writing in calculus and reflective abstraction

Journal of Mathematical Behavior21 (2002) 255–282

Writing in calculus and reflective abstraction

Laurel Cooley∗

Department of Mathematics and Computer Studies, York College, City University of New York, Jamaica, NY 11451, USA

Abstract

Reflective abstraction is a term that has been used in various, yet similar, ways in mathematics education. The useof this term is examined with a final definition honed from this group and used for the purposes of this study. For thisstudy, reflective abstraction was defined as a mechanism for the isolation of particular attributes of a mathematicalstructure that allows the subject to construct or reconstruct knowledge that is new; that is, not previously known.

With this definition of reflective abstraction as a basis, seven formal writing assignments were developed for acalculus I class. Students completed these writing assignments and returned them in a journal which was read andresponded to by the professor. In some cases, students were required to rewrite their answers.

This group of students was at a public 4-year college in a metropolitan area. The majority of the studentsspoke English as a second language. The college had recently begun implementing a writing across the curriculumcomponent and this study was born out of these efforts.

Since none of the students had studied calculus prior to this class, any construction of knowledge about thecalculus concepts was assumed to have been developed during the course of this class. The writing from thestudents demonstrated that they had reflected on the calculus concepts and constructed meaningful ideas aboutfunctions and calculus. This understanding was sometimes shown through personal analogies and sometimes ina more formal way. The students were actively engaged in writing and produced thoughtful descriptions in mostcases. The process proved to be an important conduit for information between professor and student as well as astrong tool in dissuading misconceptions and promoting reflection in their thinking.© 2002 Elsevier Science Inc. All rights reserved.

Keywords:Writing and mathematics; Language and mathematics; Cognitive theory; Conceptual knowledge; Calculus;Constructivism; Epistemology

1. Introduction

This study involved designing and integrating writing assignments into a calculus class with the goalof promoting reflective abstraction. For some time, researchers have had a strong interest in the ways that

∗ Tel.: +1-718-262-2545.E-mail address:[email protected] (L. Cooley).

0732-3123/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.PII: S0732-3123(02)00129-3

256 L. Cooley / Journal of Mathematical Behavior 21 (2002) 255–282

language and mathematics learning interact. There has also been a growing awareness of the importanceof writing and the development of mathematical concepts. Two key questions guided this study: inmathematics, can writing be helpful in the learning process? More specifically, can writing assignmentsassist students in the process of reflective abstraction?

The objective of this study was to determine if writing assignments in a calculus I class would have anyeffect in promoting reflective abstraction. Students were directed to complete writing assignments thatwere designed to focus the student on particular topics. These writing assignments were then collectedand evaluated by the professor who would return them with comments, sometimes asking the studentsto rewrite them. At the end of the semester, the journals in which the students kept all of their writtenwork were collected. These journals were evaluated for evidence of whether the process of reflectiveabstraction had taken place. Examples of these instances are given for each of the seven formal writingassignments. In addition, because the writing assignments facilitated communication between studentand professor, it became clear that this was a valuable tool for determining when students were havingmisunderstandings. Examples of these types of written responses are given as well.

2. Relevant literature

There are two main aspects to this study. The first concerns reflective abstraction and the other is writingin mathematics. Therefore, the literature focuses on these two themes, or closely related issues.

2.1. Reflective abstraction and construction of knowledge

In order to discuss abstract thought and reflective abstraction, these terms first need to be defined. Manypapers have been written on these subjects, or subjects closely related to them, with just as many varieddescriptions. Here, I will concentrate on constructivist writings, and the interpretations of these conceptsfrom this theoretical viewpoint. Then, I will offer my own synthesis which I used to inform the design ofthe writing assignments.

An originator of constructivist ideas,Beth and Piaget (1966)spoke mainly of two forms of abstraction,empirical and reflective. Empirical abstraction, the less sophisticated of the two, consists of elicitingcommon attributes from a category of objects. Empirical abstraction refers to objects that are obviousto the subject, who records certain properties in order to excerpt and analyze them, abstracting fromthe perceived objects. However, Piaget points out, in physics and a fortiori in mathematics, there is asecond form of abstraction, called “reflective abstraction.” This second type of abstraction applies tothe subject’s actions and operations as well as to the schemata which it guides the subject to construct.Reflective abstraction may be understood in two different, but associated, ways. To begin with, it maybe understood as “reflection” from a lower to a higher level (for example action to representation), oras a reconstruction, on a new level, of what is elicited from the preliminary one. Therefore, reflectiveabstraction — abstraction starting from actions and operations — differs from empirical abstraction inthat reflective abstraction is fundamentally constructive. (Beth & Piaget, 1966, pp. 188–189).

When properly understood, reflective abstraction is the mechanism which spurs development of in-tellectual thought. Reflective abstraction differs from empirical abstraction in that it deals with actionsas opposed to objects, so that it is concerned, not so much with the actions themselves, but with theinterrelationships among the actions, whichPiaget (1976, p. 300)called “general coordinations.”

L. Cooley / Journal of Mathematical Behavior 21 (2002) 255–282 257

The Piagetian process of reflective abstraction is thus understood as the following: the subject observesthe results of actions performed upon any objects resulting in logico-mathematical experience. The conse-quence of this experience is interpreted by the schemas of the actions developed by the subject. In order toobserve this output, the subject carries out other actions, using the same schemas as those the interactionof which must be considered. However, the form is new for the subject because the logico-mathematicalexperience teaches her or him something he or she was not conscious of formerly. Therefore, theabstraction by means of which the subject brings forth new knowledge involves construction. Thisconstruction, or reconstruction, replaces experience or empirical procedures for the subject at a newplane.

Twenty years later,Dubinsky and Lewin (1986)expanded on Piaget’s work, especially his concept ofgenetic epistemology. They explain that an educator who is attempting to teach new conceptual material,as in mathematics, is in effect trying to induce cognitive development. They focus on Piaget’s ideas of“equilibration” and “reflective abstraction.” Equilibration is defined as the process by which the subjecttries to understand a concept by putting it in the context of her or his overall cognitive system. Thisattempt to cognitively construct an understanding is through the process of reflective abstraction, whichPiaget defined.

In another viewpoint of abstraction,Dienes’ (1971, 1978)describes it in terms of concentrating oncommon properties. His psychological studies of children were strongly influenced by his perception ofthe nature of mathematics which he believed was a discipline created through the centuries by means ofsuccessive abstractions. Therefore, he suggests that abstraction is the main characteristic of mathematics,as well as the key point in the learning of mathematics. He offers as examples some areas of mathematics,such as number theory, which can be developed by an in-depth study of a specific situation such as thenatural numbers. He admits, however, that concentrating only on common properties can be limiting.Mathematics can have important commonalities in some cases of a concept, but not all. Instances wherecommonalities do not exist are also important for further understanding and development.

