RON TZUR, MARTIN A. SIMON, KAREN HEINZ and MARGARET KINZEL

AN ACCOUNT OF A TEACHER’S PERSPECTIVE ONLEARNING AND TEACHING MATHEMATICS:

IMPLICATIONS FOR TEACHER DEVELOPMENT1

ABSTRACT. This report presents an account of one teacher’s mathematics teaching anda perspective that underlies his teaching. Nevil was a fifth grade teacher participating incurrent mathematics education reforms in the United States. Through the account, wemake distinctions about teachers’ thinking and practice that can inform teacher educationefforts. We constructed an account by analyzing four sets of classroom observations andinterviews. We observed that Nevil decomposed his understandings of the mathematicsinto smaller components and connections among those components. He created situationsthat he believed made those components and connections transparent and attemptedto elicit those connections from the students. This account illustrates a practice that isdifferent both from traditional practice and the type of practice that we would envision as agoal for teacher development. We contribute two important aspects of mathematics teacherdevelopment from traditional to reform-oriented teaching. In particular, we describeteachers’ perspectives – assimilatory structures that constrain and afford (a) the sense theymake of professional development opportunities and (b) their potential learning in teachereducation settings.

Mathematics teacher education is a primary force in current efforts toreform mathematics teaching. Ball (1988) pointed out that for mathematicsteacher education to be effective, teacher educators must understand andbe responsive to teachers’ thinking. Based on an analysis of case studydata from work with a fifth grade teacher, Nevil, we postulate perspectivesthat underlie the practice of teachers participating in current reforms. Wethen examine how these perspectives define the challenges facing teachereducation and how understanding these perspectives can inform teachereducation efforts.

Reform documents (National Council of Teachers of Mathematics,1989, 1991, 2000) in the United States encourage mathematics teachersto decrease traditional activities of telling and showing the mathematicsstudents need to know. Instead, teachers are encouraged to increaselearning experiences that promote problem solving, reasoning, commu-nicating, and making meaningful connections among mathematical ideas.However, these documents “do not provide guidance on the specifics ofday-to-day, minute-to-minute practice” (Ball, 1996, p. 502). Quite the

Journal of Mathematics Teacher Education 4: 227–254, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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contrary, the “vision of the mathematics curriculum and the goals in theStandards are offered by NCTM without a prescription for achieving them.The approach taken is one that will empower teachers . . . to make thechanges” (NCTM, 1989, p. 251). Thus, teachers are challenged to adapttheir mathematics teaching by creating alternative, reform-oriented prac-tices. This study is part of research efforts to understand the perspectivesof teachers in transition from traditional to reform-oriented teaching.

CONCEPTUAL FRAMEWORK

Because this research involved an investigation of a teacher’s perspectiveon mathematics, mathematics learning, and mathematics teaching, webriefly describe our conceptual framework in these areas. In the context ofour research, this framework affects what we notice (Mason, 1998), whatwe take to be significant, and what we identify as challenging our currentunderstandings. For this same reason, we also articulate our understandingof teacher education and of teaching the particular mathematical contentinvolved in the case study data that we present.

A Perspective on Knowing and Learning

Our conceptual framework is based on a social constructivist perspective, acoordination of cognitive and social perspectives on knowing and comingto know (Cobb & Yackel, 1996). From a social perspective, learningis a process of enculturation to the communities in which one partici-pates (Cobb & Bauersfeld, 1995). We use this perspective to identifynorms and practices in mathematics and mathematics teacher educationclassrooms and to understand how new and modified norms and practicesare constituted. Understanding norms and practices helps us explain theinterplay between the nature of the classroom microculture (Voigt, 1995)and students’ learning.

Our cognitive perspective draws on a constructivist epistemologicalstance and theory of learning (Dewey, 1938; Dewey & Bentley, 1949;Piaget, 1970; von Glasersfeld, 1995). In particular, we conceive of know-ledge as a person’s conceptual structures and operations that are used tomake sense of and organize her or his experiential world. For abbrevi-ation purposes, we refer to both conceptual structures and operations asconceptions. We use the term perspective to postulate a broad pedago-gical structure composed of multiple conceptions that collectively organizesome aspects of a teacher’s practice. In characterizing a teacher’s pedago-gical perspective, we make no claim of having analyzed the component

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conceptions. In our work, this difference of grain size is common aswe consider mathematical understandings (conceptions) and pedagogicalunderstandings (perspectives).

Our understanding of the coordination between social and cognitiveanalyses is consistent with Cobb and Bauersfeld’s (1995) position:

This coordination does not . . . produce a seamless theoretical framework. . . . When thefocus is on the individual, the social fades into the background, and vice versa. Further,the emphasis given to one perspective or the other depends on the issues and purposes athand. (p. 8)

In our study, we focused on the perspectives underlying a teacher’s prac-tice. Therefore, we elaborate below only the cognitive component of oursocial constructivist perspective.

Our cognitive perspective on knowing and learning builds on Piaget’s(1980) fundamental notion of assimilation. This notion implies that anindividual’s current way of thinking affords and constrains (a) the senseone makes of events and processes in which one participates and (b) theconceptual advances that are possible. This view is consistent with Varela,Thompson and Rosch’s (1991) rejection of the Cartesian view of humanknowing as a connected set of mental representations of a world, includingthe world of mathematics, that exists independently of the people whoknow it. In other words, we think of the mathematics or pedagogy onesees in the world as being afforded and constrained by one’s assimilatorystructures and operations.

From this perspective, a paradox emerges with regard to how learningtakes place (Fodor, 1980; Bereiter, 1985). In order to recognize something,one must already have a structure into which that something can be assimi-lated. For example, Cobb, Yackel and Wood (1992) noted that a set ofbase ten blocks might be a clear representation of place-value relationshipsto a teacher. However, learners who do not have a concept of compositeunits (Steffe & Cobb, 1988) of ten cannot see place-value relationships inthe blocks. So how does one ever develop a more advanced assimilatorystructure?

We address the learning paradox on the basis of a constructivist view oflearning as transformation (accommodation) in the learner’s current waysof thinking (Piaget, 1985). Conceptual transformation is the result of threeinterrelated processes in the learner (Dewey, 1933; Simon et al., 1999):

1. Carrying out goal-directed activities – mental operations and physicalactions – that he or she can already perform.

2. Discriminating effects of those activities that are more and lesssuccessful relative to the goal.

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3. Abstracting new regularities in activity-effect relationships throughreflection on those efforts that brought about desired or undesiredeffects.

