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SANDRA CRESPO

SEEING MORE THAN RIGHT AND WRONG ANSWERS:PROSPECTIVE TEACHERS’ INTERPRETATIONS OF STUDENTS’

MATHEMATICAL WORK

ABSTRACT. Listening to students’ mathematical thinking is one of the trademarks ofreform-minded visions of mathematics teaching. The questions of when, where, how, andwhat might help prospective teachers learn to do so, however, remain open. This studyexamines how a mathematics letter exchange with Grade 4 students provided an occasionfor prospective teachers to learn about students’ mathematical thinking and to examinetheir interpretive practices. Analysis of the interactions between students and prospectiveteachers, and of the reflective writing of the latter, revealed changes in the patterns oftheir interpretations. I characterized these as changes in the focus of interpretation, fromcorrectness to meaning, and in the interpretive approach, from quick and conclusive tothoughtful and tentative. I also discuss factors associated with these interpretive turns.

The idea of teachers listening to and understanding students’ thinkinghas been widely promoted and supported in the education community. Inthe Professional Standards (National Council of Teachers of Mathematics[NCTM], 1991), the analysis of students’ thinking is highlighted as oneof the central tasks of mathematics teaching. In this report, the analysisof students’ thinking is seen as a resource that can help teachers makeinformed decisions in their classrooms and improve their practice. Sucha listening orientation towards teaching promotes a learning environmentconducive to and respectful of students’ own sense making and intellectualautonomy (Davis, 1996; Kamii, 1989). In contrast, when teachers do notlisten to or do not understand their students’ thinking they tend to dismissit by imposing their own formalized constructions onto the students (Cobb,1988; Maher & Davis, 1990).

Although there are various ways in which teachers can listen to theirstudents’ mathematical ideas, Davis (1996) reminded us that not allforms of listening are conducive and respectful of students’ thinking. Hediscussed three different orientations teachers might have towards listeningin the mathematics classroom. Teachers with an evaluative orientation,according to Davis, tend to listen to students’ ideas in order to diagnoseand correct their mathematical misunderstandings. Teachers with an inter-pretive orientation, on the other hand, listen to students’ ideas with the

Journal of Mathematics Teacher Education3: 155–181, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

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purpose of accessing rather than assessing students’ mathematical under-standing. Teachers with an hermeneutic orientation, in turn, continuallyand interactively listen to students’ ideas by engaging with them in themessy process of negotiation of meaning and understanding.

Despite reform efforts aiming to change the evaluative ways in whichteachers tend to listen in mathematics classrooms, the notion of teaching astelling (speaking, explaining) rather than listening (hearing, interpreting)still pervades most mathematics classrooms. Underscoring the complexityof adopting a listening approach to teaching, Ball (1993) reminded us thatlistening to students’ thinking is hard work, especially when students’ideas sound and look different from standard mathematics. “The abilityto hear [italics original] what children are saying transcends disposition,aural acuity, and knowledge, although it also depends on all of these”(p. 388). Ball (1994) also noted that teachers have powerful disincentivesto seriously consider such a change in their teaching practice. Teachers’sense of efficacy, according to Ball (1994) and Smith (1996), is at riskeach time they ask students to voice their ideas in the open. Evidence thattheir students do not understand undermines teachers’ feeling of compe-tence and ability to help students learn. By not asking students to explaintheir thinking and by not listening to their ideas, teachers run less risk ofdetecting what students do and do not know (Ball, 1994).

For prospective teachers, the idea of listening to students is not obvious.Their difficulties with unfamiliar ways of teaching and learning are wellknown. For instance, they do not see the point of encouraging “studentsto set, explore, and solve their own problems” (Wilson, 1990, p. 206).Wilson’s prospective teachers concluded – after watching a video – thatthe students were confused and that the teacher should have stepped into clarify. Similarly, Ball (1990) reported that, when prospective teacherswatch classroom episodes in which students disagree or revise their solu-tion in front of the class, many of them “assume that [the child] isembarrassed about having been wrong” (p. 13). It is these evaluative waysof seeing and interpreting that teacher educators hope to shake, uproot, andinform (Holt-Reynolds, 1995; Wilson, 1990).

One way in which I have attempted to introduce elementary preser-vice teachers to the idea of listening to and learning from students is byengaging them in an interactive mathematics letter exchange with Grade4 students. These interactive experiences are meant to provide preserviceteachers with a context in which they explore and practice listening tostudents’ ideas; at the same time, the preservice teachers can investigate thechallenges and possibilities of such an interactive approach to mathematicsteaching and learning. In the letter exchanges, preservice teachers and their

PROSPECTIVE TEACHERS’ INTERPRETATIONS OF STUDENTS’ WORK 157

students typically pose a mathematical problem, share their solutions tothe problems, and share questions, opinions, and ideas about their ownexperiences as learners of mathematics. Although preservice teachers andstudents typically meet on at least two occasions, their interactions aremainly through written letters.

One advantage of the written communication over face-to-face interac-tions with students is that they afford preservice teachers more time todecipher students’ work and to think carefully about appropriate waysto respond. Another benefit of this delayed form of interaction is thatstudents’ work can be analyzed in the company of others or revisitedand reflected upon at different points in time. These advantages, however,do not make the actual task of interpreting and inferring meaning fromstudents’ work any simpler or less problematic. Knowing what to look forand what to do with students’ mathematical utterances, whether in writtenor in oral form, is not a trivial task for any teacher.

Aware of the challenges and possibilities that this particular form ofinteraction with students would offer preservice teachers, I set out toinvestigate what they might learn from such an experience (Crespo, 1998).Changes in the interpretations of their students’ mathematical work wereone of the important developments I noticed in the preservice teachers’learning experiences. I report on the features of preservice teachers’interpretations, what they attended to and ignored of their students’mathematical work, and their developing ideas about students’ mathe-matical abilities and understandings. More specifically, this report focuseson the following questions: In the context of their mathematics letterexchanges with school students (a) how do preservice teachers interprettheir students’ work, (b) how do their interpretations change over the dura-tion of the course, and (c) what contextual factors contribute to changes intheir interpretations?

DESIGN OF STUDY

The design of the mathematics letter writing activity as a context forlearning to teach is grounded in a view of learning as situated cognition.Proponents of this theory believe that knowledge is situated in and insepar-able from the activity, context, and culture in which it is constructed.Learning, therefore, is viewed as a process of enculturation or cognitiveapprenticeship into the practices and modes of thinking of a particulardiscipline, trade, or profession (Brown, Collins & Duguid, 1989). Letterexchanges with students provided a context that resembled the interactivenature of teaching practice, but without the immediacy and pressures

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for action that characterize actual mathematics classrooms. In the letter-writing context, preservice teachers could safely and supportively playthe role of a mathematics teacher under the supervision and guidance ofexperienced teacher educators and in the company of and in collaborationwith their peers.