All of this points to an ethereal quality of reflective abstraction which makes it difficult to definein concrete terms. Something that is not easily defined may not be easily recognized. The abstractionof mathematics is one of the main obstacles to its appreciation. For example,Mason (1989)finds thatstudents of mathematics often say that they find it abstract, and hence dislike it. Yet it is the abstractionof mathematics that gives so much pleasure to mathematicians. Along the same lines,Eco (1988), indiscussing the meaning of aesthetic in the work of St. Thomas Aquinas, writes: “Aesthetic seeing involvesgrasping the form in the sensible. It therefore occurs prior to the act of abstraction, because in abstractionthe form is divorced from the sensible.” (p. 193)

Mason’s insight is useful because it helps us to understand a student’s experience of mathematics. Thestudent’s sense of abstraction is removed from or divorced from reality or meaning. In a related manner,Borasi (1984)proposes that students’ difficulties with abstraction have little or nothing to do with theprocess of abstraction. She explains, for example, that the primary objective during a history class wasnot to have students make a rapid advancement through levels of progress.Borasi (1984)notes that eventhough number theory can be treated abstractly today, this is, in part, because Cantor developed his settheory more than 2000 years after mathematicians started to study and discover the properties of thenumbers 1, 2, 3,. . . . However, students are expected to progress rapidly through mathematics, learningwhat others have done, and not necessarily participating in the abstraction process. Ultimately, Borasidefines abstraction as: “a class of situations and each situation belongs to this class because of a certainproperty it has and any other properties it might have are, for the moment, considered as irrelevant” (1984,


p. 14). In a like manner,Tall (1988)states: “abstraction (as) the isolation of specific attributes of a conceptso that they can be considered separately from the other attributes.”

Correspondingly,Mason (1989)proposes that the use of the word abstract in mathematics by bothstudents and professionals refers to a common experience, an extremely brief moment. There is a “delicateshift of attention” from seeing an expression as an expression of generality, to seeing the expression asan object or property. Thus, abstracting exists between the expression of generality and the manipulationof that expression while, for example, constructing a convincing argument. He argues that when the shiftoccurs, it is hardly noticeable. To a mathematician, this is very natural. Hence, he states, when the shiftdoes not happen for the student, it blocks advancement.

Dienes, Mason, and Borasi, all discuss abstraction in some way in terms of the common propertiesof a group of objects. Recognizing common properties may lead to specific attributes or may assist inconstituting generalizations. For example,Thurston (1990)believes abstracting is intimately linked togeneralization. He describes the general nature of the results that can be obtained through abstractionas one of its principal incentives. Another primary motive is the achievement of synthesis. However,for Thurston, the difference between simple generalization and abstraction is that generalization usuallyinvolves an expansion of the individual knowledge structure, adding more objects to the category, whileabstraction is likely to involve a mental reconstruction.

Thurston argues that abstraction thus contains the potential for both generalization and synthesis andthat these two processes define its purpose. The nature of the mental process of abstracting is, however,very different from that of generalizing and synthesizing. Abstraction, as he explains it and as similarlydiscussed by previously mentioned authors, is first and foremost a constructive process — the buildingof mental structures from mathematical structures, i.e., from properties of and relationships betweenmathematical objects. He also believes this process depends on the isolation of appropriate propertiesand relationships. It requires the ability to shift attention from the objects themselves to the structure oftheir properties and relationships.

Finally, similar to Mason’s shift of attention, Thurston explains that such constructive mental activityon the part of a student is heavily dependent on the student’s attention being focused on those structureswhich are to form part of the abstract concept and drawn away from those which are irrelevant in theintended context. In other words, the structure becomes important while irrelevant details are beingomitted, thus reducing the complexity of the situation.

Southwell (1988)explains the processes involved in reflecting on experience as association, integra-tion, validation, and appropriation. She, like all of these researchers, believes that new ideas need to beassociated with knowledge that has already been acquired. She explains that these associations need to beintegrated methodically into a new whole, and that the new ideas must be validated by the learner’s previ-ously held ideas. Then for some, though not all, the integrated ideas become a part of their value system.

Dubinsky (1991a, 1991b)describes reflective abstraction as a concept introduced by Piaget to describean individual’s construction of logico-mathematical structures during the course of her or his cognitivedevelopment. He notes two important observations made by Piaget: first, that reflective abstraction hasno absolute beginning, but rather is present at the very earliest stages in the coordination of sensorimotorstructures (Beth & Piaget, 1966, pp. 203–208); and secondly, that it continues through higher mathematicsto the extent that the development of mathematics from antiquity to the present day may be consideredas an example of the process of reflective abstraction (Piaget, 1985, pp. 149–150).

A synthesis of these ideas produced the definition of reflective abstraction that is used for this study:reflective abstraction is a mechanism for the isolation of particular attributes of a mathematical structure

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that allows the subject to construct or reconstruct knowledge that is new; that is, knowledge not previouslyknown. A feature of reflective abstraction is that it clarifies and organizes logico-mathematical experiencesin such a way as to recognize both nuances and broad generalizations among them. Any new constructionswill be associated with knowledge the subject already has. The subject orders or re-orders a class ofsituations with the characteristics of the current object so that the new knowledge fits with previousschemas, or the previous schema has been reconstructed. The new generalization occurs precisely becauseof a mental construction or reconstruction.

2.2. Writing and other reflective practices in mathematics

There have been many articles published in recent years on the use of writing in mathematics. Theauthors included here used writing in some type of reflective manner. Their results and advice were usedto develop the writing component of this study.

Mason (1989)studied students who were given problems to solve with instructions that directed theirfocus to the problem and were instructed to write their answers to the problem. The pointers directedthe students towards the “shift,” as Mason refers to it, in order to focus on abstraction and reflect on aproperty of the problem which could lead to abstraction. At some points, this may have meant focusingon the particular and the detail. At another moment, this required letting go of specific characteristics, a“drawing away” from specificity. The experienced mathematician does this without thinking, and oftenleaves the student behind. The point was to try to get the students also to make this delicate shift of focus.Mason points out that mathematicians, in hopes of helping students, often look for devices, or physicalobjects (such as diagrams or images) in which they see theirownmanifestations of their mathematicalrepresentations,their abstraction. They then offer their “representations” to students, perhaps forgettingthat students have their own perspectives, which may be different. Hence, the student would be better offif they were helped to draw their attention to what is being stressed, in order to assist them in seeing agenerality, and then expressing that generality to make the abstract shift in which the generality becomesan object. Once abstraction is recognized as a shift of attention, students may be helped by activitieswhich direct them to generalize and to express this generalization in their own terms. Then the teachercan bring the student’s focus to the process in which one can draw explicit attention to the movement inwhich the contents become objects.

Bishop (1985)discusses the idea of what he refers to as “constructive alternativism.” He believes thatthe teacher is the most important factor in the whole educational enterprise and, therefore, that it is veryimportant to research the decisions made and activities planned by the teacher.

Bishop claims that focusing on teaching activities is a significant move away from the idea of “teachingmethods.” The idea of a teaching method creates a distinction between it and the mathematical content.The notion of a mathematical activity relates to both topic and process, and is a unit of both methodand curriculum. Bishop focuses mainly on spatial activities, but states that this can be embedded in themore general category of mathematical activities, which he believes is a very important area of research.Piaget (1975, p. 16)also stated that the teachers are necessary to create situations and initial tools whichare useful problems to the child. In addition to this, he explained that the teacher should also providecounter-examples that compel reflection and reconsideration of overhasty solutions.

