONCLUSION

This article has explored the question of what it means to engage inpedagogical work in primary schools. Taking a different approach to theproject of mapping beliefs and knowledge, it drew attention to a set ofissues that commonly have remained outside the scope of analyses ofmathematics teacher education. Using insights from the work of Foucault,

80 MARGARET WALSHAW

it explored aspects of pre-service teaching in schools. In that exploration,the focus has been on the constitution of teaching identity and its compli-city within structures of power, privilege and subordination. The objectivewas to show whose experiences and what forms of knowledge count or arewithheld during the process of establishing pedagogic authority.

Looking closely into concrete situations of teaching entailed attendingto, not only the broad context of teaching experiences, but also the partcourse work plays in informing, constraining and implicating practicalwork in schools. Through pre-service teachers’ accounts of the ways theyframe their pedagogical choices, the way they view their associate’s prac-tice, and the way they perceive they are viewed by the associate, an insightis offered into multiple enactments of subjectivity. The accounts reveal thatteaching mathematics in primary school involves processes of normaliz-ation and surveillance, in which the spoken and the unspoken becomesintricately linked both to the production of teaching knowledge and to thesubjectivity of teachers. Arguably, in the spaces shared by the pre-serviceand associate teachers, issues of power and privilege feature prominently,contributing in no small way to the shaping of teacher identification.

Explanations that refocus from cognitive schema to power haveprofound implications for mathematics education. In this article, I haveoffered a theoretical and empirical direction that begins from a recogni-tion of the politics of knowledge and their reciprocally constituted effectson subjectivity. Such a direction forces a rethinking of the notion ofthe pre-service teacher who has teaching experience towards conceptu-alizing the pre-service teacher as constituted through experience. In thistheorizing, pre-service teacher identity is fractured and fragmented, andthe classroom is a place of negotiation over the real and its meanings.The concept of teacher identity, then, is best thought of as complex andmultiple, developed in response to other identities that are sometimes heldin opposition. Teaching experience then becomes much more than an issueof content knowledge and technical skills; it is, above all, a source of(micro)political engagement. Developing a sense of the pedagogical growsout of a history of response to local discursive classroom codes and widereducational discourse and practices, all of which interrupt, derail, and elidethe best intentions of the pre-service teacher.

APPENDIX: QUESTIONNAIRE

This questionnaire has been included to indicate the nature of data collection in the reportedstudy. Space for respondents’ answers has been reduced to minimize space in reproduction.


Pre-service teachers’ perceptions of mathematics teaching practice

This research will provide university staff important information about the contextsin which pre-service teachers carry out their mathematics teaching in schools. It isanticipated that the outcomes of this study may affect information which we provide toAssociate Teachers. It may also affect the nature of mathematics tasks that we set forpre-service teachers to undertake in the College.

We ask that you please do not provide any information that would lead to the identificationof the place and the people involved in your mathematics teaching practice.

We cannot use your information if it identifies you, the School, or the AssociateTeacher(s).

Roll of School (approx.)

Please provide the following general details about your teaching practice.

School Type: (e.g. state, private, integrated)

Year level of children you worked with (orappropriate age)

Usual time of day for mathematics lessons

Usual duration of lessons

Class groupings (e.g. ability, compositeclasses, only whole class teaching)

Total number of mathematics lessons occur-ring during teaching practice

Number of mathematics lessons you taughtwith Associate (e.g. having a group, orsharing teaching)

Number of mathematics lessons in whichyou had sole charge

Lesson PlanningTo the best of your knowledge what sources did your associate teacher draw from whenplanning their mathematics teaching.

Tick if this was used Source for Mathematics Lesson Planning

“Mathematics in the New Zealand Curriculum”

School policies

Official teachers’ guides

Commercially produced teachers’ guides or handbooks

Discussion with other teachers

The internet

Textbook(s)

82 MARGARET WALSHAW

Various other publications

Teacher prepared worksheets and notes from previous years

Other (please specify)

Unable to answer

Teaching Mathematics

Tick if you (i) mostly Activities used to teach mathematics

saw this (ii) sometimes

(iii) not very often

Teacher talk and exposition

Children listening

Group work and co-operative activities

Children engaged in whole class discussion

Children asking questions

Children working with hands-on equipment

Media presentations

Presentations from experts

Children researching in books and magazines

Children researching over the internet

Calculator use

Children doing a worksheet

Children writing mathematics in their own words

Field trips/science museum visits

Other

A Typical LessonOn the basis of your observations, describe a typical lesson. Include an indication of timeinvolved for each major component of the lesson.

Duration (minutes) What the teacher was doing What the children were doing

In your opinion, your associate teacher was:

Tick if you saw this Teacher experience and enthusiasm for mathematics

Highly experienced and enthusiastic

Highly experienced and not very enthusiastic

Not very experienced but enthusiastic

Not very experienced and lacking in enthusiasm for mathe-matics

Support from Associate Teacher in MathematicsCompared with the support you received from the associate in other curriculum areas, howdo you rate the support given by the associate for mathematics.


Tick Support in relation to other curriculum areas

More for mathematics

The same as other curriculum areas

Less than others

Assistance Received

Tick if received Assistance with

Resources

Ideas

Sequencing and pacing of lessons

Worksheets and/or textbooks

Teaching plans

Managing space

Involving you in year-level or syndicate math’s decision making

Developing routines and procedures

Assessment and feedback

Communicating mathematical ideas

Your Teaching

Did you reproduce any tasks or activities or ideas you had been introduced to in yourmathematics college course? Yes / No

Describe:

Did you produce any new tasks or activities, based on those from college? Yes / No

Describe:

Comment on your teaching at the beginning of the teaching practice?

Comment on your teaching at the end of the teaching practice?

Your ExpectationsDescribe the ways in which your experiences on this teaching practice met or failed tomeet your expectations.

– in the support you received for teaching mathematics.

– in the way mathematics was taught.

Comment about any aspect of mathematics teaching and learning which interested you onthis teaching practice.

84 MARGARET WALSHAW

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Britzman, D. (1991). Practice makes practice: A critical study of learning to teach. Albany:State University of New York Press.

Brown, T. (2001). Mathematics education and language: Interpreting hermeneutics andpost-structuralism (2nd ed.). Dordrecht: Kluwer.

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Cobb, P. & Bauersfeld, H. (1995). Introduction: The coordination of psychological andsociological perspectives in mathematics education. In P. Cobb & H. Bauersfeld (Eds.),The emergence of mathematical meaning: Interaction in classroom cultures (pp. 1–16).Hillsdale, NJ: Lawrence Erlbaum Associates.

Cooney, T.J. (1995). A beginning teacher’s view of problem solving. Journal for Researchin Mathematics Education, 16, 324–336.

Cooney, T.J. (2001). Considering the paradoxes, perils, and purposes of conceptualizingteacher development. In F-L. Lin & T.J. Cooney (Eds.), Making sense of mathematicsteacher education (pp. 9–31). Dordrecht: Kluwer Academic Publishers.

Davies, B. (2000). A body of writing 1990–1999. Walnut Creek, CA: AltaMira Press.Davis, R.B., Maher, C. & Noddings, N. (1990). Constructivist views on the teaching and

learning of mathematics. Journal for Research in Mathematics Education, MonographNo. 4. Reston, VA: NCTM.

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Dowling, P. (1998). The sociology of mathematics education: Mathematical myths/pedagogic texts. London: Falmer.

Dreyfus, H. & Rabinow, P. (1982). Michel Foucault: Beyond structuralism and hermen-eutics. Sussex: Harvester Press.

Ensor, P. (2001). Taking the ‘form’ rather than the ‘substance’: Initial mathematics teachereducation and beginning teaching. In M. Van den Heuvel-Panhuizen (Ed.), Proceedingsof the 25th Conference of the International Group for the Psychology of MathematicsEducation (Vol. 2) (pp. 393–400). The Netherlands: Freudenthal Institute, UtrechtUniversity.

Foucault, M. (1977). Discipline and punish. Harmondsworth: Penguin Books.Foucault, M. (1984). The care of the self (Trans R. Hurley, 1986). Harmondsworth: Pen-

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Goldsmith, L.T. & Schifter, D. (1997). Understanding teachers in transition: Character-istics of a model for the development of mathematics teaching. In E. Fennema & B.SNelson (Eds.), Mathematics teachers in transition (pp. 19–54). Mahwah, NJ: LawrenceErlbaum Associates.

Grant, T.J., Hiebert, J. & Wearne, D. (1998). Observing and teaching reform-mindedlessons: What do teachers see? Journal of Mathematics Teacher Education, 1(2),217–236.

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Klein, M. (2002). Teaching mathematics in/for new times: A poststructuralist analysis ofthe productive quality of the pedagogic process. Educational Studies in Mathematics,50(1), 63–78.

Lacey, C. (1977). The socialisation of teachers. London: Methuen.Lather, P. (1991). Getting smart: Feminist research and pedagogy with/in the postmodern.

New York: Routledge.Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life.

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

York: Cambridge University Press.Lerman, S. (1994). Reflective practice. In B. Jaworski & A. Watson (Eds.), Mentoring in

the education of mathematics teachers (pp. 52–64). London: Falmer Press.Lerman, S. (2001). A review of research perspectives on mathematics teacher education. In

F-L. Lin & T.J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 33–51). Dordrecht: Kluwer Academic Publishers.

Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington:Learning Media.

National Council of Teachers of Mathematics (1991). Professional standards for teachingmathematics. Reston, VA: NCTM.

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

Office for Standards in Education (OFSTED) (1994). Science and mathematics in schools:A review. London, UK: Her Majesty’s Stationery Office (HMSO).

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Simon, M.A., Tzur, R., Heinz, M. & Smith, M.A. (2001). Characterizing a perspectiveunderlying the practice of mathematics teachers in transition. Journal for Research inMathematics Education, 31(5), 579–601.

Sfard, A. & Kieran, C. (2001). Preparing teachers for handling students’ mathematicalcomunication: Gathering knowledge and building tools. In F-L. Lin & T.J. Cooney(Eds.), Making sense of mathematics teacher education (pp. 185–205). Dordrecht:Kluwer Academic Publishers.

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Sowder, J.T., Philipp, R.A., Armstrong, B.E. & Schappelle, B.P. (1998). Middle-gradeteachers’ mathematical knowledge and its relationship to instruction: A researchmonograph. Albany, NY: SUNY Press.

Walkerdine, V. (1988). The mastery of reason: Cognitive development and the productionof rationality. London: Routledge.

Walkerdine, V. (1994). Reasoning in a post-modern age. In P. Ernest (Ed.), Mathematics,education and philosophy: An international perspective (pp. 61–75). London: Falmer.

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Yackel, E. (2001). Explanation, justification and argumentation in mathematics education.In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the Inter-national Group for the Psychology of Mathematics Education (Vol. 1) (pp. 9–24). TheNetherlands: Freudenthal Institute, Utrecht University.

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Department of Technology, Science & Mathematics EducationMassey UniversityPrivate Bag 11222Palmerston NorthNew ZealandE-mail: [email protected]

KONRAD KRAINER

EDITORIAL

On Giving Priority to Learners’ prior Knowledge and Our Need toUnderstand Their Thinking

If I had to reduce all of educational psychology to just one principle, I would say this:The most important single factor influencing learning is what the learner already knows.

David Ausubel

Taking students’ prior knowledge into account is a prominent goal in math-ematics education, as can be seen in the demand that teachers need tounderstand what their students already know (e.g., the NCTM-citation inthe paper of An, Kulm & Wu). This need to understand learners’ thinkingalso holds true for teacher educators’ learning from (student) teachers.Therefore, in the following, when we speak about the teacher, the parallelrole of the teacher educator should be taken into consideration, too. As weknow, understanding learning as connecting new ideas to prior knowledgeneeds both a sound theoretical perspective and an attitude of mind thatviews learning as an active, socially shared construction process by thelearners and the teacher, and not as the receiving of knowledge transmittedby the teacher. However, this does not mean that teachers should not giveinputs: as teachers we have to judge critically whether the input is nottoo far away from learners’ recent knowledge and thus gives them littlechance to build from where they are. Thus teaching mathematics aims atsupporting students’ autonomous and socially shared learning processeswith regard to mathematics (Figure 1).

Mathematics ↔ Students

↑Teacher

Figure 1. The teacher supports the students’ mathematics learning process.

In this figure mathematics learning is seen as an active confrontationof the learners with challenging mathematics. To achieve this, two kindsof prior knowledge must come together in mathematics lessons: the priorknowledge of the students and the prior knowledge of the teacher in combi-nation with the materials he or she uses as a subset of the recent body of


88 KONRAD KRAINER

knowledge of (school) mathematics. Effective learning takes place whenboth kinds of prior knowledge are kept in a certain balance. It is unpro-ductive to ignore students’ recent understanding and fresh ideas, and itis equally unproductive to ignore the knowledge produced by generationsof mathematicians. Thus, teaching mathematics is a continuous dilemmasituation for teachers: on the one hand, they need to start where thestudents are and on the other, they aim at supporting students in developingan understanding of the mathematical concepts that are part of a socio-historically constructed body of mathematical knowledge. To say the samein a metaphor: teachers have to balance mathematics teaching between twoextremes, loading the students on the back of a truck and driving them athigh speed along mathematical highways (without ever knowing, whetherthey understand what is going on, where they are going, and why andhow these highways have been constructed), or leaving them with a pickand a shovel and the task of building their mathematical paths themselves(without ever knowing, how they will arrive at the highways that mark ashared body of knowledge and through which they are expected to commu-nicate with others). Consequently, mathematics teachers are mediatorsbetween a school system’s envisioned target knowledge of mathematics,defined by what a society expects from mathematics education and itscontribution to social and economic welfare, and students’ existing priorknowledge of mathematics.

Taking students’ prior knowledge seriously means building on theexisting ideas and strengths of students, picking them up where theyare. This means neither ignoring the goals that should be aimed at norabandoning our high expectations. However, it does mean that students’learning cannot be reduced to a comparison of their knowledge with expertknowledge and hence only observed and assessed in terms of “errors” and“misconceptions”. Instead, we need to focus very intensively on students’growing understanding. Measuring students against a general norm, andspeaking about their deficits is relatively easy, but it is far more difficultto take into account that students start from different backgrounds andparticipate in different communities, and to take their progress as a crucialfactor in their assessment. The same holds true for research in teachereducation. It is much easier to find deficits in teachers’ beliefs, knowledge,and so forth than to describe growth, to explain it, and to make suggestionsas to how growth might be promoted.

Teachers as individuals can hardly influence society’s envisioned targetknowledge, although it can be regarded as their professional task todiscuss goals and standards with colleagues and to try – as a part of alarger community – to have a voice in this discussion. Where the indi-vidual teacher can make a difference is in recognizing and working with

EDITORIAL 89

students’ prior knowledge. Trying to know more about students’ abilitiesand interests helps to support students, as individuals and as a group,professionally and socially.

The three research papers in this volume of JMTE inspired me to reflecton students’ prior knowledge. All three papers, in different ways, addressthis topic and yield insights into the ways teacher education programspromote (student) teachers in understanding learners’ thinking or show thatbuilding on prior knowledge might also be an attitude, deeply embeddedin a nation’s cultural heritage.

Rebecca Ambrose investigates the change in beliefs of 15 studentteachers who participated in a mathematics course and an early field-experience at an elementary school, in the context of a teacher educationproject in the US. The paper analyses the nature of the experience,considers the factors that contributed to its intensity and examines theeffects of the experience on the student teachers’ beliefs. The data showsthat close personal contact between student teachers and one child overa longer period contributed to the growth of the student teachers’ beliefs.Most of them were surprised that mathematics teaching was more diffi-cult than expected. They began to consider the importance of providingchildren with time to think when solving mathematical problems. Onestudent teacher’s response to the question what she learned from theteacher education program was “. . . not to drill it into their heads. Whenchildren learn, they need their own space and time to learn on their own.Let them have a chance first, and then see what they need help with”.This indicates that the student teacher regards building on students’ priorknowledge and on their autonomous learning as more important thanbefore, and it is assumed that the same quotation more or less describesthe participant’s situation in the above mentioned program, having hadthe early opportunity to work autonomously with a child and to share theexperiences with a partner.

Susan Empson and Debra Junk investigate the knowledge growth of13 teachers at a single elementary school in the US while implementinga student-centred curriculum in the context of a district-wide reform. Thestudy gives insights into critical components of the teachers’ knowledgebase, examining in particular their knowledge of students’ non-standardstrategies in mathematics. Teachers’ thinking was dependent on the way inwhich a topic was treated in the written curriculum materials used, whichsuggests implementation of the curriculum as a source of learning. Allteachers noted some aspect of their new-found respect for students’ mathe-matics capabilities and developed a high regard for students’ thinking. Theauthors highlight that the implementation of the curriculum “helped theteachers to formulate and address problems at the heart of their mathe-

90 KONRAD KRAINER

matical work with children”. The study shows that involving teachers insituations where they are challenged to understand students’ mathematicalthinking is a crucial starting point for promoting their knowledge andbeliefs regarding mathematics and its teaching. The teachers realize thatthey can learn from their learners, as one teacher in the study states: “We’vealways thought, well, we’re the ones that give information. But they [thechildren] give me information . . . . They bring things to my mind”.

Shuhua An, Gerald Kulm and Zhonghe Wu compare the pedagogicalcontent knowledge of 28 middle school teachers from the US and 33middle school teachers from China. The study indicates that mathematicsteachers’ knowledge in these two samples differs markedly, and exam-ines the differences between these teachers’ pedagogical content knowl-edge and its impact on teaching practice. The study highlights that bothapproaches have benefits and limitations. Regarding the use of students’prior knowledge, 45% of the Chinese teachers, but only 7% of the USteachers focused on the importance of that element of teaching. Observa-tion of classrooms showed that Chinese teachers spent at least one thirdof the time reviewing prior knowledge at the beginning or during class.This seems to be an attitude of mind, deeply embedded in Chinese cultureand education. The authors refer to the Chinese philosopher Confuciuswho stressed that, in reviewing prior knowledge; one can always find newknowledge. A further view could be that taking this time in class to connectintensively with what happened before, reviewing students’ thoughts,sharing common understandings, and thus solidifying and further devel-oping the existing knowledge, is a vital factor in promoting sustainablelearning.

All three papers in this volume indicate that it is crucial for teachereducation to give (student) teachers sufficient time to reflect on students’mathematical learning and to share the experiences with colleagues, andthat this increases (student) teachers’ awareness of the importance ofstudents’ prior knowledge and of their own learning from the students.The learners are a rich source for teachers’ learning. This suggests thatthe arrow in Figure 1 needs a second direction (Figure 2), indicating thatmathematics teaching can always be seen as an opportunity for teachersto learn. This view of teaching as a two-way street for learning fosterscuriosity and investigative attitudes in and among teachers.

Mathematics ↔ Students

�Teacher

Figure 2. The mathematics teacher and the students are all learners.

REBECCA AMBROSE

INITIATING CHANGE IN PROSPECTIVE ELEMENTARY SCHOOLTEACHERS’ ORIENTATIONS TO MATHEMATICS TEACHING BY

BUILDING ON BELIEFS

ABSTRACT. Many mathematics educators have found that prospective elementaryschool teachers’ beliefs interfere with their learning of mathematics. Often teachereducators consider these beliefs to be wrong or naïve and seek to challenge them soprospective teachers will reject them for more generative beliefs. Because of the resili-ence of prospective teachers’ beliefs in response to these challenges, teacher educatorscould consider alternative ways of thinking about and addressing beliefs, particularly thepotential of building on rather than tearing down pre-existing beliefs. Data from an early-field experience linked to a mathematics-for-teachers course provide evidence that whenprospective teachers work intimately with children, in this case trying to teach 10-year-olds about fractions, the experience has the intensity from which beliefs can grow. Most ofthe prospective teachers in the study were surprised that mathematics teaching was moredifficult than they had anticipated. They began to consider the importance of providingchildren time to think when solving mathematical problems. The change described in thestudy is incremental rather than monumental, suggesting that building upon prospectiveteachers’ existing beliefs will be a gradual process.

KEY WORDS: field experience, mathematics pre-service teacher education, prospectiveteachers’ beliefs

INTRODUCTION

Prospective elementary school teachers come to their teacher educationprograms with a variety of beliefs that are influenced by their experi-ences as students in schools. Some believe that teaching will be relativelystraightforward, consisting primarily of offering clear explanations tochildren (Richardson, 1996). They believe that their abilities to relate tochildren and manage classrooms will be paramount to their success asteachers. Weinstein (1989) characterized this orientation as an optimisticbias, because prospective teachers enter their coursework assuming thatthey already know what they need to know in order to teach (see alsoFeiman-Nemser & Buchmann, 1986). These beliefs often lead prospectiveteachers to underestimate the complexity of teaching and the kind ofknowledge that they will need to be successful. In particular, they oftenunderestimate the importance of subject-matter knowledge in teaching.


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Some mathematics education researchers have argued that teachers’subject-matter knowledge is extremely important and have documentedthe limited content knowledge of prospective teachers (Ball, 1990;Ma, 1999). Universities often require courses designed to enhanceprospective teachers’ mathematical knowledge by having them makesense of mathematics and understand the principles that underlie thearithmetic they memorized as children. Despite the promising designof these courses, prospective teachers’ beliefs about mathematics andteaching often diminish the outcomes of the courses. Some mathem-aticians have suggested that promoting changes in prospective teachers’beliefs about mathematics and mathematics teaching is critical to helpingthem to develop the content knowledge that they need to be effectiveteachers (Conference Board of Mathematical Sciences, 2001; Mathema-tical Sciences Education Board, 2001). But, teacher educators have beenmore successful in documenting the existence of beliefs that interfere withprospective teachers’ learning than in promoting belief change (Wideen,Mayer-Smith & Moon, 1998).

Often, efforts to change prospective teachers’ beliefs are initiated inmethods courses after subject matter courses have been completed andcome too late to support them in developing beliefs that will help them todevelop a deep understanding of fundamental mathematics. The researchreported here describes a program that attempted to initiate belief changeat the beginning of prospective teachers’ mathematical preparation. TheChildren’s Mathematical Thinking Experience (CMTE), part of the Inte-grating Mathematics and Pedagogy (IMAP) project, helped prospectiveteachers begin to understand the importance of subject matter knowledgein the teaching of mathematics by having them work with children in anelementary school while they were enrolled in their first mathematics-for-teachers course. For many of the prospective teachers, the field experiencewas one of their first experiences of working with children in schools and,for most, it was an intense experience that caused them to reconsider theirassumptions about mathematics teaching. I analyze the nature of the exper-ience, consider the factors that contributed to its intensity and examine theeffects of the experience on the prospective teachers’ beliefs. Specifically,it seems that the prospective teacher’s interest in relating to children provedto be a stimulus for expanding their views of teaching and affecting theirbeliefs about learning mathematics.

INITIATING CHANGE IN PROSPECTIVE TEACHERS’ BELIEFS 93

BELIEFS

Before describing the CMTE, I describe the framework I have adopted formy thinking about the process of belief change and the ways that teachereducators can promote belief change in prospective teachers. The compo-nents of this framework include several aspects of beliefs: their origins,their effects on one’s interpretations of experiences, the ways separatebeliefs combine to create belief systems, and the ways beliefs changewithin this framework. This framework provides support for the premisethat providing prospective teachers opportunities to work with children canbe a promising avenue to promote belief change.

Sources of Beliefs

Beliefs can be thought of as having one of two primary sources: emotion-packed experiences and cultural transmission (Pajares, 1992). The firstsource, emotion-packed experience, gives beliefs their “signature” quality.Many people can point to a vivid memory from which a particular beliefemerged (Nespor, 1987). For example, some prospective teachers givedetailed accounts of crying while they struggled to learn multiplicationtables. They relate these experiences to their belief that they are incapableof learning mathematics. The emotional component of these experiencesis one feature that differentiates beliefs from other forms of knowledge. Inrelating this feature to beliefs about teaching, Goodman (1988) suggestedthat these beliefs were derived from guiding images based on both positiveand negative experiences that teachers had as children.

The second source of beliefs, cultural transmission, creates beliefs thatmay be held at a subconscious level and can be thought of as resulting fromthe “hidden curricula” of our everyday lives. Culturally transmitted beliefsoften take the form of assumptions and stereotypes. For example, becauseprospective teachers’ mathematics work in school consisted mostly ofmemorizing procedures, many assume that mathematics always requiresmemorization, even though they have never heard a statement to that effect.People tend to be unaware of the culturally transmitted beliefs they hold,taking them for granted because they have neither examined nor discussedthem. These implicit beliefs may guide behavior in ways that could becharacterized as habits, with individuals doing things in particular waysthe reasons for which they are hardly cognizant.

Because of their origins, beliefs can be hard to change. It is impossibleto undo intense personal experiences or wipe out 20 years of livingin a culture. Teacher educators wishing to stimulate belief change in

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prospective teachers might pursue the avenue of creating for thememotion-packed experiences that will leave the vivid impressions thatform the basis of some beliefs. Another avenue of change is to develop acommunity that will instill positive implicit beliefs in prospective teachers.However, the duration of a teacher education program may be too short toachieve this kind of belief generation.

Effects of Beliefs

Beliefs have a filtering effect on one’s new experiences (Pajares, 1992).This filtering effect, which can make beliefs quite durable, is evidentwhen prospective teachers interpret experiences or information in theircourses in ways different from those their instructors intended (e.g., Simon,Tzur, Heinz, Kinzel & Smith, 2000). For example, one colleague hadher methods students work with kindergartners during the second weekof school. Her intention was for the prospective teachers to realize thatyoung children come to school with a great deal of informal mathematicalknowledge. After this field experience, the prospective teachers returnedto class impressed with how much the teacher had taught the children inthe first week of school (L. Clement, personal communication, January2001). Their beliefs that children’s knowledge of mathematics comesfrom formal school experiences led them to interpret the experience asevidence of teaching rather than of the children’s informal knowledge. Thispowerful filtering effect of beliefs is responsible for their important role inlearning and leads to teacher educators’ endeavors to affect the beliefs ofprospective teachers.

Belief Systems

Beliefs, whatever their source, are related to one another, forming systemsin which related beliefs are connected (Rokeach, 1968). Green (1971)pointed out that belief clusters might be held in isolation, unconnectedto other belief clusters. He wrote, “We tend to order our beliefs in littleclusters encrusted about, as it were, with a protective shield that preventsany cross fertilization among them or any confrontation between them”(p. 47). For example, some prospective teachers believe that childrenshould have opportunities to be creative. This belief might be connectedto other beliefs about art and writing and may come from childhood exper-iences in these domains. The belief about the importance of creativity forlearning may not be connected to beliefs about mathematics because theprospective teachers have not had creative experiences in mathematics.Teacher educators could help prospective teachers connect their beliefabout the importance of creativity for learning with their beliefs about


learning of mathematics so that they begin to look for the creative poten-tial in problem solving or problem posing. Green (1971) suggested thathelping people to develop well-connected belief systems should be one ofthe primary purposes of teaching.

Rokeach (1968) argued that beliefs related to a particular situation orobject, form attitudes similar to the belief clusters that Green (1971) iden-tified. For example, prospective teachers’ beliefs about teaching wouldtogether form an attitude about teaching. Rokeach pointed out that onedimension of attitudes is their degree of differentiation. Well-differentiatedattitudes are those that have a large number of parts and are well articu-lated. Attitudes formed by culturally transmitted beliefs can be undifferen-tiated and fairly simplistic. These might be in the form of “teachers shouldbe nice” or “teachers should make class interesting”. These undifferenti-ated attitudes do not encompass the complexity of the situations to whichthey apply and are similar to stereotypes in that the believer assumes thatthey can treat all situations of this type as if they were the same, instead oftaking into account the particulars of the situation.

Undifferentiated attitudes are often ones that have not been examined.Fenstermacher (1979) pointed out that the role of teacher educationprograms should be to support teachers in bringing tacit beliefs into theopen so that these beliefs can be transformed into objectively reasonablebeliefs. This process helps prospective teachers to make more principleddecisions on the basis of beliefs they believe are important, instead ofacting on the basis of the habits of unexamined beliefs and undifferentiatedattitudes. At the beginning of their course work, prospective teachers tendto have undifferentiated attitudes about teaching because they have not hadchances to refine them through the reflection that Fenstermacher discussedand may be one of the reasons that prospective teachers undervalue theirsubject matter preparation.

Changing Belief Systems

Four mechanisms for stimulating belief-system change in prospectiveteachers have been outlined above: (a) they can have emotion-packed,vivid experiences that leave an impression; (b) they can become immersedin a community such that they become enculturated into new beliefsthrough cultural transmission; (c) they can reflect on their beliefs so thathidden beliefs become overt; (d) they can have experiences or reflectionsthat help them to connect beliefs to one another and, thus, to develop moreelaborated attitudes. Many teacher education programs rely on reflection asa means for fostering belief change. Stofflett and Stoddard (1992) pointedout that this practice can serve to make prospective teachers more articulate

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and definite about the beliefs that they held before their teacher preparationstarted, but may not serve to help them form new beliefs. Vivid experiencescoupled with reflection may be required for new beliefs to form.

A fifth kind of belief change, the most dramatic, is the reversal ofexisting beliefs. A prospective teacher may change, for example, frombelieving that new material is best presented through a teacher’s lecturingto believing that new material is best developed by students’ grapplingcollectively with the material. This kind of conversion is the goal ofmany teacher educators and, all too often, they are disappointed whenprospective teachers fail to reverse their existing beliefs. McDiarmid, Balland Anderson (1989) found that the prospective teachers in their studyfailed to reverse their beliefs about teaching as telling. They found that theprospective teachers “expanded the range of options for teaching mathe-matics” (Wideen et al., 1998, p. 344) but were not converted to a wholenew way of thinking about mathematics teaching.

Prospective Teachers Beliefs About the Nature of Teaching

To build on prospective teachers’ beliefs, one must recognize that theirbeliefs about mathematics teaching and learning are part of a larger systemof beliefs that also includes beliefs about teaching, generally. For example,two central beliefs of many prospective teachers are that teachers shouldbe nice and should present instruction clearly.

Most elementary school teachers choose to enter the profession becausethey care about children (Hargreaves, 1994; Howes, 2002; McLaughlin,1991) and place strong emphasis on affective and interpersonal issues(Weinstein, 1990). Gellert (2000) found that the prospective teachers inhis study preferred to shelter students from challenging mathematicalproblems to protect them from anxiety. He attributed this attitude to thetendency of prospective teachers to see themselves in the role of nurturers.Beliefs about the importance of relationship building endure into the firstyears of teaching when new teachers often expend a great deal of energytrying to find ways to develop positive relationships with their students(Hollingsworth, 1992). In reviewing the research literature on learning toteach, Wideen et al. (1998) concluded, “Beginning teachers value socialand peer groups, positive self-concept, and helping behaviors” (p. 142).The centrality of prospective teachers’ beliefs about caring indicate thatthey will value experiences in which they can be intimately involved withchildren.

In addition to believing that nurturing is one of the primary func-tions of teachers, many prospective teachers believe that teaching entailspresenting information that student will memorize (Richardson, 1996).


They assume that explaining is a fairly straightforward enterprise (Feiman-Nemser, McDiarmid, Melnick & Parker, 1988). “Teaching itself is seenby beginning teachers as the simple and rather mechanical transfer ofinformation” (Wideen et al., 1998, p. 143). This is another undifferenti-ated attitude that eventually becomes elaborated as prospective teachersgain teaching experience. Unfortunately, as Weinstein (1989) discovered,this undifferentiated view of teaching leads prospective teachers to under-estimate the importance of their subject-matter preparation. Typically,only after they have finished their subject-matter course work and havehad teaching experience, do prospective teachers begin to recognize thecomplexity of teaching and the kind of knowledge required to do it well.

Prospective teachers’ views of ‘teaching as caring’ and ‘teaching asexplaining’ form the basis of their belief systems and will be likely to beretained while their belief systems change and develop. We hypothesizethat prospective teachers’ beliefs about children and the nature of teachingare more central to their belief systems than their beliefs about mathe-matics. Our goal in the IMAP project was to take advantage of prospectiveteachers’ interest in children. Others have found that the personal relation-ships that emerge from individual tutoring sessions have positively affectedprospective teachers’ learning in the domain of reading instruction (Worthy& Patterson, 2001). We hoped that while they worked with children inthe domain of doing mathematics, the prospective teachers’ interest intheir students would motivate them to expand their view of teaching andof mathematics when they encountered the limitations of an exclusive‘teaching as explaining’ approach. We saw their interest in children as thevehicle for motivating them to care about mathematics and mathematicsteaching and to begin to alter their views. We also expected that whenthey had experienced some teaching, their undifferentiated attitudes aboutteaching as telling would become elaborated.

Generative Beliefs for Learning Mathematics

For our project we identified several beliefs that we hoped the prospectiveteachers would develop. One is the belief that mathematics is a webof interrelated concepts and procedures. Related to this view are beliefsabout the relationship between concepts and procedures: that knowledge ofconcepts is more powerful and generative than knowledge of proceduresand that one can know procedures without understanding the underlyingconcepts. If prospective teachers begin to appreciate the importance ofconcepts in developing mathematical understanding, they might try todevelop their own conceptual understanding as well as think of ways toteach for conceptual understanding. Other beliefs are related to teaching

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and learning mathematics and include (a) believing that children bringto school a great deal of informal mathematical knowledge that can bethe basis of instruction and (b) recognizing that often the ways childrenthink about mathematics differ from the ways of adults who have beenschooled.

METHOD

Description of the Early Field Experience and the Mathematics Course

Fifteen prospective teachers volunteered to participate in the CMTE, anexperimental course, which required that they enroll in both the CMTE andtheir mathematics course. They were compensated for their participation.At the large urban regional university where the CMTE was offered, therewere twelve sections of the mathematics-for-teachers course taught by sixdifferent instructors. All instructors used the same text and the same finalexamination. The instructor of the CMTE also taught the mathematicscourse for the fifteen prospective teachers so that he could integrate thetwo experiences as much as possible. He was a professor with a researchinterest in children’s mathematical thinking. The CMTE met eight timesfor a two hour period at a local elementary school.

The prospective teachers explored number and operations in both themathematics course and the CMTE. The vision of the course was alignedwith that described by the Conference Board of Mathematical Sciences(2001), which emphasized the importance of prospective teachers’ appre-ciation for the intellectual richness of elementary school mathematics.The goals for the course were for prospective teachers (a) to make senseof “non-standard methods commonly created by students, the reasoningbehind the procedures, and how the structure of number is used inthese calculations” (Conference Board of Mathematical Sciences, 2001,Chapter 3); (b) to understand the variety of ways that the operations canbe interpreted; and (c) to be able to represent mathematical concepts in avariety of ways. The domain of the course was whole and rational numbersand included analyses of children’s invented approaches to problems.

In the CMTE, pairs of prospective teachers worked with individualchildren using specific tasks and activities designed to elicit children’sthinking; the emphasis was on problem solving rather than on symbolmanipulation. Each prospective teacher worked with a partner; one in eachpair led the problem-solving session, and the other took notes. Each partnerhad a chance to perform each role several times. The partners were encour-aged to help each other. They often exchanged ideas about which problem


to present next, what question to ask, and so on. The partners discussed theexperience afterward and considered issues that arose during the session.

During the first weeks of the course, the prospective teachers had threesessions with 6 to 9-year-old children. The goal in this phase of the CMTEwas to influence the prospective teachers’ beliefs about children’s informalknowledge and the children’s tendencies to act out story problems. Theprospective teachers provided the children with various whole numberstory problems that could be modeled using each of the four operations.For the last four weeks of the CMTE, each pair of prospective teachersworked with one 10-year-old child on fraction concepts. This part of theCMTE was designed to show that a concrete approach to this difficultconcept can help children develop understanding. The prospective teachersworked with the same child during these sessions so that they could try toteach the child over a period of time.

Participants and Site

Of the 15 prospective teachers in the CMTE, 13 were female and twowere male; five were in their first year of university course work; eightwere in their third year of university course work (six of whom had trans-ferred from 2-year colleges); and two were postgraduates completing themathematics prerequisite for the teaching-credential program. Three ofthe prospective teachers spoke English as their second language; sevenof the prospective teachers reported that they had enjoyed mathematics aschildren and felt fairly successful with it. The other eight reported thatthey found mathematics to be boring and difficult to learn when they werechildren.

The prospective teachers worked with children at a multiethnic, urbanelementary school in which 46% of the students were White, 39% wereHispanic, 10% were African American, and the remaining students repre-sented a variety of other ethnic groups. Many of the children with whomthe prospective teachers worked were bilingual. The school used a drill-based mathematics curriculum, in which emphasis was on acquisition ofstandard procedures; independent seatwork was the predominant mode ofinstruction.

Data Sources and Analysis

Data for the study came from a variety of sources including surveys,interviews, prospective teachers’ written work, and field notes. Each pro-spective teacher completed a computerized belief survey at the beginningand end of the CMTE. The survey was a pilot version of a belief surveythat was later used in a large-scale study of the CMTE (see Ambrose, 2002,

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for more information about the survey). The survey consisted of six open-ended items focused on whole number and rational number arithmetic. Tovalidate the inferences we made from the belief survey, we asked eachprospective teacher to discuss and elaborate on the responses given onthe survey that they had just completed. Two survey items were of partic-ular importance for this study. Both required the prospective teachers toconsider alternatives to standard algorithms, one in the domain of multi-digit addition and another in the domain of multidigit subtraction. Theseitems were used to determine the degree to which the prospective teachersbelieved in the importance of multiple approaches.

Each prospective teacher was interviewed at the beginning and endof the semester and the interviews were transcribed. In the initial inter-view, interviewers followed a protocol that included questions about theprospective teachers’ attitudes toward mathematics, their thoughts aboutteaching and learning mathematics, and follow-up questions about thebelief survey. The interviews were audiotaped and later transcribed. Astaff of four researchers collected field notes of the prospective teachers’problem-solving sessions with the children. Audiotapes of the problem-solving sessions were used to augment the field notes. Each prospectiveteacher was videotaped once during the semester while working with achild. After each problem-solving session, the prospective teachers wroteshort personal-reaction papers (Quickwrites) in which they shared theirinitial impressions; they wrote longer reflections as homework. Thesereflections were collected and photocopied before being returned to theprospective teachers.

My data analysis was an emergent process similar to that used ingrounded theory (Glaser, 1998) and began with intense analysis of thedata of one prospective teacher (Donna) in the group. This analysis wason-going during the semester when the CMTE was held. I read and rereadDonna’s data, looking for emergent themes, in particular, for what aspectsof the CMTE she found compelling and how these experiences affectedher beliefs. I developed a set of codes and used them to analyze data fromfour more prospective teachers to confirm or contradict hypotheses. Fromthis analysis, I saw that the sessions with the 10-year-olds had the greatesteffect, and some factors related to that experience emerged as being crit-ical to the prospective teachers’ belief change. I then analyzed those datarelating to the work with the 10-year-olds from the other 10 prospectiveteachers and coded it according to the nature of the problem-solvingsessions, the cognitive demands the sessions placed on the prospectiveteachers, and the emotional aspect of the work. In charts, I organizedsegments of coded data from the whole group of prospective teachers to


determine the extent to which various factors affected them and to onceagain confirm or contradict hypotheses.

Pajares (1992) pointed out that beliefs must be inferred; they cannot bedirectly measured. In assessing the prospective teachers’ beliefs, I lookedfor their statements and actions that indicated the beliefs they held. Basedon the same information, others might come to different conclusions aboutthe prospective teachers’ beliefs, so I offer verbatim quotes on which otherscan base their conclusions about the beliefs of the prospective teachersin the study. Characterizing the beliefs of a group of individuals can beproblematic because individuals in the group can hold different beliefs.In several cases, I discuss tendencies within the group and do not meanto imply that all the prospective teachers in the group developed identicalbelief systems. Because of the sometimes hidden nature of beliefs, eventhe individuals who hold them may be unaware of their presence. In thissense, any analysis of belief change will require interpretation, and I offermy interpretation with the recognition that there is a subjective componentto the interpretation.

THE NATURE OF THE CHILDREN’S MATHEMATICALTHINKING EXPERIENCE

Intense Teaching Experiences

Before considering the belief change that emerged from the CMTE, I beginby establishing that the prospective teachers found the experience to beintense. In their final interviews, when asked about a CMTE episode thatstood out for them, 12 of the 15 prospective teachers discussed their workwith the 10-year-olds as being the most memorable. Five discussed theexcitement they felt when their student told them he or she had learnedsomething. Julie commented, “I was so impressed that he remembered.I was just excited that I actually made an impact”. Five discussed theirconcerns that their student struggled with concepts. Holly said, “I wasn’texpecting it to be such a slow process, like with fractions. I found that shewas way behind what I thought she could understand”. Lisa spoke abouther flawed assumptions about her 10-year-old student’s understandingof English, and Donna spoke about the power of a real-world contextto support her 10-year-old student’s thinking. At some point in theirinterviews, all the prospective teachers talked about their work with the10-year-olds as being important learning experiences for them.

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The focus of the four sessions with the 10-year-old children was frac-tions. After an assessment session, the children explored, using the patternblocks, relative sizes of fractions and different names for fractions greaterthan 1. The children also solved equal-sharing problems that resultedin mixed-number answers. The tasks were designed to build conceptualunderstanding and to reduce the emphasis on the symbolic work that canlead children to misconceptions (Mack, 1995). The prospective teacherswere encouraged to make instructional decisions, during the sessions, toadapt the work to each child’s level of understanding. Some prospectiveteachers introduced the children to adding fractions whereas others spentmore time on fractions greater than 1.

In the first fractions session, the prospective teachers found that thechildren were relatively unfamiliar with fractions. The children did nothave a feel for the size of 12/13. They struggled to compare fractions.For example, many thought that 1 was greater than 4/4. The children werenot familiar with converting improper fractions to mixed numbers. Mostclaimed never to have seen improper fractions and did not know howto interpret them. They could partition wholes into parts but had troublenaming the parts they drew.

The second fraction session, which involved using pattern blocks, wasa high point for most of the CMTE pairs. The children enjoyed workingwith the pattern blocks to build representations for a variety of fractionalquantities and were successful using the pattern blocks to compare simplefractions such as 1/3 and 1/2. Many of the prospective teachers conveyedthis enthusiasm in their writing about the session. For example, Phanwrote, “Our child was on a roll. She would laugh out loud each time wegave her a fraction number”. In reflecting on this experience, Lisa wrote,

I was amazed by the progress he made in such a short amount of time. In the end hewas able to push the blocks aside and picture them [fractional quantities] in his mind . . . .Afterwards I felt very proud because I think we honestly helped him with his understandingof fractions.

By the end of the session all the children had converted some improperfractions to mixed numbers without the aid of the blocks, and most of theprospective teachers were excited by the progress the children had made.

The third fraction session surprised most of the prospective teachers.During a group discussion prior to the session, they decided to givethe children some problems identical to those they had solved in theprevious session: converting improper fractions into mixed numbers. Theprospective teachers elected to present the problems in symbolic formwithout giving the children manipulatives. They asked the children toconvert, for example, 7/6 (written in symbols) into a mixed number, and


they found that the children struggled with such review problems. Onechild claimed that 8/5 and 5/8 were the same, and another child haddifficulty drawing a representation for 3/2.

The contrast between the second and third fraction sessions impressed13 of the 15 prospective teachers. The passages in Table I are representa-tive of the reactions of the prospective teachers to their work in the twosessions. Each of the six prospective teachers quoted in the table workedwith a different child.

TABLE I

Prospective Teacher’s Reactions to Their Work with 10-Year-Olds

Prospectiveteacher

Comments following fractionsession 2 (pattern-block session)

Comments following fractionsession 3

Jane I now have the greatest feelingbecause I feel that our studentreally progressed. I feel like wereally taught him something and heunderstood.

He seemed to forget some of themore basic concepts of fractions.I was disappointed in some of hisresponses to the easier questions.

Kathy This week was so impressive. Ourstudent has improved so much overthe past week.

Today was so shocking. Last timeI walked away saying “WOW”. Itseems to me that he had learned somuch in just a week. Then today itwas like we took a backwards stepand he had forgotten everything. . . . I was very humbled today.

Ana I am amazed by the progress hemade just by using the manipu-latives. I really feel like he hasa better understanding of fractionsthan he had last time when wewere just focusing on the writtensymbols.

In an effort to make our studentcomfortable and relaxed, wedecided to begin our interviewwith easy review questions. Weasked him to draw one and a half.Pretty simple. He couldn’t do it. Ifigured that writing the numbersdown would help. It didn’t. Weassumed these tasks would beeffortless for him because heseemed to understand them soeasily in our last interview.

Gloria Wow, I am really impressed. Sheseemed to have improved since thelast time we interviewed her. Mypartner did a great job of teaching

It seemed like our student didn’tretain anything. She made the samemistake and it took her just as logto draw the fractions and compare

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TABLE I

Continued

Prospectiveteacher

Comments following fractionsession 2 (pattern-block session)

Comments following fractionsession 3

her that 11/4 is the same as 2 3/4.That was pretty intense.

8/5 and 5/8 . . . . It was a frus-trating session because it seemedlike we went backwards . . . . Thissession was so aggravating. She didnot seem to retain anything we hadworked . . . nothing seemed to sinkin.

Phan Overall the child improved a lotsince last week . . . . I think shewill learn from this experience andremember it forever.

This interview was shocking andyet very sad. She could notremember how to do just simplefractions. This had disappointed usbecause we thought she had nailedfraction problems last time we met.I learned I cannot teach a child yet.

Donna Our session went surprisingly well.I was so stoked as we taught herhow to do mixed numbers andimproper fractions and she pickedup on it and was able to write herown. She even was able to do 23/12into 1 11/12 and 10/4 into 2 2/4. Iwas amazed.

Our student had forgotten much ofwhat was taught her from the lasttime . . . . She was more confusedthat anything else today.

The terms stoked and excited used by the prospective teachers todescribe the pattern-blocks session point to the intensity of the experience.The prospective teachers’ initial reactions reflect the optimistic bias thatWeinstein (1989) identified and might strike the reader as naïve or exag-gerated. Keep in mind, however, that some of the prospective teachers were19 years old and had limited experiences working with children. Theiroriginal assumptions that they had been successful were based on whatsome might argue was a relatively limited period of time with a narrowrange of fractions (those that could be represented with the pattern blocks).Several of the prospective teachers’ comments indicated that they thoughtthat the teacher was responsible for the success of the session: “We reallytaught him something”; “My partner did a great job of teaching”.

“Today was so shocking.” “This had disappointed us.” “So aggrav-ating.” These reactions to the follow-up session express the prospective


teachers’ chagrin and surprise at their students’ lack of retention. Theirsurprise reflects the novelty of this experience for them. Experiencedteachers would be unlikely to be surprised that the children had troubleremembering a symbolic approach to converting fractions after beingexposed to it for only a short amount of time, but for these novice teachers,this result was unexpected. The experience was particularly troubling tothe prospective teachers because they felt that the children had been sosuccessful during the pattern-block activity, and many felt responsible forthat success. Their comments (“I was very humbled today”; “I learned Icannot teach a child yet”) show how personally the prospective teacherstook this experience. The emotional charge of the experience contributedto its intensity and left the kind of vivid impression from which beliefs cangrow.

Two of the prospective teachers were less affected than the others bythese two sessions and were cautious in their evaluations of what thechildren understood from the pattern-block session. Nina noted that herchild had been successful, but she was not convinced that the child’sunderstanding was completely developed. She wrote, “I was impressedby my student’s advancement from the week before. She came a longway. However, I don’t feel like she really understands what she is doingfully”. Holly was concerned by her child’s belief that there are no frac-tions in which the numerator is greater than the denominator. She wrote,“I was frustrated. I could see Stephanie’s confusion, but I don’t know ifI parted the clouds for her”. These two prospective teachers seemed notto have felt the elation or let down as intensely as their peers, and thesessions may not have engendered new beliefs for them. Their cautionin reacting to the pattern block session indicates the reflective stance thatthese two prospective teachers had throughout the CMTE. For these partic-ular teachers, the belief-system change that they exhibited may have beendue more to their ongoing reflection than to an intense experience.

Several features were in place that created the conditions necessaryfor these teaching experiences to have the intensity that they did for themajority of the prospective teachers. These features will be explored in thenext section.

Focus on Mathematics Learning Rather Than on Classroom Management

The fact that prospective teachers worked with individual children contrib-uted to the intensity of the experience. Because they were working one-on-one with the child, the prospective teachers did not face the cognitiveoverload that can accompany teaching. When prospective teachers arein student-teaching situations, they must attend to all the issues of class

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management and content, and their cognitive capacity is overwhelmedby all the stimuli they are trying to assimilate (Hollingsworth, 1989).Through their work with an individual child, the prospective teachers hadclear evidence that the child was struggling and that their teaching hadnot been entirely successful. They could concentrate their thinking on thisissue because they were not distracted by the host of management issuesthat typically preoccupy student teachers. They could not attribute thechildren’s difficulties to their behavior, attention span, or attendance. Nordid they have the option to turn to a different child to get the answers theysought; no other children could “bail out” the children or the prospectiveteachers. They had to face the fact that this was difficult material to teachand to learn.

High Cognitive Demand

Although the CMTE was stripped of some factors that occupy teachers’minds, the prospective teachers had to think about several things while theyworked with their children. They experienced what some call “knowledgein use” (e.g., Ball, 2000), that is, the knowledge that teachers must usewhile teaching. They had to consider the mathematical concept at hand,attend to what the child was doing and consider what understanding thechild had. They had to decide what question to ask or what problem toprovide in order to extend the child’s understanding and what representa-tion might help the child better understand the concept. Lisa talked aboutthe challenge of this cognitive demand, observing “trying to work withthem and think on my toes and figure out what questions to ask really fastwas hard”.

Donna talked about the difficulty of finding an appropriate vocabularythat would make sense to the child:

Donna: Sometimes you need to change your vocabulary or whateverso that it fits their world. Sometimes you don’t know whatto say. You feel like you understand – like you’re explainingthe right thing.

Interviewer: It makes perfect sense to you.Donna: Yeah. but not to the child. Sometimes you can’t explain

things.

Many prospective teachers talked about developing explanations as one ofthe most challenging aspects of their work with the children. Julie said,

I had no clue how to explain this one problem. He was just looking at me like, “Explainit”. I was just like, “I don’t know!” I didn’t even know how to explain it . . . . I know how


to find the common denominator for an addition problem, but I didn’t know how to teachit. So that was what was hard.

Kathy said, “I was trying to explain it to him, and I confused myself. It washorrible”. When they struggled to find the words to generate clear explan-ations, the prospective teachers experienced the high cognitive demand ofteaching. This was a novel experience for most of them, because few hadever been in the position of having to think on their feet in this way. Thecognitive demands of teaching coupled with the novelty of this kind ofthinking contributed to the intensity of the experience.

Connecting with Children through Mathematics

One critical feature of the work in the CMTE was the interpersonalcomponent. Working with the children on fraction concepts provided theprospective teachers opportunities to connect with children and to developrelationships, a feature evident in some prospective teachers’ commentsabout their children. Julie wrote, after her last session, “I’m so sad that Iwon’t be seeing him anymore. He had to have been the most polite child Ihave ever seen. He was so sweet . . . . I miss him already”. Although thiscomment may seem overly sentimental, the prospective teacher writing itwas one of the 19-year-old freshmen who had limited experiences workingwith children. These were the first words she wrote when reflecting on herfinal problem session, showing that this personal relationship was the mostsalient aspect of the work for her. Joe brought a present for his child to thelast interview and mentioned how much he had enjoyed getting to knowher. Tom and Alison commented that they were touched that their childcared so much about their work with her that she held on to a mathematicspaper for two weeks and brought it back to one of their sessions.

The one-on-one teaching situation was intimate in that the child wasasked to share his or her thinking, and the prospective teachers werecommitted to listening. Goldstein (1999) wrote about this kind of inter-action as involving both an intellectual component and an emotionalcomponent, requiring engagement and receptivity on the part of theteacher. As Tom noted,

Going and dealing with the student really kept me on my toes, because we had that addedresponsibility. I wasn’t just responsible for my own time and knowledge, but I was actuallygoing to be meddling with somebody else’s knowledge and time. I think that made mefocus more.

Kathy noted how important working with a child was for her: “Where elsedo you get an opportunity to sit with a child and have the child share whatshe is thinking? It was almost an honor to have the child do that. I felt

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very privileged”. The prospective teachers developed emotional ties thatcontributed to their learning because of their personal investment, which,along with the focus on the mathematics learning of one child and the highcognitive demands of this work, made the CMTE the kind of engrossingexperience with the emotional charge that leaves an impression which cangive rise to beliefs.

FRACTIONS WORK’S EFFECTS ON PROSPECTIVETEACHERS’ BELIEFS

Beliefs about Teaching Mathematics

The fractions-teaching experience affected the ways that many of theprospective teachers viewed teaching mathematics. They began to recog-nize that teaching requires more than simply presenting information tostudents. At the end of the semester, Donna commented about what shewould tell other prospective teachers: “Well, like I said before, not toexpect that a child knows what you’ve taught ‘em, because just becauseyou’ve taught ‘em doesn’t mean that they understand it”. Although Donnamay not be using the term understanding in a conceptual sense and mayinstead be talking about a student’s ability to remember a procedure,her comments reflect her recognition that teaching does not equate withstudent learning. We inferred, from the way she phrased her response, thatbefore the CMTE this prospective teacher had expected that “a child wouldknow what you’ve taught ‘em”, reflecting an initial stereotypical attitudethat teaching simply entailed presenting information to students.

Tom commented about how his views of teaching had changed: “Youthink, ‘Oh well, I’ll just tell them this and they’ll understand it’. And thenwhen you work with kids, you realize that it doesn’t work that way”.Tom’s transmission view of teaching had been expanded when he real-ized that children did not readily learn the material that he thought he hadtransmitted to them. Kathy stated,

I went into class that day thinking, “I’m so excited. I’m going to teach him this. By the endof the hour, he’s going to know it and he’ll be able to do it forever”. And it didn’t happenthat way, so I guess to just keep that in mind and to know that it’s not going to only take anhour for a child to understand a concept.

Kathy originally assumed that her student would absorb and retain theinformation she presented, and, through the CMTE, she learned thatteaching was not as straight forward as she had initially thought.

After their experiences in the CMTE, most of the prospective teacherstalked about the importance of providing children time to think, both


during individual problem-solving sessions and over long periods of time.Donna replied, in responding to the question “What did you learn from theCMTE?”

. . . not drill it into their heads. When children learn, they need their own space and time tolearn on their own. Let them have a chance first, and then see what they need help with.

We inferred from Donna’s suggestion, “Let them have a chance first andthen see what they need help with”, that she was beginning to developa more student-centered perspective toward teaching that might includea teaching-as-telling orientation but also included a concern for carefullytiming her lectures and assessing what the children knew first. Her morestudent-centered perspective was reflected when she later said,

I wanted to help her, but I also wanted her to do it on her own. I didn’t want to step in toomuch . . . . I have learned to give my child enough time on her own to try and figure out aproblem before I jump in and help her.

These comments illustrate how Donna’s attitude toward teaching becamedifferentiated. She began with a stereotypical and simplistic attitude,consisting of the belief that teaching entails presenting information. Shecontinued to hold this belief, as was evident in her comments aboutwanting to help the student. She added to that belief another belief aboutteaching as stepping back to allow students time to think. Her attitudetoward teaching grew to include two beliefs that were connected, and inthis way the attitude became differentiated.

The issue of providing children with time was a recurring theme inmany of the prospective teachers’ suggestions for people who might parti-cipate in the CMTE. Kathy mentioned, “Give them time – don’t bombardthem with questions”. Jane stated,

They don’t really learn anything when you just give them the answer. You just have to givethem time. You can’t just push them and keep asking questions, because when I did that,they were just frustrated.

Julie said, “Give them time and don’t show them everything. You have tolet them discover it for themselves, but you can help them along”.

Jane and Julie equated providing children time to think with allowingthe children to figure things out for themselves. They cautioned against“giving answers” and “showing them everything”, indicating that theyhad expanded their attitude toward teaching beyond merely presentinginformation to include facilitating children’s thinking. Nina explained herexpanded view:

Teaching is not me giving the information, and then them absorbing it, but rather givingthem the tools that they need to learn on their own. I think that’s probably the mostimportant thing that I learned.

110 REBECCA AMBROSE

The prospective teachers characterized the idea of giving children time tothink and letting them discover things for themselves as insights that theyhad gained from the CMTE. Apparently they had not started the semesterwith these ideas but developed them while working with the children. Wetook this as evidence that their attitude about teaching expanded as a resultof their experience.

All the prospective teachers recognized that teaching is not as simpleas they had expected it to be. Holly summarized her learning:

It was a lot more complex than I expected. It was also good talking to all my classmates,seeing that it’s not as clean-cut as we thought it would be.

Most came to believe that providing children with “think time” was animportant element to good teaching. Several came to believe that childrenshould have opportunities to figure things out for themselves, and afew developed faith that children could learn on their own when givenappropriate tools.

Beliefs about Multiple Solution Strategies

The prospective teachers grew to appreciate the importance of multiplesolution strategies in mathematics. Their appreciation was apparent in theircomments in interviews as well as in their belief-survey responses but wasless apparent in their work in the problem-solving sessions.

In their interviews, the prospective teachers focused on the importanceof knowing different mathematical approaches for successful teaching. Inher final interview, when asked what she had learned from the CMTE,Donna stated that she needed “to be able to know how to do things morethan just your way – that your way doesn’t work for everybody. Kidslearn in different ways”. Nina noted the importance of being flexible: “Youhave to be able to be flexible and approach math problems from differentperspectives. Each child’s learning is going to be different”. For theseprospective teachers, being familiar with different approaches would allowthem to assist different children.

Tom noted that being familiar with different approaches to problemsolving helped him in his teaching. He mentioned “getting away fromformulas. Using manipulatives and drawing pictures seemed to really helpthem make sense of what was going on”. These comments indicate aninterest in having all children use manipulatives and pictures as a wayto deepen their understanding of the concepts. He was not advocatingmultiple strategies to meet multiple needs; instead, he was arguing infavor of individuals knowing multiple strategies as a means for makingsense of the mathematics. Responses to the belief survey provided further


evidence for changes in the prospective teachers’ beliefs about multiplesolution strategies. In considering the addition segment (see Figure 1), allthe prospective teachers wanted more strategies shared at the end of thesemester than they had at the beginning of the semester.

Figure 1.

Six of the prospective teachers who originally wanted one or twostrategies shared wanted four or five strategies shared at the end of the

112 REBECCA AMBROSE

semester. At the beginning of the semester, two prospective teachers notedthe connection between the addition strategies and place-value under-standing whereas at the end of the semester, eight commented on theimportance of helping children make this connection. By the end of thesemester all the prospective teachers believed in the importance of multiplesolution strategies. For some this belief was evident at the beginning of thesemester but seems to have been strengthened; for others it had developedover the course of the semester. All the prospective teachers saw value inmultiple solution strategies, and some saw multiple solution strategies as avehicle for promoting conceptual understanding.

The prospective teachers were asked, in another segment on the beliefsurvey, which of two strategies they would prefer their students use (seeFigure 2). At the beginning of the semester, nine preferred that studentsin their classrooms exclusively use the standard algorithm for multidigitsubtraction. In supporting this choice, many wrote that it is less prone toerror, faster or “more of a sure thing”. We inferred from these responsesthat these prospective teachers were focused on the production of answers.At the end of the semester, 13 wanted their students to have access to boththe standard algorithm and an alternative approach. (Of the two remaining,one wanted her students to use the standard algorithm, and the otherwanted her students to use the alternative approach.) In explaining whythey wanted children to have access to both strategies, five mentionedthat children learn in different ways and so should have a choice ofstrategies. We inferred from their responses that although these prospectiveteachers were open to multiple solution strategies, they also believed thatcomputing correct answers is a focal point for instruction. In contrast, sixmentioned that children should learn both ways so that they would betterunderstand the concepts associated with the procedures. We inferred thatthese prospective teachers had become interested in having their studentsdevelop conceptual understanding. Julie wrote, “I would want them tounderstand the concepts behind the traditional way”. This response wasin contrast to Alison’s: “I think both are important because some kidsmay be able to solve the problem easier with the other way”; her focusis on generating correct answers rather than on understanding concepts.As was evident in the interview data, some prospective teachers wereinterested in multiple strategies to accommodate different learning styles,whereas others were interested in multiple strategies as a means to promoteunderstanding.

Although the prospective teachers talked about valuing multiple solu-tion strategies, this belief was not always evident in their work with the10-year-olds. When they had opportunities to make instructional decisions,


Figure 2.

three of the seven groups chose to teach their students the symbolicprocedure for adding fractions by finding common denominators. Whenthey did so, they neither supported their students in developing their ownapproaches for solving these problems nor presented multiple strategies.They attempted to explain the procedure for finding common denominatorsand in so doing ran into difficulties. Margie explained her experience: “Itotally confused her so much . . . . I didn’t even think of showing her withthe blocks. I was just showing her the procedure way. Then afterwards Ithought, ‘Oh God, I should have showed her this way’ ”. We were some-

114 REBECCA AMBROSE

what disappointed when we saw the prospective teachers focusing on thestandard symbolic procedure after they had discussed the value of multipleapproaches. We were relieved that at least Margie recognized that shemight have approached her instruction differently.

Beliefs about the Importance of Mathematics Understanding for Teaching

The prospective teachers’ experiences in the CMTE helped them to realizethat their understanding of mathematical concepts was essential to theirsuccess as teachers. Cindy stated,

I want to teach young children, so I didn’t think I needed to know a whole lot of actualmathematical skills and I really disagree with that now. In order to come up with a creativeway to teach it, you need to understand what you’re talking about and you need to have themath skills to do that.

This prospective teacher came to the realization that even mathematicsfor young children was complicated to teach and required conceptualunderstanding.

Many of the prospective teachers talked about how the CMTE helpedthem to appreciate the importance of the material that they were learning intheir mathematics course. Margie noted, “When we were like, ‘Why do weneed to know the meaning of this?’ and it’s so you can explain it, becauseyou can’t teach something to a student if you don’t know what it means”.

Many noted that the experience of the CMTE made the contents ofthe mathematics class more compelling. Phan noted, “We understand thatknowing math is easy but knowing how to teach math to children is hardwork”. Ana expanded on this observation:

I think that as students, we tend to think when we are learning these math concepts that it’sso obvious and easy and that it will be easy to teach them to students. But it’s not as easyas we think it is, and the students tend not to know as much as we think they know. We’renot just going to go in and knock them dead. It’s going to take work and thought.

The prospective teachers began to distinguish doing mathematics, whichthey equated with using memorized procedures and considered easy,from teaching mathematics, which they equated with understanding andconsidered more difficult.

Many shared Ana’s observation that the children did not know as muchas the prospective teachers had expected them to and remarked that withoutthe CMTE they would have doubted their instructor when he told themthat children have trouble with particular concepts or tend to think aboutproblems in specific ways. Gloria observed,

The instructor would have said, “This is how the kids are learning”, but that wouldn’t havemeant anything to me. I would have been like, “So . . .? Okay. I’ll just concentrate on doing


the math problem”. But during the CMTE, it’s like you’re right there and applying whatyou’re learning and seeing whether or not it works.

When the prospective teachers had first-hand experiences teaching math-ematics, they realized that they needed to understand it well and beganto appreciate the value of their mathematics class. As Gloria pointedout, without the practical experience, most would have focused only onmastering techniques for solving problems instead of on the mathematicalconcepts related to those techniques.

Beliefs about Children’s Informal Knowledge

The first three sessions of the CMTE, those with the 6–9-year-old children,were intended to acquaint the prospective teachers with the informalknowledge that children bring to school. By having the prospectiveteachers pose story problems to young children, we had hoped that theprospective teachers would see that the children could model the actionof the problem using manipulatives and solve a variety of story problemswithout much guidance. Most of the children that were interviewed haddifficulty with several of the story problems, perhaps because they hadnot had any exposure to using manipulatives to model story problems intheir classrooms. Although many children can readily solve such problems(Carpenter, Fennema, Franke, Levi & Empson, 1999), the children in theCMTE could not. They tended simply to add the two numbers in the prob-lems posed, regardless of the action in the problem. Although many ofthe children could solve multiplication and division problems set in storycontexts, they were so unsuccessful with some of the other problem types(comparison situations, join change unknown, etc.) that the prospectiveteachers tended to focus on the children’s difficulties rather than on theirsuccesses.

The prospective teachers found the interviews with the primary-gradechildren to be awkward because they saw each child only once and didnot have opportunities to develop rapport with the children. They also haddifficulty observing while their child was working instead of showing thechild how to solve problems. Holly noted, “It was challenging for me tohold back and not help him by hinting at ways to solve the problems he washaving difficulty with”. The children struggled to explain their thinking,probably because explanations were not asked of them in their classrooms.Ana noted, “The primary interviews didn’t make much of an impressionon me, because I didn’t expect much out of the kids, to be perfectlyhonest, and we didn’t get much”. Unfortunately, the work with the primarychildren left little impression on most of the prospective teachers, and itwas unclear how their work with the primary children affected their beliefs.

116 REBECCA AMBROSE

BUILDING ON EXISTING BELIEF SYSTEMS

The belief-system change that I have attempted to illustrate focused onthe elaboration of attitudes through the development of new beliefs andthe formation of new connections among beliefs due to an intense exper-ience coupled with reflection. We speculate that the prospective teachers’original attitudes about teaching were undifferentiated, consisting of fewbeliefs, because of their limited experiences. Inasmuch as they had not hadopportunities to reflect on their beliefs about teaching, the beliefs remainedat the subconscious level. When they struggled to teach fractions to the 10-year-olds, their beliefs about teaching became more salient to them. Theyadded to their beliefs the notions that teachers should listen to children todetermine at what point to begin instruction, provide children with timeto think, and be prepared to explain concepts in a variety of ways. This isan example of the elaboration of an attitude when new beliefs emerge andbecome connected to existing beliefs. The basis of this belief change wasan intense experience.

The struggles that many of the prospective teachers had, when theyattempted to help their children understand fractions better, could becharacterized as failed teaching experiences. Weinstein (1990) suggestedthat failed teaching experiences were critical in helping prospectiveteachers to overcome their optimistic bias about their abilities as teachers.Weinstein speculated that prospective teachers need to see that teaching isnot as easy as they had believed and that facing the challenge of teachingstudents, particularly those who struggle, would affect their beliefs. Thisseems to be the case for the prospective teachers in the CMTE. As Lisawrote, “I am beginning to find out that teaching is not as easy as is looks.It takes a lot more to be a teacher than enjoying working with children”.The prospective teachers’ CMTE experiences helped them to recognizethat mathematics teaching is much more complicated than they expectedit to be and helped them to appreciate the value of the material in themathematics course.