Context of the Study

The letter-writing experience with students took place within the imme-diate context of an elementary mathematics teacher education course. Thecourse is required for all preservice elementary teachers (Grades 1–7)enrolled in the two-year teacher education program at the University ofBritish Columbia. The teacher preparation program offers a combinationof university-based courses and school-based practica. These are organizedin the following sequence: (a) foundational courses in education for thefirst half of the first year, (b) methods courses in all subject areas duringthe second half of the first year, (c) a 13-week school practicum duringthe first term of the second year, and (d) university-based courses focusingon the social context of schooling and a chosen subject-area of specializa-tion. Typically there are no long-term school-based experiences scheduledconcurrently or connected with the university-based course work. Thereis, however, a short 2-week practicum scheduled midway through themethods courses. The practicum is not associated with any of the methodscourses but is a program-wide experience.

The mathematics methods course I was teaching was an innovativeversion of the elementary mathematics methods courses typically offeredin the university’s teacher preparation program. It was a collaborativeventure between myself, two teaching partners, and three collaboratingteachers. My teaching partners were an assistant professor in the collegeof education and a fellow graduate student who, like me, focused herdissertation on one aspect of the course we were teaching (see Nicol,1997).

The course’s mathematics letter-exchange project, in turn, was acollaborative enterprise between myself and a local full-time teacher, whowas completing her Master’s degree in our department. For the collab-orating teacher the experience was meant to provide her students withan authentic audience with which to communicate mathematically (seePhillips & Crespo, 1996). In my case, the letter exchange was meantto provide a context for preservice teachers to explore and investigatestudents’ ways of thinking and communicating in mathematics.

Furthermore, the letter-writing and reading project was an integral partof the mathematics methods course. It was meant to serve as a laboratory


setting for preservice teachers to explore and try the ideas discussed duringthe methods classes. Interactions with students, in turn, served as the focusof class discussions and reflective journal writing. Most preservice teacherscorresponded with one student, although a few of them corresponded withtwo students. The letter writing activity spanned the whole methods course(11 weeks) and took place each week during one of our two 1.5-hourmeetings. Similarly, the letter writing activity was an integral part of themathematics classes of the Grade 4 students. They read and wrote theirletters during regular mathematics time with the assistance and encourage-ment of their teacher and peers. In both classes, students and preserviceteachers worked within groups of four, an arrangement meant to encouragethem to collaboratively read and write the letters.

Data Sources

For the study I used the data from thirteen preservice teachers (from agroup of 20). The data consisted of (a) all written work associated with thecourse, (b) videotaped class sessions (used for descriptive purposes), (c)six mathematics letters preservice teachers received and wrote in conjunc-tion with three common class problems they sent to the students, (d) oneteacher-directed letter from the students about the role of calculators andcomputers in their classroom, (e) journals about deliberations and reflec-tions on the interactions with students (reflection-on-action), and (f) a casereport written at the end of the course about the learning experiencesrelated to the work with the students. The preservice teachers’ writtenassignments – journal entries and case reports – were the main sourcesof data.

Data Analysis

The analysis of preservice teachers’ journal entries focused on identifyingand contrasting their patterns of interpretation at the beginning and the endof their letter exchanges. Preservice teachers’ case reports also provideddata that helped document changes in the interpretations of students’mathematical work. Secondary sources of data (e.g., mathematics letters,instructor’s journal, class videos) were occasionally referenced in order tohighlight and explore contributing factors to the changes in the participa-ting preservice teachers’ views, discourse, and practices. The data citedthroughout the following section are referenced by a code of letters andnumbers referring back to the data source and the timeline associated withit. For instance, a letter exchange referenced [ml1] means that the data isfrom the first mathematics letter the indicated participant wrote. A quotefrom a preservice teacher’s journal that is referenced [MJ1] means that

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it is a mathematics journal entry from the first week of class. Finally, areference coded [CR, p. 1] indicates that the quote is from the participant’scase report and is found in page 1 of the report.

EXAMINING PRESERVICE TEACHERS’ INTERPRETATIONS

This section highlights patterns in preservice teachers’ initial and laterinterpretations. The contrast between initial and later interpretations isfirst highlighted with a description of the contextual features of the courseactivities associated with the time period. The description is followed byan analysis of a sample letter exchange and reflective journal entry of onestudy participant, Sally. Examples from other preservice teachers are citedthroughout the text to provide further evidence of change in the inter-pretations the preservice teachers made about their students’ mathematicalwork.

A Beginning Letter Exchange

The Horse Problem (Burns, 1987), a problem we had worked on anddiscussed during our second week of class, was a common problem preser-vice teachers chose to send to their students in their first letters. Theproblem reads: “A man bought a horse for $50. He sold it for $60. Thenhe bought the horse for $70. He sold it again for $80. What is the financialoutcome of these transactions?” This deceptively simple-looking problemnever fails to cause uncertainty and confusion as to how it might be inter-preted and solved. Usually students generate alternative ways of solving itand come up with reasonable-sounding explanations for right and wronganswers. In response to the problem, common answers students offer anddefend are that the man makes $30, or $20, or $10; breaks even; or loses$10 or $30. Eventually, students narrow the solutions down to two: make$20 or make $10. Usually, proponents of either solution cannot easilydiscern why or where the competing solution may be flawed. Typicallythese solutions read:

Earned $20: The man bought the horse twice, which means hespent $50 and $70, or $120 buying the horse. He also sold thehorse twice, which means he made $140 from selling the horse.Subtracting what he made from what he spent, 140− 120, theresult is that the man made $20.

Earned $10: The man bought the horse for $50 and sold itfor $60, leaving him with a $10 profit. But then he bought the


horse again for $70, so that he used up his $10 profit, and hewas even. Then, he sold it for $80 and made another $10. In theend, the man made only $10.

The problem generated a lot of interest and a lively discussion aboutthe different answers and solution methods. In their journals, preserviceteachers wrote about being intrigued by “the number of ways in whichpeople went about solving the horse problem,” and “how others explainedhow they got their answers.” Some seemed excited about the prospect ofallowing students to discuss their solution strategies and explanations in amathematics class, whereas others thought that “kids should ultimately beshown the right answer and how it is arrived at.” This and similar experi-ences seemed to make preservice teachers curious about students’ thinkingand how they would justify their answers. It was therefore with greatanticipation that the preservice teachers awaited their students’ responsesto their first letters. The first letter exchange between Sally (preserviceteacher) and one of her students (Samantha) reads as follows:

Sally:Dear Samantha:. . . I’ll give you an example of a question we did [in our class] for youto try. When you write back, will you solve it in your letter, and write everything youthink down, even the mistakes? That will help me see how you think in math. Here is thequestion:

A man bought a horse for $50.He sold it for $60.Then he bought it for $70.He sold it again for $80.