Bishop emphasizes the personal nature of the meaning of any new mathematical idea. A new ideais meaningful to the extent that it makes connections with the individual’s present knowledge. It mayconnect with the individual’s knowledge of other topics and ideas in mathematics and may also connect

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with knowledge of other subjects outside of mathematics. The ultimate problem for teaching mathematicsthen is not that of rigor, but of the development of meaning, of the existence of mathematical objects.The goal of teaching then is in sharing and developing mathematical meaning. Rather than spending amajority of time thinking about content, knowledge, and mathematical topics, teachers should be thinkingabout the student activities in class. A focus on these mathematical activities can improve the situationand put the activity of the student at the center of the teacher’s concerns.

Bishop argues that communication is a key element in the mathematics classroom. He is not referringto a new construct, but the type of communication where mathematical meaning is discussed. Meaningand understanding are about the connections one has between ideas. Communication in the mathematicsclassroom is, therefore, about sharing mathematical meanings and connections. We can only share ideas byexposing them. Important vehicles for this are talk, symbolism, uses of diagrams, examples in differentcontexts, analogies and metaphors, and written accounts and descriptions. Interactive communicationbetween the teacher and student is necessary to learn about the students’ analogies, contexts, examples,etc. and enable those to be exposed and shared. The ideas of interactive communication will encouragestudents to take more part in sharing of mathematical meaning.

Along these same lines,von Glasersfeld (1996)writes that in order to teach abstract ideas, the instructormust generate experiential situations in which the students themselves can make the necessary abstrac-tions. In order to encourage such abstractions, the instructor must be successful in establishing a commonlanguage with the students, that is a language of carefully negotiated and coordinated meanings, or as hecalls it, a consensual domain. He also states that, from the constructivist view, it is not helpful to assumethe students’ answers are simply wrong and that their misconceptions must be replaced by the “correct”conceptions. In order for a new conception to become operative in a student’s thinking, it must be relatedto others that are already there. A simple and efficient way to do this is when the new structure is built outof elements with which the students are familiar. Further, students need to be shown that there are elementsin their experience that can be related differently from the way they habitually relate them. To make suchchanges desirable to students, they must be shown that the new way provides advantages and thinking thatreaches beyond passing exams and getting good grades. He finds it counterproductive to dismiss a studentas wrong and then show them the right way to proceed. He believes this disregard to a student’s effortinevitably kills motivation. He recommends reviewing the method the student uses to give the teacher aclue to a conceptual connection that is either missing, or, perhaps, has been attained and can be built upon.

On their use of journal writing,Ellerton and Clements (1990)note a common thread in the research thatpeople of all ages who are studying mathematics have difficulty in reflecting on the deeper meanings ofwhat they are studying. If teachers of mathematics provide sensitive and constructive advice to studentson how to make journal entries more reflective, if they are able to find the time to read and respond to theentries, and the learners make regular and thoughtful entries, then they find the journal writing processto be very valuable.

Southwell (1988)carried out a study to investigate the relationship between experience and reflectionon that experience. In particular, she studied third year teacher education students. Southwell states thatone of the critical issues in learning mathematics which has not been adequately covered is the balancebetween theory and practice or the interplay between experience and actual acquisition of concepts.Techniques devised to enhance the reflective process need to be applied to mathematical problem solvingand to mathematics education research. She states that although there is a general acceptance of thenecessity for reflection in order for learning to be effective, not many have attempted to define or describejust what reflection is or to develop reflective strategies.


Waywood (1992)reports on the experimental use of writing in secondary mathematics classrooms. Theexperiment included about 500 students from diverse socio-economic backgrounds and had been runningfor 4 years. There were many students for whom English was a second language. The experiment was aprocess of refinement through action and reflection.

While there are many different ways to keep a mathematics journal, the heart of the process used byWaywood (1988)is the use of prose to review, reflect on, and integrate concepts. His idea has been largelyencouraged through the “Writing across the curriculum movement” and then in the “Writing to learn”movement. However, Waywood notes, these movements have not had a large impact in the mathematicsclassroom. Yet it is clear that mathematics is richer than just a collection of formulas, and writing mustbe for us a way of accessing this richness.

The journal writing was analyzed by a rubric with the labels “recount,” “summary,” and “dialogue.”The point was not to just recognize differences, but to determine how these styles of writing relate to thelearning of mathematics. The teachers’ experience of the students who wrote journals led to the followinghypothesis: “The mode of journal writing reflects a stance towards learning on the part of the student.”(p. 35). His assertion is that there is a link between different dispositions that students have towardslearning and the different ways that students have of organizing their writing.

In the recount mode, the key feature is a reporting of what happened. It is a passive observation. Thesummary mode has the essential quality of summarizing the codifying of content. It may be for preparingfor an exam or more generally to gain a mastery of content. The dialogue mode has the basic feature ofa “to and fro” or interaction between several different ideas. This signals a more creative stance towardsknowledge, knowledge being what is created or recreated.

The student understanding of the journal writing task proved to be essential. A first rationale for thejournal tasks was that they seemed to cover what intelligent learning is about; the tasks included summary,questioning, collecting examples, and discussing. However, experience with students trying to carry outthese tasks led to the view that these words were only tokens for the changing interpretations in students’minds. Each of these tasks was interpreted differently, depending on the students’ stance towards learningand their experience with journal writing.

Students intend to do what is required of them, so what they do reflects their understanding of whatthey think they were asked to do. Students did make a shift from happenings to relations.

In this experiment, teachers became more aware of themselves as communicators and paid moreattention to how the learning was organized. They became aware of and began to address elements oflearning that had not traditionally been part of the mathematics instruction.

Waywood offers four conclusions based on this experience. They are:

1. A clear understanding of what is intended by “keeping a journal” must be communicated.2. Class and homework time has to be given to supporting the use of the journals.3. The journal work must be seen to be as highly valued as more traditional aspects of mathematics

learning: journals need to be assessed and reported on.4. Ideally journals need to be introduced at the level of departmental policy.

Richards (1990)reported asking her students to keep a record of what language, reading, writing,speaking, and listening, they used in 1 week’s time in their mathematics class. She also had a colleagueobserve the class, focusing on the modes of communication that were evident. There was a surprisinglylarge range of language used. For example, from writing alone, instances of the following types wereobserved:

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Summaries: of findings, processes used; learning done.Translations: of definitions, information; concepts and how they are applied.Definition: of terms used; mathematical areas.Reports: on an area of mathematics; their work and what they’ve done.Personal writing: feelings; conversational reports and responses; letters of response.Labels: for diagrams accompanying explanations; numerical representations.Instructions: for solving a problem (steps involved).Notes: from books; from other children’s books; from peers’ tutor sessions; from teacher

tutor sessions; ideas to follow through.Lists: of findings, knowledge; words; symbols/terminology; ideas; questions; goals;

content.Descriptions: of procedures; conversations.

This names only some of the types of writing observed. Richards assesses that the extent of this listcertainly draws attention to the significance of language factors in mathematics learning. While thecommonly held view is that mathematics is the least dependent on language factors, it could be arguedthat in learning mathematics, students experience the most difficulty.

Richards asks an important question about how students are able to learn to encode their mathematicalexperiences. Who tells them how to write mathematics? Examples of student writing that appear to beliterally incorrect, but looked at from an intentional point of view actually have meaning, occur withconsiderable frequency in her first year mathematics classes. Unfortunately, Richards notes, it is herexperience that students who are sloppy are most often those who do not have or cannot express, aconceptual understanding. They are also most often the students who make mistakes; they appear to usnot to be rigorous.