Note that most of the prospective teachers continued to hold ontotheir beliefs that teaching involves explaining things to children, eventhough they spoke of the importance of giving children time to think forthemselves. This was evident in the prospective teachers’ actions in theproblem solving sessions when they presented children with the standardalgorithm for fraction addition. We concluded from their actions that, for atleast the six prospective teachers in these three partnerships, their actionsindicated a “teaching as telling” belief along with a belief about mathe-matics learning as the acquisition of standard symbolic procedures. Wedid not conclude that our efforts to help them change their beliefs werewasted. Instead we interpreted these examples as evidence that prospective


teachers do not let go of old beliefs while they are forming new ones. Theirnew beliefs about the importance of multiple solution strategies, or theirknowledge of these approaches, or both of these, were not strong enoughto compel them to introduce multiple approaches for adding fractions. Wewere encouraged that the prospective teachers could be critical of theirown actions, and we concluded that while their beliefs grew, several newbeliefs which might compel them to make different teaching decisions inthe future, would be added to their belief systems. Their lack of success inteaching the addition of fractions may accelerate their acquisition of newbeliefs while they experience the limitation of their existing belief system.The mismatch between the prospective teachers’ comments in their inter-views and surveys and their decisions with their students may be construed,not as evidence of conflicting beliefs, but as evidence of evolving beliefs.Although belief change had been initiated, the IMAP team recognized thatthe prospective teachers had not developed all the beliefs we would likethem eventually to develop. Over time, the prospective teachers’ beliefsystems may continue to change when they have more experiences uponwhich they have opportunities to reflect. More beliefs may be added totheir belief systems, and their beliefs about teaching-as-telling may stillexist but may be weaker or less central.

Providing prospective teachers with intense experiences that involvethem intimately with children poses a promising avenue for belief change.Coupling these experiences with reflection allows the beliefs that arisefrom these experiences to be examined and refined. The CMTE cameearly in the teacher preparation program. Given the incremental natureof belief change, teacher educators might consider creating several suchexperiences throughout the teacher preparation program, especially whileprospective teachers are doing their subject matter preparation, to ensurethat the beliefs become well connected and that attitudes become differen-tiated. One intense experience is a starting point but seems insufficient tocatalyze all the belief change that teacher educators might desire. Variousexperiences are required to help prospective teachers develop new beliefsabout mathematics, and teacher educators would be wise to recognize thatthese beliefs will probably coexist with, rather than replace, the beliefs thatpreceded them.

ACKNOWLEDGEMENTS

This work was supported through the Interagency Educational ResearchInitiative (IERI) Grant G00002211. Any opinions expressed herein arethose of the author and do not necessarily reflect the views of IERI.

118 REBECCA AMBROSE

The author wishes to thank her colleagues, Bonnie Schappelle, RandyPhilipp, Lisa Clement, Jennifer Chauvot, Cheryl Vincent, and JudySowder, for their help with this article.

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University of California-DavisOne Shields AvenueDavis, CA 95616USAE-mail: [email protected]

SUSAN B. EMPSON and DEBRA L. JUNK

TEACHERS’ KNOWLEDGE OF CHILDREN’S MATHEMATICSAFTER IMPLEMENTING A STUDENT-CENTERED CURRICULUM

ABSTRACT. Our study investigated the knowledge 13 elementary teachers gained imple-menting a student-centered curriculum in the context of district-wide reform. Participantscomprised all the teachers in grades three, four and five at a single elementary school. Webelieved that investigating teachers’ responses to fictional pedagogical scenarios involvingnonstandard algorithms would yield insights into critical components of their knowledgebase. We looked in particular at teachers’ knowledge of children’s mathematics. We foundthat teachers were in the midst of creating a knowledge base focused on children’s math-ematics and grounded in knowledge about alternative conceptual trajectories through theelementary curriculum. Teachers’ knowledge of nonstandard strategies supported by thecurriculum materials was stronger and more coherent than their knowledge of students’novel nonstandard strategies. Strong mathematical knowledge was not necessarily asso-ciated with strong knowledge of children’s mathematics. Teachers’ thinking varied by atopic’s treatment in the written curriculum materials used, suggesting implementation ofthe curriculum as a source of learning.

KEY WORDS: children’s mathematics, elementary, teacher knowledge, teacher learning

INTRODUCTION

As educational reform in the U.S. grows, greater numbers of teachersthan ever before have become involved in teaching curriculum programsthat require new kinds of mathematics knowledge (Ball, Lubienski &Mewborn, 2002; Floden, 1997; Richardson & Placier, 2001; Sherin, 2002).Some research has found that implementing these programs providessignificant opportunities for teacher learning, especially when accom-panied by professional development (Featherstone, Smith, Beasley, Corbin& Shank, 1995; Hull, 2000; Remillard, 2000; Sherin, 1996). But, witha few exceptions, teachers’ knowledge in these contexts remains largelyunexplored. Our study investigated the knowledge teachers gained throughimplementing an innovative, student-centered curriculum, and examinedthe implications of this knowledge for characterizing a knowledge base forteaching mathematics.

Because we start with the premise that teachers’ knowledge is situatedin practice (Ball et al., 2002), we examine teachers’ knowledge in the


122 SUSAN B. EMPSON AND DEBRA L. JUNK

context of teachers’ thinking about children’s mathematical thinking. Wefocused on teachers’ knowledge of children’s mathematics because, as abody of information, it epitomizes some of the major characteristics anddilemmas of the new knowledge policy-setting documents (e.g., NationalCouncil of Teachers [NCTM], 2000). For example, teachers are requiredto know and be able to form relationships among a variety of commonlyused student-generated strategies for multidigit problems in addition to theusual standard algorithms. Teachers also need to be prepared to make senseof both common and novel strategies during lessons, as well as before andafter lessons (Even & Tirosh, 2002).

The research took place in the context of district-wide1 reform in math-ematics instruction in Austin, Texas (Batchelder, 2001). This reform wasa result of a grant from the National Science Foundation to implementa student-centered, Standards-based (NCTM, 1989, 2000) mathematicscurriculum, Investigations in Number, Data, and Space (TERC, 1995–98), at the elementary level. By “student-centered”, we mean a curriculumdesigned to elicit and build on students’ ways of understanding math-ematics. By “Standards-based”, we mean a curriculum designed to beconsistent with the vision for mathematics instruction put forth by NCTM(1989, 2000).

Our questions included: (1) What knowledge do teachers who haveimplemented a student-centered curriculum use to make sense of students’non-standard strategies? (2) How might teachers’ acquisition of thisknowledge be linked to the use of the new curriculum materials?


In this study, we looked specifically at teachers’ knowledge of students’non-standard strategies for multidigit operations. We focused on teachers’knowledge of children’s mathematics, but also considered teacher beliefs,as they were expressed in teachers’ responses about children’s mathe-matics. We defined non-standard strategies as student-generated strategiesinvolving number relationships and informal as well as formal models ofoperations (e.g., Carpenter, Franke, Jacobs, Fennema & Empson, 1998;Fuson, Wearne, Hiebert, Murray, Human, Olivier, Carpenter & Fennema,1997). For example, a non-standard strategy for adding 47 and 26 couldinvolve adding 40 and 20 to get 60, 7 and 6 to get 13, then combining thoseintermediate sums to get 73. Research has amply documented that childrenreadily generate strategies to solve problems involving number operations(Kilpatrick, Swafford & Findell, 2001). Further, compared to standardalgorithms, students’ non-standard strategies for multidigit operations are

TEACHERS’ KNOWLEDGE OF CHILDREN’S MATHEMATICS 123

for the most part conceptually, rather than procedurally, driven (Fusonet al., 1997; Kamii, 1994). Thus, we reasoned, teachers’ knowledge ofstudents’ nonstandard strategies could provide information about teachers’conceptual knowledge of the mathematics of multidigit operations forteaching.

In this section, we review briefly research on teacher knowledge andbeliefs, then describe the framework we use to conceptualize teachers’knowledge for teaching mathematics.

Research on Teachers’ Knowledge

Prior research on U.S. elementary teachers’ subject-matter knowledge, inparticular, has found that many teachers hold narrow, procedural under-standings of algorithms and have difficulty generating multiple, flexiblerepresentations of operations beyond addition and subtraction (Ball, 1990;Kennedy, 1990; Ma, 1999; Post, Harel, Behr & Lesh, 1991; Simon, 1993).For instance, teachers in a National Center for Research on Teacher Educa-tion study (NCRTE, 1992) did not know why zeros should be used as“place holders” in the standard algorithm for multidigit multiplication, andU.S. teachers in Ma’s (1999) study generated few if any models for frac-tion division. Mewborn (2000) reported similar results across a range ofresearch on pre-service and in-service teachers’ subject-matter knowledgeat all levels. These findings suggest that many teachers do not understanddeeply the mathematics they teach. Much of this research, however, hasfocused on teachers who are learning to teach, or those who teach with afocus on computational fluency, rather than those who have been involvedin instructional reform.

In contrast to studies documenting deficits in teachers’ knowledge,some research suggests that teachers’ knowledge can become deeperand more flexible as teachers implement programs designed to developstudents’ conceptual understanding of mathematics, as opposed to compu-tational fluency alone (Featherstone et al., 1995; Hull, 2000; Sherin, 1996;Schifter, 1998; Sowder, Philipp, Armstrong & Schappelle, 1998). Forexample, Hull (2000) found that after two years of implementing a student-centered, standards-based middle-school curriculum at seven sites acrossTexas, teachers’ knowledge of proportional reasoning increased, in somecases dramatically. Written pre- and post-assessments documented growthin accuracy, number of strategies, and variety of problem representationsin teachers’ knowledge of proportional reasoning. Schifter (1998) andSowder and colleagues (1998) have argued that teachers who concentratedon understanding children’s thinking may also develop broad, deep mathe-matical understanding. It is not clear, however, how much or what kind


of knowledge for teaching mathematics teachers may learn under thesecircumstances.

Research on Teachers’ Beliefs

Teachers’ beliefs about the nature of mathematics and children’s learningplay a significant role in teaching (Thompson, 1992). Several researchershave argued that knowledge and beliefs must both change if teachers are tochange how they teach (Fennema & Nelson, 1997), although it is not clearif changes in beliefs precede changes in knowledge, or vice versa (Franke,Carpenter, Levi & Fennema, 1998; Philipp, Clement, Thanheiser, Schap-pelle & Sowder, 2003). In any case, these researchers have emphasized,changes in beliefs accompany changes in teacher knowledge.

Framework for Teachers’ Knowledge

Researchers have proposed that knowledge for teaching mathematics hasseveral important characteristics. First, knowledge of mathematics is inte-grated with knowledge of teaching and learning mathematics (Ball et al.,2002; Philipp et al., 2003; Schifter, 2001). We suggest, in particular, thatteachers’ knowledge of concepts, procedures, and mathematical practicesneeds to be integrated with knowledge of children’s thinking. We refer tothis integrated knowledge as knowledge of children’s mathematics.

Consider the topic of division as an example of how knowledge ofmathematics can be integrated with knowledge of children’s thinking.Mathematicians define the division of a by b in terms of the multiplicationof a by the multiplicative inverse of b, 1/b, if it exists.2 This definitionmakes no distinction between dividing into b groups and groups of b.However, informally, division can be defined in several ways, includingsituations in which a total needs to be partitioned into a given numberof equally-sized groups, or into groups of a given size (Greer, 1992).Although this distinction does not exist in formal mathematics, childreninitially interpret these different situations as two distinct processes, andlearn only later that they are united by a single mathematical operation.Knowledge about how these two informal models of division developin children’s early problem solving can be used by teachers to integratechildren’s informal mathematics with more formal mathematics.

Second, Ma (1999) argued that highly developed teacher knowledge isbroad, deep, and thorough. In her framework, breadth of knowledge refersto connections across several topics of similar conceptual complexity;depth of knowledge refers to connections across the longitudinal devel-opment of a single topic; and thoroughness refers to connections acrossseveral key topics and their longitudinal development. Depending on the


content of the knowledge base, however, we suggest there are differencesin the nature of these connections within and across topics. Whereas Ma’s(1999) content framework emphasizes the structure of fundamental math-ematics, ours emphasizes what we refer to as children’s mathematics.3

For example, building on Bruner’s (1960) notion of structure, Ma (1999)posited that mathematics knowledge for teaching consists of proceduraltopics, conceptual topics, and basic principles. Basic principles, such as“inverse operations”, and “rate of composing a higher value unit”, exem-plify mathematical structure, because they afford mathematical connec-tions to a multitude of related topics, and they represent elementary butpowerful ideas that will be revisited in advanced mathematics.

Thus, we suggest that teachers’ knowledge of children’s mathematicsshould also be broad, deep, and thorough. Building on the notion thatthe integration of children’s thinking and mathematics provides a criticalfoundation for teachers’ knowledge, we use breadth to refer also to knowl-edge of a variety of child-generated strategies and their interrelationshipsfor a given type problem; depth to refer to mathematical justification fora given child-generated strategy; and thoroughness to refer to the integra-tion of knowledge of children’s thinking for a given set of problems in atopic area with knowledge of the longitudinal development of children’sthinking about that topic.

METHOD

Participants

We interviewed 13 teachers about what they had learned while imple-menting Investigations in Number, Data, and Space (TERC, 1995–1998), focusing in particular on how teachers made sense of students’nonstandard strategies for multidigit operations. Participants comprised allthe teachers in grades three, four and five at a single elementary school,located in an urban district in Texas that had adopted Investigations as itselementary mathematics program. The implementation of Investigationswas part of a larger district initiative, described above (Batchelder, 2001).All participants had been teaching Investigations for one or two years atthe time of the study. They represented a range of experience: four of theteachers were first-year teachers, and five had 15 or more years’ experi-ence. One teacher had begun her career before the advent of New Math,and so had experienced in one way or another every major curriculummovement conceived in the past 40 years. The student body was demo-


graphically typical of the district; over half of the students came fromlow-income families.

Teachers had participated in up to four district-led workshops eachschool year on implementing Investigations in Number, Data, and Space.These workshops were designed to help the teachers become familiar withthe organization of the text materials and learn the mathematics by doingand discussing activities (Batchelder, 2001). There was little examinationof student work beyond what was presented in the curriculum materialsdialogue boxes.

Instrument

All teachers were interviewed once. The interview consisted of five ques-tions, beginning with an open-ended question about what and how teachershad learned implementing Investigations (Table I, Question 1). The inter-view ended with an open-ended question about which Investigations unitsteachers thought had contributed the most to student and teacher learning(Table I, Question 5). These two questions were designed to elicit teachers’reports of knowledge gained and, in particular, their beliefs of the role ofimplementing Investigations curriculum materials in their learning.

TABLE I

Interview Questions

Type of question Question (without probes)

1. Open-ended question aboutlearning

(Given prior to interview) I would like you to describea time when you learned something as a result ofteaching Investigations. I’m interested in hearingabout something you learned that stands out in yourmind as being important to teaching mathematics. Iwould like to hear what you learned, and how youthink you learned it.

2. Multidigit multiplicationscenario: Commonnostandard strategies

One goal of Investigations is to get students to solveproblems in many different ways. Suppose that youwere teaching multidigit multiplication. What are atleast three different strategies that children might useto solve 18×25?

3. Multidigit division scenario:Novel nonstandard strategywith mistake

A student was solving 144÷8 (show card). She said,“I know, I can just split it in half. So I will keepdividing by 2. I need to do that 4 times, since2+2+2+2 is 8.” As she talked, she wrote:


TABLE I

Continued

Type of question Question (without probes)

How would you respond to this student?

4. Multidigit subtractionscenario: Novelnonstandard strategy

Your class is working on subtraction with regrouping.A student says she has come up with a simple method.She explains that 6–9=–3 (“negative 3”), and 30–10=20, and –3+20=17. Does this strategy makesense? Why or why not?

5. Stimulated recall of learning (Provide list of appropriate grade units.)

a) Which units do you think helped your studentsreally learn important mathematics?

b) Which units, if any, do you think helped YOU learnmore about mathematics?

To focus the inquiry, we posed three scenarios in which teachers wereasked to speculate about how they would respond to students’ nonstandardstrategies during instruction (Table I, Questions 2–4). We modeled ourscenarios on items from A Study Package for Examining and TrackingChanges in Teachers’ Knowledge, published by NCRTL (Kennedy, Ball& McDiarmid, 1993). The first scenario (Table I, Question 2) askedteachers to generate at least three strategies for multidigit multiplica-tion that students might use. This question was designed to measure thebreadth of teachers’ knowledge of nonstandard strategies, their conceptualunderstanding of those strategies, and their knowledge of their relativedevelopmental sophistication. If one of the strategies listed included the


standard U.S. algorithm for multidigit multiplication, the teacher wasasked to describe an additional strategy.

The next two scenarios presented nonstandard strategies for teachersto evaluate. They represented the kinds of scenarios a teacher in a student-centered instructional setting could encounter, and assessed the knowledgeteachers would to bring to bear to make sense of such strategies. Specific-ally, the second scenario (Table I, Question 3) was designed to investigatethe depth of knowledge teachers used to interpret a student’s novel divi-sion strategy, including whether teachers distinguished between additiveand multiplicative composition in division, and the types of connectionsteachers made between informal and formal models of division. Thestrategy was based on a an intuitive model of division based on repeatedhalving, which requires keeping track of the number of partitions madeeach time. In the scenario given to the teachers the student repeatedlyhalves 4 times which results in 16 partitions rather than 8. This strategywas consistent with a partitive model of division, because it involved split-ting 144 into 8 equal groups – in contrast to a measurement model, whichwould have involved finding how many groups of 8 in 144.

The third scenario (Table I, Question 4), which we adapted froma questionnaire item in the NCRTE (1992) study package, involved astudent’s nonstandard strategy for subtraction with regrouping that incor-porated negative numbers. The item was intended to assess how teachersdetermined the validity of a novel nonstandard strategy for a specificnumber combination, and, more generally, for any number combina-tion. In designing all the items, we included strategies that we knew,through experience or empirical research, that elementary mathematicsstudents could generate independently of direct instruction. All three scen-arios were designed to assess the connections teachers made betweenstudent-generated strategies and mathematics.

Data Collection and Analysis

We interviewed all 13 teachers in March of the school year. Six of theteachers were just finishing their first year of implementing Investigations,and the rest were finishing their second year. All the interviews were audiorecorded and supplemented by notes. Most took place during teachers’planning periods and lasted about 45 minutes.

We transcribed all interviews and analyzed them by first looking forthemes, then refining those themes into codes. We revisited and revisedour codes several times to establish consistency within and across teachers.Codes were based on well-established findings in the literature concerningchildren’s thinking, Investigations-based strategies, and our own inter-


pretations. For example, codes addressed the kinds of strategies teachersdescribed, models of operations, and nature of justification. Half of thedata was coded by both of us to ensure reliability. There were veryfew disagreements about our coding of problem-solving strategies thatwere well documented in the literature, such as using partial products tocompute a multidigit product. Disagreements tended to concern details ofteachers’ thinking, such as whether the use of paired addends was a kindof doubling strategy, and were easily resolved through discussion.

Limitations

Clearly there are limitations to using a single interview to assess teachers’knowledge. Although we made use of teachers’ self reports of theirlearning activities, for instance, we were unable to assess with confidencehow teachers’ knowledge changed. Because we collected no classroomdata, we had no information about how teachers would actually respondto the kinds of strategies emulated in the interview scenarios. Nonethe-less, there is precedent in research on teacher knowledge for using itemsformatted as were our interview items (Ball, 1990; Kennedy et al., 1993;Ma, 1999). Although the interview represents a single point in time, ourinterview protocol provided multiple opportunities for teachers to thinkabout the fictional strategies, and consequently we were able to docu-ment detailed, plausible responses. We do not claim that these responsescharacterize adequately each teacher’s knowledge, but we do claim that,collectively, they yield a valid analytic depiction of a knowledge base indevelopment for teaching mathematics.

FINDINGS

In this section we report and present evidence for our main findings.These findings are organized by broad claims about teachers’ knowledge,including the extent to which it was integrated, and its relationship tocurriculum use. We discuss areas in which teachers’ knowledge of mathe-matics was more and less integrated with knowledge of children’s thinking,the role we believe the curriculum played in the development of thisknowledge, and teachers’ beliefs about children’s mathematics.

Analysis of Multidigit Multiplication Elicited Integrated Knowledge ofChildren’s Mathematics

Teachers’ knowledge of children’s mathematics was broadest and deepestin the topic multidigit multiplication. The evidence for this claim includes


the variety of nonstandard strategies teachers generated for multidigitmultiplication, their explanations of these strategies, and, for someteachers, the developmental progression among the variety of strategiesthey identified. All the strategies that teachers described were consistentwith children’s invented strategies documented elsewhere (e.g., Baek,1998), and several teachers framed their comments in terms of what theyhad observed children doing. Together, these two observations suggest weelicited teachers’ knowledge of children’s mathematical thinking, ratherthan only knowledge of mathematics.

When teachers were asked to generate three different strategies studentswould use to solve 18×25, all but one of the 13 teachers described at leastthree nonstandard strategies. As a group, teachers described six distinctstrategies beyond direct modeling and repeated addition (not includingthe standard algorithm for multiplication), suggesting breadth in teachers’knowledge of children’s mathematical thinking. These strategies also drewon conceptual features of the number system and the operation of multi-plication, indicating an integration of teachers’ knowledge of children’sthinking with knowledge of mathematics. We describe these strategies inorder of increasing mathematical sophistication.

The most basic strategy, after direct modeling and repeated addition,involved chunking the addends or successively doubling. Eight teachersdescribed how children would chunk 25s into 50s or 100s, then add,and another teacher said children would know that 10 25s is 250, thenrepeatedly add onto that. These teachers all mentioned the use of 25 as a“landmark” number, that is, an important “number we can use . . . to helpus tell where we are when we are counting or calculating with numbers”(Russell & Rubin, 1997, p. 27). Several teachers pointed out that childrenmight relate these numbers to money, facilitating chunking strategies.

Four teachers reported that children might round one of the factors toa multiple of 10 to make the multiplication easier, then compensate bysubtracting the extra factors. For example, Ms. Barill4 said “they wouldround up 18 to 20 and say, ‘20 times 25 is 500. Now I know that 20 is 2more than 18, so I need to take away 2 times 25 which is 50’ ”.

Finally, a majority of the teachers described breaking the multidigitmultiplication down into two, three, or four partial products, making useof the base-ten structure of the numbers and the distributive property.Using the terminology found in Investigations, teachers referred to thesestrategies as “cluster strategies”. For example, Ms. Rojas computed fourpartial products:

Okay, so I know that we do the 10 times 20, the 8 times 20 and then they would do the10 times 5 and the 8 times 5 (see Figure 1) . . . . And even in some of the children who


may mathematically be even at a lower stage, if 20 still seems even bigger for them, they’llbreak that down even smaller and do 10 times 10, 10 times 10 twice as opposed to the 10times 20 . . . . They would go ahead and multiply 10 times 20 and get the 200. And they’ddo the 8, 2 times, ah, the 20 times 8, excuse me, would be 160. And then they would do10 times 5, would get the 50, and the 8 times 5 would get 40. And then they would turnaround and add ‘em all up. Which is what I am going to attempt to do. And get 450.

Figure 1. Ms. Rojas’s nonstandard algorithm for multiplication.

Though Ms. Rojas’s strategy is isomorphic to the steps in the standardalgorithm for multiplication, her explanation of the strategy is couched interms that relate to how children might actually work through this problem,and demonstrates a flexible understanding of multiplication as an oper-ation. Specifically, the conceptual organization of the strategy suggeststhat she, along with the eight other teachers who described this familyof strategies, understood the implicit use of the distributive property tocompute partial products.5 This finding is in stark contrast to prior researchdocumenting teachers’ limited understanding of “why the numbers ‘moveover’ [and] what the number in the partial product on the second rowmean[s]” (NCTRE, 1992, p. 30).

Several teachers went beyond identifying a variety of strategies todescribe longitudinal connections among children’s strategies. Theseteachers appeared to have in mind a model for the genesis of multidigitmultiplication in children, from repeated addition, to chunking, to theuse of partial products in cluster strategies. Their observations were inaccord what has been documented about the development of children’smultidigit strategies elsewhere (Ambrose, Baek & Carpenter, in press;Baek, 1998). This literature describes strategies involving chunking asthe children’s first move away from repeated addition toward efficiencyin solving multidigit multiplication.


Essentially, these early invented strategies involve decomposing themultiplicand additively, and using that decomposition to chunk the multi-plier. As teachers here reported, children do not necessarily choose effi-cient chunks, although it is the teachers’ goal for students that they movetoward using cluster strategies based on multiplicative decomposition offactors. Several of these observations extended the information providedby Investigations, and so suggest that some of the teachers were engagedin extending their models of children’s thinking based on practical inquiry(Franke, Carpenter, Levi & Fennema, 2001; Richardson, 1990).

Analysis of Novel Division and Subtraction Strategies Elicited LessIntegrated Knowledge of Children’s Mathematics

When asked to interpret students’ novel nonstandard strategies, teachersdrew on knowledge that was, in comparison to knowledge of multi-digit multiplication, less integrated. In particular, teachers who wereable to make sense of students’ novel strategies within the constraintsof the interview tended to call on knowledge of mathematics discon-nected from knowledge of children’s thinking. Individual exceptions to thisfinding, however, provide instructive examples of teachers’ knowledge ofchildren’s mathematics. Evidence for these claims comes from teachers’responses to students’ strategies for multidigit division and subtraction.

Teachers found the division item more difficult than the multiplicationitem. When presented with a student’s novel strategy for dividing 144 by 8(see Table 1 above), which contained a mistake, only four out of the thir-teen teachers were able to make sense of the strategy within the constraintsof the interview. Three of these four teachers did so using explanations thatdrew on formal mathematics concepts, but did not connect with knowl-edge of children’s thinking. In particular, they recognized the student haddecomposed the divisor additively, when it should have been decomposedmultiplicatively, a significant insight (Simon, 1993). For example, Ms. Lee,who described herself as “a very mathematical person”, concluded that

[Decomposing the divisor multiplicatively] is a valid way to do it . . . if you’re at the pointwhere you can break down the number . . . where you can say [if dividing by 12], 12 is 2times 2 times 3. Yeah. Then it will work. (Solves 24 divided by 12 by dividing successivelyby 2, 2 then 3.) So if they understand factoring, then it’s a valid way to do it, but if they’renot at that point, then I would stop with, ‘It didn’t work.’

Ms. Lee further noted that if breaking the divisor down additively – as thefictional student had – worked, then dividing 144 by 7 then 1 (i.e., 7+1instead of 2+2+2+2) should also yield the correct answer. Other teachersarrived at a conclusion similar to Ms. Lee’s by referring to inverse opera-tions. Ms. Nichol worked her way back up the chain of repeated division by


two by repeatedly doubling the divisors, and discovered that the net resultwas that the student had divided by 16 instead of 8. Ms. Devine noted thatthe student

kept cutting in half, in half again, in half again, until you got two plus two plus two plustwo, when in essence that’s not really what you’re doing. This is really two times twotimes two . . . They have to know that division and multiplication are, you know, together.They’re opposites.

These teachers’ explanations demonstrated critical understanding of math-ematics, but little integration of mathematics and children’s thinking. Theyused the definition of division as the inverse of multiplication correctly topoint out why the strategy does not work. However, this type of knowledgedoes not provide a way to understand the possible origin of division in thischild’s thinking.

Ms. Tilden was the only teacher to use a partitive model of division –in which the goal is to partition the total into a given number of groups– consistently throughout her response. Her response, more than the otherteachers, preserved the integrity of the child’s strategy, because it referredto the steps the child had already taken to solve the problem, and reinter-preted them in light of what could have been the child’s goal – to partition144 into 8 groups. Talking about the first division by two, Ms. Tilden said,“Okay, there are two 72s in 144. And I would just ask her, ‘Well doesthis mean there are two 36s in 144? Or four 36s in 144?’ And I meanat this point I would probably be asking her to draw pictures, you know,have some kind of visual to show me what that would look like” (drawsFigure 2). Then later: “Maybe she’s saying there are two 18s in 36, sohow many 18s are there in 144? I’m just trying to figure out how theconversation would work . . .”.

Figure 2. Ms. Tilden’s model of student’s division strategy.


This explanation built on the child’s mathematics. Ms. Tilden’sapproach to helping the student map out all of the groups that were createdwith each successive halving of the partial quotient provided a way toconnect repeated halving and partitive division, an informal model that isreadily accessible to students. To make this connection, the student neededa way to keep track of the number of groups created by halving, since theprocess caused the number of groups to double each time, not increase bytwo, and should have stopped when eight groups have been created (thepartitive meaning of 144 ÷ 8).

Ms Tilden’s response suggests the power of children’s mathematicsas a knowledge base for teaching. Rather than treating the mistake as amisconception to be suppressed, Ms. Tilden found a way to build on it.Interestingly, unlike some of the other teachers, she did not discuss themistake in the child’s partitioning of the divisor. It is not clear, however,whether the distinction between multiplicative and additive partitions issalient in her interpretation of the strategy, since the mistake could beattributed to not keeping track of the number of groups, instead of whetherto add or multiply the twos. In other words, a child who is using repeatedhalving to divide may not be thinking ahead about when to stop.

The rest of the teachers used neither mathematical principles norchildren’s thinking to make sense of the strategy, and consequently couldnot explain why dividing by two four times in succession was not equiva-lent to dividing by eight. For example, Ms. Puma said she did not considerrepeated halving to be division – “she’s splitting it in half each time, butit’s not really divided” – suggesting she held an isolated understandingof division. Four teachers interpreted division primarily in measurementterms, that is, in terms of the number of eights that would fit into 144.Measurement division is a useful, accessible model of division for youngchildren (Carpenter, Fennema, Franke, Levi & Empson, 1999); however,its use in this context is not consistent with the fictional student’s strategy,and so limits the opportunity for a teacher to engage the child’s emergentthinking about the relationship between repeated halving and division.

The subtraction strategy (see Table I above) was easier than the divisionstrategy for teachers to interpret, yet, as with the division strategy, teachers’knowledge of mathematics and the extent to which it was integrated withknowledge of children’s thinking influenced the quality of the explanationsteachers offered.

Although most of the teachers were convinced of the strategy’s validityafter working a few well-chosen examples, one, Ms. Lee, constructed ajustification for the strategy. In reference to 36–19, Ms. Lee said, “It’sbasically the inverse of regrouping. Instead of regrouping and taking a 10


from here [the tens’ place], you are subtracting this [the ones’ column] andgetting negative three, which means you’re going to have to eventually . . .

take three away from [the tens’]”. In other words, Ms. Lee argued that thenegative difference in the ones’ place represented a deficit to be accountedfor in the tens’ place.

Additionally, Ms. Tilden and Mr. Garcia also argued based on principlesnot dependent on specific number combinations that the strategy general-ized. For instance, Mr. Garcia said it should work for combinations otherthan the ones he tried because

what you are dealing with is negative and positive numbers in essence and you are takingit one place at a time. You are dealing with the ones place and then coming up with ananswer and then dealing with the tens’ place and coming up with an answer . . . . Becausewhen you can deal with owing or negativeness, you realize that numbers can be owed aswell as had.

In essence, Mr. Garcia argued that, in problems that involve column addi-tion or subtraction, one can compute each column independently of theothers, and combine the partial sums or differences at the end. The strategyof treating ones and tens separately, then combining the results, is anapproach that has been noted in children’s strategies for addition as wellas subtraction (Carpenter et al., 1999; Fuson, 1992).

Curriculum Materials Influenced the Development of Teachers’Knowledge of Children’s Mathematics

Teachers’ knowledge of mathematics and children’s thinking was rela-tively well integrated for multiplication, but much less so for division, arelated topic, and subtraction. We speculate that differences in the generalquality of teachers’ responses to the multiplication item, compared to thedivision item in particular, may be explained in terms of the opportuni-ties for teacher learning created by teaching the curriculum. Teachers’self-reports of their learning and analysis of the curriculum materialscorroborate this claim.

First, most teachers reported that they had learned nonstandardstrategies for multidigit multiplication as a direct result of teaching Investi-gations – from interacting with students, preparing for lessons, or partici-pating in professional development workshops. Seven teachers specificallydescribed how they had increased their knowledge of multidigit multiplic-ation beyond knowing how to execute the standard algorithm. Each onereported that, before teaching Investigations, he or she knew only one wayto multiply multidigit numbers. Mr. Garcia, for example, said:

I was always very quick mathematically, mentally. And I would do it just in the way that Iwas taught . . . But through clustering what you can do is break a number down into more


familiar parts and numbers that are easier to work with . . . . By watching kids and doingwhat they do I’ve become better at breaking down numbers and understanding how to dealwith large numbers.

These teachers’ self reports were corroborated by their ability to generatetwo or more examples of students’ nonstandard multidigit multiplicationalgorithms later in the interview. Further, their descriptions of nonstandardstrategies for multidigit multiplication tended to be the most mathematic-ally detailed of any the teachers gave.

Second, in our analysis of Investigations units, we noted differencesin the treatment of multidigit multiplication and division in grades fourand five, where these topics are central to the curriculum. The activities incurriculum units such as “Arrays and Shares”, “Landmarks in the Thou-sands”, “Packages and Groups”, “Mathematical Thinking at Grade Five”,and “Building on Numbers You Know” develop multidigit multiplicationthrough skip counting, breaking apart numbers into more familiar parts,solving clusters of related problems,6 building towers of multiples, andusing landmark numbers, such as multiples of ten, to estimate products.These activities appear repeatedly and with increasing sophisticationthroughout these curriculum units, yielding a coherent, developmentaltreatment of multidigit multiplication. In contrast, there is less overallemphasis on the development of multidigit division. Further, althoughboth partitive and measurement interpretations of division are describedand illustrated, the kinds of strategies students learn for multidigit multi-plication, such as the use of skip counting and multiples towers, favora measurement interpretation when translated to multidigit division. Thecurriculum also encouraged teachers to reformulate numeral-only divisionsentences such x ÷ y as “how many ys are in x”, a further reinforce-ment of the measurement interpretation. The net result of this differentialtreatment of multiplication and division appeared to be the creation oflimited opportunities to learn about children’s informal partitive strategiesfor division.

Teachers Gained New Beliefs About Children’s Mathematics

Overall, 10 out of the 13 teachers expressed what they described as newbeliefs about children’s mathematical thinking that were a direct resultof teaching Investigations. The majority of these beliefs – expressed byseven teachers – had to do with students’ ability to solve problems ontheir own and to generate new mathematics. Mr. Jaimez noted that “kidsdo have an intuitive sense of math” and “can really be in charge of theirmathematical learning”. Ms. Capa said, “We’ve always thought, well,we’re the ones that give information. But they [the children] give me


information . . . They bring things to my mind”. These beliefs were alsoevidenced in teachers’ responses to the instructional scenarios, especiallyin how teachers valued the nonstandard strategies. Two teachers describedspecifically how meaning needed to be built for operations and proceduresbefore children learned to compute efficiently.

Teachers valued students’ novel strategies in particular, even when theydid not have the knowledge to assess thoroughly the mathematical orpedagogical viability of these strategies. Although these kinds of beliefsdo not require sophisticated domain knowledge, they also do not engagethe potential in students’ invented strategies to ground new mathema-tical understanding in children’s mathematics. For example, although twoteachers emphasized the importance of making productive use of students’mistakes in other parts of the interview, neither teacher did so for thedivision strategy, which contained a mistake; lack of specific knowledgeof children’s mathematics may have limited teachers’ ability to act on thisbelief is this situation.

The data suggest teachers had acquired a predisposition to elicitstrategies from children, and to expect a variety of responses. Theybelieved quite strongly that they, as well as their children, were bene-fiting from this kind of approach to mathematics. All these beliefs wereconsistent with instructional frameworks based on current mathematicslearning research (e.g., Hiebert, Carpenter, Fennema, Fuson, Wearne,Murray, Olivier & Human, 1997), and so suggest movement toward beliefsthat reinforced teachers’ use of children’s mathematics as a foundationfor instruction and as a context for their own learning. However, withoutsubstantial knowledge of children’s mathematics, it may be difficult, if notimpossible, fully to actualize these kinds of beliefs.

DISCUSSION

We began this study with two questions: (1) What knowledge do teacherswho have implemented a student-centered curriculum designed to supportteacher learning use to make sense of students’ nonstandard strategies?and, (2) in what ways might teachers’ acquisition of this knowledge belinked to the use of the new curriculum materials?

We found that, collectively, the teachers exhibited a fair amount ofknowledge of children’s mathematics in a single topic area, although thestory is not one of uniformly high knowledge across topic areas. Teachershad broad, and in some cases deep, knowledge of nonstandard strategiesfor multidigit multiplication represented in the curriculum materials, butnot of less common strategies. We also found some teachers’ knowledge of


mathematics was disconnected from knowledge of children’s thinking, andthis disconnection influenced their hypothesized responses to children’sthinking. These teachers treated a mistake in thinking about division as awrong idea to be corrected. One teacher, using knowledge of children’sthinking about division, saw, within the child’s mistake, productive, albeitinformal, thinking about division, and described an interaction that couldbuild on her thinking.

We also found that teachers’ knowledge of children’s mathematicsappeared to be, in large part, a result of implementing Investigations inNumber, Data, and Space. Knowledge of children’s mathematics was,for these teachers, most robust in multiplication, a topic that was treatedexplicitly in the curriculum materials in terms of its mathematical organi-zation and the development of children’s thinking. We speculate thatthis attention to children’s mathematical thinking allowed the curriculumwriters to design tasks that elicited fairly predictable patterns of reasoning,and to alert teachers to the variety and meanings of those strategies. Onceteachers realized their students could indeed generate their own strategiesto solve problems, it appears that they were motivated to understand andextend these strategies as much as possible. Several teachers, such asMs. Rojas, related personally to the mathematical empowerment they felttheir students were experiencing from implementing the curriculum andinteracting with students.

Significantly, there were limitations in teachers’ knowledge that were aconsequence, we believe, of limitations in teachers’ opportunities to learnfrom the curriculum, as described above, and from each other. Consider,for example, the fact that the Chinese teachers in Ma’s (1999) study whoexhibited Profound Understanding of Fundamental Mathematics (PUFM)had, on average, 18 years of experience teaching in ways that empha-sized conceptual understanding and gave rise to PUFM. During this time,teachers regularly planned together and discussed their teaching with eachother. The teachers in our study, who ranged in total teaching experience,had at most two years’ experience teaching in a way that could, we haveargued, give rise to knowledge children’s mathematics. Thus, experienceswith this type of teaching and opportunities to plan and discuss with peerswere limited for these U.S. teachers in comparison with Chinese teacherswith PUFM.

How might these limitations be addressed? A persistent theme,throughout our findings, was teachers’ reliance on what students said anddid to help them see new ways of thinking mathematically. When teacherswere not sure how to interpret a strategy, they often expressed a desire totalk further with the fictional student. More teachers singled out students


over curriculum materials and workshops as a source of learning. If dataabout students’ thinking are marshaled so that teachers may debate andanalyze what their students understand, students represent a potentiallypowerful learning resource for teachers.

We suggest, in particular, that teachers’ knowledge of mathematicsneeds to be developed in tandem with knowledge of children’s thinking(Philipp et al., 2003; Schifter, 2001), and that this learning should be basedon teachers’ interactions with their own students when possible (Franke,Kazemi, Shih & Biagetti, 1998). In addition to using knowledge frame-works of common developmental patterns in children’s thinking as a basisfor learning mathematics (e.g., Carpenter et al., 1999), our findings suggestthat instances of novel or mistaken thinking can also serve as catalysts tolearning mathematics. Although teachers struggled, as individuals, withthe division item, for example, they had the collective mathematical andpedagogical knowledge to make sense of it. We can imagine that, had theteachers had opportunities to listen to and to discuss each other’s responsesin a formally organized school-based forum, teachers’ understanding couldhave been measurably enhanced. As teachers become more attuned tostudents’ invented strategies beyond those that are common, collectiveexamination of novel or mistaken strategies could facilitate the kind ofmathematics learning called for in the literature (Ball et al., 2002; Even &Tirosh, 2002).

For many teachers, implementing Investigations fostered a desire tocontinue learning because, we maintain, the curriculum helped themformulate and address problems at the heart of their mathematical workwith children. At the end of the interview, Mr. Jaimez said, “What Iwould like to do is take some time, if I can, to spend some time lookingat these problems and, I mean, I’d love to figure out some more aboutit”. Perhaps it is not surprising that the learning most salient to teachersdid not take place in district-wide workshops. The most pressing instruc-tional dilemmas teachers face with reform have more to do with decidingwhether or not a student’s nonstandard algorithm for multidigit subtractionis legitimate in the classroom context of discussing and extending manyother students’ strategies, than with following a curriculum script (Junk, inprogress).7 Solutions to instructional dilemmas cannot be provided in theform of curriculum materials or knowledge frameworks, although thesetypes of resources play a facilitative role, as the current study suggests.

One could argue that our study shows, once again, how weak U.S.elementary teachers’ knowledge of mathematics is. After all, the teachershad participated in district-sponsored professional development sessionson the implementation of Investigations in Number, Data, and Space four


– eight times each, yet had difficulty making sense of alternative strategiesfor subtracting and dividing. We reject this interpretation, however,because we identified critical differences between teachers in our studyand teachers in prior studies (e.g., Ma, 1999; Mewborn, 2000). Further, weinterpret teachers’ difficulties as an indication of the complexity of the kindof knowledge teachers are expected to have and to use in order to imple-ment reforms in mathematics education. We have argued, in particular,that teachers’ integrated knowledge of children’s thinking and mathe-matics, in combination with productive beliefs, differs from knowledgeof mathematics in isolation from knowledge of children’s thinking.

CONCLUSION

The most astonishing result of this study is that every teacher noted someaspect of their newfound respect for children’s mathematics capabilities.If nothing else, the teachers had acquired an epistemological dispositionthat acknowledged the existence of a multiplicity of strategies, and wasfounded on a high regard for children’s thinking.

However, one thing that Investigations did not do – and, as a set ofcurriculum materials, cannot be expected to do – was to prepare teachersto detect and build on the mathematical potential in students’ nonstandardstrategies, in addition to what was presented in the curriculum. Thereis a great deal of intellectual work involved in learning to hear and tounderstand the mathematical significance of children’s thinking that goesbeyond acquiring the specifics of knowledge of mathematics. It is anongoing process that requires formal, school-based structures for profes-sional learning be put in place, such as those reported by Ma (1999)for Chinese teachers and by Lewis (2000) for Japanese teachers. Neitherformal preparation in mathematics nor mathematics teaching methodscourses can fully prepare teachers to engage with student thinking in thisway.

There has been a great deal of attention to research on teacher knowl-edge that is focused on teachers’ insufficient understanding of the mathe-matics. In fact, several teachers in this study commented ruefully on theirown mathematics preparation. Noting that she used to hate mathematics asa student, Ms. Rojas said, “But now I’m like, ‘I could’ve been an awesomemath student . . . had somebody taught it this way!’ ” Our study suggeststhat the implementation of an innovative curriculum fostered knowledge,beliefs, and values that were to a notable extent shared by the teachers inone school. The challenge for U.S. schools is to recognize the learning thatteachers like Mr. Jaimez and Ms. Rojas are eager to take on, and to organize


school-based learning structures that build on teachers’ daily work withchildren.

ACKNOWLEDGEMENTS

We gratefully acknowledge Linda Levi and Corey Drake for their feedbackon earlier drafts of this manuscript. We also thank three anonymous JMTEreviewers and Editor Peter Sullivan for their numerous helpful suggestionsfor improvement. The first author thanks the Department of Curriculumand Instruction, University of Wisconsin-Madison, for support as a visitingscholar during the time the majority of this report was written.

NOTES

1 In the U.S. schools are grouped into administrative and policy-setting units called“districts”. Districts can correspond to cities (so, for example, Austin, Texas has one schooldistrict, the Austin Independent School District), though – especially in the case of largercities – not necessarily (for example schools in San Antonio, Texas are grouped into severaldistinct districts). The decision to use Investigations for elementary mathematics instruc-tion was made at the district, rather than school, level.2 For example, in the set of integers, only two elements have a multiplicative inverse (1and –1). In the set of rationals (all a/b with b not equal to 0), every element but 0 has amultiplicative inverse. One way to think about rational numbers is as an extension of theintegers so that the operation of division is closed (i.e., for all a and b in the set of rationals,with b not equal to 0, a ÷ b yields a unique rational).3 We do not consider these two content frameworks to be mutually exclusive; in fact webelieve they can inform each other. However, we conjecture they orient teachers to funda-mentally different kinds of teacher-student interactions, and perhaps ultimately, differentkinds of student outcomes.4 All teachers’ names are pseudonyms.5 The other 4 teachers did not use strategies that demonstrated understanding of thedistributed property.6 E.g., 3×3, 6×3, 12×3; i.e., problems “arranged in such a way that if you figure out oneof the facts, it may help you find a clever strategy to solve some of the others in the cluster”(“Arrays and Shares”, p. 22).7 Junk, D.L. (in progress). Teaching mathematics and the dilemmas of practice. Unpub-lished doctoral dissertation, University of Texas at Austin.

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Science and Mathematics Education1 University Station, D5705University of Texas at AustinAustin, TX 78712-0382USAE-mail: [email protected]

SHUHUA AN, GERALD KULM and ZHONGHE WU

THE PEDAGOGICAL CONTENT KNOWLEDGE OF MIDDLESCHOOL, MATHEMATICS TEACHERS IN CHINA AND THE U.S.

ABSTRACT. This study compared the pedagogical content knowledge of mathematicsin U.S. and Chinese middle schools. The results of this comparative study indicatedthat mathematics teachers’ pedagogical content knowledge in the two countries differsmarkedly, which has a deep impact on teaching practice. The Chinese teachers empha-sized developing procedural and conceptual knowledge through reliance on traditional,more rigid practices, which have proven their value for teaching mathematics content. TheUnited States teachers emphasized a variety of activities designed to promote creativityand inquiry in attempting to develop students’ understanding of mathematical concepts.Both approaches have benefits and limitations. The practices of teachers in each countrymay be partially adapted to help overcome deficiencies in the other.

KEY WORDS: pedagogical content knowledge, mathematics teaching, student’scognition, teacher’s knowledge, unit fraction

During the past several decades, there has been increased attention tocomparative studies in mathematics education, especially with respect tothe movement of reforming mathematics education in the beginning ofthe 21st Century. According to Robitaille and Travers (1992), compar-ative study provides opportunities for sharing, discussing, and debatingimportant issues in an international context. Stigler and Perry (1988)observe:

Cross-cultural comparison also leads researchers and educators to a more explicit under-standing of their own implicit theories about how children learn mathematics. Withoutcomparison, teachers tend not to question their own traditional teaching practices and arenot aware of the better choices in constructing the teaching process (p. 199).

In 1996, U.S. eighth and twelfth graders scored below average in math-ematics when compared with other countries in the Third InternationalMathematics and Science Study (TIMSS) assessment (Silver, 1998). In1999, U.S. eighth-grade students scored slightly above the internationalaverage in mathematics and science performance according to the ThirdInternational Mathematics and Science Study-Repeat (TIMSS-R) whencompared with students in 37 participating nations. This report indicatedthat there has been improvement in the U.S. in mathematics education.However, to compete globally and achieve a top rank internationally,


146 SHUHUA AN ET AL.

mathematics education in the U.S. would benefit from continuing toexamine international mathematics education practices and research.

Teachers and teaching are found to be one of the major factors relatedto students’ achievement in TIMSS and other studies. According to theNational Council of Teachers of Mathematics (2000), “Effective teachingrequires knowing and understanding mathematics, students as learners,and pedagogical strategies” (p. 17). In an era of globalization and infor-mation, teachers’ knowledge in mathematics is becoming more complexand dynamic (Fennema & Franke, 1992). New aspects of teaching, suchas knowledge of technology, must be mastered. However, the balance andintegration of pedagogy and content knowledge, referred to as pedagogicalcontent knowledge (Shulman, 1987; Pinar, Reynolds, Slattery & Taubman,1995), should be the most important element in the domain of mathematicsteachers’ knowledge. Pedagogical content knowledge addresses how toteach mathematics content and how to understand students’ thinking. Thisincludes, taking into consideration both the cultural background of thestudents as well as their preferences for various teaching and learningstyles. The purpose of this study was to examine the differences betweenChina and the United States of teachers’ pedagogical content knowledgein mathematics at the middle school level.


Shulman’s Model of Pedagogical Content Knowledge

According to a Chinese saying, if you want to give the students one cupof water, you (the teacher) should have one bucket of water of your own.Shulman (1985) believes that “to be a teacher requires extensive and highlyorganized bodies of knowledge” (p. 47). Elbaz (1983) has the same view,“the single factor which seems to have the greatest power to carry forwardour understanding of the teacher’s role is the phenomenon of teachers’knowledge” (p. 45).

Shulman (1987) has stated further that pedagogical content knowledgeincludes knowledge of learners and their characteristics, knowledge ofeducational contexts, knowledge of educational ends, purposes and values,and their philosophical and historical bases. Pedagogical content knowl-edge refers to the ability of the teacher to transform content into forms thatare “pedagogically powerful and yet adaptive to the variations in abilityand background presented by the students” (Shulman, 1987, p. 15). TheNetwork of Pedagogical Content Knowledge In the current study, pedago-gical content knowledge is defined as the knowledge of effective teaching

MATHEMATICS TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE 147

which includes three components, knowledge of content, knowledge ofcurriculum and knowledge of teaching. This is broader than Shulman’soriginal designation. Knowledge of content consists of broad mathematicsknowledge as well as specific mathematics content knowledge at thegrade level being taught. Knowledge of curriculum includes selecting andusing suitable curriculum materials, fully understanding the goals and keyideas of textbooks and curricula (NCTM, 2000). Knowledge of teachingconsists of knowing students’ thinking, preparing instruction, and masteryof modes of delivering instruction.

Although all three parts of pedagogical content knowledge are veryimportant to effective teaching, the core component of pedagogical contentknowledge is knowledge of teaching. Figure 1 suggests the interactiverelationship among the three components and shows that knowledge ofteaching can be enhanced by content and curriculum knowledge.

Figure 1. The network of pedagogical content knowledge.


Ma’s study (1999) calls for teachers to have “profound understandingof fundamental mathematics”. However, as indicated in Figure 1, profoundcontent knowledge alone is not sufficient for effective teaching. Aneffective teacher must also possess a deep and broad knowledge ofteaching and curriculum or profound pedagogical content knowledge. Withthis knowledge, teachers are able to connect their knowledge of content,curriculum, and teaching in a supportive network. In this network, threetypes of knowledge interact with each other and are able to make trans-formations from one form to another around the central task of teaching.Ultimately, these components together address the goal of enhancingstudents’ learning. As shown in Figure 1, this network of knowledge isimpacted by teachers’ beliefs. Ernest (1989) and Fennema and Franke(1992) also reveal the importance and impact of teachers’ beliefs on theirknowledge. Different educational belief systems produce different attrib-utes of pedagogical content knowledge. In turn, profound pedagogicalcontent knowledge plays an important role in shaping teachers’ beliefs andin determining the effectiveness of their mathematics teaching (An, Kulm,Wu, Ma & Wang, 2002).

Teaching can be seen as either a divergent or a convergent process. Adivergent process of teaching is one that is based on content and curriculumknowledge but is without focus and ignores students’ mathematicalthinking. A convergent process of teaching focuses on knowing students’thinking, which consists of four aspects: building on students’ mathe-matical ideas, addressing students’ misconceptions, engaging students inmathematics learning, and promoting students’ thinking mathematically.Together, these four aspects of convergent teaching comprise the notion ofteaching with understanding, which is an essential to effective teaching(Carpenter & Lehrer, 1999). Under a convergent process, students, nottextbooks and curriculum, are the center of teaching. Throughout theconvergent teaching process, an effective teacher attends to students’mathematical thinking: preparing instruction according to students’ needs,delivering instruction consistent with students’ levels of understanding,addressing students’ misconceptions with specific strategies, engagingstudents in activities and problems that focus on important mathematicalideas, and providing opportunities for students to revise and extend theirmathematical ideas (Kulm, Capraro, Capraro, Burghardt & Ford, 2001).

There are two kinds of teaching beliefs regarding students’ learning:learning as knowing and learning as understanding. A teacher who holdsthe belief of learning as knowing often assumes that mathematics is learnedand understood if a concept or skill is taught. This type of learning usuallyis achieved at a surface level. Teachers are often satisfied with students’


knowing or remembering facts and skills but are not aware of students’thinking or misconceptions about mathematics. This divergent teachingprocess often results in fragmented and disconnected knowledge.

A teacher who holds the belief of learning as understanding realizesthat knowing is not sufficient and that understanding is achieved at thelevel of internalizing knowledge by connecting prior knowledge througha convergent process. In this process, the teacher does not only focuson conceptual understanding and procedural development, making surestudents that comprehend and are able to apply the concepts and skills, butalso consistently inquires about students’ thinking. Teachers who use thisconvergent process develop systematic and effective ways to identify anddevelop their students’ thinking. These ideas are summarized in Figure 2,showing that with profound knowledge of students’ thinking, teachers canenhance students’ learning substantially, leading to content mastery.

Figure 2. Two types of learning.

Although there has been some research comparing Chinese and U.S mathematics teachers’ content knowledge (e.g., Ma, 1999), there hasbeen very little research comparing their profound pedagogical contentknowledge. Ma’s work (1999) focuses on comparing elementary teachers’content knowledge, without fully accounting for cultural contexts, goalsand teacher beliefs. Her study documented that Chinese teachers withmore mathematical training seemed to know more than U.S. mathematicsteachers. Many of Ma’s examples did provide indications about howteachers apply their mathematics knowledge in teaching, but stopped shortof a systematic study of how mathematical and pedagogical knowledgewas integrated with knowledge of students’ thinking.

The current study focuses on the middle-school level, addresses mathe-matics teachers’ pedagogical content knowledge within a cultural context,and explores how this knowledge is used by teachers to understand anddevelop students’ mathematical thinking. The question that provided the


focus for this study was: What are the differences in teachers’ profoundpedagogical content knowledge between middle school mathematicsteachers in China and the United States?

METHODOLOGY

Subjects

The subjects were 28 mathematics teachers in fifth- to eighth-grade levelsfrom 12 schools in four school districts in a large metropolitan area inTexas and 33 mathematics teachers in fifth- and sixth-grade levels from 22schools in four school districts in a large city in Jiangsu province in easternChina. In order to examine the teachers’ profound pedagogical contentknowledge at the middle school level (particularly in the area of fraction,ratio, and proportion), this study included U.S. teachers from fifth to eighthgrade, because U.S. mathematics curricula in these grades are similar tothose in fifth and sixth in China.1

Criteria for inclusion of teacher volunteers in the study were: (1) currentteaching of mathematics in fifth to eighth grades; (2) teaching in schooldistricts that have characteristics typical of each nation’s public schoolswith respect to the students’ ethnic, economic, and cultural diversity; (3)having at least three years of teaching experience at the fifth to eighthgrade levels; and (4) willing to provide the data relevant to the reliabilityand validity of this study, including classroom observations and interviews.

The U.S. teachers all had bachelor’s degrees; three had master’sdegrees. They had an average of 24 hours of mathematics course workand an average of 13 years teaching experience. Only one participanttaught fifth-grade mathematics, 14 of them taught sixth grade, 7 wereseventh-grade teachers, and 6 were eighth-grade teachers. It should benoted that the U.S. teachers who participated in this study only teachmathematics. All of the Chinese teachers had three-year education degreesat Normal schools after ninth grade; 23 also had three-year universitydegrees, including 10 who majored in fields other than mathematics. Theaverage number of hours in mathematics courses for the Chinese teacherswas 15. For example, with a three-year degree at university, a teacherhad calculus, modern algebra, elementary mathematics methods, history ofelementary mathematics education, and Olympic elementary mathematics.Their average length of teaching experience was 9 years. Six of them werefifth-grade teachers, and 28 taught sixth grade. As in the case of the U.S.teachers, all Chinese teachers in this study only taught mathematics.


The U.S. schools were located in both urban and suburban areas andthe school populations ranged from 800 to 1300 students. The ethniccomposition of the schools was similar and, on average, consisted of 30%African American, 27% Caucasian, 27% Hispanic and 16% Asian. Schoolsin China were located in urban areas and the number of students in eachschool ranged from 1000 to 1200. All students were from the same ethnicgroup.

Procedures

Data were collected with an author-constructed Mathematics TeachingQuestionnaire, an author-constructed Teachers’ Beliefs about MathematicsTeaching and Learning Questionnaire, and interviews and observationswith selected teachers. Both questionnaires were prepared first in English,and then translated into Chinese. This article will focus on an analysis ofthe Mathematics Teaching Questionnaire.

Mathematics Teaching QuestionnaireThe questionnaire consisted of four problems that were designed toexamine teachers’ profound pedagogical content knowledge in topicsof fractions, ratios, and proportion (see Figure 3). Each of the fourproblems focused on one aspect of teachers’ knowledge of students’cognition, with attention to assessing teachers’ knowledge and strategiesfor, namely, building on students’ mathematics ideas, identifying andcorrecting students’ misconceptions, engaging students in learning, andpromoting student thinking.

Classroom ObservationsAfter reviewing and analyzing the responses to the questionnaires, fiveteachers from each country were selected for observation to confirmthat their teaching matched their responses on the questionnaire. Theywere chosen so as to represent a range of education background, lengthof teaching experience, and level of responses to the questionnairesfor classroom observations. The observations were conducted at a pre-arranged date and time. Field notes and audiotape recordings were madeduring the classroom observations using an Instructional Criteria Observa-tion Checklist that was constructed as a guide. The Checklist was adaptedfrom criteria used for analyzing the instructional quality of mathematicstextbooks (AAAS, 2000). The observation criteria included specific activ-ities in the categories: building on student ideas in mathematics, beingalert to students’ ideas, identifying student ideas, addressing misconcep-tions, engaging students in mathematics, providing first-hand experiences,


Mathematics Teaching Questionnaire

Problem 1Adam is a 10-year-old student in 5th gradewho has average ability. His grade on the lasttest was an 82 percent. Look at Adam’s writtenwork for these problems:

3

4+ 4

5= 7

92

1

2+ 1

1

2= 3

2

5

a. What prerequisite knowledge might Adam not under-stand or be forgetting?

b. What questions or tasks would you ask Adam, in orderto determine what he understands about the meaning offraction addition?

c. What real world example of fractions is Adam likely tobe familiar with that you could use to help him?

Problem 2A fifth-grade teacher asked her students towrite the following three numbers in orderfrom smallest to largest:

3

8,

1

4,

2

3

Latoya, Robert, and Sandra placed themin order as the follows.

Latoya:1

4,

2

3,

3

8Robert:

2

3,

1

4,

3

8Sandra:

1

4,

3

8,

2

3

a. What might each of the students be thinking?b. What question would you ask Latoya to find out if your

opinion of her thinking is correct?c. How would you correct Robert’s misconception about

comparing the size of fractions?

Problem 3You are planning to teach procedures for doingthe following type of fraction multiplication.

a. Describe an introductory activity that would engage andmotivate your students to learn this procedure.

b. Multiplication can be represented by repeated addition,by area, or by combinations.Which one of these representations would you use toillustrate fraction multiplication to your students? Why?

c. Describe an activity that would help your students under-stand the procedure of multiplying fractions.

Problem 4Your students are trying to solve the followingproportion problem:The ratio of girls to boys in Math club is 3:5.If there are 40 students in the Math club, howmany are boys?

a. Describe an activity that you would use to determinethe types of solution strategies your students have usedto solve the problem. Here are two students’ solutionsto the problem:

June’s solution:3

5= x

40

There are 24 girls, so there 16 boys.Kathy’s solution:

3

8= x

40

There are 15 boys.b. What question would you ask Kathy to determine if she

could justify her answer and reasoning?c. What suggestion would you provide to June that might

help her correct her approach?d. What strategy would you use to encourage your students

to reflect on their answers and solutions?

Figure 3. Mathematics teaching questionnaire.


promoting student thinking about mathematics, guiding interpretation andreasoning, and encouraging students to think about what they had learned.

InterviewsAfter each observation, an interview was conducted using a set of interviewquestions. The objective of the interviews was to examine teachers’ beliefsabout the goals of mathematics education and the impact of their beliefs ontheir teaching practices, to investigate the teaching approaches they use inthe classrooms, to learn how the teachers prepare for instruction and howthey determine their students’ thinking. The interview questions exploredfurther the teachers’ pedagogical content knowledge and its importance intheir teaching.

Data Analysis

QuestionnaireA constant comparative data analysis method was used in the analysis ofthe Mathematics Teaching Questionnaire. In all, 18 different categorieswere identified which included the responses to the four problems. Theresponses were categorized into groups and assigned a descriptive code.Two researchers used the resulting codes to analyze the responses inde-pendently. Both sets of codes were compared, and then, through discussionwith the third researcher, the disparities were reconciled to reach valu-able agreements on the responses. Table I lists the categories and theirdefinitions. In Table II, the 18 categories are grouped according to thefour components of pedagogical content knowledge in the conceptualframework.

Interviews and ObservationsTranscriptions were made of the interviews. The transcriptions were codedusing the 18 response categories and the four components of pedagogicalcontent knowledge. The responses to interview questions and the fieldnotes and checklists from the observations were also analyzed through theuse of concept mapping to clarify key teacher ideas and beliefs. Thesedata confirmed that the teachers’ responses to the Mathematics TeachingQuestionnaire were consistent with their actual classroom practices andtheir knowledge and beliefs about effective teaching.

RESULTS

The responses of the U.S. and Chinese teachers to the MathematicsTeaching Questionnaire are presented in Table III, based on the 18


TABLE I

Categories for Describing Teachers’ Responses to Pedagogical Content KnowledgeQuestions

Category Brief definition

1. Prior knowledge: Know students’ prior knowledge and connect it to newknowledge.

2. Concept or definition: Use concept or definition to promote under-standing.

3. Rule and procedure: Focus on rule and procedure to reinforce the knowl-edge.

4. Draw picture or table: Use picture or table to show a mathematical idea.

5. Give example: Address a mathematical idea through examples.

6. Estimation: Solve problems using estimation.

7. Connect to concrete model: Use concrete model to demonstrate mathema-tical idea.

8. Students who do not understand prior knowledge: Students lack inunderstanding of prior knowledge.

9. Provide students opportunity to think and respond: Promote students tothink problems and give them chances to answer questions.

10. Manipulative activity: Provide hands-on activities for students to learnmathematics.

11. Attempts to address students’ misconceptions: Identify students’ miscon-ceptions.

12. Use questions or tasks to correct misconceptions: Pose questions orprovide activities to correct misconceptions.

13. Use questions or tasks to help students’ progress in their ideas: Posequestions or provide activities to increase the level of understanding forstudents.

14. Provide activities and examples that focus on student thinking: Createactivities and examples that encourage students to ponder questions.

15. Use one representation to illustrate concepts: Apply repeated addition toaddress the meaning of fraction multiplication, or use area to address thegeometrical meaning of fraction multiplication.

16. Use both representations to illustrate fraction multiplication: Apply bothrepeated addition and area to address the meaning of fraction multiplica-tion.

17. Unintelligible response: Provide response that is not relevant to thequestion.

18. Incorrect: Provide a wrong answer.


TABLE II

Categories for Describing Four Aspects of Teaching to Pedagogical Content KnowledgeQuestions

Pedagogical content Essential components Categoryknowledge Number

Building on students’ 1. Connect to prior knowledge 1,8

math ideas 2. Use concept or definition 2

3. Connect to concrete model 7

4. Use rule and procedure 3

Addressing students’ 1. Address students’ misconceptions 11

misconceptions 2. Use questions or tasks to correct misconceptions 12

3. Use rule and procedure 3

4. Draw picture or table 4

5. Connect to concrete model 7

Engaging students 1. Manipulative activity 10

in math learning 2. Connect to concrete model 7

3. Use one representation (area) 15

4. Use both representations (area & repeated 16

addition)

5. Give example 5

6. Connection to prior knowledge 1

Promoting students’ 1. Provide activities to focus on students’ thinking 14

thinking about 2. Use questions or tasks to help students’ progress 13

in their ideas

mathematics 3. Use Estimation 6

4. Draw picture or table 4

5. Provide opportunity to think and respond 9

categories and four components of pedagogical content knowledge. Sinceeach teacher’s responses could be coded into more than one of the 18categories, the resulting total percentage is greater than 100 for eachproblem.

On the one hand, the results show that both groups of teachers haveextensive and broad pedagogical content knowledge and are able to applyvarious methods to help students learn mathematics. However, there aresome important differences in each of the four components of teaching


TABLE III

Percentage of U.S. and Chinese Teachers for Each Category of Response to Problem 1to 4

Pedagogical content Essential components Problem Category US Chinaknowledge number number % %

Building on students’ 1. Connect to prior knowledge

math ideas Forget prior knowledge 1.a 1 46 27

Does not understand prior 1.a 8 11 55

knowledge

2. Use concept or definition 1.b 2 29 51

Concept 29 21

Unit fraction 0 30

3. Connect to concrete model 1.c 7 93 42

4. Rule and procedure 1.b 3 25 76

5. Unintelligible response 1.b 17 12 3

Addressing students’ 1. Address students’ 2.a 11 86 97

misconceptions misconceptions

2. Use questions or tasks to 2.c 12 61 100

correct misconceptions

3. Use rule and procedure 2.c 3 11 42

4. Draw picture or table 2.c 4 29 30

5. Connect to concrete model 2.c 7 26 12


Engaging students in 1. Manipulative activity 3.a 10 39 18

math learning 2. Connect to concrete model 3.a 7 36 64

3. Use one representation (area) 3.b 14 64 28

4. Use both representations 3.b 15 11 67

(area & repeated addition)

5. Give example 3.a 5 4 91

6. Connect to prior knowledge 3.a 1 7 45

7. Unintelligible response 3.c 17 25 24

Promoting students’ 1. Provide activities to focus on 4.d 14 68 94

students’ thinking students’ thinking

about mathematics 2. Use questions or tasks to 4.b 13 57 100

help students’ progress in

their ideas

3. Use estimation 4.c 6 4 6

4. Draw picture or table 4.a 4 14 15

5. Provide opportunity to think 1.b 9 61 79

and respond



for understanding between the U.S. and Chinese teachers. A discussion ofeach difference between the two groups of teachers follows.