How much money does he have in the end, or did he lose money? Lift for answer after youtry it [Sally had included the answer key under a post-it note] [ml1]

Samantha:Dear Sally: Thanks for the letter, I really enjoyed it. The first time I tride [tried] the horseproblem I came up with 70. This is how I got 70, start with 80− 10. The second time Itride [tried] the horse problem I got 20 because you start with $80− $10 and look at thedifferents [difference] between $70 and $50 witch [which] = $20. [ml2]

Sally:Dear Samantha:. . . Thanks foryour answer to the horse question. Thanks for including allyour attempts. I will explain 2 ways to get the answer [ml2].

Bought – $50} He made $10

Sold – $60} $10 (60− 50) +$10

Bought – $70} He made $20

Sold – $80} $10 (80− 70)

Or try using a number line.

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Samantha:Dear Sally: Thank you for your letter, I do understand the horse problem. Thank you forexplaining it !! [ml3]

After each letter writing session preservice teachers were required towrite in their journals about the insights, surprises, and questions that theirstudents’ letters had evoked. Sally’s journal entry read as follows:

Here is how [one of] my students responded to the Horse Question:1st time she got 70 by subtracting 10 from 80. I’m not sure why she did this, and to behonest, I was so pressed for time in reading and responding to 2 letters, that I didn’t thinkto ask her. From there, she must have looked at the answer, and seen that she didn’t havethe same thing, and tried again. This time it seems she was not really sure how I got $20,and “grasped” to get this answer. She explained that she started with $80, subtracted $10,and looked at the difference between $70 and $50 which is $20. I wrote back explaining 2ways of approaching the solution, and asked her to let me know if she didn’t understand.[Sally, MJ3]

Early Interpretations

Focusing on correctness.Despite indications that the preservice teacherswere inspired by the idea of listening to students’ mathematical thinking,once faced with their students’ work, they tended to focus on the correct-ness of their students’ answers. “Yes! They got it,” “This kid screwedup,” “Wow, was she ever off,” were some of the comments they madealoud while reading their students’ responses. Such comments high-light preservice teachers’ tendency to accept students’ right answers asevidence of understanding and to see students’ wrong answers as signsof confusion or carelessness. These tendencies were also apparent in thepreservice teachers’ initial journal entries in which they wrote little aboutthe meaning of students’ work. They noted, for example, whether studentswere successful or unsuccessful answering their questions and made nospecific inferences about what or how the students were or were notunderstanding.

Sally’s written exchanges and later journal entry illustrate the inter-pretive challenge that this form of interaction with students presented tothe preservice teachers. Quite often preservice teachers found themselveshaving to formulate a response letter with little idea as to what the studentdid or said. It is also important to note that Sally’s journal entry is quite


elaborate in comparison with the beginning journal entries from mostpreservice teachers in the study. Yet, Sally’s analysis focused mostly onthe correctness of the student’s work. Consider for instance that rather thanexplore and analyze the intriguing work she received from her student,Sally saw the answer as evidence of the student’s lack of understanding.Sally’s presumption of her student’s lack of understanding is apparent inher decision to respond by showing two ways of getting the correct answer,without acknowledging or using the work the student had already done.Another indication of Sally’s assumption is her statement that the student“must have looked at the answer” in order to solve the problem correctly.

Noteworthy in Sally’s interpretive comments is her admission that shedid not quite understand the student’s solution. She seemed genuinelypuzzled by the students’ work. Yet, her subsequent analysis focused onexplaining how the student could have come up with the right answerdespite producing an explanation for her work that, in Sally’s eyes, wasconfused and possibly made up. Rather than exploring and speculatingwhat the student’s work might mean or suggest, Sally closed other possibleinterpretations when she conclusively stated that the student “must have”looked at the answer.

Samantha’s work, however, need not be so easily dismissed. A secondlook at her work could suggest an alternative interpretation. One mayconsider, for instance, that her solution is counter-intuitive, that is, it usesnumbers that are not included in the problem (e.g., $10). This suggests thatshe is not mindlessly plugging in numbers as Sally insinuated. Taking thisperspective makes the student’s work seem more sensible and reasonableeven though at first glance it may seem indecipherable or inaccessible.Samantha’s solution also raises a few interpretive questions. For instance,one may ask, why did she use the $50, $70, and $80, and not the $60?Where did the $10 come from – the difference between $50 and $60 or thechange between $70 and $80? Also, what is her $70 referring to – the $70in the problem or the $70 that resulted from the first subtraction (80− 10)?

This sort of exploration may lead to other plausible interpretations ofthe student’s solution. For instance, one could reasonably say that her“made $70” solution addresses Sally’s first question of “how much moneydoes he have in the end?” – a question about the resulting balance. Areasonable answer to this question is that in the end the man has $80 in hispocket. The student’s “made $70” answer, however, may have consideredthat in the process of selling and buying the horse, the man had to borrow$10 in order to make up the $70 needed to buy the horse the second time.So, in the end, the man ends up with $80 minus the $10 he borrowed. Inthe second attempt, the student may be addressing Sally’s second question

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“did he make or lose money?” – a question about the resulting profit.The student may have figured that she also needed to subtract the $50the man started with from the $70 he had in the end. Though this is onlya conjectured explanation for how the student may have worked out theproblem, considering this as a possibility serves to open the avenues ofinterpretation, which in turn can suggest other ways of responding to andprobing the student’s solution to the horse problem.

Similar to her students’ mathematical work, Sally’s work is not as clear-cut or single-minded as my early discussion might suggest. Her work isalso open to multiple interpretations and raises important interpretive ques-tions about what she intended to see and was able to see in her student’swork. For instance, there are many indications that she was intending toattend to her students’ thinking and not only to the correctness of the work.In her letter, she asked her students to “write everything you think down,even the mistakes” so that she would be able “to see how you think inmath.” In her response letter she offered not one but two different solutionmethods to the horse problem. This suggests a concern for encouragingstudents to see multiple solutions to problems. In light of this, one also hasto wonder why Sally chose to send her students the answer key (with onlythe answer) in the first place.

It is also interesting to note that in her analysis Sally could re-stateher student’s solution word-by-word. She realized that she did not reallyknow why the student subtracted 10 from 80. Did she mean that she hadno real data or proof as to why she did this, or that she did not see whythe student did this? It is also interesting that Sally chose not to speculate,in writing, about her students’ reasons for subtracting 10 from 80. Yet atthe same time, Sally shared her conjecture or perhaps suspicion of howthe student came up with the right answer. With this I wish to point outthat I am not claiming that correctness is the only issue to which Sallywas attending in her beginning interpretations. I do claim that correctnesswas a prevalent focus of attention when analyzing the student’s work atthe beginning of the course. This claim is based on the examination ofnot only Sally’s work but also of other preservice teachers’ writings. Thisfocus on correctness is also more apparent when contrasted with preserviceteachers’ later interpretations.