Similarly,Davis and Jones (1990)write that we believe that students’ developing sense of mathematicalrigor goes hand in hand with their ability to use mathematical language precisely, consistently, andcreatively. In other words, we believe that the development of rigor is conditional upon a consistentlanguage that is capable of expressing it. Conversely, the need for an appropriate mathematical languageseems to grow out of a need to express and to be rigorous.

The experiences of these researchers were taken into consideration in the development of these writingassignments and how the writing experience would be carried out.

3. Methodology

3.1. The participants

This study involved 25 calculus I students at a public 4-year college located in a large metropolitancity for whom this was the first calculus class. Eighty percent of these students had a language other thanEnglish as their primary language, including Cantonese, French, Hindi, Mandarin, Russian, Spanish,and Uzbek. Therefore, the writing component was really two-fold for this particular group. While theprimary goal was to focus on communication between the professor and students that would encouragea reflective process on the part of the students and provide them more personal attention to their needs,there was also a goal of promoting the use of English. The college presently has a “Writing Across the


Curriculum” component and these kinds of activities are in the process of being incorporated throughouta large number of classes.

3.2. The research question

Can writing exercises in an introductory calculus course promote and/or foster reflective abstraction?

3.3. Development of the writing assignments

In this study, reflective abstraction is a mechanism for the isolation of particular attributes of a math-ematical structure that allows the subject to construct or reconstruct knowledge that is new; that is,knowledge not previously known. A feature of reflective abstraction is that it clarifies and organizeslogico-mathematical experiences in such a way as to recognize both nuances and broad generalizationsamong them. Any new constructions will be associated with knowledge the subject already has.

With this in mind, the questions were designed so that, in some cases, they were phrased in order tofocus on common properties among different objects in order for students to construct or reconstructtheir knowledge. Based onMason’s (1989)studies using what he describes as the “shift of attention” andThurston’s (1990)article, where he illustrates the dependence between constructive mental activity of thestudent and the attention the student has on those structures which are to form part of the abstract conceptwhile being drawn away from those which are irrelevant. In addition, a common theme throughout theliterature was the connection between the distinguishing the common properties among a group of objectsand abstraction.Thurston (1990)emphasized generalization as a principal incentive for abstraction. Withthis in mind, some questions begin with specific cases and then move to more general cases in orderto invite attention to possible generalities. Any indication by the students in their writing of makinggeneralities, pointing out nuances, discussing a new construction or a reconstruction was consideredevidence of reflective abstraction.

These are certainly not the only possible questions. This process is quite different than any kind ofassessment typically used in a mathematics class. I would argue that it is a more difficult process usedhere. In addition to having students answer questions correctly in the form of solving math problems,they are also expected to explain, discuss and generalize their answers. There is a process of learning andrefinement also for the professor in developing these types of questions.

3.4. The writing assignments

While the students were given seven formal writing assignments that were related to their mathemat-ics work, they were also given a “math biography” assignment the first day of class in order to set atone about the importance and also to familiarize them with writing. The math biography and the firstformal writing assignment which asked them to comment on their computer lab (as well as a few otherpoints) were written so that there could be, basically, no wrong answers possible. The point of these twowas to introduce the students to writing and allow for the professor to give plenty of positive feedbackto whatever was written. From there, each writing assignment became a bit more conceptual than theprevious one. Also, there was less structure about specifics for each question, requiring the student tointerpret the question, and made it open to that interpretation. This allowed the students to write aboutany connections they had and it also made it impossible for a student to copy another student’s response,


since they were all so unique. The writing assignments as given to the students are listed in the followingsections.

3.4.1. Math biography assignmentYour first writing assignment is your own math biography. Please write several paragraphs about your ex-

periences with mathematics in the past. What was your school system like? Where did you go to school be-fore here? Do you have a favorite teacher? How about a least favorite teacher? What was your favorite math-ematics class or subject? Please list all the mathematics classes you had in high school as well as any collegemathematics classes you have previously completed. Why are you taking this class? What is your major?What are your plans for the future? Please feel free to include anything else that you feel is important.

3.4.2. Formal writing assignments

#1 (a) How is the work we are completing in the classroom related to work completed in the MapleLab?(b) What is the relationship between calculus and the graphing work we have done in the lab?Instead of using algebraic solutions, we are plotting graphs, why?(c) Is there anything that we have done so far that is not clear to you?(d) What was the most interesting thing we have done so far? Have you learned anything?(e) Generalize the translations we have been doing to any function. Why does translating workfor any function?

#2 (a) Explain what a function is and give examples. Explain the different ways a function can berepresented.(b) Explain what a piece-wise defined function is and give at least one example.

#3 (a) Given the following two composite functions, determine which part isu(x) and which part isf(u).

y = (x2 + 3)4 y = 1/√

x − 2(b) Write one or two paragraphs explaining exactly how you recognized which part isu(x) andwhich part isf(u) for these two composite functions.(c) Write instructions for a friend how “in general” one determinesu(x) andf(u) for anycomposite functions.

#4 Write one or more paragraphs on the significance or meaning of the following limits. You mayuse examples or definitions in your descriptions if you desire, but you must explain the limits inyourownwords. A definition copied from the book will not be accepted unless it is clearlyexplained in your own words.(a) limx→1 2x + 1 = 3(b) limx→4 f (x) = 7(c) limx→a f (x) = L

#5 Given the following, answer the questions below:limn→∞ an = L limx→a f (x) = L

(a) Discuss the similarities and/or differences of these two kinds of limits.(b) What do these limits signify, in your own words?


(c) Give examples for when each type of limit does exist and when it does not exist and explainwhy.

#6 Which type(s) of function(s) require the following rules in order to determine the derivativefunction? Give examples and explain why each rule applies.(a) Chain Rule (b) Product Rule (c) Power Rule (d) Quotient Rule

#7 (a) Give several examples of functions and graphs of functions which contain a point or pointsthat have a horizontal tangent. Specify these points and explain why they have horizontaltangent lines.(b) Explain why the derivative is equal to 0 at these points listed in 7(a).(c) What is the connection between the derivative being equal to 0, the extrema points, and theinstantaneous rate of change? Explain in paragraph form, use examples, and write in your ownwords.(d) Give at least three different examples of functions with their graphs that contain at least onepoint that is not differentiable. Specify these points and explain why they are not differentiable.(e) How is continuity of a function related to the derivative? Explain in paragraph form and giveexamples.(f) Does continuity imply differentiability? Does differentiability imply continuity? Explainyour answers.(g) What else can you say about any of the topics above?