Building on Students’ Ideas about Fractions

Understanding and ForgettingAs shown in Table III, 46% of the responses of the U.S. teachers toProblem 1(a) indicated that, in their view, Adam forgot the prerequisiteknowledge of finding common denominators, while 55% of the responsesof the Chinese teachers had an opposite view, namely, that Adam did notunderstand the prerequisite knowledge of finding common denominators.For example, Mrs. Ross mentioned that Adam forgot prior knowledge:“Adam does not remember that to add, you must add like denominators,and fourths and fifths are not alike. He does not remember to makeequivalent fractions using the lowest common denominator”.

Mrs. Wang pointed out that Adam did not understand the fractionbecause he seemed to separate numerator and denominator into inde-pendent parts. Furthermore, Mrs. Sheng, Mrs. Wang, and Mr. He indicatedthat only with like units can two numbers be added, such as 3 books + 5books = 8 books. The Chinese teachers realized that three books cannot beadded to 4 desks because “book” and “desk” are different units. A fractionis a number, so only with like unit fractions can two fractions be added;therefore with unlike unit fractions, two fractions cannot be added. Adam’smistake indicated that he did not understand the concept of like units andcould not see the connection between whole number and fraction, whichmay misdirect him to think fraction as something else but not a number.

Here “forgot” and “did not understand” have two distinct meaningsfor the teachers. The teachers who said the student “forgot” did not knowtheir students’ thinking about fraction addition and did not understand thechallenges students are likely to encounter in learning fraction addition.They appeared to believe that learning simply consists of knowing or notknowing; that is, remembering or forgetting. In contrast, teachers whosaid that the student “did not understand” showed evidence of knowingstudents’ thinking about fraction addition.

A large percentage of Chinese teachers connected prior knowledge ofwhole number addition to fraction addition, that is, that numbers with likeunits can be added. This means that no matter what numbers they are(such as whole numbers, decimals and fractions) as long as the numbershave the same unit, they can be added together. This connection not onlyhelps students to see a fraction as a number, but also helps students tounderstand and to use the rule of fraction addition easily. Understandingalso means that students are able to internalize a concept and use it in


different situations, such as understanding like units in whole numberaddition and applying like units in fraction addition. Internalizing andconnecting knowledge about like units into a coherent whole provides aclose link that makes learning easier and leads to mastery.

Use of Models to Develop Concepts and ProceduresThe results in Table III show that the 93% of the U.S. teachers tended tobuild on students’ ideas of addition of fractions with various approaches,by focusing on the connection with concrete or pictorial models. Incontrast, only 42% of Chinese teachers used concrete models. Most ofthem used definitions or the unit fraction concept to develop students’knowledge of addition of fractions by emphasizing procedural develop-ment and the following of rules.

The U.S. teachers used a wide variety of concrete and pictorial models,including pizzas, pies and cakes, a Hershey bar, an egg carton, crayonboxes, measuring cups, sports, money, time, and fraction pieces. The useof concrete models and pictures helps students to visualize and explorea mathematics concept and to connect learning with their experiences.This learning will be meaningful and will make sense to students. Thisapproach is supported by research showing that mathematics should servestudents’ needs to make sense of experience arising outside of mathematicsinstruction (Fennema & Romberg, 1999).

For Chinese teachers, conceptual understanding is very important inlearning. As shown in Table III, only 29% of the U.S. teachers empha-sized concepts in developing students’ fraction ideas, while 51% Chineseteachers focused on concepts to build understanding. Of these, 30% usedthe unit fraction concept. The Chinese teachers believed that the unit frac-tion is a critical concept in learning fractions and is more easily understoodby children. For example, Mrs. Li, a sixth grade teacher, suggested thefollowing four questions when responding to part (b) of Problem 1:

1. What is the unit of the fraction?2. What fractions can be added directly? Give the problems 1/5 + 2/5, 1/3

+ 2/3, and 5/6–1/6 to Adam. Can you give a picture problem for this?3. What does it mean to find the common denominator? Why do you need

to find the common denominator? And how do you find the commondenominator?

4. How do you add fractions with unlike denominators?

Mrs. Li provided an example and explained it with Figure 4 andFigure 5: “Look at the example 1/3 + 1/2”. 1/3 + 1/2 does not have acommon denominator; in other words, the unit fractions are unlike, so


numerators cannot be added directly. Figure 4 shows that 1/3 and 1/2 donot have equal sized parts.

Figure 4. Circles for comparing unit fractions.

To help the students find the like unit fractions, Mrs. Li explainedfurther: “To divide the unit ‘1’, a circle, into 6 equal-sized parts, the unitfraction becomes 1/6 for both circles”. The unit fraction is now the same(see Figure 5).

Figure 5. Circles for like unit fractions.

The questions that Mrs. Li asked, involving the unit fraction concept,were intended to help Adam understand the meaning of fraction addition.

The example by Mrs. Li shows that applying the unit fraction conceptin teaching fractions makes the concept of fraction addition more rigorousand meaningful than the part-whole relationship. Using unit fractionsto build conceptual understanding connects fractions to students’ priorknowledge of the concept of whole number and helps students to constructfractions in a continuous and systemic way. In addition, it places numeratorand denominator in the context of a number and it also links numer-ator and denominator by multiplication and repeated addition, which arecomponents of the prior knowledge of fractions.

In contrast, in answering part (b) of Problem 1, U.S. teachers tended touse a concrete model to build the concept of fraction addition. For example,Mrs. William used the fraction pieces to help Adam. She said:

Have Adam use fraction pieces to show 3/4 and 4/5 and ask: “What must we do in order tocombine 3/4 and 4/5? Let’s use the fraction pieces to work together to find what size piecesfourths and fifths share, then find equivalent fractions and add”.


This approach helps students to visualize the size of fractions andlook for equivalent fractions, but it emphasizes less the “why” of findingequivalent fractions, and it puts fractions in a separate and non-continuousdomain. With this approach, questions such as “Why can equivalent frac-tions be added?” “What are the connections between addition of fractionand the concept of ‘parts of whole?’ ” could not be answered clearly.

Sowder et al. (1998), agreed that the notion of a fraction as a quantity,as a number, is important for understanding. In contrast, Kerslake (1986)argued that some teachers and children have difficulty conceiving of afraction as a number and considered it either as two numbers or not asa number. Some Chinese teachers in this study also believed that a studentmight not think of a fraction as a number at all. But this misconceptionof fraction can be corrected by understanding the unit fraction concept.One of the key reasons for students having misconceptions and confu-sion about fractions is the way fraction concepts are taught. Teachers ortextbooks in the U.S. typically introduce fractions as parts of a whole.This concept separates a fraction into two parts, and increases confusionbetween numerator and denominator. This difference between the U.S. andChinese teachers in developing fraction ideas may produce the disparity inthe knowledge of students’ thinking, which is illustrated in the teachers’responses to part (a) of problem 1.

Table III indicates that in their responses to part (b) of Problem 1, 76%of Chinese teachers focused on procedures and rules to build students’ideas about fractions. In contrast, far fewer U.S. teachers (25%) believedthat using procedures and rules were effective in building fraction ideas.This result is not surprising, since Chinese teachers focus on developingskills as an integral part of learning. The history of mathematics develop-ment in China has had a great impact on mathematics teaching in China.Under the influence of the classic work Arithmetic in Nine Chapters, themain characteristic of Chinese mathematics is the development and prac-tice of accurate and efficient means of computation and to apply these inreal life. In general, Arithmetic in Nine Chapters has defined the tradi-tional mathematics style as useful in applications and calculation (Li &Chen, 1995). In addition, under the influence of the Chinese examinationsystem, in order to help most students to pass the examinations, teachersnot only pay attention to students’ conceptual understanding, but also workextremely hard to build students’ proficiency in computation and solvingnon-routine problems. Chinese teachers believe that developing proficientprocedural skills helps to reinforce what students have learned and allowsthem to transfer skills easily to new knowledge. Most importantly, it aidsstudents’ confidence in their ability to understand mathematics.


Addressing Students’ Misconceptions

Models and ConnectionsTable III shows that, in their responses to Problem 2, 86% the U.S. teachershad understood each student’s thinking on the comparison of fractions. Inresponse to part (c), they used various activities, graphs, manipulatives,and procedures to help students to correct misconceptions, focusing onthe use of concrete models. For example, Mrs. Nelson would show Robertphysical examples of a candy bar. By seeing the sections of 2/3, 1/4, and3/8, Robert would be able to conceptualize concretely each fraction.

The U.S. teachers helped students correct a misconception aboutcomparing fractions by using experience with a variety of models. Theuse of various models helps students build ideas about abstract mathema-tical concepts. This approach is supported by NCTM (2000) that says thatconcrete models provide students with concrete representations of abstractideas and support students in using representations meaningfully.

In contrast, Chinese teachers dealt with students’ misconceptions by avariety of activities, but they focused on developing the explicit connectionbetween the various models and abstract thinking. For example, to correctRobert’s misconception about ordering fractions, Mrs. Jian presented thefollowing approach:

Have Robert cut two equal-sized ropes, one in 7 pieces, and one in 2 pieces. Then have himcompare a section from each rope to find out which one is longer. Help him summarize therules: take one part from each rope, the one with the short part has a larger denomin-ator, and the fraction is smaller. Therefore, compare fractions by not only looking at thedenominator; a large denominator does not mean the fraction is larger.

In this example, Mrs. Jian not only used a concrete model to help Robertbuild understanding, but also connected the model to abstract ideas and therules for ordering fractions.

Use of QuestionsAsking questions is one of the effective ways to engage students’thinking and learning. Probing questions involving misconceptions canguide students in identifying errors by themselves and develop a deepconceptual understanding. Carroll (1999) found that probing questionsare effective in identifying student errors through engaging students inreasoning and thinking processes. In addition, questions assess learning,promote discussion, and provide direction for teachers in planning.

Posing questions in mathematics teaching is another feature of Chineseeducation, which is a reflection of Arithmetic in Nine Chapters. In thebook, teachers are urged to develop and use sequences of questions,answers, and principles during planning and instruction. In this study,


the questioning strategies were displayed extensively in the responses ofChinese teachers. As shown in Table III, 100% of Chinese teachers wereable to use questions or tasks to correct the misconceptions posed inProblem 2, compared with 61% of the U.S. teachers. We take an examplefrom Mrs. Wu: “Can you directly order fractions by comparing numeratorsonly while the numerators and denominators are all different? Have Latoyause the same-sized paper to fold 1/4, 2/3, 3/8, and then compare these threepieces”. Furthermore, Chinese teachers not only asked focusing questionsto identify each student’ thinking, but also understood student’s thinkingin different ways.

As indicated in Table III, 79% of the U.S. teachers did not poseappropriate questions in order to identify students’ thinking and helpstudents progress their mathematics ideas on comparing fractions, whileonly 39% of Chinese teachers had similar problems. Questions asked byU.S. teachers consisted of “How did you order these?” and “Explain whyyou put the fraction in that order” and provide a chance for Latoya to think,but do not directly lead her to recognize her misconception.

Engaging Students in Mathematics Learning

Use of RepresentationsThe results show that there are differences in the way the U.S. and Chineseteachers engage students in mathematics learning. Most U.S. teacherssuggested engaging and motivating the students to learn the procedureof multiplication through various activities, such as manipulatives, andpictorial representations. In their responses to Problem 3, as shown inTable III, 64% of the U.S. teachers would prefer to use one representation– area to illustrate fraction multiplication – while 67% of Chinese teachersuse two representations – area and repeated addition.

By applying manipulatives, such as cutting a paper circle, singing afraction song, playing with money, using base ten blocks, or drawing andcoloring areas, the U.S. teachers sparked their students’ interest in frac-tion multiplication and engaged students in a meaningful and concretelearning process. This “learning by doing” approach encourages studentsto acquire knowledge through inquiry and creative processes and fostersstudents’ creativity and critical thinking. The use of manipulatives in frac-tion multiplication is supported by Sowder et al. (1998), who reported theeffectiveness of the use of paper folding to learn fraction multiplication.This study reported that most U.S. teachers used area representations toillustrate fraction multiplication.

In contrast, most Chinese teachers used both area and repeated addi-tion to illustrate fraction multiplication, and they understood when and


how to use each representation. For example, Mrs. Yian provided a clearexplanation of how to use different representations for multiplication: (a)If a fraction multiplies a whole number, using repeated addition is easierfor students to understand, (b) If a fraction multiplies a fraction, usingthe area graph is better for visualization, (c) For fractions that are mixednumbers use both methods. This example shows that Mrs. Yian knowswhich representations can be helpful for students in solving multiplicationproblems in particular situations.

Connections in Introductory ActivitiesChinese teachers connected concrete models and stories related tostudents’ life more frequently than did the US teachers in their responsesto part (a) of Problem 3. As shown in Table III, 91% of Chinese teachers inthis study would engage and motivate their students to learn the procedureof multiplication by giving examples, which connect to concrete models(64%) and students’ life experiences, such as examples of stories, in addi-tion. By providing examples related to students’ real life, manipulatives,and concrete models, Chinese teachers were able to make a connectionbetween manipulatives and the strategy of solving problems, and buildunderstanding for students through developing rigorous procedures. Forexample, Mr. Wang designed an introductory activity:

In the class, 56 students were divided into four groups with 14 students in each group.The teacher has a student divide a colored paper into 4 pieces, so each group gets 1/4 ofthe paper. To illustrate the procedure of fraction multiplication, let’s use group one as anexample first: In order to share 1/4 of the paper among 14 students in group one, 1/4 of thispaper will be divided equally into 14 pieces. How much paper will each student in groupone get? How do you write this expression? It should be expressed as

1

4× 1

14Figure 6. 1/14 of 1/4 paper.

Mr. Wang continued to explain the procedure of multiplication fractions:

If all 4 groups do the same, every student in class will get one part of the paper. How muchpaper will each student get? To find 1/14 of 1/4, a student can divide 1/4 into 14 parts,taking one part of it, which means the student divide one paper into 4 × 14 parts and haveone of 4 × 14, the result can be expressed as:

1

4 × 14× 1 = 1 × 1

4 × 14Figure 7. One part of 1/14 of 1/4 paper.


Therefore,

1

4× 1

14= 1 × 1

4 × 14= 1

56Figure 8. The equivalent of the whole number 1 multiplying a fraction.

For two students, they will get 2/14 of 1/4 paper, i.e.

1

4× 1

14× 2 = 1 × 4

2 × 14= 2

56Figure 9. Two parts of 1/14 of 1/4 paper.

So two students get 2/56 of paper.

Mr. Wang concluded:

Now we can arrive at a conclusion: when multiplying fractions, the numerator will be theproduct of numerators, the denominator will be the product of denominators.

1

4× 2

14= 1 × 2

4 × 14= 2

56Figure 10. The rule of multiplying fractions.

At last, Mr. Wang applied the above conclusion to direct students to solvethe part (c) of Problem 3:

3

4× 2

3= 3 × 2

4 × 3= 6

12Figure 11. The solution of part (c) of Problem 3.

This activity addressed visually the connection between a concretemodel and procedure of fraction multiplication, which provides clear stepsfor fraction multiplication and also promotes students’ engagement inlearning. Notice that the notation from the above is different from thatof U.S. notation. In Chinese textbooks, the multiplication of fractions isrepresented differently: dividing1/4 of the paper into 14 pieces is writtenthe same as “to find 1/4’s 1/14” (Chinese language expression) and itsexpression is 1/4 × 1/14, while in U.S. it would be “to find 1/14 of 1/4”and its expression is 1/14 × 1/4. The Chinese way of defining fractionmultiplication seems to have the same order between the meaning andexpression (i.e., first having 1/4, and then having 1/4’s 1/14) which comple-ments student thinking more effectively, while the U.S. way of definingfraction multiplication tends to produce confusions for students.


Although 36% of responses from U.S. teachers focused on connectingto the concrete model, and 39% of them related to manipulatives activities,the U.S. teachers again often ignored developing the connection betweenmanipulative activities and abstract thinking. For example, Mrs. Parkerwould direct students to, “Cut a circle (paper) into 3 pieces. Take 2/3.Cut each one of these 2 pieces into 4 and take 3 of each one. You have 6out of 12 pieces”. She failed to connect the manipulative explanation andprocedure of multiplying fractions. The lack of such connection would failto build a bridge for students to understand why they use the manipulativeand how the activity would help them to use the procedure in doing themultiplication. Only 4% of responses of U.S. teachers showed exampleswith connecting concrete models or manipulatives to the procedure ofmultiplying fractions.

Use of Prior KnowledgeIn the introduction of new concepts, using prior knowledge not only helpsstudents to review and reinforce the knowledge being taught, but also helpsthem to picture mathematics as an integrated whole rather than as separateknowledge. It develops generalizations and helps students to solidify whatthey have learned and allow them to transfer the knowledge to new situ-ations (Suydam, 1984). Linking the new and prior knowledge in contextwill also help students know why and how to learn the new topic and graspnew knowledge with better understanding. This is supported by NCTM(1989), “Connection among topics will instill in students an expectationthat the ideas they learn are useful in solving other problems and exploringother mathematical concepts” (p. 84). Furthermore, NCTM (2000) pointsout, “Because students learn by connecting new ideas to prior knowledge,teachers must understand what their students already know” (p. 18).

In this study, 45% of the Chinese teachers focused on the importance ofdetermining students’ prior knowledge; in contrast, very few U.S. teachers(7%) gave examples for prior work to help students learn fraction multi-plication. For example, Mrs. Zhong presented the following introductorystory, which connects addition:

The monkey’s mother bought a watermelon and cut it into 9 pieces; every monkey ate 2/9of the pieces. How many pieces did four monkeys eat? How do you express this problemin addition? How do you express this problem in multiplication? Which method is easier?

This example helps students transfer prior knowledge of repeated addi-tion to multiplication with better conceptual understanding of fractionmultiplication.

In reviewing prior knowledge, one can, according to Confucius, alwaysfind new knowledge (Cai & Lai, 1994). Mathematics education in China


has been following this idea for classroom teaching for centuries. Obser-vation of classrooms in this study showed that Chinese teachers spent atleast one-third of the time reviewing prior knowledge at the beginning orduring class. One of the teachers said that she only teaches a new lesson 5to 10 minutes every day; the rest of the time is spent reviewing and rein-forcing the knowledge. The review process not only promotes continuityand attains a more comprehensive view of topics previously covered, but isalso a diagnostic tool that helps teachers to identify student strengths andweaknesses and provides valuable insight for future instruction (Suydam,1984).

Promoting Students’ Thinking about Mathematics

Visual Activities and Abstract Thinking

The results in Table III show different emphases on approaches to promotestudents’ thinking: U. S. teachers tend to use charts and tables, concrete orpictorial models, and manipulatives, while Chinese teachers tend to buildstudents’ abstract thinking using procedures. For example, in response toProblem 4, Mrs. Flores displayed this chart for a proportion activity:

Girls: 3 6 9 12 15 18 21 24

Boys: 5 10 15 20 25 30 35 40

Using a chart to write equivalent ratios shows a pattern and makesproblems easier for students to understand.

In Chinese texts, simple equation and direct/indirect variation are intro-duced at the sixth grade in 11th & 12th elementary math textbooks(Jiangsu, 1998). The similar content areas in U.S. 6th grade textbook(Glencoe, 2000) used by teachers in this study was also introduced to6th grade students. However, teachers’ responses in this study showed thatChinese teachers tend to use the algebraic approach in solving proportionproblems more than do U.S. teachers. For example, Mrs. Wang used thefollowing approach:

The ratio of girls to boys is 3:5, which means the girls are 3/5 of boys and boys are 5/3 ofgirls. So if the ratio of boys to girls is more than one, it will be a direct variation. Let thenumber of boys be x; the number of girls will be (40-x), so the ratio of girls to boys = x:(40-x). Therefore, from the following proportion:

3

5= x

40 − x

We can find the number of boys.


Mrs. Zhen explained the procedure without using proportion:

Since girls are 3 parts and boys are 5 parts, the total is 8 parts with 40 students. Every parthas 40/8 = 5 students. So boys will be 5 parts time 5 students, i.e., 25 students.

Furthermore, Table III shows that 43% of U.S. teachers in this studyprovided general questions that probably would not provide insight intostudents’ thinking. A question such as “Do you have the problem set upcorrectly?” could prompt a student to look at problem again, but placesless focus on students’ misconception. In contrast, all Chinese teachers inthis study used probing questions at various levels, which help teachersto explore students’ thinking directly in different ways and encouragestudents’ thinking deeply and critically. Mrs. Wang would ask her students:

What measurements are being compared in the ratio of 3:8? Girls are being compared tothe whole.What measurements are being compared in the ratio of x: 40? Boys are being compared tothe whole.How can we use unequal ratios to make a proportion? How can we make changes in orderto get two equivalent ratios?

As a basis for understanding questions she asked, Mrs. Wang would directher students to solve the problem using two different ways:

Method 1. Let x be the number of boys, so 5/8 = x/40Method 2. Let x be the number of girls, so 3/8 = x/40

Effective teachers “know how to ask questions” (NCTM, 2000, p. 18) andhow to use these questions to enhance the students’ thinking.

Self-reflection and CommunicationNCTM (2000) views communication as an essential part of mathematicsteaching and learning: “Reflection and communication are intertwinedprocesses in mathematics learning” (NCTM, 2000, p. 61). In this study,the U.S. teachers tended to provide students with opportunities to discussand share their ideas. Mrs. Baker “always asks students to explaintheir answers”. Mrs. Larson would have students engage in “discussion,reworking, discourse, and remembering and learning from their mistakes”.Chinese teachers in this study also encouraged students’ thinking andcommunication, but focused on developing reflection. Chinese teachersencouraged their students to think about problem solving, to substi-tute their answers into the original equations, and to check to see if itmakes sense. Reflection occurs when students check and analyze theirwork and thinking and helps students to reorganize knowledge and find


their errors by themselves. Importantly, reflection develops a deep under-standing and fosters good learning habits and has been viewed as a criticallearning strategy constantly taught in mathematics classrooms in China.The importance of reflection was noted by Fennema and Romberg (1999)who stated that reflection plays an important role in solving problems,and a critical factor of reflection is that teachers recognize and valuereflection. In this study, Chinese teachers Mrs. Wang and Mrs. Lu encour-aged students to be a “mathematics doctor”, which means to reflect andto examine the errors in problem solving. They also valued reflectionby giving praise to students who do well on checking procedures andanswers.

In this study, neither group of teachers used estimation very often. Forexample, in their response to Problem 4, only 4% of the U.S. teachersand 6% of Chinese teachers mentioned using estimation to help students’thinking. Estimation is an important skill in the thinking and reasoningprocess. It fosters a good number sense, helps students to think and reasonlogically, and develops proficient skills in computations and problemsolving.

DISCUSSION AND CONCLUSIONS

Four Aspects of Analysis of Pedagogical Content Knowledge

Comparative study can increase our understanding of how to produceeducational effectiveness and enhance our understanding of our owneducation and society (Kaiser, 1999). However, it is difficult to conduct avalid comparative study between different cultures without setting essen-tial components as a norm in analysis of data. This study includedcomparisons and contrasts of teachers’ pedagogical content knowledgebetween the U.S. and China by using four aspects of pedagogical contentknowledge of students’ thinking: building on student ideas in mathematics,addressing students’ misconceptions, engaging students in mathematicslearning, promoting student thinking about mathematics. Each of theseaspects consists of several essential components (see Table II). By analysisof these essential components, this study provided suggestions as to howto set up dimension and scope for further cross-cultural comparativestudies. Although the four aspects in this study are only a part of pedago-gical content knowledge, namely, knowledge of students’ thinking, thisstudy addressed a critical way of assessing teachers’ knowledge regardingstudents’ thinking.


Importance of Pedagogical Content KnowledgeTeacher knowledge of mathematics is not isolated from its effects onteaching in the classroom and student learning (Fennema & Franke, 1992).Teachers’ pedagogical content knowledge combines knowledge of content,teaching, and curriculum, focusing the knowledge of students’ thinking.It is closely connected with the content knowledge, connected with theway of transformation of content knowledge in the learning process and inthe way in which teachers know about the students’ thinking. This studyindicated that deep and broad pedagogical content knowledge is importantand necessary for effective teaching. Teaching for understanding includesa convergent process in which teachers build students mathematics ideasby connecting prior knowledge and concrete models to new knowledge,focusing on conceptual understanding and procedure development. Inaddition, teachers should be able to identify students’ misconceptionsand be able to correct misconceptions by probing questions or usingvarious tasks. Teachers should engage students in learning by providingexamples, representations and manipulatives. Finally, effective teachingrequires the effort of promoting students’ thinking by a variety of focusingquestions and activities. The results of this study supported the idea thata balance is needed between the use of manipulatives and developingprocedures. Although manipulatives develop conceptual understandingof mathematics, procedural learning is an essential learning process forreinforcing understanding and achieving mathematical proficiency and isa necessary step for problem solving. Without developing firm under-standing and skill with procedures, students will not be able to solveproblems efficiently and confidently. Last and most importantly, atten-tion to learners’ cognitions is a key component in teachers’ pedagogicalcontent knowledge and effective teaching. Knowledge of students’ mathe-matical thinking helps teachers to enhance their own knowledge of contentand curriculum, prepare lessons thoroughly, and teach mathematics effec-tively. Without knowledge of students’ thinking, teaching cannot producelearning; it may instead be like “playing piano to cows” (a Chineseidiom).

Conclusion

The results of this study indicated that mathematics teachers’ pedagogicalcontent knowledge in the two countries differed markedly and this has adeep impact on teaching practice. The Chinese system emphasizes gainingcorrect conceptual knowledge by reliance on traditional, more rigid devel-opment of procedures, which has been the practice of teaching and learning


mathematics content for many years. The United States system emphasizesa variety of activities designed to promote creativity and inquiry to developconcept mastery, but often has a lack of connection between manipulativesand abstract thinking, and between understanding and procedural develop-ment. Both approaches have shown benefits and limitations in teachingand learning mathematics, and also illustrate the different demands onteachers’ pedagogical content knowledge.

This study cannot necessarily be generalized to all mathematicsteachers in the United States and China because the samples included onlyone city from each country, with 23 schools from China and 12 schoolsfrom the U.S. However, this is an internal comparative study and, with acentralized education system in China, one city may represent the wholesystem of education in China. With a locally controlled education systemas in the U.S., one city may not reflect the whole United States. There-fore, the results cannot necessarily be applied to teachers in the UnitedStates. Nevertheless, these results do point to the importance, from aninternational perspective, of pedagogical content knowledge and to theessential components that can promote further understanding of effectivemathematics teaching.

NOTE

1 7th and 8th graders in China have already learned Algebra I.

REFERENCES

American Association for the Advancement of Science (2000). Middle grades mathematicstextbooks: A benchmarks based evaluation. Washington, DC: Author.

An, S., Kulm, G., Wu, Z., Ma, F. & Wang, L. (October, 2002). A comparative study ofmathematics teachers’ beliefs and their impact on the teaching practice between theU.S. and China. Invited paper presented at the International Conference on MathematicsInstruction, Hong Kong.

Cai, X.Q. & Lai, B. (1994). Analects of confucius. Beijing: Sinolingua.Carpenter, T.P. & Lehrer, R. (1999). Teaching and learning with understanding. In E.

Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding(pp. 19–32). Mahwah, NJ: Erlbaum.

Carroll, W.M. (1999). Using short questions to develop and assess reasoning. In L.Stiff (Ed.), Developing mathematical reasoning in grades K-12: 1999 NCTM yearbook(pp. 247–253). Reston, VA: National Council of Teachers of Mathematics.

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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). New York: The Falmer Press.

Fennema, E. & Franke, M.L. (1992). Teachers knowledge and its impact. In D.A. Grouws(Ed.), Handbook of mathematics teaching and learning (pp. 147–164). New York:Macmillan Publishing Company.

Fennema, E. & Romberg, T.A. (1999). Mathematics classrooms that promote under-standing. Mahwah, NJ: Lawrence Erlbaum Associates.

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Li, J. & Chen, C. (1995). Observations on China’s mathematics education as influenced byits traditional culture. Paper presented at the meeting of the China-Japan-U.S. Seminaron Mathematical Education. Hongzhou, China.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: LawrenceErlbaum Associates.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluationstandards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standardsfor school mathematics. Reston, VA: Author.

Pinar, W.F., Reynolds, W.M., Slattery, P. & Taubman, P.M. (1995). Understandingcurriculum. New York: Peter Lang.

Robitaille, D.F. & Travers, K.J. (1992). International studies of achievement in math-ematics. In D.A. Grouws (Ed.), Handbook of mathematics teaching and learning(pp. 687–709). New York: Macmillan Publishing Company.

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Department of Teacher Education Shuhua AnCalifornia State University1250 Bellflower BoulevardLong Beach, CA 90840-2210USAE-mail: [email protected]

Department of Teaching Gerald KulmTexas A&M University Zhonghe WuLearning and CultureCollege Station, TX 77843USAE-mail: [email protected]

[email protected]

TERRY WOOD

EDITORIAL

IN MATHEMATICS CLASSES WHAT DO STUDENTS’ THINK?

In this issue of the Journal of Mathematics Teacher Education, one

central theme of the contributions is the realization of the importance

that teachers (or teacher educators) understand and respond to stu-

dents’ thinking whether for mathematical or pedagogical purposes.

Another primary theme is the necessity for teachers and/or teacher

educators to learn from practice. The words that are attached to these

themes make them appear to be well-accepted amongst mathematics

educators and are thought to be the ‘common wisdom’ of professional

development and practice. But these ideas are still problematic in tea-

cher education, simply because the meaning of, for example ‘under-

standing students’ thinking’, has not been discussed and common

meaning fleshed out within the mathematics education community.

Consequently, teachers and teacher educators’ while engaged in profes-

sional development may use the same words, but may talk past each

other as the meanings held by each are quite different. We need to

think about ways in which we might sustain an international dialogue

that would allow the development of common meanings.

Doerr and Thompson in their paper, describe the ways teacher edu-

cators learn through preparing their students to teach. Using multime-

dia case study of practice and a three-tier research design, Doerr and

Thompson examined teacher educators’ learning about their teacher

preparation students as they were learning about middle school stu-

dents from the multimedia case study. They found that teacher educa-

tors’employed two different implementation strategies. The strategy of

one group of teacher educators was open-ended and explored the

issues raised in the case study, while the other group’s strategy focused

on the mathematical content in the case study.

Kazemi and Franke provide information on practicing teachers’

learning during work that is done collectively as they developed deeper

Journal of Mathematics Teacher Education 7: 173–174, 2004.

� 2004 Kluwer Academic Publishers. Printed in the Netherlands.

understanding of their own students’ mathematical thinking. In this

paper, the focus is on documenting key shifts in teachers’ participation

across the year as evidence of teacher learning. Kazemi and Franke

found that the first shift in participation occurred when teachers as a

group learned to attend to the details of children’s thinking. A second

shift in participation occurred as teachers began to develop possible

instructional trajectories in mathematics their children might make.

This result raises interesting questions: What do these instructional tra-

jectories look like? Can they be developed by teachers? And, if so,

how does this occur?

In the last paper, to identify probable conditions for teacher learn-

ing, Steinberg, Empson and Carpenter examined the case of one tea-

chers’ change. They identified the conditions conducive to learning as:

membership in a discourse community that consisted of expert others,

engaging in processes of reflectively generating, debating and evaluating

new knowledge and practices and ownership that ‘‘the problems of

teaching that change is meant to address need to be problems that

teachers want to solve and feel capable of solving.’’ These conditions

were dependent on the teachers’ focusing on student thinking in her

practice of teaching.

Finally, Christine Keitel reviews Simon Goodchild’s book, Students’

goals. A case study of activity in a mathematics classroom. Goodchild

investigates students’ activities in a Year 10 (age 14–15) mathematics

classroom in a British secondary school in order to (according to

Keitel) ‘‘raise again the most basic and challenging questions: What do

we really know about our students’ learning in classroom practice of

mathematics? What do they tell us, if we listen to them?’’ In his book,

Goodchild presents the underlying main research objectives as:

What are the goals, rationales, purposes, and interpretations towardswhich students work in their regular mathematics activity?

What are the features of the classroom, arena and setting, which com-prise the socio-cultural context in which students engage in mathemat-ics activity?’’

Keitel’s review is enticing and intriguing leaving no doubt that

Goodchild’s book is an important read for those interested in the stu-

dents’ view of mathematics learning and corresponding classroom con-

textual characteristics.

174 TERRY WOOD

HELEN M. DOERR and TONIA THOMPSON

UNDERSTANDING TEACHER EDUCATORS ANDTHEIR PRE-SERVICE TEACHERS THROUGH

MULTI-MEDIA CASE STUDIES OF PRACTICE

ABSTRACT. The challenges facing those who seek to prepare mathematics teachersare well established in the literature. Most of the research to date has focused on theperceptions and understandings of pre-service teachers, but not on the perceptions andunderstandings of teacher educators. In this study, we explore how four teacher educatorsunderstand their pre-service secondary teachers as the pre-service teachers attempt tomake sense of teaching through the investigation of a multimedia case study of prac-tice. We found that the teacher educators adopted two different implementation strategies:one strategy tended to be open-ended and exploratory; the other was more focused onthe teacher educators’ goals of anticipating student understanding and developing mathe-matical content knowledge for teaching. We also found that, in using the case study, teachereducators elicited pre-service teachers’ thinking about the complexities of the teacher’s rolein small group work, about the value of explicitly revealing the teacher’s reflections on thelessons, about the role of planning and preparation, and about the limits of pre-serviceteachers’ abilities to understand and appreciate students’ thinking and to extend lessonideas. Both teacher educators and their pre-service teachers gained perspectives on therole of a teacher’s mathematical content knowledge. These results imply that multimediacase studies of practice can serve as vehicles for revealing the knowledge and practice ofteacher educators, as they engage in supporting the professional development of pre-serviceteachers.

KEY WORDS: case methods, multimedia case studies, pre-service teacher education,secondary mathematics education, teacher educators

INTRODUCTION

Mathematics teacher educators face several practical problems in theirwork with pre-service teachers: finding sufficient high quality classroomsfor placements, developing robust understandings of mathematics amongpre-service teachers, supporting their reflections on students’ mathematicalthinking, and developing images of practice that go beyond “tellingclearly.” The number and intricacy of theories attempting to model mathe-matics teaching has increased substantially over the last two decades(Koehler & Grouws, 1992). Such theoretical work appears to be promisingin informing the practical work of teacher educators. However, much of thecurrent research in mathematics teacher education focuses on pre-service


176 HELEN M. DOERR AND TONIA THOMPSON

teachers’ mathematical content knowledge, their beliefs, and their pedago-gical content knowledge. There is not much research that that focuses onthe teacher educators who work with these pre-service teachers as theystruggle with the complexities of learning to teach. This study intends tocontribute to the emerging literature on the knowledge of teacher educatorsby examining the understandings of teacher educators as their pre-serviceteachers attempt to make sense of teaching through the investigation of amultimedia case study of practice.


The current literature on the work of teacher educators includes accountsof self-study of practice and development of professional knowledge inmathematics education and other fields (e.g., Geddis & Wood, 1997;Hudson-Ross & Graham, 2000; Kitano, Lewis & Lynch, 1996; Onslow& Gadanidis, 1997; Tzur, 2001) and a few empirical studies thathave examined the knowledge and professional development of teachereducators (John, 2002; Kremer-Hayon & Zuzovsky, 1995). The self-studies provide us with some glimpses into the experiences of teachereducators and the development of their professional knowledge. Geddisand Wood (1997) discuss the strategies an expert teacher educator (thesecond author of their study) used to transform the subject matterknowledge held by pre-service teachers into the practical knowledgeneeded for teaching. The teacher educator confronted a core dilemmain teacher preparation: namely, how to provide pre-service teachers withinitial routines and strategies for managing the behavior and learning inthe classroom, while at the same time taking a critical perspective onthose strategies and grasping the inherent ambiguity and complexity ofpedagogy. The teacher educator focused on lesson planning as a way ofmanaging this dilemma by engaging the pre-service teachers in developingand articulating a range of ways to represent their mathematical knowledgefor the purpose of teaching and in evaluating these different representa-tions. In this way, reflecting on planning became a mechanism the teachereducator used to transform the subject matter knowledge of his pre-serviceteachers into useable knowledge for teaching. As a result of his self-analysis of professional development, Tzur (2001) postulated a non-linear,interconnected four foci model of development for teacher educators: asa student, as a mathematics teacher, as a mathematics teacher educator,and as a mentor of mathematics teacher educators. Ways of thinking andparticipating in a community are activities that occur at each focus. The

UNDERSTANDING TEACHER EDUCATORS 177

ways of thinking at one focus become the explicit target of reflection andinteraction from another focus.

While examining teacher educators’ learning in an inquiry-basedapproach to professional development, Kremer-Hayon and Zuzovsky(1995) found that the teacher educators shifted from the use of personalknowledge sources to professional knowledge sources and increasinglyencouraged their students to develop independent thinking. John (2002)investigated the personal histories as well as the practical knowledgeand understandings of six teacher educators as they worked with studentteachers. This study included a detailed description of two of the teachereducators, one in mathematics and one in science, and revealed a deeplybiographical and intensely practical account of their knowledge andexperiences. Across all six cases, John found that four dimensions charac-terized teacher educators’ knowledge: intentionality, practicality, subjectspecificity, and ethicality. For the intentionality dimension, John describedthe goals and intentions of the teacher educators in his study:

[They] all strongly argued for the preparation of teachers by means of improving theirstudent teachers’ capacity for professional judgment and decision making rather than byproviding extensive practice of skills in a single classroom – or even several. They alsocalled for the exercise of insight, strategic understanding and critical thinking rather thaneffective performance of learnt skills. (p. 336)

As with the teacher educator in Geddis and Wood’s (1997) study, theseteacher educators wanted their pre-service teachers to move beyond theimmediacy of practice to achieve insight and engage in critical thinking.The practicality dimension (John, 2002) referred to the role of prac-tical action and reflection-on-action that was found among the teachereducators. Consistent with Tzur’s (2001) emphasis on reflection and inter-action, the teacher educators in John’s study engaged their pre-serviceteachers in practical discourse on actual practice, urged them to learnfrom their own and each other’s practice, and tried to improve theirunderstandings of children as learners. The role of reflection in preparingteachers and in supporting teachers’ continued professional developmenthas been widely accepted in the research literature on teacher educationsince Schön’s work in the 1980’s (e.g., Clarke, 2000; Jaworski, 1994;Mewborn, 1999; Schön, 1983, 1987; Wheatley, 1992).

As we argue in the next section, the use of case studies of practice(or case methods) with pre-service teachers has been shown to have twoimportant characteristics for teacher educators that are directly relatedto the dimensions of intentionality and practicality. First, when studyingcases, individuals are engaged in articulating their ideas and providingreasons for their conclusions. Cases provide a site for analysis, reflec-


tion, and reasoning about practice. As such, they provide an opportunityfor teacher educators to reach their goals and intentions for building pre-service teachers’ capacities for professional judgment, decision-making,insight, and critical analysis. Second, a case study of practice is deeplyembedded and grounded in the particularities of the context of practice.The complexities of the setting, the nuances of meanings, the ambiguitiesof partial information, the differences in perspectives, and the conflicts ingoals are all present in the realities of practice. Case studies are a means ofcapturing these multiple dimensions of practice for the novice practitioner.Since cases are situated in the particulars of practice, they provide teachereducators with an opportunity to focus on practical actions and reflection-on-actions. These two characteristics of cases, namely as sites for analysisand as situated in practice, are well-aligned with what researchers haveidentified as the needs and goals of teacher educators for their pre-serviceteachers and provide the conceptual framework for this study.

What is less well understood is how and why teacher educators woulduse cases to meet their intentions for the learning of their pre-serviceteachers. The overarching goal for this research study is to understand theknowledge and practice of teacher educators as they used the artifacts of amultimedia case study of practice in order to understand and support theprofessional development of their pre-service teachers. In particular, weexamine:

− How do teacher educators implement the use of a multimedia casestudy of practice?

− What issues for teaching do teacher educators address as their pre-service teachers investigate the artifacts of a case study of practice?

− What understandings emerge for the teacher educators as their pre-service teachers investigate the artifacts of a case study of practice?

We see the implementation choices, the issues raised during the imple-mentation, and the emerging understandings of the teacher educators asrevealing the teacher educators’ knowledge and practice. Since we areusing case studies of practice as a site for understanding teacher educators’knowledge and practice, we will first situate the use of cases more broadlywithin the field of education.

Case-based instruction in education has used both text and video cases(Barnett, 1991, 1998; Copeland & Decker, 1996) and, more recently,multimedia tools (Herrington, Herrington, Sparrow & Oliver, 1998;Lampert & Ball, 1998; Masingila & Doerr, 2002). Broad and varyingappeals have been made for the use of case studies for teachers’ profes-sional development. These appeals range from the potential for providingparadigmatic exemplars of practice to providing a means of understanding


theoretical principles, while bridging the gap between theory and practice(Sykes & Bird, 1992). As we have noted above, two important character-istics of cases, namely as a site for analysis and as situated in practice,appear well aligned with the needs and goals that teacher educators havefor their pre-service teachers.

As a site for analysis, several researchers have suggested that theprimary purpose of a case study is to support the development of crit-ical analysis and informed decision-making. The intent of such use isto present an opportunity to “exercise and develop skills of educationaldecision making” (Merseth, 1990, p. 54). Other researchers have empha-sized the use of case study methodology as a means of linking teachingand learning theory to practice (Lundeberg, Levin & Harrington, 1999;Shulman, 1986; Shulman, 1992). It has been argued that pre-serviceteachers gain an understanding of teaching through the use of cases byallowing them to apply theoretical and practical knowledge to specificschool contexts (Lundeberg, 1999). Case-based instruction has been usedto foster observation skills, identify relationships, and construct organ-izing principles (Doyle, 1990; Merseth, 1991; Shulman, 1992; Wasserman,1994). Such experiences may strengthen “observational, interpretative, andcritical skills” (Florio-Ruane & Clark, 1990, p. 24). Reflective reasoningmay occur as pre-service teachers think about what a case teacher’s actionmay mean or as they imagine themselves taking the same action (Masingila& Doerr, 2002; Moje & Wade, 1997). In other words, cases provide a sitefor pre-service teachers to reflect on the actions of another teacher.

A second key characteristic of a case study of practice is that it issituated in practice, with all its concomitant complexity, ambiguity, andincomplete information. As Feltovich, Spiro and Coulson (1997) haveargued, the knowledge base of teaching is an ill-structured domain and,as such, is best learned by a criss-crossing of the landscape through thestudy of cases of practice. Case-based learning has been suggested as oneeffective way of presenting this “advanced knowledge” in teaching (Spiro,Coulson, Feltovich & Anderson, 1988). It is precisely within the complex,ambiguous and partially understood context of practice that teachers haveto make reasoned judgments and decisions for action. As Merseth (1999)has argued, cases can present a realistic picture of a classroom. Thismay allow pre-service teachers to comprehend the relationship betweenteaching strategies and learning outcomes. Some researchers have arguedthat cases illuminate the contextual dependence of the act of teaching(Harrington, 1995; Kleinfeld, 1992). Harrington (1995) contends thatpedagogical reasoning is cultivated by providing novice teachers with theopportunity to evaluate contextual knowledge. Through the use of case


studies, teacher educators can encourage pre-service teachers to thinkdeeply about concrete experiences and to reflect on the actions taken inspecific contexts.

The complexities of classroom environments present particular prob-lems for the novice teacher whose limited teaching experience andprofessional knowledge make it difficult to observe effectively thecomplexity of interactions that occur, often with great rapidity, in a typicalclassroom. Beginning teachers are often primarily concerned with issuesof classroom management and the planning of lessons. While these areimportant priorities, they may impede the novice teachers’ abilities tofocus on the subtleties of understanding student thinking, or the nuancesof facilitating group work and class discussion, or the trade-offs inherentin teachers’ decisions (Scardamalia & Bereiter, 1990). Not surprisingly,beginning teachers in professional development programs often seek fromtheir teacher educators specific strategies, guidelines, explanations, and“rules of thumb” to apply in understanding and responding to the complexinteractions that occur in a typical classroom. Learning through casesstudies, it is argued, promotes teachers’ understanding of the complexitiesof practice and of the need to become more analytical about the data ofclassroom practice (Wassermann, 1993).

It is this paradigm – that case studies are sites for analysis and aresituated in the particulars of practice – that frames the use of cases inthis research study. Teacher educators are faced with the challenging taskof meeting the practical needs and concerns of their pre-service teacherswhile, at the same time, supporting their professional development alonglines that will deepen their mathematical knowledge, develop their under-standing of children’s reasoning, and enhance their ability to reflect upontheir decisions and actions in the classroom. The multimedia case studyused in this research is intended to provide shared access to the artifactsof practice for pre-service teachers’ analysis and reflection. The case studysimultaneously provides us, as researchers, with a site for analyzing theknowledge and practice of teacher educators as they, in turn, use thecase materials with their pre-service teachers. The goal of this study isto examine how teacher educators use these artifacts to understand andsupport the development of their pre-service teachers as they attempt tounderstand the complexities of practice and as they engage in articulatingtheir ideas about that practice.


METHODOLOGY

This qualitative case study is part of a larger research project on the use,by mathematics teacher educators, of multimedia case studies to supportthe professional growth and development of pre-service secondary mathe-matics teachers. We describe the multimedia case study materials, theparticipants in this study and their implementation of the case materials,and then our data sources and methods of analysis.

Case Study Materials

The multimedia materials used in this research study were intended tocapture the artifacts of practice in a seventh grade class of 23 studentsin an urban public school over the course of one and a half class sessions(Bowers, Doerr, Masingila & McClain, 1999). The purpose of the casestudy lesson was to engage middle school students in the collabora-tive analysis of data in an effort to make a mathematically viable groupdecision about the most important factors to consider when purchasinga pair of sneakers. The problem involved identifying eight factors thatmight be used when deciding on the purchase of a pair of sneakers. Sixgroups of students were asked to order the factors from most importantto the least important. After each of the groups ranked the list of eightfactors, the teacher challenged the students to develop a ranking systemto aggregate all of the groups’ lists into one final, ordered list of factors.The case study teacher then called upon groups of students to presenttheir ranking systems to the class (see Doerr and English (2003) for anextended discussion of student strategies for this task). The case studymaterials included background information on the school, the teacher’slesson plans, the teacher’s anticipations of the lesson and her reflectionsafter the lesson, video of the whole class discussions and small groupinteractions, a scrolling transcript that was linked to the video, copies ofstudent work, and related mathematical activities.

The facilitator’s guide suggested some ways in which the case studycould be used and also contained a set of discussion questions that orga-nized the issues of teaching and learning mathematics around four themes:planning, facilitating group work and whole class discussion, under-standing student thinking, and mathematical content and context. Thesefour themes were intended to guide and support the reflection and analysisby the pre-service teachers without overly constraining or dictating howteacher educators could use the case study.


TABLE I

Experience of Teacher Educators and Numbers of Pre-service Teachers

Site Number of Year of Previous Previous

pre-service experience experience experience

teachers as a teacher with text or with multi-

educator video cases media cases

Site A 7 5 yes no

Site B 5 8 yes no

Site C 12 16 yes no

Site D 6 5 no no

Participants and Implementation

The four teacher educators participating in this study were experiencedteacher educators, two at small colleges and two at mid-sized universities.The pre-service teachers were graduate and undergraduate students in thefinal stages of their preparation for full-time student teaching. Most wereconcurrently involved in classroom observations and had some limitedteaching experiences. All the participants in this study had a copy of theCD-ROM “Ranking Data to Make Decisions: The Case of the SneakersPurchase” (Bowers, Doerr, Masingila & McClain, 1999). Each teachereducator also had a copy of the facilitator’s guide. The teacher educatorsused the materials for a minimum of three to four class hours over atleast a two-week period. The number of pre-service teachers, the yearsof experience of the teacher educator, and the teacher educators’ previousexperience with video or text-based cases and with multimedia-based casesis shown in Table 1, for each of the four sites.

At two of the sites (A, C), the teacher educators introduced the casestudy by having the pre-service teachers watch the video portion of theCD-ROM while they were in a lab setting. At the other two sites (B,D), the teacher educators introduced the case study by posing to the pre-service teachers the same sneakers purchase problem that was given tothe seventh grade students. The pre-service teachers solved the problem ingroups and presented and discussed their solutions. The teacher educatorsthen asked the pre-service teachers to view the lesson taught on the CD-ROM and to respond to assigned questions. At site B, this was done in alab setting with the pre-service teachers; at site D, the CD-ROM was usedoutside of class time. Site D was the only site where the teacher educatordid not directly use the CD-ROM during class meeting time. Rather, the


pre-service teachers viewed the CD-ROM on their own and were assignedwritten questions related to the case study to complete outside of class.These reflective pieces then served as points of discussion during classsessions. At sites A, B, and C, the pre-service teachers also wrote responsesto assigned questions and viewed portions of the case-study outside ofclass time.

Data Sources and Analysis

Questionnaires, interviews, and written assignments from all four sitesconstituted the data corpus that allowed us to analyze how the teachereducators used the artifacts of practice to understand and support theprofessional development of their pre-service teachers. We conductedobservations of the methods classes at site D when the case study wasused. The participant-observer data from this site allowed us to analyzethe teacher educator in his interactions with the pre-service teachers andto examine, in greater detail, the interpretations of the issues of practiceby each of the pre-service teachers at that site. The data from this siteconsisted of essays, written responses to assigned questions, field notes bythe researcher as a participant observer, questionnaires distributed at theend of the course, and semi-structured interviews with both the pre-serviceteachers and the teacher educator.

The teacher educators and the pre-service teachers completed separatequestionnaires that were designed to identify (a) how the case study wasused and the goals of the teacher educator, (b) the background and experi-ences of the pre-service teachers, and (c) the salient issues in the case studyfor the teacher educator and the pre-service teachers. The teacher educatorquestionnaire included these questions:

For my class, the most valuable part(s) of this case study was . . .

We spent much of our time analyzing and discussing . . .

The pre-service teacher questionnaire included these questions:

For me, the most valuable part(s) of this case study investigation was . . .

I wished we had spent more time in class analyzing and discussing . . .

As a result of investigating this case study, what things (if any) did (will) you do differentlyor pay more attention to in your student teaching?

These questions were intended to elicit from the respondents the identifi-cation of the issues that were the focus of class discussions and that wereof value to them as they used the case study materials.

We conducted semi-structured interviews with the teacher educators toprobe the issues that were raised in the questionnaires and to understand


better how the teacher educators perceived the relationship of those issuesto the professional growth and development of the pre-service teachers.The pre-service teachers completed several written assignments based onstudy questions from the facilitator’s guide and wrote an essay on thecharacteristics of effective teaching. In an attempt to gather a consistentdata set, these assignments were the same across all four sites.

Data Analysis

The analysis of this data was conducted in three stages, using inductivequalitative methods (Strauss & Corbin, 1990) to identify the issues thatemerged from the investigation of the case study for the teacher educatorsand their pre-service teachers. In the first stage, we coded the responses ofthe teacher educators and pre-service teachers to the open-ended questionson their respective questionnaires. As a result of this coding, we identifiedemerging issues for the teacher educators and the pre-service teachers.We analyzed the pre-service teachers’ written work for instances of theseissues and other possible issues that may not have been addressed in theresponses on the questionnaires. This analysis included the interviews withthe pre-service teachers at site D. Disagreements between coders wereresolved by referring to other instances of the code within the data in orderto clarify the meaning of a particular code across several contexts.

In the second stage of analysis, we analyzed the teacher educators’interviews, seeking elaborations and instances of the issues from theinitial coding, identifying new issues that may not have emerged in theresponses on the questionnaires, and describing key issues from the teachereducators’ perspectives. During this stage of analysis, we also comparedthe issues raised by each teacher educator and his pre-service teachersin order to identify those issues that may have been a concern from theperspective of the teacher educator but not the pre-service teacher or viceversa. This analysis included the field notes taken at site D that reflectedour observations of the interactions between the teacher educator and hispre-service teachers.

In the third stage of our analysis, we compiled profiles on the use ofthe case study by each of the teacher educators. These profiles consistedof a description of the implementation of the case study at each site andidentified those issues that were critical for the teacher educator and forthe pre-service teacher. These profiles then became the basis for categor-izing the implementations and the themes for organizing the issues thatthe teacher educators elicited from the case study as they used it withtheir pre-service teachers. The results of this analysis are presented in thefollowing section. We have identified the data sources from interviews,


questionnaires, or classroom observations. For purposes of anonymity, allresults are reported in a single gender. Each of the teacher educators isreferred to with a male pronoun and each of the pre-service teachers witha female pronoun.

RESULTS

We report the results of our analysis in two parts. First, we briefly describethe major differences in implementations that occurred across the foursites. Second, we elaborate the four themes that emerged for the teachereducators and their pre-service teachers through the use of the case study.Within each of these themes, we present the specific issues that wereaddressed by each teacher educator and his pre-service teachers and wedescribe the understandings that emerged for the teacher educator.

Implementation

At two of the sites (A and C), the teacher educators were open-ended intheir implementations of the multimedia case study materials; they beganby having students engage directly with the case materials. At site A, theteacher educator described his pre-service teachers as “very autonomous”learners and simply directed them to “use it how you want to, but if Iwere you, I wouldn’t watch the follow up reflection yet, today at all, untillater when we’re using this” (Interview). This teacher educator encour-aged his students to investigate the materials from their own perspectives,with only the suggestion that they refrain from looking at the case studyteacher’s reflections until after they had discussed the lesson in the case.Similarly, at site C, the teacher educator began his use of the materials witha five-minute orientation to the mathematical content on the case and thenassigned each student two different questions from the facilitator’s guide.The students then worked through the case study materials, looking forinformation and insights into their assigned questions. The following classwas spent discussing general impressions of the case study and hearingeach student’s responses to her assigned questions.

At the other two sites (B and D), the teacher educators began their use ofthe case materials by having their pre-service teachers work on the mathe-matics problem (the sneakers problem) that was central to the case study.At both sites, this decision was directly related to the goals that the teachereducator had for his students. At site B, the teacher educator wanted touse the multimedia case study because of the opportunity it presented todevelop among his pre-service teachers “this idea of thinking about and


anticipating students’ thinking” (Interview). In order to accomplish this,the teacher educator felt that he had first to engage the pre-service teacherswith their own thinking about the mathematical content. At site D, a centralcourse goal for the teacher educator was to have the pre-service teachersexperience challenging mathematical content similar to what they mightencounter in teaching typical middle and high school mathematics. Theteacher educator felt that the case study demonstrated effectively to thepre-service teachers a situation where the mathematics might appear to berote but in fact requires a deeper understanding of mathematical relation-ships. Both of these teacher educators began their pre-service teachers’investigation of the multi-media case by investigating the mathematicalcontent of the case.

Emergent Themes

Our intention in the analysis of the data was to understand the knowledgeand practice of the teacher educator in terms of his experiences and hisperceptions with respect to his particular pre-service teachers. The fourthemes that emerged from the analysis were: (1) the case study teachers’interactions with groups of students; (2) supporting reflective practice; (3)the role of planning and preparation for lessons; and (4) understandingmathematical content. The first theme was found at all four sites, thesecond theme at sites A, B, and C, and the third and fourth themes at sitesB and D.

The Teacher’s Interactions with Groups of Students

At site A, the teacher educator described his pre-service teachers’ concernsover the case-study teacher’s interactions with one of the small groups ofstudents. These pre-service teachers felt that the teacher did not understandwhat the group was doing and needed to spend more time with that group.They felt that the teacher did not probe deeply enough into their thinking,as she did with some other groups. Furthermore, they found that this lackof probing did not match the teacher’s professed belief in the value oflistening to students in order to move the lesson forward. The teachereducator saw this analysis by his pre-service teachers as reflecting theirunderstandings of issues of fairness and equity in classroom interactions.

At site B, the teacher educator also found that, as at site A, hispre-service teachers focused their attention on the case study teacher’sinteractions with one of the small groups. These pre-service teacherssensed that the case-study teacher was more passive in her actions withthis group and that she did not pursue or continue with questions. Thisled the pre-service teachers to speculate as to why that might be. As with


site A, these pre-service teachers were not able to come to any definitiveconclusion as to the reasons why the case study teacher acted as she did. Atsite B, the teacher educator led a discussion of “what role the teacher mighthave in interacting with the students, what kind of information maybeteachers should be keeping in mind and in relationship to her lesson plan”(Interview). In this way, the teacher educator encouraged an analysis ofthe role and the thinking of the case study teacher among his pre-serviceteachers.

At sites C and D, the teacher educators focused on the complexitiesof managing small group instruction. At site C, the teacher educator indi-cated that his pre-service teachers’ recognized the complexities of groupwork and that they desired to learn more about how to accomplish whatthey saw in the case study. One of his students commented: “I’m goingto try to find somebody that knows how to do that so I can watch it morebecause it looks pretty complicated” (Questionnaire). The teacher educatorfelt that the case study helped his pre-service teachers to understand thecase study teacher’s intentions when she interacted with small groups ofstudents. The teacher educator found that his pre-service teachers came torecognize that the teacher’s interactions were purposeful and that this wasan explicit role of the teacher: “They sensed that that was a very purposefulthing on the part of the teacher that was made possible because of thecareful preparation in which the teacher thought about what she wantedthe students to do and thought about the kinds of expectations” (Interview).At the same time, the teacher educator found that the pre-service teacherswere “anxious [because] they didn’t know enough yet to do anything morethan appreciate it [the teacher’s role]” (Interview). The teacher educatorfelt that his pre-service teachers did not yet know enough to construct aplan and implement it themselves, but that many hoped to work in theirstudent teaching with a teacher who did.

At site D, the pre-service teachers had been introduced to cooperativelearning as part of a foundations course and had unanimously describedthe role of a teacher during cooperative grouping as being one of a facil-itator or mediator of learning. For the teacher educator, the case studyprovided a realistic setting where the pre-service teachers could observeeffective group work, which they had not seen in their field experiences.One pre-service teacher commented: “Before I had viewed the CD-ROM,I hadn’t really known of a situation where groups worked” (Observation).Another class participant explained it this way: “I have yet to see a teachercombine each group’s findings together and have the class go back andfind an answer given the whole class’s input. She took it (cooperativelearning) one step further” (Observation). Cooperative learning had been


introduced as a teaching and learning strategy but was yet to be observedas an effective structure in an actual classroom. The case lesson becamea realistic setting that the teacher educator could use for discussion withthese novice teachers.

Supporting Reflective Practice

The multimedia case study provided two potential opportunities for theteacher educators in supporting the reflective practice of the pre-serviceteachers. As we argued earlier, the lesson in the case materials is a contextfor the pre-service teachers to observe and analyze what has occurred in aclassroom after it has happened. This is an opportunity for the pre-serviceteachers to reflect on the actions of the case study teacher. The case mate-rials also include the reflections on action by the case study teacher. Thisprovides an opportunity for the teacher educators to use the reflections onaction of an experienced teacher in order to support the development ofreflective practice among the pre-service teachers.

At site A, the teacher educator described his program as one wherethey “stress reflection” (Interview). This teacher educator did not usethe case study teachers’ reflections, but instead engaged his pre-serviceteachers in a discussion about the case study teacher’s interactions withthe small groups. As their analysis of the case progressed, the pre-serviceteachers became somewhat more tentative in their judgments about thecase study teacher. They wanted to see all the other groups, not just thethree selected for the case study; they wanted to know more about whathappened next; they expressed a need to have “the whole picture.” Theynoted that “she has so much to do” and that “it is hard to run a classroomin that way” (Interview). The teacher educator saw these responses in termsof how the pre-service students were beginning to “see the complexities”of the classroom (Interview). At least three of the pre-service students sawthese complexities in terms of their own emerging practices and expressedtheir concerns that they will need to “think about ways of explaining andstudent interpretation” (Questionnaire). The teacher educator supportedthe reflective reasoning of his pre-service teachers by engaging them inreflecting on the actions of the case study teacher.

At site B, the teacher educator used the case study teacher’s reflectionswith his pre-service teachers. The teacher educator was surprised to findthat three of his students were critical about a lack of depth in the teacher’sreflections: “it was fine for her own reflection, but if she were doing thisfor other people to make sense of . . . they thought the teacher should haveelaborated some points much better” (Interview). The teacher educatorelaborated: “I was a little bit disturbed or perturbed by this somewhat of


a tendency to expect other people to tell them everything” (Interview).Particularly startling was one pre-service teacher’s comment: “I did notfind them [the teacher’s reflections] overly helpful to me as all students andgroups of students are different” (Questionnaire). This pre-service teacherwould appear to be so focused in the particulars of practice that she doesnot see how reflecting on one setting could be of value in understandinganother setting. Two of the pre-service teachers at site B did find valuein the teacher’s reflections. One indicated that it “allowed me to comparemy reaction to the lesson to her reaction to the lesson” (Questionnaire).Another felt the reflections clarified “what she was thinking when shewas planning the lesson and what she was thinking during the lesson”(Questionnaire). Both of these pre-service teachers indicated that the casestudy teacher’s reflections would help them in preparing and teaching theirlessons.

At the third site, C, the teacher educator noted that his pre-serviceteachers found the reflections “very focused and purposeful” (Interview).One of the pre-service teachers observed that the case study “gave mea more general but paradoxically more specific idea of what goes intoeffective teaching practices” (Questionnaire). The teacher reflections inother video-based materials that this teacher educator had used were“generic comments” and “almost in the order of endorsements” (Inter-view). In contrast, the reflections in this case study showed reflective,careful planning and the teacher’s thinking about what she thought herstudents were doing. The teacher educator commented: “We teach them[pre-service teachers] about teaching, but we don’t actually show themteachers practicing, reflecting, and evaluating” (Interview). The artifacts ofthe teacher’s lesson plans and her reflections went beyond what the teachereducator had found in other video materials, observation experiences, andreadings. For the pre-service teachers, the reflections appeared to providean important link to the relationship between the planning and anticipationand the actual outcomes of a lesson. As one pre-service teacher wrote: “Sheprovided insights on what she had expected and what actually happened”(Questionnaire). Another observed that the reflections “let you know whatthe teacher was trying to do and what the teacher accomplished. It gave youinsights into the lesson” (Questionnaire). The teacher educator commentedthat the case study had “incredible depth” and noted that for him “it takes along time to unpack it as information and then you have to repackage it intoyour own thinking for professional training” (Interview). He anticipatedhaving even better results with his pre-service teachers now that he hadexperienced the depth of the case study materials.


The Role of Planning and Preparation

At sites A and C, the teacher educators did not use the case study artifactsto emphasize the role of planning and preparation for lessons, althoughat both sites this issue was addressed by the pre-service teachers in theirwritten assignments and questionnaires. At site B, the teacher educatorfelt that anticipating students’ thinking was an important component of theplanning and preparation for teaching a lesson. The teacher educator sawthat anticipating students’ thinking “was a challenging task for the teacher”and that his pre-service teachers had difficulty with that task and “did notsee the value in thinking about student thinking ahead of time” (Interview).The teacher educator saw the teacher in the case study as well preparedand as having thought out the approaches that the students might take.However, his pre-service teachers felt her plans (like her reflections) were“not detailed enough” (Interview). This surprised the teacher educator andreveals a mismatch in the perspectives of the teacher educator and the pre-service teachers as to what constitutes appropriate elements in a lessonplan.

Since the case study lessons lasted only one class session and part of thenext days’ session, the teacher educator at site B involved his students inplanning mathematical lessons that could follow these lessons and wouldbring out the ideas of weighted averages. The teacher educator felt thatthe mathematics of this concept was significant content that his pre-serviceteachers needed to understand both mathematically and pedagogically. Theactivity of developing a next lesson revealed to the teacher educator thepre-service teachers’ “beliefs in action” (Interview). He found that theywould suggest, for example, that the lessons needed to mix different modesof instruction or types of assessment, but that they could not articulatewhy this should be done. The pre-service teachers could not articulate“the criteria that they should use to decide what kinds of activities theyshould use” to develop the mathematical ideas (Interview). The teachereducator reflected that he had not anticipated how revealing of the pre-service teachers’ thinking this planning activity would be, that it wasvaluable for him to see them engage in this type of planning and that hewas then thinking about how to bring this in as a component in the methodscourse.

At site D, an important goal for the teacher educator was for the pre-service teachers to plan lessons with relevant and motivating contextsfor the mathematical content. The pre-service teachers understood thesneakers problem to be relevant to students’ lives; they identified this asbeing a motivation for learning. One pre-service teacher observed: “they[the 7th graders in the case study] were interested and they wanted to solve


the problem . . . they could bring to the solution their own understandingsince they have all bought sneakers before” (Observation). Another pre-service teacher reasoned that, because of the students’ familiarity with thecontext of the problem, students were able to focus on applying problem-solving skills. One pre-service teacher used the idea of real-world data inher own teaching as a result of viewing the case:

The lesson that I taught last week was on ratios, proportions, and percents and I wantedto show how it applied to the real world like and I kind of took that from the CD-ROM.For my ratio, proportions, and percent unit, the students’ final project dealt with buyinga house, putting a down payment on it, and computing simple interest. The problemsinvolved information from a real world situation. (Observation)

This pre-service teacher valued the use of a real context and incorporatedit to her developing practice. She shared its effectiveness with the groupand felt that the students in her class “really enjoyed the lesson” (Obser-vation). While computing house payments can be considered a real worldapplication of mathematics, it falls short of being a relevant one for middleschool students. This pre-service teacher may have been constrained bythe curriculum in her field placement or perhaps she had not understoodthe distinction between real world contexts and relevant middle schoolcontexts when constructing her problem. The teacher educator did not takethe opportunity to explore these aspects of context with the pre-serviceteachers.

Understanding Mathematical Content

At site B, the teacher educator intended to “emphasize the value and impor-tance of anticipating students’ thinking” with his pre-service teachers(Interview). The teacher educator had spent a class session focusing onthe mathematics of the case study and found that his pre-service teachershad difficulty “thinking about how else middle school kids might solve”this problem and that “it was much harder than what I thought it wouldbe for them” (Interview). This teacher educator, while knowing that thelack of understanding of school mathematics by pre-service teachers hasbeen well addressed in the research literature, was nevertheless surprised atthe extent to which this the lack of understanding was displayed by his pre-service teachers. This dismay occasioned his reflection on the developmentof his own perceptions on the difficulties that pre-service teachers havewith respect to mathematical content:

But they seem to have difficulty. I used to think some of that may be because of simply theirlack of experience working with kids. And then I went from there to maybe it’s becauseof their lack of content knowledge, because sometimes anticipating the different ways kidsmight solve [a problem] is really just reflection of how you might solve differently the same


problem. The stronger the mathematical understanding the person [has then] he or she cancome up with several different ways of solving problems. So, I thought that may be anotherreason. But now I’m moving more towards this cultural interpretation or cultural reasonsfor pre-service teachers and in-service teachers having difficult anticipating children’sthinking. It’s not something that they are used to doing and they don’t necessarily seethe value. (Interview)

Investigating mathematical knowledge in the highly contextualized situ-ation of the case study revealed this teacher educators’ developing under-standing of limitations of pre-service teachers’ mathematical knowledge.Understanding and anticipating children’s thinking had been an over-arching goal for this teacher educator. His reflections on the difficultiesand dilemmas that he faced with his pre-service teachers indicated a shift inhis way of thinking from focusing on the pre-service teachers’ necessarilylimited experiences with children, to weaknesses in their mathematicalunderstandings, and finally to cultural beliefs and values about thinkingabout children’s thinking.

