Seeing More Than Right and Wrong Answers

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 of reform-minded visions of mathematics teaching. The questions of when, where, how, and what might help prospective teachers learn to do so, however, remain open. This study examines how a mathematics letter exchange with Grade 4 students provided an occasion for prospective teachers to learn about students’ mathematical thinking and to examine their interpretive practices. Analysis of the interactions between students and prospective teachers, and of the reflective writing of the latter, revealed changes in the patterns of their interpretations. I characterized these as changes in the focus of interpretation, from correctness to meaning, and in the interpretive approach, from quick and conclusive to thoughtful and tentative. I also discuss factors associated with these interpretive turns.

The idea of teachers listening to and understanding students’ thinking has been widely promoted and supported in the education community. In the Professional Standards (National Council of Teachers of Mathematics

[NCTM], 1991), the analysis of students’ thinking is highlighted as one of the central tasks of mathematics teaching. In this report, the analysis of students’ thinking is seen as a resource that can help teachers make informed decisions in their classrooms and improve their practice. Such a listening orientation towards teaching promotes a learning environment conducive to and respectful of students’ own sense making and intellectual autonomy (Davis, 1996; Kamii, 1989). In contrast, when teachers do not listen to or do not understand their students’ thinking they tend to dismiss it 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 their students’ mathematical ideas, Davis (1996) reminded us that not all forms of listening are conducive and respectful of students’ thinking. He discussed three different orientations teachers might have towards listening in the mathematics classroom. Teachers with an evaluative orientation, according to Davis, tend to listen to students’ ideas in order to diagnose and correct their mathematical misunderstandings. Teachers with an interpretive orientation, on the other hand, listen to students’ ideas with the

Journal of Mathematics Teacher Education 3: 155–181, 2000.

© 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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SANDRA CRESPO purpose of accessing rather than assessing students’ mathematical understanding. Teachers with an hermeneutic orientation, in turn, continually and interactively listen to students’ ideas by engaging with them in the messy process of negotiation of meaning and understanding.

Despite reform efforts aiming to change the evaluative ways in which teachers tend to listen in mathematics classrooms, the notion of teaching as telling (speaking, explaining) rather than listening (hearing, interpreting) still pervades most mathematics classrooms. Underscoring the complexity of adopting a listening approach to teaching, Ball (1993) reminded us that listening to students’ thinking is hard work, especially when students’ ideas sound and look different from standard mathematics. “The ability to 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 disincentives to seriously consider such a change in their teaching practice. Teachers’ sense of efficacy, according to Ball (1994) and Smith (1996), is at risk each time they ask students to voice their ideas in the open. Evidence that their students do not understand undermines teachers’ feeling of competence and ability to help students learn. By not asking students to explain their thinking and by not listening to their ideas, teachers run less risk of detecting 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 well known. For instance, they do not see the point of encouraging “students to set, explore, and solve their own problems” (Wilson, 1990, p. 206).

Wilson’s prospective teachers concluded – after watching a video – that the students were confused and that the teacher should have stepped in to clarify. Similarly, Ball (1990) reported that, when prospective teachers watch classroom episodes in which students disagree or revise their solution in front of the class, many of them “assume that [the child] is embarrassed about having been wrong” (p. 13). It is these evaluative ways of seeing and interpreting that teacher educators hope to shake, uproot, and inform (Holt-Reynolds, 1995; Wilson, 1990).

One way in which I have attempted to introduce elementary preservice teachers to the idea of listening to and learning from students is by engaging them in an interactive mathematics letter exchange with Grade

4 students. These interactive experiences are meant to provide preservice teachers with a context in which they explore and practice listening to students’ ideas; at the same time, the preservice teachers can investigate the challenges and possibilities of such an interactive approach to mathematics teaching 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 to the problems, and share questions, opinions, and ideas about their own experiences as learners of mathematics. Although preservice teachers and students typically meet on at least two occasions, their interactions are mainly through written letters.

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

Aware of the challenges and possibilities that this particular form of interaction with students would offer preservice teachers, I set out to investigate what they might learn from such an experience (Crespo, 1998).

Changes in the interpretations of their students’ mathematical work were one 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’ mathematical abilities and understandings. More specifically, this report focuses on the following questions: In the context of their mathematics letter exchanges with school students (a) how do preservice teachers interpret their students’ work, (b) how do their interpretations change over the duration of the course, and (c) what contextual factors contribute to changes in their interpretations?

DESIGN OF STUDY

The design of the mathematics letter writing activity as a context for learning to teach is grounded in a view of learning as situated cognition.

Proponents of this theory believe that knowledge is situated in and inseparable from the activity, context, and culture in which it is constructed.