As Bickhard (1991) noted, such an explanation of learning does not requirethe learner’s use of the more advanced structure to recognize a situationprior to constructing that structure. It only requires the learner’s abilityto (a) use activities available through current structures that bring abouteffects, (b) reflect on and differentiate between those effects, and (c) reflecton and differentiate among classes of activity-effect relationships. Thisview of learning can be illustrated in people’s independent constructionof increasingly sophisticated strategies while playing games. An exampleis learning to play the game “24.” In this game, one is competing withothers to quickly figure out how to use four given numbers and the fourbasic operations to arrive at 24. Through reflection on previous attempts,a player comes to realize the usefulness of considering factors of 24. Forexample, given the numbers 2, 3, 4, 9 one uses 9

3 and 4 × 2 to produce 3and 8, respectively. Having described the cognitive component of our viewof knowing and learning, we turn to key features of our understanding ofteaching.

A Perspective on Teaching

Our perspectives on teaching build on the perspectives on learning we havearticulated. A significant part of the teacher’s responsibility is to promoteand support the students’ development of more powerful mathematicalconceptions. This is accomplished through a reflection-interaction cycle(Simon, 1995). We assume that the teacher must make a clear distinctionbetween her or his knowledge and the learners’ conceptions in order togenerate useful hypotheses about students’ learning. Through reflection onstudents’ actions and language, the teacher infers students’ current concep-tions. In particular, the teacher attempts to understand students’ currentunderstandings of the mathematical area to be addressed. Based on theseinferences, and on the teacher’s own understanding of the mathematics, theteacher specifies a hypothetical learning trajectory (HLT).

The HLT is composed of three interrelated parts: the teacher’s goals forplausible advances in the students’ current conceptions, problem situationsthat can create opportunities to promote such advances, and hypothesesas to the processes by which the learning might occur (Simon, 1995). Togenerate or adjust learning opportunities, the teacher designs activities thatshe or he hypothesizes students can initiate and carry out on the basis oftheir current conceptions (Tzur, 1999). When the goals for the studentsare conceptual, the teacher engages them in activities available to them

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already; in particular, he or she encourages and orients their reflection onactivity-effect relationships as a means to promote the intended conceptualadvance.

We take for granted that the teacher’s hypotheses as to the path ofstudents’ learning are only that, hypotheses. Thus, while interacting withstudents the teacher continually modifies the goals and/or learning oppor-tunities on the basis of ongoing reflections on students’ evolving activity.When a period of teacher-student interactions (e.g., a lesson) is over, theteacher reflects on the students’ current conceptions and plans for thenext period of interactions. This perspective of teaching as a reflection-interaction cycle implies that the teacher’s knowledge – conceptions ofmathematics, perspectives on mathematical activity and representations,and teaching-learning processes of particular mathematical content – isconstantly changing.

A Perspective on Teacher Education

Our perspectives on mathematics teacher education are based on ourperspectives on mathematics teaching in two ways. First, mathematicsteaching as we conceive of it defines the goal for our teacher educa-tion efforts. Second, we attempt to adapt the processes that we havearticulated for mathematics teaching to the teaching of mathematicspedagogy (Simon, 2000a). From our perspective on mathematics teaching,a teacher’s understanding of students’ current understandings plays acrucial role in the teacher’s ability to promote learning. Cooney andKrainer (1996) emphasized that this role applies equally to promotingdevelopment of children’s mathematics and of teachers’ teaching.

A central problem for research, then, is to articulate develop-mental landmarks in teachers’ practices. This orientation “necessitatesan emphasis on understanding teachers’ thought processes” (Cooney &Krainer, 1996, p. 1162), that is, the complex perspectives that structureteachers’ attention and awareness (Mason, 1998). In line with this orienta-tion, recent research focused on certain aspects of those perspectives.For example, Steinbring (1998) identified three components of teachers’epistemological knowledge about mathematics. Grant, Heibert and Wearne(1998), Lloyd and Wilson (1998), and Vacc and Bright (1999) studiedthe relationship between teachers’ beliefs about and understanding ofmathematics and the nature of change in the teachers’ practice. Ourstudy was designed to further our understanding about the interconnectednature of mathematics teachers’ practices and the complex perspectivesthat underlie their practices. At this time of radical shift in teaching, thisstudy informs a reconceptualization of mathematics teacher education by

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exploring pedagogical perspectives of a teacher in transition. In particular,this work contributes to articulating goals for teacher education and tounderstanding assimilatory structures that teachers use to make sense ofteacher development opportunities.

A Perspective on Knowing and Learning Division

In the analysis section, we focus on data from a set of lessons in whichNevil’s goal for the students was for them to understand the long-divisionalgorithm most frequently taught in the United States. We articulate ourthinking about teaching this mathematical content to make explicit theconceptual framework that structured our analysis of the data.

Before an understanding of the long-division algorithm can be fostered,students must have a conception of division. Because this conception isprerequisite to understanding long-division, we provide a brief outline forthis part of our framework. A conception of division may be developedin the context of realistic problems that can be solved by separation of anoriginal quantity into equal groups. For example, the teacher might presenttasks such as “How many boxes are needed to pack 32 candies, if each boxcontains exactly 8 candies?” Such tasks do not require a conception of divi-sion. Students can solve them using their established conception of number– composing and counting same-size groups – and manipulative tools suchas Unifix� cubes. Students’ reflection on such grouping activities can leadthem to distinguish a set of problems for which the grouping activity isappropriate. Initially, students would abstract two separate relationships,partition and quotition. We use the two terms to indicate the actions usedby the learner. Partition indicates distribution into a given number of sets;quotition indicates the making of given-size sets. Later, the students wouldconstruct connections among these ideas into a concept of division.

To foster students’ (re)invention of computational algorithms for divi-sion, we would engage them in solving division problems with base tenblocks. An example of such problem would be, “If you want to place192 marbles in 6 boxes equally, how many marbles will each box hold?”In using base ten blocks we assume that the students already understandhow these blocks represent a place-value, base ten number system. As thestudents create the initial quantity and begin to separate it into 6 equalgroups, we would engage them in writing down their steps. After theactivity with the blocks had become a routine, we would also ask themto predict the result of their next step. For example, we would ask, “Couldyou tell how many you will have left after equally distributing the 19 “tens”into the 6 boxes?” Through reflection on their division activities with baseten blocks, and their activities to predict the next steps, we expect the

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students to abstract relationships among the quantities involved. Relation-ships between the product of the divisor and the partial quotient, or thedifference between that product and the dividend can serve as the basis forunderstanding the canonical algorithm.