At site D, the mathematical reasoning of the seventh grade studentsin the case study impressed both the pre-service teachers and the teachereducator. For the teacher educator, the mathematical content was congruentwith his goals for the methods course. Having the pre-service teachersexperience challenging mathematical content had been a continuingobjective. The teacher educator set aside portions of class sessions forthe pre-service teachers to solve typical middle and high school prob-lems with the intent of taxing their mathematical capabilities. The teachereducator felt that the case study demonstrated effectively to the pre-service teachers a situation where the mathematics might appear to berote but in fact required a deeper understanding of mathematical rela-tionships. Initially, the pre-service teachers saw the case study problemas simplistic, requiring little mathematical knowledge. However, afterviewing the students’ solutions and the case study teacher’s facilita-tion of the lesson, the pre-service teachers had a greater appreciationof the required mathematical knowledge. The pre-service teachers weresurprised by the extent to which their learning of mathematics had notprepared them for seeing the mathematical relationships in the sneakersproblem.

This was further illustrated during a follow-up discussion to an assignedwritten case study question. The question addressed the mathematicalrelationship between ranking using the sums of each group’s rank andranking using the reverse sums of each group’s rank. Upon initial consid-eration, none of the six pre-service teachers believed there to be a mathe-matical relationship between the two approaches. One pre-service teacherclaimed:


I just said that there was very little mathematical relationship except traits that were morefrequent in one would have higher values than in the other method. So there might be somecorrelation but it is really not mathematically direct. (Observation)

The class evaluated the proposition that a mathematical relationship didexist and worked collaboratively to reveal it. After working through anexample with both approaches to ranking, one pre-service teacher shared:

What I am thinking is somewhere there should be some total, and the reverse rank plus therank should come up with a total and have some significance on that total because they arejust the opposite. (Observation)

The class discussion encouraged the pre-service teachers to think moredeeply about the problem. The teacher educator’s use of collaborativedialogue around the case issue supported these changes in understanding.

The teacher educator was surprised at how the mathematics in thecase study revealed content weaknesses among his pre-service teachers.Since this teacher educator had limited familiarity with the research onpre-service teachers’ subject matter knowledge, he was surprised whenthese secondary mathematics education majors demonstrated a lack ofknowledge about concepts that the teacher educator felt that the pre-service teachers ought to have learned in high school. The teacher educatoralso reported that, after he had viewed the case study classroom video,he initially thought that the case study teacher herself had had difficultyknowing whether or not the students should divide by 6 or 8 when rankingthe criteria using averages. This issue of the correct divisor comes aboutwhen the seventh grade students in the case study are confused as towhether to divide by 6 (the number of lists) or by 8 (the number of itemson each list). The teacher educator felt that the case study teacher seemednot to know whether it would make a difference if 6 or 8 divided each sum.

The case materials provided an important opportunity for the pre-service teachers to externalize their insufficient mathematical knowledgeand situate it in an actual classroom context. This was particularly memor-able for the teacher educator since the pre-service teachers’ solutions to thesneakers problem were representative of each of the solutions presented bythe seventh grade student groups. The same discussion around ranking byaverages and the determination of the correct divisor to be used occurredamong the pre-service teachers. The teacher educator was surprised thatthe pre-service teachers’ solutions were so similar to the solutions of theseventh grade students and was amazed that one pre-service teacher wascompletely off track in her solution. The mathematics of the case studymaterials afforded an opportunity for both the teacher educator and his pre-service teachers to explore the limitations of the subject matter knowledgeof the pre-service teachers within the context of a real classroom.


In summary, we found that these teacher educators used the investiga-tions of the artifacts of a case study of practice to support the developmentof their pre-service teachers’ thinking in four areas: (1) the complexities ofthe teacher’s role in small group work, (2) the value of explicitly revealingthe teacher’s reflections on the lessons, (3) the role of planning and prepar-ation, and (4) the limits of pre-service teachers’ abilities to understandand appreciate students’ thinking and to extend lesson ideas. Both teachereducators and their pre-service teachers gained perspectives on the role ofa teacher’s mathematical content knowledge.

DISCUSSION

The results of this study give some insights into the knowledge and prac-tice of teacher educators as they seek to manage a core dilemma in theirpractice: namely, their goals and intentions to support pre-service teachersprofessional development in ways that lead to analysis and critical thinkingwhile at the same time meeting the pressing needs of pre-service teachersfor practical strategies for the classroom. We have organized the discus-sion of these results in terms of the four emerging themes presentedabove, while highlighting some of the differences among the teachereducators’ knowledge and practices and their emerging understandings oftheir pre-service teachers.

At all four sites, the teacher educators and their pre-service teachersfocused on elements of teacher-student interactions in group situations.At two of the sites (A, B), the discussion appeared to be driven by thepre-service teachers’ concerns with the nature and quality of the casestudy teacher’s interactions with one small group of students. The teachereducators used this as an occasion to support a more critical analysis on thepart of the pre-service teachers by analyzing why the case teacher mightbe interacting with the students in this particular way and the relationshipbetween her beliefs about listening to students and the perceived lack ofprobing with a particular group of students. The teacher educators foundthat their pre-service teachers became more aware of the limitations oftheir perspectives and this, along with their analysis of the interactions,suggested that the pre-service teachers were beginning to appreciate thecomplexity and difficulty in understanding classroom interactions.

At the other two sites (C, D), the teacher educators focused the pre-service teachers’ attention on the complexity of the role of the teacher andthe relationship between her plans and purposes and her subsequent inter-actions with students. The case study teacher’s interactions with groupsof students provided an example of practice that the pre-service teachers


had not yet seen in any classrooms that they had experienced. The teachereducators seemed satisfied that their pre-service teachers had taken animportant step in appreciating the planning, knowledge, and skills thatenter into effective interactions with groups. One of the teacher educatorsrecognized, as did his pre-service teachers, that the analysis of the casestudy alone had not prepared them to plan and implement effective groupinteractions. Like the teacher educator in Geddis and Wood’s (1997) study,this teacher educator recognized the needs that pre-service teachers havefor immediate and practical strategies for the classroom. But in this case,the teacher educator supported his pre-service teachers in taking firststeps to recognize the complexities that are involved in planning andorchestrating effective group interactions.

The results of this study suggest that we need to discuss the constructof reflective practice (the second emerging theme) at two levels: first,the activity of the teacher educators themselves reflecting on their prac-tice and, second, the activity of the teacher educators in supporting theirpre-service teachers in reflecting on their emerging practices. As Tzur(2001) argued through his self-analysis of teacher educator development,reflecting and interacting with others are a central activity in the develop-ment of a teacher educator. Analogously, John (2002) found that teachereducators uniformly argued for the value of reflection in supporting pre-service teacher professional development. The results of this study wouldsuggest that the meaning of such reflection in the preparation of pre-service teachers varied substantially among the four teacher educators whoparticipated in the study.

The use of a multimedia case study of practice provided severalopportunities for the teacher educators to reflect on their own practice.At site B, the teacher educator reflected on his own developing theoryfor understanding the limitations of pre-service teachers’ subject matterknowledge. At site C, the teacher educator simultaneously found the casestudy teacher’s reflections to provide a valuable resource for illuminatingthe practice of reflection for his pre-service teachers while recognizing thatthe depth of the case study materials provided him with a resource thathe needed to unpack and repackage with his own thinking about profes-sional development. At site D, the teacher educator came to recognizethat the mathematical experiences of his pre-service teachers left themunder-prepared to transform that knowledge into effective teaching. Sincemultimedia case studies of practice were a new medium for all of theseteacher educators, one could view all their experiences with the materialsas provoking reflection on their own practice.


The activity of the teacher educators in supporting their pre-serviceteachers in reflecting on their emerging practices appears somewhatcomplex to understand, particularly because it introduced a new elementinto this practice, namely, reflecting on an experienced teacher’s reflec-tions. How the teacher educators described and characterized the reflec-tions of their pre-service teachers varied across sites. At site A, where theteacher educator stressed the role of reflection in the pre-service teacherpreparation program, the teacher educator saw his pre-service teachers ascritical and analytical in their thinking, especially about the role of the casestudy teacher in interacting with groups. However, this teacher educatordid not explicitly use the case study teacher’s reflections in any way. Thismay have simply been due to time constraints, as the use of the materialshad to be integrated into an already existing preparation program, or itmay be that the teacher educator already found his pre-service teachers tobe sufficiently reflective about practice. This teacher educator may havefelt, therefore, that there was less to be gained by having his pre-serviceteachers examine the reflections of the case study teacher.

The results at Sites B and C are striking in their contrast: at the first site,the teacher educator was dismayed by some of his pre-service teachers’views that the case study teacher’s reflections were inadequate, especiallygiven his own position on the value of those reflections. This served asevidence to the teacher educator that his pre-service teachers wanted to betold about teaching, which we interpret as a lack of reflection on the part ofthese pre-service teachers. At this site, the case study teacher’s reflectionsdid not appear to further the reflective activity of most (but not all) of thepre-service teachers. In contrast, at site C, the teacher educator and thepre-service teachers uniformly found the reflections valuable because theyprovided the pre-service teachers with insight in to the reasoning of thecase study teacher. This provided the teacher educator with a means ofmanaging the dilemma, referred to by Geddis and Wood (1997), of howto meet the immediate concerns of pre-service teachers for strategies ofpractice while, at the same time, giving them an opportunity to examinethe reasoning and rationales behind the practices. While it is not clearhow to account for the differences between site B and site C, it should benoted that it was a small number of pre-service teachers (three out of five)who were critical of the depth and usefulness of the case study teacher’sreflections.

Finally, contrary to the findings of John (2002), the teacher educatorat site D did not emphasize the role of reflection with his pre-serviceteachers. This may well be another instance of a lack of familiarity bythis teacher educator with the research literature on the role of reflection


in preparing teachers; on the other hand, it might be an indication that thisteacher educator’s primary concerns with content and planning may haveexcluded his explicit development of the professional language and activityof reflection with his pre-service teachers. The pre-service teachers didengage in writing about and discussing various aspects of the case studyand we would see this as reflective activity. At this one site we did not findlanguage about this as reflective activity.

At sites B and D, the issues for the teacher educators around planningand preparation were interconnected with the issues around mathematicalunderstanding. The teacher educators used the artifacts of the case study tofocus on the mathematical content, the mathematical context, and the useof student thinking about the mathematics by the case study teacher. It wasat these two sites that the teacher educators began their implementationsof the case materials with an exploration of the mathematical problemthat was central to the case. As we reported earlier, the teacher educatorat site B emphasized the importance of anticipating student thinking asa key component in planning for a lesson and engaged his pre-serviceteachers in developing plans that would extend the mathematical ideasin the lesson that they had seen. We see that for this teacher educator,understanding students’ ways of thinking about the mathematical contentand building on students’ mathematical ideas are essential elements to bereflected in the planning and preparation for lessons by the pre-serviceteachers. The teacher educator found that the pre-service teachers haddifficulty in anticipating middle school students’ responses and that theirown planning for follow-up actions revealed that they were not able toarticulate rationales for pedagogical strategies. The pre-service teachers’notions of planning did not appear to include an understanding of studentthinking as a component of lesson planning or the articulation of reasonsfor teaching activities. We interpret this as an instance of the desire of theteacher educator for his pre-service teachers to develop insight and analysiswith respect to the tasks of teaching, while the pre-service teachers mayhave been more concerned with the immediate practicality of what to doin the next lesson.

At site D, the role of the mathematical context played a central partin the planning that the teacher educator encouraged among his pre-service teachers. Through the case study lesson, this teacher educatoralso encountered the limits of his pre-service teachers’ knowledge ofschool mathematics. Through an analysis and discussion of the mathe-matical solutions that the pre-service teachers themselves developed, theteacher educator was able to move his pre-service teachers forward intheir appreciation of the depth of apparently elementary concepts and


the importance of the teacher’s mathematical content knowledge. What isparticularly striking at this site is that the investigations of the case mate-rials precipitated a realization of the limitations of the pre-service teachers’understanding of the mathematical content for both the teacher educatorand the pre-service teachers. Through the use of the case, the teachereducator moved from having pre-service teachers simply do mathematicalproblems and create lesson plans around mathematical content, to facili-tating dialogue around the teaching of mathematics and the knowledge ofmathematics necessary to teach it.

CONCLUSIONS

Overall, we argue that the multimedia case study provided a site foranalysis and reflection that was situated in the context of practice and, assuch, it served as a resource for teacher educators to use with pre-serviceteachers in articulating their ideas about issues related to teaching. Therewere two patterns of implementation by the teacher educators who usedthe materials: one was open-ended and began directly with investigationof the issues in the case and the other began with a focus on the mathe-matical content of the case materials. All the teacher educators gainedinsights into the strengths and limitations of the thinking of their pre-service teachers as they made observations, reflections and analyses ofthe case study materials. All the teacher educators used the case studyto investigate aspects of the role of the teacher in facilitating effectivegroup work. The teacher educators were able to move beyond the surfacefeatures of what the case study teacher did to posing more critical questionsabout the reasoning behind her actions, as evidenced by the interpretationsgiven by the pre-service teachers and the concomitant discussion of theirperspectives.

While the investigation of the case study materials occasioned thereflection of the teacher educators on their work, the support of reflectionamong the pre-service teachers varied across sites. Two of the teachereducators used the materials to support the development of their pre-service teachers’ understanding of mathematical content along severaldimensions: planning, anticipating student thinking, and understandingmiddle school mathematics more deeply. The use of the case materialsrevealed some of the limits of pre-service teachers’ abilities to under-stand and appreciate students’ thinking and to extend lesson ideas. This,in turn, prompted the teacher educators to reflect upon their course goalsand the difficulties pre-service teachers encounter in making meaningfrom the course content. The results of this study suggest that multi-


media case studies of practice can serve as a vehicle for revealing theknowledge and practice of teacher educators as they manage the dilemmasof supporting pre-service teachers’ professional development. The differ-ences found among teacher educators in the support and development ofreflective practice, particularly through the use of an experienced teacher’sreflections, imply that this is an area in need of further research. Theuse of the interrelationship between planning and preparation and theunderstanding of mathematical content appeared powerful for some ofthe teacher educators in meeting their goals for the professional devel-opment of their pre-service teachers. Finally, this study suggests that theuse of multimedia cases provided an opportunity for teacher educators toreflect on their understandings of the professional development of theirpre-service teachers.

ACKNOWLEDGEMENTS

This material is based upon work supported by the National ScienceFoundation (NSF) under Grant No. 9725512. Any opinions, findings andconclusions or recommendations expressed in this material are those of theauthors and do not necessarily reflect the views of the NSF.

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Syracuse UniversityMathematics Department215 Carnegie HallSyracuse, NYU.S.A.E-mail: [email protected]

ELHAM KAZEMI and MEGAN LOEF FRANKE

TEACHER LEARNING IN MATHEMATICS: USING STUDENTWORK TO PROMOTE COLLECTIVE INQUIRY

ABSTRACT. The study describes teachers’ collective work in which they developeddeeper understanding of their own students’ mathematical thinking. Teachers at one schoolmet in monthly workgroups throughout the year. Prior to each workgroup, they posed asimilar mathematical problem to their students. The workgroup discussions centered on thestudent work those problems generated. This study draws on a transformation of partici-pation perspective to address the questions: What do teachers learn through collectiveexamination of student work? How is teacher learning evident in shifts in participationin discussions centered on student work? The analyses account for the learning of thegroup by documenting key shifts in teachers’ participation across the year. The first shift inparticipation occurred when teachers as a group learned to attend to the details of children’sthinking. A second shift in participation occurred as teachers began to develop possibleinstructional trajectories in mathematics. We focus our discussion on the significance ofthe use of student work and a transformation of participation view in analyzing the learningtrajectory of teachers as a group.

KEY WORDS: children’s mathematical thinking, professional development, school-wideinquiry, sociocultural theory, student work, teacher learning

A large body of literature has demonstrated that supporting teachers tomeet the ambitious and complex visions of mathematics reform is difficult(e.g., Borko et al., 1992; Jaworski, 1994; Kazemi & Stipek, 2001; Schifter,1998). Because of the inherent complexity of understanding how and whatteachers learn, Wilson and Berne (1999) have called on researchers tostudy professional development rooted in teachers’ own practice. Organ-izing teacher learning around the study of student work is one particularway in which professional development can be situated in practice (Ball &Cohen, 1999; Lin, 2002; Little, 1999).

In this article, we describe an approach to professional development inwhich teachers used their students’ mathematical work as a focus for theircollective inquiry (Franke et al., 1998; Richardson, 1990). The purpose ofour analysis is to account for the learning of teachers as a group. To dothis, we draw on sociocultural theories of learning that define learningas the transformation of participation (Rogoff, 1997). In the researchreported, we address the following questions: What do teachers learn


204 ELHAM KAZEMI AND MEGAN LOEF FRANKE

through collective examination of student work? How is teacher learningevident in shifts in participation in discussions centered on student work?Our findings lead us to conjecture about the use of student work as amediator of teacher learning.

BACKGROUND

A number of recent publications have advocated using student work asa tool for professional development (e.g., Ball & Cohen, 1999; Little,2002). This use of student work has the potential to influence profes-sional discourse about teaching and learning, to engage teachers in a cycleof experimentation and reflection and to shift teachers’ focus from oneof general pedagogy to one that is particularly connected to their ownstudents. Whether these opportunities are realized depends on the actualuse of student work in professional activity. Empirical research on theuse of student work is limited, given its relatively new emergence as amechanism for promoting professional development. It is a component ofJapanese Lesson Study1 and alternative versions (Fernandez, Cannon, &Chokshi, 2003) and is featured in protocols developed by the Coalitionof Essential Schools and Harvard Project Zero (Blythe, Allen, & Powell,1999). Examining student work is also a component of several new case-based approaches to professional development in mathematics such asDeveloping Mathematical Ideas (Schifter, Bastable, & Russell, 1999), theAlgebraic Thinking Toolkit (Driscoll, 1999) and the QUASAR cases oncognitive demand (Stein, Smith, Henningsen & Silver, 2000).

In a recent review of 26 published reports and papers in the UnitedStates, Little (in press) found only a handful of studies that constructeddetailed observational records of teacher interactions around student work.Those studies suggest that simply bringing together teachers to “look atstudent work” did not necessarily open up opportunities for learning. Howstudent work was used, the ways classrooms were represented in teachertalk, and the norms and habits of professional discourse influenced thepotential impact on teacher learning and knowledge (see also Crockett,2002; Crespo, 2002). As the use of student work in various professionaldevelopment communities grows, the need to examine how student workis used to focus teacher inquiry heightens. We contribute to this need byoffering a way of documenting teacher learning that examining studentwork supports.

TEACHER LEARNING 205


Our analyses are guided by a situated view of learning. Understandinglearning, as it emerges in activity, is paramount to such a perspective(Greeno & Middle School Mathematics Through Applications Project,1998). This perspective centers on how people engage in routine activityand the role things such as tools and participation structures play inthe practices that evolve (Wertsch, 1998). We apply a transformationof participation view, as described by Rogoff (1997), Lave (1996) andWenger (1998), to account for a group’s collective examination of studentwork.

Rogoff (1997) explains a transformation of participation view oflearning by contrasting it with two other models of learning: acquisitionand transmission. Both models assume a boundary between the world andindividual; the former posits that individuals receive information trans-mitted from their environment while the latter posits that the environmentinserts information into the individual. The transformation of participa-tion view takes neither the environment nor the individual as the unitof analysis. Instead, it holds activity as the primary unit of analysis andaccounts for individual development by examining how individuals engagein interpersonal and cultural-historical activities. Rogoff (1997) providesthe following explication:

. . . a person develops through participation in an activity, changing to be involved in thesituation at hand in ways that contribute both to the ongoing event and to the person’spreparation for involvement in other similar events. Instead of studying a person’s posses-sion or acquisition of a capacity or a bit of knowledge, the focus is on people’s activechanges in understanding, facility, and motivation involved in an unfolding event or activityin which they participate (p. 271).

The shifts in participation do not merely mark changes in activity orbehavior. Shifts in participation involve a transformation of roles and thecrafting of new identities, identities that are linked to new knowledgeand skill (Wenger, 1998). Lave (1996) states, “. . . crafting identities is asocial process, and becoming more knowledgeably skilled is an aspect ofparticipation in social practice . . . who you are becoming shapes cruciallyand fundamentally what you ‘know’ ” (p. 157). Our use of this conceptualframework in this article is an example of what Rogoff (1997) terms theinterpersonal level of analysis in that it “focuses on how people commu-nicate and coordinate efforts in face-to-face interaction . . .” (p. 269).This focus on the interpersonal level leads us to give primacy to theway the practice of the teacher group evolved, while we keep individualcontributions and the larger social context of professional development


practices in the background. Analyzing teachers’ collective engagementwith student work, then, reveals not only their deeper knowledge aboutstudent thinking and mathematics but also their developing professionalidentities as teachers. This attention to participation and identity hasfurther implications for the kinds of practices that teachers pursue withone another and with their students.

METHODOLOGY

This study uses data from a workgroup of ten teachers who met regularlyacross the academic year in order to document the way the group’s prac-tice developed during that first year. It is beyond the scope of the paperto provide a full account of why it has been sustained to the present(see Franke & Kazemi, 2001). Our main contribution in this article isto provide an analytic frame for understanding teacher learning as shiftsin participation. We engaged teachers from one elementary school in on-going professional development that consisted of two main components:(a) facilitated workgroup meetings centered on students’ written or oralmathematical work; and (b) observations and informal interactions withteachers in their classrooms.

The design of the professional development was modified based onpilot work using a Cognitively Guided Instruction [CGI] (Carpenter,Fennema, Franke, Levi, & Empson, 1999) model of professional devel-opment at two other elementary schools in the 1996–1997 school year.We did not follow the CGI approach by conducting workshops with theteachers and presenting them with the frameworks, nor did we designactivities using videos or worksheets for them to make sense of howthe typologies for problem types and strategies related to one another.Instead, we introduced CGI principles and terminology as teachers madeobservations of their own students’ mathematical thinking. However, wedid provide teachers with common problems to use in their classes thatconsisted of CGI word problem types (e.g., join change unknown2 or amissing addend problem, see Table III for examples).

Setting and Participants

The study took place at Crestview Elementary School (all names arepseudonyms) in a small urban school district. Data were collected duringthe 1997–1998 school year. This school was selected because it was adistinguished school in the state and noted in the district for its higher


mathematics test scores, relative to other schools in the district. (Highscores in this district meant performing at the thirtieth percentile.)

In 1997–1998 the school converted to a year-round calendar whichresulted in four cross-grade heterogeneous “tracks,” each consisting of13 teachers. To accommodate enrollment, at any one time, three tracksattended school while one was on vacation. The student body, roughly1300 students, was primarily Latino (90% Latino, 7% African Amer-ican, 3% Asian American). The transiency rate was approximately 30%.Over 90% of the student body received free or reduced cost lunch. Eachclassroom was bilingual, but students were transitioned to mainly Englishinstruction in the upper grades.

The group met about once a month throughout the school year afterschool. Only seven meetings were used for data analysis because the firstmeeting was an introductory meeting, and teachers did not bring studentwork. School administrators and support teachers were also invited to themeetings. The principal helped support the occurrence of these meetingsby giving up one faculty meeting per month. Thus teachers did not have anextra meeting to attend. The principal and resource teacher attended threemeetings. The principal joined each meeting briefly either at the beginningor the end, and she spoke regularly with the research staff.

Workgroups

Student work from teachers’ classrooms guided the substance and direc-tion of discussions at each workgroup meeting. Prior to the meetings,teachers used a common problem that they could adapt for their studentsin their class. For each meeting, teachers selected pieces of student workto share with the group. During the first year, our research team (theauthors and two university colleagues) chose these problems ahead oftime. The order in which we posed the problems was not set before westarted but unfolded based on our reflections on what was happening inthe workgroups.3 The mathematical domains we chose to focus on duringthe workgroup reflected those that the teachers were working on in theirclassrooms, such as place value, addition and subtraction, multiplication,and division. Problem types were given to teachers prior to the workgroupsand are shown in Table I.

Classroom Visits

To provide ongoing support to the teachers, build relationships, and collectdata, we visited the teachers in their classrooms as a means of learningmore about student thinking and teachers’ practices. Typically, we visitedonce and frequently twice, between each workgroup meeting. The visits


TABLE I

Student Problems Presented in Order of Workgroup Meetings

Problem type Problem

W1. Join Change Ashley has 9 (46)a stickers. How many more stickers does she

Unknown (JCU) need to collect so she will have 17 (111) stickers altogether?

W2. Measurement 1. There are 31 children in a class. If 4 children can sit at a

Division table, how many tables would we need?

2. There are 231 children taking a computer class after

school. If 20 students can work in each classroom, how

many classrooms would we need?

W3. Multiplication 1. Mrs. North bought 13 pieces of candy. Each piece of

candy cost 4 pennies. How many pennies did she spend on

candy altogether?

2. Mrs. North bought 15 boxes of animal crackers. Each box

cost 47 cents. How much money did Mrs. North spend

altogether?

W4. Computation 9 + 7 = ______ 20 + 17 = ______

28 + 34 = ______ 29

+ 16____

W5. Compare Rosalba has 17 (101) bugs in her collection. Hector has 8 (62)

bugs in his collection. How many more bugs does Rosalba

have than Hector?

W6. Choice of missing 1. Yvette collects baseball cards. She has 8 (67) Dodger cards

addend (JCU), in her collection. How many more Dodger cards does

measurement Yvette need to collect so that she will have 15 (105) Dodger

division, or baseball cards altogether?

subtraction 2. Yvette has 34 (274) baseball cards. She wants to put 10

baseball cards in each card envelope (or box). How many

envelopes will she need to put away all of her cards?

3. Yvette had 34 (274) baseball cards. She sold 11 (89) of

them. How many does she have left?

W7. Final Meeting Overview of all problems and strategies; no new problems

posed.

Note. a The numbers in parentheses indicate the larger number size we provided.


TABLE II

Study Participants

Name Grade Teaching Experience

Rosalba K 8

Jazmin K 1

Miguel 1 0

Paula 1 3

Adriana 1 3

Rose 2 3

Patrick 2 0

Kathy 3/4 0

Anna 3/4/5 0

Natalie 3/4/5 9

were not structured formal observations, but, rather, informal visits.4 Dueto the focus of this particular article, we will not provide direct analysisof our classroom data. However, the role of classroom visits will beapparent in the way the facilitator made use of her knowledge of teachers’classrooms in shaping conversations.

PROCEDURE

The research described in this article involves cross-grade workgroupmeetings run by the second author5 with 10 teachers from one of the tracksin the school. The teachers represented a range of grade levels and teachingexperience (see Table II). During the workgroup meetings we encouragedteachers to adapt the problem by changing the number size and contextif they felt the changes would be appropriate for their students. We askedteachers, however, to keep the structure of the problem the same. At thebeginning of each meeting, teachers briefly reflected in writing about thepieces of student work they had selected to share with the group. Theyalso indicated what problem they actually posed to their class and whythey made any changes.

The facilitator then invited teachers to share the variety of strategiesthat they observed in their classrooms. Teachers could comment on howthey adapted the problem, how their students reacted to the problem, andspecific ways in which their students attempted to solve it. As the strategieswere described, the facilitator recorded them on chart paper so the group


could revisit them later in the meeting. The facilitator consistently pressedteachers to describe the details of the students’ strategies. The facilitatoralso introduced common strategies into the discussion if it appeared thatteachers had not seen them in their own classrooms.

In order to explore what the strategies revealed about student thinking,the facilitator then asked teachers to compare the relative mathematicalsophistication that the strategies demonstrated. For example, a strategythat involves counting by ones from nine to 17 is less sophisticatedmathematically than one involving a derived fact, for example 8+8 is 16,8 is one less than 9 so it would be 17. As teachers voiced their ideasabout the strategies, we introduced terminology from the CGI frame-works such as direct modeling and derived facts to label the strategies intoworking frameworks that revealed the development of students’ thinking(see Table III). Occasionally, the facilitator brought in her knowledgeof research on children’s thinking by elaborating on observations thatthe teachers made. The group discussed the mathematical principles thatunderlie the various strategies and what they revealed about students’mathematical understandings. The facilitator redirected questions aboutparticular strategies and their place within the framework to the group fordiscussion or encouraged teachers to investigate them when they returnedto their classrooms. The working frameworks served as a source forcontinued deliberation, reflection, and elaboration in subsequent meetingsas teachers continued to pose problems to their class and learn about theirstudents’ thinking.

ANALYSIS

Data Sources

In the larger study, data collection occurred across two settings: the work-groups and classrooms. We documented all the interactions we had withteachers in the workgroups, in their classrooms and in informal interac-tions. The data analyzed for this article consist of: (a) seven workgroupmeeting transcripts from audio recordings; (b) written teacher reflections;(c) copies of student work shared by the teachers; and (d) end-of-the-yearteacher interviews.

Data Analysis

The data were collected during a single school year and were managedand analyzed systematically. We drew on case study and grounded theoryapproaches (Merriam, 1998; Strauss & Corbin, 1988). In the analytic


process, we made initial conjectures while analyzing existing data and thencontinually revisited and revised those hypotheses in subsequent analyses.The resulting claims and assertions are thus empirically grounded and canbe justified by tracing the various phases of the analysis.

We, the authors, transcribed the audiotapes from each meeting,collating the written comments made by teachers at each meeting, creatinglogs and noting major themes. During several initial readings of the tran-scripts and summaries, we asked questions of the data that centered onbuilding an understanding of how teachers were talking about student workand what kinds of mathematical and pedagogical issues were raised. Wecreated two broad categories that reflected issues raised in the workgroup:(a) understanding student thinking and mathematics, and (b) examiningrelations between students’ mathematical thinking and classroom practice.

We then identified a number of more descriptive themes that consis-tently emerged and re-emerged across the year, creating focused codesfor each of these themes that reflected the content of the conversations.We used the focused codes to code all the transcripts (see Table IV forfocused codes). The codes were not mutually exclusive and were appliedto exchanges or segments of conversation. An exchange was defined as aunit of conversation centered around the same issue. Thus, if the facilitatorasked a teacher to describe a strategy and then there were several turnsin which the strategy was detailed, the entire exchange was coded ratherthan individual turns. Some exchanges had multiple codes. We created atable (see Table IV) following each of our codes across the year. We wrotememos that kept track of the way the focused codes revealed the learningtrajectory of the group in relation to the larger themes that were of interestto this study. This was because we were interested in understanding theway teachers’ talked about student work and whether there were changesor developments in their interactions across the year. We used the analyticcommentaries to articulate how discussions about mathematics, studentthinking and pedagogy evolved over the course of the year. We identi-fied the trajectory of two major shifts: attention to children’s thinking anddeveloping instructional trajectories in mathematics. Finally, we selectedexchanges that were illustrative of the the development of the two majorshifts.

FINDINGS

Two major shifts in teachers’ workgroup participation emerged from ouranalyses. The first shift in teachers’ participation centered around attendingto the details of children’s thinking. This shift was related to teachers’


TABLE III

Problem Types and Strategies

Problem Direct Modeling Counting Derived Facts

Join ChangeUnknownAshley has 9stickers. How manymore stickers doesshe need to collectso she will have 17altogether?

Makes a set of 9counters. Makes asecond set of coun-ters, counting “9,10, 11, 12, 13, 14,15, 16, 17,” untilthere is a total of 17counters. Counts 8counters in secondset.

Counts “9 [pause],10, 11, 12, 13,14, 15, 16, 17,”extending a fingerwith each count.Counts the 8extended fingers.“It’s 9.”

“9 + 9 is 18 and 1less is 17. So it’s 8.”

Separate ResultUnknownThere were 24children playingsoccer. 7 childrengot tired and wenthome. How manychildren were stillplaying soccer?

Makes a set of24 counters andremoves 7 of them.Then counts theremaining counters.

Counts back “23,22, 21, 20, 19,18, 17. It’s 17.”Uses fingers tokeep track of thenumbers of stepsin the countingsequence.

“24 take away 4 is20, and take away 3more is 17.”

MeasurementDivisionThere are 31children in a class.If 4 children cansit at a table, howmany tables wouldwe need?

Makes a set of 31counters. Measuresout four counters ata time until all thecounters have beenused. Counts 7 pilesof 4 counters and 1piles of 3 counters.“We need 8 tables.”

Skip counts by 4suntil 32, “4, 8, 12,16, 20, 24, 28, 32”Uses fingers to keeptrack of the groupsof four. “We need 8tables.”

“4 times 6 is 24.7 more to get to31. So that’s 2 moregroups of 4. That’s8 tables. But onetable only has 3kids.”

MultiplicationMrs. North bought13 pieces of candy.Each piece of candycost 4 pennies. Howmany pennies didshe spend on candyaltogether?

Makes 13 piles of4 counters. Countsthem all up by ones.“It’s 52 cents.”

Skip counts by 4s,13 times, “4, 8, 12,16, 20, 24, 28, 32,36, 40, 44, 48, 52.It’s 52 cents.”

“10 times 4 is 40. 3times 4 is 12. 40 +12 is 52.”


TABLE III

Continued

Problem Direct Modeling Counting Derived Facts

CompareRosalba has 17 bugsin her collection.Hector has 8 bugsin his collection.How many morebugs does Rosalbahave than Hector?

Makes a row of 17counters and a rowof 8 counters nextto it. Counts the 9counters in the rowof 17 that are notmatched with theset of 8.

There is nocounting analogof the matchingstrategy.

“8 + 2 is 10 and 7more is 17. It’s 9.”

Note. Adapted from: Carpenter, Fennema, E., Franke, M.L., Levi, L., & Empson, S.B.(1999). Children’s mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heine-mann.

attempts to elicit their students’ thinking and to their subsequent surpriseand delight in noticing sophisticated reasoning in their students’ work. Thesecond shift in teachers’ participation consisted of developing possibleinstructional trajectories in mathematics that emerged because of thegroup’s attention to the details of student thinking. The particular mathe-matical focus was related to the understanding of place value evident instudents’ ability to decompose and recompose numbers efficiently. In thecourse of presenting our findings, we also highlight the mediating role ofthe facilitator and student work in the learning of the group.

Shifting Participation Towards Attention to Children’s Thinking

The first major shift in teachers’ participation occurred in how theyattended to the details of students’ mathematical thinking. The wayteachers engaged around student work shifted early in our work together;teachers found ways to interact with students about their strategies and todocument those interactions for the purpose of sharing in the workgroup.Teachers came to the first workgroup meeting uncertain and unaware ofthe different ways their students solved the workgroup problems (seeTable IV). As teachers continued to try a variety of problems, the focusof the group shifted again towards giving details about what the teacherspercieved as more complex student-generated algorithms. Using examplesfrom the data, we show how the substance of workgroup exchangesshift as teachers’ engagement with the student work shifts from one ofuncertainty about students’ thinking to one of active engagement with


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student strategies and a recognition of the mathematical competencies thestrategies revealed.

Initial participation: Being unaware of the details of students’ strategiesFor the first workgroup meeting, teachers had posed a join changeunknown or a missing addend word problem (e.g., 7 + ___ = 11) to theirstudents. The workgroup meeting began with the facilitator asking theteachers how their students solved the problem. During the course of thismeeting, teachers examined their students’ work, trying to determine howtheir students had solved the problem. The first five teachers, who sharedtheir students’ strategies, pointed to the incorrect strategies their studentsused and were surprised that the problem was difficult for their students.Although three out of 16 strategies shared were correct ones, the teacherpresenting the work were unsure as to how the students had completed theproblem. The teachers then tried to interpret what the student had donebased on what was written on the paper. For example, one of Miguel’sstudents had simply written “1 2 3 4 5” on his paper. The group generatedseveral possibilities about how the student had solved the problem. Thegroup concluded that the student had counted out loud “7, 8, 9, 10, 11”but wrote “1 2 3 4 5” to keep track of his count. Miguel, however, couldnot verify that strategy because he had not seen how the student solved theproblem or heard him talk about it.

Generating strategies for eliciting student thinkingIn the first two workgroups, teachers worried that eliciting student thinkingwas a difficult practice. Many teachers relied on their students’ writtenwork and did not see it as important to engage their students in conver-sation about their strategies. Teachers interpreted the direction to pose aproblem to their students in ways that inhibited their ability to talk to thestudents by giving it as a test, sending it home for homework, asking asubstitute to give the problem, or giving it as independent work and notcirculating to talk to the students. Many teachers worried out loud that theyhad missed their students’ thinking while others thought that the difficultystemmed from students’ inability to articulate their thinking.

Given the general uncertainty about students’ thinking, the facilitatormade a purposeful move in the first two meetings to engage the teachersin brief discussions about how to elicit student thinking. In fact, as TableIV indicates, much of the facilitator support occurs in the first half of themeetings, modeling and providing support for teachers’ interactions in theclassroom. The facilitator recognized that eliciting students’ ideas wouldrequire a change in the ways teachers talked with their students. As a


group, the teachers began to notice that the student work did not speakfor itself. It could provide a trace of student thinking, but interactions withstudents were important. In order to raise this issue for everyone in thegroup to consider, the facilitator drew on her knowledge of what a fewteachers were already doing in their classrooms and provided new ideasabout questions that might help children explain their thinking.

In the first workgroup meeting [W1], the facilitator explained thatstudents often make very general comments when they are not used toexplaining their thinking. Those comments do not reflect a lack of strategybut a lack of experience. Some teachers agreed. Sara added to what thefacilitator said by explaining the resistance she experienced when shebegan to ask children how to explain their thinking.

Sara: Or sometimes too I found that some of my students were,I guess just like, they felt intimidated. They had their rightanswer, but then you ask them, “how did you get it?” It waslike, “I got it, why do you want to know?” They didn’t wantto say how they got it. (W1: 10/28/97)

Sara’s comment made public the idea that students would have to learnthat explaining their thinking was both valued and a necessary part of doingmathematics. To help teachers start conversations with their students, thefacilitator suggested asking questions such as, “What numbers were youthinking of in your head?” “What number did you start with?” Kathythen shared a strategy she used by explaining that sometimes students canshow what they did, even if they cannot describe it verbally very well. Forexample, students can demonstrate how they counted. For many teachersin the group, eliciting student thinking was a novel practice.

Detailing strategiesThe impact of the facilitator’s press for details was evident in subsequentworkgroup meetings. Some teachers began to draw on annotations theymade on the student work to help them remember what their studentshad said. The following exchange demonstrates Miguel offering a studentstrategy with more specificity than he did in the first workgroup. Theproblem involves figuring out how many tables of four are needed for 16children (See Table I). He refers to the notes he made.

Miguel: So afterwards, one kid really impressed me. John, after he didit – he showed me the answer, and I wrote down what he said.He laid out one crayon and he put four crayons around it. Andhe represented one table with four students. So he put anothercrayon out and set the second table. And he put four students


there, and the third crayon and he represented four students.Until he got up to 16. He counted up to 16 with the crayons.(W2: 12/9/97)

Detailing more sophisticated student strategiesWhen teachers posed the multiplication and computation problems intheir classes, a subtle shift occurred in the kinds of strategies teachersfocused on detailing. The teachers began to recognize that some strategieswere quite intricate and amazing because these strategies were not consis-tently emerging from all classrooms. In the January meeting, four teachers(Kathy, Jazmin, Miguel, and Natalie) brought work that evidenced moresophisticated reasoning. For the multiplication problem, what is the totalcost of 15 boxes of animal crackers at 47 cents each?, (see Table I) teachersshared a variety of students’ strategies. In Kathy’s class, some studentsadded 15 sevens first and then the same number of 40s. Natalie saw similarstrategies in her class and she described the ways that students kept trackof the 40s and sevens.

Natalie: This one [he counted the sevens] by threes, and this one bytwos. He went 14, 14, 14, 14, 14, 14. Since there were onlythree left because it was an uneven number, the last three hemade 21. And then over here with the fours, did 8 and 8. Andthen put the eights together to make 16. So he’s got theserafters going out.

Patrick: It’s really wild.

Facilitator: We actually see this strategy a lot. When we let kids inventtheir own ways – this is one of the most common ways thatthey do it. Naturally, without us prompting at all, they use thiskind of arrow notation. (W3: 1/20/98)

Repeatedly in transcripts, especially in meetings 3 through 5 (see TableIV), teachers expressed their amazement as these strategies were shared,“That’s wild!” “How neat!” “Wow!”, suggesting that they were surprisedand intrigued by students’ invented algorithms. In the fourth workgroup,Kathy shared strategies for 28 + 34 and 20 + 17. For 28 + 34, one of herstudents, Ricardo, had added eight and four to make 12. Then he addedthe 12 to 30 to make 42 and then 20 more to make 62. For 20 + 17, headded 7 to 20 to make 27. And then he added the remaining ten to make37. The teachers were amazed but now able to follow both the strategiesfairly easily. However, they needed some help with the third one Kathydescribed.


Kathy: The last one, it started to make sense to me why he was doingthis because the last one is 29 plus 16. He switched it to 28 plus 17. Whichmakes it even to add up to 40. He could add two out of the 17 to make 30.And then add 15. Do you know what I mean?

Teachers: Wait, what! Say that again. I don’t understand.Kathy: You can break the seven apart into a five and a two.Facilitator: Oh, and he took the two from here and made this a –Kathy: 30. And add 15 to this to make 45.6

Teachers: Ohhhhhh!Paula: That went over my head!Patrick: That’s amazing![A little chatter about that strategy.]Patrick: Like he had to reason – but he had to reason – see, what’s so

weird about that is he had to – he’s got the 15 and the two onthe bottom group there which are easy to add. You make 30and 45. I mean, but he had to think that the 28 could become29 and the 17. I’m sorry, the other way around.

Kathy: He can borrow it.Patrick: But he broke up the seven into five and two. Now how did

he think though, I need a seven. So I’ll make 16, 17 and 29,28. That’s like, he’s going way ahead of his thinking. That’s– I mean the rest of it makes sense when you see it likethis, but he had to think all that way. He’d be a great chessplayer.

[Teachers laugh] (W4: 3/3/98)

Paula’s comment, “that went over my head,” and Patrick’s choice ofnoting “what’s so weird about the strategy” suggested that the child’sreasoning was atypical and required teachers to slow down and follow itclosely (many of them saying in unison, “wait, wait! Say that again”). Thesharing of the invented algorithms created both amazement and levity – theteachers were poking fun at their own need to go over the strategies slowly.This further supports the claim that these strategies were not familiarto them. It is important to note that the group was not dismissing thestrategies, at least not publicly. Patrick’s comment that Ricardo would bea great “chess player” – because he is planning ahead – reflected Patrick’spositive regard for the child’s thinking.

Observing sophisticated reasoning in primary gradesThe shift in noting children’s sophisticated reasoning is an importantmarker of teacher learning because the comments teachers offered in thediscussions showed they were impressed with their own students’ thinking.


If students in their own classes were using these strategies, then they allshould be able to encourage such thinking. This shift in the type of observa-tions made in the later workgroup meetings contrasted with what teachersnoted in the first meeting. In that first meeting, in five separate exchanges,teachers focused on how children were unsuccessful with the problem orwere using cumbersome strategies. It is also significant to the trajectory ofthe group that these observations of sophisticated reasoning were not justlimited to students in the upper grades. The kindergarten and first gradeteachers also observed new competencies in their students. In the secondmeeting, Miguel declared twice how stunned he was that his first graderswere able to solve a division problem and felt he had underestimated them.And Jazmin explained below during the third meeting:

Facilitator: Some of the kids can count by twos.Miguel: In kindergarten – wow that’s great.Facilitator: Tell them what happened today when you did the two crayons

for two cents each.Jazmin: And then what if I bought three crayons. And they said six.

Six cents.[Teachers impressed, “Wow!”]Jazmin: Because we thought they would need a lot more visuals and

manipulatives. And some of them were like, we don’t need it.Facilitator: One student today made up a problem. Five pieces of gum.

Five cents each.Jazmin: Yeah. Eduardo. And then he had to go back because he put

his final answer as five. And I said, “Well how many piecesof gum do you have?” And he said, “Five.” And I said, “Howmuch do they all cost?” He said, “Five cents.” Well, you haveto go back and indicate that because he couldn’t count byfives. He went back and he put the five markers for each one.And then he went back and counted them. Each time he puthis markers, he would go back and recount them to make surehe had five. Long process. But he got it right. (W4: 1/20/98)

The teachers began to re-evaluate their contentions that particular typesof problems belonged to particular grade levels. It is important to note thatwhen Jazmin came to the first workgroup meeting, she was the only teacherwho did not bring student work. She expressed then how she was afraid topose the first problem to her students, thinking that they would be unableto solve the problem.


Making progress: Seeking help and sharing successes of elicitationThe claim that the group was learning to attend to children’s reasoningis further supported by the struggles and successes that teachers madepublic in the workgroup. We see consistent emphasis on detailing acrossthe workgroup meetings – the coding reveals between 4 and 12 exchangesin each meeting to which teachers brought student work (see Table IV). Inexchanges in each workgroup meeting, teachers return to voicing both theirpuzzles and experiments with helping children articulate their thinking.But, eliciting student thinking was not a straightforward task for everyonein the group. Teachers shared two different kinds of observations in thegroup. The first related to some people’s practice of modeling or explicitlyshowing strategies. The second was working on eliciting strategies in thefirst place. Next, we provide examples of those.

Noting the impact of teacher modeling on children’s strategiesThe structure of the workgroup meetings, as we mentioned earlier,remained similar throughout the year. And the bulk of each meeting wasspent on describing as many different strategies that students were using.During the fourth meeting, two teachers, Miguel and Paula, made anobservation or raised a concern about how their own decisions to modelstrategies first affected the range of strategies their students subsequentlyused. We cite these two observations as examples that it mattered to thegroup that teachers were attending to children’s thinking. Two teachers,who believed it was their job to show children strategies first, began toquestion whether that was necessary.

Miguel: I’m kind of concerned now because I’ve been teaching mykids one method of adding large numbers. Use numbers like11 plus 6. I say, take the 11 put it in your head. 6 in your hand.12, 13, 14, 15, 16, 17, 18. What worries me now from whatyou’re saying, I feel like maybe I’m stunting their ability togroup. (W4: 3/3/98)

Miguel’s reference to children’s ability to group is related to theongoing conversation the group had been having about children’s under-standing and use of place value to solve computational problems (an issuewe develop further below).

Paula: I find that when I say – if I’m explaining these things to mykids, if I don’t say anything at all, and just put it [the problem]up there and say, okay, this is what you need to do. Solve itand be able to tell me how you got the answer, they do muchbetter than if I try to kind of prep them on what they’re doing.


[Other teachers nod to indicate they have noticed the samething.] (W4: 3/3/98)

In the fifth and sixth meeting, there were six exchanges among theteachers about the various sophisticated strategies that children across thegrade levels were using. We argue that it was these exchanges that madepublic the fact that some teachers were noting a greater range of strategiesin their classrooms than others. In the fifth workgroup, Anna shared hercontinued frustrations with eliciting student thinking.

Anna: And then I just have another question. I always feel like whenI come in here, I have my stuff and I haven’t had time tolook at it. So what I’m wondering is any suggestions on like– because I’m trying to get them to write out what they do, soI don’t need to sit and talk to everybody. And it’s not totallyworking yet, and I haven’t made time to talk to people. So Icome in and feel like, you know what, I have all this stuff andI still don’t really know what they did. So anyone have anysuggestions? (W5: 3/24/98).

In contrast to the beginning of the year, Anna received a barrage ofhelp from the teachers, and notably not just from the facilitator. A numberof teachers suggested different ways she could select a few students toobserve more closely each day, either by selecting them to share theirstrategies at the board or asking them questions independently. They gaveher suggestions about how to select which students to talk to and whatkinds of questions to ask. They also provided several ways of structuringthe class period so that students could work on different tasks, whileenabling Anna to work with a small group of students.

The sharing of strategies introduced the idea that students had strategiesof their own, distinct from teachers’ attempts to teach strategies. Beingable to detail students thinking implicitly meant that teachers’ participa-tion with their students in the classroom was changing. In the penultimatemeeting, teachers who had struggled to elicit strategies came back to sharesuccesses:

Patrick: A couple of my kids did a subtraction algorithm. Just 15minus 9. 15 minus 8 equals 7. And those were the only kidswho got it right. I wanted them to elaborate on how they didit. How did you know to subtract? I’m just really beginningto get them to explain, even to understand strategies and toexplain them. So it’s going to be awhile before they’re able toarticulate it. But I’m encouraged because they’re starting to


understand that they can articulate how they knew something.(W6: 5/5/98)

..............................................................Anna: . . . I was always having trouble sitting, making time to sit

down with them and really listen to what they’re saying.Yesterday, I finally did it. Harvey said that he went 5, 10, 15.And then he had 3 fingers, 3 boxes. So of course, I’m like,it’s a strategy! And I talked to him! And I listened! And we’rewriting it down! And I was so excited . . .. (W6: 5/5/98)

Patrick and Anna’s contributions to the discussion exemplify thatpaying attention to the details of children’s thinking did emerge as anormative aspect of what it means to contribute to the workgroup. Atthe beginning of the year, there was doubt in the group that studentscould explain their thinking. At first, prompted by the facilitator and thensupported by several teachers’ experimentation in their classroom, someteachers began to share ideas about eliciting students’ thinking. By theend of the year, the teachers were sharing the kinds of conversations theyhad with students that uncovered their thinking and the tasks they usedto enable children to express their reasoning. In meetings 4, 5, and 6, asTable IV shows, there were marked exchanges where particular teachersmade the kinds of declarations that Patrick and Anna made.

In sum, the first major shift in participation that emerged from ouranalyses was a shift towards attending to children’s thinking. The contentof the exchanges shifted towards attention to the details of students’strategies. Initially, the facilitator played a key role in pressing teachersto note the details of children’s strategies. This probing also created a needfor teachers to elicit children’s thinking in their class. Because the contentof exchanges shifted, the discourse of the group began to shape a particularstance about the role of teachers, namely, that (a) teachers’ work involvesattending to children’s thinking; (b) teachers make public their efforts toelicit student thinking; and (c) teachers recognize students’ mathematicalcompetencies.

Shifting Participation Towards Developing Instructional Trajectories inMathematics

Our analysis of the data revealed that teachers did not only learn to attendto the details of students’ strategies, but also learned that the practiceof detailing children’s strategies provided opportunities to recognize thatstudents had powerful mathematical ideas. This recognition supported ashift in the group’s practices to discuss possible instructional trajectories.We present our analysis of the data to show how the group’s discussions led


to the identification of particular mathematical ideas for this trajectory. Itwas how teachers interacted with each other around the student work, againsupported by the facilitator, that mathematical goals came into sharperfocus for the group. We will show how teachers made use of the detailsof children’s mathematical thinking as they began reconsidering what theywanted to accomplish as teachers in mathematics. Workgroup discussionsinvolved: (a) attending to children’s knowledge of place value through afocus on the tens structure of the number system, (b) how to build onstudents’ mathematical thinking, and (c) how to relate students’ mathe-matical understanding to classroom tasks. These discussions contributedto teacher discourse that increasingly centered on instructional trajectoriesin mathematics. As we examined the workgroup participation in relation tothose ideas we saw that while the conversations evolved, the evolution wasnot linear. As indicated in Table IV, the issues do not build, then peak atone point in time, become resolved and completely disappear. Rather, ideasabout instructional trajectories enter into conversations at different pointsduring the year and they come up repeatedly. We found that the instruc-tional trajectories developed in relation to other aspects of the teachers’experiences in the workgroup and in the classroom.

Attending to children’s knowledge of place value through a focus on thetens structure of number systemThe facilitator’s moves, early in the workgroup meetings, helped tosurface a mathematical direction for the workgroup conversations. Aswe described earlier, Table IV shows five separate exchanges in the firstmeeting during which teachers shared students’ unsuccessful or cumber-some attempts at the problem. Two of the teachers, however, identified astrategy that took advantage of the tens structure of our number system.Kathy explained that two of her students “estimated” to find the differencebetween 48 and 111. They added 60 to 48 to get 108 and then added threemore to get to 111. More commonly she saw her students use tallies tocount up from 48 to 111 without organizing them into rows of ten. Kathynoted that students who used that strategy often miscounted. A few ofRose’s students used base ten blocks to create a set of 48 by putting out 4ten blocks and 8 units. Yet she too noticed that some went on to count byones while only a few actually used the tens blocks to solve the problem.The students who used tens to count up from 48 to 111 were generallymore successful.

The facilitator capitalized on the sharing of those strategies to intro-duce the idea of using tens and ones to solve the problems. She askedthe teachers to generate direct modeling strategies for solving problems


by ones and by tens. After some of strategies were generated, she thensuggested ways teachers could support students to progress from directmodeling by ones to direct modeling by tens.

While the idea of using tens in direct modeling was introduced early,the teachers had not begun to explore the use of tens in students’ inventedalgorithms. This makes sense in light of the fact that, with the excep-tion of one strategy, teachers did not observe any invented strategies.The teachers initially thought that the invented strategies were linked tostudents’ “exposure” to them or to how “smart” students were. For manymembers of the workgroup, the invented strategies were not typical ofthe way they themselves, let alone their students, would have solved theproblem.

Recognizing students’ use of ten to solve problems also meantconnecting that idea to teachers’ conventional views on place value.During the second meeting, Miguel expressed his fear of teaching placevalue, a topic he had heard from other teachers was notoriously difficult.The facilitator responded by characterizing the strategies that were sharedduring that meeting as evidence of place value understanding, pushing theidea that place value was not about identifying the hundreds’ place. Sheexplained,

But a lot of what the kids are doing with this problem is place value. Figuring out howmany 20s are in 231 is place value. You can think about division as putting things ingroups. Place value is putting thing in groups, but it’s putting things into groups of ten. . . .

I have 89 pieces of candy, and I want to give ten to each teacher, how many teachers can Igive them to? . . . It gives them a context in which they have to figure out how do I take 89and break it down into tens and ones (W2: 12/9/97).

Miguel interpreted place value instruction and understanding aschildren’s ability to identify hundreds, tens, and ones column. The facilit-ator used Miguel’s concern to open up the idea that children’s work withgroups of tens, through the problem contexts that teachers were posingto their students, was already laying a foundation for understanding placevalue.

Beginning to think about how to build on students’ thinkingSome exchanges in the workgroups involved hypothetical discussionsabout how to help students move on in their ideas. In the first three meet-ings, in four separate exchanges, the facilitator encouraged teachers toconsider place value understanding as they thought about the next stepswith students. For example, in the second workgroup, the problem wasto find how many classrooms were needed to allow 231 children to takecomputer classes, if only 20 children could be in a room. The facilitatorasked teachers how they would support students who were direct modeling


using ones. She directed the teachers to think about using an element of thestudent’s strategy instead of imposing or asking students to do somethingthat was not connected to their initial strategy. The facilitator providedideas for teachers to take up and try in their classroom. She providedsuggestions for questions that teachers could use when they interacted withstudents and helped teachers consider how to help students advance theirstrategies. Thus, while the group was learning to pay attention to the detailsof their thinking, the facilitator was already prompting teachers to thinkabout how they might respond to help students advance their strategies.

Relating students’ mathematical understanding to classroom tasksThe discussions about building students’ place value understanding inthe classroom peaked in the fourth and fifth workgroups. There were 14exchanges combined across both workgroups in which a teacher posed aquestion or an issue about practice to the group.

The fourth workgroup was a watershed – the teachers posed straight-forward computation problems, and Kathy came to the group with a hostof sophisticated strategies. The facilitator, in a surprised tone, asked her,“how come you’re getting all of these strategies all of a sudden?” Kathylaughed and said “they (the kids) went on vacation!” But she went on todescribe some of her general uses of problem solving while Patrick askedher questions about the materials she made available to students and whatshe emphasized. Drawing on a number of linked exchanges, the excerptsbelow show how Kathy responded:

Kathy: I’ve just consistently done word problems every day. Plusmental math every day. And encouraging them to solve it twoways. Show their work. . . . They have options to use tens andones [base ten blocks]. And they used those at the beginningof the year to help them count. . . . They have the option touse that or they can draw a picture, whatever they wanted.And so they got a lot of work with that. And then a lot ofthem would still go to the algorithm. And I guess wheneverthey brought it to me, I would say, “what is this?” . . . I meana couple of times in front of the class, I’ve said people haveshared their strategies. . . . And I’ll say, “What did you reallyadd?” So there’s just a lot of reinforcement of, “What arethese numbers? Is this really a two or is this a 20?” (W4:3/3/98)

The tone of the discussions shifted again because teachers began to bemore interested in how to create opportunities for students to generatetheir own efficient strategies. They shared their own attempts and looked


to members of the groups that were more successful eliciting students’strategies. Mental math activities had been adopted by a number ofteachers, and they used their experiences with that task in their classroomas they responded to questions. In the next exchange during the fourthworkgroup, Patrick wanted his students to start using a “bar” to representten, rather than ten individual marks. Kathy responded by suggestingparticular ways he might be able to move students towards that benchmark.The facilitator’s role in this conversation was quite limited and providesevidence that the teachers were developing and sharing their expertise inthe workgroup. It also provides evidence that detailing students’ strategieswas being coupled with developing a mathematical instructional trajectory.

Patrick: It would be nice if somebody would just make a one bar. I’mwaiting for the kid that makes a ten bar, that draws a line andsays that’s a ten bar. And then makes the three for the 13. AndI’ve been trying to reinforce that, and they’re just not doingthat. And I even write these visually. I write numbers on theboard with bars and dots. And I say, what is that? And they’llsay 43. And they’ve got it. And I’ll do 26 and 43. How muchis it? It’s 69. And they can do that or whatever the answerwas. But then they won’t do it themselves.

Kathy: Do you give them numbers that are even ten, like adding andsubtracting? Because I was at the same point as you. Like, Idon’t know when I was doing that. I was doing a lot with thebars.

Patrick: Like 30 minus ten.Kathy: Yeah. And basically using numbers that were even ten and

then larger numbers. And so they sort of –Patrick: Can you give me an example?Facilitator: 120 minus 60.Kathy: Yeah, something like that. Or even stay under 100. Like 80

minus 20. Where they see that using the tens might be easier.Just so they get used to using them. I think I did a lot of that,and then they used the tens more. But some of them still goback to the ones. I mean, even some of them where you cancircle it. Can you group these instead of making slash marks?And they don’t use the strategy (W4: 3/3/98).

In the exchange, Kathy appeared confident in giving suggestions toPatrick. Yet she also had questions about helping students reach bench-marks which she shared with the group later on in this meeting. She notedthat some students in her class still make tally marks when they encounter


large numbers. Later in the meeting, she asked for suggestions abouthelping them make their strategies with larger numbers more efficient.

During the next meeting, the group returned to mental math activitiesand discussed ways they implemented mental math in their classrooms.Patrick, again, prompts the group to explain how the task is structured inthe classroom.

Patrick: How do you do the mental math thing because I’ve seen Karla(another teacher not in this workgroup but in the school) doit? How do you do it?

Adriana: I do it exactly like she does. I got the idea from her! [Laughs]Facilitator: This idea is spreading.Kathy: My kids love it! We have professor of the day.Patrick: Can you just model it for me? You have a group of kids and

you say, “Who wants to do a problem?” (W5: 3/24/98)

For the next 18 turns, Adriana and Kathy explained the logistics of howto organize a mental math activity as Patrick asked them more questionsabout how to do this. This talk also produced new ideas about using studentthinking. For example, towards the end of the exchange, the facilitatorraised the question of whether the students have an opportunity to ask eachother questions, knowing this is a strategy that supports dialogue in theclassroom. Later in the meeting, Patrick asked what the goal was of havingstudents solve problems. Kathy responded, “To show their thinking . . . Icare about what you’re doing in your head.” Other teachers contributed toKathy’s response, and the discussion then moved to how teachers struc-tured the time when students shared strategies in front of the class. Threeteachers described their various classroom management strategies whilePatrick and Anna asked questions.

These technical questions about tasks show that teachers were exper-imenting in the classroom, and moreover, because several teachers hadbegun to use similar structures, they could compare the impact on studentthinking. It was not just a matter of sharing the latest technique because theteachers wanted to help their students begin to develop efficient compu-tational algorithms. The shift in detailing strategies and noting students’sophisticated strategies also motivates a shift in making practice moretransparent. The classroom has to be represented and cannot be directlyviewed in the workgroup meeting. Patrick’s question about mental mathmarks the need to explain practice; his question emerges as a questionabout ‘tell’ me how you’re doing mental math because ‘mental math’ itselfis not transparent (see also Little, 2002).


Summary of Shifts over Time

We have been concerned thus far with the shifts evident in the natureof teachers’ participation over time. By using exchanges from the work-group conversations, we have shown how the group’s attention shifted withrespect to children’s thinking, the mathematical ideas at work in children’scomputational strategies, and the respective questions about practice thatthose observations generated. Figure 1 summarizes the trajectory of theworkgroup over the course of the year.

Figure 1. Summary of shifts in participation across Year 1.

DISCUSSION

This paper is an effort to document teacher learning through shifts inparticipation in regular workgroup meetings focused on examining studentwork. The workgroup was a setting where teachers shared student thinkingand made public their classroom practices. By struggling to make sense ofand to detail their students’ thinking, the teachers’ participation developedthe intellectual practices of the workgroup. We focus our discussionon the significance of the use of student work and a transformation ofparticipation view in analyzing the learning trajectory of the group.

Student Work as a Tool for Learning

A tool or artifact can provide a means through which participants ina community of practice negotiate meaning (Wenger, 1998). The work-group conversations revolved around student work, an artifact of students’mathematical thinking, which then opened a window into each teacher’s


classroom. The common problems that teachers posed to their studentsallowed teachers to focus on shared meaning, build common groundand negotiate crossing the boundaries of the workgroup meetings andtheir classrooms. The artifacts supported the development of a sharedlanguage that, in turn, contributed to the construction of workgroupmeeting practices.

We agree with Ball and Cohen (1999) that “simply looking at students’work would not ensure that improved ways of looking at and interpretingsuch work will ensue,” (p. 16). Because the use of student work is beingadvocated in current conversations about professional development, it isimportant to underscore the role that student work played in our work.All the student work came from teachers’ own classrooms and thus eachteacher could speak to how the work was generated and had opportuni-ties to return to their classrooms to clarify their understanding of studentthinking or to extend it. They all had instructional practices that theymade explicit and on which that they could build. During the workgroupmeetings, the facilitator and the teachers used what was present or not instudent work to initiate discussions of student thinking, mathematics, andpedagogy. Centering the activity on teachers’ own student work allowedfor conversations that deepened as well as challenged teachers’ notionsabout their work as teachers. They developed more detailed knowledgeof their own students’ mathematical thinking and began to articulatebenchmarks in the learning trajectories for their students and instructionaltrajectories to support their work. The student work also allowed theteachers’ to begin to see themselves as mathematical thinkers when theywere willing to struggle through student strategies they did not understand.

Learning as Participation

Understanding learning as changing participation is significant to ouranalysis in this paper. We tracked changes in teacher learning by examiningshifts in the practices of the workgroup. The teachers’ experiences withtheir students and their shared experiences with their colleagues influ-enced the form and direction of the workgroup meetings. “When indi-viduals participate in shared endeavors, not only does individual devel-opment occur, but the process transforms (develops) the practices of thecommunity” (Rogoff, Baker-Sennett, Lacasa, & Goldsmith, 1995, pp. 45–46). Although we have not attempted here to describe individual teacherchange across the workgroups, it is evident from our analyses that certainteachers were actively contributing their methods of experimentation tothe discussions (e.g., Kathy) while others struggled more to elicit theirstudents’ thinking in the first place (e.g., Anna). It is important that


some teachers experimented successfully with the workgroup problems intheir classrooms while others struggled. The questions, confusions, andsuccesses teachers shared made certain ideas public that helped shapethe focus and trajectory of the group. In this article, we have made thetrajectory of the group the main focus of our analysis.

We wish to consider how a transformation of participation perspectivestrengthens our understanding of teacher learning. Clearly, we can assesschanges in teacher knowledge by relying on pre/post measures of indi-vidual teachers’ knowledge and beliefs. We do not argue that examiningindividual teachers developing knowledge and beliefs is unimportant. Infact, these are key resources for a developing community (e.g., Even& Tirosh, 2002; Leinhardt & Smith, 1985). However, we believe thatby attending to shifts in participation, we can understand the followingaspects of teacher learning: (a) how teachers working together supportedthe development of each other’s thinking and the practices they used intheir classrooms; (b) how and when teachers asked each other for help andcontributed to discussions in the workgroup because of their own experi-mentation in the classroom; and (c) how teachers looked at the strategiesstudents in other classrooms used and then used those as markers for whatto expect from their students.

These aspects of the workgroup practice contributed to the developmentof a particular kind of intellectual and professional community for thisgroup of teachers. Their shared experience was beginning to develop ideasabout instructional trajectories for developing student math concepts, atleast with respect to students’ fluency with place value. That is not to saythat they agreed with one another or had reached a consensus about futuredirections or their roles within the classroom and the school. The knowl-edge and beliefs that teachers constructed, however, emerged from theircontributions to the creation and continual development of the practiceof workgroup meetings and their classroom communities. It is importantto understand how teachers participate in developing practice in order toknow how to help support teachers’ engagement with student thinking,mathematics, and pedagogy. Paying attention to the kinds of shifts thatmay take place as teachers first begin to work together can also help usidentity key markers of generative professional learning structures withinschools.

Central to a transformation of participation perspective is that shifts inparticipation are in service of new roles and identities. Our analysis of thefirst year of workgroup data leads us to conjecture about the new kinds ofidentities teachers may be forming through the ways classroom teachingwas portrayed in the discussions (Little, 2002). First, teachers were exper-


iencing new ways of working together around a particular focus towards along-term goal – building children’s understanding of the tens structure ofour number system in order to develop their fluency with computation andtheir understanding of operations. Second, the group was exploring newways of being – teachers elicit and listen to children’s mathematical ideas,interpret them, and use resources to decide where to go next to developideas. Third, the teachers were finding ways to experiment within theirown classrooms and use the workgroup as a place to further reflect ontheir experimentation.

We have only seen the beginning of the teachers’ development of thesepractices. As the teachers worked together to begin to create a communityof learners around the teaching and learning of mathematics, they werealso beginning to create a set of norms about what it meant to teach at theirschool. Continued longitudinal work will help us understand the extent andsignificance of teachers’ changing roles and identities.