3.5. The procedure

The math biography was completed on the first day of class, in order to set a tone on the significanceof writing. Three of the writing assignments were completed in class and the others were done outside ofclass. Each of the math biographies received written feedback and were returned the following class. Allwriting assignments were returned within two classes. In addition to the seven formal writing assignments,there were many informal writing assignments. The formal writing assignments were kept in a journalwhich was collected after each assignment and reviewed.

von Glasersfeld (1996)argues that student responses should not be simply dismissed as wrong by theprofessor and the student shown the right way to proceed. He suggests reviewing the method used bythe student in order to get a clue of the conceptual connection in order to build on that.Bishop (1985)also discussed the importance of allowing students to make connection of their own. With these per-spectives in mind, the written comments and questions to each writing assignment had several goals.These included showing counter-examples to incomplete reasoning, pointed questions to bring anotherperspective, questions to get them to expand further, and encouragement for complete well thought outanswers as well as simple corrections for small errors. They were never told an answer was wrong, butrather asked to explain further, or directed to consider an example which would allow them to draw theirown conclusions. In some cases, students were asked to rewrite or write further and return the journal.This would then be commented on a second time and returned. All of the comments and question writtenby the professor were with the purpose of keeping the focus on the important properties of the problem,while allowing flexibility with the descriptions and connections the students were making.Ellerton andClements (1990)had noted that if mathematics teachers provided advice on how to make journals more



reflective, if they read and respond to the entries, and the students make regular and thoughtful entries,the journal writing process was very valuable. This model was therefore applied here.

As mentioned earlier, the main goal was to promote reflective abstraction. Experimentation with writingin calculus was piloted by the author earlier, but not so extensively. AsRichards (1990)also found in herstudies of writing in mathematics, in these pilot studies, it was encountered that when students were notable to express themselves clearly in writing on a particular concept, they also were not able to work withthe concept as an object. It was also found in these previous trials that writing was a very personal way tocommunicate with the students and that they responded positively. Both in the previous trials and in thisstudy, students would often approach the instructor to discuss what had been written in response to theirwriting, even if it had merely been an underlined sentence or a stray mark. They were very attentive toanything written directly to them. It was decided this would be a productive way to get their attention andto document their progress (or lack thereof). Speaking to them, answering questions in class, or even goingaround to their desks while they worked in class did not have the same mindful responses from them.

3.6. The assessment

The writing assignments were worth 5% of their grade and it proved to be enough to register considerableimportance with them, since all of the students handed in all of the writing assignments. In addition, therewas a weekly Maple Lab assignment. Students completed interactive lab notebooks which included awriting component as well. This study, though, focuses solely on the journal writing.

The writing assignments were designed to invite reflective abstraction; to find generalizations anddifferences; to organize classifications; and to use all of these to construct or reconstruct knowledge.However, given that these students in particular come with such a broad range of background experiences,the comments to their writing were given in such a way as not to dismiss an answer given in any way, butto broaden it, if possible. The responses were given in an individual way so that students would feel freeto use whatever references they had and to describe their understanding in a way that was natural to them.That included not marking grammar or misspellings. If a student obviously put very little thought intowhat was written or had copied something out of the book, they were asked to rewrite the assignment, butthat was rare. The goal was to get the students to discuss the calculus concepts in question in an abstractmanner and to allow them to relate these concepts to their previous constructs.

For the purposes of this study and based on the previous works, reflective abstraction is a mechanism.This mechanism can be used to isolate particular characteristics of a mathematical structure. This allowsthe student to construct or reconstruct knowledge that is new; that is, knowledge not previously known.Reflective abstraction may be used to clarify and organizes logico-mathematical experiences, not onlyby their nuances, but also by recognizing broad generalizations among them. Therefore, if studentswrote about new constructions, demonstrated by recognizing nuances or generalities, reconstructing withprevious objects or schemas, or organizing the common properties of a group, this new knowledge wasconsidered a product of reflective abstraction and that reflective abstraction had taken place.

4. Analysis

There were initial responses (the first response given to a question) clearly demonstrating that studentshad used reflective abstraction in that they were generalizing and grouping specific attributes of functions



or focusing on particulars. Some made references and analogies to previous experiences. It is safe toassume that any knowledge of calculus and function as related to calculus was a new construction sincenone of them had studied calculus previously. Other demonstrations of reflective abstraction came when astudent rewrote an answer and articulated a reconstruction of what had previously been written. This maybe due to a comment that caused a reflection on different aspects or it may have been due to somethingworked on in class or it could have been from something completely different. The point being that thewriting, and the comments to their writing, initiated some student reflection as demonstrated in theirresponses.

Given the individual nature of reflective abstraction and the construction of knowledge, this particulartype of study does not call for quantitative results. Rather, a description of some of the typical typesof writing that students offered follows, as well as some very striking responses. Examples of writingthat demonstrate an occurrence of reflective abstraction are given below. In addition, student work thatestablishes difficulties is also included to show their usefulness. The writing served the purpose of aconduit for information between the students and professor, alerting the professor to misunderstandingsor trouble points for students. This also proved to be very helpful.

Student responses are given verbatim, with any spelling or grammatical errors. Any underlining, paren-thesis, or quotations in the student responses are taken directly from the students’ own writing. Commentsin [. . . ] are offered by the author if it was deemed that the student writing would not be understood bythe reader.

4.1. First responses demonstrating reflective abstraction

#1 (a) How is the work we are completing in the classroom related to the work completed in theMaple Lab?(b) What is the relationship between calculus and the graphing work we have done in the lab?Instead of using algebraic solutions, we are plotting graphs, why?(c) Is there anything that we have done so far that is not clear to you?(d) What was the most interesting thing we have done so far? Have you learned anything?(e) Generalize the translations we have been doing to any function. Why does translating workfor any function?

Chi: How does translating work? It work the same for all function. For example, within the functionif we subtract, the graph shifts to the right. This is because thex that used to give a certainy nowhas moved to the right by whatever subtracted. It has to be bigger to get same effect. The same istrue for adding onlyx has to be smaller now. If we add outside the parenthesis, this just makesybigger,x stays same. So, graph goes up. The same is happening if we subtract outside, they getssmaller. When we multiply a number, the graph will become narrower than the original. becauseif we multiply, the function will shrink sincey gets much bigger for the samex. Therefore, thegraph becomes narrow unless it is multiplied by a fraction then gets wider sincey is smaller.Negative signs changey to negative or positive depending original. So, any function, these arealways true.

Chi has been able to generalize the translation properties across functions as a class. He can reason whythe properties hold and has developed an inter-relation between the algebraic process and the geometricinterpretation.


#2 (a) Explain what a function is and give examples. Explain the different ways a function can berepresented.(b) Explain what a piece-wise defined function is and give at least one example.

Mary: Believe it or not, all this material is related. Mostly by the fact they all have to do with function— even though they be a little different than the next — and by the fact that in each of themour main goals are to find their domain and range — and to graph them accordingly. Piecewisefunctions are just defined different in different parts of the domain. All of them have each valuein domain going to just one value in range. Otherwise, ifx goes to more than one range value,we can’t tell whichy we are talking about, so it cannot be a function if this happens.

Mary has been able to connect that all the functions have commonalities and that the domain and rangeof these functions, while different, are each related to the graphs of those functions. She also writes veryclearly about how a function is defined.

Maria: Function can be in lots of different forms. A point is a function. Or a graph, more than onepoint or an equation. It’s a function if there is only oney for each of thex’s. So a circle is nota function since there is twoy’s for onex sometimes. Piecewise functions are just the same,except the domain is given and more than one equation is used. But only a function if oney foreachx. Piecewise are sometimes not functions, too if there isx that goes to 2y’s or more.