METHODOLOGY

This study was part of the Mathematics Teacher Development (MTD)Project, a 4.5-year inquiry into the development of prospective and prac-ticing elementary mathematics teachers (henceforth referred to as the“teachers”). By development, we mean professional development towardsthe establishment of teaching practices consistent with current reformrecommendations. We structured the project using the teacher develop-ment experiment (TDE) methodology articulated by Simon (2000b). Thismethodology involves promotion of teacher development and study of thatdevelopment in a cyclical fashion similar to the constructivist teachingexperiment (Cobb & Steffe, 1983). Thus, each intervention with theteachers is followed by an analysis of the teachers’ current perspectives,and each analysis session leads to a formulation of the next intervention.

The TDE consists of two components: whole class teaching experi-ment(s) (Cobb, 2000) and accounts of practice (Simon & Tzur, 1999). Thewhole class teaching experiment of the MTD Project consisted of 5 coursesin consecutive semesters. Each course was taught by the second author andinvolved 9 teachers and 10 prospective teachers. The first course and partof the second focused on promotion of the teachers’ conceptions of keyideas in geometry and ratio. The last three-and-a-half courses focused onpromotion of the teachers’ pedagogical understanding. In particular, wewanted them to learn how to analyze children’s thinking and how to deviseand implement tasks that promote certain conceptual advances in students.

The study reported in this paper derived from the account of practicecomponent of the MTD research program. An account of practice is anadaptation of case study methodology and is the researchers’ explicationof a teacher’s practice. Practice refers to what the teacher does and theperspectives that underlie what he or she does. Using several data sets, wegenerate an account through intensive team-work of line-by-line analysisof transcripts while listening to the interviews or observing classroomvideotapes, as well as reflecting on our own field notes regarding teachers’participation in the courses and work in their own classrooms. We identifysections of data that pertain to what we consider to be the central challengefacing teachers who are participating in the reform. This challenge is tocreate alternatives to direct instruction that can promote students’ learning

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of particular mathematical ideas. With this focus in mind, MTD researchteam members focus on how the teacher promotes students’ learningof particular mathematical ideas, contributing either tentative inferencesabout the teacher’s practice or questions about possible interpretations ofthe data. Those inferences and questions serve as the basis for hypothesesthat we generate and continually revise to explain the ensemble of the data(Glaser & Strauss, 1967).

We consider that our hypotheses constitute an account of practice whenthey offer a coherent explanation for what previously had been, for theresearch team, a diverse and often unrelated set of observations. Eachaccount represents our commitment to comprehend how the teacher organ-izes her or his experiential reality with respect to teaching mathematics.We assume that everything a teacher does makes sense from her or hisperspective. It is our challenge to infer from our data a perspective fromwhich this might be so. An account pertains to a period of time in theteacher’s teaching during which we consider her or his practice to beessentially consistent.

In an account of practice, we explain the teacher’s perspective fromthe researchers’ perspective, a stance that involves a subtle but importantdistinction. Our accounts may differ significantly from what teacherswould articulate about their practices. We structure our accounts usingparticular conceptual lenses, often not shared by the teachers, that defineour focus and guide our interpretations. We recognize the paradox withinwhich we work. To conceptualize the teacher’s practice we must “setaside” our current view of practice to consider perspectives different fromour own. Yet, our current view of practice is fundamental to what wenotice. For a detailed discussion of accounts of practice see Simon andTzur (1999).

In the MTD Project, we typically generated 3 data sets per semesterfor each teacher. The account of Nevil’s practice, reported here, emergedfrom our analysis of four data sets. In particular, we analyzed one data setgenerated before the instructional program and three data sets generatedearly in the program, prior to observable impact of the program on hispractice. In a single data set, we interviewed Nevil about his plans priorto the observed lessons, observed and videotaped at least two consecutiveand related mathematics lessons, and interviewed him after each lesson todocument his reflection about that lesson and plans for the next lesson.Most of the data discussed in this paper were taken from the extendedfourth set that consisted of four consecutive observations and five inter-views. We chose this data set because it was central to our ability to make

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sense of Nevil’s practice and because we considered it to represent andillustrate his teaching over the period of time spanning all four data sets.

We present this account of Nevil’s practice because we believe thiscareful analysis can contribute to understanding the perspectives ofteachers involved in the reform. As stated above, we consider under-standing teachers’ perspectives at different points in their developmentto be a critical component of the knowledge base for effective teachereducation.

ANALYSIS OF NEVIL’S PRACTICE

This section consists of four parts. First, we provide an overview of Nevil’sunderstanding of mathematics and his orientation to teaching. Next, wesummarize the sequence of his four lessons on the long-division algorithm.In the third part, we identify five aspects of the lesson sequence thatinitially we found difficult to explain. Finally, we articulate our account ofNevil’s practice and indicate how it explains the five aspects of the lessonsequence that we could not explain initially.

Overview of Nevil’s Mathematics and Orientation Toward Teaching

Nevil taught in a district that provided a list of desired grade-leveloutcomes in mathematics; the identification and creation of learning activ-ities for the students was left to the teacher. Nevil was pleased to workunder these guidelines. Nevil’s mathematical understanding was strong. Inparticular, his work in the first MTD course showed a solid understandingof multiplicative reasoning related to ratio. He utilized his considerablemathematical knowledge in designing instruction for his fifth-graders.His instruction usually was preceded by careful analysis of the intendedmathematics.

Nevil was committed to minimizing his telling and showing studentsthe mathematics to be learned. He strove to create a collaborative climatein which students participate in all activities. Rather than focusing onstudents’ abilities to execute mathematical procedures, Nevil strove toreveal basic ideas and connections between these ideas and correspondingpaper-and-pencil algorithms. For example, in Excerpt 1 Nevil articulatedhis goals for the division unit. In Excerpt 2 he explained the need foranalysis of the content to be taught.

Excerpt 1 (Pre-set interview)

N: [I want my students to] understand what the [division] algorithmis all about, that it is not just, we put the 2 up here, we multiply,

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we subtract, we get an answer, we bring something down, we dothis. I want them to see why we are doing it that way, that thereis a purpose to this algorithm, . . . that there is something actuallygoing on with these numbers and how they relate to each other.

Excerpt 2 (Interview about lesson 1)

N: Last year I didn’t [teach division] this way. I did the, “here’s a[long] division problem. Let’s review how to do it, and let’s doa bunch of them, and then let’s have a test on Friday” kind ofapproach . . . Some of the kids could do it, others never got it. . . Then the next time we did division was with fractions . . . and Itried desperately to help the kids understand what was going on.. . . I didn’t do any of the stuff that I learned where you [just] invertthe fraction and multiply. So I spent two or three weeks goingback to division with whole numbers and trying to re-teach theconcept of division . . . and they never understood. . . . So I foundthat I had to go back and really understand what division was allabout. . . . [This is how] I came up with . . . the rationale for this[year’s] approach – to examine what division is first, and try some[division situations] with the Unifix cubes and to see it, and then tobegin analyzing this long-division algorithm that they have alreadylearned and to see if they have been making connections and if Ican help strengthen those connections

In order to monitor his students’ progress, Nevil attended to their contri-butions and noted differences in their mathematical competence. Thesedifferences were rooted in his analysis of the intended mathematics. Forexample, in one interview he clearly distinguished between the mathe-matical understandings of two students. He said that one student could useUnifix cubes to represent both partitive and quotitive division; the othercould show a partitive representation only and was unable to make senseof her partner’s quotitive representation. In another interview, Nevil madefiner distinctions among four different quotitive solutions that some of hisstudents used and two different partitive solutions.