ACKNOWLEDGEMENTS

The research reported in this paper was supported in part by a grantfrom the Department of Education Office of Educational Research andImprovement to the National Center for Improving Student Learning andAchievement in Mathematics and Science (R305A60007-98). The opin-ions expressed in this paper do not necessarily reflect the position, policy,or endorsement of the Department of Education, OERI or the NationalCenter.

Correspondence concerning this article should be sent to the first authorat [email protected]. The authors wish to thank the faculty,administrators and staff at Crestview Elementary School who welcomedus into the school and allowed us to share their questions, frustrationsand successes. We are also indebted to the collaboration of Jeff Shihand Stephanie Biagetti. We appreciate the valuable comments of severalanonymous reviewers.

NOTES

1 Japanese Lesson Study is a form of professional development practiced most commonlyamong elementary school teachers in Japan. It is a process through which teachers analyzeand develop classroom lessons together.2 Join Change Unknown is a term used by CGI researchers to categorize addition andsubtraction problem types. The word, “join,” refers to a joining or combining actionin the word problem. “Change unknown,” refers to the location of the unknown in the


word problem, in this case an unknown addend. See Carpenter et al. (1999) for the fullcategorization scheme.3 We began with a join change unknown (JCU) or a missing addend problem to introduceteachers to children’s mathematical thinking. Many adults see the problem as a subtractionproblem, but children will often use an adding or joining strategy to find the missingaddend (Carpenter et al., 1999). For that reason, researchers refer to addition and subtrac-tion problems by indicating whether there is a joining or separating action. Most teachersacross the grade levels were working on addition and subtraction in the first trimester, andwe wanted them to consider how the two operations are related since many textbooksseparate the study of addition from subtraction. We moved next to multiplication anddivision contexts. To many teachers, especially in the primary grades, we suspected that thedivision and multiplication contexts would appear too difficult since their students wouldnot have started instruction in those areas. However, we wanted them to have opportunitiesto see that all of the children from kindergarten through grade five would be able to solvemultiplication and division problems. We would then be able to talk about direct modelingstrategies (in which students represent each number and model the action in the problem)and how they cut across mathematical operations.

We returned to the addition and subtraction context but through computation problemsas the teachers approached standardized testing time. We wanted to have conversationsabout how students could continue to use the same strategies that they used in wordproblem contexts for computational problems. We also wanted teachers to have an oppor-tunity to see how writing problems vertically versus horizontally would affect the kindsof strategies students used. Finally, we hoped teachers would notice how that students canuse a variety of strategies to solve computational problems. We then moved on to compareproblems to help teachers think about how action in the wording of the problem may makea problem more or less difficult for students to solve. In the final two meetings of the year,we revisited earlier problems to have some closure about the principles we had learnedthroughout the year.4 Our informal conversations with teachers during these visits were consistently focusedon their students’ thinking and the evolving frameworks that we constructed in the work-groups. However, we did model how to elicit student thinking through our informalinteractions with their students and the questions we asked students during those inter-actions. On occasion, we suggested additional problems teachers might want to try withtheir students. We also shared interesting strategies that we encountered as we talked withtheir students.5 The research reported in the article is part of a larger study of teacher learning andchange. There were four workgroup meetings taking place in the school, each led by adifferent member of the research team. The authors of this article led workgroup meetingscentered on student work. We chose to focus our findings and discussions around oneworkgroup using student work for pragmatic reasons.6 An easier strategy may have been to just take one from 16 and add to 29 to make 30.Then add 30 and 15. Kathy, however, reported that this student first changed the numbersto 28 and 17.


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ELHAM KAZEMI

University of WashingtonCollege of EducationBox 353600122 MillerSeattle, WAU.S.A.E-mail: [email protected]

MEGAN LOEF FRANKE

University of Calfornia, Los AngelesLos AngelesU.S.A.

RUTH M. STEINBERG, SUSAN B. EMPSON and THOMAS P. CARPENTER

INQUIRY INTO CHILDREN’S MATHEMATICAL THINKINGAS A MEANS TO TEACHER CHANGE

ABSTRACT. In the context of U.S. and world wide educational reforms that requireteachers to understand and respond to student thinking about mathematics in new ways,ongoing learning from practice is a necessity. In this paper we report on this process forone teacher in one especially productive year of learning. This case study documents howMs. Statz’s engagement with children’s thinking changed dramatically in a period of onlya few months; observations and interviews several years later confirm she sustained thischange. Our analysis focuses on the mathematical discussions she had with her students,and suggests this talk with children about their thinking in instruction served both as anindex of change, and, in combination with other factors, as a mechanism for change. Weidentified four phases in Ms. Statz’s growth toward practical inquiry, distinguished by heruse of interactive talk with children. Motivating the evolution of phases were two sorts ofmechanisms: scaffolded examination of her students’ thinking; and asking and answeringquestions about individual students’ thinking. Processes for generating and testing knowl-edge about children’s thinking ultimately became integrated into Ms. Statz’s instructionalpractices as she created opportunities for herself, and then students, to hear and respond tochildren’s thinking.

KEY WORDS: discourse community, elementary mathematics, practical inquiry, teacherchange, teacher learning, teacher reflection

Mathematics educators have articulated a vision for teaching mathematicsthat includes engaging students in problem solving, mathematical argu-mentation, and reflective communication (NCTM, 1991, 2001). Calls forinstructional reform in mathematics have been accompanied by demands,in many countries, for radical changes in teaching practices. Many teachershave learned to teach in ways consistent with calls for reform (Cobb, Wood& Yackel, 1990; Cobb & McClain, 2001; Fennema et al., 1997; Hiebert,Carpenter, Fennema et al., 1997; Hiebert & Wearne, 1993; Jaworski, Wood& Dawson, 1999; Schifter & Fosnot, 1993; Sullivan & Mousley, 2001).Without attention to how teachers learn, however, our understanding ofinstructional reform is seriously incomplete (Franke, Carpenter, Levi &Fennema, 2001; Hammer & Schifter, 2001; Richardson & Placier, 2001;Schön, 1983; Sherin, 2002).

A small but growing body of research has focused on teacher learningas practical inquiry into the problems of teaching (Jaworski, 1998, 2001;


238 RUTH M. STEINBERG ET AL.

Lampert, 1985; Richardson, 1994; Tabachnick & Zeichner, 1991). Thisresearch has found that teachers who engage in practical inquiry are ableto change their teaching in ways that are sustainable and self generative(Franke, et al., 2001). There has been little research, however, on theprocess of change towards inquiry-oriented practice.

In the current study, we focus on one teacher’s use of practitionerknowledge and research-based knowledge as she learned to integratepractical inquiry into her teaching. We focus in particular on the mathe-matical discussions she had with her students, and argue that this talk withchildren about their thinking during instruction served both as an index ofchange, and, in combination with other factors, as a mechanism for change.We concentrate on this latter feature of teacher-student talk because, wecontend, it provides insight into the nature of generative change (Franke etal., 2001) in teaching.

This teacher’s mature teaching can be characterized as an integra-tion of inquiry and instruction, in which both she and students learned.Although the process of change we have documented does not necessarilyrepresent the path to practical inquiry that all teachers should take, it lendsuseful insight into how a teacher can combine practitioner knowledge andresearch-based knowledge to ask and answer questions profitably aboutteaching and learning.

The teacher, Kathy Statz,1 taught mathematics using the preceptsof Cognitively Guided Instruction (CGI) (Carpenter & Fennema, 1992;Carpenter, Fennema, Franke, Levi & Empson, 1999; Carpenter, Fennema,Franke, Levi & Empson, 2002). CGI is a research and professional devel-opment program founded on the fact that children enter school with a richstore of informal knowledge that provides a basis for engaging in problemsolving. We draw on previous research that documents levels of teachers’engagement with children’s thinking in order to track Ms. Statz’s learning(Fennema et al., 1996; Franke et al., 2001; Simon & Schifter, 1991). Wego beyond documenting the fact of change to describe how she progressedfrom one level to the next, initially by reflecting on instruction as questionsabout her students’ thinking were posed for her and, later, by posing andanswering such questions herself. Changes in her practice accompaniedthese changes in her stance towards teaching.


Research suggests that teachers learn a great deal from teaching, but thecontent of that learning varies from teacher to teacher (Richardson, 1990;Richardson & Placier, 2001). Conditions that appear to be most condu-

INQUIRY INTO CHILDREN’S THINKING 239

cive to learning include: 1) membership in a “discourse community” thatprovides tools for framing and solving the problems of teaching (Ball,1996; Cobb & McClain, 2001; Stein, Silver & Smith, 1998; Wenger,1998); 2) processes for reflectively generating, debating and evaluatingnew knowledge and practices (Ball, 1996; Jaworski, 1988; Wilson &Berne, 1999; Wood, 2001); and 3) ownership of change, so that the prob-lems of teaching that change is meant to address are problems that teacherswant to solve and feel capable of solving (Loucks-Horsley & Steigelbauer,1991; Simon & Schifter, 1991).

None of these conditions, alone or in combination, assures ongoingteacher learning. Perhaps the most important is teachers’ own stancetowards practice as inquiry (Jaworski, 1994; Schifter & Fosnot, 1993;Tom, 1985). This inquiry can take several forms. It can be exercised ininteraction with students and the curriculum (Sherin, 2002) or removedfrom classroom interactions, in reflection on action (Mewborn, 1999;Schön, 1983, 1987; Wood, 2001). Little (1999) noted that the “systematic,sustained study of student work, coupled with individual and collectiveefforts to figure out how that work results from the practices and choicesof teaching” may be one of the most powerful sites for teacher inquiry(p. 235). Student thinking is not the only focus possible, but it is onethat has proven productive for teachers and students (Carpenter, Fennema& Franke, 1996; Carpenter, Fennema, Peterson, Chiang & Loef, 1989;Schifter, 2001; Steinberg, Carpenter & Fennema, 1994; Tzur, 1999).

Teachers who change in ways that embrace new knowledge and beliefsabout children’s problem solving do not necessarily sustain that changeor continue to change. Franke et al. (2001) found that the most profoundchange among a group of 22 CGI teachers occurred for those who engagedin practical inquiry into children’s thinking. Those 10 teachers, more thanthe rest, thought of the research-based framework for children’s problemsolving as “their own to create, adapt, and investigate” (Franke et al.,2001, p. 683). Franke et al. (2001) called this learning “generative change”because teachers used what they knew to generate new knowledge throughpractical inquiry, and saw this inquiry as part of their identity as profes-sionals. In particular, 1) these teachers believed understanding children’sthinking was central to their work, and 2) their knowledge of children’sthinking went beyond the frameworks first presented to teachers in staffdevelopment four to eight years earlier.

Not all teachers who use problem solving in teaching (e.g., NCTM,2000) take a stance of inquiry toward their practice. There are many profi-cient teachers whose instruction is based on problem solving but who donot engage in practical inquiry. However, as Franke and colleagues (2001)


argued, practical inquiry is a powerful means to the continued improve-ment of practice. The case we report here is an example of a teacherwho not only taught in reform-oriented ways, but also developed a stanceof inquiry towards her practice. We examine in closer detail the processof change towards what Franke and colleagues (2001) called generativelearning and the mechanisms that appeared to stimulate this change.

Levels of Teacher Change in Cognitively Guided Instruction

The teacher in this study taught mathematics using Cognitively GuidedInstruction (CGI). In this approach to professional development, teachersare encouraged to use research-based knowledge about children’s mathe-matical thinking to make instructional decisions (Carpenter et al., 1999). Itdiffers from most curricular interventions in that lessons are not prescribedfor the teachers. Instead, teachers plan for instruction using what theyknow about their own students and their general knowledge of children’sproblem solving. CGI consists of research information about the develop-ment of children’s thinking, portrayed through problem-type frameworksthat emphasize semantic differences among problems and solution strategyhierarchies. Teachers learn which semantic features of problems are easiestfor children to understand, and which features are more difficult. In whole-number addition and subtraction, for example, problems involving actionson sets (e.g., joining two sets together) are generally easier than problemsinvolving relationships between sets (e.g., comparing the sizes of two sets).Similar analyses have been developed for multiplication and division, andthe development of base-ten concepts and multidigit strategies for additionand subtraction. (See Carpenter et al. [1999] for more information.)

A longitudinal study of a sample of 21 first through third-grade teachersinvolved in CGI professional development documented five distinct levelsof teachers’ use of children’s thinking (Fennema et al., 1996). Franke etal. (2001) revised the levels to reflect the integration of teacher beliefs andpractices, and called the classification scheme engagement with children’smathematical thinking (Table I). The levels are useful for characterizingteacher change, for they go beyond dichotomizing teachers’ practice intoreform-oriented or not. Fennema et al. (1996) found that teachers whoengaged with children’s thinking at levels 3, 4a, and 4b of the scale taughtin ways that were distinctly different from teachers at levels 1 and 2. Thekey distinctions hinged on students’ opportunities to solve and discussproblems. Further, student outcomes were higher in the classrooms ofteachers at the top three levels (see also Carpenter et al., 1989). We reviewthese levels here and use them to describe Ms. Statz’s growth.


TABLE I

Levels of Engagement with Children’s Mathematical Thinking from Franke, et al., 2001,p. 662 (Copyright 2001 by the American Educational Research Association; reproducedwith permission from the publisher)

Level 1: The teacher does not believe that the students in his or her classroom can solve

problems unless they have been taught how.

Does not provide opportunities for solving problems.

Does not ask the children how they solved problems. Does not use children’smathematical thinking in making instructional decisions.

Level 2: A shift occurs as the teacher begins to view children as bringing mathematical

knowledge to learning situations.

Believes that children can solve problems without being explicitly taught astrategy.

Talks about the value of a variety of solutions and expands the types of problemsthey use.

Is inconsistent in beliefs and practices related to showing children how to solveproblems.

Issues other than children’s thinking drive the selection of problems andactivities.

Level 3: The teacher believes it is beneficial for children to solve problems in their

own ways because their own ways make more sense to them and the teacher wants the

children to understand what they are doing.

Provides a variety of different problems for children to solve.

Provides an opportunity for the children to discuss their solutions.

Listens to children talk about their thinking.

Level 4A: The teacher believes that children’s mathematical thinking should determine

the evolution of the curriculum and the ways in which the teacher individually interacts

with students.

Provides opportunities for children to solve problems and elicits their thinking.

Describes in detail individual children’s mathematical thinking.

Uses knowledge of thinking of children as a group to make instructionaldecisions.

Level 4B: The teacher knows how what an individual child knows fits in with how

children’s mathematical understanding develops.

Creates opportunities to build on children’s mathematical thinking.

Describes in detail individual children’s mathematical thinking.

Uses what he or she learns about individual students’ mathematical thinking todrive instruction.


At level 1, teachers mostly used direct instruction, and, as measured byinterviews and beliefs scales, did not believe children could invent theirown strategies to solve problems. There was little to no opportunity forchildren to solve problems. At Level 2, teachers began to believe childrencould solve problems on their own, but were inconsistent in implementingthis belief. At Level 3, teachers believed children should solve problemsusing their own strategies because it led to deeper understanding. Childrenwere presented with a variety of problems to solve and discuss. Teacherslistened to children’s thinking, but did not necessarily build on it. Althoughthey understood the problem-types and solution-strategy frameworks, theywere not aware of individual children’s thinking in detail. Nonetheless, thislevel marked a departure from the teacher-centered instruction that charac-terized levels 1 and 2. At Level 4A, teachers believed children’s thinkingshould drive the curriculum. Children’s thinking was elicited and teacherscould describe that thinking in detail. However, decisions about how tobuild on that thinking were made at a global level, for the whole class.At Level 4B, teachers believed the curriculum should be driven by whatindividual children know. They knew what problems individual childrencould solve, what strategies children used, and how children’s strategies fitwith understanding mathematics. Teachers used this knowledge to build onindividual children’s thinking in instruction. Fennema et al. (1996) arguedthat instruction at Levels 3, 4A and 4B “epitomize the process standardsof the reform movement” (p. 429). Thus, teachers who reach levels 3 andabove teach in ways that are consistent with U.S. calls for mathematicseducation reform (National Council of Teachers of Mathematics, 1991;2000).

Fennema et al. (1996) found 19 out of 21 teachers teaching at Level 3or higher at the end of a four-year intervention; that is, 90% of the studyteachers taught using reform-oriented practices by the study’s conclusion.Nine of those teachers began the study at Level 1, and seven teachers atLevel 2. Twelve teachers’ instruction was classified as Level 3 by the endof the study, suggesting that attaining Level 4A or 4B is not common, evenamong teachers who change.

The case study we report here deepens our understanding of howteachers may attain a level of teaching in which instruction is basedon teachers’ generative knowledge of individual children’s mathematics(Level 4b in the scale).


METHOD

This study was conducted in collaboration with one fourth grade teacher,Ms. Statz, who has worked as a classroom teacher and mathematicsresource teacher for grades K-5. We collected data at three points in time.At the first point, the first author acted as an observer participant for afive-month period, in Ms. Statz’s third full year of teaching. At the secondpoint, the following year, Ms. Statz was observed by the second author,over a period of several months. These data are used to ascertain whetherMs. Statz maintained the changes documented here, and are only brieflyreported in this paper. At the third point, several years later, Ms. Statzwas interviewed about her growth as a teacher, looking back on her careerbeginning with her pre-service teacher education.

Data Collected in Ms. Statz’s Third Year of Teaching

At the time of the observations, Ms. Statz had been teaching three years,all of which were with fourth grade classes. She had implemented CGIfrom the first year, after learning about it in her University certificationprogram. That year’s class consisted of 21 students from a racially, ethni-cally, linguistically, and economically diverse population of families. Athird of the class was new to the school. All the children’s names reportedin this paper have been changed.

PROCEDURES

Classroom observations. Thirty-four complete mathematics lessons wereobserved by the first author, over a five-month period. Lessons wereaudiotaped and parts were transcribed. Field notes were taken on teacher-student interactions, students’ solution processes, class organization, andthe teacher’s knowledge of, and efforts to build on, children’s thinking.Nine children were randomly selected as target students and their solutionprocesses were documented regularly by observing them and asking themhow they solved the problems. The other children were observed on arotating basis.

Teacher’s meetings with researcher. The first author met with Ms. Statz 13times during the first year of data collection, usually once a week for 30–40minutes. All meetings were audiotaped and transcribed, and includeddiscussions about Ms. Statz’s knowledge of her children’s thinking, herdecision making processes regarding content and classroom organization


and their relation to children’s thinking, and the researcher’s observa-tions of specific solution strategies in interviews with students or in classobservations.

Student assessments. Each child was interviewed at length, at the begin-ning and at the end of the study, on solution strategies for word problems ina variety of content areas. Students’ mathematics journals were examinedregularly by the researcher.

Data analysis. Observations and interview decisions were made inresponse to the situations arising in the classroom and in the teacher-researcher discussions. Themes consistent with teacher change and theCGI framework for teachers’ engagement with children’s thinking weremarked, such as teachers’ knowledge and beliefs, building on thinkingin instruction, and teacher-identified dilemmas and resolutions (Denzin &Lincoln, 2000).

Career Interview. Ms. Statz was interviewed by the second author severalyears after the primary data were collected. The interview was designed toelicit the story of her teaching career, in terms of formative events, suchas high points, low points and turning points. The interview was adaptedfrom interviews (Math Stories) used by Drake (under review)2 to elicit thestories for teachers involved in reform.

RESULTS AND DISCUSSION

In her third year of teaching, Ms. Statz changed dramatically in her knowl-edge of children’s thinking and the use of interactive talk to enhance thisknowledge and build on her students’ understanding. In this section, wedescribe how Ms. Statz made the transition from good reform-orientedteaching, corresponding to level 3 in Franke et al.’s (2001) scale, tooutstanding teaching, corresponding to level 4b, that incorporated practicalinquiry as a way to continue to acquire knowledge and gain insights intoteaching. We documented four distinct phases of change. We frame thechanges Ms. Statz experienced in her third year of teaching by describing,based on her own reports, Ms. Statz’s earlier and later years of teachingand teacher learning.


Early Teaching Years

Ms. Statz reported that she left her teacher certification program with acommitment to working with a belief in all children’s potential to succeed.She began her teaching career in a fourth-grade classroom. She describedher first year teaching as one of two low points in her career because shehad no materials to teach mathematics other than a commercial textbookseries. She tried to use it, but felt “lousy” when she taught expositorylessons. She felt her students were not learning.

Ms. Statz decided to abandon the textbook in the middle of the year andto begin using the framework for problem types and solution strategies,the basis for CGI, to plan instruction. She had the support of her principaland a mathematics resource teacher, Ms. J, who was herself experienced atbuilding on children’s thinking. During this time, Ms. Statz gladly acceptedhelp from Ms. J, “who would say things like . . . ‘Don’t tell [the students]that. Let them find it out.’ ”

Ms. Statz reported that, with help from Ms. J, she changed her mathe-matics instruction a great deal during that first year, but described thesecond year and beginning of the third as “more of a plateau.” Theremaining stretch of the third year – the time we report on here – wasdescribed by Ms. Statz as “a big jump.”

Third Year Teaching: A Year of Change

In her third year of teaching, Ms. Statz experienced intense growth inunderstanding children’s thinking, knowledge of the content area, andbeliefs about her role as a teacher. Concurrent with this learning, Ms. Statzdeveloped a stance of inquiry into her teaching practice and its relationshipto children’s thinking. In this section, we document these changes, andconsider possible mechanisms of growth. These mechanisms include thetypes of situations that prompted Ms. Statz to perceive a need for change,and how her concerns and internal debates inhibited or contributed tochange.

PHASE 1: LEVEL 3 ENGAGEMENT WITHCHILDREN’S THINKING

In November of the school year, Ms. Statz’s instruction was consistent withcalls for reformed classrooms (NCTM, 1989) and CGI’s Level 3 of engage-ment with children’s thinking (Table I): students solved challenging wordproblems using their own strategies; the teacher gave students opportuni-ties to present and talk about their strategies; and children recorded how


they solved problems in mathematics journals. No textbook was used. Ms.Statz’s goals for her students coincided with several of those of the reformmovement:

Being able to write about math. And being able to verbalize what they’re doing andthinking about math . . . Being able to feel comfortable enough about math to share whatthey’re talking about. And to develop an appreciation for each other. That people solvemath problems differently and that’s okay.

Ms. Statz often chose topics for word problems that related to a story thechildren had read in class or to events in the children’s lives (e.g., sellingthings in the school store, counting the number of names on a child’s cast).

Talk with Children about their Thinking

Although students had opportunities to solve problems, Ms. Statz didnot build on or extend children’s solutions strategies in her interactionswith students. After children solved problems, there were short discus-sions of each problem, lasting 5 to 10 minutes, in which four or fivedifferent strategies were presented. Although Ms. Statz encouraged thechildren to talk about their strategies, she seldom challenged them tojustify them, think of alternative solutions, or relate their strategies to moreadvanced strategies. She listened to what they said, and accepted it withlittle questioning.

For example, one day late in the fall, Dan showed the class how hehad solved a word problem by calculating 68 + 37. He drew 37 tallies andcounted by ones from 68, using the tallies to keep track. Ms. Statz thencalled the next child. She did not question Dan’s strategy or relate it tomore advanced strategies that used base-ten concepts (such as adding thethree tens and the seven ones) presented by other children.

Ms. Statz’s belief in accepting and encouraging a variety of strategiesfrom children was so strong she perceived her role in helping children toprogress as passive. Later in the school year, Ms. Statz reflected on herinteractions with students at that phase: “I would just accept what was puton the board and that was all ‘good’, that’s fine. Go have a seat. Nextperson.”

Further, Ms. Statz’s knowledge of individual children’s thinking wasnot well integrated with the research-based framework for children’sthinking that was the basis for CGI. In December we asked Ms. Statz toclassify students in her class according to the strategies she predicted theywould use, in our clinical interviews with them, to solve a set of problems.She accurately predicted the strategies of students who “direct modeled”3

to solve problems, but did not predict the more sophisticated strategies fiveof her children used. Although she knew who used more or less advanced


strategies, when it came to strategies beyond direct modeling, Ms. Statzdid not describe or classify these strategies appropriately.

Not surprisingly, this state of knowledge influenced Ms. Statz’s inter-actions with students. For example, in October, Ms. Statz assisted a childto solve a problem in a way that acknowledged neither his understandingnor the solution-strategies framework. David was trying to solve a Joiningproblem with an Unknown Change (“Raji has 17 dollars. He wants to buya pet snake that costs 33 dollars. How many more dollars does he needto earn to buy the pet snake?”) by counting up from 17 to 33. He waskeeping track of how many numbers were added by going back and forthbetween each number sequence: “first is 18, second is 19, third is 20” andso forth until he got to “ninth is 26.” He lost track of his double counthere and stopped. His strategy was appropriate to the additive semanticstructure of the problem but the method of keeping track appeared tomake more demands on his working memory than he could handle. Ms.Statz’s first attempt to help David encouraged him to continue to thinkabout the problem additively, but did not address his specific difficultyof keeping track. She suggested, instead, that he add 10 to 17, and thenasked if it would be enough. Before David could respond, another child,Nick, said he had solved the problem by subtracting 17 from 33. Ms. Statzsuggested to David that he could use a strategy similar to Nick’s and solvethe problem by separating 17 counters from 33. Although this strategy is“concrete” it does not directly model the semantic structure of the problem,which is additive. It re-represents an additive semantic structure in termsof subtraction and so is a more difficult strategy (Carpenter et al., 1999).When Ms. Statz asked him to solve the problem using subtraction, shedid not realize this strategy required knowledge of the inverse relationshipbetween addition and subtraction that David may not have had.

Beginning Inquiry. Discussion of episodes like this one began to help Ms.Statz reflect on how she used knowledge of children’s thinking in instruc-tion. In a conversation with the participant researcher a few days later,Ms. Statz said she was often not sure how to respond to children, likeDavid, who were struggling with a specific strategy. She then recalled astrategy she had recently seen in which a child used tallies to keep track ofa count. She realized it would have been an appropriate strategy to suggest,because it would have addressed his specific difficulty of keeping track andwould have allowed him to build on the additive structure that he saw in theproblem. Thus, this conversation facilitated new connections for Ms. Statzamong different episodes involving interactions with children’s thinking.In the next section, we document how, as the first author and Ms. Statz


continued to have these kinds of conversations, Ms. Statz became increas-ingly dissatisfied with her use of children’s thinking and began a transitionfrom Level 3 towards Level 4A engagement with children’s thinking.

PHASE 2: MOTIVATION FOR TRANSITION FROM LEVEL 3TO LEVEL 4A ENGAGEMENT WITH CHILDREN’S THINKING

In this phase of change, Ms. Statz became dissatisfied with what she knewabout children’s thinking. She realized that many of her students’ strategieswere basic or even wrong and did not necessarily show well developedunderstanding. In response to this dissatisfaction, Ms. Statz formulatedquestions for teacher inquiry that guided her in later phases. She thought,in particular, about her knowledge of students’ strategies and her roleas a teacher in facilitating more sophisticated strategies and deepeningchildren’s understanding. Little action to answer her questions took placein this phase. Rather, this phase was distinguished by “reflection-on action”(Schön, 1987), as Ms. Statz stepped back from her practice and began toidentify personal dilemmas.

The transition from satisfaction to dissatisfaction with Level-3 typeengagement with children’s thinking appeared to have been triggered bydiscussions with the first author about the researcher’s problem-solvinginterviews with the children. Ms. Statz sat in on some of the interviews andwas intrigued, often surprised, and sometimes troubled by how childrensolved the problems.

For example, one child, Pang, wrote down every single number between398 and 500 to solve a Join Change Unknown problem (“Robin has 398dollars. How many more dollars does Robin have to save to have 500dollars to buy a new bike?”). Surprised by this strategy, Ms. Statz checkedPang’s journal and discovered she was using similar strategies there too:

The way she solved this is kind of strange . . . Can I go see what she’s got in her journal?. . . Yeah, she’s doing similar things.

Ms. Statz became especially concerned about seven children who wereusing the standard subtraction algorithm incorrectly in the interviews.Ms. Statz had not introduced this algorithm in class; instead, she encour-aged children to generate their own conceptually grounded strategiesfor multidigit addition and subtraction. However, for problems involvingregrouping, these seven children always subtracted the smaller digit fromthe larger one, an example of a buggy algorithm when the smaller digitis the minuend. When asked if she had seen the children using this kindof strategy in class, Ms. Statz replied, “That’s something that surprised


me during these interviews.” Because she believed children should alwayssolve problems with understanding, and not by rote, she was taken aback:

The borrowing with regrouping really disturbs me now. That’s all that I’ve been thinkingabout now that we’ve been interviewing the kids and we see that they don’t have it. Andthey don’t; they’re not even coming up with a good way of explaining it. It doesn’t makesense to them.

Ms. Statz’s growing knowledge of the basic character and inadequacy ofher students’ strategies, combined with a strong belief against groupingchildren by ability or even by type of errors, presented a dilemma for her:how to accommodate a wide range of children’s thinking without resortingto ability grouping or remediation. Maintaining her belief in the centralityof student-generated strategies to the development of understanding andconfidence, she started to think about how she could assist these childrento grow mathematically without directly telling them how to solve prob-lems. She re-examined her belief that the teacher should accept whateverstrategies children chose to solve problems, and began to believe sheneeded to be more active in helping children progress – especially thosechildren who used incorrect strategies or still counted by ones to solveproblems involving quantities in the hundreds.

Ms. Statz’s deliberations about children’s buggy subtraction algorithmsillustrate these nascent changes. She discussed with the first author thefact that the children did not connect their paper-and-pencil algorithms totheir knowledge of working with base ten blocks. For the first time, shedebated whether to show children how paper-and-pencil algorithms couldbe modeled using base-ten blocks to make sure children understood:

I was even struggling with the idea of getting the overhead projector and doing it for thewhole class. Here are the base ten blocks, here is my marker and this is what we are doing.

But maybe we should try it, let them construct it themselves first?

I am struggling with how to go about doing that. Give them lots of take away problems andtry to go around to each person individually? That is the hard part. That’s what I’m tryingto figure out, how to do that.

In summary, in this phase, Ms. Statz became aware of the inadequacy ofher knowledge of children’s strategies and some of the consequences forher students. As she learned more about their thinking by sitting in on theparticipant researcher’s one-on-one problem-solving interviews with herstudents, dilemmas arose regarding her teaching, because she saw evidenceof rote or underdeveloped understanding. This phase was an intense onefor Ms. Statz, characterized by uncertainty and unresolved questions; shereported she thought about these issues a lot, “even when I first wake up.”


The net result was that Ms. Statz developed a deeply felt motivation tolearn more about the children’s thinking.

PHASE 3: LEVEL 4A ENGAGEMENT WITHCHILDREN’S THINKING

This phase was characterized by the integration of practical inquiry intoMs. Statz’s practice. Motivated by a need to know more about her students’thinking, Ms. Statz began to assess systematically individual children indepth and decide how to use the information in planning instruction.Because Ms. Statz was interested in specific questions such as how Davidused counting on to solve Join Change Unknown problems, and how tohelp Pang go beyond counting by ones to find triple-digit differences, sherequired more talk with individual students in the context of instruction.

Talking with Children about their Thinking. In January, Ms. Statz began tospend much more time with individual children at their desks. Previously,these sessions usually lasted no more than a minute. Now they often ran formore than 10 minutes per child or a pair of children: “What I am noticingin these last couple of weeks is that I am spending less time with all kidsand more time with particular kids.”

Ms. Statz was especially interested in children who were using inef-ficient or incorrect strategies. She began to probe their thinking more, tohelp them use base-ten blocks to represent quantities in the hundreds andthousands, and questioned them in ways that built on their thinking. Forexample, to solve 378 + ? = 600, Mark started to represent the 378 withbase-ten cubes. Ms. Statz asked him if he could do it in his head, andhe replied that he did not think so. So Ms. Statz scaffolded a strategy thatfollowed his use of base-ten materials, but focused on operating on numberrelationships instead of base-ten blocks. She asked him how much wasneeded to get from 378 to 380 and he immediately gave the answer 2. Thenshe asked how much was needed to get from 380 to 400; he answered 20.And to get from 400 to 600, he knew it would take 200. Mark then totaledthe addends on paper, to get 222.

This time spent talking with individual students was fruitful for Ms.Statz. Her growing knowledge of their thinking helped her to adapt instruc-tion to their needs and motivated her to consider children’s thinking inspecific content areas:

I definitely think I am noticing more than I noticed before. And not only just noticing it butknowing where to take it and how to push further and how to question more. . . .


This period was also stressful to Ms. Statz, because as she learned moreabout her students’ thinking, her questions grew:

I think I am more frustrated now about teaching math. . . . I am more exhausted . . . becauseI am then spending more time thinking about it. Although the things that I think are comingout of it are good, I am seeing what needs to be done. I am spending more time with kidswho need specific things. I think it is making a difference . . . Now there are all these otherquestions that are on top of it.

After a few weeks of focusing on specific children and interacting exten-sively with them, Ms. Statz felt more confident in her knowledge ofindividual children, but she was frustrated she did not have enough timeto do this kind of in-depth work with everyone. A new dilemma arose:

But sometimes I feel like I’m spending too much time and neglecting the rest of theclassroom. I’ve felt like that, more frustrated almost, this year because I’ve needed moretime with each kid.

Ms. Statz wanted the benefits of talking extensively with children abouttheir thinking to extend to all. Thus, she started thinking of new ways toorganize her class that would allow her to spend more time with individualsbut ensure that all children were working and progressing.

When the participant researcher suggested that Ms. Statz would be ableto talk to more children if they worked in pairs, Ms. Statz rejected the idea,because she believed working in pairs would not be beneficial to studentswho used significantly different strategies to solve problems. Two dayslater, Ms. Statz devised a solution that accommodated her concern. Sheasked the children to “choose partners who solve problems in a similarway to you.” Each pair got a sheet with the problems and a space fortwo strategies. Each child could use his or her own strategy or the paircould generate two strategies together. The children were fairly accuratein choosing partners who were solving problems in a similar way. Ms.Statz had time to work with individuals while the other children workedwith their partners. Having ten pairs to work with instead of 20 individualsmade the class more manageable for Ms. Statz.

To help children move forward to using more sophisticated strategies,Ms. Statz told them she would ask both children from a pair to explainhis or her partner’s strategy at discussion time. Most children were able toexplain their partner’s strategy. When a child had difficulties, the partnerexplained the strategy. More children were called to the board in this way,during the discussion time, and giving opportunities to more children toshare strategies helped to solve another issue that concerned Ms. Statz.

Ms. Statz started thinking of how to choose problems to help childrenprogress:


I wouldn’t say it is any easier (to come up with the problems for the children). In fact, it maybe more challenging to decide what type of problems to use . . . I think I am thinking moreof the problems . . . and the kids who are doing the problems that I am writing specificallyfor them.

In summary, in this phase Ms. Statz learned about her students’ solutionstrategies by talking with children individually about their thinking. Thenew need to spend much time with individuals stimulated her to think ofhow to change the class organization. She began to experiment, an activitythat continued in the fourth phase. She continued to generate new questionsand look for solutions. This phase corresponded to Franke et al.’s (2001)Level 4A classification of engagement with children’s thinking. Ms. Statzinteracted with students individually to learn more about their thinking.She acquired detailed knowledge of what her students understood andthe kinds of strategies they could use. But decisions about what to teachwere still largely driven by the global notions of children’s thinking andthe curriculum. In the final phase of change, Ms. Statz began to build onindividual children’s knowledge in instruction.

PHASE 4: TRANSITION TO LEVEL 4B ENGAGEMENT WITHCHILDREN’S THINKING: BUILDING ON CHILDREN’S

THINKING IN WHOLE-GROUP DISCUSSIONS

In this phase Ms. Statz considered how her teaching practices influencedchildren’s thinking, and how what she learned about children’s thinkinginfluenced her teaching practices. She began to experiment with usingthe knowledge she gained from working with the children individually toguide whole-class discussions.

Ms. Statz wanted to increase the number of children who presentedstrategies to the group, yet felt that the sharing time was not productivefor many children because they were not listening to or thinking about thepresenters’ strategies. Ms. Statz was especially concerned about childrenwho might not understand the more sophisticated strategies. To addressthis concern, she started to get students more actively involved in thediscussions: she would typically stop the child who was presenting andask the class or specific children what they thought the child’s next stepwould be. Ms. Statz asked other questions, in addition, such as “Can youtell what she did?” “How is her strategy different than somebody else’s?”“How can we make this strategy easier? Clearer?”

The children became more involved during discussions and Ms. Statzincreased the discussion time considerably, from 5–10 minutes early in theyear up to two consecutive lessons of 45 minutes of discussion for one set


of problems. The discussions in this phase lasted on average 21.8 minutes(based on 18 observations), whereas discussions in phases 1, 2 and 3 lastedan average of 7.9 minutes (based on 16 observations).

Ms. Statz’s efforts to elicit children’s thinking in discussions andinvolve the audience in responding to this thinking led to a breakthroughin her engagement with children’s thinking. She started to use the whole-class discussions to help individual children progress. This practice markeda change in instructional goals and orientation toward the use of interactivetalk:

I never used the sharing strategies as a time to move the kids along. That was just a timefor the kids to be able to talk about their strategies . . . And maybe that’s why they’re beingmore focused on it (now). Because I’m including more of them in the discussion. I havegiven the kids more time to discuss their strategies. And I have used the information that Iam getting from their strategies to move other kids. In the past I would just have [student]show the class that problem and that would be all, but not use it as a teaching moment, toteach the rest of the class about renaming fractions or whatever it was.

Ms. Statz remembered individual children’s strategies or difficulties thatwere elicited while working one-on-one with children and addressed themin whole-class discussions. The way in which Ms. Statz began to usewhole-class discussions to help individual children move from strategiesbased on counting by ones to strategies that incorporated base-ten conceptswas especially striking.

The whole-group discussion of the following problem typifies thesekinds of discussions: “Ellen has 287 books. How many more books wouldshe need to have 400 books?” First Ms. Statz called on Anne, who usuallysolved problems like this one by writing all the numbers between 287and 400 and counting them by ones. This time, Anne began to solve theproblem, with help of her partner, by adding 200 to 287 to get 487. Theteacher stopped her and asked the class what problem Anne was facing atthat point. Some children said that Anne had 87 too many. Ms. Statz askeda few children what they would do to continue and then asked Anne ifthat was what she did. She helped Anne and the rest of the class to do thecalculation 200 − 87, needed for the next step, by counting down 80 by10s to 120, and subtracting 7 from 120 by taking away 5 then 2 more.

When Ms. Statz called on Jared, he was hesitant to share his strategywith the class, because, he said, “It will take me years.” He drew tallymarks and wrote next to each single number: 288, 289, 290, 291, . . . Theteacher stopped him when he got to 310 (in his notebook he had drawntallies all the way to 400) and asked him what he needed to get from 300to 400. Jared said 10 10s or 2 50s. Other children suggested one 100 andJared added the 100 to the 13 tallies he made to count from 287 to 300.Jared concluded by saying the strategy was “really easy.”


Another child presented the beginning of his solution, writing “287 +100” vertically. Ms. Statz stopped him and asked Anne again what 287plus 100 was. Then she asked a few of the children, who tended to useones instead of tens or hundreds to calculate multidigit sums, a seriesof problems in which 100 was added (387+100, 487+100 . . . 987+100).The child who solved the problem showed the next step: 387 + 10. Ms.Statz returned to Jared and asked him to solve the problem. When he haddifficulty, she asked another child from the group she was targeting thatday. The next steps, involving 397 plus 3, then 100 plus 10 plus 3, werehandled by Ms. Statz in a similar manner.

The fourth child who came to the board solved this problem usingthe standard subtraction algorithm (400 − 287 written vertically, withregrouping from right to left). He, as well as other children, explained theconceptual underpinnings of each step. For example, Mary explained thatshe saw 400 as 40 tens: if you take 1 ten, 39 tens are left, so you just write39 tens and a 10 on top.

In this example, Ms. Statz used whole-class discussion to help specificchildren, such as Anne, to construct base-ten concepts and to use moreadvanced strategies. Her agenda that day involved helping these childrenby building on her knowledge of their strategies. Before this lesson, Ms.Statz rarely, if ever, attempted to build on children’s thinking in whole-class discussions. Afterwards, she regularly used whole-class discussionslike this one to assist individual children.

Whole-Group Discussion and Inquiry into Mathematics. As Ms. Statzcontinued to grow in her use of group discourse to advance children’sunderstanding, she broadened her inquiry into new domains, such as multi-digit multiplication and division, and fractions. With these investigationscame new uses of interactive talk to build content in ways that were greaterthan the sum of the individual contributions to a discussion.

For instance, to elicit children’s fraction thinking, Ms. Statz began bygiving partitive and measurement division problems that have fractionsas answers, and asked children to solve them using their own strategies(Baker, Carpenter, Fennema, & Franke, 1992). Although Ms. Statz did notinstruct the students in specific strategies, all students were successful insolving the problems and depicting fractional quantities using drawingsand diagrams.

Near the end of March, Ms. Statz, with much excitement, told the firstauthor what had happened in class. The day before, the children wrote andsolved their own word problems. Kanisha, a student who routinely countedby ones to solve multidigit problems, wrote a problem similar to the kind


of problems they have been solving: “There were 20 cakes and there were7 kids. How much cake will each kid get?” Kanisha solved the problemusing an unconventional partitioning strategy based on repeated halving.To begin, she gave each child two whole cakes. She then partitioned theremaining six cakes in half, and wrote the numbers 1 through 7 on the firstseven halves to designate 1 half for each of seven children (Figure 1).

Figure 1. Kanisha’s repeated halving strategy for sharing 6 remaining cakes among 7children.

Continuing, she halved each of the remaining halves, to create enoughfourths for seven people. She wrote the numbers 1–7 to indicate giving onefourth to each of the seven children. A half and a fourth remained in the lastcircle. Ignoring the fourth, Kanisha divided the half into seven pieces andagain wrote 1–7 on each. (Ms. Statz later helped her create an appropriaterepresentation of equal pieces). Thus, at this point, each sharer had onehalf, one half of a half, and one seventh of a half. Ms. Statz reminded herto share the remaining fourth among the seven children and Kanisha didthat. Therefore, each child also got an additional one seventh of a fourth.Ms. Statz then helped Kanisha decide the sizes of the fractional pieces.She helped her see that, since the half was divided into seven pieces, eachpiece was 1/14: “We said, if there are 7 slices in one-half, how many pieceswill be in a whole cake?” Kanisha knew it was 14. In a similar manner, shefigured that each of the seven pieces in the fourth was 1/28. Thus Kanishagave as an answer for her problem: 2 + 1/2 + 1/4 + 1/14 + 1/28.

Although this kind of strategy is not common in the standard teachingof fractions, it is common in classrooms where teachers encourage childrento generate their own strategies (Empson, 1999). As the following accountof the whole-group discussion suggests, strategies based on non-standardpartitions of sharing situations, such as this one, can be mathematicallyrich (Streefland, 1991).

Ms. Statz accepted Kanisha’s strategy and was excited by her achieve-ment: “The fact that Kanisha did that was fabulous. I thought it was reallygood. But then what came out of it (in the whole class discussion) . . .

was really cool.” After Kanisha had presented her strategy to the class,Ms. Statz asked the children what the result would be if the fractions werecombined. She thus used Kanisha’s strategy as a basis for posing a new


problem to the class. Ms. Statz prompted the children by asking them howmany twenty-eighths were in one-fourteenth. Then they figured how manytwenty-eighths were in one-fourth and one-half and found that Kanisha’sfractions combined to make 2 and 24/28 cakes.

Another girl then said she saw an easier way to solve the problem:divide the six cakes that were left into seven pieces each (because therewere seven children sharing) and each child gets one seventh from eachcake for a total of 2 and 6/7 cakes for each child. Ms. Statz used thetwo apparently different answers as an opportunity to explore the idea ofequivalent fractions, which was new to the children. She again posed anew problem to the group by asking how they could decide whether thetwo amounts were the same or not. Ms. Statz did not see immediately howto help the children answer this question meaningfully and she resorted to asymbolic technique based on reducing 24/28 to 6/7 by finding the greatestcommon factor. However, the following year, Ms. Statz used opportunitieslike this one to elicit children’s informal justifications about how equiv-alent fractional amounts were related (e.g., Empson, 2002, Figure 6; thisexample of a student’s work came from Ms. Statz’s classroom).

In this example, we see how Ms. Statz not only reacted to and builton children’s strategies spontaneously in discussion, but also used theproblem and children’s different strategies as a basis for new, morechallenging problems. Through Ms. Statz’s orchestration of the group’sdiscussion, the class explored mathematics topics that went beyond eachchild’s effort.

In summary, in this phase Ms. Statz’s knowledge of children’s thinkingcontinued to grow. We classify her engagement with children’s thinkingas Level 4b on the CGI scale, because she used knowledge of specificchildren’s thinking to inform classroom interactions. She found ways tobuild on the children’s strategies in whole-class discussions and not justindividually.

Continued Inquiry

As Ms. Statz continued to grow in her use of group discourse to advancechildren’s understanding, she broadened her inquiry into new mathematicsdomains. With these investigations came new uses of interactive talk tobuild content in ways that were greater than the sum of individual contri-butions to discussion. The following year, the second author documentedMs. Statz’s continued inquiry into children’s thinking in the domains offractions and multidigit multiplication and division (e.g., Baek, 1998;Empson, 2002). Later, Ms. Statz became involved in inquiry into children’salgebraic thinking (Carpenter, Franke, & Levi, 2003).


Retrospective Reflection

In a career interview several years after the events reported here, Ms.Statz was asked about her growth as a teacher. She attributed her growthin teaching to two main factors: a second pair of eyes in the classroomfocused on children’s thinking, and the freedom to experiment withinstruction based on children’s thinking.

A second pair of eyes focused on children’s thinking. When asked todescribe the most important turning points in her teaching, Ms. Statzmentioned the times, such as in the year reported here, when she had asecond person in her classroom, who was knowledgeable about children’sthinking and who could see and describe things to her that would haveotherwise gone unremarked. Interactions with people like the mathematicsresource teacher, or researchers such as us, provided some of the “rawmaterial” for Ms. Statz to reflect on her students’ knowledge. Her priorbeliefs in the value of children’s thinking provided some of the motivationfor this reflection.

Freedom to experiment. Ms. Statz credited the freedom she had to givechildren problems to solve, to talk to her children about their thinking, andto experiment with interactions with students for the growth she experi-enced as a teacher. In her current role as a mathematics resource teacher,she worried that an emphasis on teaching using printed curriculum mate-rials – even standards-based curriculum programs – may prevent teachers’deep engagement with children’s thinking. Because of an increasedemphasis, in many districts, on following these programs, Ms. Statz saidthat teachers do not have the same kinds of opportunities to experiment andfind out what their children know and can learn. They feel that they do nothave the freedom to have discussion sessions that last 45 minutes becausethere is so much to cover. Ms. Statz believed that, unless teachers are ableto have lengthy discussions with children about their thinking, they willnot be able to learn from their teaching – or at least not the same kinds ofthings about children’s thinking that she had learned.

DISCUSSION

This case study documents how Ms. Statz’s engagement with children’sthinking changed dramatically in a period of only a few months. In Phase1, children talked about their strategies, and Ms. Statz listened, but rarelychallenged children to extend their thinking or referred to their strategies


in later discussion. In Phase 2, as the participant researcher shared infor-mation with Ms. Statz about how individual children were thinking, Ms.Statz realized there was a discrepancy between what she knew about herstudents’ problem solving and how her students actually solved problems.She realized, in particular, that some students used mistaken strategies andothers used very basic strategies, and was concerned by this information. InPhase 3, Ms. Statz began to make time in her instructional routine to talkto children in more depth about their thinking, for her own benefit. Sheconcentrated on students whose thinking she believed to be problematicin some way, and struggled with how to support these students’ learningby building on their thinking, rather than imposing her own. Her solu-tion involved engaging students in talking to other students who solvedproblems in a similar way. Finally, in Phase 4, Ms. Statz began to useinformation gathered in one-on-one interactions to build on children’sthinking in group discussions. Instruction was guided by knowledge ofindividual children’s thinking. Ms. Statz continued to benefit from talkingwith children about their thinking, but now that talk was also designed tohelp children advance.

The extent of the change over the course of the study is especiallystriking, given the fact that, at the beginning, Ms. Statz already usedmany reform-oriented ideas in her teaching and believed that childrenshould construct their own knowledge. Her engagement with children’sthinking corresponded, at the beginning of the study, to Level 3 in Frankeet al.’s (2001) scale. By the end of the her third year of teaching, Ms.Statz’s engagement with children’s thinking was characterized by gener-ative learning, and corresponded to Level 4b engagement with children’sthinking.

We argue that the primary driving force behind the process of changewas Ms. Statz’s need to know more about children’s mathematicalthinking, and her pursuit of this knowledge in interaction with students.This need was founded on her beliefs about the importance of student-generated strategies, first fostered in her pre-service teacher-educationcourses, and on her realization of gaps in her knowledge of her students’thinking. It was nurtured by her participation in the discourse communityof CGI teachers and researchers. As she organized her interactions withchildren to learn more about their thinking, new dilemmas arose abouthow to increase children’s opportunities to express their thinking and tolearn by listening to other children’s thinking. These dilemmas, in turn,led to solutions that allowed Ms. Statz to continue to learn about children’sthinking while children learned of each others’ thinking. Ms. Statz beganto use whole-group discussions not just as displays of thinking, but also


as arenas for building on thinking. Each of these reflection cycles wasgrounded by increasingly detailed knowledge of children’s thinking. Ulti-mately, learning about children’s thinking was integrated with classroomparticipation structures to elicit and build on children’s thinking. In thisway, Ms. Statz’s learning about children’s thinking became generative.

CONDITIONS FOR TEACHER CHANGE

In outlining the conceptual framework for this study, we discussed threeconditions for teacher change based on the research literature. We revisitthese conditions here, and speculate about their contributions to Ms. Statz’schange as a teacher.

1) Membership in a discourse community. The CGI framework forchildren’s thinking provided a basis for conversation and other kindsof interactions about a phenomenon central to Ms. Statz’s work as ateacher: children’s mathematics learning. By virtue of interactions with“old timers” (Lave & Wenger, 1991; Wenger, 1998) such as Ms. J andthe participant researchers, Ms. Statz became an increasingly knowledge-able member of this discourse community. Because the tools for thinkingabout her teaching provided in this discourse community intersected withthe problems that were most pressing for her as a teacher, Ms. Statz wasmotivated to use and adapt these tools for herself. Although there is nosingle point at which Ms. Statz can be said to have joined this discoursecommunity, her opportunities for engaging in it were multiple and occurredin a number of contexts.

2) Processes for reflectively generating, debating and evaluating newknowledge and practices. The processes for producing new knowledgeand practices identified in this case study were, for the most part, infor-mally organized. Other than during her pre-service teacher education, Ms.Statz did not participate, in her early teaching years, in formally organizedlearning opportunities, such as professional development workshops inmathematics. However, these informal processes were powerful for her,perhaps because, in partnership with old timers in the CGI discoursecommunity, she was able to formulate and address some of the mostpressing practice-based dilemmas.

We identified two sorts of processes in particular for generating newknowledge and practices at work in Ms. Statz’s third year of teaching.The first process involved the participant researcher making available newinformation to Ms. Statz about her students’ thinking, through conver-


sation, examination of students’ written work, and one-on-one problemsolving interviews observed by Ms. Statz. The process of examiningher students’ thinking in partnership with the participant researcher (a“second pair of eyes”) served as a kind of scaffolding to Ms. Statz’s owninquiry into children’s thinking by placing in the foreground aspects ofher students’ thinking she had previously not seen. The second processinvolved Ms. Statz’s independent inquiry into students’ thinking. As shetook ownership of questions about children’s thinking, as well as theoutcomes, she became engaged in practical inquiry. This inquiry includedquestions about class organization such as how to assist strugglingchildren, how children learn from each other, and how to conduct mean-ingful discussions. Thus, processes for generating and testing knowledgeabout children’s thinking became integrated into Ms. Statz’s teaching asshe created opportunities for herself, and then students, to hear children’sthinking.

In contrast, at the beginning of the study Ms. Statz did not learn fromher students in the manner in which she did later, even though she gavethem opportunities to solve problems in their own ways and to talk abouttheir strategies. We suggest a change in Ms. Statz’s perception of her roleas a teacher, from passive to active, provided a motivation to learn moreabout what her students were doing in order to use that knowledge to helpthem advance. She realized that, even as a teacher who valued children’sinformal thinking, she could have goals that called for children’s thinkingto progress.

We speculate this passive role is a common step in teachers’ develop-ment. It seems there is a need at the beginning of teacher change to “stepback,” and not intervene in children’s problem solving very much (Jacobs& Ambrose, 2003). After becoming convinced children can generate theirown solution strategies, teachers become active again but in a different wayfrom before, by: helping students develop their own strategies; helpingstudents who do not understand the meaning of the problem; helpingthem express their solutions in multiple forms; asking probing ques-tions; and leading discussions that build on children’s ideas and stress themathematics content of those ideas.

3) Ownership of change. Ms. Statz’s transition to practical inquiry isevidence of ownership of the change she experienced. That Ms. Statzmade this transition may be due, in part, to the nature of the participantresearcher’s interactions with her. The participant researcher did not giveMs. Statz ready-made activities, but encouraged her to make her own


decisions about instruction. Reflecting later on the process she experiencedin this collaboration, Ms. Statz commented:

You allowed me to voice my concerns. And you were somebody to listen to the things thatI had problems with. You gave suggestions. Yet you also said: ‘It’s up to you. Do it yourway. Try it your way. It’s up to you with your class.’ I guess I learned to stop asking foradvice and I learned to start thinking on my own. Because I knew you would say, ’Whatdo you think?’ So then I was already doing some of the thinking and trying it out on youmore.

Because of the participant researcher’s insistence that Ms. Statz knew herown students best, Ms. Statz developed a predisposition to ask and answerher own questions, which led to a sense of professional autonomy. Further,without the freedom to experiment with the curriculum, cited by Ms. Statzas key to her development as a teacher, she may not have developed thispredisposition, no matter what the participant researcher’s stance was.

IMPLICATIONS FOR PRACTICE

What have we, as researchers and teacher educators, learned fromconducting this study? How has the collaboration between teacher andresearcher-teacher educator helped us educate other teachers? We presentsome insights based on the first author’s subsequent experience withteachers from about 80 schools in Israel, many of whom made majorchanges in their teaching and pre-service education. Some adjustmentswere necessary in adapting teachers’ use of CGI to Israeli classrooms,because of larger class sizes (35–40 students), a different culture, andthe national curriculum. The findings of this study informed the profes-sional development work in Israel in three ways. In her work withteachers, the first author focused on 1) eliciting and interpreting children’sthinking, 2) building on children’s thinking in one-on-one interactions,and 3) building on children’s thinking in group discussions. The teacherdevelopment program included study of children’s solution strategies, useof challenging problems, encouraging a variety of solutions, discussionof classroom organization, and examination of teachers’ beliefs aboutthe kinds of problems children can solve without direct instruction instrategies. Special importance was attached to understanding and aimingfor the highest levels of teacher development in Table I. It is a difficulttask for teachers to obtain a clear picture of a student’s current level,to understand his/her difficulties and to help in a manner that builds onhis/her thinking. Because it was not feasible to have a “second pair ofeyes” in each classroom, the teacher development included analyses of


written or videotaped examples of individual students and teachers inter-acting with students from participating teachers’ classrooms. A secondimportant topic is how the teacher can stimulate class discussions thatbuild on children’s thinking and help them progress. The experience withMs. Statz was an important catalyst in bringing this topic to the forefrontof the teacher development. For example, one useful activity was to askthe teachers to bring four examples of children’s strategies on a problemand to think how they could build a discussion around them. This activityhelped the teachers understand what sorts of questions they could ask andwhat mathematical ideas to emphasize.

CONCLUSION

An enduring problem in teacher change is the tension between inductingteachers into new instructional practices and respecting teachers’ profes-sional autonomy. In this study, these tensions were represented, respec-tively, by CGI and Ms. Statz’s personal teaching dilemmas. Part of Ms.Statz’s learning concerned learning problem types and solutions strategies;but the other, more important part had to do with learning how to usethis knowledge and how to generate this kind of knowledge by/for herselfin practice – that is, to conduct practical inquiry into children’s thinking.Ultimately, this practical inquiry was integrated into her interactions withchildren and became generative. The result was a body of knowledgefor Ms. Statz that was richer and more complex than CGI’s research-based framework for children’s thinking because it was informed bythe “concrete particulars” (Lampert, 1985) of her own practice-baseddilemmas, and driven by her growing knowledge of her own students’thinking. Ms. Statz reported that she began teaching with a strong beliefin the value of children’s thinking. Many teachers have such a belief, butwithout specific knowledge of children’s thinking, they may not be ablefully to implement it.

We conjecture that mechanisms that help teachers see their students’thinking in new ways, combined with the freedom to respond to, andexperiment with, this information about children’s thinking are key tothe development of practical inquiry in teachers. More specifically, aturning point for Ms. Statz was the realization that talk with children abouttheir thinking was valuable, not only because it provided opportunitiesfor students to articulate their thinking, but also because it provided acontext for her to ask and answer questions about children’s thinking forherself. As Ms. Statz learned how to use the information gathered in theseinteractions, she began to influence the direction of these conversations


through specific questions about cognitively, socially, and mathematicallyappropriate extensions of individual children’s thinking.

Although Ms. Statz accomplished remarkable change during the courseof the study, the process was difficult. The experience of phases of uncer-tainty and conflict that had no obvious solutions was emotionally trying.Yet Ms. Statz was open to seeing and responding to these dilemmas, eventhough she was not sure what would result. We believe this openness isattributable to a school atmosphere that was open to teachers’ experimenta-tion, and the emphasis by the participant researcher on Ms. Statz’s capacityto ask and answer questions about her practice.

Finally, whatever form participation in a discourse community takes,we believe it must emphasize the teacher’s professional autonomy. Thisemphasis reinforces the capacity of teachers for practical inquiry, andprovides a means for the discourse community itself to adapt and remainvital in response to new perspectives.

ACKNOWLEDGEMENTS

We would like to thank Kathy Statz, the teacher who collaborated in thisstudy, for her major contributions; Ellen Ansell, Linda Levi and DebraL. Junk, for their feedback on earlier drafts of this manuscript; threeanonymous JMTE reviewers and editor Peter Sullivan for their suggestionsfor improvement; and Lou Her for translating interviews from English toHmong for two students. The first and second authors thank the Universityof Wisconsin-Madison for support as visiting scholars in the WisconsinCenter for Education Research and the Department of Curriculum andInstruction, respectively, during part of the time this study was conductedand written. The research reported in this paper was supported in part bythe National Science Foundation under Grants No. MDR-8955346 andMDR-8954629. The opinions expressed in this publication are those ofthe authors and do not necessarily reflect the views of the National ScienceFoundation.

NOTES

1 Her real name, used with her permission.2 Drake, C. (under review). Mathematics stories: The role of teacher narrative in theimplementation of mathematics education reform.3 Children initially direct model story problems, representing each object in a storyproblem one for one in the strategy, and acting out the semantic structure of the storywith these objects. For example, to solve a Join Change Unknown problem, such as “Lucy


had 7 dollars. How many more dollars does she need to buy a puppy that costs 11 dollars?”by direct modeling, a child would represent the first set of dollars with 7 objects (e.g.,counters, tallies), and join other objects to the set until there was a total of 11 objects.The child would then count the set that was joined to the initial set for the answer to thestory problem. Strategies beyond direct modeling include counting, deriving facts, andimposing a different semantic structure on the problem (see Carpenter et al., 1999 for moreinformation).

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RUTH M. STEINBERG

Kibbutzim College of EducationNamir St 147Tel Aviv 52507IsraelE-mail: [email protected]

SUSAN B. EMPSON

University of Texas at AustinU.S.A.

THOMAS P. CARPENTER

University of Wisconsin-MadisonU.S.A.

BOOK REVIEW

Simon Goodchild (2001) Students’ Goals. A case study of activity in a

mathematics classroom. Norway: Caspar Forlag. ISBN 82-90898-29-0

pbk

INTRODUCTION TO THIS BOOK

In times of ‘maths wars’, controversial debates and noisy disputes

among educationalists and politicians about setting benchmarks, defin-

ing more rigorous standards and calling for more testing, it is refresh-

ing to read such a gentle but profound and serious book, which

fundamentally questions our knowledge about mathematics classroom

practice and the appropriateness of our scientific approaches or policy

measures to increase effectiveness of practice. The author, an experi-

enced teacher, an engaged teacher educator and a modest but very

ambitious researcher, dares, vigorously, to raise again the most basic

and challenging questions: What do we really know about our stu-

dents’ learning in the mathematics classroom practice? What do they

tell us, if we listen to those who are usually ignored or may be dealt

with as ‘objects’ of our research? What do they experience as learning

mathematics or being engaged in mathematical activity and what

meaning do they ascribe to this?

Simon Goodchild tries to find new ways to reconstruct mathematics

classroom practice from different perspectives, in particular by finding

appropriate means to explicate students’ conceptions of and goals for

learning mathematics in school. He approaches students as those who

have to be heard, as the autonomous subjects of their learning and

experts of classroom practice that is shaped by the constraints they

encounter there.

To read this book is a fascinating experience. The author, very

frankly, with great honesty and self-critique, opens to the reader his

efforts and his struggle during the research process. The research

account is not easy to read – one has to go backwards and forwards

sometimes to follow arguments – but it does capture immediately the

reader’s full interest when working one’s way through the book. Simon

Goodchild speaks about his journey of discovery. This is the same for

this reader, when she experiences how unusual approaches are related



to familiar and ‘‘famous’’ theories; unusual ways are used to analyse

and to reinterpret events or situations one might consider as fully

known already, and ‘‘common sense’’ knowledge is contrasted to

sometimes exotic and surprising references one can encounter through-

out the book.

To become more concrete in his study, Simon Goodchild investi-

gates students’ activities in a Year 10 (age 14–15) mathematics class-

room in an English secondary school. Mathematical practices and the

underlying goals, rationales, beliefs and conceptions – which he per-

ceives as gained by participating in this practice – are reconstructed.

These are analysed to find patterns of interpretation, meaning and

importance that are attributed to this practice and to mathematics as a

result of teaching and learning in this classroom. In his Introduction,

the author summarises his work as follows:

This monograph is an account of the reconstruction of a model of the mathemat-

ics classroom arising from experience as a researcher during a year in which Iattended nearly every lesson of an unexceptional year 10 mathematics class. It ishoped that it will bring illumination to common events and episodes from a fresh

perspective, thereby developing understanding of the practice of the mathematicsclassroom. The account is structured with the intention of developing a model thatrepresents the complexity and organic coherence of the classroom (p.10).

The underlying main research objectives are presented as the princi-

pal questions:

1. What are the goals, rationales, purposes, and interpretations towards whichstudents work in their regular mathematics activity?

2. What are the features of the classroom, arena and setting, which comprisethe socio-cultural context in which students engage in mathematics activity?(p. 93).

The research report catches one’s interest because of two peculiar

characteristics: On the one side, it is the – still rare – focus on the per-

spectives of the students, which demands a specific design and conduc-

tion process of the study itself. On the other side, it is the theoretical

background the author is re-constructing and extensively and compre-

hensively unfolding in a detailed account that is unusual compared to

literature reviews known in many research studies. He recalls, compares

and connects prominent theoretical approaches coming from cognitive

psychology, epistemology or sociology, educational studies, philosophy,

pedagogy, experimental sciences such as cognitive sciences, which for

some time have been a focuspoint in discourses in mathematics educa-

tion, and complements them with some that are not yet ‘mainstream’.

On a macro-level, he searches for differentiation instead of simi-

larity, with the intention to develop different theoretical layers of

270 BOOK REVIEW

perspectives which are justifiable and grounded, while looking from

different angles and levels to offer an unknown enrichment of

accounts of classroom reality. This allows him appropriately to

locate and justify his research approaches on the one hand and on

the other hand to create a diversity of tools for his microanalysis

and in-depth interpretation that might be applicable in other

research studies as well. On the micro-level, he develops fine-grained

analyses of students’ perceptions and goals by applying a variety of

qualitative research approaches.

The mathematics classroom practice is re-constructed by using

interpretative methods of observation, hermeneutic and discourse

analysis of coded transcripts of students’ discussions during their

engagement with a mathematical task or activity, and complemented

by thorough and in-depth reflections on the unstructured conversa-

tions and open interviews with individual students during and after

routine activities.

Before going into a detailed discussion of the key ideas of this book,

and presenting some personal experiences in using parts of the research

account with my student teachers at different stages of their studies, I

want to give a brief overview of the research account and the structure

of its presentation.

OUTLINE OF CHAPTERS

The first three chapters are concerned with presenting the theories

within which the research is framed. Chapter 1: The Process of coming

to know describes the personal position and the learning process of the

author, which leads him to undertake the study and justify the refer-

ences he draws upon. He starts to examine learning and cognition

from three anthropological perspectives: facets of radical or social con-

structivism, contributions of activity theory to the socio-cultural theory

of cognition, and of social practice theory to cognition in everyday prac-

tice. Here he reflects critically on a considerable and diverse literature

related to these perspectives, going back to their roots as well as call-

ing on some studies that could be considered as side tracks or devia-

tions from the main theories; however, bringing forward surprising

interpretations.

Chapter 2: Goals discuss differences in concepts of students’ learning

goals as interpreted in constructivism and social practice theory and

extends to some of the existing research evidence relating students’

motivation, approaches, and orientations. The author explores

271BOOK REVIEW

research contributions on students’ metacognitive activity, metacogni-

tion, and meta-learning.

Chapter 3: A Framework for Explaining Activity in Mathematics

Classrooms, draws principally upon the work of three well known edu-

cators – the Norwegian mathematics educator Stieg Mellin Olsen, the

American educationalist Walter Doyle and the English psychologist

and mathematics educator Richard Skemp who are so different in their

approaches in my view that I would never have tried to relate them. In

referring to and bringing together contributions of these ‘idols’ or

‘model researchers’, an attribute that he explains and justifies in

detailed argumentation, the author develops a theoretical framework

that identifies and allows him to elaborate the three layers of classroom

activity he is going to apply: ‘‘the social arena of the classroom, the

student’s setting as she engages in the task set and the private concep-

tions of the individual students’ interpretation of her activity’’ (p. 10).

In Chapter 4: Methodology arid Research Design. The research pro-

cesses and the methodology employed to explore activity in the mathe-

matics classroom are reported together with careful discussion of the

various methods of qualitative and quantitative research and a con-

vincing and smooth unfolding of the development of the design of the

study.