Maria is able to articulate clearly the definition of function, how this can manifest itself and when thefunction relationship does not hold. She also sees the piecewise defined function as an extension of hernotion of function.


y = (x2 + 3)4 y = 1/√


Leon: To determine theu(x) andf(u) for any composite function, you first extract the inside function(u(x)). To do this, which you would do by taking the part of the function which could standon its own as a substitution forx within a function of f(x). After extracting that part of thefunction and placing it asu(x) the outside function (f(u)) would be the entire function with theu(x) part substituted by “u.” That way you can tell which function has been composed in theother function.

Leon is able to unpack composite functions into separate functions and to see the operator as a functionitself. He speaks ofu(x) as a substitution forx in f(x), categorizing this information in his schema forfunctions.

Xue: u(x) always inside of af(u) function, We can see theu value is in the square root, prendency[parenthesis] andf(u) is to put theu(x) inside of its function, which is function inside of function.[. . . ] Each one [u(x) or f(u)] could stand by itself and be functions.


Xue is building groups of situations in her schema of function, now adding this description of compositefunction.

Gilda: For example, in a fruit, the seed is always in the “inside” of the fruit, so if you’re looking fora seed you would have to keep biting into the fruit in order to obtain the seed. In this casef(u)would be the apple andu(x) would be the seed. The only way you would getu(x) is if you gotf(u) = the apple & bit it.

Gilda is taking her experience with composite functions and connecting it with her own experiencesfrom the past. She is building a schema for composite function which has a representation of theseed inside the fruit. Gilda’s complete answer is given inFigs. 1–4along with the comments of theprofessor.


Sumita: As the difference betweenx-values and 1 become less then the difference betweenf(x) and 3also become less.

Sumita has developed a dynamic interaction between the domain and the limit and uses this in her personaldefinition.

Hilda: x is moving towards a number. When this happens, they values are also moving towards anumber. That number isL. How close do the numbers get?x can get infinedtly [infinitely] closeto the number 1 or 4 ora. Theny values move infinedtly close toL. x or y can sometimestouch the numbers, but only if the numbers are inside the domain or the range. Sometimes thenumbers are not and sometimes they are, depends on which function. Sometimes there is a holeor something at thex value or at they value, but still has limit. Depends.

Hilda writes about her connections between the limit, the domain and the range. She appears to havesome flexibility with the notion on limit, depending on which functions she is working with.


(a) Discuss the similarities and/or differences of these two kinds of limits.(b) What do these limits signify, in your own words?(c) Give examples for each limit of when it does exist and when it does not exist and explainwhy.

Lin: If ε is big theM will happen fast. Ifε is small then theM will happen slow.

Lin also has a dynamic interaction here, betweenε andM, and has incorporated her idea of fast and slowinto her schema of the sequence limit.


Fig. 1. Gilda: Question 3.

Babul: In both the domains are approaching values. Asx or n approachesa or infinity, the distancebetweeny values andL gets smaller if the limit is true. If the distance betweeny andL getsbigger, or jumps around, the limit is not true. This is true for both limits. One is sequence,njust gets bigger. It’s like the horizontal tangent.



Babul has been able to make some important generalizations and connections between the two limits. Heis constructing his schema for limits and including the horizontal tangent.

#6 Which type(s) of function(s) requires the following rules in order to determine the derivativefunction? Give examples and explain why each rule applies.(a) Chain Rule (b) Product Rules (c) Power Rule (d) Quotient Rule

Maya usedy = (3x − 2x2)(5 + 4x) as an example for the product rule. She finds the derivative usingthe product rule and then writes:



Maya: It is possible to simplify the equation then take the derivative by using the power rule. However,it makes the problem more complicated, and therefore, we use the product rule.

She then gives the example of(3x − 2)/(2x − 3) and finds the derivative using the quotient rule. Shethen writes:



Maya: It is possible to write the function in the form ofuv, such as(3x − 2)(2x − 3)−1 then use theproduct rule. We can use both rules. [. . . ] But it is necessary by looking at the functions anddetermine which rule will solve the problem easily and in short time.

Maya is able to determine the commonalities among functions and writes them in their various forms ofproducts or quotients. She can then use this to determine which is most efficient. She is able to make theshift of attention and use whichever classification is suitable.


Shiela: We can say that the power rule is the simple case of the chain rule. [. . . ] There is no basicdifference between chain rule and power rule. Both methods are used to take the derivativesand results are the same. However, for some functions by using power rule we can save time.In the same way, for some functions by using chain rule we can make the problem simple andeasy.

Shiela has constructed her schema for the derivation rules and for functions and is able to draw on theseas is most efficient for the situation. She has made connections between the chain rule and power ruleand can group them in the same category or separate them, as needed.

Carol: All these rules can be used for the same function sometimes. For example, I can use the chainrule or the power rule forx3 + 6x2 since I can just letx = the composed function, the one onthe inside that’sy = x. But that’s stupid since it takes more work. But you could if you wantedto any time. But if I havey = (3x + 4)5 then only the chain rule works unless I multiply it allout and that would be stupid this time. When we have a function dividing a function, I use thequotient rule but sometimes I use the other ones too. So if I have sin(x)/3x2, I could use quotientor product rule whichever I want. They both work. Sometimes I use one rule “inside” anotherrule. Like if I havexsin(x4) I use the chain rule inside the product rule. At first, I was confusedabout what rule to use then I see it not that hard.

Carol is able to distinguish between the different functions to see that the rules can be used interchangeablyin certain situations. She also has realized that some rules are more efficient in certain circumstances.She is able to coordinate the different functions with the different rules and make a decision based on herunderstanding of the different rules.

Maribel: The chain rule for example uses the equation of d/dx[f (u)] = f ′(u) u′ and like all theother above rules can be used with composite functions. [. . . ] Each rule can be used witheach of the two types of functions I have mentioned [composite functions and trigonometricfunctions] because they can always be combined and use more than one rule to solve anequation. However if the function, for example, is just a function with an exponent you knowto use the power rule. [. . . ] The relationship between the two rules (power and chain) is thatthe power rule is a special case of the chain rule.

Maribel is repeatedly able to generalize all the rules across all functions. She is able to group them togetherif needed, or use them individually, using the characteristics of the particular function to decide.

Mia: As you see in above [Her example uses both the chain rule and the product rule to find thederivative of the same function], you get same result. But not every equations we can use bothchain rule and power rule to get same result. Some equations have to use only power rule, productrule, or quotient rule. Chain Rule and Power Rule are similar and differ in some ways. If you usechain rule in some equations, the algebraical process might be longer or complicated but if youuse power rule, you can get the answers easily. But no matter which rule you use, you end upwith same result.

Mia has generalized the rules and found that they can be interchanged, but it may not always be useful todo so, depending on the function in question. While she is building her schema for derivative, she doesnot express that the power rule is necessarily a special case of the chain rule.


#7 (a) Give several examples of functions and graphs of functions which contain a point or pointsthat have a horizontal tangent. Specify these points and explain why they have horizontaltangent lines.(b) Explain why the derivative is equal to 0 at these points listed in 7(a).(c) What is the connection between the derivative being equal to 0, the extrema points, and theinstantaneous rate of change? Explain in a paragraph form, use examples, and write in yourown words.(d) Give at least three different examples of functions with their graphs that contain at least onepoint that is not differentiable. Specify these points and explain why they are not differentiable.(e) How is continuity of a function related to the derivative? Explain in paragraph form and giveexamples.(f) Does continuity imply differentiability? Does differentiability imply continuity? Explainyour answers.(g) What else can you say about any of the topics above?