In all, we conceived of Nevil as a skillful mathematical thinker. Hewanted students to make connections between their everyday experiencesand mathematical algorithms, and he was competent in making distinc-tions among individual students’ mathematical actions. He reflected onhis teaching and endeavors to help his students develop understanding.Towards this end, he also chose to participate in the MTD teacherdevelopment program.

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Nevil’s Division Unit

In this part we present our analysis of the data set from the division unit.The four lessons that we observed are numbered 1–4. In the interview priorto Lesson 1, Nevil described the previous lesson in which the studentsworked on an introductory task.

Excerpt 3 (Pre-set Interview)The kids started out with 16 Unifix cubes and I asked them . . . to showtheir partner what 16 divided by 2 would look like. . . . I am trying toget the students to understand that division has to do with beginningwith a number of things and then dividing it into sets. . . . Some of thestudents have a really good understanding of what division is all about,but I feel that there are still some that might not quite get it.

Nevil had a clear goal for his students’ learning: He wanted them toform a specific idea about division that can support their understandingof the algorithm. The first lesson that we observed proceeded along theplan that Nevil described to the researcher during the pre-set interview(last part of Excerpt 2). First, he engaged pairs of students in workingwith Unifix cubes on the task, “Could you show 20 divided by 4 with thecubes?” Based on Nevil’s responses in both interviews, before and after thelesson, his goal was that the students come to see, or revisit, both aspectsof division: quotitive and partitive. Nevil interacted with the pairs, both toassess the aspects of division they used and to promote their ability to useone, and preferably both, aspects. After 25 minutes, Nevil concluded thatmost students were able to see at least one of the two aspects of division theway he did – breaking a set into smaller equal sets. He put the Unifix cubesaway and moved to the second part of the lesson. Assuming that studentshad already been taught the algorithm in fourth grade, he asked them tosolve, individually, three long-division examples that he wrote on the board(140 ÷ 7, 568 ÷ 8, 5454 ÷ 6) and observed their execution of the algorithmfor about 10 minutes. Nevil had volunteers put their computations for thethree examples on the board. He then initiated the third part of the lesson,a 30-minute, whole-class discussion about the meaning of the steps of thealgorithm as performed in the example 140 ÷ 7. In summary, the lessonconsisted of a task with the cubes, pencil-and-paper computation, and adiscussion of the meaning of the steps used in the computation.

Nevil began the whole-class discussion by eliciting students’ reasoningabout the location of digits in the quotient of the first computation (140 ÷7), that is, why they put the 2 above the 4. Many of them explained thatthe 2 stood for 2 tens, hence 20. This explanation seemed to fit with hisexpectations. He then asked them to discuss the reason for the subtraction

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step (14 – 14 = 0). Students’ responses included, “to find the remainder,”“to find what is left,” “to check the answer,” or “to make the problemeasier, because the number is smaller.” The students’ responses did notseem to be what he wanted. After asking them several times to visualizethe absent Unifix cubes in order to explain, “Why do we subtract?” Nevilasked them to continue thinking about the reason for the subtraction stepfor homework.

In the interview after Lesson 1, Nevil explained that he was frustratedwith the students’ inability to grasp what for him seemed obvious. Forhim, one uses subtraction in the long-division algorithm to account for thequantity that had already been distributed into equal groups. Specifically,he explained to the interviewer that in the example 140 ÷ 7 one subtracts14 – 14 and gets zero because by distributing two tens (20) to each of theseven groups one had accounted for all 14 tens (140) of the initial quantity.In his reflecting on the lack of success of the lesson, Nevil consideredthat students might not recognize the relationship between the algorithmand the solution with cubes because of how place value is handled in thealgorithm.

Excerpt 4 (Interview about Lesson 1)

N: If we are going to multiply maybe it would make more sense tothem to see that we are really multiplying by 20 . . . that we writeout the whole number and then we are comparing. . . . I think Iwould ask them to explain to me visually what we’ve done so far,with – if they could use Unifix cubes or something.

R: So [at] that point you might bring the Unifix cubes back for themto work or you just [ask them to] imagine it?

N: I think I would ask them to imagine it. And to tell me that we’vemultiplied 20 times 7, and we’ve gotten 140, well that means thatwe’ve accounted for everything that was in that original set. Andin this next step, what we are doing is comparing the original setto the, the first number which [in this case] is 20 times 7, 140.

Nevil planned and implemented Lesson 2 along the lines of Lesson 1,trying to get the students to see the reason for the subtraction step. Hebegan by reviewing the initial steps of the algorithm. This time he empha-sized the multiplication step 20 × 7 = 140 and even recorded the productas 140 rather than as 14. From there, he moved on to the subtraction step.However, when the students’ responses to his repeated questions, “Why arewe subtracting?” did not satisfy him, he changed the manipulatives that heasked them to visualize, from Unifix cubes to base ten blocks. In the nextinterview, he explained that such a change might help the students see the

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base ten aspect of distributing the initial quantity into a certain number ofgroups, that is, the distribution first of the hundreds, then the tens, and thenthe ones.

Following this change, Nevil continued his inquiry about the subtrac-tion step for the two first examples (140 ÷ 7, 568 ÷ 8), but still thestudents did not seem to respond as he expected. Toward the end of thelesson, he posed a new example, 100 ÷ 6. In the interview that followedLesson 2, Nevil explained why he posed the new example. He said thateven with the change to base ten blocks, “We had gotten as far as we weregoing to get with this idea of subtracting and what does it mean, and weweren’t going to get beyond from the students.” He made it clear that hisintention in posing the new example was to begin working on the issue ofremainder, not to inquire further into their understanding of the subtractionstep. Nonetheless, we observed that he incorporated his interpretation ofthe subtraction step into the discussion of the new example:

Excerpt 5 (Lesson 2)

N: So what did we take care of? What shape of base ten blocks didwe take care of?