Another letter exchange between Miriam (preservice teacher) and herstudent (Beth), further illustrates preservice teachers’ restrictive attentionto the correctness of their student’s mathematical work. Beth provided thefollowing solution to the horse problem:


Dear Miriam: Here is how I figured your math question out. A man bought a horse for fiftydollars and sold it for sixty so he gained 10 dollars and he bought it for seventy dollars sohe lost ten and sold it for eighty dollars so he gained ten dollars. [ml2]

Miriam responded as follows:

Dear Beth: . . . Thank you foryour letter. . . . I am glad you tried the problem that I gaveabout horses. Many people in my class had the same answer as you, but the answer shouldbe $20. Here is how I did it. [ml2]

Buy 50 Sell 60

+70 +80

120 140

140− 120 = $20

Afterwards Miriam wrote in her journal:

Beth tried the horse problem that I gave her and said that it was easy, but in fact, she waswrong. I was at a total loss because I did not want to come out and say that she is wrongbecause she mentioned how easy the question was. Even in our class there were a numberof people who had the same answer as her ($10) when we first got the question. I told Beththis and explained that it is not the correct answer, then I told her how to use one way tosolve the problem. I can understand where her reasoning led her to have the $10 as profit.Instead, it’s supposed to be $20. [Miriam, MJ3]

Miriam’s analysis of her student’s solution focused, like Sally’s, on thewrongness of the student’s work without delving into what the studentmay or may not have understood. Both preservice teachers responded bytelling their students how to find the correct answer. Although, in contrastto Sally, Miriam claimed she “can understand” her student’s solution,her lack of elaboration, along with the response she made to correct herstudent’s thinking, suggest otherwise. Sally and Miriam’s inattention tothe details in their students’ work were not isolated cases nor confined tostudents’ wrong answers. Preservice teachers whose students had reachedthe correct answers also seemed unaware of and inattentive to what theycould learn from analyzing their students’ work. Some simply commentedthat their student: “did a good job at explaining how she got the answer”(Linda), was “successful” answering their question (Mitch), or “seemsadvanced in math and writing compared to others’ letters” (Terry). Others,like Thea, simply noticed that their students’ letters “were short andcontained little information.”

Making quick and conclusive claims.Quick judgment was a secondprevalent feature of preservice teachers’ early interpretations of theirstudents’ mathematical work. Early on preservice teachers did not hesitateto make conclusive claims about their students’ understanding or lack of

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understanding based on the correctness of the work they received. Very fewpreservice teachers raised questions about their students’ understandingof the specific mathematics. Those who did, as in the case of Sally andMegan, did so out of suspicion that the student “must have looked at theanswer” before solving the problem, (Sally) or that the student “did notwork out the problem herself” (Megan).

In addition to making claims about the students’ mathematical under-standing or lack thereof, preservice teachers’ interpretations also focusedon their students’ mathematical abilities and attitudes. Based on thestudents’ questions, comments, and writing style, preservice teachers madeclaims about their students’ personal characteristics and dispositions aslearners of mathematics. Consider Rosa’s interpretive comments afterreading her student’s first letter.

After receiving the first letter I can tell that Paul is having problems in math. He enjoysaspects of math that he understands, but does not like it when he cannot understand things(e.g., division). This is only natural. I have been trying to think of ways that might helpPaul with his difficulties. With the help of the readings, I have come up with a numberof possible routes: a math journal (so that the teacher is in a better position to help Paul);asking questions like those presented in the article “asking questions” which will help Paulthrough a problem step by step; more group work (i.e., discussion); more math games(seeing as Paul thinks these games are fun); and using manipulatives and/or illustrations towork through a problem. [Rosa, MJ2]

Rosa’s analysis of her students’ comments is both plausible and contest-able. Rosa, however, had already begun to devise strategies to alleviateher student’s alleged difficulties with mathematics. Students’ comments,such as the statement from Rosa’s student that “as for math I need morepractice,” often caught the preservice teachers’ attention. Interestingly, thepreservice teachers seemed to relate to these types of comments moreeasily than to the actual mathematical work the students were providing.Their previous experiences with mathematics seemed more readily avail-able for aiding interpretations of such comments. Megan, for example,concluded: “I can tell that the students are trying to impress me,” and thereason she provided was “because they both stated in different words thatmath is great.”

As Rosa’s and Megan’s interpretive comments illustrate, preserviceteachers were noticing clues not only in students’ mathematical workbut also in the comments, questions, and suggestions the students wereproviding. From the letter’s length to its decorations to its personal under-tones, preservice teachers were often drawing and expressing inferencesand conclusions about their students’ personal characteristics. Compar-isons among students were often referenced in such interpretations. Thesecomparisons were more prominent, however, in those preservice teachers


who corresponded with two students. Lesley, for example, concluded thefollowing about the differences she observed in her two students’ firstletters.

Lynn’s letter seemed more focused on personal questions but this could be because she isbored in math or doesn’t know the English terms to use [Lesley knew Lynn was an ESLstudent]. Parker was full of advice, which tells me he seems to have his own ideas aboutgood and bad teachers and the methods that work better. [Lesley, MJ3]

Students’ brief responses to the preservice teachers’ questions and theirlack of elaboration on their answers were often interpreted as a lack ofinterest in mathematics and a sign of their lack of mathematical abilities.Her student’s brief letters and responses, according to Nilsa, gave her“grounded reasons to believe that Gina is at the bottom of her class inmath.” Likewise, Linda derived a similar interpretation: “If I wanted togeneralize, I could make the assumption that, based on his writing andspelling abilities (which are dreadful), he’s not good at math, but I thinkI will wait a bit on that.” Not much later, however, Linda stated: “I don’tthink he has much of an understanding of how to work with numbers eitherin his head or on paper/with manipulatives.”

Often, preservice teachers attended to the spelling in their students’letters. “I think what struck me first about her letters was nothing to do withmath, it was her spelling – it was atrocious!” said Nilsa of her student’s firsttwo letters. Similarly, Linda commented about her students’ communica-tion skills. “I’m not sure how things are going to go with David. He doesn’twrite as much as Shelley, nor does he spell very well (that’s the Englishteacher in me coming out!).” Students’ misspellings coupled with theirmiscommunication exerted a powerful, though often inadvertent, influenceon the preservice teachers’ conclusions about their students’ mathematicalabilities and attitudes.