Bella: The connection between the derivative being zero, even though it doesn’t imply that there is amin or a max; however, when there is a min or a max it does imply that the derivative could bezero (or undefined). And therefore the instantaneous rate of change (velocity) is also found bywhat the tangent line is. Therefore if the tangent line is zero, the instantaneous rate of change isalso zero, since the derivative equals the instantaneous rate of change.

Bella broadly discusses the relationships between these concepts, making their connections, and reasonswith the conditions of their existence.

Maria: So, continuity doesn’t imply differentiability, however differentiability does imply continuitysince differentiability means that you can have a tangent at that point and thereforehasto becontinuous.

Maria discusses the relations between continuity, differentiability, and the tangent line and points to theirgeometric interpretation.

Henry: The derivative is equal to zero at these points, because the derivative is defined as the change in(y) with regards to (x). Since the line is horizontal at these points, (tangent line) then there is nochange in the (y) value. There is zero change in the (y) so the derivative is zero. The derivativeat the maximum and minimum points is equal to zero as the tangent lines at these points arehorizontal, therefore no instantaneous rate of change in (y).

Henry has constructed his understanding of derivative based on the relation between slope, change inyto x, with horizontal lines and instantaneous rate of change. He is demonstrating a cooperation of theseideas to form his schema of derivative.

Gilda: Thus, you see that continuity does affect differentiability b/c if a function is discontinuousat a certain pt. How can we find the derivative? Then I’d say that continuity doesn’t implydifferentiability but differentiability does imply continuity.

Gilda is discussing the relation between continuity and her schema for derivative to reason that herunderstanding of the derivative necessitates continuity.


Sarah: Any thing that has a motion, in order to achieve the highest or lowest speed it has to be stop atcertain point. In the same way, when a function is increasing then decreasing with in an intervalit has to be zero at certain point.

Sarah is using her previous knowledge of motion to construct her understanding of extrema values.

4.2. Reflective abstraction demonstrated in a rewritten response

Some of the best documentation of promoting the reflective abstraction was in asking students torewrite a response. Their original responses would be commented on with questions, examples, andcounter-examples. They would then be asked to read what was written, think it over, and rewrite theirresponse. Therefore, the comparison of the original writing and the rewritten response would allow for ademonstration of a reconstruction by the student.


y = (x2 + 3)4 y = 1/√


Hans: (original writing) In the above equationsu(x) is the inside function and is the direct value ofx.In the first equation(x2 + 3)4, the direct value ofx is found from(x2 + 3). f(u) is the outsidefunction and dictates what is done to the inside value.

Hans: (rewritten version) In the equationy = (x2 + 3)4, theu(x) value isx2 + 3 which is the insidefunction. Theu(x) determines what is first done to the equation. Thef(u) is u4 and is the outsidefunction and directs what is done to the inside function.

[. . . ] To do these functions first determine the inside function which will direct the firstoperation to be carried out on the equation. The outside function is next determined. This is theoperation that is done to the inside function which condenses the overall equation.

Hans is able to discuss the composite function as two separate functions. In the second writing, he isable to describe them as “condenses the overall equation” and describe the interactions between the twofunctions.

Susan:(original writing) This means we can plug in any positive or negative numbers or 0 in theequation, consequently we get the value of y which is the range.

Susan:(rewritten) However, the range is [−4,∞). That means we can get any values that are between[−4,∞). But we cannot get anyy-value that is less than−4. The range will be whatever comesout of the function, depending on which function it is.

In Susan’s original response, there was a vagueness to the idea of range, stating only that it is the value ofy. In her second response, she is able to make a stronger statement about the range and to actually find it.

Carol: (original writing) Theu value is contained and the other is not. You need to look for eachone.


Carol: (rewritten) One function has been input to other function. So what used to come out of theufunction now goes into the other function. It’s like machine with another little machine in it.Now the little one makes the bigger one run. One function is folded into the other one and eventhough they were two now that they are one.

In Carol’s rewritten work, she is showing that she is developing the idea of composite function fromsomething very limited to discussing the domain and range (using the term input). She is also makingconnections with what is familiar to her with comparing a function with a machine.


Linc: (original writing) By limit I mean, the value that they values will get closer and closer to, butnever actually equal that value. [. . . ] This means that while thex value off(x) get closer to thevalue 4 from the right and the left they values (f(x)) get closer to the value 7 but will neveractually be equal to it.

After making several pointed questions and giving him examples, in the next writing assignment, Linctries to compensate and points out repeatedly, four separate times, thaty may actually be equal toL ormay not.

Linc: (rewritten answer) However in “b” when they values get closer to the limit or even equal thelimit the x values, get closer or may equal a specific value depending on the function. (He repeatsthis point several times in his answer.)

4.3. Examples of misunderstandings as written by students


(a) Discuss the similarities and/or differences of these two kinds of limits.(b) What do these limits signify, in your own words?(c) Give examples for each limit of when it does exist and when it does not exist and explain why.

Rajendra: They are similar because both functions have the same limit.

This comment was interesting since the student was interpreting that the limit of the sequence and thelimit of the function were both approaching the same value, since they were both approachingL.

#6 Which type(s) of function(s) require the following rules in order to determine the derivativefunction? Give examples and explain why each rule applies.(a) Chain Rule (b) Product Rule (c) Power Rule (d) Quotient Rule

Racquel:The chain rule is usually used in composite functions.


The key word here is “usually.” Some students would start off by making sweeping generalizations intheir writings and would find that this often would not hold. They had a tendency to want to generalizebroadly as much as possible. In response, they would be given counter-examples to show that their rulewas not universal. Some students would then go in the other direction and would write words suchas “usually” or “most of the time” or “sometimes” in cases where the generalizations did hold morebroadly.

Hector: For the functiony = x2 + 2x, thechain rulecould not be used as there was no part of thefunction that could be defined as the inside or outside function.

There were a number of similar comments from students who did not make the connection between thechain rule and the power rule.

Joan: The different between the chain rule and the power rule is the chain rule has to find the derivativeof the inside and outside function and the power rule don’t.

Joan’s response was after correctly finding the derivative using both methods for the same function. Thiscould be an example of someone who is not able to clearly express themselves.

#7 (a) Give several examples of functions and graphs of functions which contains a point or pointsthat have a horizontal tangent. Specify these points and explain why they have horizontaltangent lines.(b) Explain why the derivative is equal to 0 at these points listed in 7(a).(c) What is the connection between the derivative being equal to 0, the extrema points, and theinstantaneous rate of change? Explain in a paragraph form, use examples, and write in yourown words.(d) Give at least three different examples of functions with their graphs that contain at least onepoint that is not differentiable. Specify these points and explain why they are not differentiable.(e) How is continuity of a function related to the derivative? Explain in paragraph form and giveexamples.(f) Does continuity imply differentiability? Does differentiability imply continuity? Explainyour answers.(g) What else can you say about any of the topics above?

Dani: When a function is continuous, it definitely has a derivative. For instance with polynomials:y = x3 − 2x + 15, the function is continuous everywhere, therefore it is differentiable becausethere’s no jumps, holes or cusps. If the function is continuous, it must be differentiable becauseit is continuous for all the points, and, on the other hand, if a function is differentiable, it has tobe continuous, too.