S: 100.N: Yeah, we couldn’t do the 100 right. . . . So we need to trade it in

and get rows. Yeah?S: Um, since there is [inaudible] 60 minus 100 is forty and forty

[inaudible]N: Okay 40 is still in my bag. Still got 40 left. Yes?S: Yeah.N: Okay, now what do I do? I figured out how many tens will be in

each of the 6 groups, right? How many rows. And there is onlyone [row] in each, okay. But that accounts for 60 of the original100. Agreed? So I have 40 left in my original pile. That is why Isubtracted, to find that out. Okay. Because I don’t have the cubesin front of me. I have to figure it out with the numbers so now Ihave 40 left.

In Excerpt 5, we noticed that once Nevil decided he could not elicitfrom the students the reason for the subtraction step, his teaching changed.He explicitly presented his interpretation of that step through the nextexample. At the end of Lesson 2, Nevil assigned for homework the taskof explaining the connection between the algorithmic solution for a newexample (200 ÷ 6) and the process with base ten blocks.

Nevil planned to begin Lesson 3 with students using base ten blocksto demonstrate 100 ÷ 6. Nevil explained that he wanted the students to

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actually use the blocks. His reason was that the blocks serve “as a concreteway of showing what division is all about . . . a kind of a concept builder.. . . Instead of [focusing] on the algorithm and the numbers [the kids are]actually dividing something by something else.” Then, he would again putthe blocks away and ask them to visualize their solution with the blockswhile connecting this solution to steps in the algorithm, particularly thesubtraction step. As it turned out, Lesson 3 consisted mostly of actualmanipulation of the base ten blocks in small groups and of groups reportingtheir solutions to the class. Contrary to Nevil’s intention that students showthe connection to the algorithm via partitive solutions, a majority of thestudents solved the example in a quotitive way. Nevil tried to lead thosestudents to consider a partitive solution by asking them, “Is there any otherway to do this problem?” However, they responded with more quotitivesolutions. Toward the end of the lesson he called on a student (Carol)whom he knew had used a partitive solution. Through leading questions,he elicited a description of the process of trading base ten units and puttingthem into 6 equal groups, while he specified the connection between herdescription and the algorithm, orally and by writing on the board.

Nevil began Lesson 4 by asking students to work the example 100 ÷5 using base ten blocks. He then asked them to consider the connectionbetween division with the blocks and the algorithm, first in small groupsand later in a whole class discussion. Again, the first two groups presentedquotitive solutions, so he called on Carol, the student who presented thepartitive solution in the previous lesson:

Excerpt 6 (Lesson 4)

N: Okay. Now I am asking the people who did it in five groups,and you found 20 in each group. I am asking you. Did youfind any relationship to this algorithm?

Carol: None.N: None. I don’t believe that. I believe that if you would have

thought – you would have found some kind of connection.You are all very intelligent people and we have been doingthis [for] several days and you have been doing a lot ofthinking about this. And I don’t believe that you didn’t findany kind of connection.

Excerpt 6 indicates Nevil’s expectation that students who used apartitive solution would see the connection to the algorithm. We note thatNevil seldom blamed students. He usually responded to students’ diffi-culties by taking responsibility for the difficulties and striving to figure outan alternative way of teaching that would reveal the intended mathematics

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to the students. This rare response seemed to indicate a high level of frus-tration. Nevil had tried all the alternatives that he could generate, includingthe work with the base ten blocks that he believed made the connectionsobvious. Yet, in spite of all his attempts to reveal to the students whatseemed so obvious to him, they did not “get it.”

Following the exchange presented in Excerpt 6, Nevil asked thestudents to work on another example (97 ÷ 5) with the base ten blocks.Then, by heavily paraphrasing contributions from a small group that usedthe partitive solution, he explained in detail the process of trading base tenblocks, while pointing to specific connections to steps in the long-divisionalgorithm. For example, he pointed out that algorithmically asking howmany times 5 goes into 9 corresponded to putting one ten in each of thefive groups and that the number 1 above the 9 represented that ten. Whentalking about the connection to the subtraction step, Nevil repeated severaltimes the idea that the subtraction did not mean taking something away, butrather that portions of the initial quantity were moved into the five groups.

Developing an Account of Nevil’s Practice

As we initially examined the data set described above, we noticed partic-ular aspects that we could not explain with the perspectives that we hadavailable. A useful account of Nevil’s practice would have to provide asound interpretation of the following:

1. Nevil began the division lessons by asking the students to show 16divided by 2 and 20 divided by 4 with Unifix cubes. His goal was“to get the students to understand that division has to do with begin-ning with a number of things and then dividing it into sets.” From ourperspective, in order to complete the task of representing division withthe cubes, the students needed to have already constructed the meaningof division.

2. During the first two lessons, Nevil deliberately chose not to makethe manipulatives available to students during the discussion ofthe long-division algorithm. Rather, he invited students to visualizemanipulative solutions to inform their interpretation of the algorithm.

3. For homework, Nevil assigned the same question the students couldnot answer during Lesson 1, and proposed and implemented essen-tially the same plan for Lesson 2.

4. Initially, Nevil did not focus on the multiplication step, but proceededdirectly to the subtraction step.

5. In the latter stages of the lesson Nevil began to use leading questionsand explicit statements of what he previously tried to elicit from thestudents.

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In our first attempt to make sense of these aspects of Nevil’s practice,we observed the following characteristic that seemed to remain unchangedthroughout the lesson sequence. Nevil expected the students to see whatseemed obvious to him, the relationship between dividing blocks andthe long-division algorithm. This was extremely clear when he becamefrustrated about Carol and her classmates’ inability to see the connectionbetween a partitive solution with the blocks and the steps in the algorithm.He did make modifications in his approach: He changed the manipulativesto be visualized, changed from visualizing to manipulating base ten blocks,changed the example, and moved from eliciting ideas to stating his ideas.However, all of these efforts seemed to be grounded in the notion that if thestudents had experienced division using the physical manipulatives, theyshould see how the long-division algorithm paralleled the manipulativesolution. We came to understand Nevil’s perspective as follows.

Nevil attempted to deeply understand the mathematics that he teaches.He viewed mathematics as a connected, logical set of ideas. He wasaware of those situations that clearly demonstrated, for him, particularmathematical relationships. Cobb (1989) pointed out that an experi-ence of perceiving mathematical truth is an inherent part of developingmathematical understandings.