In Chapter 5: The Arena, Chapter 6: The Setting, and Chapter 7:

Conception, the three layers of students’ activity is each considered sep-

arately in discussing and interpreting classroom events and interviews

with individual students. In Chapter 5, the classroom culture and the

rationales of students are explicated, supported by appropriate exam-

ples from data, In Chapter 6, the individual students’ setting is

described by investigating examples of teacher intervention with indi-

vidual students, interaction between students, resources, the mathemat-

ical content as experienced by students and purposes for learning

mathematics. In Chapter 7, conceptions of students are explored by

investigating their individual interpretations and modes of awareness

of their learning processes in classroom practice.

Chapter 8: Mathematical Activity in Classroom Practice constitutes

the culmination of the whole research enterprise and is the main chap-

ter of the monograph. This chapter provides the reader with the syn-

thesis of the content of goal statements of the preceding chapters,

from each of the three perspectives of cognition discussed in Chapter

1, to demonstrate that the three different angles together give a more

holistic picture of the mathematics classroom.

In an endnote, an additional analysis of archived extra-data is pro-

vided. This is used to check plausibility and trustworthiness of the

272 BOOK REVIEW

interpretation and analysis of the data and confirm the unexceptional-

ity of the classroom events collected. It includes a general critique of

the employed methods, a discussion about limitations and about what

could be the value of the research presented here. Value is seen in the

development of new frames for existing knowledge and complementing

this knowledge by the development of new instruments and tools for

research on classroom practice and students’ goals.

SOME OF THE MAIN THEMES AND IDEAS OF THE BOOK

In this section, I address some of the key ideas that call for special

attention. The author substantiates his position that knowledge is

intrinsically social: understanding means entering a social practice

and learning means behaving according to the ‘rules’ of this prac-

tice. For example, in paraphrasing Scribner and Cole (1981) he

states that:

the main outcome of experience in mathematics classrooms is to learn how to domathematics classroom practice. In the classroom, mathematics is set within aspecialised ‘classroom’ discourse, which allows students to locate and follow cuesand signals, skip over peripheral text and apply a variety of resources to bring the

highly stylised tasks to some form of resolution. Activity within the classroom isnot mathematics and for all its pretence it is not about the students’ current orfuture experience of the world outside the classroom. Success in classroom practice

does not prepare a student for the practice of mathematics or any other activityoutside the classroom. Success may reveal a student’s potential to learn a particu-lar type of practice. Engaging in reflective activity may arise when students ‘expect’

this to be part of the practice (p. 227).

This leads him to conclude that, as ‘‘expectations are dialectically

constituted in gap closing’’, according to Gattegno’s maxim: ‘‘only

expectations are educable’’ (p. 227).

A major constraint is attributed to the role of testing and test per-

formance as part of the social arena. The author conjectures that in

the context of tests and examinations, which characterise routinized

tasks, any development of approaches used in classroom that is not

sustained will be unlikely to make a lasting impact on students

because their expectations will return to ‘normal’. He argues that

despite all intentions to use problem solving approaches for learning,

the type of test questions call for surface attention and induce a sur-

face approach to studying and therefore, produce only superficial

and low level understanding which is finally applied to all activities

in classroom practice. Most mathematical schoolwork does not con-

stitute any challenge and uses only a minimum of skills, marked by

273BOOK REVIEW

repetitiveness and boredom. Using terms of Mellin–Olsen, classroom

mathematics is experienced by students as ‘blind activity’ and ‘dead

labour’. He notes that most often so-called ‘real-world problems’ are

considered by students as a word gambit without any relation to

reality or meaning.

Insisting on the perspective that mathematics is a tool for gaining

power, status and worth, he explains why mathematical activities

are perceived by students as having factual and symbolic power,

but no meaning. By unfolding the complexity of the mathematics

classroom practice and revealing the various, partly conflicting

‘models of behaviour’ and rationales and their effect on daily prac-

tice, he demonstrates that the identified predominant rationales and

the social constraints in arena and setting for teachers and learners

not only make teachers unable to constructively realize curricular

goals, but create conflicts with pedagogical conceptions propagated

in the new curriculum. These include ‘learner-centeredness’, ‘partici-

pation of learners’ and ‘active and productive learning mathematics

by conceptual understanding and communication’, problems of

classroom practice with which teachers are overburdened and need

support.

One hypothesis is the conviction that conceptions and goals of

mathematics learning, as patterns of behaviour and thinking, are

embedded in their social, cultural, and class discourses and can be

appropriately analysed as ‘texts’. These ‘texts’ manifest themselves not

only in the specificity of the words they use (and their meanings), but

also how and in which context they use the words, in which kind of

practice they are created and in the specific stories they tell. The char-

acteristics of these collective patterns are that they are effective in daily

life, learned from others and shared with others. Therefore, they

should be best unfolded in a collective discourse among students in

which they become explicit, first unconsciously, but by continuous

reflection more consciously.

The author structures his study very clearly and convincingly.

Advanced organizers and post organizers emphasizing either his spe-

cial arguments or his main results in the text, the reader is directed

to the different perspectives he employs in his study. The establish-

ment of a process of increasing self-reflection and professional con-

sciousness on the part of the researcher himself, pointing to his

own presumptions and patterns that guide interpretations of prac-

tices, represents not only an effective means to reveal the process of

research, in its most important steps of analysis and its constant

refinement, but also a means to integrate the reader into the

274 BOOK REVIEW

research process. This allows consequences to be drawn for the

political, pedagogical and professional debates on innovation and

change in classroom practices and what is reinforcing or hindering

developments.

In his last paragraphs the author summarizes his findings from an

elaborated point of view and relates them to the value of qualitative

approaches in mathematics education as a scientific discipline. He

shows that the partiality and inconsistency that occurs in the daily life

of students and teachers belongs to their professional and personal

lives; they constantly have to reconcile conflicting goals and percep-

tions. In particular he points out how the power attributed to mathe-

matics in curicula and by assessment modes that focus on procedural

discourses works to reduce opportunities to learn mathematics. Best

intentions of teachers together with conflicting cultural models or

rationales can turn empowerment actions into disempowerment of

pupils and the difficulties to reconcile conflicts can be increased by

them. Without awareness of such accounts of classroom practice,

teachers cannot raise alternative expectations or search for collective

reflections on changes in their practices. The author formulates this

very well with respect to one of the most prominent cultural models

emerging from his analysis:

Students may perceive the tasks of the mathematics classroom as remote fromtheir ‘activities’ that take place outside the classroom. The tasks set in the class-room are peculiar to that arena, and are presented in a specialized language withsignals that cue particular responses. Students construct their own viable models

of classroom mathematics and engage in their tasks from these models. At thesame time they can dismiss any challenges to their model’s viability because theirmodel is extensible rather than corrigible, and thus adapts to take account ‘new

rules’. Students do not seek a rational basis for the rules because much of theirexperience of classroom mathematics has taught them not to expect consistency,coherence or rationality (p. 222).

In my view, Goodchild’s study has moved the dominance of mathe-

matics from a common sense position to a rigorous and theoretical

understanding of its dimensions and of how it plays itself out in the

mathematics classroom in terms of creating mathematical opportuni-

ties for learners.

But in contradiction to the foreword of Paul Ernest, I would not

state that Simon Goodchild’s findings, as descriptions of practices in

reality, are only depressing, for me they are very much enlightening

and lead to further valuable studies to probe discourse analysis beyond

the pedagogic level and explore complex interaction between pedagogy

and politics.

275BOOK REVIEW

USING THE STUDY WITH TEACHER EDUCATION

STUDENTS

This study is very relevant for any approach to classroom practice

research. The interpretations are carefully elaborated and connected,

they may allow practitioners to use the methodological instruments for

analysing their own practices and or they may represent means that

can empower readers autonomously to reflect alternative interpreta-

tions and patterns for action in daily practice.

During my first reading, I tried to share my experiences with some

of my students. In two of my major seminars I exposed parts of the

book to two groups of very different students:

1. Student teachers at the beginning of their undergraduate studies in

an introductory course.

2. Student teachers in the final term of their studies starting their

master thesis.

With very different tasks in mind, I confronted the first group with

some of the excerpts of students’ interviews from Goodchild’s study

and asked them to recall their own experiences in school time and

search for similar situations, or make notes about other students’ expe-

riences. As they usually are asked to write a biographical essay, they

complemented their excerpts with examples of stories about getting

stuck, complete lack of understanding, hard debates with other stu-

dents about getting meaning and significance of a given task and so

on. In short, the excerpts encouraged them to look for their own sto-

ries and remember their school mates’ struggle as well as their own.

They were mostly fascinated that a researcher and teacher are giving

students a voice and really were interested to listen to them – they

argued that this might be an almost necessary condition for becoming

a teacher, but almost nobody among them had experienced such a ‘lis-

tening’ mathematics teacher.

The other group of student teacher worked partly on data from

another research project (Learners’ Perspective Study, 2004) and were

asked to analyse this data with the tools and theoretical approaches

developed and summed up by Goodchild (Summary of constructivist

account of learning on p. 22, the summary of the ‘socio-cultural per-

spective’ or ‘activity theory’ on p. 30, and the summary of ‘practice

theory’ on p. 41 as well as the comparative discussion of all three of

them on p. 41). We explored a possible application of these theoretical

accounts and the related tools as one means for different elaborations

of interpretations and what was considered as equally valid for them

276 BOOK REVIEW

compared to those based on other theories. I wanted also to further

confront them with interpretations from the other ‘‘layers’’ of tools

that Goodchild developed in parallel as contrat or complement. The

experience that all layers give another interpretation but also have

their own right and justification, was, in the first instant, a surprising

experience and later a serious difficulty. Nevertheless this proved

enriching for them, as they usually fall into the trap of looking for

only one theoretical construct to be used throughout their whole story.

We had hot discussions about theories and ‘‘how to apply them’’,

about different research designs which had to be developed later on

and finally were very much influenced by these debates. Most of my

students shared my enthusiasm about the book and confessed to hav-

ing learnt very much about themselves as students and as future teach-

ers.

In the first instance, the overwhelmingly applauding foreword of

Paul Ernest made me as a reader cautious and critical – I have to con-

fess that I was much irritated when I had to read this enthusiastic

appraisal – why to exaggerate so much? However, curiosity persisted.

After finishing my experiences with the book, I have to agree that

what Paul Ernest says about the study and its author is very much

substantiated and fully justified.

REFERENCE

Learners’ Perspective Study (2004). The Learners’ Perspectives Study – An Interna-

tional Research Collaboration on Mathematics Classroom Practice http://www.

edfac.unimelb.edu.au/DSME/lps (last update February 20, 2004).

Christine KeitelFreie Universitat BerlinGermany

277BOOK REVIEW

THE INTERNATIONAL COMMISSION ON MATHEMATICAL

INSTRUCTION (ICMI) – THE FIFTEENTH ICMI STUDY: THE

PROFESSIONAL EDUCATION AND DEVELOPMENT OF

TEACHERS OF MATHEMATICS

INTRODUCTION

This document announces a new Study to be conducted by the Interna-

tional Commission on Mathematical Instruction. The focus of this

Study, the fifteenth to be led by ICMI, will be the professional educa-

tion and development of mathematics teachers around the world. The

premise of this Study is that the education and continued development

of teachers is key to students’ opportunities to learn mathematics.

What teachers of mathematics know, care about, and do is a product

of their experiences and socialization both prior to and after entering

teaching, together with the impact of their professional education. This

impact is variously significant: in some systems, the effects of profes-

sional education appear to be weak or even negligible, whereas other

systems are structured to support effective ongoing professional educa-

tion and instructional improvement. The curriculum of mathematics

teacher preparation varies around the world, both because of different

cultures and educational environments, and because assumptions about

teachers’ learning vary. Countries differ also in the educational, social,

economic, geographic, and political problems they face, as well as in

the resources available to solve these problems. A study focused on

mathematics teacher education practice and policy around the world

can provide insights useful to examining and strengthening all systems.

We recognize that all countries face challenges in preparing and

maintaining a high-quality teaching force of professionals who can

teach mathematics effectively, and who can help prepare young people

for successful adult lives and for participation in the development and

progress of society. Systems of teacher education, both initial and

continuing, are built on features that are embedded in culture and the

organization and nature of schooling. More cross-cultural exchange of

knowledge and information about the professional development of

teachers of mathematics would be beneficial. Learning about practices

and programs around the world can provide important resources for



research, theory, practice, and policy in teacher education, locally and

globally. Study 15, The Professional Education and Development of

Teachers of Mathematics, is designed to offer an opportunity to

develop a cross-cultural conversation about mathematics teacher edu-

cation in mathematics around the world.

Because the professional education of teachers of mathematics

involves multiple communities and forms of expertise, the Study also

explicitly welcomes contributions from individuals from a variety of

backgrounds. Mathematicians and school practitioners are particularly

encouraged to submit proposals for contributions.

The Study will proceed in three phases: (a) the dissemination of a

Discussion Document announcing the Study and inviting contribu-

tions; (b) a Study Conference, to be held in Brazil, 15–21 May 2005;

and (c) publication of the Study Volume – a Report of the Study’s

achievements, products and results.

First is this Discussion Document, defining the focus of the Study

and inviting proposals for participation in a Study Conference. We

welcome individual as well as group proposals; focusing on work

within a single program or setting, as well as comparative inquiries

across programs and settings. In order to make grounded investiga-

tions of practice in different countries possible, we invite proposals in

three formats: papers, demonstrations, and interactive work-sessions.

Details are provided below.

Second, a Study Conference will be held in Brazil in May 2005,

bringing together researchers and practitioners from around the world.

The Conference will be deliberately designed for active inquiry into

professional development of teachers of mathematics in different coun-

tries and settings. Some sessions will offer paper presentations; other

sessions will engage participants in direct encounters with particular

practices, materials and methods, or curricula.

Third, a Study Report – the Study Volume – will be produced, repre-

senting and reporting selected activities and results of the Study Confer-

ence and its products. This Report will be useful to the mathematics

education community, as well as for other researchers, practitioners,

and policymakers concerned with the professional education of teachers.

WHY CONDUCT A STUDY ON THE PROFESSIONAL

EDUCATION OF MATHEMATICS TEACHERS?

Three main reasons underlie the decision to launch an ICMI study

focused on teacher education. One reason rests with the central role of

280 DEBORAH LOEWENBERG BALL AND RUHAMA EVEN

teachers in students’ learning of mathematics, nonetheless too often

overlooked or taken for granted. Concerns about students’ learning

compel attention to teachers, and to what the work of teaching

demands, and what teachers know and can do. A second reason is

that no effort to improve students’ opportunities to learn mathematics

can succeed without parallel attention to their teachers’ opportunities

for learning. The professional formation of teachers is a crucial ele-

ment in the effort to build an effective system of mathematics educa-

tion. Third, teacher education is a vast enterprise, and although

research on mathematics teacher education is relatively new, it is also

rapidly expanding.

The timing is right for this Study. The past decade has seen substan-

tial increase in scholarship on mathematics teacher education and

development. A growing number of international and national confer-

ences focus on theoretical and practical problems of teacher education.

Publication of peer-reviewed articles, book chapters, and books about

the development of teachers of mathematics is on the rise. Centers for

research and development in teacher education exist increasingly in

many settings. A Survey Team led by Jill Adler will report on the

development of research on mathematics teacher education as part of

the program at the tenth International Congress on Mathematics Edu-

cation (ICME-10) in July 2004 in Copenhagen. In addition, it is signif-

icant that the past decade has also included the launching of a new

international journal (in 1996): the Journal of Mathematics Teacher

Education (JMTE) is published by Kluwer, and edited by an interna-

tional team of scholars. Seven volumes later, JMTE hosts a thriving

international discourse about research and practice in teacher

education.

Mathematics teacher education is a developing field, with important

contributions to make to practice, policy, theory, and research and

design in other fields. Theories of mathematics teachers’ learning are

still emerging, with much yet to know about the knowledge, skills, per-

sonal qualities and sensibilities that teaching mathematics entails, and

about how such professional resources are acquired. The outcomes of

teacher education are mathematics teachers’ practice, and the effective-

ness of that practice in the contexts in which teachers work. Yet we

have much to learn about how to track teachers’ knowledge into their

practice, where knowledge is used to help students learn. And we have

more to understand about how teacher education can be an effective

intervention in the complex process of learning to teach mathematics,

which is all too often most influenced by teachers’ prior experiences as

learners, or by the contexts of their professional work.

281THE FIFTEENTH ICMI STUDY

Study 15 aims to assemble from around the world important new

work – development, research, theory, and practice – concerning the

professional development of teachers of mathematics. Our goal is to

examine what is known in a set of critical areas, and what significant

questions and problems warrant collective attention. Toward that end,

the Study aims also to contribute to the strengthening the international

community of researchers and practitioners of mathematics teacher

education whose collective efforts can help to address problems and

develop useful theory.

SCOPE AND FOCUS OF THE STUDY

This Study focuses on the initial and continuing education of teachers

of mathematics. Our focus is the development of teachers at all levels,

from those who teach in early schooling to those who teach at the sec-

ondary school. (In this Discussion Document, we use ‘‘primary’’ to

refer to teachers of students of ages 5–11; ‘‘middle’’ to refer to ages

11–14, and ‘‘secondary’’ for ages 14 and older.) Teacher development

is a vast topic; this study focuses strategically on a small set of core

issues relevant to understanding and strengthening teacher education

around the world.

The Study is organized in two main strands, each representing a

critical cluster of challenges for teacher education and development.

In one strand, Teacher Preparation and the Early Years of Teaching,

we will investigate how teachers in different countries are recruited

and prepared, with a particular focus on how their preparation to

teach mathematics is combined with other aspects of professional or

general academic education. In this strand, we will also invite contri-

butions that offer insight into the early phase of teachers’ practice.

In the second strand, Professional Learning for and in Practice, we

will focus on how the gap between theory and practice is addressed

in different countries and programs at all phases of teachers’ develop-

ment. In this strand, we will study alternative approaches for bridg-

ing this endemic divide, and for supporting teachers’ learning in and

from practice. This strand may be explored at any of the develop-

mental stages – preservice, early years, and continuing practice – of

teachers’ practice. In both strands, we seek additionally to learn how

teachers in different countries learn the mathematics they need for

their work as teachers, and how challenges of teaching in a multicul-

tural society are addressed within the professional learning opportu-

nities of teachers.


Table 1 provides a graphic representation of the scope and focus of

the Study. The table makes plain that for Strand 1, the focus will be

on the preservice and early years of teaching only; the Study will not

focus on issues of recruitment, program structure and curriculum for

experienced teachers. However, Strand II, focused on professional

learning in and from practice, may be studied at all phases of teachers’

development.

Strand I: Teacher Preparation Programs and the Early Years of

Teaching

This strand of the Study will examine a small set of important ques-

tions about the initial preparation and support of teachers in countries

around the world, at the preservice stage, and into the early years of

teaching. How those phases are structured and experienced varies

across countries, as does the effectiveness of those varying structures.

Questions central to the investigation of initial teacher preparation and

beginning teaching will include:

(a) Structure of teacher preparation: How is the preparation of teach-

ers organized – into what kinds of institutions, over what period of

time, and with what connections with other university or collegiate

study? Who teaches teachers, and what qualifies them to do so?

How long is teacher preparation, and how is it distributed between

formal study and field or apprenticeship experience? How is the

preparation of teachers for secondary schooling distinguished from

that of teachers for the primary and middle levels of schooling?

TABLE I

Scope and Focus of the study

Strands Phases of teacher

development

Initial teacher education

(preservice and early years

of teaching)

Continuing practice

Programs of teacher

education (recruitment,

structure, curriculum, first

years)

yes no

Professional learning for

and in practice

yes yes


(b) Recruitment and retention: Who enters teaching, and what are the

incentives or disincentives to choose teaching as a career in partic-

ular settings? What proportion of those who prepare to teach

actually end up teaching, and for how long? How do teachers’ sal-

aries and benefits relate to those of other occupations?

(c) Curriculum of teacher preparation: The Study seeks to probe a

small set of key challenges of teacher preparation curriculum and

investigate whether and how different systems experience, recog-

nize, and address these issues. Two such issues are:

� What is the nature of the diversity that is most pressing within

a particular context – for example, linguistic, cultural, socio-

economic, religious, racial – and how are teachers prepared to

teach the diversity of students whom they will face in their clas-

ses?

� How are teachers prepared to know mathematics for teaching?

What are the special problems of subject matter preparation in

different settings, and how are they addressed? Is interdisci-

plinarity in teacher education commonplace, and if so, how is

managed? How do faculty in education interact with faculty in

mathematics over issues of teacher education?

In addition, we invite proposals that identify and examine other spe-

cific central challenges for the curriculum of teacher preparation.

(d) The early years of teaching: What are the conditions for beginning

teachers of mathematics in particular settings? What supports

exist, for what aspects of the early years of teaching, and how

effective are they? What are the special problems faced by begin-

ning teachers, and how are these experienced, mediated, or

solved? What is the retention rate of beginning teachers, and what

factors seem to affect whether or not beginning teachers remain in

teaching? What systems of evaluation of beginning teachers are

used, and what are their effects?

(e) Most pressing problems of preparing teachers: Across the initial

preparation and early years, what are special problems of teaching

mathematics within a particular context and how are beginning

teachers prepared to deal with these problems?

(f) History and change in teacher preparation: How has mathematics

teacher preparation evolved in particular countries? What was its

earliest inception, and how and why did it change? What led to

the current structure and features, and how does its history shape

the contemporary context and structure of teacher education?


Proposals for this Strand may offer descriptions accompanied by

analyses of practices, programs, policies, and their enactment and out-

comes. This is a scientific Study, and thus, we seek papers based on

systematically gathered information and analyses.

In order to maximize the range of systems of teacher preparation

about which we can learn through this Study, we seek proposals from

a variety of countries. The Study’s investigation will be improved if

the countries represented on the Program differ in size, population

diversity (language, culture, race, socio-economic), performance in

mathematics, centralization of curricular guidance and accountability,

and level of societal and economic development.

Contributions to Strand I will be organized into a coherent section

of the Study, with an overview and one or more analytic comparative

commentaries to extend what can be learned from the individual cases

and studies.

Strand II: Professional Learning for and in Practice

This strand of the Study adds substantive focus, in complement to the

first. Whereas the first Strand examines programs and practices for

beginning teachers’ learning, the focus of the second relates to teach-

ers’ learning across the lifespan. This strand’s central focus is rooted in

two related and persistent challenges of teacher education. One prob-

lem is the role of experience in learning to teach; a second is the divide

between formal knowledge and practice. Both problems lead to the

central question of Strand II: How can teachers learn for practice, in

and from practice?

Researchers and practitioners alike know that, although most

teachers report that they learned to teach ‘‘from experience,’’ experi-

ence is not always a good teacher. Prospective teachers enter formal

professional education with many ideas about good mathematics

teaching formed from their experience as pupils. Their experience

learning mathematics has often left them with powerful images of

how mathematics is taught and learned, as well as who is good at

mathematics and who not. These formative experiences have also

shaped what they know of and about the subject. These experiences,

along with many others, affect teachers’ identities, knowledge, and

visions of practice, in ways which do not always help them teach

mathematics to students.

Moreover, teacher education often seems remote from the work of

teaching mathematics, and professional development does not neces-

sarily draw on or connect to teachers’ practice. Opportunities to


learn from practice are not the norm in many settings. Teachers

may of course sometimes learn on their own from studying their stu-

dents’ work; they may at times work with colleagues to design les-

sons, revise curriculum materials, develop assessments, or analyze

students’ progress. In some countries and settings, such opportunities

are more than happy coincidence; they are deliberately planned. In

some settings, teachers’ work is structured to support learning from

practice. Teachers may work with artifacts of practice – videotapes,

students’ work, curriculum materials – or they may directly observe

and discuss one another’s work. We seek to learn about the forms

such work can effectively take and what the challenges are in

deploying them.

Strand II of the Study asks how mathematics teachers’ learning may

be better structured to support learning in and from professional prac-

tice, at the beginning of teachers’ learning, during the early years of

their work, and later, as they become more experienced. Central ques-

tions include:

(a) What sorts of learning seem to emerge from the study of practice?

What do teachers learn from different opportunities to work on

practice – their own, or others’? In what ways are teachers learn-

ing more about mathematics, about students’ learning of mathe-

matics, and about the teaching of mathematics, as they work on

records or experiences in practice? What seems to support the

learning of content? In what ways are teachers learning about

diversity, about culture, and about ways to address the important

problems that derive from social and cultural differences in partic-

ular countries and settings?

(b) In what ways are practices of teaching and learning mathematics

made available for study? How is practice made visible and

accessible for teachers to study it alone or with others? How

is ‘‘practice’’ captured or engaged by teachers as they work

on learning in and from practice? (e.g., video, journals,

lesson study, joint research, observing one another and taking

notes).

(c) What kinds of collaboration are practiced in different countries?

How are teachers organized in schools (e.g., in departments) and

what forms of professional interaction and joint work are

engaged, supported, or used?

(d) What kinds of leadership help support teachers’ learning from the

practice of mathematics teaching? Are there roles that help make

the study of practice more productive? Who plays such roles, and


what do they do? What contribution do such people make to

teachers’ learning from practice?

(e) What are crucial practices of learning from practice? What are the

skills and practices, the resources and the structures that support

teachers’ examination of practice? How have ideas such as ‘‘reflec-

tion,’’ ‘‘lesson study,’’ and analysis of student work been devel-

oped in different settings? What do such ideas mean in actual

settings, and what do they involve in action?

(f) How does language play a role in learning from practice? What sort

of language for discussing teaching and learning mathematics –

professional language – is developed among teachers as they work

on practice?

Examining how some systems and settings organize teachers’ work

or their opportunities for continued learning close to the work of

teaching can offer images and resources for grounding the ongoing

development of professional practice educatively in practice.

DESIGN OF THE STUDY

The Study on the Professional Education of Teachers of Mathematics

is designed to enable researchers and practitioners around the world to

learn about how teachers of mathematics are initially prepared and

how their early professional practice is organized in different countries.

In addition, the Study takes aim at an endemic problem of profes-

sional education – that is, how learning from experience can be sup-

ported at different points in a teacher’s career, and under different

circumstances. Toward this end, the study is designed to invite a vari-

ety of kinds of contributions for collective examination and delibera-

tion at the Conference: research papers; program descriptions

accompanied by analysis; conceptual work; demonstrations of practice;

and interactive work on important common problems of teacher edu-

cation and teacher learning.

The Study Conference will be organized to be different from a con-

ventional research meeting. Although research papers will be part of

the program, substantial time will be designed for direct engagement

with artifacts and materials of practice, for critique and deliberation,

and for collective work on significant problems in the field. The Pro-

gram Committee will design the Conference using the proposals we

receive, and add, as needed, commentators, activities, and other

resources so that the Conference enables participants to work together


at the meeting, and to generate new insights, ideas, and questions

important to the professional education of teachers of mathematics

around the world. We anticipate that participants will be organized

into working groups that will meet regularly across the Conference,

affording the opportunity for joint discussion, work, and possible

plans for future collaborative activity. Working groups’ ideas will be

shared across the Conference; we will experiment with useful formats

for such exchange of ideas generated in the course of the Conference.

We also envision innovative plenary activities to provide common

experiences for collective examination, discussion, and learning. Partic-

ipation in the Study Conference is by invitation only, as is detailed

below.

CALL FOR CONTRIBUTIONS TO THE STUDY

The Study is designed to investigate practices and programs of mathe-

matics teacher education in different countries, and to contribute to

an international discourse about the professional education and devel-

opment of teachers of mathematics. The International Programme

Committee welcomes high-quality proposals from diverse researchers

and practitioners who can make solid practical and scientific contribu-

tions to the Study. New researchers in the field are encouraged to sub-

mit proposals, as are those actively engaged in curriculum

development for teacher education or professional development in any

setting. Mathematicians – who play a crucial role in preparing and

supporting teachers who are not specialists of the discipline – are

urged to submit proposals and to participate in the Study. To ensure

a rich and varied scope of resources for the Study, participation from

countries under-represented in mathematics education research meet-

ings is encouraged.

The conference will be a working one where every participant will

be expected to be active. As is the normal practice for ICMI studies,

participation in the Study conference is by invitation only, given on

the basis of a submitted contribution. Proposed contributions will be

reviewed and selections made based on the quality of the work, as well

as to increase the diversity of perspectives offered, and the potential to

contribute to the advancement of the Study. The number of partici-

pants invited to participate will be limited to approximately 120 peo-

ple. The Study Volume, to be published after the conference in the

ICMI Study Series, will be based on selected contributions and reports

prepared for the conference, as well as on the outcomes of the confer-


ence. The Study Website (http://www-personal.umich.edu/~dball/icmi-

study15.html), accessible also after the conference, will contain selected

examples of practice in teacher education, or teachers’ learning. A

report on the Study and its outcomes will be presented at the 11th

International Congress on Mathematical Education to be held in Mex-

ico in 2008.

The International Programme Committee (IPC) for the Study invites

submission of contributions on specific questions, problems or issues

related to this Discussion Document. Proposals for contributions are

invited for three formats: (a) papers; (b) demonstrations; (c) interactive

work-sessions. Submissions should reach the Programme Chairs by

e-mail (at the addresses below) no later than October 15, 2004, but ear-

lier if possible. All submissions must be in English, the language of the

conference. To avoid confusion or loss of proposals, please label elec-

tronic attached files: hyour surname_your given namei_ICMI15_prop.

doc.

The contributions of those invited to the conference will be made

available to other participants among the conference materials or on

the conference website (http://www-personal.umich.edu/~dball/icmi-

study15.html). However, an invitation to the conference does not

imply that a formal presentation of the submitted contribution will be

made during the conference or appear in the Study Volume published

after the conference.

It is hoped that the conference will attract not only ‘‘experts’’ but

also some ‘‘newcomers’’ to the field with interesting and refreshing

ideas or promising work in progress. Unfortunately, an invitation to

participate in the conference does not imply a financial support from

the organisers, and participants should finance their own attendance at

the conference. Funds are being sought to provide partial support to

enable participants from non-affluent countries to attend the confer-

ence, but it is unlikely than more that a few such grants will be avail-

able.

Papers should be no longer than 2000 words and five single-spaced

pages at most. Papers will be organized into thematic sessions by the

Program Committee. Papers should report on analysis of practices and

programs of mathematics teacher education in particular settings, with

attention to the main questions and foci of the Study as discussed

above. For example, one paper might report on special practices of

helping beginning primary teachers learn mathematics for teaching.

Another might analyze how teachers in a particular setting work

together on studying student work in geometry, and use that systemat-

ically to improve their teaching of geometry. Invited are: research


reports; conceptual-analytic or theoretical papers grounded in

examples of practice; and descriptions, accompanied by evidence

appropriate to the claims of the paper. Camera-ready copy for inclu-

sion in the materials for the Conference is required. All submissions

should be in English, the language of the Study Conference, and

should use Times 14-point font. Please also write a 200-word abstract

that includes the main goal of your paper, demonstration, and work-

session, and what its main elements will comprise. Paper proposals

without abstracts will not be reviewed.

Demonstrations are sessions in which particular materials,

approaches, or practices will be shared, examined, and critically dis-

cussed. We encourage sessions that will make as vivid as possible

the materials, approaches, or practices to be demonstrated. Such

sessions may engage participants actively in examples; may use arti-

facts of practice, such as videotapes, examples of teachers’ work, or

actual materials. For example, if a group of teachers studies video-

tapes of their teaching, a session might be designed to provide Con-

ference participants with an opportunity to experience, firsthand,

what opportunities for learning this might offer, as well as what

some of the challenges might be. Proposals for demonstrations

should include the goals of the session, what will be demonstrated

and how it relates to the foci of the Study, a clear plan for the ses-

sion itself, capacity for participation in the session, and any special

requirements (technology, space, other) for the session. Proposals

for demonstrations should be no longer than 1200 words, or three

single-spaced pages, at most, and should additionally include a 500-

word summary of the approach or practice that will be demon-

strated, and what participants will do in the session. Proposals with-

out summaries will not be reviewed. This summary must be in

camera-ready form for inclusion in Conference materials, using

Times 14-point font. If artifacts are used, they must be made acces-

sible in English, the official language of the Study. Proposals for

demonstrations should make clear the theoretical foundations of the

practices to be demonstrated.

Interactive work-sessions are sessions in which a common problem

of mathematics teacher education will be worked on by a group of

researchers and practitioners attending the Conference. Proposals for

work-sessions should include a clear description of the topic to be

worked on, a clear explanation of the theoretical or conceptual issues

to be addressed, a detailed plan for the work-session, the artifacts or

materials that will be used to provide a context for the collective

work, and who will lead the session. For example, an interactive


worksession might be designed to center on how to assess teachers’

learning; another might be structured to engage participants in the

development of tasks that involve the use of mathematics in the

work of teaching. Proposals for work-sessions should be no longer

than 1200 words and three single-spaced pages at most, and should

additionally include a 500-word summary of the problem and how

the session will engage participants in work on the session. This sum-

mary must be in camera-ready form, with Times 14-point font, for

inclusion in the Conference materials. Proposals without summaries

will not be reviewed.

Proposals will be read and evaluated on the basis of the following

criteria: (a) clear links to the Study’s goals; (b) explicit fit with

Strand I or II; (c) clearly structured and written, with attention to

writing for others who may not share the same assumptions, experi-

ence, or knowledge; (d) attention in the design of the paper, demon-

stration, or interactive worksession to the cross-cultural nature of

the Study and the Conference. Successful proposals will be devel-

oped to be sensitive to the cross-cultural differences while also

designed to profit from those other differences; (e) potential to con-

tribute to the quality of the Study overall. This implies that some

very good proposals may not be accepted if they do not add in the

same way as others do to the overall scope and diversity of the

Study.

More details regarding formatting of proposals in all three catego-

ries will be available on the Study 15 website at http://www-per-

sonal.umich.edu/~dball/icmistudy15.html, which will be regularly

updated with information about the Study and the Study Confer-

ence.

STUDY TIMELINE

� Proposals for participation in the Study should reach the program

co-chairs (see below) by October 15, 2004.

� Proposals will be reviewed and decisions made about inclusion in

the Conference Program by November 20, 2004. Notifications about

these decisions will be sent by November 30, 2004 to all those who

submitted proposals.

� The Study Conference will be held in Brazil, from 15-21 May

2005.

� The Study Volume will be published by 2007, and a report of the

Study and its results will be made at ICME-11 in 2008.


INTERNATIONAL PROGRAMME COMMITTEE AND

CONTACTS

The study is co-chaired by Deborah Loewenberg Ball and Ruhama

Even. Their contact information is listed below. Please direct all inqui-

ries concerning this Study to both co-chairs.

The members of the International Programme Committee (IPC) are:

DEBORAH LOEWENBERG BALL

(CO-CHAIR IPC)

4119 School of Education610 E. University Ave.University of MichiganAnn Arbor, MI 48109-1259USAE-mail: [email protected]

RUHAMA EVEN

(CO-CHAIR IPC)

Department of Science TeachingWeizmann Institute of ScienceRehoboth 76100IsraelE-mail: [email protected]

JO BOALER

Stanford UniversityUSA

CHRIS BREEN

University of Cape TownSouth Africa

FREDERIC GOURDEAU

Universite LavalCanada

MARJA VAN DEN HEUVEL-

PANHUIZEN

Utrecht UniversityNetherlands

BARBARA JAWORSKI

Høgskolen i Agder (Agder Univer-sity College)Norway

GILAH LEDER

La Trobe UniversityAustralia

SHIQI LI

East China Normal UniversityChina

ROMULO LINS (CHAIR OF THE

LOCAL ORGANISING COMMITTEE)

State University ofSao Paulo at Rio ClaroBrazil

JOAO FILIPE MATOS

Universidade LisboaPortugal

HIROSHI MURATA

Naruto University of EducationJapan

JARMILA NOVOTNA

Charles UniversityCzech Republic

ALINE ROBERT

IUFM de VersaillesFrance


Ex-officio members:

BERNARD R. HODGSON

Secretary-General of ICMIUniversite LavalCanada

HYMAN BASS

President of ICMIUniversity of MichiganUSA


PETER SULLIVAN

EDITORIAL

SOME WAYS OF KNOWING MATHEMATICS AND SOME

IMPLICATIONS FOR TEACHER EDUCATION

An earlier JMTE editorial (Sullivan, 2003) considered the impor-

tance of prospective teachers learning the nature of the discipline they

would teach. The argument was put, using the case of science, that

many fields are too broad for teachers to learn beforehand all the

knowledge they will need. It was proposed that, on one hand, teachers

need to know how to find new knowledge for themselves, and on the

other hand, to be aware of the nature of the discipline of science. This

is particularly relevant when considering a trend, in Australia at least,

toward the teaching of contextualized science. I heard recently a report

of a teacher in a grape growing region building a curriculum on the

science of wine making. This can only work efficiently and effectively,

generally, if teachers can not only acquire, by themselves, the neces-

sary knowledge of the relevant science, but also can present this

knowledge so that students are not merely learning about wine grow-

ing but are learning about the fundamental processes, methods and

principles of science. Science teacher education might well emphasize

such issues. Similarly mathematics teachers need also to be able to

learn aspects of mathematics as well as appreciating the nature of the

discipline they are teaching. This issue of the journal addresses the

nature of the discipline of mathematics in a way that has obvious

implications for initial and continuing teacher education. Together the

contributors argue that mathematics is contestable, changeable, ideo-

logical, political, and integrally connected to our students’ affective

responses.

Of course, when the school mathematics curriculum emphasized the

procedures involved in arithmetic and algebraic calculations then

teachers had particular expectations about the mathematical results

and even the optimal methods. However the more we are interested in

the use of mathematics and the more we focus on the processes for

creating the mathematics, the less we can be certain that the products



are unique or optimal. This has implications for teaching practice.

Recently I was planning a trip from Paris to Glasgow, being aware

that I already had a non-refundable Paris to London ticket that I

could choose to use, and I had unlimited access to web-based informa-

tion and bookings. The problem requires various currency conversions

(both between Euros and Pounds and between each and Australian

Dollars, in my case), and some percentage work (it seems the web sys-

tems do not calculate the taxes and charges until the booking is made),

yet the real challenge is in the logistics (for example, how much does it

cost by bus from Heathrow to Luton airport, and how long does it

take?). As it happens, I told this story to a teaching colleague and she

decided to use it as the basis of a mathematics lesson, although using

a more local journey. The teacher subsequently reported that the les-

son degenerated into chaos, mainly because she used selected informa-

tion in posing the problem, and the students were unwilling to accept

the constraints imposed by the way the problem was posed and

wanted to solve a real problem using as much information as they

could gather. Among a range of issues this raises is that our confidence

in the solution is a function of the information used to pose a prob-

lem, assuming accuracy in calculation.

In this context, readers are invited to solve a problem in the arti-

cle by Irit Peled and Sarah Hershkovitz, Evolving research of mathe-

matics teacher educators: The case of non-standard issues in solving

standard problems. In the article, the authors discuss the use of

some pupils’ responses to a proportionality problem as a focus for

teacher development experiences. They report on rich experiences

that resulted from consideration of some unanticipated, but reason-

ably justified, results. I initially accepted what I understood to be

the constraints of the problem and calculated an answer. When I

first read the alternate responses considered in the article, I accepted

the perspectives as interesting but felt that the justification went

beyond the constraints of the problem. I mentioned this to a profes-

sional mathematician colleague who specializes in models of applied

situations. He argued that the response that I thought aberrant was

not only contextually acceptable but also mathematically acceptable

as well, claiming that it is only in schools that we expect solutions

to be restricted only to readily available information. In this way

mathematics becomes contestable and negotiable. The point is that

not only must teachers listen to students’ explanations of their inter-

pretation of situations and strategies used, but also be open to pos-

sible acceptability of unanticipated responses. It is an element of the

nature of mathematics.

296 PETER SULLIVAN

A further perspective on the nature of mathematics is evident in the

article by Rainy Cotti and Michael Schiro Connecting teacher beliefs to

the use of children’s literature in the teaching of mathematics. The arti-

cle examines the ideological potential of mathematics even, in this

case, for very young children through story books. They argue that

teachers can variously adopt either Scholar Academic, Social Efficiency,

Child Study or Social Reconstruction positions in their teaching, and

that the positions they adopt influence not only the choice of content

but their media and teaching approaches. Cotti and Schiro argue that

teacher educators need to assist teachers to appreciate their underlying

ideological positions and that such awareness will contribute to more

effective dialogue between teachers, especially those working and plan-

ning together. This political dimension of mathematics has been con-

sidered by Howson (1980), for example, who described examples in

Chinese mathematics textbooks, the effect of which was to criticize

capitalism, and in mathematics texts from Church schools in the early

19th Century that reinforced religiously bigotry of the era. Interest-

ingly, it is possible that more appreciation of the political potential of

mathematics may increase the interest in its study. There was an exam-

ple from the late 1980s at a time when there were people who pro-

posed mandatory AIDS testing. Using junior secondary level

mathematics it was possible to show that the rate of false positives

while low, was such that there would be many more people who would

be told they had AIDS but did not, than there would be people in the

overall population who did have AIDS. The Cotti and Schiro article

stresses ideological dimensions in approaches to teaching, but is also

relevant in considering the nature of the mathematics itself.

The article by Leone Burton ‘‘Confidence is everything’’ – Perspec-

tives of teachers and students on learning mathematics adds a further

dimension to this discussion. Burton examines upper secondary teach-

ers and students perspectives on confidence in learning mathematics.

Both groups argued that confidence is important. Interestingly even

though neither teachers nor students considered the meaning of the

term to be problematic, they each used the term confidence in quite dif-

ferent ways. While teachers saw confidence as either a prerequisite to,

or a product of success, it could be inferred from the students’

responses that confidence to them was more related to feeling that they

could understand, and that they would have the opportunity to

explore, perhaps with their peers, strategies, possibilities and alterna-

tives. In this, confidence, and perhaps other affective constructs,

changes from an independent affective variable to part of the essence

of mathematics and its teaching. This connection is also made by

297EDITORIAL

Kilpatrick, Swafford and Findell (2001) who described five dimensions

of mathematics – conceptual understanding, procedural fluency, strate-

gic competence, adaptive reasoning, and productive disposition – and

saw each of these as part of the content and nature of mathematics.

Together the three articles present a compelling case for explicit con-

sideration of the nature of mathematics as part of teacher education,

and that such consideration can include the fallibility of mathematics,

the complexity of contexts, openness to acceptance of various strate-

gies and results, the ideological potential of mathematics teaching, and

that there is more to mathematics than its skills and procedures.

Interestingly, the book Thinking mathematically: Integrating arithme-

tic and algebra in elementary school by Thomas Carpenter, Megan Loef

Franke and Linda Levi, that is reviewed by Alison Price, complements

these articles. Of course no one is suggesting that reconsidering the

degree of certainty of mathematical results detracts in any way from

its capacity to categorize, describe, predict, simplify and model aspects

of the world. In part, Price summarizes these authors as proposing

that teachers should be willing to go beyond merely inviting students

to articulate their idiosyncratic strategies and insights and seek to for-

malize those insights by examining general statements, modes of pre-

sentation, and interpretation of results.

Mathematics teaching is more challenging than it was, but it is

becoming more interesting. Likewise, mathematics teacher education.

REFERENCES

Howson, A.G. (1980). Socialist mathematics education: Does it exist? Education

Studies in Mathematics, 11(3), 271–285.

Kilpatrick, J. Swafford, J. & Findell, B. (2001). Adding it up: Helping children learn

mathematics. Washington, DC: National Academy Press.

Sullivan, P. (2003). Incorporating knowledge of, and beliefs about, mathematics into

teacher education. Journal of Mathematics Teacher Education, 6(4), 293–296.

298 PETER SULLIVAN

IRIT PELED and SARA HERSHKOVITZ

EVOLVING RESEARCH OF MATHEMATICS TEACHER

EDUCATORS: THE CASE OF NON-STANDARD ISSUES IN

SOLVING STANDARD PROBLEMS

ABSTRACT. This study describes the learning of researchers who engage in mathe-

matics teacher education as an integral part of their practice. As teacher educators

working with teachers on the subject of proportional reasoning, the authors reflected

on teachers’ solutions to a standard problem and analyzed answers that would con-

ventionally be considered incorrect. This exploration showed that some incorrect

answers made sense, were based on problem situation analysis, and brought atten-

tion to the fact that conventional formal answers were given without much delibera-

tion on their meaning in the situation. This insight prompted a second research

phase in which teachers discussed and explained alternative solutions, and developed

deeper analysis of problem situation in solutions that had been correct in the first

place. The importance of making teachers aware of the nature of alternative solu-

tions was further exhibited in a third research phase in which teachers evaluated chil-

dren’s answers to the same problem. The pedagogical insight that emerged stressed

the importance of making teachers aware of the tension between an almost auto-

matic application of a mathematical model, and of analyzing problem situation dur-

ing problem solving. In addition, the researchers developed better understanding of

the mathematical challenge associated with the proportional reasoning problem, a

stronger awareness of the role of sensitivity to their learners (the teachers), and of

the role of reflection.

KEY WORDS: learning through teaching, problem situation, problem solving, pro-

portional reasoning, teacher education

This article is written from our perspective as researchers who are also

teacher educators, describing how our knowledge grew as we engaged

concurrently in both teaching and research. It is composed of two lay-

ers: a meta-level analysis that examines the sequence of three research

phases and our decisions and planning of each new research phase,

building on the results of the preceding phase, and a specific analysis

that presents the research results of each phase. Our meta-level analy-

sis involves a description of conditions that created a learning oppor-

tunity for the researcher-educator, while the specific analysis describes

what pedagogical knowledge was learned by the researchers and by

the participating teachers.



BACKGROUND AND FRAMEWORK

The Learning of a Mathematics Teacher Educator

The mathematics educator is often described as being in a constant

learning mode, to the extent that an elderly relative of ours used to

remark:

You learn and you learn – when will you know?

Teachers’ and teacher educators’ learning occurs as an ongoing inte-

gral part of their practice and through their reflection on and in their

practice. However, in order to learn from practice, they need to be

actively engaged in a task that promotes reflection, or be aware of the

importance of reflecting on their own teaching and, as a result, acti-

vate a reflective mode. Models of the nature of teacher reflection were

suggested in many works (e.g., Lerman, 2001; Steinbring, 1998). Other

works stress the general importance of reflection in action and on

action (Schon, 1983) and the role of awareness (Mason, 1998) and

thus suggest that teacher learning models also apply to the learning of

teacher educators. Jaworski (1994) gives a reflective account of her

learning as a teacher educator investigating teaching through class-

room observations, showing how these models help her analyze her

own learning process (e.g., p. 73).

Based on the works mentioned above and on the model developed

by Zaslavsky and Leikin (2004), Zaslavsky, Chapman and Leikin

(2003) propose a framework for the professional growth of mathemat-

ics educators, emphasizing the crucial role of reflection in the ongoing

learning process of mathematics educators. Their model has three

embedded layers showing three facilitator–learner cycles. In each cycle

a task is assigned by an educator (facilitator) to a learner. In the inner

layer, the facilitator is the mathematics teacher and the learners are

her students. In the following layer, the facilitator is a mathematics

teacher educator and the learners are mathematics teachers, while in

the third layer, the facilitator is a mathematics teacher-educator educa-

tor and the learners are mathematics teacher educators.

Focusing on the learning participants’ growth, Zaslavsky et al.

(2003) distinguish between direct and indirect learning. Direct learning

occurs through reflection of the learners on their solutions to tasks

and indirect learning occurs through the facilitator’s observations of

the learners’ action and her reflection on them. Thus, when teachers

participate as learners in in-service programs, they are often assigned

tasks that facilitate more direct learning processes. However, teachers

300 IRIT PELED AND SARA HERSHKOVITZ

and teacher educators learn indirectly from their own teaching when

they design and implement tasks for their students.

Although there are many similarities between the learning of teachers

and teacher educators, the teacher educator bears, as suggested by Ja-

worski (2001), greater responsibility for connecting theory and practice.

Therefore, the teacher educator has to make a stronger effort to learn

and develop. Jaworski (2001) also details the nature of the teacher edu-

cator action as facilitating the connection between theory and practice

by developing effective activities that, in turn, promote teachers’ ability

to create effective mathematical activities for her own students.

In this study, we, as teacher educators who are also researchers,

assume an even greater responsibility for learning, in the sense that we

not only engage in interpreting research, but also in doing research

and increasing the body of existing knowledge. In reflecting on our

research and analyzing its phases, we use the Zaslavsky et al. (2003)

model to describe how our learning as teacher educators and research-

ers grew in the course of this study.

SPECIFIC RESEARCH TOPIC: TENSION

IN PROBLEM SOLVING

With the strong shift towards learning with understanding, teacher

educators aim at helping teachers adapt new goals and different class-

room norms for problem solving activities. In spite of years of efforts

to make problem solving more meaningful through a reform in mathe-

matics education goals (NCTM, 1989), different studies present evi-

dence that meaningless solutions still persist. In a collection of articles

Greer (1997), Reusser and Stebler (1997), Yoshida et al. (1997), and

Verschaffel, De Corte and Burghart (1997), show that children and

pre-service teachers do not use realistic considerations in solving a spe-

cial set of word problems. For example, when they are given a run-

ner’s best result in the 100 ms, they use multiplicative relations to find

his time on a much longer track without casting doubt on the implicit

assumption that he can keep up the same speed.

Another (frequently quoted) example for meaningless problem solving

behavior is the ‘bussed soldiers’ problem used in a national assessment

test (Carpenter, Lindquist, Matthew and Silver, 1983). In this problem,

children are asked to calculate the number of buses needed for transport-

ing a given number of soldiers, and most children simply divide the num-

bers, disregarding the fact that the answer should be a whole number.

301EVOLVING RESEARCH OF MATHEMATICS TEACHER EDUCATORS

According to Silver, Shapiro, and Deutsch (1993), children find it difficult

to make sense of the remainder in a division problem and to connect

problem situations with their mathematical knowledge.

Children’s behavior in solving these different problems is also

explained in the light of existing classroom norms (Gravenmeijer, 1997;

Hatano, 1997), suggesting that a norm change could help. This explana-

tion was indeed evident in a teaching experiment conducted by Verschaf-

fel and De Corte (1997). Another focus is on changing the traditionally

sterile nature of the problems (Greer, 1993; Nesher, 1980). With this idea

in mind, DeFranco & Curcio (1997) asked children to solve the bus

problem meaningfully by adding a request to call the bus company and

order the buses. However, an issue that emerges following such changes

is whether children are still engaged in mathematical thinking or just use

pragmatic considerations (Cheng and Holyoak, 1985).

With such focus on realistic considerations we might forget that one

of the main goals of problem solving is to give children tools with

which they can mathematize (i.e., apply a mathematical model) situa-

tions, perceiving them through mathematical lenses and focusing on

problem structure (Greer, 1993, 1997).

It follows that, on the one hand, we would like teachers as well as

children to use realistic considerations yet, on the other hand, we want

them to identify the structure of the mathematical problem. The first

goal focuses on the special constraints of the situation, while the sec-

ond implies ignoring the special context features. Thus there is poten-

tial for creating goal tension and conveying conflicting messages.

Indeed, in a retrospective article on their research, Verschaffel, Greer

and De Corte, (2002) recognize this conflict and suggest re-negotiating

the didactical contract and making it explicit, so that children know

how much reality and what degree of precision they should bring into

consideration when modeling a described situation (p. 273).

It is the teacher educator’s task to help teachers change the class-

room environment in a way that promotes meaningful problem solv-

ing. In this work, we deal with a more complex challenge of trying to

work with teachers on problems they correctly consider as standard

(routine) and that involve a straightforward application of a mathe-

matical model. We show the possible differences between teachers and

children in their conception of such a problem and conclude by sug-

gesting that teacher education programs should make teachers aware

of these tensions and more sensitive to their students.

The study was conducted in the course of our work with different

groups of pre-service and in-service teachers. It is a common practice

for us to investigate an issue in one group and then to reflect on our


action, design new tasks and to try them in another group. In this

sense, the study is an example of our typical practice. This paper pre-

sents an account of the insights we developed in a sequence of evolving

phases examining the solutions and the attitudes to alternative solu-

tions of a standard problem with a number of different groups of pre-

service teachers and in-service teachers.

METHOD

The Tasks

The Mathematical Problem

The central mathematical task in this study is a task we composed and

named The School-Bus Problem. By assigning it to teachers and with

appropriate discussions, we later turned it into a pedagogical task.

The School-Bus Problem: When the sixth grade classes went on a trip, one of thetwo buses broke down and the children had to crowd on the seats of the second

bus. The children were distributed in the bus in equal crowdedness1. Three chil-dren were sitting on every 2 seats. The back-seat of that bus had 4 seats. Howmany children were sitting on it?

The density or, in this case, crowdedness constraint requires that the

ratio between children and seats is kept the same. The answer, 6 children

on the back-seat (4 seats), can be calculated in several ways such as using

a doubling strategy (the back seat can be composed of two pairs of seats,

so double the number of children, or: twice the number of seats, so dou-

ble the number of children) or solving the equation 3/2 = x/4. Now that

we have figured the answer and are convinced that it is the right answer,

what do we say to the solution presented in Figure 1 that concludes that

the answer is 7 children?

A solution of this kind puzzled us, just as it might have puzzled the

reader, and prompted further investigation described in this article.

How It All Began

In the course of analyzing word problems used in teaching propor-

tional reasoning, we asked teachers and pre-service teachers to solve

several word problems. In addition to a variety of proportional rea-

soning problems, the task included the missing-value School-Bus prob-

lem (requiring a computation of a certain value) and also the

following equivalence version (asking whether the ratios are equal) of

a similar school-bus situation:


When the sixth grade classes went on a trip one of the buses broke down and thechildren had to crowd on the seats of the two other buses. In the first bus 4 childrenwere sitting on every 3 seats. In the second bus 3 children were sitting on every 2seats. Were the buses equally crowded? If not, specify which bus was more crowded?

Participants and Setting

The nature of the solutions of the School-Bus problem triggered a

study that consisted of a sequence of phases in which we examined

teachers’ and students’ answers to the problems and elicited reflective

discussions. Ultimately, the study consisted of three different settings

with different groups of participants:

First Phase: Pre-service and in-service teachers were given a set of

proportional reasoning problems including either the School-Bus prob-

lem or the equivalence School-Bus problem.

Second Phase: Pre-service and in-service teachers were asked to solve

the School-Bus problem (missing value) and discuss their solutions. A

solution (suggested by teachers in the first phase) using situation con-

siderations and not applying a proportion model was ready to be used

as a discussion trigger, in the event that no teacher suggested a solu-

tion of this type. The pre-service teachers were given an assignment

that asked the teachers to interview children on that same task.

Third Phase: Using the children’s answers from the interviews, ele-

mentary school and secondary school teachers were asked to solve sev-

eral problems, including the School-Bus problem, and then to evaluate

a variety of the children’s answers.

In each phase, we examined the teachers’ answers and/or their reflec-

tions on alternative solutions provided by colleagues and by children.

We observed how they became aware of issues they had not noticed

before, and how their views evolved as they discussed and reflected on

these different solution approaches. Most of the data were collected

through written tasks. Teachers’ individual solutions and reflections

Figure 1. Is this a good alternative solution for the School-Bus Problem?


were given in writing and collected. Detailed notes were taken of class

discussions that followed teachers’ individual work, and all the draw-

ings that were done during these discussions were also collected.

RESULTS

Phase 1: A Simple Challenge Getting Complex or Incorrect Answers

that Make Sense

A few unexpected answers given by some teachers in the following two

groups triggered this entire study. One was a group of pre-service sec-

ondary school special education teachers (n=14), that participated in

a course on children’s psychological development of mathematical con-

cepts. The other was a group of elementary school teachers (n=28)

that participated in a weekly mathematics workshop over the course

of the entire school year. These teachers were given a set of ratio and

proportion problems with the purpose of investigating (and later dis-

cussing) the effect of context. One of the problems was either the miss-

ing value School-Bus problem or the equivalence problem.

Specific Analysis

The problems were given and were answered in a written form. The col-

lected answers were categorized according to the mathematical model

used in the process of solving the problems. It was found that 8 out of

14 (57%) pre-service teachers and 23 out of 28 (82%) teachers used pro-

portional or multiplicative reasoning and solved the School-Bus prob-

lem or the missing value problem correctly. The remaining teachers

answered incorrectly. A few of the solutions caught our attention

because they were more detailed than others and were accompanied by

some drawings of the situation. One of the teachers’ more detailed solu-

tions is presented in Figure 1, and two of the pre-service teachers’ solu-

tions that were accompanied by drawings are presented in Figure 2.

These figures include two solutions to the School-Bus problem that

result with the answer 7 children. They differ in the way the seats are

drawn, either with or without spaces between the seats. The implicit

problem assumption is that there are no spaces (otherwise, the propor-

tion model does not hold). This point has to be declared explicitly when

solvers assume otherwise, or when a question about it is raised.

Meta-level Analysis

Our original interest in the distribution of answers related to investi-

gating which context creates more difficulty in applying proportional


reasoning. In analyzing the solutions, our first reaction was to

regard as incorrect all answers that did not yield the proportional

reasoning answer (6 children). However, drawings and detailed

explanations in some of these incorrect answers were quite convinc-

ing and we could not give an immediate counter-argument or pin-

point their flaws.

This situation was astonishing. An answer (7 children as in Figure 1)

that we were sure was wrong suddenly seemed to make sense and we

wondered how teachers would react to this solution. We assumed (and

later verified our assumption) that solutions of this kind could also be

suggested by children, and that teachers might be dismissing them

without much deliberation. At this point, we decided to deviate from

our original research and pedagogical plans to expose teachers to diffi-

culties in proportional reasoning problems and to investigate this new

issue with some other groups of teachers.

Using Zaslavsky et al.’s (2003) model, we can say that we experi-

enced indirect learning of the challenge we presented to our students

(the teachers). We observed how they responded to the task and

started to adjust our teaching. These observations facilitated our

understanding of the mathematical task, making us realize that the

mathematical challenge was more complex than we anticipated. In this

role as facilitators, we practiced sensitivity to our learners, becoming

aware of possible sensible alternative answers. Thus, our learning con-

cerned the mathematical challenge and sensitivity to students, two of

the three elements in Jaworski’s teaching triad (1994) describing the

components of the act of teaching.

Figure 2. Examples of pre-service teachers’ solutions to the School-Bus problem using

drawings (explicit situation analysis).


As teacher educators and researchers, we decided that we would like

to learn more about our new insights and whether teachers can benefit

from them. Therefore, in the succeeding phase we designed tasks in

which teachers would discuss alternative problem solutions.

Phase 2: Deeper Analysis Evoked by Alternative Solutions

Following the previous experience, the School-Bus problem was pre-

sented and discussed in several teacher workshops. In all these work-

shops, most teachers solved the problem formally, using proportional

reasoning. Moreover, in most of these, at least one teacher suggested a

different solution, frequently the solution presented in Figure 1 (7 chil-

dren). If that did not occur naturally, we presented this solution as

‘‘someone’s answer’’ and asked for reactions.

Three examples of typical discussions follow. The first two examples

involve two groups of elementary school teachers who participated in a

56-h summer workshop. The teachers were asked to solve two problems

involving density, one being the School-Bus problem (missing value).

Example 1: Difficulties in Representing the Situation

The group consisted of 31 teachers. Most teachers used proportional

reasoning. During their work, the instructor observed that one teacher,

Leona, made a drawing of the seating arrangement. She drew the two

pairs of seats one next to the other, made check marks to mark the

children, added a child on the space between the two pairs of seats,

and concluded that the total number of children was 7.

Leona was asked to present her answer to the group. Many of the

teachers thought that her answer was wrong. When asked to explain

their assertion they said ‘‘It’s wrong because this is a ratio problem’’.

Challenged to come up with a more convincing argument, one of the

teachers explained: ‘‘It says that they have the same density, which

means the same ratio’’.

The instructor asked the teachers to explain what density meant in

this case, and to try to draw the seating as it would have looked,

according to their (ratio) solution. This turned out to be a very diffi-

cult task. It was only with a combined group effort that a final

detailed drawing was made. At first, one teacher made a general draw-

ing as depicted in Figure 3a (this drawing would have actually been

fine, had it been accompanied with an explanation of the average

amount of seat that each child gets). Following a discussion that lead


the teachers to specify that density in this case meant 2/3 of a seat per

child (the question whether it was an exact value per child or an aver-

age did not come up), they tried to figure out how to draw the seat

assignment accordingly. Their chosen division strategy was quotition-

ing, knowing that each part is 2/3 of a seat. In spite of the difficulty

they encountered in implementing this action, no one suggested divid-

ing the 4 seats into 6 equal parts (although they knew the answer).

They made some unsuccessful trials, e.g., tried to mark intervals of 2/3

of a seat, but used inaccurate intervals ending up with the fifth and

sixth part beyond the given 4 seats (Figure 3b). Following a coopera-

tive effort, Natalie succeeded in dividing the 4 seats into parts of 2/3

seat per child (Figure 3c).

At the end of this exchange, Natalie reflected: ‘‘I could have a child

in class that would answer like that (Leona), and then I would have to

understand what he says so that I could relate to it’’. The session con-

tinued with a discussion elaborating Natalie’s point on the importance

of understanding how children perceive the situation.

Example 2: The Convincing Power of an Alternative Solution

The group consisted of 35 elementary school teachers. As described in

the previous example, most teachers used proportional reasoning (mainly

a doubling strategy) and produced the answer of 6 children. Sharon drew

two ‘‘groups’’ each representing 3 children getting 6 altogether, but after

some deliberation she added a 7th child, as shown in Figure 4.

Rachel responds: ‘‘That’s wrong. The kid who sits in the middle [at

the end of each pair of seats towards the middle] has now [when there

are 7] two kids squeezing him. In the case of the 2 [seats with 3 chil-

dren] they only squeeze him from one side’’.

The arrows in Figure 4 show what Rachel meant (The arrows were

added because her answer was quite hard to understand, and were not

(b)

(c)

(a) 3 children 3 children

1 2 3

1 2 3 1 2 3

4 (5) (6)

Figure 3. Steps in collective group efforts to construct a seat assignment.


a part of Sharon’s drawing). Some other teachers reacted, but no one

elaborated on why the most common answer, 6, was the right answer.

The instructor asked them how they envisioned the seating. Most

teachers did not make any drawing.

Rachel deliberated for a long time and then came up with the draw-

ing shown in Figure 5a. Sharon made a new version of her drawing,

as shown in Figure 5b.

The two were asked to copy their drawings onto transparencies and

present them to the group. The instructor asked for a vote on these

two options. Five teachers (Sharon and four other teachers who had

answered 6 earlier) voted for Sharon’s answer (7 children), and 12

voted for Rachel’s answer (6 children), with the remaining 18 teachers

abstaining. The discussion continued but no teacher came up with a

convincing argument one way or the other.

Example 3: Difficulties Envisioning the Meaning of Density

in the Situation

The participants were 10 pre-service secondary school math teachers

taking a course on the didactical and psychological aspects of teaching

mathematics as a part of their teacher certification requirements.

In solving the School-Bus problem, one student said that he was

deliberating between the answers 6 or 7 children. Another student,

(a)

(b)

Figure 4. Sharon’s initial drawing (arrows show Rachel’s claim).

(a) Rachel’s drawing:

(b) Sharon’s drawing:

Figure 5. Two alternative teacher drawings of bus seating.


Holly, agreed and offered explanations for the two answers: Holly: ‘‘It

can be 6 because if on each 2 [seats] they put 3 [kids], then 3 plus 3 is

6. But it is also possible they put 7 kids [on 4 seats] (presented in

Figure 6), because we can think of the 4 seats as 3 pairs of seats

(points at what she means by pairs, as shown in Figure 6). On each

pair you seat three [getting 9 temporarily], and if you take out the

doubles [2 kids that were counted twice] you get 7 kids’’.

Other students claimed that Holly’s answer was incorrect, ‘‘because

if you use a seat twice then the density changes’’. One of the students

suggested amending the situation by arranging the seats in a circle …The class concluded that they needed a clear definition of density. One

student suggested ‘‘I would define it as the ratio. Two seats for 3 chil-

dren, the ratio is two to three’’. They continued by constructing a

mapping table:

Seats Children

2 3

4 ?

In the table they could see that the ratio between the lines is 2, and

someone said: ‘‘The number of seats increased times two, so the num-

ber of children should increase similarly’’. At that point, the instructor

elicited the following discussion:

Instructor: Can we also look at the ratio between seats and children?

It is [going from the number of seats to number of chil-

dren] times 1½, so could we also use it with the 4 [seats]?

What would it mean? We tend to do it automatically. You

mentioned that we should think logically. Children [who

solve this problem] are thinking logically and use drawings.

Abby: There’s a length given [for the seats] and I divide it into

the number of children and see what width each child gets.

Figure 6. A pre-service teacher’s solution (Holly).


Ron: But then you get half a person.

Abby: But we do have 3 children on 2 seats, so no one is getting a

whole seat. It’s like in real life when someone is sitting with

you [on the same seat].

Instructor: Can we define the density here as the width [of a seat] per

person?

Ron: But then we again get the ratio [that we discussed earlier].

Instructor: If each child gets 2/3 of a seat, then when we have 4 seats

and we want to keep the same density, how many children

should we put there?

Class in unison: Six.

It should be noted that the above discussion took place in a group

of pre-service secondary school teachers who were also third year math

students. They, nonetheless, had to struggle with the meaning of den-

sity, especially with its specific meaning in the given situation. Some

attempts to analyze the situation, such as putting the seats in a circle,

seem pseudo-realistic.

Specific Analysis

Most teachers solved the School-Bus problem using multiplicative or

proportional reasoning. Yet, in most groups, at least one teacher sug-

gested the solution with the ‘‘between the seats’’ arrangement that pro-

duces the answer ‘‘7 children’’. In class discussions, the main argument

for the proportional reasoning answer (6 children) is circular: ‘‘because

this is a ratio and proportion problem’’.

Meta-level Analysis

Our own understanding of the mathematical challenge, the School-Bus

problem, grew as we conducted more discussion groups. In some of

the first group discussions, as in the case of example 2, we ourselves

were not entirely sure how to react to the 7-children solution. As a

result, we let some of the groups discussions end on an inconclusive

note.

With further discussions, several ideas came up. For example:

(1) The meaning of ratio as the amount of seats per child came up in

earlier arguments following our request to try to draw the seating

arrangement. Example 1 details the group effort to figure out the

number of children on 4 seats given that each child gets 2/3 of a

seat. Still, it is only in further group discussions that this idea was

used to contradict the 7-children solution using the following

argument: Since 7 children on 4 seats end up with less seats per


child than 3 children on 2 seats, then the answer 7 has to be

incorrect.