Dani is stating that she does not have a clear concept of differentiability or continuity or their relationship.

Harold: To be continuous, a function has to have an (x) value that defines a (y) value on the graphof the function at every point. If the function is discontinuous then the (y) value at that pointdoes not correspond on the graph to an (x) at this point. There therefore can be no change inthe (y) value if there is no (y) value to change, at these points. The derivative therefore cannotexist.


Harold has constructed a geometric interpretation of continuity and his understanding of the points thatmake a graph with his schema on derivative to bring together their relationship. However, he misses thefunction that can have bothx andy in the domain and range and still be discontinuous.

5. Conclusions

There was an interesting trend in reviewing the journals: there were more indications of reflectiveabstraction with each writing assignment. In other words, the journals as a whole contained more examplesof students classifying and organizing information, making connections and discussing relationshipsbetween concepts, generalizing across categories, noticing particulars in certain circumstances whiledrawing away in others. It appears that they became better at the process as the semester progressed. Ibelieve they did and the reasons why are varied.

One reason students demonstrated more reflective abstraction may be that they were better at writing.Many of them seemed anxious in the beginning of the semester about having to write because Englishwas not their first language and they thought that they would be graded on grammar and/or spelling.After several assignments, they realized this was not the case and that what they wrote was the importantissue. They became freer about sharing what they thought and knew better how to express themselves inthe mathematical sense. In addition to developing their talents for writing mathematics, they were alsolearning more mathematics in the process. Therefore, they had more structures available to draw on intheir process of constructing knowledge as the semester moved forward.

The writing assignments themselves became more demanding, and required more responsibility fromthe students. They were designed to start simply, with the math biography and description of their MapleLabs. The comments from the professor would ask for more information or to clarify or simply to respond.This laid the groundwork for explanations and greater thoughtfulness. The goal of each writing assignmentwas to elicit a conceptual response from the student. With each writing assignment there was more toreflect on. The goal was to promote the use of reflective abstraction as a means of promoting greaterunderstanding and knowledge construction.

The communication that the journal writing provided between professor and student was a very strongtool for determining student understanding or lack of understanding of particular topics. Each writingassignment is a diagnostic assessment, in which the professor can focus on the particular results ofeach student. This drew the students’ attention. No other activity in class produced such an attentivereaction. As Bishop mentioned, teaching activities are important in the mathematics class, as opposed tothe teaching methods. While he focuses on spatial activities, writing stimulates the students to use theirpersonal experience, allows for a steady flow of communication between student and teacher, and givesa personal aspect to the education of each student.

As Waywood found, it was extremely important for the writing assignments to be commented on andreturned in a prompt manner. First, if students needed to rewrite a section, it was necessary to allow themtime to do so before the next writing assignment. Also, students wanted the assignments returned in atimely fashion to see how they had done. It would not have made sense to comment if they were havingtrouble on a topic and not return it until weeks after that topic was finished in class. Therefore, to keepit all as part of the learning process, it had to stay within a timely exchange. At points, this was arduousand time consuming. However, it was not as much work or as time consuming as I thought it would beand the results made it more than worthwhile.


There were several strong outcomes from this experiment with writing in calculus.

• The students made it evident that they wanted to answer well and that the writing assignments wereimportant to them. They responded to the professor’s comments and demonstrated a seriousness ofpurpose with their written responses. They were able to give a kind of self-expression that is virtuallyimpossible in a typical homework assignment or in the classroom setting. This attention to detailcan promote the reflective abstraction process that they need in order to construct or reconstruct theknowledge for calculus.

• The communication that is initiated through this process of writing was beneficial not only to thestudents, but also to the professor. Students were able to have a type of diagnostic analysis of theirown understanding via the comments given by the professor. The professor in turn received a strongbasis for understanding what kinds of knowledge the students have constructed, and which particularareas needed attention.

• The process of writing, comments, and further writing produced evidence of the process of reflectiveabstraction for many of the students. It is not clear, nor could it ever be absolutely clear, if thequestions asked of the students or the thinking they do in order to write a response are the catalysts orif the writing assignment is a conduit by which they can demonstrate that the process has occurred.However, in either case, the writing assignment gave them opportunities to articulate their newconstructions.

6. Discussion and recommendations

In contrast to humanities classes, for example, in a typical college mathematics course there is verylittle room for discussion. Granted, it can be very difficult to have a genuine exchange about very newand rigorous concepts. However, students are still expected to be able to discuss such ideas. If there is notdiscussion in class, it is not clear exactly how students can learn to reflect, ask questions, and discuss themathematics they are learning. In other words, upper level mathematics, in particular, is taught more asif the professor has the information, and that information is dispensed in a manner that, ultimately, willmake it understandable to the students. However, in a philosophy class, for example, there is a constantexchange of ideas and discussion of the meaning of what they are learning. Thus, philosophy majors areknown for being able to discourse about what they are learning, whereas mathematics students often havedifficulties articulating what they are learning. I would argue that this is because they never learnhowtoarticulate these ideas, nor how to discuss or debate these ideas. They may not be able to ask meaningfulquestions since they do not know how to put their questions into words. Journal writing offers a conduitfor discussion. It helps students to articulate their thinking and to reflect on what it is they are learning.

Reading through the journals, I noticed that the writing at the end of the semester was more expository,had more examples, and was better organized than the writing at the beginning of the semester. If suchprogress in the way that students can express their mathematical ideas can be noticed in just one semester’stime, it follows that an even markedly higher improvement might be noticed over one or more years. Awriting component of some sort throughout a mathematics department could produce students that notonly reflect on what they are writing, but who are also better able to use this writing in such importantmatters as proof. This constant reinforcement of explaining, justifying, and writing is a very valuable toolin higher mathematics, not to mention in critical thinking in general.


Bearing all of this in mind, the philosophy of this writing in mathematics study was to promote reflectiveabstraction. This would entail, as indicated by Dubinsky and Lewin, a process of cognitive development.In order to abstract, the students would need to construct or reconstruct their knowledge on a new planeand become aware of constructs that they were not previously conscious of.

Given the broad agreement demonstrated by the authors listed previously who had written on reflectiveabstraction, such cognitive development is not easy to promote. The writing assignments and commentswritten in response to the students were developed to promote what Mason refers to as “the delicate shiftof attention” that comes so easily to people who have a strong proclivity for mathematics. Questions weremeant to invite students to generalize and synthesize where needed and to focus on key particulars inother places. As Tall described abstraction as an isolation of specific attributes and as Borasi refers to itas a class of situations, all promote a way of clarifying and organizing logico-mathematical experiencesin such a way as to recognize both nuances and generalizations. These authors also agree that any newconstructions must be associated with knowledge the student already has.

In the future, the writing-rewriting model should be further explored. In the end, I found that, while thismodel required more from the professor, it produced more satisfying results. It would be useful to havestudents respond to the previous comments each time they hand in a new assignment, thus expanding thecommunication. One very pleasant discovery in this study was the willingness of the students to writeand participate in the writing activity. They were in general very cooperative and open in their writing.They gave a very honest effort in carrying out the writing assignments, in that they tried to articulate whatthey were learning in a meaningful way. We were able to open the door to communicating mathematicsin a more interactive way.

References

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