Once we have made a mathematical construction and have used it unproblematically, weare convinced that we have got it right – it is difficult to imagine how it could be any otherway. Mathematical objects are, for all intents and purposes, practically [emphasis original]real for the experiencing subject. (p. 33)

From our perspective, Nevil saw the world through his own mathe-matical conceptions, unaware of the role that his conceptions played inhis experience of the world. For Nevil, his identification of division inthe world around him was a perception of one aspect of objective reality.Thus, he assumed that division could be seen by all in the distributionof physical objects into groups. Further, for him, the connection betweendivision in physical representations and the steps of the algorithm wasperceivable and apparent. Note that we do not claim that Nevil held anyformulated positivistic view of mathematics; we only infer that a positiv-istic view was implicit in his perspective on mathematics learning whichmight have been partly or wholly implicit as well. That is, learning meantcoming to see first-hand particular aspects of mathematical reality and theconnections among them. Our sense is that for Nevil, coming to understandmathematics was often like identifying difficult-to-see physical entities inthe environment. For example, to find a house from the top of the localmountain, one needs to have particular reference points that orient one’sperception, such as locating the nearby water tower. In this view, then,

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learning is a gradual process of coming to see additional aspects of the webof mathematical relationships; aspects seen previously serve as points ofreference for what is yet to be seen. Accordingly, first-hand manipulationof physical objects is critical for learning, because it shows the intendedmathematics in the concrete relationships among objects and affords theconnection to the abstract, symbolic operations with numbers.

Consistent with his perspective of mathematical knowing and learningpostulated above, Nevil considered teaching to be about assisting studentsin seeing the intended mathematics. Towards this end, he employed threeinterrelated activities. He decomposed his mathematics into perceivablepieces and connections. He created conditions that afforded students’perceptions of these pieces and connections. He monitored whether ornot the students perceived the mathematics by determining whether whatthey reported matched the mathematics as he knew it. Nevil continuallyadjusted the first two activities based on the results of the previous activity.We illustrate the activities with examples from the data presented earlier.

In planning the lesson sequence, Nevil decomposed the long-divisionalgorithm into sub-procedures to be considered. Underlying this strategyseemed to be his appraisal that the students would not see the whole rela-tionship, but that breaking the task into smaller steps would make therelationship to the blocks solution easily perceivable. His monitoring ofthe students’ contributions led him to decompose the mathematics in thealgorithm further. For example, he decided that a discussion of the multi-plication step and a greater focus on the place-value meaning would makeit easier for students to see the relationship to the subtraction step.

We can see Nevil’s second strategy, the creation of conditions thatafforded students’ perceptions, in his initial activity with Unifix cubesand the subsequent assignment of three long-division computations. Byjuxtaposing the two situations, he intended to allow the students to seethe correspondence between the concrete situation and the algorithm. Hismonitoring of the students’ contributions led also to a revision of the initialconditions. Instead of asking them to visualize Unifix cubes, he asked themto visualize base ten blocks. He later modified the conditions further byengaging the students in actually working with the base ten blocks.

The examples above highlight that Nevil’s focus in teaching was onorienting students’ perceptions. This is illustrated further in Excerpt 7.

Excerpt 7 (Interview after Lesson 3)

N: I’m wondering if they are not understanding what I want them tounderstand because I haven’t come up with a way [that allows]them to see it. . . . I’m approaching it from the way I saw it, and Isaw it [only] recently. . . . At some point trying to come up with a

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way to teach this I had to understand it more than I already did.. . . So I’m trying to come up with a way that will help them be ableto see what I want them to understand.

Excerpt 7 indicates that, for Nevil, what was problematic in teachingwas how to create situations in which students could see the mathematicsfor themselves. This approach is different from traditional teaching interms of types of learning situations the teacher considers to be appro-priate. In this case, the teacher focused on students’ active involvement inthe exploration of concrete representations of the intended mathematics sothey would discover the mathematics rather than being told or shown.

In Nevil’s assessment activity, we identified two related key features:carefully listening to students’ responses and analyzing what they learnedof the intended mathematics. Nevil’s listening was oriented to whetherstudents’ contributions matched his way of understanding the mathe-matics. If there was a match, he concluded that the students had grasped theintended mathematics, as his acceptance of their responses to the meaningof the “2 above the 4” (in 140) indicated. If there was no match, heconcluded that they had not grasped it; hence more decomposition andrefining of conditions was needed.

Nevil’s decisions about what and how to teach next were based on whathe had identified as lacking. Thus, he monitored students to determine whatthey still needed to learn and not to identify what they did know. Thismakes sense from the perspective of the account that we have articulated.If the mathematical relationships are pre-existing and perceivable as is, byall, Nevil was concerned with what was not yet seen. A focus on whatand how the students already understood fit with a different perspectivein which perception is understood as afforded and constrained by extantconceptions.

Having articulated our account of Nevil’s practice, we now use it toexplain the five observations presented at the beginning of this section.The first observation referred to Nevil’s task of using Unifix cubes to show16 ÷ 4 and 20 ÷ 4 as a means to promote students’ understanding of divi-sion. This puzzled us because of our perspective that a student must haveabstracted a rudimentary notion of division in order to be able to representit on the basis of a context-free division expression. However, that wouldnot be a goal that Nevil would formulate. His implicit sense that divisioncould be seen by all, as is, led him to inquire not how a concept of divisioncould first be constructed, where no such concept previously existed, butrather how he could orient the students to see division more fully. Thus, byworking with the Unifix cubes he was giving them an opportunity to seedivision.

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The second observation referred to Nevil’s deliberate choice to putaway the Unifix cubes before the discussion of the meaning of the stepsin the algorithm. Nevil’s goal was to promote a particular connection,the connection between the idea of division and the algorithm. For Nevil,the cubes provided a context from which the idea of division could begleaned. Once he believed that this had occurred, he worked on a connec-tion between the idea of division and the algorithm. Only a realizationthat students had not gleaned the idea from the cubes was grounds forbringing out the concrete materials once again. We infer such a realizationto be implicated in his decision to bring out the base ten blocks in the thirdlesson.

The third observation was about Nevil’s plan for the second lesson andthe questions he posed to students. The plan and questions were essen-tially the same as what, by his judgement, had not worked in the firstlesson. To explain this, we focus on what we infer to be Nevil’s implicitpresumption that the intended connections were visible to all. Accordingto such a perspective, the modification needed was one designed to makethe connections more obvious. As discussed above, for Nevil this involvedfurther decomposition of the mathematics and modifications in the condi-tions created to foster the perception. When these changes met withlimited success, he was unable to question his overarching perspective,that is, to consider that the students were unable to see the mathematicalrelationships that were apparent to him.