A Later Exchange

During the analysis of the data I began to notice that the pattern in preser-vice teachers’ interpretations began to change around our 5th week ofclass. It was during that week that our class began studying the topicof measurement. We worked on various problems on area and perimeter,and on surface area and volume. A journal prompt titled Area of Interestinvited preservice teachers to explore the relationship between perimeterand area before they selected a problem that dealt with this content for theircorrespondents. This particular prompt challenged preservice teachers’own understandings of these concepts and encouraged them to clarify theirideas on these topics. Rosa, for example, wrote:

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I must admit that I went to our text to get a more complete understanding of what area andperimeter are. I tried doing some of the activities suggested in the book to apply what Iknew about these concepts. I then went back to the handout to see if this information couldhelp me in answering the questions: e.g.: “is it possible to have two figures that have thesame perimeter but different areas” and “what about two figures that have the same areabut different perimeters?” I found myself doodling on scrap paper to try to answer thesequestions. I drew a square and a long, skinny rectangle, both of which had a perimeter of16; thus 2 shapes can be different yet have the same perimeter. I then drew a grid to answerthe second question and found that it too was possible. [Rosa, MJ4]

Their own explorations into the concepts of area and perimeter, in turn,made preservice teachers curious about their students’ thinking and under-standings of these concepts. Sally, for instance, constructed her own task(based on a perimeter-and-area activity suggested in our course’s textbook)to try out with her students. Interestingly, Sally’s problem had an open-ended design similar to that of the journal prompt I had provided. Herarea-perimeter problem and students’ responses read as follows:

Sally:Samantha: If you have 24 square tiles of equal size, how many ways can you arrange themto make different size rectangles? Do you think they will all have the same perimeter? Whyor why not? Do you think they will all have the same area? Why or why not? Now find thearea and perimeter for each rectangle and record the information.

Length Width Perimeter Area

Rgle 1

Rgle 2

Rgle 3

etc. . . .

Do you notice anything about the rectangles’ perimeter If so, what? Do you noticeanything about the rectangles’ area? If so what? [ml3]

Student 1 (Samantha):


Rgle 1 6 4 20 24

Rgle 2 8 3 18 24

Rgle 3 4 6 16 24

etc. . . . 24 1 50 24

12 2 28 24

I don’t agree that they all have the same perimeter. Because I tride [tried] it and they don’thave the same perimeter. I agree that they all have the same area because they all have 24.I think it was easy because you explained it well. [ml4]

Student 2 (Jordan):



Rgle 1 6 4 20 24

Rgle 2 8 3 18 24

Rgle 3 4 6 16 24

etc. . . . 24 1 24 24

12 2 28 24

I new [knew] when to stop because you have to use a[n] equal number like 2 and not 5because you will run out of tiles. No, I don’t think they all have the same Perimeter. I thinkit was pretty easy. [ml4]

Afterwards Sally wrote in her journal:

Today we got responses back from our students on the area/perimeter questions. Samanthaand Jordan worked on the question together and did a good job, though a couple of timesthey seemed to have forgotten a side when adding up the perimeters of some rectangles. Ithought it was particularly interesting to note that they correctly found the perimeter of a[rectangle] length of 6 and width of 4 to be 20, but then found the perimeter of a rectangleof a length of 4 and width of 6 to be 16. It seems to me that they forgot to include oneof the sides of 4 in the addition of sides, but it’s very interesting that they didn’t initiallymake the connection that a 6×4 rectangle has the same dimensions, including perimeter,as a 4×6 one. [Sally, MJ5]

Changing Interpretations

Focusing on meaning.After the area/perimeter exploration, most preser-vice teachers’ journal entries were noticeably more analytical of themathematics involved in the students’ responses. One example is foundin Sally’s comments, for example, which show that – differently from herhorse problem journal entry – her focus was not solely on the correctnessbut also on the meaning of the students’ work. Notice that even thoughSally thought that the students “did a good job” solving her problem shecontinued to investigate her students’ work. She highlighted some of theinteresting features she noticed. She speculated (rather than concluded)that the students “seemed” to be forgetting a side when adding up thelengths of the sides, a common error students make when they deal withperimeters of regular shapes. She also noticed that her students did notrecognize the identical properties of the 6x4 and 4x6 rectangles that relateto the commutative property of multiplication and the conceptual relation-ship between the dimensions and the linear and surface measurements ofrectangular shapes.

Another example can be found in letter exchanges between Thea andher students on the same topic. This example is particularly interestingbecause, as noted earlier, Thea’s initial interpretive comments focused onthe form (e.g., length), not the content, of her students’ work. Yet, like

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Sally, Thea had become curious about her student’s understanding of areaand perimeter. She said, “I’m really interested to hear what the studentsare learning about area and perimeter, so I can’t wait for my next letter.”Particularly, she said, because “I know that this can be tricky because Iwas trying to figure out how to do it myself.” Here are the questions Thea(preservice teacher) posed and the responses she received from her student(June).

Thea:Our teacher told us that you were going to learn about area and perimeter this week. Howis it going? Are you having trouble? Pretend that I have no clue what area and perimetermeans. Can you explain what they mean in your own words? What isarea? What isperi-meter? Try to explain these words to me. [ml4].

June:I think that area means space inside the shape and perimeter means distance around theshape. For example: [ml5].

Afterwards Thea wrote in her journal:

I asked my student to explain to me what area and perimeter meant as if I didn’t know it.[“Pretend that I have no clue what area and perimeter means. Can you explain what theymean in your own words? What is area? What is perimeter?”]. Her response:

Area means space inside the shape and perimeter means distance around the shape. (Thenshe provided a picture of a square divided up into 42 smaller squares, each square markedfrom 1–42 on the inside, on the outside the squares were counted from 1–28). So she wrotethe perimeter is 28 and the area is 42.

I was quite pleased with her response because she made the effort to explain it in herown way rather than copying something from a text. From her response I think that sheunderstands what area and perimeter means. I was looking forward to how she woulddefine area (perimeter seems to be less of a problem for most people), as I myself am notquite sure how I would define it for the students. June said area means the space insidethe shape. This seemed to me to be a good response but that lead me to wonder how shewould define volume, or distinguish between the two. I’m also puzzled about why she didnot put cm2 for the area. Does she not understand this part of it yet, or did she just forget?Either way it seems to show that she does not have as complete an understanding about area


and perimeter as we would like our students to have. Squaring whatever unit you use tomeasure area is a vital concept to grasp if you want to understand area completely. [Thea,MJ5]

This excerpt further illustrates the changes in preservice teachers’discourse about their students’ mathematical work. Thea commented, asother preservice teachers did, on her student’s work as being “interesting,”and provided a much more elaborate and detailed analysis of her student’smathematical work. Notice that, although Thea was impressed with herstudent’s response and thought it showed that the student understood theconcepts, she continued to look beyond the surface of the student’s answer.As a result, Thea began to raise questions about the completeness andinterconnectedness of her student’s ideas about area: “[I] wonder how shewould define volume or distinguish between the two?” She also noticedthat the student did not indicate a unit of measurement in her definition ofarea, and she found this puzzling. It is interesting to note that, rather thanmaking a quick evaluation, Thea considered alternative reasons that couldexplain the student’s omission of the unit of measurement in her definitionof area: “Does she not understand this part of it yet, or did she just forget?”

Overall, preservice teachers’ later interpretations became moredetailed, more exploratory, and less conclusive. Analytical commentsbecame more prominent and frequent in the preservice teachers’ discourse.The comments revealed greater attention towards the meaning of student’smathematical thinking rather than surface features. Examples of suchcomments included: “Something that I found interesting in Jess’ answer isthat she knew that she wanted the largest area of land but she did not statethis fact” (Carly); “I noticed that he changed all the units from meters tocentimeters” (Megan); “How did she get 90 cm2? All her perimeters werein meters, and all her areas were in centimeters2” (Nilsa).