(2) In one of the group discussions on the equal-distance idea that is

suggested by the 7-children solution, some teachers claimed that it

is irrelevant because it is unrealistic. In reality, children are not

small dots, and therefore this solution is incorrect. This prompted

us to ask: What if the problem was about tiny ants (the size of these

dots)? This question led to our own understanding that the density

criterion would still require that each ant should get the same

amount of seat, and that although the equal-distance criterion

makes sense and seems fair, it is not the conventional criterion for

density.

Why was it so difficult for the teachers (and for ourselves) to come

up with a reasonable argument that would, on one hand, support the

application of a proportional reasoning model, and on the other hand,

show the flaws in applying any other mathematical model?

We found a partial answer to this question through deeper analysis

of the proportional reasoning answers. The data available at that point

included the first phase groups and the group discussions. Analysis of

the first phase data showed that 93% (26 out of 28) of the teachers

and 86% (12 out of 14) of the pre-service teachers either gave a formal

solution without any explanations or a solution with minor explana-

tions. Only 7% of the teachers and 14% of the pre-service teachers

analyzed the situation explicitly. A very similar picture emerged during

class discussions. Most teachers immediately wrote formal expressions

without any deliberations, explanations or drawings. Drawings

appeared in the initial solution mainly in cases that offered a non-pro-

portional reasoning solution. When teachers were explicitly encouraged

to represent the situation, they found it to be a very difficult task, and

only by group cooperative efforts did they manage to deal with it.

The nature of our learning in this phase was different from the first

phase. Using Zaslavsky et al.’s model (2003) we can say that it was

more direct. Since we were interested in the effect of teachers’ discus-

sion of alternative solutions and also in learning more about the task

(the mathematical challenge) itself, then, in a way, as researchers we

‘‘gave’’ a research task to ourselves as teacher educators. Thus, our

intentional reflection on the conduct and the effect of the discussions

produced direct learning on the benefits of the pedagogical activity of

the teacher educator.

Following the different discussions among teachers and pre-service

teachers, and noting the difficulty they had in relating to the problem


situation, we wondered how teachers would react to actual children’s

answers and whether their reaction would depend on their own

answers. We expected to get a variety of answers and were interested in

exploring teachers’ reactions to the different answers. We decided to

construct this set by asking pre-service teachers to interview children,

and after acquiring a repertoire of answers, we planned asking teachers

to evaluate the children’s answers.

Children’s Answers

Following class discussions, we asked the participating pre-service sec-

ondary school teachers to collect the students’ answers to the missing

value version of the School-Bus problem. Since the purpose was to

assemble a repertoire of answers and make teachers aware of them, no

special sampling was carried out and the students were free to inter-

view children at any grade and ability level. Most pre-service teachers

interviewed high school students (mainly tenth graders), others inter-

viewed middle school students (seventh and eighth graders), so the

grade range was from seventh to twelveth grade, with a total of 22 stu-

dents. Some answers were similar to the teachers’ answers, some were

more sophisticated and others were more realistic. Figure 7 presents

(through typical examples) the different types of answers. Most

answers came from a number of children in different grades.

The first four answers use formal mathematical thinking with or

without partial situational analysis. The last three answers use mainly

a situation analysis. The formal answers use a variety of mathematical

models: additive, multiplicative, and proportional reasoning.

Phase 3: Evaluating Children’s Answers

With the repertoire of children’s answers, some of which matched the

original answers given by teachers and student teachers, the next step

was to study teachers’ attitudes towards this variety of answers.

Two groups of in-service teachers participated in this phase: elemen-

tary school teachers and secondary school teachers. As a part of an

in-service teachers workshop the participants received a written ques-

tionnaire in which they were asked to (individually) solve some prob-

lems including the School-Bus problem (missing value), and then

relate to the children’s answers. They were asked to mark the chil-

dren’s answers as correct or incorrect and evaluate the explanations as

good, average or poor. The questionnaire included all types of chil-

dren’s answers that appear in Figure 7.


Since there were no formal teacher answers other than proportional

reasoning solutions (e.g., no additive solutions), the teachers’ answers

were categorized into two groups: those that used proportional reason-

ing (PR), and those that used situational analysis (SA). The PR category

was further split into the various strategies (this categorization would

help us later in explaining some of the evaluations).

Answers Answer features

1 On a pair of seats three children are sitting,

that’s one more. So on four seats five children

will sit [one more than 4].

Additive reasoning

2 Four seats are composed of two pairs of seats. So

we have 3 kids plus 3 kids that’s 6.

A doubling strategy

3 The answer is 6 children, because 6/4 is equal to

3/2. It’s the same ratio.

Proportional reasoning

4 Each kid on a pair of seats actually has two-

thirds of a seat, so if we take the 4 seats and

divide by 2/3 we get the number of children

(divides) that’s 6.

(note: This exceptionally good and insightful

answer was given by a gifted 7th grader)

Proportional reasoning

with meaning of the ratio

(seats per child)

5 In order to seat them in the same density, we

need to know if the kids are fat or thin. I would

say that on four seats we can put between 4 and 8

kids, depending on the kids, as I explained

earlier, in order for them to be as crowded as

those on the pair of seats.

Use of realistic

considerations

6 I mark each kid on a pair of seats this way:

and now I combine two such seats, so there’s

between them a place for another child, so we get

that it would be right to put 7 kids there:

Combination of a

doubling strategy with an

unconventional (equal-

distance) definition of

equal crowdedness.

7 The back seat of a bus has 5 seats. Use of personal

situational knowledge.

Figure 7. A repertoire of children’s solutions to the school bus problem.


As seen in Table I, most teachers (94% of elementary school teach-

ers and 82% of high school teachers) used PR. They used different

strategies or arguments in their calculations. The most frequent answer

in both groups was the doubling strategy. Some of the teachers formu-

lated this explanation as a mapping rule: In each pair of seats there are

3 children, in four seats there are 6 children. Because of their similarity,

the mapping rule explanation was categorized together with the dou-

bling strategy. Some teachers related in their solutions to the situation.

Next, we explored the possible relationship between teacher answers

and their reactions to student answers. Our hypothesis was that teach-

ers would reject students’ answers that did not fit with their way of

solving the problem. Specifically, we thought that teachers who solved

the problem formally would not accept (as correct) solutions that used

situation analysis. In addition to that, the children’s repertoire included

a new type of situational analysis, an answer claiming that it would be

fair to consider children’s weight in assigning them to seats. This realis-

tic consideration is close in nature to the realistic considerations men-

tioned in the introduction. We expected teachers to treat it differently,

perhaps more negatively, than they would treat the 7-children answer.

Table II depicts the percentage of teachers who expressed a positive

attitude towards a certain answer accepting it as a correct answer, or

towards an explanation accepting the explanation as an average or a

good explanation. Because the elementary school teachers and the high

school teachers reacted similarly, we merged the results of the two

groups. The evaluations are presented separately for teachers who gave

a PR answer and teachers who used SA (except for one case, all the

rest gave the 7-children answer).

Because the groups are small (especially the group of teachers that

gave a situational answer), we refer to the percentages without com-

paring them statistically. The results should be regarded as an example

of teachers’ reactions, not necessarily representing a general trend.

Specific Analysis: Teacher Answer Effect on the Evaluation

In general, teachers’ reactions were quite as expected, showing an

interaction effect between teachers’ answers and their evaluation. In

looking at reactions to PR answers, it was noted that, while most of

the PR teachers evaluated PR answers as correct, only about a third

of the SA teachers accepted PR answers as correct.

The opposite ‘‘order of attitude’’ can be observed when we look at

the reactions to SA answers: Larger percentages of SA teachers evalu-

ated the SA 7-children and weight (fat-thin) answers as correct answers

relative to the PR teachers’ evaluations.


TABLE

I

Teachers’Answ

erDistribution

Teachers

Situationalanalysis

Proportionalreasoning(A

nsw

er:6children)

Fat/thin

Inbetween

Answ

er:7

Doubling

2/3=

4/6

2/3

seat

Noexplanation

Elementary

schoolteachers(n=

18)

–1

13

11

2

6%

72%

6%

6%

11%

117

6%

94%

Highschoolteachers(n=

39)

16

14

11

43

3%

15%

36%

28%

10%

8%

732

18%

82%


Better Evaluation for 7-Children than for Weight

Larger percentages in both PR and SA groups evaluated the answer

7-children as a correct answer than the weight answer. Similarly, the

explanation for 7-children was more positively regarded than the

explanation for the weight answer.

Looking at the picture in general, we can see that in a majority of

cases the children’s explanations were viewed positively, even when

the answers were considered to be wrong. However, the weight

answer generated a wider variety of reactions. Some teachers reacted

favorably while others thought that a child that gives such an answer

is actually trying to evade dealing with the problem or acts mischie-

vously.

The following are positive comments on the 7-children explanation:

– Here is an interesting idea, another point of view of the situation.

However, it looks like the density is not the same.

– It is a nice drawing, perhaps it is right.

– It is difficult to argue that this child is not right

– I did not think about it but he is actually right. It is possible to put

another child between the two seats. Although there is no mathemati-

cal calculation (except counting), it is still a nice answer.

Positive reactions to the weight explanation:

– Although such an answer is not considered to be correct, it makes

sense.

– He is a creative thinker and takes into account sensible data that is

not given in the problem

– It is indeed important to take into account the children’s size.

Negative reactions to the weight explanation:

– There aren’t so many heavy children. If there is one, he can be

ignored.

– This child just doesn’t want to think.

Unexpected Reactions

Somewhat unexpected was the fact that a few of the PR elementary

school teachers evaluated a PR answer as wrong. As it turns out, these

were cases in which the teachers used a doubling strategy and the chil-

dren’s proportional reasoning was more sophisticated. For instance,

one of these teachers accepted the answer ‘‘6 children’’ but did not

accept the child’s explanation: 3/2 = 6/4. Another teacher did not like

the explanation each child actually has 2/3 of a seat.


TABLE

II

PercentageofTeachersWhoUse

ProportionalReasoningthatGaveaPositiveEvaluationofDifferentTypes

ofChildren’sAnsw

ersand

Explanations

Students’answ

ers

Teachers’answ

ers

7children

Fat/thin

Additive

Proportion

Answ

erExplanation

Answ

erExplanation

Answ

erExplanation

Answ

erExplanation

Proportion

21

40

13

30

327

46

45

n=

49

43%

82%

27%

61%

6%

55%

94%

92%

Situation

77

56

14

35

n=

888%

88%

62%

75%

13%

50%

38%

62%


There were also surprises in the opposite direction, such as the case

of a high school teacher who favored a child’s explanation that used a

higher level of PR than her own. She remarked ‘‘Great answer, nice

work, this child even calculated the approximate part of a seat each child

may get. It’s correct and very impressive’’.

Negative Evaluation of the Additive Answer

Most of the teachers did not accept the additive reasoning answer (5

children on 4 seats) although about half of them thought that there was

some sense in the explanation. This was the least favorable answer.

Meta-Level Analysis

Both situational answers caused some doubt in teachers’ minds,

although seemingly of a different nature. Unlike the 7-children solu-

tion, that is wrong because it uses an incorrect density criterion (equal

distance instead of equal amount of people per seat or seats per per-

son), the realistic answer taking the children’s weight into account

actually claims implicitly that the conventional criterion is unfair, and

suggests using a more subjective crowdedness criterion. As teacher

educators, we need to create new tasks that will involve teachers in

further analysis of the nature of these answers.

Our learning in this phase can be viewed as a continuation of the

previous phase. In our role as researchers posing a research question,

we facilitated our learning as teacher educators, making our own

learning more direct.

DISCUSSION

We embarked on this study almost accidentally. A Hebrew expression,

based on the Biblical story of Samson, suggests that sometimes a fierce

encounter can turn sweet. Similarly, in our story, teachers’ answers

that were considered incorrect led to an interesting investigation with

some pedagogical gains.

In the first part of this discussion, we analyze how we learned and

what we learned about the nature of educator-researcher learning pro-

cess. In the second part, we discuss what we learned about the investi-

gated pedagogical issue.

How We Learned: the Learning of a Researcher Teacher Educator

Our attention in the original study was drawn to some solutions employed by

teachers in the course of solving a mathematical proportional reasoning


problem. These solutions were considered by us as incorrect, yet they

were accompanied by convincing arguments. Becoming aware of our

doubt in evaluating the answers, we kept on investigating and learning

about the nature of these answers by designing tasks that focused on

this new issue and trying them with different groups of teachers and

pre-service teachers.

As presented in Figure 8, the study can be characterized by shifting

the research focus from the original topic, effect of context in propor-

tional reasoning problems, to a new topic. Another shift occurred within

the new topic, formal and situational answers, in moving from an unin-

tentional research phase to a planned research phase (involving a move

from indirect learning to direct learning, as discussed further on).

Our learning began with being sensitive to our students, teachers

and pre-service teachers that participated in our original research. Yet

it was only later, while reflecting on our actions and employing a

meta-level analysis of the study, that we became aware of the crucial

role of this sensitivity. This role was recognized by Jaworski (1994),

who analyzed the act of teaching, representing it by a triad that

includes sensitivity to students as one of its three components.

In an effort to observe these changes in a more general theoretical

framework, we use an analysis of the issues and operations of mathe-

matics educators suggested by Jaworski (1999, 2001), and a model that

describes educator-learner relations and growth of knowledge devel-

oped by Zaslavsky et al. (2003).

Jaworski (1999, 2001) details three levels of action and reflection for

teachers and teacher educators that include: (1) Engaging in prepara-

tion of effective mathematical activities for students. (2) Engaging in

the action of teaching and making decisions about ways of teaching.

(3) Helping teachers develop expertise in preparing mathematical tasks

and making pedagogical decisions. Jaworski (2001) sees the teacher

educator as operating at all levels, since her operating at Level 3 actu-

ally involves dealing directly with issues at Level 2, which requires atten-

tion at Level 1 (p. 302). This view is realized in the following analysis

of our operation.

Reflecting on our action in this study, we can characterize it as deal-

ing with all different levels. At first, when we inquired about teachers’

proportional reasoning knowledge and gave them a mathematical

problem, we were engaged in a Level 1 issue. The results led us to ask

a Level 2 question about teachers’ reactions to situational answers.

However, because of our role as teacher educators, and since we iden-

tified a pedagogical issue, it was natural for us to design tasks with

a double goal, a research goal and a pedagogical goal. At level 3, as


teacher educators, we were trying tasks, presenting the School-Bus

problem to different groups of teachers. We observed how the discus-

sion developed, and whether teachers gained some insights from it. At

the same time, as researchers, we designed the tasks in a way that

enabled us to collect data on the nature of teachers’ solutions and on

their evaluations of children’s solutions. Our learning as teacher educa-

tors and researchers was analyzed using the model proposed by Za-

slavsky et al. (2003). Following an attempt to use this model in

describing the learning processes in this study, and keeping in mind

the abovementioned level analysis (Jaworski, 2001) of operation, we

realize that we need to regard the learning model more flexibly. Rather

than identify ourselves as teacher educators in the model, we view our

roles as changing in the course of the study.

When (in the first phase) we gave our teachers a task that involved

the solution of a mathematical problem and then evaluated their own

solutions, we were acting as teachers giving a mathematical challenge

to their students. When (in the second and third phase) we asked

teachers to discuss further emerging pedagogical issues, and when we

asked them to evaluate children’s answers, we were acting as teacher

educators. However, we were also, all along the study, acting as

researchers, specifically designing tasks as a basis for answering

research questions. As a result of the added research dimension and

because it was planned with the intention that we learn from it, our

teaching became a direct learning opportunity.

This research dimension brings us back to Jaworski’s (2001) levels.

Jaworski suggests that teacher educators operating at Level 3 can be

considered to be employing ‘‘educative power’’ (along the lines of

‘‘mathematical power’’ and ‘‘pedagogical power’’ describing actions in

lower Levels). Perhaps, then, in designing tasks to broaden our theo-

retical knowledge, we can say that ‘‘researchive power’’ is used.

To summarize what we learned on the meta-level: We recognized

the role and importance of our action components, sensitivity to our

Phase 1 Phase 2 Phase 3

Unintended

Intended

Research on formal and situational solutions.

Research on proportional reasoning problems.

Figure 8. Changing the focus of intended research.


students and the use of a challenging task, and became aware of the

structure of our own research and learning process. We see it now as a

part of a more general phenomenon of learning from and while teach-

ing, and identify it as a special case of simultaneously researching and

teaching resulting in a more direct opportunity to learn.

WHAT WE LEARNED IN SPECIFIC LEVEL: MATHEMATICAL

AND PEDAGOGICAL KNOWLEDGE

The Mathematical Task

The example used in this study, the School-Bus problem, is a standard

problem. Problems of this type are conventionally used as an applica-

tion of proportional reasoning. It is, therefore, usually expected that

children will directly identify it as a case of proportion and not delib-

erate too much on the situation. In the first phase of the study (and

again in further phases), when this problem was assigned to teachers,

most solved it formally as expected. Yet some pre-service and in-ser-

vice teachers used situational considerations in analyzing the problem,

and did not apply proportional reasoning. Some of the children who

were (later in the study) interviewed by our pre-service teachers gave

more situational answers.

The Teacher-Education Task

The second phase of the study involved discussions and evaluations of

the mathematical task solutions. Although most teachers thought that

certain situational answers were wrong, they could not offer a convinc-

ing argument for their claim. For example, the most frequent reaction

to the drawing of 7 children depicted as dots and equally spaced, was:

This is a ratio problem and this solution does not use ratio.

An analysis of the characteristics of teachers’ answers revealed a

possible explanation for their initial inability to produce a sound argu-

ment. As it turned out, most teachers gave a formal answer using some

mathematical expression without further explanation and without an

explicit effort to relate to the situational meaning of the solution.

This behavior changed in the course of class discussion, when teach-

ers began coming up with drawings for their proportional reasoning

solutions. Thus, somewhat paradoxically, meaningful analysis of one’s

own solution occurred as a result of trying to understand the alterna-

tive (situational) solution. The request to compare this solution to

their formal solution, forced teachers to engage in better understand-


ing of the meaning of density and of the meaning of equality of ratios.

The mathematical challenge turned out to be more complex, creating a

more interesting learning opportunity than originally planned.

In the course of our research we, too, gained some mathematical

knowledge and developed relevant arguments as a result of repeated

engagement with the task and its solutions. Our evaluation of the

alternative solutions underwent several changes. At first we believed

these solutions to be wrong, then we became less confident and

thought that perhaps there were considerations we did not take into

account (as was found in realistic research problems), and perhaps

these answers were actually correct. Eventually, we arrived at the con-

clusion that under the conventional definition of density, these alterna-

tive solutions were indeed incorrect.

The main benefits of comparing different approaches in a class dis-

cussion lie, in this case, not in choosing ‘‘the best solution’’ but in

helping one to understand better one’s own solution. In the case of stan-

dard application problems, the importance of the discussion is in mov-

ing beyond an almost automatic fitting of a mathematical model to

better understanding why that model is relevant and how it is realized

in the specific situation.

Additional reasons for promoting such discussions on standard

problems come from the analysis of the teachers’ answers and attitudes

depicted in the third phase. This analysis shows that some teachers did

not use proportional reasoning, and that some teachers treat answers

that are better than theirs as incorrect. On the other hand, it also

shows that when teachers focused on the detailed explanations of chil-

dren’s answers, they exhibited a positive attitude towards the different

approaches. Perhaps, then, in a class environment that encourages

children to suggest different approaches, children will be able to

explain their answers, good ideas will not be suppressed, and, we hope,

not only children but also teachers will learn from these discussions.

It is interesting to compare the insight gained by situation analysis

in the teachers’ discussion, to the learning that developed in a case

involving children’s problem solving. The case is detailed by Hatano

and Inagaki (1998) and mentioned by Romberg and Kaput (1999). In

this example, first graders are working with a comparison problem that

asks how many more boys there are than girls in a given situation.

Most children answer correctly, immediately applying a subtraction

mathematical model. However, one child claims that subtraction is

used when one takes a subset away from a set, while in the problem

situation it is physically impossible to take girls away from a set of

boys. At first, the rest of the children cannot explain why their solution


is correct. Then, following a discussion and a representation of the sit-

uation, all children learn more about the meaning of subtraction, real-

izing how they can extend the take-away meaning. When we compare

this episode to ours, we see that in both cases teachers and children

apply a mathematical model without much deliberation, and in both

cases, their (correct) solution becomes more meaningful following a

discussion involving further analysis of the situation.

A good example of meaningful discussions in standard (routine) prob-

lems can be found in Lampert’s recent book (2001). On the cover of the

book, a girl is solving a problem written on the board: Car goes 40 mph.

How far in 3½ hours? This is a standard transportation problem. It can

be solved in one step by recognizing that it is a constant speed problem in

which one simply multiplies the speed by the time to get the distance.

But, in the picture we can see a number line and the girl seems to be deep

in thought. In her book, Lampert gives a detailed and insightful account

on different, often unexpected, complexities that emerge in the course of

solving this and other similar problems. Complexities (situational or

mathematical) that come to the surface in the specific class environment

created by her as these fifth graders’ teacher, lead to worthy discussions

and to the construction of meaningful mathematics.

It seems that teachers are ready to promote understanding. In their

individual reactions to children’s answers, many teachers expressed

readiness to accept the explanations that came with the alternative per-

spectives, even when they thought that the answers were incorrect.

Although this was not their reaction in a real class situation, many of

their comments expressed genuine admiration for children’s creative

ideas. These reactions indicate an open-minded atmosphere and consti-

tute a positive starting point from a teacher-education perspective.

With our request for openness, we should add a word of caution.

Although we ask not to reject a solution just because it uses realistic

considerations, we would not want teachers to get the impression that

every solution should be accepted. We observed some tendencies to

take our message to extremes when teachers thought that an additive

solution to the bus crowdedness problem was wrong, and yet accepted

an additive explanation.

The issues raised in this work suggest that teacher education pro-

grams need to make teachers aware of the existing tensions between

applying a mathematical model and using situational considerations,

and of the dangers of applying a mathematical model without fully

understanding why it fits. Group discussions can show teachers that

standard problems are not as straightforward and let them experience

a change in their own conceptions. Teachers will, we hope, create a


similar environment for their own students, helping them develop a

deeper mathematical understanding.

The process of investigating the above pedagogical issue shows that

a researcher educator can contribute to research in mathematics educa-

tion by employing ‘‘researchive power’’. This includes being sensitive

and aware, ready to seize research opportunities that occur while prac-

ticing teaching and also being able to design relevant challenging tasks

and integrate the results of their implementation into a coherent peda-

gogical statement.

ACKNOWLEDGEMENT

We wish to thank Analucia Schliemann, Roza Leikin, and Orit Zaslav-

sky for their helpful comments.

NOTE

1 The study was conducted in Israel. In Hebrew the same word is used for density

and crowdedness. In the article, we use both English words as they fit the context.

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Irit PeledSchool of Education,University of Haifa,Mount Carmel,31905 Haifa,IsraelE-mail: [email protected]

Sara HershkovitzCenter for Educational TechnologyTel-Aviv 61394IsraelE-mail: [email protected]


RAINY COTTI and MICHAEL SCHIRO

CONNECTING TEACHER BELIEFS TO THE USE OF

CHILDREN’S LITERATURE IN THE TEACHING OF

MATHEMATICS

ABSTRACT. This article presents examples that illustrate how teachers use chil-

dren’s literature in the teaching of mathematics. The examples are related to four

curriculum ideologies that have influenced mathematics education in the USA for

the last 75 years. It discusses why it is relevant to help teachers understand the ideo-

logical positions that influence their use of children’s literature during mathematics

instruction, summarizes the four ideological positions, and presents results of a study

of how teachers’ ideological positions relate to their use of children’s literature in the

teaching of mathematics. The study examines two research questions: ‘‘Can an

instructional tool be developed that will highlight for teachers the different ways in

which they and others use children’s literature to teach mathematics?’’ and ‘‘Can that

instructional tool stimulate teacher discussion and reflection about their own beliefs

and the ideological nature of the instructional environment in which they learned (as

students) and teach (as teachers)?’’ Study results indicate that both questions can be

answered in the affirmative.

KEY WORDS: children’s literature, curriculum ideologies, curriculum integration,

mathematics instruction

In courses and workshops for pre-service and experienced elementary

teachers, we stress the use of children’s literature as one of the power-

ful tools available to help in teaching mathematics.

Recently we heard teachers discussing a children’s book entitled The

M & M Counting Book (McGrath, 1994). It is a familiar book that

can be found in the children’s section of bookstores in the USA. It is

a book in which vivid pictures of M & Ms are used to illustrate math-

ematics problems for young children, and in which children are enticed

to do things like count the number of M & Ms on a page. M & Ms

are small ‘‘round’’ chocolates encased in brightly colored candy shells.

(Other popular books about commercially available candies include

Reese’s Pieces Peanut Butter Counting Book (Pallotta, 1998) and Her-

shey’s Milk Chocolate Bar Fractions Book (Pallotta, 1999).) What was

interesting is that four of the teachers used the book in their class-

rooms, but for different purposes.



One teacher said she used The M & M Counting Book because it

provided her children with pictures that allowed them to visualize and

understand mathematics concretely. Under her instruction, her stu-

dents used real M & Ms with the book to model addition and subtrac-

tion as joining and taking away respectively. She said the book helped

her students gain insight into those operations and how they relate to

each other.

Another teacher said she used The M & M Counting Book because

it provided her students with a chance to practice mathematical skills.

She had her students use M & Ms to solve and practice basic mathe-

matics facts and she also used M & Ms with the book for the purpose

of reinforcing and rewarding her students’ appropriate mathematical

behaviors.

A third teacher used The M & M Counting Book for yet a different

purpose: because students were interested in reading it and because it

seemed to help them naturally develop mathematically. She used M &

Ms with the book because students enjoyed eating, counting, arrang-

ing, and playing games with them. She facilitated these activities

because they seemed to stimulate the children’s intellectual and social

growth.

A fourth teacher said she used The M & M Counting Book because

it provided her students with an example that they could relate to of

how big business and our ‘‘capitalist’’ culture is constantly at work

trying to make us into consumers. She said she used the book to help

her students understand how they were constantly being ‘‘brain-

washed’’ by industry to spend money and consume material goods.

She did not use M & Ms with the book; she used pebbles that she col-

lected from a stream near her house.

Here were four primary school teachers, all at the same meeting, dis-

cussing the same book, but for widely differing educational goals. One

teacher was interested in understanding, another in skills, yet another

in growth, and a fourth in values. All four teachers were delighted that

others were also using the book and supported each other’s use of chil-

dren’s literature to aid in the teaching of mathematics. However, all

four seemed surprised and uncomfortable with the ways in which the

other three teachers were using The M & M Counting Book, although

they did not say why.

It was immediately apparent to us that the teachers’ different pur-

poses could be related to curriculum ideologies currently debated in the

USA under the mantra ‘‘math wars’’ (a widespread heated disagree-

ment in the USA among educators, politicians, and the general public

about the purposes and instructional methods of mathematics educa-

330 RAINY COTTI AND MICHAEL SCHIRO

tion). And it was no surprise that the teachers discussing The M & M

Counting Book were at a loss to understand each other’s perspectives

on the use of the book with children. Many books have been written

about the use of children’s literature in teaching mathematics (e.g.,

Braddon, Hall & Taylor, 1993; Burns, 1992; Schiro, 1997; Welchman-

Tischler, 1992; and Whitin & Wilde, 1995), and journals such as

Teaching Children Mathematics have published many insightful articles

over the years. While these books and journals sometimes discuss ped-

agogical issues (such as the role of teachers and the nature of learn-

ing), only a few examine broader ideological issues in any depth (such

as the purpose of schooling and the nature of knowledge most worth

teaching). Virtually none relate their discussion to the ongoing debates

in the USA about the purposes of mathematics education.

As a result, presenting how ideological perspectives influence educa-

tors’ use of children’s literature became a focus of our mathematics

methods courses and workshops. This involved relating ideological

positions on the use of children’s literature to larger ideological debates

about the teaching of mathematics that are currently taking place in

the USA. To aid this endeavor, a Mathematics and Children’s Literature

Belief Inventory was created as a tool to facilitate teachers’ understand-

ing of their own ideological positions and those of other teachers. In

this article the focus is on ideological versus philosophical positions. As

in the field of curriculum theory, the term ideology is used to refer to

teacher beliefs that influence classroom action, while the term philoso-

phy is used to refer to more classical systems of theoretical thought

(e.g., O’Neill, 1981; Schiro, 1978, 1992; Zeichner, 1993).

This article begins with a discussion of why it is relevant to help

teachers understand the major ideological positions that influence how

teachers use children’s literature in the mathematics classroom. Next,

the ideological positions that are introduced to the teachers are pre-

sented. Then a study is described in which teachers’ ideological posi-

tion in relation to the use of children’s literature in the teaching of

mathematics is explored.

Empowering Teachers

There are three main reasons why mathematics teacher educators

might want pre-service and experienced teachers to understand the

ideological pressures that influence their thought and classroom behav-

ior. Teachers who understand their own ideological orientations might

more effectively accomplish their intentions and learn a wider range of

methods of using children’s literature in teaching mathematics. Teach-

331CONNECTING TEACHER BELIEFS TO CHILDREN’S LITERATURE

ers who understand the ideological orientation of colleagues (and their

communities) might better communicate and work with them. In addi-

tion, teachers who understand the ideological pressures exerted on

them by society might be able to put those pressures in perspective,

and to minimize the influence of those pressures. Examples of each of

these reasons constantly arise during work with teachers on how to

use children’s literature in the classroom to aid the teaching of mathe-

matics. Examples for each reason follow.

First, in our courses on Teaching Elementary School Mathematics

and our seminars on Mathematics and Literacy over the last 6 years,

we observed that helping teachers understand their own ideology and

the range of ideological options available to them could enable them

to accomplish more effectively their own instructional intentions. For

example, examining teachers’ lesson plans in which children’s literature

was used to teach mathematics frequently provided opportunities to

show goal statements that were inconsistent with activities intended for

children. Objectives related to ‘‘skill’’ development (characteristic of

one ideology) were often accompanied by activities related to helping

children ‘‘understand’’ mathematics (characteristic of another ideol-

ogy). When teachers saw inconsistency in their instructional plans, it

assisted them in clarifying their intentions and in designing instruc-

tional plans, consistent with their intentions. Knowledge of the range

of ideological positions that exist also often allowed teachers to select

intentions and activities that they were not previously aware of or

inclined toward.

Second, in our courses on Teaching Elementary School Mathematics

and our seminars on Mathematics and Literacy over the last 6 years

we observed that when teachers understand the range of ideological

perspectives that colleagues (and their communities) can hold, this can

enable them to better communicate and work with their colleagues

(and their communities). For example, a number of our teachers

reported frustration while working with colleagues on curriculum com-

mittees that were dealing with issues of when and how to introduce

children’s literature into the mathematics curriculum. When our teach-

ers became able to identify their colleagues’ ideological perspectives

and to express their ideas using the linguistic and conceptual frame-

works of colleagues who opposed their ideas, they discovered fewer

frustrations and greater success in accomplishing their intentions. (As

one teacher expressed: knowing why people oppose what you want to

accomplish and knowing how to speak the language of your oppo-

nents has real advantages during frustrating discussions in curriculum

meetings.) Knowledge of the ideologies can, similarly, enable teachers


to contribute more effectively to the public debate about educational

issues.

Third, we have seen how helping teachers understand the ideological

pressures exerted on them by society can help them put those pressures

in perspective and minimize, as warranted, the influence of those pres-

sures. For example, after teachers were introduced to the curriculum

ideologies discussed in this article, a frequent response to their new

understanding was to identify university faculty and school teachers

they have known who have different ideologies. A number of our pre-

service teachers also reported feeling liberated upon discovering other

options available to them than the ones presented to them by some

faculty members. This has been particularly true of faculty members

who, students report, have promoted the use of children’s literature

primarily as a way of bringing about social justice.

Teacher Ideology, Children’s Literature, and Teaching Mathematics

Among the major ideological positions that influence how educators

in the USA use children’s literature in the mathematics classroom are

the Scholar Academic, Social Efficiency, Child Study, and Social

Reconstruction positions (Schiro,1978, 2004; Schubert, 1996; and

Zeichner, 1993). The Scholar Academic position primarily values the

understanding of the mathematical content identified by mathematical

scholars. The Social Efficiency position mainly values skills needed to

function productively in society. The Child Study position, above all,

values mathematical meanings constructed by children that they dis-

cover as a result of experiencing their environment. Most of all, the

Social Reconstruction position values mathematics as a tool useful in

bringing about social justice. Note that all positions recognize the

need to help children develop mathematical understanding, skills,

meanings and values, but that different emphases have a major

impact on the pedagogical orientation and instructional practices of

educators.

These ideological positions emerge from an analysis of philosophi-

cal, political, and social influences on educators’ beliefs and behaviors

during the 20th century. This framework for examining teacher beliefs

is supported by the professional literature on curriculum positions in

the USA. Table I shows the alignment of the ideological classification

schemes of six curriculum theorists. All theorists have identified posi-

tions similar to the Scholar Academic, Social Efficiency and Child

Study ideologies. Two schemes do not identify the Social Reconstruc-

tion position; however, this is not surprising because the Social Recon-


TABLE

I

ComparisonofIdeologicalPositions

Schiro(1978)

Schubert(1996)

McN

eil(1977)

Posner

(1992)

Fenstermacher

&Soltis(1992)

Zeichner

(1993)

Scholar

academ

ic

Intellectual

traditionalist

Academ

icTraditional

andstructure

ofthedisciplines

Liberationist

Academ

ic

Socialeffi

ciency

Socialbehaviorist

Technological

Behavioral

Executive

Socialeffi

ciency

Childstudy

Experientialist

Humanist

Experiential

andcognitive

Therapist

Developmentalist

Social

reconstruction

Critical

reconstructionist

Social

reconstructionist

Socialreconstructionist


struction position was just re-emerging as a distinct position around

1990. It should be noted that in 1986 and 1987 Schubert distinguished

only three positions while in 1996 he revised his classification to

include four positions. It should also be noted that, even though the

terms used to label positions may be different, the underlying positions

are the same. Terminology has changed radically over the last century.

For example, different labels for the Child Study ideology over the last

century in the USA have included Child Study (1890s), Progressive

Education (1910–1950), Open Education (1965–1980), Developmentalist

(1970–1990), and Constructivist (1990–present).

The four ideologies are conceived to be ideal types around which

educators cluster. The interpretation of the ideologies in this article is

based on the ideological climate in the USA. It needs be noted that

the nature of the ‘‘ideological battles’’ in the USA is different from

those in other parts of the world. Many international (non-USA) stu-

dents in our curriculum theory courses report that, in comparison with

their homelands experience, they are surprised at the extent to which

these ideological positions are distinct from each other in the USA

and the ferocity with which ‘‘ideological battles’’ (recently popularized

by the terms ‘‘math wars’’ and ‘‘reading wars’’) are fought in the

USA. In the USA ‘‘ideological battles’’ over education are fought not

only among academics but also among the general public in newspa-

pers, on the Internet, during public elections, and in our state legisla-

tures. Battles have resulted in textbooks that are written with certain

ideological stances being banned by certain states, teacher tests with

certain ideological stances being mandated by particular states, state

curriculum being rewritten to reflect specific ideological positions, and

elected officials campaigning on platforms aligned with particular ideo-

logical stances. For example, the early 1990 version of the Massachu-

setts Mathematics Curriculum Standards was written to reflect the

1989 NCTM Curriculum and Evaluation Standards, which leaned

toward the Child Study Ideology with its emphasis on ‘‘constructiv-

ism.’’ At the end of the 1990s, with a change in Massachusetts state

government, the new State Director of Instruction had the Mathemat-

ics Standards rewritten with the mandate that all use of the word

‘‘constructivism’’ or its roots be removed from the Standards. The

range of positions and energy underlying the ‘‘math wars’’ can be

gauged by doing an Internet search with a search engine such as Go-

ogle using the term ‘‘math wars.’’ Such a search recently returned

126,000 hits. In this article we attempt to describe the ideological posi-

tions in an unbiased manner, using the rhetoric of adherents of the

different positions.


While these ideological battles are in the forefront of heated debates

in the USA, it is also possible to identify similar ideologies in other

countries. For example, Cheung (2000) used a similar ideological

framework to examine teacher ideologies in Hong Kong, while histo-

ries of mathematics education, such as Howson’s (1978) history of

mathematics education in Great Britain, identify projects aligned with

specific ideologies (e.g., the Nuffield Project with the Child Study ide-

ology and the Continuing Mathematics Project with the Social Effi-

ciency ideology).

Scholar Academic Position

Scholar Academics believe that, over the centuries, our culture has

accumulated important knowledge that has been organized into the

academic disciplines found in universities. The purpose of education is

to help children learn the knowledge of the academic disciplines – in

this case, mathematics. While acquiring an understanding of mathe-

matics, children should learn important knowledge including mathe-

matical information, the conceptual frameworks of mathematicians,

and how to prove mathematical conjectures. Teachers should be math-

ematical mini-scholars with a deep understanding of mathematics who

can clearly and accurately present it to children, often using direct

instruction or ‘‘guided inquiry’’.

In the 1890s, The Committee Of Ten (National Education Associa-

tion, 1894) decided that the content of mathematics should be one of

the foundations of the school curriculum. During the middle of the

20th century organizations such as the School Mathematics Study

Group (SMSG) worked to rebuild the school mathematics curriculum

so that its content mirrored the most up-to-date mathematical under-

standings of professional mathematicians. Recently, some states, such

as Massachusetts in the USA, decided that understanding mathematics

was so important that children would have to pass a mathematics test

in order to graduate from high school.

Amanda Bean’s Amazing Dream (Neuschwander, Woodruff & Burns,

1998) was used by a teacher in one of our classes to help her students

understand multiplication ‘‘numerically as repeated addition of the

same quantities’’ and ‘‘geometrically as rows and columns in rectangu-

lar arrays,’’ in a way that was consistent with the Scholar Academic

ideology. She read the entire book to her students and then afterwards

revisited certain pages. On the page with five boxes that each con-

tained four lollipops, she described how the ‘‘repeated addition’’ model

of multiplication worked and how to write a multiplication problem to


represent the situation. She then showed her students the page with

the picture of nine cakes, each of which contained four candy flowers,

and had her students describe and write a multiplication equation that

represented the situation. She repeated this process with several other

pages in the book, highlighting the ‘‘repeated addition’’ model of mul-

tiplication. She gave her students worksheets with pictorial representa-

tions of ‘‘repeated addition’’ multiplication problems and had them

write an equation for each problem. When her students finished the

activity, she asked them to describe their understanding of multiplica-

tion, and she rephrased each description to make sure that the correct

interpretation of multiplication was presented. The next day, she

repeated the whole process with the ‘‘rectangular array’’ model of mul-

tiplication. The focus on obtaining an accurate understanding of multi-

plication and the ability to write an equation that accurately

represented multiplication were central to this teacher’s use of Amanda

Bean’s Amazing Dream. In this case a children’s book provided a con-

text for illustrating, conveying and helping children acquire mathemat-

ical concepts.

Teachers have also been observed using information books, such as

If You Look Around You (Testa, 1983) to convey geometry concepts,

or The Story of Money (Maestro, 1995) to convey measurement con-

cepts, in ways that are consistent with the Scholar Academic ideology.

Social Efficiency Position

Social Efficiency advocates believe that the purpose of schooling is to

efficiently meet the needs of society by training youth to function as

future mature contributing members of society (Schiro, 1978). Their

goal is to train children in the skills and procedures – of mathematics

– that they will need in the workplace and home in order to live pro-

ductive lives. Teachers manage instruction by selecting and using edu-

cational strategies designed to help children acquire the mathematical

behaviors prescribed by their curriculum. Instruction is guided by

behavioral objectives and reinforcement, and may require repeated

practice by students to gain and maintain mastery of mathematical

skills and procedures.

Early in the 20th century Bobbitt (1913, 1918) argued that the

school curriculum should consist of those skills that children need to

function as future contributing adult members of society. During the

middle of the century, Gagne (1963, 1965) urged educators to scientifi-

cally determine and sequence behavioral objectives that would embody

the mathematical process skills that children need to acquire. At the


same time Skinner (1968) provided the behaviorist psychology that

guided the endeavors of Social Efficiency advocates to create compe-

tency-based, individually prescribed instruction designed to shape

desired mathematical behaviors in children (Schiro, 1978). Recently,

advocates of the Social Efficiency ideology have re-focused the educa-

tional debate on ‘‘skills’’ versus ‘‘understanding’’ or ‘‘meanings,’’ as

the issues of testing for child, teacher, and school accountability have

intensified in the USA (Schiro, 2004).

An experienced teacher from one of our classes was observed behav-

ing in ways consistent with the Social Efficiency ideology. She used

Amanda Bean’s Amazing Dream to help her students begin to memo-

rize their multiplication facts. She read the entire book to her students

and ended by highlighting the following statement made by Amanda

Bean on the last page of the book: ‘‘Today I will start to learn the

multiplication facts.’’ The teacher then showed her students flash cards

that she had made that represented situations that arose in the book.

Each portrayed, on one side, a multiplication problem that represented

a situation in the book and, on the other side, had the equation and

its answer. The teacher went through the cards with her students, ask-

ing them to call out in unison answers to the problems. She then dis-

tributed a set of teacher-made flash cards to each student and had

them memorize the answers to problems. They first worked alone, and

then worked in pairs to test each other. At the end of the lesson the

teacher had her students take a short written quiz in which they had

to write answers to multiplication problems that were on the flash

cards. Over the next several weeks, the problems in each child’s set of

flash cards were extended to cover all of the multiplication facts. To

encourage and to reward the children, the teacher constructed and

posted on a bulletin board a ‘‘Multiplication Superstars’’ chart with

each child’s name on it. A star was placed next to the names of chil-

dren who earned 100% on quizzes or worksheets of multiplication

facts. In this example a children’s book presented a stimulus for learn-

ing a mathematical skill: the automatic recall of multiplication facts.

Child Study Position

Child Study proponents believe that schools should be enjoyable

places where children develop naturally according to their own innate

natures. The goal of education is the growth of individuals, each in

harmony with their own unique intellectual, social, emotional, and

physical attributes. Child Study advocates believe that mathematical

experiences should center on the needs and interests of the child.


Given an appropriate instructional environment and encouragement, it

is believed that children will become powerful users of mathematics

and constructors of mathematical meanings. Teachers facilitate learn-

ing and growth by creating rich and responsive intellectual, social,

physical and emotional environments in which they use a variety of

instructional methods and concrete materials to promote children’s

construction of meanings. Real world contexts are often used to

enhance the mathematical connections that children make and to pro-

mote mathematical communication and problem solving.

In the 1890s, Parker proclaimed, ‘‘The centre of all . . . education is

the child.’’ (1894, p. 383). He wanted children, rather than content, to

be the focus of education. During the middle of the 20th century, Edu-

cation Development Corporation designed pattern blocks, tangrams,

mirror cards and attribute blocks to allow children to ‘‘mess around’’

with mathematics in ways that would allow them to enjoy themselves

as they created their own mathematical meanings (Elementary Science

Study, 1970). Today, advocates of the Child Study ideology argue for

developmentally appropriate practice, constructivism, and making the

child’s needs and nature central to instructional planning. They assert

that the most worthwhile mathematical knowledge that children

acquire in school are those personal meanings that they themselves

construct out of their experiences with their environment, whether or

not the mathematical algorithms and concepts they construct corre-

spond to the traditional mathematical knowledge valued by society

(Schiro, 2004).

Child Study teachers focus on creating situations in which children

can construct or discover their own mathematical meanings. Chil-

dren’s books provide a rich, meaningful, real world or fantasy con-

text that can stimulate and motivate children’s learning. Amanda

Bean’s Amazing Dream is an example of a book in which the main

character sees the value of mathematics, and discovers and uses it in

her everyday world. We observed a group of pre-service teachers

from one of our mathematics methods classes use the book in ways

consistent with the Child Study ideology. These teachers had children

use manipulatives to re-enact the story and discover the multiplica-

tion problems along with Amanda. The teachers did this through

asking their children open-ended questions about what they noticed

as they used manipulatives to construct the problems embedded in

the story, and what they thought would happen as Amanda pro-

ceeded through her dream.

After reading Amanda Bean’s Amazing Dream, one of the aforemen-

tioned teachers had students take a walk through their school to


discover things they could put in the form of a multiplication problem.

When they returned to their classroom, each child drew an example of

what he or she saw for a class multiplication book. The teacher then

had children connect the story to daily activities and events in their

lives outside of school, and each child ultimately created his or her

own multiplication story. One of this teacher’s goals in using the book

in this way was to encourage her students to see how mathematics

learned in school relates to ‘‘everyday experiences’’ encountered out-

side of their classroom.

In addition, we have observed that teachers in our classes who advo-

cate the Child Study ideology frequently select children’s books that

come with manipulatives, such as Pattern Block City (Planet Dexter,

1995), and use children’s books along with manipulatives.

Social Reconstruction Position

Social Reconstruction proponents are conscious of the problems of

our society and the injustices done to its members, injustices such as

those originating from racial, physical, gender-based, cultural, social

and economic inequalities. They assume that the purpose of schooling

is to facilitate the construction of a new and more just society that

offers maximum satisfaction to all of its members. By learning mathe-

matics, children acquire powerful tools that allow them to understand

better the problems of society and, more systematically and knowl-

edgeably, to find ways of acting in order to make it the most just, fair

and productive society possible. Teachers collaborate with students

during instruction as they teach mathematics by immersing students in

situations in which mathematics can help them confront and improve

real social crises that are meaningful and relevant to their lives.

In the early 1930s, Counts (1932) dared teachers to confront and

analyze the problems of society, and, based on those analyses, he

dared schools to create a new social order that was more just and

equitable than the existing one. The problems Counts referred to

included worker exploitation, poverty, racism, crime, and socio-

economic class conflict. During the middle of the century, educators

helped children learn mathematics so that they could understand and

act to improve, such social problems as school segregation, unequal

treatment of the sexes, environmental pollution, and nuclear prolifera-

tion. Teachers attempted to build a better society by teaching children

the mathematics they would need to become active social change

agents focused first on understanding and then on improving society.

Today, educators write about the socioeconomic, ethnic, linguistic,


racial and gender inequities of mathematics instruction, and urge

teachers to work to transform their instruction and society to eliminate

the inequities so that all children will have equal opportunity to suc-

ceed in learning mathematics (Zaslavsky, 1996; National Council of

Teachers of Mathematics, 2000). They continue to be concerned with

social problems and view working toward equity in mathematics edu-

cation as one means of attaining a more fair and just society (Schiro,

2004).

Teachers who are advocates of the Social Reconstruction ideology

use children’s literature in mathematics in a number of ways. We

observed one of our student teachers use Amanda Bean’s Amazing

Dream to highlight a member of an at-risk population engaged in

powerful mathematics, in an attempt to counteract inequities that exist

in mathematical achievement among at-risk populations. The main

character in the book is a non-Caucasian female who enjoys mathe-

matics and learns multiplication to enable her to do one of her favorite

things, counting, much faster. The teacher followed up the use of the

story with activities in which children created multiplication problems

based on issues connected to cleaning up their community environment

and helping people in the community who were less fortunate. First,

they collected trash, including cans, to clean up the neighborhood.

Then they used multiplication problems to determine the amount of

money they earned from the returnable cans they collected, and gave

the money to the community’s food pantry. In doing so, the teacher

had her students deal with issues of environmental exploitation and

poverty, both high priorities for advocates of Social Reconstruction.

While teaching third grade, one of the authors of this article used

The Great Kapok Tree (Cherry, 1990) to highlight the problem of the

destruction of the rainforests and relate it to the conservation of envi-

ronmentally sensitive areas in her students’ community. After reading

and discussing the book with her students, she had them use informa-

tional books, such as Fifty simple things kids can do to save the Earth

(Earthworks Group, 1990) and newspapers, to find statistics and other

information that would help them understand the problem of water

pollution faced by their community. Next, children spoke with local

government officials who had been invited to their classroom and used

their mathematical analyses while discussing and promoting solutions

to the problems. One of the goals of this endeavor was to show how

mathematics could empower people to improve themselves and their

society.

Many experienced teachers in our classes have reported using books

that promote specific products that appeal to children, such as The


M&M counting book and The gummy candy counting book (Hutchings

& Hutchings, 1997), to enlighten children about the ways society influ-

ences what they buy. At the beginning of this article, we cited one

instance of this.

THE STUDY

The research questions posed in this study were: (1) ‘‘Can an instruc-

tional tool be developed that will highlight for teachers the different

ways in which they and others use children’s literature to teach mathe-

matics?’’ and (2) ‘‘Can that instructional tool stimulate teacher discus-

sion and reflection about their own beliefs and the ideological nature

of the instructional environment in which they learned (as students)

and teach (as teachers)?’’ The instructional tool developed is called

The Mathematics and Children’s Literature Belief Inventory. The main

purpose in using the inventory was not the collection and analysis of

data on the numbers of pre-service and experienced teachers who hold

a particular ideology. Rather, the inventory provided a starting point

for examining the use of children’s literature during mathematics

instruction, and the ways in which ideological concerns (about such

things as the purpose of schooling and the nature of teaching) could

influence that instruction.

First, the population involved in the study is described. An account

of what was done in courses to highlight how teacher beliefs influence

their use of children’s literature during mathematics instruction fol-

lows. Then the development and testing of the Mathematics and Chil-

dren’s Literature Belief Inventory is discussed and observations about

the use of the Inventory during instruction are presented. Finally, limi-

tations and future research questions raised by the study are provided.

Population

The study took place at two universities adjacent to urban areas in the

Northeast United States, one located in the state of Massachusetts and

one located in the state of Rhode Island. Both have schools of educa-

tion that operate within larger institutions. The study was conducted

in undergraduate and graduate level courses on teaching mathematics.

The population of students in these courses included pre-service and

experienced teachers. There were 109 pre-service teachers in four intro-

ductory level courses on teaching elementary school mathematics.

There were 18 experienced teachers who taught at grade levels from


pre-school through high school in a graduate level course on mathe-

matics and literacy.

Course Instruction on Ideologies and Children’s Literature

Two foci of our courses on methods of teaching elementary school

mathematics are: (1) how to use children’s literature to aid in the

teaching of mathematics and (2) the different ideological positions that

influence this endeavor.

Children’s storybooks that contain mathematics are presented;

teachers read at least five children’s mathematics storybooks and

examine several teacher resource books that contain ideas about how

to use children’s literature in the classroom (e.g., Braddon, et al., 1993;

Sheffield, 1995); and instruction is provided on the use of children’s lit-

erature during mathematics instruction (Schiro, 1997).

Next, the Mathematics and Children’s Literature Belief Inventory

(described in the next section) is administered and teachers construct

graphs of their ideological preferences in relation to the use of chil-

dren’s literature in teaching mathematics. As they finish their graphs,

the instructors ‘‘tell the fortunes’’ of teachers by reading their graphs,

describing to each teacher the things that they do and do not believe

in, the famous educators they agree and disagree with, and the other

teachers in the room with whom they agree or disagree. Teachers

examine each other’s graphs, discuss similarities and differences in

their graphs, and discuss educators’ beliefs about why and how to use

children’s literature in the teaching of mathematics.

A short lecture is then presented on the nature and history of the

four ideologies, and how each ideology can influence classroom prac-

tice. Teachers then continue to discuss their ideological preferences,

whether or not they recognize the existence of the different ideological

positions in their professional lives, and the advantages and disadvan-

tages of each ideology. Each teacher is asked to affirm whether or not

the Inventory was accurate in its depiction of his or her ideology and

beliefs about teaching.

Later in the courses, teachers construct, teach and reflect on math-

ematics lessons in which they read a children’s storybook and have

children engage in related mathematical activities. Their reflections

relate their ideological beliefs and goals to the activities in which

children engage, and examine relationships between anticipated and

achieved student outcomes. Teachers are encouraged to use any of

the four ideologies to construct and teach their lessons, and they can

stress understanding, skill acquisition, meaning development, or value


construction in their lessons – so long as they clarify their inten-

tions.

Finally, this instruction on curriculum ideologies and children’s liter-

ature is generalized to all mathematics education, during which such

things as the purposes of mathematics instruction and the types of

knowledge valued during mathematics instruction are examined, while

referencing the previous work on ideologies. Teachers also participate

in discussions of how the ideological positions are reflected in different

mathematical organizations (such as the National Council of Teachers

of Mathematics and Mathematically Correct, which endorse different

ideological positions). Teachers also attempt to identify the ideological

positions of school textbooks and the curriculum guidelines of state

governments and national organization (for example, the National

Council of Teachers of Mathematics curriculum guidelines take a dif-

ferent ideological position than those of the state of Massachusetts).

These discussions highlight the underlying ideological reasons for the

controversy currently taking place in the USA over what mathematics

should be taught in schools.

The Mathematics and Children’s Literature Belief Inventory

A Mathematics and Children’s Literature Belief Inventory was created

for two primary reasons. First, it was constructed to help pre-service

and experienced teachers understand the different ways in which chil-

dren’s literature could be used during mathematics instruction and the

pedagogical implications of using children’s literature in different ways.

Second, it was developed because we were dismayed at the low level of

discussion about the use children’s literature in the teaching of mathe-

matics. Most of the discussion was confined to ‘‘activities to do with

children’’ that were related to specific storybooks, and the responses of

children to those books and activities. The need to push thought and

discussion to a more reflective level was apparent.

The inventory includes six sections, one each on instructional pur-

poses, teaching, learning, knowledge, childhood and evaluation. One

section of the inventory is presented in Figure 1. Teachers rank order

the statements in each section of the inventory from the statement they

like most to the one they like least. Analysis of teacher responses on the

inventory results in a graph of their ideological preferences. An example

of a completed graph that portrays a teacher who primarily favors the

Child Study ideology is presented in Figure 2. (Note that the height of

the line in each horizontal section of the graph indicates the degree to

which that ideology is preferred.) The full inventory, as well as instruc-


tions describing how to administer it, graph the results, and interpret

teacher graphs is available from the web site http://www2.bc.edu/�schiro/mclbi.html. Mathematics teacher educators are granted permis-

sion to reproduce and use the inventory with their classes.

The Mathematics and Children’s Literature Belief Inventory was

created by simplifying and transforming a more general Curriculum

Belief Inventory that was originally designed in 1975.

The more general Curriculum Belief Inventory was used during the

first session of courses on curriculum theory and philosophy to introduce

students to the ideological debates about education that have taken

place in the USA over the last 100 years; to help students begin to under-

stand their own curriculum ideologies; and to help students see how their

beliefs fit into the larger debate about the nature and purposes of US

education. It was used again at the end of curriculum theory courses to

help students clarify their beliefs, after they spent a semester reading

about curriculum ideologies and thinking about their own beliefs. The

more general Curriculum Belief Inventory was used with more than 1000

students. After completing the inventory, having their ideology graphed,

and learning about the ideological positions and the history of those

positions during a fourteen week long graduate school course, students

have only rarely indicated that an inaccurate reading was given.

The Curriculum Belief Inventory has been used in two research

studies. One study (Schiro, 1992) involved one hundred experienced

educators (with a mean of 14.2 years of professional experience). It

looked at educators’ perceptions of the changes in their curriculum

belief systems over time. Two findings of that study were that educa-

Part 1: Purpose for using children’s mathematical literature in schools.

___ Children’s literature should be used in the mathematics class to provide children withthe ability to perceive problems in society, envision a better society, and act to changesociety so that the lives of all people are improved.

___ Children’s literature should be used in the mathematics class to help prepare childrento be mature productive members of society who will use mathematics in ways thatwill meet the needs of business, government, industry, their communities, and theirfamily.

___ Children’s literature should be used in the mathematics to convey to children themathematical understandings that mathematicians have developed over the centuries.

___ Children’s literature should be used in the mathematics class to provide enjoyableand stimulating child centered contexts in which children can discover mathematicsthat is developmentally appropriate.

Figure 1. Part 1 of mathematics and children’s literature belief inventory.


tors shifted their ideological stance and that the major events that ini-

tiated such shifts in ideology were changes in job from teacher to

administrator, changes in grade level taught, and changes of school in

which a teacher was employed. Reliability and validity of the instru-

ment and the results of the study were determined by having all partic-

ipants in the study examine and comment on the accuracy of their

ideology graphs that resulted from taking the inventory, and having

them read the findings of the study and comment on them. All partici-

pants in the study agreed that their ideology graphs correctly reflected

their beliefs and that the general results of the study were accurate.

Figure 2. Example of a completed graph for the mathematics and children’s literature

belief inventory.


Another study (Cotti, 1997) looked at how the pedagogical beliefs

of 44 pre-service teachers in a master’s degree teacher education pro-

gram changed from the time they began the program until they com-

pleted it. In this second study the Curriculum Belief Inventory was

given to the pre-service educators four times: upon entry to the pro-

gram, after completing one semester of course work and prior to stu-

dent teaching, after completing student teaching and prior to their last

semester of course work, and at the end of the last semester of their

program (which included the course on curriculum theory mentioned

above). At the end of the program, the pre-service teachers were inter-

viewed. During the interview the pre-service teachers were shown four

graphs of their ideology that were constructed from the data collected

at the above mentioned times, and were asked if the graphs were accu-

rate representations of their curriculum philosophy at the time the

data were collected. All 44 master’s candidates indicated that their

graphs were accurate representations of their beliefs at each of the

times of data collection. In addition, responses to questions during the

interview supported the accuracy of the ideology graphs. Major find-

ings of this study include: that upon entry into teacher education pro-

grams pre-service teacher beliefs are disconnected from classroom

practice; after program coursework teacher beliefs reflected the ideo-

logical orientation of their teacher education program; that ideological

stance was affected by cooperating teacher beliefs and practices during

student teaching experiences; and that an orderly progression of devel-

opmental stages could be identified that described how beginning

teachers developed their ideological orientation.

In the fall of 2000, the more general Curriculum Belief Inventory

was transformed into the Mathematics and Children’s Literature Belief

Inventory used in this study. It was transformed by changing general

statements (about the purpose of education, teaching, learning, etc.)

into specific statements about the use of children’s literature in the

teaching of mathematics. The original Curriculum Belief Inventory

was also simplified by deleting several sections of the inventory (on

freedom and slogans) so that the inventory took less time to complete

and also so that only the most critical dimensions of purposeful teach-

ing were included. The six sections of the inventory now include

instructional purposes, teaching, learning, knowledge, childhood, and

evaluation. These six sections are included (a) because we and our

teachers view them as critical areas of concern to the practice and dis-

cussion of mathematics instruction, (b) because the positions of each

of the four ideologies explored by the inventory can be clearly distin-

guished within each of these sections, (c) because teachers usually have


strong positive or negative feelings about several of the statements

within each section, and (d) because our teachers can, with little

instruction on our part, understand the distinctions between the state-

ments within each section of the inventory.

FINDINGS

As noted previously, the research questions posed in this study were:

(1) can an instructional tool be developed that will highlight for teach-

ers the different ways in which they and others use children’s literature

to teach mathematics; and (2) can that instructional tool stimulate

teachers’ discussion and reflection about their own beliefs and the

ideological nature of the instructional environment in which they

learned (as students) and teach (as teachers). Our main purpose was to

explore these questions and not simply to collect and analyze data on

the numbers of pre-service and experienced teachers who hold a partic-

ular ideology. The results follow.

First, the Mathematics and Children’s Literature Belief Inventory

does highlight for teachers the different ways in which they and others

could and do use children’s literature to teach mathematics. The

graphs of both pre-service and experienced teachers demonstrated that

most teachers do take an ideological position that indicates that they

are primarily in favor of using children’s books during mathematics

instruction in a manner that is in accordance with an identifiable ideo-

logical position (See Table II). Use of the inventory indicated that 95%of the pre-service teachers (104 out of 109) and 94% of the experienced

teachers (17 out of 18) were primarily in favor of using children’s

books during mathematics instruction in a manner that is in accor-

dance with a single identifiable ideological position or, in a small num-

ber of instances, a combination of two ideological positions.

In addition, teacher graphs resulting from use of the inventory

revealed that 90 of the pre-service teachers (83%) and 12 of the experi-

enced teachers (67%) identified the Child Study ideology as their pri-

mary ideological position. Such a large percentage of teachers with a

Child Study orientation is not surprising, given that the ideological

orientation of the teacher education faculty at both institutions in

which this study took place are primarily Child Study and that major

themes inherent in the education programs at both institutions are

constructivism and developmentally appropriate practice. That the

ideological orientation of faculty has an influence on the ideologies of

their students is consistent with previous research (Cotti, 1997).


Our data also revealed that six of the pre-service teachers (6%) and

two of the experienced teachers (11%) identified with the Social Recon-

struction ideology; that one pre-service teacher (1%) and none of the

experienced teachers identified with the Scholar Academic ideology;

and two of the pre-service (2%) and none of the experienced teachers

identified with the Social Efficiency ideology.

Within both pre-service and experienced groups, there were teachers

whose ideological orientation can best be described as mixed, a deter-

mination made as a result of nearly equal emphasis on two or more

ideologies. There were five pre-service teachers (5%) and three experi-

enced teachers (16%) in that category. It is important to note that

three of the experienced teachers who had nearly equal emphasis on

two ideologies worked with special needs children, and their graphs

showed an emphasis on both the Child Study and Social Efficiency

ideologies. They explained this by saying that at heart they were Child

Study but that their schools place an enormous emphasis on behav-

ioral objectives and preparing special needs students to fit into society.

For example, one of the three experienced teachers whose inventory

revealed an almost equal emphasis on Child Study and Social Effi-

ciency stated that she was not surprised by the outcome because of her

position as a special education teacher who had been overwhelmed

with information and teaching strategies concerned with training her

students to become contributing members of society.1

Within both pre-service and experienced groups there were a small

number of teachers whose ideology graphs indicated that they did not

have an ideological preference. There were five pre-service teachers (5%)

and one experienced teacher (6%) in this category. All of these teachers

were asked if their ‘‘seemingly random’’ graphs were accurate descrip-

tions of their belief systems, and all affirmed that their graphs were accu-

TABLE II

Distribution of Teacher’s Ideological Preferences

Child

Study

Social

Reconstruction

Scholar

Academic

Social

Efficiency

Dual

ideology

preference

No ideology

preference

Preservice

Teachers

(n = 109)

90 6 1 2 5 5

Experienced

Teachers

(n = 18)

12 2 0 0 3 1


rate descriptions of their beliefs and that they did not hold a well-defined

and systematic set of beliefs about mathematics instruction.

The validity of the teacher ideology graphs in representing teacher

beliefs was verified after the ideologies were presented. This was

accomplished during class by asking teachers to indicate if the graphs

of their ideologies accurately depicted their beliefs. All teachers indi-

cated that their graphs correctly depicted their beliefs.

All the experienced teachers (100%) recognized the ideological posi-

tions and could point to instances of them in their everyday endeavors

involving other teachers, parents, school administrators, and instruc-

tional materials. Most pre-service teachers recognized the ideological

positions and they could identify former teachers who taught in accor-

dance with different positions or faculty who attempted to convince

them of the value of different positions. This included identifying ele-

mentary school teachers that they had had who leaned toward the

Child Study ideology, Arts and Science faculty members whose teach-

ing reflected the Scholar Academic ideology, and faculty members

within their teacher education programs who tried to influence them to

adopt either the Child Study or Social Reconstruction ideologies. Stu-

dents who had taken courses in special education beyond the introduc-

tory level identified faculty teaching courses in that area who

promoted the Social Efficiency position. Both pre-service and experi-

enced teachers were able to identify the ideological positions of Inter-

net sites they visited in coordination with course work (such as http://

wgquirk.com, http://www.mathematicallycorrect.com, and http://

www.rethinkingschools.org).

Second, the instructional tool did stimulate teachers’ discussion and

reflection about their own beliefs and the ideological nature of the

instructional environment in which they learned (as students) and

taught (as teachers).

All classes, on their own initiative, discussed the ideological orien-

tation of faculty members in their respective higher education insti-

tutions. According to pre-service and experienced teacher

discussions, the ideologies of most of the faculty within their institu-

tions, outside of the teacher education departments, are Scholar

Academic. Pre-service and experienced teacher discussion further

identified the ideologies of faculty teaching in the elementary educa-

tion program in one teacher education department as primarily

Child Study, and in the other as a combination of Child Study and

Social Reconstruction. Teachers raised this issue without prompting,

and determined consensus on this by raising their hands in agree-

ment or disagreement.


In addition, a large number of pre-service teachers reported a feeling

of relief, freedom and empowerment upon discovering their ideological

stance, discussing with fellow students the ideological stance of some

of their college instructors whose ideological stance was different from

theirs, being shown how their ideological stance and that of their fac-

ulties fit into a larger historical picture, and being assured that it is

acceptable to follow their own ideological direction. Helping

pre-service teachers deal with the ideological messages conveyed to

them by faculty and their institutions became an unexpected, yet sig-

nificant part of class discussions.

Experienced teachers reported that insights resulting from the inven-

tory about themselves, the schools in which they teach, and the col-

leagues with whom they teach had an effect on their professional lives.

Usually this involved figuring out how to work with colleagues with

different ideologies. After taking the ideologies inventory and discuss-

ing it in class, one teacher in her third year of teaching at a private

girls’ school resigned from her job.2 She resigned because she realized

that the reason why she was so unhappy teaching at her school was

because its overarching ideology was Social Efficiency while hers was

Child Study. Soon after the end of the school year the teacher

reported that she had obtained a teaching position in another private

school whose ideology was primarily Child Study.

Both pre-service and experienced teachers had difficulty distinguish-

ing between the different types of knowledge (understanding, skills,

meanings, and values that are being argued about during the current

controversies in the USA over what mathematics should be taught in

schools) until they were discussed in our courses. This raised an impor-

tant issue that was pursued in class discussions as teachers were

assisted in connecting different conceptions of knowledge to the use of

children’s literature in teaching mathematics. While most of the pre-

service and experienced teachers were aware of the ‘‘reading wars’’

being fought in the USA between ‘‘phonics’’ and ‘‘whole language’’

proponents, they were unaware of the current ‘‘math wars.’’ As further

inquiry was made into the teachers’ understanding of ‘‘educational

philosophy,’’ the extent of their knowledge became clear. The ‘‘philos-

ophy’’ that they knew was the ‘‘philosophy and history of great men’’

(Plato, Aristotle, Dewey, etc.). They had little understanding of how

‘‘philosophical issues’’ might relate to practical curriculum and instruc-

tional phenomena.