The fourth observation referred to Nevil’s initial lack of focus on themultiplication step. Our interpretation is that Nevil was using his ownanalysis of division, with blocks, as the basis for teaching about themeaning of the algorithm. In his own exploration with the blocks, themultiplication step was assimilated into his conception of multiplicationand division as being an organization of quantities into equal-size groups.For Nevil, arranging the 14 stacks of 10 blocks into groups was recorded byplacing the digit in the quotient and by multiplying 14 x 10. It did not occurto him to ask about the meaning of the multiplication because the studentshad just explained (in the context of explaining the 2 in the quotient) thelink to arranging the 14 stacks of blocks into groups. Therefore, it wassufficient to go directly to the subtraction step as a comparison between theoriginal amount and the amount already distributed to groups. Note that weare not claiming that Nevil consciously rejected a focus on the multiplica-tion step. Rather, due to the way he had assimilated the algorithm into hisconception of multiplication and division, it was not a salient part of hisoriginal decomposition of the mathematics. After the students repeatedlyfailed to give responses that he considered satisfactory, he re-examined the

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intended mathematics for opportunities for further decomposition. This re-examination led to an explicit focus on the multiplication step before thesubtraction step.

The fifth point was that Nevil became more explicit in indicating tothe students the ideas he previously tried to elicit from them. We suggestthat because of his feeling that he had done all in his power to decomposethe mathematics and to create proper conditions for perceiving it, Nevilresorted to showing them directly what he wanted them to see. He wasfrustrated with his inability to indirectly orient their seeing. However, hewas committed to revealing to them what he considered to be an importantand accessible piece of mathematics.

DISCUSSION AND IMPLICATIONS

Characterizing a Perspective

The development of the account of Nevil’s practice has contributed to ourpostulation of a more general construct, a perception-based perspective(Simon et al., 2000), which we take to be fundamental to Nevil’s practice.This perspective consists of four interrelated stances:

1. A platonic view of knowledge, that it exists as part of an objectivereality accessible to all.

2. A view that mathematics makes sense and that mathematical ideas areinterconnected.

3. A view of mathematics learning as an active process of first-handexperiencing of mathematics as it exists.

4. A view of mathematics teaching as creating situations that reveal themathematical ideas and as orienting students’ attention to key aspectsof those situations.

Thus, the teacher’s role is to create opportunities for students to perceive,first-hand, intended aspects of mathematics.

Contrasting Three Perspectives

A perception-based perspective is a complex set of pedagogical concep-tions. For further clarification, we contrast it below with traditionalperspectives and with perspectives that we refer to as conception-based.We emphasize that traditional, perception-based, and conception-basedare our characterizations of teachers’ practices, and not how the teacherswould describe their practices.

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Traditional approaches to mathematics teaching can be characterizedby teachers’ attempts to transmit particular mathematical ideas to students.Like a perception-based perspective, traditional approaches include aplatonic view of knowing. However, in contrast to traditional approaches,teaching based on a perception-based perspective emphasizes studentscoming to see mathematical ideas and relationships through their ownexperience of the mathematics. Hence, the teacher’s primary role is notto directly transmit the intended ideas to students, but to orchestrateconditions that engage students in actively seeing and connecting thoseideas. We conjecture that transition from a traditional to a perception-based perspective can be an important step toward development of aconception-based perspective. We will return to this point in the nextsection.

A conception-based perspective is based on the notion of assimila-tion, that is, that humans have no access to a reality independent of theirways of experiencing it. One can only perceive what is part of one’sexperiential reality (von Glasersfeld, 1995). Thus, individuals’ conceptionsplay a major role in what they perceive, what they learn, and how theylearn it. The perspective that we described in the conceptual framework isan example of a conception-based perspective. From such a perspective,mathematics is thought of as a web of conceptions that humans abstractthrough reflection. Learning is the building up and the continual trans-formation of one’s conceptions. Teaching, promoting intended conceptualadvances, entails two major components. The first is the creation oflearning tasks for which students set goals and use activities available tothem to accomplish the task. The second is orienting students’ reflectionto identify patterns in their activity and the effects of that activity. Theseideas are further elaborated in Dewey (1933) and Simon et al. (1999).

A perception-based perspective on teaching contrasts with aconception-based perspective. Whereas the former assumes access to apre-existing, objective mathematical reality, the latter explains learning interms of building on the distinctions that learners can make. These distinc-tions are afforded and constrained by their current conceptions and theirgoals for the activity in which they are engaged.

The construct of perception-based perspective that we postulated didnot result only or directly from the account of Nevil’s practice. Yet, thisaccount helps to elucidate the construct. Because we could not attributelimitations of Nevil’s teaching to some lack of mathematical understand-ings, we were compelled to reflect on and explain aspects of his practicethat initially did not make sense to us. By doing so, we realized thatamong the four stances of the perception-based perspective, the last two

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– learning as an active process and teaching as decomposing the mathe-matics and creating conditions to reveal it – serve an epistemologicalcore similar to the core of a traditional approach. In other words, theaccount of Nevil’s teaching accents the nature of constraints put forthby the epistemological stance of a perception-based perspective and thelimitations of an approach to teacher development that focuses on fosteringbetter mathematical understandings among teachers.

A Perception-Based Perspective as a Step Toward a Conception-BasedPerspective

There are three parts to our reasoning behind the conjecture that aperception-based perspective can be a step towards a conception-basedperspective. First, a platonic epsitemological core is shared by both tradi-tional and perception-based perspective. Hence, the change from tradi-tional to perception-based perspective does not require a major paradigmshift. Second, our analysis of the MTD course data (Simon et al., 2000)has yielded the following conclusion. In the MTD courses, the design ofinstruction for the combined group of practicing and perspective teacherswas based on our conception-based perspective. However, the courseexperiences seemed to be assimilated and accommodated by the teachersinto perception-based perspectives. These first two points suggest thattraditional perspectives might evolve into perception-based perspectives.The third point involves consideration of a perception-based perspective asa step towards a conception-based perspective. From a conception-basedperspective, seeing mathematical relationships is key. If teachers can cometo explore why seeing particular relationships is problematic for somestudents and not for others, they may begin to develop understandings ofassimilation that can support a conception-based perspective.

Implications of a Perception-Based Perspective

We present three implications of a perception-based perspective for teachereducation. First, the construct provides teacher educators with a frame-work within which to (a) anticipate how teachers might interpret partic-ular teacher development opportunities and (b) understand the teachers’responses after the fact. We illustrate this point with two examples. As partof teacher development programs, opportunities to interview and reflecton students’ mathematical activity might be interpreted by teachers as achance to look closely at what mathematical relationships students havenot perceived, as opposed to an opportunity to focus on students’ under-standings of the mathematics. Another type of opportunity for teachersis a course in which they learn mathematics in a reform-based situation.

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Teachers who experience a new-found understanding of the mathematicsmight conclude certain problems, manipulatives, or software environmentsmake particular mathematical relationships obvious. Thus, because theydo not make a distinction between their assimilatory conceptions and thestudents’, they may attempt to simply transport what was done with themto their own classrooms.