Questioning and revising claims.Preservice teachers’ generalized claimsabout their students’ mathematical attitudes and abilities also began tochange about midway through the methods course. When reading preser-vice teachers’ first narrow interpretations I, along with my teaching part-ners, sought to encourage them to reconsider their claims by suggestingother perspectives and challenging the evidence they had used to drawtheir conclusions. In our responses to their journal entries we oftenasked them questions such as: “What other interpretations apply?”; “Whatevidence have you collected?”; “Are you suggesting that there is a connec-tion between a student’s writing abilities and their math abilities?” Wealso introduced a 2-column format (description/interpretation) for writingjournal entries in the hope of helping them distinguish between describing

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and inferring students’ thinking, and for helping them become more awareof the evidence they were using to make their interpretations.

Although the descriptive/interpretive tool helped focus preserviceteachers’ attention onto their students’ mathematical thinking, it wasthe contradictions and surprises preservice teachers found in students’work that challenged their conclusive and evaluative claims. Miriam, forexample, who initially doubted her students’ mathematical abilities, soonfound that another student had referred to Beth as a “genius friend.” Linda,who admitted to “read[ing] David’s [letters] first” with the thought of“get[ting] through the bad first and save the good (her second student’sletter) for after,” could not contain her amazement when her student’sfourth letter showed a change in his writing patterns. “David wrote!. . . It was an amazing letter,” wrote Linda in disbelief. Similarly Megan,who thought she had her two students’ mathematical abilities and atti-tudes figured out, received contradictory evidence in their working onthe “staking your claim” (area-perimeter) problem she had sent to bothstudents.

I thought this problem would be challenging, but I never thought Jake would find it sodifficult. . . . I was quite shocked that Jake had such difficulties with this problem becausehe has mastered [the] majority of problems I have given him in the past. None of myproblems in the past have dealt with perimeter and area. I am not sure how much workJake has done in geometry in the past, therefore I may have been expecting too much fromhim based on past letters. Area and Perimeter may be Jake’s weak area in Math. I am notsure though. As far as I know he may have not been feeling well on the day he answeredmy question. . . . I must admit that I thought Mary would have more difficulties with thisproblem than Jake, because of past responses to letters. [Our instructor] stated in classtoday that people are weaker and stronger in different areas. This has proven to be true inthe letters I received this week. [Megan, MJ5].

The face-to-face meetings with students provided many of the preser-vice teachers with contradictory data, which led them to question andrevise their earlier convictions and claims about their students’ mathe-matical attitudes and abilities. Some preservice teachers, for example,were able to see that students who seemed very talkative and outgoingin their letters became shy and uncooperative during group work sessions.Others saw their students, who in their letters seemed uninterested andunmotivated to do mathematics, show a different side when interacting inperson.

Meeting and working with their students in person helped some preser-vice teachers look at their students from a different perspective. “We wentover some of the questions I had asked her in previous letters,” commentedNilsa, only to find that “she had no trouble with them.” “I think she is moreof a talker than a writer,” Nilsa rationalized afterwards. Like Carly and


Nilsa, other preservice teachers began to formulate alternative explana-tions for their students’ previous “lackings.” A few examples follow: “He’san enthusiastic and intelligent student but his organization skills are a bitweak” (Marcia); “He finds it difficult to express through writing” (Rosa);“She writes very slow. . . it took her a long time to write a sentence. . . nevertheless she had no trouble finishing and figuring out the problem Igave her” (Miriam).

The final course assignment, the case report, provided another catalystfor preservice teachers to change their previous interpretations. Lesley, forexample, related the following:

In my journal, I wrote several times regarding how it seemed that “Lynn” had moredifficulty explaining and clarifying her math thinking on paper than “Parker.” . . . WhenI compared my journal reflections with the letters, I was very surprised to discover thatParker is the one who least explained his thoughts and problem solving skills. [Lesley, CR:pp. 6–7]

Similarly, Linda became aware of her questionable interpretive prac-tices when she reviewed her journal entries and her responses to thestudent she had perceived to be of lowest ability. In her case report, Lindadiscussed how David’s messy work and misspellings played a role in herdeveloping perception of him as a less capable student. Her perceptions,Linda was dismayed to find out, affected her communication with herstudents. She reported that she found substantial differences in her “writingstyle, questioning, length of letters, and problems posed.” She wonderedwhether and how these differential communications with her students inturn affected her students’ mathematical work. This and the precedingexamples show that preservice teachers had begun to see contradictionsin their initial claims and to look for alternative explanations, which oftenincorporated considerations for the content, context, and format affectingtheir students’ mathematical performance.

EXAMINING INFLUENTIAL FACTORS

I have highlighted the main features of preservice teachers’ beginninginterpretations of their students’ mathematical work and the ways in whichtheir interpretations changed over the course of their interactive experi-ences with students. In this section, I examine what might have helped thepreservice teachers begin to see more than right and wrong answers in theirstudents’ mathematical work and try to illuminate the reasons behind theinterpretive turns.

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Introduction of Challenging yet Accessible Problems

Ball (1988) shared her deliberations on the kinds of tasks she typicallychose for prospective teachers in an initial teacher education course. Shediscussed the importance of choosing a task that was unfamiliar, intriguingthough inviting, in order to ensure prospective teachers’ genuine engage-ment with the mathematical content they were studying. The NCTMProfessional Standards (1991) also highlighted the role that a carefullyselected task can play in promoting students’ learning of mathematics.Similarly, in this study, the introduction of challenging yet manage-able mathematical problems seemed to have played a role in preserviceteachers’ patterns of interpretations.

The introduction of unfamiliar mathematical tasks, that is, tasks thatchallenged and extended preservice teachers’ own understanding of mathe-matics, helped them move away from their evaluative interpretations.Notice that in the case of the horse problem, the familiarity of themathematical content and format of the problem may have played a rolein preservice teachers’ superficial and unproblematic interpretations oftheir students’ work. It is interesting to note that once the mathematicaltasks became less familiar, such as the area/perimeter problem, preser-vice teachers’ interpretations became less certain, more exploratory, andincreasingly respectful of their students’ mathematical sense making. Touse Duckworth (1987) and Schön’s (1983) notion, preservice teachersbegan to challenge themselves to “give reason” to students’ mathematicalwork. That is, they began to raise questions and attempt to understand theirstudents’ seemingly impenetrable and indecipherable work.