One teacher’s comments during a class discussion summarize the

type of reflection teachers expressed. This teacher reflected on the ideo-

logical position she aligned with according to the inventory. She noted


that her beliefs did indeed agree with the Child Study position, but

that she found it equally important to recognize the potential for

applying the strategies and purposes of the other ideological positions

in using children’s literature to teach mathematics. She stated further

that to operate from only one position would prove to be ‘‘lacking in

discretion’’ with respect to how best to meet individual students’

needs.3 Such balanced perspectives represent the level of development

we hope our teachers will attain.

LIMITATIONS AND FUTURE RESEARCH

The Mathematics Curriculum Belief Inventory provides a quick picture

of teacher ideology over seven variables (purpose of schools, nature of

knowledge, view of learning, etc.). Actual results of the inventory do

not categorize teachers as falling within only one ideology but portray

their beliefs across all variables and within the context of a category

system of ideal types (in which teacher tendency toward ‘‘idealized’’

positions rather than complete agreement with those positions is por-

trayed). For this research, the more complex picture of teacher beliefs

given by the Inventory has been simplified and teachers have been cat-

egorized as having one ideology (when they are primarily aligned with

one ideology), a combination of ideologies (when they are fairly

equally aligned with two ideologies), or as having no preference (when

ideological alignment cannot be determined). The Inventory, how to

score it, and the method of analyzing it are provided at www2.bc.edu/

�schiro/mclbi.html. Simplification of the more complex portrayal of

teacher ideology is one limitation of this research.

Another limitation is that several teachers commented that they

would have preferred to give two statements the same ranking within

each section of the Mathematics Curriculum Belief Inventory. The

inventory did not allow this option.

Further, while this research focused on connecting teacher beliefs to

the use of children’s literature in the teaching of mathematics, it is

located in the following broader contexts that need to be examined in

the future.

Teachers use more than just children’s literature while teaching

mathematics, and additional questions must be asked and answered.

These include: ‘‘What is the relationship of teacher ideology to the

teaching of mathematics, independent of the type of instructional

material used?’’ and ‘‘Do teacher beliefs vary with the use of


textbooks, computers, manipulatives, academic games, and other mate-

rials?’’

Teachers also work within a social and political context. What is the

relationship of teacher ideology to the beliefs about education of the

larger communities to which they belong and the political movements

within those communities? This is particularly relevant in the USA

where curriculum is not planned nationally by governmental or profes-

sional organizations (that ‘‘might’’ be knowledgeable of research about

mathematics teaching and unaffected by city, state, and national poli-

tics). Rather, curriculum planning is a local responsibility (as deter-

mined by the US Constitution) in which state and local communities

determine educational priorities through highly politicized elected

organizations that are easy prey to political movements such as the

‘‘math wars’’ (Kilpatrick, 2001). How do changes in national, state,

city, and school priorities affect teacher ideology and instructional

practice? How is systematically planned curriculum change within

schools, cities, and states affected by reoccurring social and political

debates about education, such as those related to the ‘‘math wars’’ in

the USA? How does the sociology and politics of mathematics educa-

tion in the USA relate to that of other countries – and what are the

benefits and detriments of different national systems? And how do

ideological and political debates about mathematics education relate to

similar struggles in other content areas (such as the dispute over pho-

nics versus whole language in literacy education) in the USA and other

parts of the world? Many of these topics have been addressed outside

of an ideological context – what would research uncover if the rela-

tionship of such research to ideological issues were also explored?

CONCLUDING COMMENTS

Many factors influence how teachers use children’s storybooks to help

children learn mathematics. These include the nature of a book’s story,

the mathematics embedded in the book, mathematical activities (both

commercial and teacher made) that build on the ideas presented in the

book, and teachers’ ideological orientations. The professional literature

has tended to focus on the influence of the first three of these factors.

This article described four curriculum ideologies that can influence

how teachers use children’s literature in the mathematics classroom. It

also described one way of helping pre-service and experienced teachers

relate their beliefs to these curriculum ideologies and examine their

beliefs about using children’s literature during mathematics instruction.


It involved the use of an inventory (that can be downloaded from the

website www2.bc.edu/�schiro/mclbi.html) that teacher educators can

use to introduce teachers to the ideologies. It related how ensuing class

discussions and teacher reflections led to an examination of how ideol-

ogy influences all mathematics instruction. Finally, it presented limita-

tions and directions for future research.

It is not enough to teach pre-service and experienced teachers appro-

priate content and effective pedagogy, without including an examina-

tion of broader ideological issues. Knowing their own ideological

orientation can help teachers more effectively to design instruction

consistent with their viewpoints. Knowing the ideological pressures

being placed on teachers in today’s education conscious society can

help them understand the influence and implications of those pres-

sures. Knowing the ideological orientation of colleagues can help

teachers better understand, communicate with, and work with those

colleagues. And since all of us grow as teachers and are likely to shift

our ideological orientations several times during our lives (Schiro,

1992), knowing the ideological options available to us can help us

choose which orientation we might wish to have at a particular time.

NOTES

1Unpublished student reflection paper, Massachusetts institution, Spring, 2002.

2Unpublished student reflection paper, Massachusetts institution, Spring, 2002.

3Unpublished student reflection paper, Rhode Island institution, Spring, 2002.

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Champion Hall Michael SchiroBoston CollegeChestnut Hill, MA 02467USAE-mail: [email protected]


LEONE BURTON

‘‘CONFIDENCE IS EVERYTHING’’ – PERSPECTIVES OF

TEACHERS AND STUDENTS ON LEARNING

MATHEMATICS1

ABSTRACT. This paper problematises ‘confidence’ as a frequently used explanation

for performance in the mathematics classroom. I report on an interview-based study

of how some English ‘Advanced’ level (16+) students who have chosen to study

mathematics, and their teachers, speak about confidence with respect to the learning

of mathematics. I outline what constitutes confident learners for these teachers and,

differently, for their students and what the students feel teachers could do to improve

the students’ confident state. I discuss the implications of this for the education of

prospective teachers of mathematics.

KEY WORDS: classroom climate, confidence in mathematics learning, students,

teachers

The quote in the title of this paper is from an interview with a teacher

participating in a study I did in the UK with teachers and ‘Advanced’

level2 mathematics students of 17 years of age. This teacher is reflect-

ing a widely held view that performance in mathematics and confi-

dence go hand-in-hand. Success in mathematics breeds confidence.

Confidence in mathematics breeds success. This is despite there being

no agreement on what confidence is, how it can be recognised or mea-

sured. Nonetheless, there is a considerable research literature in psy-

chology that addresses issues of ‘confidence’ through self-esteem,

perceived self-efficacy and attributional style (for a comprehensive

review, see Malmivuori, 2001). Some of these studies have argued that

it is possible to improve students’ academic performance by manipulat-

ing their styles of attribution (e.g., Wilson & Linville, 1982). Other

research has used Seligman’s theory of learned helplessness to make

similar points (Abramson, Seligman & Teasdale, 1978). In schools, the

psychologistic notion of ‘ability’ is linked to achievement, even to the

labelling of low-achievers as ‘low-ability’, and ‘confidence’ is some-

times used as an indicator. Kathleen Lynch and Anne Lodge (2002)

have challenged this pervasive genetic view of ‘ability’, taken from

psychology, whereby, persistently, a link is made with ‘confidence’ in

ways that can affect a pupil’s self-image and consequent choices. The



outcomes, in terms of mathematics, can also constitute generalised

observations of potential links with race and gender. For example:

‘‘white males and African-American females [demonstrate] higher

achievement and more confidence than white females and

African American males, respectively’’ (Stanic & Reyes, 1995, p. 273).

In other words, instead of locking achievement in mathematics to indi-

vidual ‘confidence’, it is legitimate to ask how far low achievement in

mathematics is a consequence of structural and curriculum factors

which successfully diminish many students’ interest and motivation

and are then mis-labelled as ‘lack of confidence’.

Social psychological research on gender and education has suggested

that girls have lower self-esteem than boys, especially after adolescence

and that this is a consequence of differential expectations by teachers

(Freiberg, 1991). The gendered relationship between confidence and

achievement in school mathematics has been argued throughout the

research literature (see Clute, 1984, p. 56) but without the construct of

‘confidence’ being taken as problematic. For example,

[O]ne noteworthy finding that emerged from the Fennema and Sherman studies

(1977, 1978) was that males in Grades 6 through 12 consistently showed greaterconfidence than females in their ability to learn mathematics. Initially those differ-ences were not reflected in differences in achievement; however, for the older stu-

dents, confidence in mathematics was a good predictor of performance for femalesbut not for males. (Leder, 1990, p. 19)

In the USA, Chipman, Brush and Wilson (1985), reviewing the work

up to the mid-1980s, found that confidence (which they claimed was

coincident with enjoyment gained from mathematical experiences) was

a crucial indicator for female participation in secondary and tertiary

mathematics. In a recent book on beliefs edited by Leder, Pehkonen

and Torner (2002), the psychological approach continued to dominate;

interestingly, ‘confidence’ did not appear in the Index. However, the

Editors, in their first chapter, used a model developed by Bar-Tal

(1990) to describe beliefs. In this model, ‘‘confidence is one of four char-

acteristics of beliefs’’ (p. 3). In a chapter in this book, Op ‘T Eynde, de

Corte and Verschaffel pointed out that ‘‘research on this topic has not

yet resulted in a comprehensive model of, or theory on, students’

mathematics-related beliefs’’ (p. 15). The work cited draws on the theo-

retical frameworks and methodologies of ‘mainstream’ social psychol-

ogy, sharing a number of common assumptions about the nature of the

self and treating as unproblematic the relationship between the individ-

ual and society, and between beliefs and behaviour. John Mason, writ-

ing in this Journal, pointed out that: ‘‘From an enactivist perspective,

the distinction between belief and behaviour is artificial’’ (2003, p. 288).

358 LEONE BURTON

A further question emerges from more recent indications that, in

some countries including the UK, the gap in mathematical perfor-

mance between girls and boys at the end of compulsory schooling

has almost been eliminated. Does this mean that ‘confidence’ is no

longer as crucial to mathematics attainment as once it was assumed

to be? Or, since this apparently increasingly equitable state does not

appear to have had a major impact on the male/female ratio opting

for higher levels in mathematics, does ‘confidence’ only become an

issue when students are in the post-compulsory phase of education?

For example, males continue to dominate and to gain more of the

higher-grade passes at ‘Advanced’ level in England and Wales

(Mendick, 2003).

In discussing the Australian experience, Helen Forgasz and her col-

leagues pointed out that:

It is important to address some of the bigger issues that affect the participation

and achievement of both males and females. Critical to this is solving the problemof appropriately qualified teachers at every level from early childhood to univer-sity. (Forgasz, Leder & Thomas, 2003, p. 258)

To ensure that teachers are appropriately qualified for the purpose of

mathematics education for all, it would appear legitimate also to ask

for a better understanding of ‘confidence’ as a feature of the structural

and curriculum factors that must be addressed.

The findings of the research cited are contradictory and, in particu-

lar, definitions of ‘confidence’ vary across different studies and between

different constituencies within a given context, e.g., teachers, male and

female students and researchers. The relationship between ‘confidence’

and learning (e.g., of mathematics), on which this thesis is based,

appears to be more complex than the research findings quoted seem to

indicate. The thesis was challenged, for example by Kenway and Willis

(1990), in respect of the subtlety of interactions between influences that

are undetectable in statistically-based studies and the complexity, and

shifting nature, of how these influences operate. A similar challenge

was made by Malmivuori (2001) who presented a complex model of

affect in mathematics learning and, like John Mason in the quote

above, pointed to the interrelationship for learning between affect, cog-

nition and the social (see particularly her Chapter 3). She described

self-appraisals, self-evaluations or self-beliefs, as ‘‘the core in the

affectcognition interplay in their (the students’) personal and

self-regulatory mathematics learning processes’’ (p. 123) and proposed

changes in pupils’ self-confidence or self-efficacy, as measures of these

central affects. To these challenges, I would wish to add the absence of

359PERSPECTIVES OF TEACHERS AND STUDENTS

the voices of learners and the implications of potential conflict between

teachers’ and learners’ constructions.

Degrees of confidence, and their respective strength, therefore, have

not only been a central issue for mathematics teachers in the judg-

ments that they make about pupil performance in their classes, but

also for researchers who investigate both affective and cognitive out-

comes for their students. Evans (2000) pointed to the ‘‘substantial

measure of agreement about the affective variables that might be

expected to influence thinking and performance in mathematics in

older students and adults’’ (p. 44). High confidence, one of these, has

been associated with achievement, low confidence (often discussed as

anxiety, see Buxton, 1991; Evans, 2000) with poor performance and/or

aversion to the subject. In undertaking the study on which I am draw-

ing here, I saw ‘confidence’ as a label for a confluence of feelings relat-

ing to beliefs about the self, and about one’s efficacy to act within a

social setting, in this case the mathematics classroom. McLeod and

McLeod (2002, p. 115) pointed out that: ‘‘Interest on beliefs has cen-

tred on their role in linking affective and cognitive processes’’. How-

ever, I believe that it is not straightforward to separate affect from

cognition. Learners construct themselves and consequently feel differ-

ently under different conditions and these feelings are intimately con-

nected with both the social settings and the learning experiences within

them. Since I understand learning as being a product of this interplay

between affect and cognition within social contexts, I was interested to

find out, in the mathematics classroom, whether female and male

‘Advanced’ level students and their teachers associated various mean-

ings with confidence and what these were. My main purpose in under-

taking this research, therefore, was to investigate the ways in which

confidence was understood and interpreted by students, engaged in

mathematics by choice, and their teachers. That is, I was making an

assumption that, for my purposes, ‘ability’ could be interpreted in

terms of a combination of success (i.e., achievement) in compulsory

schooling up to the age of 16+ and motivation enough to make the

choice to continue with mathematical studies. ‘Advanced’ level stu-

dents in the UK have emerged from compulsory schooling and chosen

to continue with mathematics, in the case of this study in the same

school, as one of the subjects that they offer for entry to university or

adult life. I wished to make use of the meanings associated with ‘confi-

dence’ by the teachers and students in order to reflect on their implica-

tions for prospective teachers and thus future students.

Every learner’s meaning-making is very personal but mediated

through the cultural climate of the classroom. What meaning learners

360 LEONE BURTON

make of their experiences is fundamental to their learning as well as

their engagement in mathematics. As Bruner pointed out in 1990: ‘‘It

simply will not do to reject the theoretical centrality of meaning for

psychology on the grounds that it is ‘vague’. Its vagueness was in the

eye of yesterday’s formalistic logician. We are beyond that now’’

(p. 65). I was interested in how ‘confident’ behaviour was understood

in mathematics classrooms and whether, potentially, the meaning given

to confidence in learning mathematics conflicted with its meaningful

application in other subjects or environments. I was also interested in

the possibility that ‘confidence’ itself, like mathematics, may have pow-

erful connotations, which have implications for academic performance

of students in schools rather than such performance being seen simply

as a function of ‘expectations’. Whether students displayed classroom

behaviours identified by their teachers as ‘confident’ or not, I expected

that they would use the description ‘confident’ and be able, and will-

ing, to discuss it in different contexts. Teachers, too, have criteria that

they use to ascribe learning behaviours as ‘confident’ and to identify

pupils in their classes as in ‘high’ or ‘low’ categories. An investigation

of the discourses would, I expected, provide diverse meanings and res-

onances of this term for students, and their teachers, in mathematics

classrooms (Burman & Parker, 1994) and perhaps give an insight into

how young women and men were defining themselves, and being

defined, with respect to the learning of mathematics.

In this paper, I am focussing upon the perspectives of the teachers

on confidence and contrasting them with those of the students (for a

paper dealing with why most of these students reject mathematics for

future study, see Burton, 2001). Instead of leaving such issues unchal-

lenged within teacher education and remaining in the domain of ste-

reotypes, my purpose is to raise and query assumptions that are

central to the education of future teachers.

THE STUDY

I decided to conduct an interview-based study as I wanted to engage

with the discourses used by students and their teachers with respect to

the learning of mathematics, particularly ‘confidence’. Consequently,

issues that caused strong feelings might be raised. Interviews seemed to

me the best format for dealing with such and for gaining access to qual-

itatively richer data than other possible methods might provide. The

interviews were semi-structured in that I had a schedule of issues; the

interviews began and ended in the same way but I allowed the


discussion to dictate what came up and in which order (see Appen-

dix A). Where possible, I interviewed (self-chosen) pairs of students

because of experience that this provides a situation that they find more

relaxed. Also, in the interplay of assertion, contradiction and opinion

between the pair, a range of pertinent issues seems more easily to be

provoked within a paired interview making it more conducive to con-

versation than individual interviews. One interviewer with two intervie-

wees also minimally diminishes the power of the interviewer. I have no

reason to believe that this study differed from earlier ones, in these

respects. In only one school, School 1, were the teachers interviewed as

a pair (in the room used for interviewing but during lunch break). In

the other schools, the teacher interviews were dictated by timetables

and were individual. The teacher interviews were all over 1 hour in

length, the pupil interviews roughly 30 minutes each. My focus for the

interviews was ‘confidence’ but this was approached through general

discussion with the students, the first request being to describe a ‘nor-

mal’ or ‘typical’ maths. lesson, whereas the first question to the teachers

was ‘‘Is confidence important to success in learning maths?’’ That is, I

did not immediately introduce ‘confidence’ to the discussion with stu-

dents but I did with the teachers. This was, in part, related to the other

foci of the study. At no point did I see the interviews as being an

opportunity for me to challenge the opinions or feelings of either the

students or the teachers, other than to ask why or how. I was gathering

information on their perspectives and saw my research role as one of

reception, clarification and absorption although that is not to deny the

power of the interviewer to direct, emphasise and, in other ways, con-

trol an interview. After all, mine was the original agenda. One can only

try to minimise one’s own perspectives and interests and be rigorous in

revealing and exploring where those of the interviewees differ.

To preserve their anonymity, the students were invited to provide a

pseudonym that was the only name used in the interview and in subse-

quent transcription and analysis. In the case of the teachers and the

schools, each school was numbered, as were the teachers.

In a large English industrial city, the schools where the interviews

took place volunteered themselves as a result of a city-wide invitation

to secondary schools with Advanced level mathematics students in

post-16 classes (Years 12 and 13). Two of the schools, 1 and 4, were

single sex (one of each). Thirty sixteen-year-old students, who consti-

tuted all the Year 12 pupils studying ‘A’ level mathematics in the four

schools were interviewed in self-chosen pairs, all, except one, single

sex. Two male students were interviewed individually. Thirteen of the

students were female (six pairs and one female in the mixed-gender

362 LEONE BURTON

pair). The one single sex (female) independent (private) school

enhanced the number of females. It is noticeable that, of the 19 stu-

dents interviewed in the two mixed schools, only five were female (all

in School 2). I interviewed two teachers in each of three of the schools

and one teacher in the fourth school. In all four schools, these were

the only teachers who taught the ‘Advanced’ level students (see

Table I).

The data were sorted into statements, coded and analysed by topics,

beginning with those in the schedule of the interview (Appendix A) but

then broadened to include those raised by the interviewees. The process

led to a set of themes and issues specified in the data by the intervie-

wees and substantiated by their comments (see Appendix B). I agree

with Regan de Bere (2003) that: ‘‘discourses are articulated in order to

argue in favour of the credibility of one interpretation of events over

another’’ (p. 115). In this paper, I am drawing on the discourses used

by students and teachers around confidence in learning mathematics.

With Walkerdine (1990), I see cognition as inseparable from, and impli-

cated in, the context constituted through the discursive practices.

WHAT IS CONFIDENCE? – THE TEACHERS’ VIEWS

The teachers responded to my question: What is confidence? by

describing it in terms of the behaviours that contributed to their image

of a confident student; they had no difficulty in identifying their confi-

dent students. They spoke about willingness. In their terms, willingness

to ‘‘have a go’’,3 ask questions and challenge them, was evidence of

confidence. I believe that such behaviour must be understood in the

TABLE I

School number Pupils Teachers

1 (all females) 4 · 2 (females) 2 (pair)

2 (mixed) 2 · 2 (females) 2 (individuals)

5 · 2 (males)

1 · 2 (mixed)

3 (mixed) 1 · 2 (males) 1

1 (male)

4 (all males) 1 · 2 (males) 2 (individuals)

1 (male)


context of the style of pedagogy reported by the students as being uni-

form across all classes, when they were asked to describe a ‘normal’ or

‘typical’ maths lesson (see Appendix B and note the contradiction

between the teachers and the pupils). This style was predominantly,

although not always, teacher-led even where the classes were very

small and a lot of the work was done in a group or groups.

In one class, everyone, teacher and students, spoke of one lesson as

being outstanding in their memory of a lesson that they had particu-

larly enjoyed. It was referred to as the Mastermind lesson and its pur-

pose had been to explore permutations and combinations. Since one

pupil, Louise, had said, in her interview:

She had a Mastermind game with the different colours and you had to work out

how many options you had for the position of the colours in the thing and thatwas getting us to do it physically rather than just like being – and there you gothat’s it – just believe me when I say that we actually had to do it ourselves andwe had to account in little groups for what we thought the reason was for how

many options you can have.

and Emma, her pair, added: ‘‘We keep asking her as well if she can do

more lessons like that because it actually helped us to remember the

theory as well as the idea of permutations’’, I mentioned this to the

teacher who recalled that lesson and wondered why she hadn’t done

more like it. The fact that the lesson remained in the memories of all

those who had participated in it, and that the teacher agreed that this

style of lesson had not been repeated, reinforces the interpretation of

the power, as well as rarity, of such an enquiry approach.

In that same school, the teachers spoke of what they called an

‘‘induction programme’’ to enhance confidence that, according to these

teachers, ‘‘has already been diminished by the teacher from the Pri-

mary school’’ (Teacher 2, School 1) with:

Lots of games, lots of practice books, lots of participation. When you ask a ques-tion or you do something which you know everybody’s got, that’s the child you

pick on, the ones that aren’t sure or wouldn’t put their hands up, and say ‘Couldyou help with this?’, ‘Could you do this?’, ’Can you help so and so?’ and helpsmore with the class participation. Never tell a child it’s wrong, ‘Well, has anybody

else got another idea?’ and the hands come up fast and forward and by and largethey like maths except for the odd few. (Teacher 2, School 1)

The teachers agreed about this induction programme (which, if the

interviewed students experienced it, would have taken place 5 years

earlier). Yet despite this, the impression that all the students gave of

their school experience was of class-work dominated by the teachers

(and, for confirmation from other research, see Lyons, Lynch, Close,

Sheerin & Boland, 2003). It was, however, the case that both teachers

364 LEONE BURTON

and students spoke of a pedagogic style at ‘Advanced’ level that,

although still teacher dependent, was more informal and participatory

than that experienced earlier in the school. Teachers reinforced the

impression of transmission teaching by saying, in response to a ques-

tion about their teaching style:

You just plod through five books and have an exam at the end (Teacher 1, School 2).So far it’s been mainly textbook based and worksheets (Teacher 1, School 4).

I tend to do more work from the front than I should do…I would like to spend moretime with pupils doing work which is not based around a teacher-led activity. But Ifind that I tend to slip back into this too often (Teacher 2, School 4).

So, despite the teachers speaking of their students’ willingness to chal-

lenge and question, in describing their teaching style they appeared to

interpret this kind of behaviour as responsiveness to them, not the adop-

tion of agency, and consequent independence of learning. In terms of

‘voice’, the voice that is heard, used, and the students are expected to

reflect, is most likely to be the voice of the teacher as is borne out by the

data from the students (see below). As Volmink also pointed out:

The reality is that most students do mathematics in an environment where theycome to accept, after some agonising and sometimes devastating experiences, that

the only thing that counts is what the teacher wants and what the teacher knows.(Volmink, 1994, p. 58)

Quotes from the students and teachers in this study demonstrate the

kind of classroom where showing confidence may be behaving in cer-

tain specified ways or meeting the particular demands or expectations

of the teacher; this may be through, for example, the familiar class-

room pattern of initiation, response, evaluation (see Mehan, 1979)

between teacher and individual student. Although one of the teachers

described confidence in terms of ‘‘an acceptance of responsibility for

tackling the mathematics’’, acceptance of responsibility is dependent

upon the style of pedagogy and the epistemology. Indeed, in the terms

of this quoted teacher, acceptance of responsibility for tackling the

mathematics is not the same, epistemologically, as the kind of respon-

sibility identified by Wood (1994), in an ‘inquiry’ primary classroom in

the USA, where students held a belief ‘‘that it is their responsibility to

make sense of mathematics and to make it understandable to others’’

(p. 149). Confidence under these conditions may require different

behaviours, social, communicative and respectful, rather than the dem-

onstration of enthusiastic, individual responses to a teacher’s questions

(hands up, etc., see below).

Referring to what was called ‘‘over-confidence’’, one of the teachers

said ‘‘I wouldn’t equate confidence necessarily with being the one who


is always willing to give the answer’’ (Teacher 1, School 2). Another

observed a gender difference:

Boys I think tend to exhibit that [over-confidence] more than the girls do. I’ve

come across very few girls who, I thought, were over-confident about their abili-ties. They tend to be very shrewd about what they can do and what they can’tdo…whereas boys, it’s all bravado quite a lot of it. (Teacher 2, School 4)

Teachers responded to the question of how they recognised a confident

learner by saying:

Volunteer answers, hands up, seating in the class. I think where they sit in theclass tells you quite a bit about a confident learner…[they] gravitate to the middle.(Teacher 2, School 2)

The ones who read the questions, pick their pen up, and start writing or their

hands shoot up into the air. (Teacher 1, School 1)

It’s the body language. (Teacher 2, School 1)

All of the teachers in this study used the notion of innate ability, were

judgmental of their students’ potential, speaking for example of the

need to ‘‘cope with the majority’’ and of their attitudes and behaviour.

Indeed, most of the interview with one of the teachers was devoted to

speaking of behavioural problems related to getting students to work

in the required manner: ‘‘some of the lads are very childish, very

immature.’’ Another teacher was judgmental in speaking of the diffi-

culty in persuading the ‘A’ level students to talk: ‘‘If they do speak up,

it’s usually rubbish…they’re not very good at all, with the exception of

one who is outstanding, one of the best I’ve taught’’. Although I have

not explored these aspects here, the teachers did link confidence with

behaviour, and, because of this, there is a similarity with the findings

reported by Lyons et al.:

Teachers generally attributed students’ improvements in mathematics to having aninnate ability and being encouraged and supported by the teacher. On the other

hand, teachers did not hold themselves responsible for any observed deteriorationin students’ mathematical performance. Here students’ own attitudes, behaviour orlack of ability were deemed to be the main causal factor. (Lyons et al., 2003, p. 276)

In summary, teachers defined confidence in similar behavioural terms

and spoke of ‘‘willingness’’. Their expectations were that the ‘confi-

dent’ learner was the one who demonstrated those conventional, com-

petitive, masculine classroom behaviours that, in the literature, have

come to be associated with confidence (see, for example, Fennema &

Peterson, 1985). Despite three of the teachers claiming that their teach-

ing style was discursive and/or problem-based, discursive appeared to

mean question/answer and problem-based was book dependent. This

meant that their statements, as with all interviews, have to be

366 LEONE BURTON

understood in the context of their pedagogy and epistemology. I dis-

cuss the implications of this below.

WHAT IS CONFIDENCE? – THE STUDENTS’ VIEWS

Unlike the teachers, the pupils described confidence as to do with how

they felt and they were very localised in their comments. Many of

them invoked a sense of what they called ‘‘can do’’, or agency, in the

mathematics classroom. Fred said: ‘‘If you’ve got the question in front

of you, you think ‘I can do it’ and then, if you get stuck on it, you

think ‘Well, I think I can sort it out myself without any help’’’. Louise

and Clare, like many of the students, expressed ‘‘can do’’ differently in

that they thought it had to do with understanding (and see Boaler,

1997). Louise said: ‘‘When you can see the light at the end of the tun-

nel, when it becomes clear and when you think I can actually under-

stand this. That has an amazing uplift on your confidence when you

see things and you think ‘yeah, I can do this’’’. Clare, indicating the

importance to her of creating her own version, offered: ‘‘More under-

standing than memory because, if you understand how to do some-

thing, then you’ll be able to go back and do it again.’’ Emma made an

interesting distinction: ‘‘If I don’t understand, then I can’t, although I

might be able to do it. Then if I don’t understand it I get confused as

well so I have to have both, be able to do it AND understand it.’’ So,

getting answers correct and having both knowledge and understanding

confirmed ‘‘can do’’ (see categories in Appendix B).

But confidence, they thought, was also a factor of working with oth-

ers; David, as did most of the students, supported collaboration when

talking about confidence (see Appendix B):

I think if you discuss it with somebody else it will make you more confident. Ifyou can talk to others and see if they’re struggling, and if you are doing something

well and they’re not, that will give you more confidence in your ability to do it.Because there is obviously going to be things that you’re better at and somebodyelse thinks that they’re better at than you. If you do a test or something and you

get a low result or something like that, that’s all you get and if you don’t talk toanyone about it, that will lower your confidence. But if you discuss it and if it’ssomething that you have done well and they haven’t, that will build up your confi-

dence. You think ‘Oh, well, I have done that well’ .

However, David, in that quote, is also making reference to a class-

room that has competition embedded in it, when he says: ‘‘if it’s some-

thing that you have done well and they haven’t’’. Other male students

made competitive comments. For example, another David said:

‘‘We’ve got one really clever kid in our class and every time if I do


something better than him or I can get ahead of him, ‘Yeah!’ Competi-

tion helps your confidence.’’ And Steven described his class: ‘‘There is

always boys on the one side and girls on the other. So the boys all

stick together and the girls have to stick together and if they don’t get

anything, well that’s it.’’ Emma compared her mixed primary school

and her all-girls secondary school:

there was a lot of competition between the boys where the girls tended to help

each other more. The boys who were there would out-do each other and I think Ibenefited more coming to an all-girls’ school because girls, there is that element ofcompetition within certain subjects but with things like maths and science, girls

will tend to go: ‘Well, if you don’t understand, I’ll help you.’

Many female students report their dislike of competitive classrooms.

Lyons et al. identify the conflict between socialisation and expected

mathematical behaviours:

In our society, the cultural code that is most celebrated for boys is one that em-

phasises dominance, both of other boys and of girls … while co-operation, caring,and passivity remain highly prized, albeit increasingly contested, feminine val-ues...Fennema (1996) suggests that doing advanced-level mathematical problemsolving requires girls (and boys) to be independent, active, questioning and

rule-breaking. Yet, for girls to behave in this way, is to step outside typicallysocialised ways of behaviour. (Lyons et al., 2003, p. 18)

Elsewhere (Burton, 1989), I have pointed out that the construal of

independence can also be different for different genders.

But both females and males addressed the importance of classroom

working style for the building up of their confidence. They wanted to

work together, with their friends, and only 6 males mentioned the need

for time to work individually, but of these, 4, together with 10 other

males and 6 females discussed the need to work together as an impor-

tant feature of the classroom. That constituted 20 of the 30 students

who spoke about the effects on their confidence of being able to work

collaboratively. All of the students responded to the request to

describe a most enjoyed lesson by mentioning problem solving or

coursework, which appeared to be experienced in a more collaborative

style. These ideas, introduced by the students, were not part of the

teachers’ discourse on confidence.

There was an acceptance by many students in all four schools that,

to have success at mathematics, you needed a combination of confi-

dence and work; they said that confidence and success feed each other

and are in a continuous relationship (see Appendix B). Katherine

explained the idea that many of them expressed: ‘‘I think you have to

have a little bit of success to get the confidence, but I don’t really

know because I think you have a little bit of success and realise that

368 LEONE BURTON

you can do it and then that builds your confidence up.’’ As for hard

work, they all, like Mark, accepted that: ‘‘You have to work hard all

the time; make sure you put the effort.’’ Jodie put the two together:

‘‘For success, you need work and understanding – you either under-

stand it or you don’t. It’s hard work and it’s difficult.’’ Andrew con-

firmed some of the complexity:

Yes, I think they are locked together in a way that, if you’re a confident person,you’re confident in yourself and the teachers that you have and your friends. You

can become successful because you tend to let things, you don’t let things get ontop of you and you’ve got things there.

I asked the students what teachers should and should not do to

encourage an environment that would boost confidence. They wanted

teachers to facilitate discussion, teamwork, a light-hearted approach, a

relaxed classroom where you are not afraid of making errors. They

told anecdotes to support the fact that they did not want to be put

down, persistently asked the same question, made to look a fool or

feel patronised, put into a position where others laugh at you, be

‘‘thrown in at the deep end’’. Again, these kinds of behaviours are

consistent with a competitive classroom culture that values achieve-

ment over understanding. The students felt that teachers should

explain well, should not rush the work, should know what they are

talking about and should be sensitive to students who are struggling to

understand. None of this is surprising, nor even unexpected. It is, how-

ever, seriously disturbing that students at the most advanced end of

the school are still making these kinds of comments about teachers

and their classroom environment. I address this below.

In summary, and unlike the teachers, when the students spoke about

confidence, they concentrated on feelings and how the classroom could

function to make those feelings better, or worse. They drew attention

to their desire for a collaborative working style and they spoke of hav-

ing a ‘‘can do’’ feeling, reinforced by success, that is getting correct

answers and having both knowledge and understanding. They asserted

that success was dependent upon confidence and work.

IS CONFIDENCE IMPORTANT? DISCUSSION

Both teachers and students agreed on the importance of confidence

and, in this, their views were consistent with the literature. However,

they talked about confidence in very different ways. For the teachers,

it was recognised mainly through behaviour. As Elizabeth Fennema


and Penny Peterson pointed out: ‘‘It is not easy for teachers to identify,

or at least to verbalize, teaching behaviors. It is much easier for teachers

to focus on children’s behavior’’ (1985, p. 30, italics in original). This

was what the interviewed teachers did. They did not discuss the com-

plexity of the learning/teaching situation, nor talk about the different

forms of confidence that they had observed in their classes, other than

a few references to over-confidence. Their approach to confidence in

the mathematics classroom reflected the kinds of stereotypes in society,

discussed in the gender literature, of the independent male, the depen-

dent female. Their comments were also consistent with the literature

already discussed that does not treat confidence, or ‘ability’ as prob-

lematic constructs. There is a cause for concern when confidence is

connected to behaviour in that the notion of willing behaviour, shown

through hands up in response to questions, or volunteering an answer,

immediately taking up pen to write, showing commitment in body lan-

guage (see quotes in teacher section above), is confirmatory of much

observed male confident behaviour. Behavioural interpretations, such

as those of the interviewed teachers, can be misleading but can also

reinforce a particular kind of classroom culture that validates competi-

tion. Many students did not appreciate a competitive classroom cul-

ture and spoke of their preference for collaboration.

For the students, confidence was a product of how they felt to

which the contributory factors were how they worked, discussion,

challenging activities. They approached confidence in a much more

layered way, recognising its multi-faceted nature. To help to construct

and reinforce their confidence, they wanted a discursive environment

where they would be willing to question, challenge as well as ask, be

asked and answer, be expected to problem solve. They had a sophisti-

cated view of this classroom and a very few experiences in mathemat-

ics to back up this view, although they did describe experiences in

other subject lessons.

LESSONS FOR TEACHER EDUCATION

What are the lessons for teacher education that we can draw from this

study? Certainly in the UK, teacher education students are either

straight from school or university or, quite frequently, adults returning

to study. Many are not comfortable with notions about learning that

differ from widespread experience of transmission teaching, attainment

testing, and success gained often in the absence of understanding a

broader picture. Teacher educators, therefore, may be confronted with

370 LEONE BURTON

students who have a combination of concerns about their own learning

of mathematics with related worries about their understanding. Depen-

dence on a book, or the inevitable worksheets, is part of displacing the

responsibility for learning from their own shoulders, onto either the

book or the students. Either way, such a dependency culture does not

create the kinds of conditions where students feel excited, interested

and challenged to learn more mathematics.

So, if ‘‘confidence is everything’’, or, at least, if confidence motivates

and/or sustains learning and learning nurtures confidence, what might

teacher educators do to work on those feelings and practices that stu-

dents say constitute confidence, for them, in order to help to generate

or support them? In this respect, prospective teachers are students and

their experiences can be compared to those of the mathematics stu-

dents in the study. In a recent paper in JMTE, Andrea Lachance and

Jere Confrey discussed the building of teacher communities. They

wrote: ‘‘what supports the development of community among second-

ary mathematics teachers is similar to what we would define as a

‘good’ mathematics program for students’’ (2003, p. 133). I agree with

them, as I believe would the students in this study, and the reasons

they give for why creating teacher (and classroom) communities sup-

port effective teaching and learning. Fran Arbaugh, in the same issue

of JMTE, discussed self-efficacy, or confidence, in relation to teacher

development. She quoted a teacher saying something very similar to

one of the students I quoted above: ‘‘Sometimes it gives you the confi-

dence if someone else did the same thing you did and they had success

with it also. [It] also builds up your confidence that you’re doing it the

right way – a successful way’’ (2003, p. 153). This similarity between

the quotes of students and teachers could result from the teachers, in

Fran Arbaugh’s study, not being positioned as teachers, but behaving,

or feeling, like students. In my study, the teachers were reflecting upon

the learning of their students.

To see what features might help to construct classrooms that promote

confidence, we need to listen to what the students in the study were say-

ing. Whereas the teachers regarded ‘confidence’ as individual and

behavioural, the students were emphasising coming to know in a collab-

orative pedagogical climate, a community that questioned, conjectured,

built argument, challenged, justified and reflected. The feelings that are

provoked by such a climate were what they described as ‘confident’ and

resulted from the kind of pedagogic setting created by the teacher. The

teachers, too, recognised that such a classroom would be more consis-

tent with what the students preferred than a classroom built around

bookwork supported by worksheets and dominated by teacher/student


questioning. It is impossible to say if such recognition was rooted in

their beliefs about motivation, or how they understood mathematical

knowledge and its creation or learning. But they did acknowledge the

difficulties they had in creating and maintaining such a classroom.

It seems to me, therefore, that the results of this study direct teacher

educators towards two main features. The first is to manage their own

classrooms in a way consistent with meaning-making thereby provid-

ing a model to their teacher education students of what such a class-

room looks and feels like. The second feature is to provide students

with support and guidance in creating such classrooms for themselves

since it was the teachers who commented on their own difficulties with

implementing this style of classroom. Indeed, those of us who have

worked in such classrooms as part of teacher education courses can

attest to the success that they have in changing the attitudes and

expectations of the students. However, it is not easy to organise a

classroom where the mathematics is not prescribed but is generated

through the activities of the students and where it is the responsibility

of the teacher to help draw that mathematics out of the activities, help

the students to interrogate the many different forms of it which they

offer, and expect student involvement in the process of questioning,

justifying, challenging and reflecting.

These results pose teacher educators with dilemmas around the iden-

tification of the differing classroom needs both of their students, and

of their students’ subsequent students. The students in this study were

asking for opportunities to acquire mathematical meaning, rather than

be expected only to reproduce the meaning of others. This, in turn,

repositions learners from being dependent on their teachers to adopt-

ing agency for their own learning. The literature is very helpful in

resolving the dilemmas and identifying the responsibilities.

The agency of mathematical learners and how they come to create

and work with their meanings, their authorship (see Povey & Burton,

1999), is directly linked to differences between others’ meanings, that is

the codified, paradigmatic knowledge they are expected to learn and

reproduce and their personal meanings created and devised through

their own activities. As learners develop their own meanings, they find

their voices as mathematical authors and come, critically, to evaluate

these voices. For example, Raffaella Borasi referred to what she called

‘a successive draft approach’ to mathematics learning that ‘‘would

encourage students to pursue the solution to novel mathematical prob-

lems by attempting alternative approaches and critically examining

and building on their tentative results, whether correct or not’’ (1996,

p. 33). These are the very experiences for which the students were

372 LEONE BURTON

asking when they spoke about what made them feel confident. Experi-

enced in this way, the purpose of schooling in mathematics shifts from

the acquisition of knowledge ‘objects’ to the use of a reflective process

of coming to know within a learning community where discourse is

paramount. Measurement of success is then calculated not in the

reproduction of quantities of externally authored, disconnected facts

or skills, but in the mathematical ways through which learners demon-

strate their knowledge and skills in authoring their mathematics and

their discursive skills in argumentation and critique.

Barbara Jaworski (1999) pointed out that ‘‘the Purdue team, Cobb,

Yackel and Wood, took as central to their teaching experiment in sec-

ond grade mathematics lessons an analysis of processes of negotiation

and sharing of meaning in the construction of classroom mathematics’’

(p. 169) as did she, herself, in Jaworski (1994) (also see, for example,

Brown & Walter, 1990; Burton, 1999; English, 2002 and other refer-

ences below). Such an approach to learning is closely connected to

feelings of self-efficacy and confidence, as has been pointed out by

these, and other, students (see, for example, Burton, 2001), and teach-

ers (as above).

The notion of ‘confidence’ was very present in the discourses of both

teachers and students just as my review (above) pointed to its presence

in the literature. It might be said, therefore, that, in constructing the

interviews in the way that I did, I ‘led’ the interviewees to use the word

‘confidence’ as a way of talking about success in the learning of mathe-

matics and I am very conscious of the ‘researcher’s always-present’

responsibility when interviewing. In response to this, I would say that

neither the teachers nor the students required or sought explanations or

definitions of this term from me. On the contrary, I was looking for

elucidation from them. In the teachers’ case, asked to describe ‘confi-

dence’, they spontaneously adopted the conventional articulations that

link ‘confidence’ with successful performance. In the students’ case,

either it was they who introduced ‘confidence’ in response to questions

about how they learnt or they responded to a question from me about

when, and under what circumstances, they did, or did not, feel ‘confi-

dent’. However, the ways in which they described what, for them, con-

stituted confident and unconfident behaviour was not what I expected

and was certainly different from how the teachers spoke of confidence.

My argument is that, indeed, ‘confidence’ is part of the accepted class-

room discourse of teaching and learning mathematics and that, conse-

quently, it takes little encouragement from a researcher, for teachers

and students to enter that discourse. Nonetheless, my study shows that

teachers and students construe that discourse in very different ways.


Whereas, for teachers, ‘confident’ was a way of describing the behav-

iour of those who are successful performers at mathematics, for stu-

dents, speaking about ‘confidence’ opened a complex discussion of

preferred learning and teaching styles, experiences, understanding and

feelings. Although the students acknowledged the connections they saw

between confidence, work and success, that discourse seemed to relate

to that of the teachers quite closely. However, in discussing ‘confi-

dence’, they spoke spontaneously about working collaboratively, under-

taking open problems in ways that gave them agency, and disliking

competition so that the linguistic term seemed to operate, for them, as

a marker for describing more complex issues.

CONCLUSIONS

If we listen to the voices of the students who spoke with me, they are

confirming that confidence and success are closely intertwined. But

their descriptions of the environment necessary to such positive out-

comes rejected dependency. They wanted (although not in these terms)

agency, authorship, collaboration and reflection. Teachers, it seems to

me, have a responsibility to be more critical of the behavioural infor-

mation on which they appear to lean and try to ensure that these stu-

dent needs are met. Teacher educators have a responsibility to feature

these needs in their mathematics courses and emphasise their nature

and how they can be created and nurtured. The link between confi-

dence and success in mathematics appears to be robust for teachers

and students. But, it is too easy to label pupils as ‘confident’, or not.

Much more demanding on teachers is to create a classroom in which

the nature of confidence and its frailty is acknowledged and the condi-

tions put in place to help it flourish.

APPENDIX A

Students

The interview began with the first two requests below and included the remainder

but not necessarily in the same order. The interviewees dictated the flow and order.

The final question concluded the interview.

� Describe a normal/typical maths lesson.

� Describe a lesson particularly enjoyed. Why?

� Lesson particularly disliked. Why?

� When, and why, chose to take mathematics at ‘A’ level?

374 LEONE BURTON

� Describe maths.

� What attracts you to it?

� When you are talking to people who don’t do maths, what do they say to you

when you say you are doing ‘A’ level maths?

� More maths after ‘A’ level?

� Feel about maths? Useful subject to study? Why?

� If you get stuck on something in class, what do you do?

� Would you say you are a confident person? What does it mean to say you are/

are not confident? At all times or only under certain circumstances? When? In

maths?

� How does it feel to be confident? In maths?

� When you don’t feel confident, how does it feel? In maths?

� Is there anything that a teacher can do to make you more/less confident?

� It is said that boys are confident in maths and girls aren’t. Is this the case in

your experience?

Teachers

The interview began with the first three questions, included the other topics, but not

necessarily in the same order, as well as points raised by the teachers. The intervie-

wees dictated the flow and order. The final question concluded the interview.

� Is confidence important to success in learning mathematics?

� What is confidence?

� How do you recognise it? (In particular, in the case of the pupils I am meeting).

� Specific to maths?

� Personal, or departmental, strategies to boost confidence?

� Teaching style?

� People say that boys are confident in maths and girls less so. Is that your expe-

rience?

APPENDIX B

STUDENT CATEGORIES

Please note that the only categories included here are those that attracted more than

three comments. Also note that, as students were interviewed in pairs, there was fre-

quently agreement by both of the pair to a particular comment. The categories relate

to the discussion themes listed in Appendix A, not all of which are addressed in this

article.

Normal maths lesson

1. Working from books/exercises/blackboard examples/checking homework (all

students in all four schools).


Lesson enjoyed

(three students in one school commented on lesson similarity making it hard to think

of a particular lesson enjoyed):

1. active (all 4 schools, 15 specific comments made but pair often confirmed with-

out commenting); (Mastermind lesson, School 1, all students);

2. coursework (eight comments in two schools).

N.B. Teacher-led discussion (one comment); teamwork (one comment); oral les-

sons/group discussions (one comment); all of these students, who were from

three different schools, confirmed that lessons of this sort did not happen in

mathematics.

Lesson not enjoyed

1. failure to understand (six comments in three schools);

2. repetitive (six comments in two schools);

3. doesn’t relate (three students in two schools).

Choice of ‘A’ level

1. favourite subject/enjoyment (five comments, two schools);

2. GCSE (16+ examination effect) (10 comments, all 4 schools);

3. critical filter to university (13 comments, all 4 schools).

Other’s reactions

1. difficulty/cleverness/mad (13 comments, all four schools);

2. bad feelings (boring/shock/lack of enjoyment/memory) - (six comments, three

schools).

‘A’ level maths10 comments, 4 negative, 6 positive (2 schools) specifically on course.

1. maths is Useful (seven comments, all four schools);

2. maths is Connected (nine comments, all four schools);

3. maths is Fragmented (four comments, in one school);

4. understanding for yourself (three comments, two schools);

5. right/wrong (three comments, two schools);

6. different ways of getting answers (three comments, two schools).

Maths after ‘A’ level?No (20 students); Don’t know (one student); Going into computing/science (five

students).

Confidence

1. importance of (three comments, two schools);

2. understanding, as well as knowledge (nine comments, two schools);

3. can do (five comments, all four schools);

4. right answers (six comments, all one school);

376 LEONE BURTON

5. working together (14 comments, all four schools);

6. working with friends (16 comments, all four schools);

7. working on your own (two supportive comments, one school; four comments

indicating that class is organised to work on their own, but in fact they work

with their friend, two schools).

Lacking confidence7 comments, all by male students, in two schools.

Competition

1. Positive comments by three male students in one school.

Feelings15 comments, 3 schools.

Success

1. genetic (six comments, three schools);

2. product of confidence and work (16 comments, all 4 schools);

3. teaching/support/working atmosphere (five comments, two schools).

Teachers should/should not21 positive comments, all 4 schools; 18 negative comments, all 4 schools.

Girls23 comments, all 4 schools, themes: confident, clever, concentrate, ask for help, lis-

ten, co-operative, willing to try.

TEACHER CATEGORIES

Please note that seven teachers were interviewed and the themes arose either from

my questions, or the student comments. Where a teacher has commented on a

theme, I indicate by teacher/school.

Confidence important?

All teachers agreed that confidence is important.

What is confidence?

1. willingness to try, move on (all teachers in all four schools);

2. an understanding plus questioning, able to talk (Teachers School 2, School 3);

3. body language (Teachers, School 1, School 2, School 4).


Do anything to enhance confidence?Year 7 Programme (School 1).

1. structure of course, feedback, use ‘‘simpler’’ examples (School 2);

2. problem solving book (School 3);

3. deliberate teaching strategies, e.g. hints, written comments, no departmental

strategy (School 4).

Teaching style

1. transmission (Teacher 1, School 1; Teacher 1, School 2; Teachers, School

4 – regretfully, Teacher 1);

2. example based – discursive (Teacher 2, School 1; Teacher 2, School 2);

3. textbook-based (all four schools);

4. problem-solving book/puzzles (Teacher, School 3).

Discussion

1. ‘‘class participation’’ (Teacher 2, School 1);

2. using discussion points, in book, expecting them to talk (Teachers, School 2);

3. ‘‘work in groups’’ / ‘‘do something on their own’’ (Teacher, School 3).

Boys/Girls

1. male competition (Teacher, School 2);

2. female uncertainty (Teacher, School 2);

3. need to discipline boys (Teacher, School 3);

4. girls show responsibility (Teacher, School 4).

ACKNOWLEDGEMENTS

I would like to express my appreciation of the comments on this paper

by the anonymous reviewers and the Editor of this Journal, as well as

the critical readings by Helen Forgasz, Christine Hockings and

Heather Mendick.

NOTES

1 This paper was first given as a regular Lecture at the ICMI - 2nd East Asia Regio-

nal Conference on Mathematics Education and 9th South East Asian Conference on

Mathematics Education, May, 2002, Singapore.2 The end of compulsory schooling in England and Wales is marked by the General

Certificate of Secondary Education taken, usually, at 16 plus years. Students may

then follow an optional programme towards an ‘Advanced’ level qualification which

378 LEONE BURTON

is recognised for entry to university. The students in this study were all on such a

two-year programme.3 Direct quotations taken from the transcripts are in quotes, except where offset,

and are reproduced exactly as spoken.

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BOOK REVIEW

Thomas P. Carpenter, Megan Loef Franke and Linda Levi (2003).

Thinking mathematically: Integrating arithmetic and algebra in the ele-

mentary school. Portsmouth, NH: Heinemann. ISBN 0-325-00565-6.

INTRODUCTION

‘Thinking Mathematically’ is both one of the most exciting books for

primary school teachers that I have read recently, and in some ways

one of the most frustrating. As the title suggests, the book addresses a

way in which it is possible to work with children in elementary school

on the inter-related nature of arithmetic and its generalisation into

algebra.

The book is a development of the ideas from the authors’ (Carpen-

ter et al., 1999) previous book Children’s Mathematics written in con-

junction with Elizabeth Fennema and Susan B. Empson. In that first

text the reader was introduced to Cognitively Guided Instruction as a

method of working with elementary children on problems solving,

mathematical communication, and teaching for understanding. ‘Think-

ing Mathematically’ takes these ideas, and the children, further, into

the world of generalised arithmetic and algebra. For readers of ‘Chil-

dren Mathematics’ the format will be familiar. The text is clearly set

out, with shaded areas containing examples and quotations from the

teachers who were involved in the project, and a CD is included con-

taining video clips of class teaching and individual interviews with chil-

dren explaining their mathematical thinking.

It starts from the premise that ‘for many students and adults, arith-

metic represents a collection of unrelated and arbitrary manipulations

of numbers and equations and algebra is perceived as a separate col-

lection of meaningless procedures that are only tangentially related to

arithmetic’ (p. 133). Carpenter, Franke and Levi draw on their work

over five years with a group of elementary school teachers and chil-

dren in the United States. The study has focused on encouraging chil-

dren to think more fluently about the arithmetic they encounter in

elementary school. Their intention is that through reading this book,

and using the related CD, teachers will be enabled to develop the



understanding and skills to carry out such teaching in their own class-

rooms.

THE CONTENT OF THE BOOK

The content is exciting for those of us who believe that even very

young children are able to think logically and mathematically when

given the right contexts. Following an introduction in Chapter 1 which

discussed the purpose and structure of the book, Chapter 2 offers ideas

for the introduction and development of the concept of equality. This

emphasises the need to teach children that the equals sign indicates

equality on either side rather than being an instruction to carry out a

calculation and write an answer. A set of benchmarks describe the

stages children may pass through in their development of this concept,

which could be used by teachers to evaluate their children’s responses.

The causes of misconceptions about the equals sign are discussed and

ways of teaching that avoid or remediate such misconceptions are sug-

gested. For example, they suggest that not all equations are written in

the form 3 + 5 =h, since children then interpret the equals sign as

an instruction to add, but to consider equations such as 3 + ? ¼ 8,

3 + 5 ¼ 6 + 4, etc. They also suggest that teachers avoid the use of

the equals sign ‘as a shorthand for a variety of purposes that do not

represent a relation between numbers’, such as J J J J J ¼ 5

(p. 20).

This discussion of children’s understanding of equality moves, in

Chapter 3, into the development and use of relational understanding

as children are asked to solve equations such as 43 + 28 ¼ h + 42

without the need to calculate. One teacher, Ms K., is shown teaching a

single pupil, Emma: together they work on a series of calculations

which move Emma from using calculation to using relational thinking

to solve such problems. The authors then discuss the use of true and

false to label equations, starting with simple statements such as 12 ) 9 ¼3, then moving on to statements which encourage the use of relational

thinking such as 27 + 48 ) 48 ¼ 27, and 54 + 17 ¼ 17 + 54.

Chapter 4 focuses on the development of conjectures as children are

asked to make their implicit knowledge explicit. At this stage the con-

jectures are articulated in everyday language e.g. ‘when you add a

number to another number and then subtract the number you added,

you get the number you started with’ (p. 54), or ‘when you add two

numbers, you can change the order of the numbers you add and you

will still get the same number’ (p. 55). A summary of conjectures

384 BOOK REVIEW

about basic properties of number operations follows, including addi-

tion and subtraction involving one, multiplication and division involv-

ing zero and one, and commutativity.

Equations with multiple and repeated variables are presented, in

Chapter 5, through contextual tasks such as the distribution of a set of

mice across two cages. While the total number of mice remains the

same, the distribution across the cages can be represented by a range

of additions: 5 + 2 ¼ 7, 1 + 6 ¼ 7 etc. and the use of symbols, such

as x + y ¼ 7, to record these variables is discussed. The authors

emphasise to the reader that ‘the primary goal in giving students these

number sentences is not to teach students efficient ways to solve alge-

braic equations; it is to engage them in thinking flexibly about number

operations and relations’ (p. 73). On the CD, children are shown solv-

ing more complex equations such as 3 · p + p + 2 ) p ¼ 17. Some,

like Susan, use trial and error, ‘I first tried 4 … That was too small so

I tried a bigger number, 6. That was too big so I tried 5…’ (p. 73).

Erika ‘is more flexible in her thinking’ arguing that ‘Well, there’s a p

and you take away a p so it’s like it was never there. So it’s like 3

times p plus 2 equals 17, and 15 and 2 is 17. So 3 times p is 15, and

that makes p 5’ (p. 74). The authors assert that the children are not

being taught solutions to such problems but encouraged to explore

and discuss relationships to create their own solutions.

The use of symbols is extended in the following chapter to represent

conjectures, leading on to early ideas on justification and proof. Levels

of justification are discussed and examples show these different forms

of justification used by the children. The children are encouraged to

produce generalisations even when they are not able to prove them.

The final chapter addresses the ordering of multiple operations and

if…, then… statements. Towards the end of this chapter the text

moves from examples of work with the children to address the tea-

cher’s own level understanding of how to prove, for example, ‘why we

invert and multiply when we divide fractions’ (p. 128).

So, the book and CD clearly show some young children, beginning in

kindergarten, engaging with these abstract ideas; starting from arithme-

tic and moving through the language of generalisation towards the use

of algebraic symbolism. As such it is an exciting and stimulating read.

However, while celebrating the clarity of the text and the use of real

examples from the classroom to further illuminate this, I had several

problems with the book. One is mainly practical, relating to the CD,

while the others are more fundamental and relating to the intended

readership.

385BOOK REVIEW

ISSUES RELATED TO THE USE THE BOOK

Using the CD in Connection with the Book

The book is supported by an excellent CD, showing examples of chil-

dren working one-to-one with a researcher and in classroom situations.

However there is an assumption that the reader will have easy access to

the material on the disc as the introduction recommends that a ‘reading

of the text would not be complete without viewing the accompanying

episodes’ (p. xii). The disc requires the use of QuickTime 5.0 (or higher)

which can be downloaded free via the internet, and easily carried out at

the university. However I found that at home, without the use of Broad-

band, the downloading was difficult, expensive and time consuming.

This would be a real disadvantage for many primary school teachers

who may not have easy access to fast internet connection at school and

might, at home, find the hassle too much and give up. If the software is

free, would it not have been possible to include it on the CD?

Also, the need to watch an extract from the disc at a specific time

meant that I, as a reader, was tied to the office – not my favourite

place for reading. While the video extract was enjoyable and would be

useful for discussion in groups, for the individual reader a fuller tran-

script of the incident in the text would have allowed great freedom of

use. This raises the question of intended readership.

Who is this Book for?

This book could be used by primary/elementary teachers themselves,

or used by teacher educators in pre-service or in-service work. The

audience for the book is clearly identified by the authors as the ele-

mentary school teacher, who is addressed throughout. In the introduc-

tion (p. xi) the authors state that ‘the goal is to help you understand

your own students’ thinking so that you can help them to make sense

of the mathematics they are learning.’ But, while recognising that

many adults do not have the relational understanding of arithmetic

and algebra described here, the authors do not give evidence that

teachers will acquire this understanding through the reading of the

book.

The inclusion on page vii of the forward of the equation

½ðk� 2Þ þ 1� þ ½ðh� 2Þ þ 1� ¼ ðkþ hÞ � 2þ 2 ¼ ðkþ hþ 1Þ � 2

which aims to describe ‘when you add an odd number to another odd

number, the answer is an even number… represented compactly’

386 BOOK REVIEW

(p. vi) would, I suspect, ensure many elementary school teachers would

immediately close the book and find something better to do.

There has been considerable work carried out in the UK, and else-

where, on student teachers’ understanding of mathematical subject

knowledge, which finds that many student teachers are particularly

insecure in their own understanding of algebraic concepts and are

often afraid of algebraic notation. For example, in England, Rowland

et al. (2000) found only 43% of the postgraduate pre-service primary

teachers at their university were secure in reasoning and mathematical

argument while in Belgium, van Dooren et al. (2003) found a signifi-

cant subgroup of pre-service primary teachers were unable to use alge-

braic methods even after 3 years of teacher education.

The authors have worked with the teachers cited in the book over a

period of time and show exciting teaching and excited children engag-

ing with issues of symbolism, justification and proof. However the

assumption that an individual primary/elementary school teacher could

engage with the ideas alone, through the reading of the book, remains

to be shown. Given the research evidence it seems likely that many

teachers, in British primary schools and perhaps in those of other

countries, would be unable, or unwilling, to engage with the mathe-

matics required to understand this book on their own.

In-service Teacher Education

Could the book therefore be used for those working with in-service

teachers? Barkai et al. (2002) studied in-service elementary teachers in

Israel – giving them statements such as ‘the sum of any five consecu-

tive integers is divisible by 5¢ – to prove or refute. They found ‘a sub-

stantial number of teachers applied inadequate methods to validate or

refute the propositions … many teachers were uncertain about the sta-

tus of the justifications they gave’ (p. 57). They conclude that ‘it seems

essential that professional development programmes attempt to

enhance elementary school teachers’ algebraic knowledge to determine

the validity of their students’ conjectures’ (p. 63–64). So, there is a

need for such in-service work: does ‘Thinking Mathematically’ provide

a resource for these programmes? The answer has to be both yes and

no!

The book provides a valuable resource for in-service teacher educa-

tion, to enable teachers to understand the powerful algebraic founda-

tions of the informal calculation methods invented by children in

their classes. With the support of teacher educators it would be pos-

sible to use the book and the CD to demonstrate what it is possible

387BOOK REVIEW

for children and teachers to do in the classroom, while at the same

time giving the teachers a range of activities to support and develop

their own algebraic confidence and competence. But for this purpose

I, as an in-service teacher educator, would require more that we are

offered here.

Continued professional development of teachers in Britain is

mostly at Master’s level and would require the teachers to under-

stand much more of the research background to the project than

we are offered here. Little is said about the research base, the num-

ber of teachers involved, the number of children, the time spent on

the project in each classroom. There is no reference at all to the

sort of research project it is. What was the teachers’ involvement?

Are they involved in the research or being researched upon (Jawor-

ski, 2003)? There is no data offered to indicate the success of the

project other than by the descriptive accounts given in the text. The

reader is given the impression that all teachers and all children were

able to benefit from this approach but there is no hard evidence of

this. Were there teachers who dropped out of the project? Were

there children who could not form conjectures or understand those

formed by their peers? In many of the CD excerpts we see one

child answering confidently, but what of the rest of the class? With-

out the answers to these questions we have to take the evidence of

the book on trust.

There is also very little reference to research literature. Only three

references are given in the entire book; one of these to the National

Council of Teachers of Mathematics (2000) ‘Principles and Stan-

dards for School Mathematics’. This is in stark contrast to the for-

mat of the authors’ previous work, ‘Children’s Mathematics’, which

contains a very useful appendix on ‘The Research Base for Cogni-

tively Guided Instruction’ supported by 22 references to published

articles on the subject. Children’s Mathematics was also available

with a Workshop Leader’s guide for Professional Development pro-

grammes. Perhaps these resources are being developed. If so they

could, I believe, be a valuable resource for mathematics education

community.

Preservice Teacher Education?

My initial response to the question of the book’s relevance to initial

teacher education was a negative one. Could the students I work with

engage with this material in a meaningful way? I doubted it. However

I also heard myself thinking – How do you know? What evidence do

388 BOOK REVIEW

you have? These questions gave direct rise to the development of a ses-

sion on one of our pre-service programmes.

The students on this programme are all teaching assistants in

schools, studying part-time to gain qualified teacher status. As such

they are generally mature students, with non-standard entry qualifica-

tions and many entered the programme with serious insecurities and

negative attitudes to mathematics. This is a four year programme and

the students with whom I was working are in their third year: 50 stu-

dents, organised into two teaching groups. They have generally strug-

gled with the mathematics at their own level, although, since they have

considerable experience of working with children in school, often sup-

porting children with learning difficulties, their understanding of chil-

dren’s learning is very good.

The session was originally planned to address early algebra in

terms of number patterns and incomplete number sentences of the

form 7 + h ¼ 12. However, following my reading of the book, and

in discussion with a colleague, we decided to try out some of the

ideas in the book. Starting from concepts of equality, then adding 0

to any number, the students developed conjectures and wrote alge-

braic equations for these. They created equations for the commuta-

tive and associative rules and recognised these as rules they had

learnt in their first year course. For example, starting from a given

addition e.g.

56þ 27 ¼ 83

they talked about how they solved

56þ 28 ¼

then generalised this verbally e.g. ‘if you add two numbers together to

get a total, when one is added to the second number the total will be

one more’, and were able to write an equation such as:

xþ ðyþ 1Þ ¼ ðxþ yÞ þ 1

For some students this gave new insight into the use of brackets. For

the first time they realised why they needed the brackets, to represent

their spoken language. They then realised that this did not only apply

to +1 but to the addition of any number resulting in

xþ ðyþ zÞ ¼ ðxþ yÞ þ z

389BOOK REVIEW

which they recognised as the associative law for addition. They

expressed amazement that this law, which they had learnt but not

really understood earlier in the programme, was relevant to aspects of

the mathematics they taught in school, and also that they had been

able to generate the algebraic equation for it so easily.

We also discussed what age the children they were teaching would

be able to engage with the ideas and they were very vocal about the

relevance to the children in their classes (aged 4–11 years). In the

National Numeracy Strategy (DfEE, 1999), which they use to plan

and teach in school, the concept of using the answer to one calculation

to find the answer to a similar problem is explicitly taught. But the use

of an explicit conjecture to form a generalisation for this understand-

ing was new to them. Many students commented on how being

encouraged to express the generalisation in words increased their own

understanding and would be useful with children.

At the end of the session almost all the students said that this had

been one of the best mathematics lessons they had ever had. It had

allowed them to make some early links between arithmetic and algebra

which they had not previously understood, and they were enthusiastic

about the way in which these ideas could encourage children to suc-

ceed in mathematics at a higher level, in a way in which they them-

selves had not been to in secondary school. This session, while not

specifically using the text or video clips from the book, addressed the

basic concept of integrating arithmetic and algebra at an elementary

school mathematics level, and demonstrated that, for these students at

least, the integration was helpful to improve their own understanding

of the arithmetic ideas and the progression into algebraic ideas. Fur-

thermore, they could see the potential for this integration when work-

ing with elementary aged children. In future sessions I would wish to

use the text and the video clips more explicitly to show how this

potential could be realised.

This is, for me, the beginning of new ideas for working with all our

students. Will it work for the one year Postgraduate Certificate in

Education students? What about the more traditional BA students,

many of whom come straight from school with less experience of chil-

dren’s learning? These are things I will have to explore, and must leave

the reader to explore in their own context.

I have therefore come to the conclusion that the book, despite its

shortcomings, provides much that will be of interest to those

involved in mathematics teacher education at primary/elementary

level, and look forward to seeing how these ideas can be used in

390 BOOK REVIEW

teacher education and in schools to increase children’s mathematical

understanding.

REFERENCES

Barkai, R., Tsamir, P., Tirosh, D. & Dreyfus, T. (2002). Proving or refuting arithme-

tic claims: the case of elementary school teachers. PME 26 proceedings, Vol. 2

(pp. 57–64), Norwich, UK.

Carpenter, T., Fennema, E., Loel Franke, M., Lebi, L. & Empson, S. (1999). Chil-

dren’s Mathematics: Cognitively Guided Instruction. London: Heinemann.

DfEE (1999). The National Numeracy Strategy: Framework for teaching mathematics

from Reception to Year 6. London: DfEE/ CUP.

Jaworski, B. (2003). Research practice into/influencing mathematics teaching and

learning development: Towards a theoretical framework based on co-learning part-

nerships. Educational Studies in Mathematics, 54(2&3), 249–282.

National Council of Teachers of Mathematics (NCTM) (2000). Principles and Stan-

dards for school mathematics. Reston, VA: NCTM.

Rowland, T., Martyn, S., Barber, P. & Heal, C. (2000). Primary teacher trainees’

mathematical subject knowledge and classroom performance. In T. Rowland &

C. Morgan (Eds), Research in mathematics education, Vol. 2, London: British Soci-

ety for Research into Learning Mathematics.

van Dooren, W., Verschaffel, L. & Onghena, P. (2003). Preservice teachers’ preferred

strategies for solving arithmetic and algebra word problems. Journal of Mathemat-

ics Teacher Education 6(1), pp. 27–52.

Oxford Brookes University Alison PriceHarcourt Hill CampusOxford, OX2 9ATUK

391BOOK REVIEW

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