Second, as a description of teachers’ perspectives on mathematicsteaching, the construct can inform teacher education instructional designby specifying broad understandings of teachers – understandings they canuse to engage in teacher development tasks. This point is illustrated in thenext section.

Third, the construct perception-based perspective and its contrast withconception-based perspective define a direction for teacher development.That is, if we take perception-based and conception-based as descriptionsof two locations on a developmental spectrum, we can take as a goal forteacher development the change from the former to the latter. Thinkingof a transition from perception-based to conception-based as a develop-mental step depends on a claim that a conception-based perspective is morepowerful for mathematics teaching than a perception-based perspective.We now offer a brief argument to justify this claim. The reader shouldkeep in mind that our conception-based perspective, as outlined in theconceptual framework for this article, structures not only our postulationof these perspectives, but also our understanding of the limitations of aperception-based perspective.

In order to discuss the relative power of these two perspectives,it is necessary to revisit the fundamental difference between them. Aperception-based perspective is based on a view of knowledge as a“mirror” of a reality that is external and universal (Cobb, Yackel & Wood,1992). A conception-based perspective is based on the notion that wecan only know our own experiential reality, which is determined by ourassimilatory conceptions to that point. However, even those who we wouldcharacterize as having a conception-based perspective operate, at times, asif there were a universally accessible reality. For example, people usuallyassume that others see the same physical objects that they do, or give thesame meaning to commonly used words, unless there are some indicationsthat this is not the case (von Glasersfeld, 1995). What is different aboutthe conception-based perspective is that individuals who have developedthat perspective have the possibility, at any time, to step back from thisassumption of a universally accessible reality to question the differencesin learners’ experiential realities.

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In mathematics education, a perception-based perspective may beadequate for thinking about some situations. When learners have alreadyabstracted two potentially related conceptions, a teacher who holds aperception-based perspective like Nevil’s might be able to design alesson to promote connections between the two conceptions. That is, theexpectation that students see the connections in a well-chosen situationis reasonable, because, from a conception-based perspective, the ideas tobe connected are available in the learners’ experiential reality. However,a perception-based perspective does not address the challenge of engen-dering a new conception. For example, a perception-based perspectivedoes not explain why a child does not see division in an arrangementof objects, nor how the child comes to understand division. Moreover,a perception-based perspective does not assist the teacher in thinkingabout how such conceptions might be promoted when, as in Nevil’s case,students’ do not perceive the intended mathematics despite the teacher’sdeep understanding and decomposition of the mathematics. In contrast, aconception-based perspective can be the basis for inferring how learners’current conceptions limit the mathematics they can see in certain situ-ations, and for thinking about how these conceptions might afford activitiesand goals that can foster intended abstract ideas.

Summary

Teachers’ perspectives consist of intricately related webs of understand-ings of which little is known, particularly as these perspectives haveevolved in relation to current mathematics education reform efforts. In theaccount of Nevil’s practice we postulated an assimilatory perspective thatmay be useful to consider in teacher development efforts with teachers whoare participating in current reforms. By focusing on teachers’ assimila-tion, we invite rigorous consideration of how teachers make sense of whathappens in their classrooms and what they encounter in teacher develop-ment situations. Research of this type can contribute to the establishmentof a broader set of distinctions about teachers’ conceptions, and henceserve as a significant part of a knowledge base for mathematics teacherdevelopment.

The account of practice presented in this report describes one adapta-tion of teaching that is based on, but not determined by, a perception-basedperspective. In our work, we have observed that teachers develop diversepractices that can be characterized as perception-based (Heinz et al., inpress). An important implication for teacher education of the particularpractice characterized in the account of Nevil is the need to help teachersdistinguish between their mathematics and students’ mathematics. Nevil

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represents a subset of teachers who are at least beginning to have somepowerful mathematical understanding. Nevil used his mathematical under-standings in a pedagogically significant way to set goals for students’learning. On the other hand, teachers must set aside their mathematicsin an attempt to see the mathematical situation through the eyes of theirstudents.

The account of Nevil is particularly illustrative because Nevil under-stands the mathematics he is teaching, is able to set mathematical goalsthat focus on understanding, and is aware of his students’ failure to under-stand – all key attributes for carrying out a reform agenda that emphasizesproblem solving, reasoning, and making connections. The account demon-strates the limited range of options available to Nevil and his inability toquestion the fundamental assumptions of a perception-based perspective.As we suggested earlier, the ability to identify these assumptions and tosee them as modifiable is the beginning of a paradigm shift.

Looking Ahead

Based on the work that we have been describing, we would argue theneed for research that investigates the promotion of teacher developmenttowards a conception-based perspective. Promoting such a paradigm shiftis an ambitious undertaking even in teachers who use significant mathe-matical understandings, like Nevil. In our own research, we have onlybegun to explore this possibility. Here we describe the two approaches thatwe have used in our initial attempts in hope that it will spark other ideasand further work.

One approach has been to create opportunities for teachers to comparetwo students in the context of similar mathematical tasks. We use thisapproach to orient teachers’ thinking to what students bring to the task.Working with teachers such as Nevil, who deeply understand the mathe-matics involved, enables them to potentially make useful comparisonsbetween the students’ thinking and their own.

Our second approach has been to set up a task environment that wouldbe used with elementary students and ask the teachers to anticipate howthe students’ work in that environment might progress. The teachers areencouraged to role-play the students. In the MTD Project, this activity withteachers builds on a shared notion that has developed during earlier coursework, that it is important to monitor whether what is attributed to studentsis the students’ voice or the teacher’s voice.

Although we are cautiously optimistic that such approaches can effectsome change, we do not know what reasonable expectations are nor howto maximize the impact of our efforts in this direction. Currently, we are

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analyzing more data from case studies and from the whole class teachingexperiment that might shed light on activities of the teacher educator thatcontributed to promotion of teachers’ sensitivity to students’ thinking. Thedata consisted of teachers’ struggle to understand why, for example, astudent could not make sense of a certain situation that they took forgranted. We think that through this work some teachers, such as Nevil,might have begun to develop a distinction between the child’s and theteacher’s voice, that is, their different mathematical experiences.

NOTE

1 The research was conducted as part of the activities of the NSF Project No. REC-9600023, Mathematics Teacher Development. All opinions expressed Are solely those ofthe authors.

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Ron TzurMartin A. SimonCurriculum and Instruction Department,The Penn State University,270 Chambers Bldg,University Park, PA 16802,USAE-mail: rxt9@psu.edu

Karen HeinzCollege of Science and Technology,Department of Computing and Mathematical Science,Texas A&M University,Corpus Christi, TX 78412,USA

Margaret KinzelDepartment of Mathematics and Computer Science,Boise State University,Boise, ID 93724-1555,USA