The relationship between the type of mathematical task and preserviceteachers’ interpretations makes, in retrospect, a lot of sense. Consider,for instance, that familiar problems are more likely to yield familiarresponses, which preservice teachers are likely to recognize as similarto their own mathematical work and therefore be less inclined to ques-tion and explore. Unfamiliar problems, on the other hand, are likely toyield unfamiliar responses from students, therefore increasing the likeli-hood that preservice teachers inquire into students’ mathematical thinking.This is not to say that this is an unproblematic relationship. Recog-nizing their own ways of thinking in students’ work helped the preserviceteachers recognize meaning and analyze students’ work, whereas receivingunfamiliar work from students could also make it difficult for preserviceteachers to recognize meaning in such work. Nevertheless, is importantfor teacher educators to consider the careful selection and introduction ofmathematical tasks, not only to ensure prospective teachers genuine mathe-


matical inquiry, but also to promote a respectful and inquiring orientationtowards the analysis of students’ work.

Content and Form of Reflective Writing

Journal writing is a popular activity promoted in teacher preparationprograms as a means to engage preservice teachers in critical reflectionand analysis of their own teaching and learning of subject matter. Teachereducators and researchers, however, have found it difficult to move preser-vice teachers’ writing from a focus on personal and emotional aspects oflearning-to-teach towards matters of general and subject-specific pedagogy(Richert, 1992). They also found it difficult to move preservice teachers’writing toward analysis, synthesis, deliberation, or reflection. Most preser-vice teachers’ writing takes the form of reporting and summaries oftheir experiences (Anderson, 1992). The content and form of preserviceteachers’ writing, therefore, is an important factor to consider and examine,particularly if writing plays such a prominent communicative and reflectiverole as in this study.

The introduction of the descriptive/interpretive journal writing toolduring the third week was an intervention that seemed to have aided preser-vice teachers’ interpretive turns. Although not all preservice teachers usedthe double-column format to the same extent, its introduction provideda model for reporting and reflecting on students’ mathematical work. Inaddition, this tool seemed to have helped focus preservice teachers’ writ-ings onto the examination of students’ work. Such a focus was important ifwe consider that the value of reflecting on students’ work may initially nothave been apparent to preservice teachers. Notice that by the time preser-vice teachers made their journal entries, they had already responded totheir students’ work. These after-the-fact interpretations may have seemedtoo late to inform their already-sent responses. In turn, the value offocusing their reflective writing on further interpreting the students’ workmay not have been readily apparent to preservice teachers until they wereasked to do so.

Being directed to focus the writing upon the students’ work helpedpreservice teachers become more self-conscious and analytical. Theyreported that spelling out their analysis of their students’ work: helpedthem to “move beyond superficial considerations to a deeper more criticalanalysis” (Sally) and become “more insightful” (Terry); and allowed them“in time to (estimate) guess a little better why students answered ques-tions in a certain way” (Megan). Writing and reflecting on students’ workwith an explicit focus and format played an important role in developingpreservice teachers’ interpretations.

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The case study at the end of the term provided another context forpreservice teachers to engage in reflective writing and to revisit andreinterpret their students’ mathematical work. This assignment allowedpreservice teachers in this study to gain further insights that they mightnot have had otherwise. Revisiting the students’ data and their own journalreflections on the data provided an occasion for preservice teachers toreconsider and revise their convictions about their students’ attitudes andabilities as well as to uncover flaws in their own interpretations. Accessto tangible records and the process of revisiting such records was animportant aid to the process of meaning making and self-examinationinvolved in reflective writing (Wassermann, 1993).

Interactive Experiences With Students

Field experiences and field-related experiences are common activities inteacher preparation programs. Although the practice setting has long beenconsidered the authentic place for learning to teach, it has become increas-ingly clear that the practice setting may not necessarily be the best northe only place for learning to teach (Feiman-Nemser & Buchmann, 1986).Finding ways to help prospective teachers make problematic (and an objectof investigation) what they experience in the practice setting has been thefocus of much of the recent research and practice in teacher education. Theincorporation of an unconventional field-related experience can thereforeprovide insights into important features to consider when designing fieldexperiences for prospective teachers.

Different from traditional forms of interactions, written interactionsprovided the opportunity to disregard managerial or disciplinary issuesand school and curricular pressures which are often the focus of preserviceteachers’ attention. In addition, delayed interactions with students madeit possible for preservice teachers to engage in the collaborative analysisof students’ mathematical thinking with other preservice teachers. At thesame time, this form of interaction brought to the foreground concernsabout students’ abilities to communicate their ideas in writing. This, inturn, helped many preservice teachers begin to make problematic the actof interpreting and making sense of students’ written work.

The delayed interaction with students also provided the time and oppor-tunity for preservice teachers to question and extend their own mathe-matical ideas. Examining their students’ responses sometimes served toengage preservice teachers in further mathematical explorations of theproblems they had posed (as was shown earlier in Thea’s case). There werealso occasions when their attempts to make sense of a student’s responseled preservice teachers to make some mathematical discoveries of their


own. For example, the analysis of her student’s work led Miriam to reviseher own solution to a problem she had sent to her student. Her problemasked to find a rectangular shape that would give the maximum area for agiven perimeter. After reading her students’ explanation for why the squarewould give the most amount of area, Miriam began to delve more deeplyinto the definitions and classifications of shapes, particularly whether andwhy a square would also be considered a rectangle.

The opportunity to interact with two students, on the other hand,provided experiences that those interacting with one student did not haveso readily available. These preservice teachers often gave their studentsthe same problem and therefore were able to obtain more data about howtwo different students responded to their mathematical problems and ques-tions. Writing to two students also raised pedagogical challenges, whichcorresponding with one student did not. Reading and writing two lettersin the same amount of time that others had for reading and writing onewas a source of difficulty and concern for these preservice teachers. Sally,for example, worried about being too rushed to carefully think and reflectduring the writing of her letters. Megan, in turn, worried about providingequal time and quality of attention to each student. As a result, someof these preservice teachers focused their investigations primarily on thedifferences between the two students’ mathematical work.

The face-to-face and more immediate interactions provided opportu-nities for preservice teachers to contrast and compare their students’mathematical performances in different media and in different settings.Preservice teachers, therefore, were able to examine the role that structureand setting played in their students’ mathematical work. This, as it turnedout, became an important source of tension and deliberation in preser-vice teachers’ journals. They had constructed conjectures and had madeassumptions about their students as learners of mathematics based on thestudents’ written work. Meeting their students in person provided anothersource of data to confirm, elaborate, and challenge those assumptions.

CONCLUDING NOTE

Learning about students’ thinking was not among the items the preserviceteachers expected to learn in a mathematics methods course. The ideaof listening to students’ mathematical ideas was received with a mixtureof intrigue and excitement. There were those who did not expect theanalysis of students’ work to be problematic – “a matter of just tappinginto their thinking.” Others conceived the task of interpretation as a matterof deciding whether or not the student was “on the right track.” There were

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only a few preservice teachers who initially wondered whether studentswould be able and willing to communicate their thinking and whether theythemselves would be able to understand it.

I have, however, documented the ways in which preservice teachersbegan to reflect on and change their interpretations of their students’mathematical work. I have characterized preservice teachers’ interpreta-tions as taking two important interpretive turns. The first turn refers to achange in the focus of interpretation, from correctness to meaning. Thesecond turn can be thought of as a change in the interpretive approachitself, from quick and conclusive to more deliberate and tentative. The firstinterpretive turn saw changes in preservice teachers’ interpretations froma limited focus on the correctness of the students’ solutions towards addi-tionally attending to, exploring, and recognizing mathematical meaning instudents’ solutions to problems. The second interpretive turn saw preser-vice teachers who initially made quick and conclusive judgments abouttheir students’ mathematical abilities and attitudes begin to make morethoughtful and provisional interpretive claims.

These interpretive changes became apparent both in preserviceteachers’ written interactions with students and in their reflective writing.Their weekly journal entries became lengthier, more elaborate, and moredetailed concerning the students’ work. In addition, their interpretationsbecame more analytical and exploratory of the mathematics in theirstudents’ work. I offered three factors as important contributors to thereported interpretive changes: (a) the introduction of challenging butaccessible mathematical problems, (b) the content and form of reflectivewriting, and (c) interactive experiences with students. Although I did notregard these as the sole contributing factors, I did recognize their prom-inent role in overturning the patterns of interpretations of the preserviceteachers in this study.

The study brings to the forefront the importance of attending tothe interpretive discourse and practices teacher candidates bring to theirteacher preparation courses. It also shares some insights into the waysin which a course-based interactive experience with students providedan occasion for preservice teachers to make their interpretations explicitand the object of their investigations. Furthermore, the study builds uponDavis (1996) conceptual categories of teachers’ orientations to listening inmathematics classrooms by using and elaborating the categories of evalu-ative and interpretive listening as useful ways to characterize and analyzepreservice teachers’ interpretations. Noticeably absent from this study,however, was Davis’ third category of hermeneutic listening. I found noevidence of this form of interpretation in the data collected for this study.


One has to wonder whether this orientation to listening is accessible tonovice teachers and what kind of experiences might bring about such atransformation in their interpretive practices.

The study also reports evidence of change in the focus and approachof preservice teachers’ interpretations. This is particularly salient whenwe consider that the participants had weak mathematical backgrounds.This raises questions about the role that preservice teachers’ often incom-plete understanding of mathematics plays in the kinds of interpretationsthey make. This is a question about the relationship between content andpedagogical content knowledge which in this study came to the fore in thediscussion on how the introduction of open-ended and exploratory typesof tasks seemed to affect the interpretations preservice teachers made. Yet,there were occasions in which preservice teachers’ analysis of students’work generated further study of the mathematics involved. This is to saythat this reversed relationship also warrants further study: How does theanalysis of students’ mathematical work affect the growth of preserviceteachers’ mathematical understanding?

In addition, the study underscores the importance that preserviceteachers’ evaluative interpretations be challenged. The research onteachers’ expectations has long discussed the dangers of teachers’ over-generalized and conclusive inferences about students’ mathematical abili-ties (e.g., Good, 1987). This work has shown that the expectations teachershave of students who they believe to be more capable tend to be morerespectful and demanding than the expectations they hold for studentswho they perceive to be less capable. Brophy and Good (1974) found thatteachers who form these unexamined ways of seeing and judging developrigid and stereotyped perceptions of their students based on prior recordsor on first impressions. This is also true for preservice teachers.

Another danger of ignoring the conclusive and evaluative discourse thatpreservice teachers might bring to their mathematics teacher educationcourses is the message that it carries about the certainty and predicta-bility of teaching practice and students’ thinking. This kind of discourseserves to limit teacher education students’ ability to conceptualize newand richer images for mathematics teaching and learning. Ball and Chazan(1994), for example, discussed the pernicious effect of the evaluative andjudgmental discourse in current descriptions and discussions of math-ematics teaching practice. Such a discourse, closes rather than openspractitioners’ conversations about mathematics teaching practice. Further-more, “the common syntax of shoulds and should haves distorts practicewith a stance of implied clarity” (Ball & Chazan, 1994, p. 4). An alter-native syntax “of could and might,” as they proposed, would not only

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better represent the uncertain nature of teaching practice, but also helpwiden the scope of preservice teachers’ explorations into their students’mathematical understanding.

Aside from considering the dangers of ignoring preservice teachers’ways of seeing, talking, listening, and acting towards their students, it isalso useful to consider what kind of experiences might invite them to re-frame their interpretive lenses. This study offers a particular pedagogicalintervention teacher educators might consider when they offer opportuni-ties to investigate students’ thinking. Where and how such an interventionwould fit within, and build upon, the range of experiences available ina teacher preparation program is a larger, though important, question toconsider when designing such experiences, especially because one singleencounter of this type of experience in an isolated course is unlikely tohave the kind of impact that teacher education programs hope to have inthe preparation of beginning teachers. We need a more concerted strategyin order to help prospective teachers begin to seriously consider Davis’(1996) vision of “teaching as listening,” where teachers see an open-endedinquiry approach towards their students’ mathematical work as a viableand desirable teaching practice.

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Ball, D.L. (1988). Unlearning to teach mathematics.For the Learning of Mathematics,8(1), 40–48.

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Ball, D.L. (1993). With an eye on the mathematical horizon: Dilemmas of teachingelementary school mathematics.The Elementary School Journal, 93, 371–397.

Ball, D.L. (1994, November).Developing mathematics reform: What don’t we know aboutteacher learning – but would make good working hypotheses?Paper presented at theconference on Teacher Enhancement K-6, Arlington, VA.

Ball D.L. & Chazan, D. (1994).An examination of teacher telling in constructivistmathematics pedagogy: Not just excusable but essential. Unpublished manuscript. EastLansing: Michigan State University. [Cited with author’s permission].

Brophy, J. & Good, T. (1974).Teacher-student relationships: Causes and consequences.New York: Holt, Reinhart, and Winston.

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Crespo, S. (1998). Math penpals as a context for learning to teach: A study of preserviceteachers’ learning.Dissertation Abstracts International59-05A: 1530.

Davis, B. (1996).Teaching mathematics: Toward a sound alternative. New York: GarlandPublishing.

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

Nicol, C. (1997). Learning to teach prospective teachers to teach mathematics.DissertationAbstracts International58-06A: 2168.

Phillips, E. & Crespo, S. (1996). Developing written communication in mathematicsthrough math penpals.For the Learning of Mathematics, 16(1), 15–22.

Richert, A.E. (1992). The content of student teachers’ reflections within different structuresfor facilitating the reflective process. In T. Russel & H. Munby (Eds.),Teachers andteaching: From classroom to reflection(pp. 171–191). London: Falmer Press.

Schön, D.A. (1983).The reflective practitioner: How professionals think in action. NewYork: Basic Books.

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513G Erickson HallMichigan State UniversityEast Lansing, MI 48824-103E-mail: [email protected]

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