Learning, therefore, is viewed as a process of enculturation or cognitive apprenticeship into the practices and modes of thinking of a particular discipline, trade, or profession (Brown, Collins & Duguid, 1989). Letter exchanges with students provided a context that resembled the interactive nature of teaching practice, but without the immediacy and pressures

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SANDRA CRESPO for action that characterize actual mathematics classrooms. In the letterwriting context, preservice teachers could safely and supportively play the role of a mathematics teacher under the supervision and guidance of experienced teacher educators and in the company of and in collaboration with their peers.

Context of the Study

The letter-writing experience with students took place within the immediate context of an elementary mathematics teacher education course. The course is required for all preservice elementary teachers (Grades 1–7) enrolled in the two-year teacher education program at the University of

British Columbia. The teacher preparation program offers a combination of university-based courses and school-based practica. These are organized in the following sequence: (a) foundational courses in education for the first half of the first year, (b) methods courses in all subject areas during the second half of the first year, (c) a 13-week school practicum during the first term of the second year, and (d) university-based courses focusing on the social context of schooling and a chosen subject-area of specialization. Typically there are no long-term school-based experiences scheduled concurrently or connected with the university-based course work. There is, however, a short 2-week practicum scheduled midway through the methods courses. The practicum is not associated with any of the methods courses but is a program-wide experience.

The mathematics methods course I was teaching was an innovative version of the elementary mathematics methods courses typically offered in the university’s teacher preparation program. It was a collaborative venture between myself, two teaching partners, and three collaborating teachers. My teaching partners were an assistant professor in the college of education and a fellow graduate student who, like me, focused her dissertation on one aspect of the course we were teaching (see Nicol,

1997).

The course’s mathematics letter-exchange project, in turn, was a collaborative enterprise between myself and a local full-time teacher, who was completing her Master’s degree in our department. For the collaborating teacher the experience was meant to provide her students with an authentic audience with which to communicate mathematically (see

Phillips & Crespo, 1996). In my case, the letter exchange was meant to provide a context for preservice teachers to explore and investigate students’ ways of thinking and communicating in mathematics.

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


159 setting for preservice teachers to explore and try the ideas discussed during the methods classes. Interactions with students, in turn, served as the focus of class discussions and reflective journal writing. Most preservice teachers corresponded with one student, although a few of them corresponded with two students. The letter writing activity spanned the whole methods course

(11 weeks) and took place each week during one of our two 1.5-hour meetings. Similarly, the letter writing activity was an integral part of the mathematics classes of the Grade 4 students. They read and wrote their letters during regular mathematics time with the assistance and encouragement of their teacher and peers. In both classes, students and preservice teachers worked within groups of four, an arrangement meant to encourage them to collaboratively read and write the letters.

Data Sources

For the study I used the data from thirteen preservice teachers (from a group of 20). The data consisted of (a) all written work associated with the course, (b) videotaped class sessions (used for descriptive purposes), (c) six mathematics letters preservice teachers received and wrote in conjunction with three common class problems they sent to the students, (d) one teacher-directed letter from the students about the role of calculators and computers in their classroom, (e) journals about deliberations and reflections on the interactions with students (reflection-on-action), and (f) a case report written at the end of the course about the learning experiences related to the work with the students. The preservice teachers’ written assignments – journal entries and case reports – were the main sources of data.

Data Analysis

The analysis of preservice teachers’ journal entries focused on identifying and contrasting their patterns of interpretation at the beginning and the end of their letter exchanges. Preservice teachers’ case reports also provided data 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 to highlight and explore contributing factors to the changes in the participating preservice teachers’ views, discourse, and practices. The data cited throughout the following section are referenced by a code of letters and numbers referring back to the data source and the timeline associated with it. For instance, a letter exchange referenced [ml1] means that the data is from the first mathematics letter the indicated participant wrote. A quote from a preservice teacher’s journal that is referenced [MJ1] means that

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

EXAMINING PRESERVICE TEACHERS’ INTERPRETATIONS

This section highlights patterns in preservice teachers’ initial and later interpretations. The contrast between initial and later interpretations is first highlighted with a description of the contextual features of the course activities associated with the time period. The description is followed by an analysis of a sample letter exchange and reflective journal entry of one study participant, Sally. Examples from other preservice teachers are cited throughout the text to provide further evidence of change in the interpretations the preservice teachers made about their students’ mathematical work.

A Beginning Letter Exchange

The Horse Problem (Burns, 1987), a problem we had worked on and discussed during our second week of class, was a common problem preservice teachers chose to send to their students in their first letters. The problem reads: “A man bought a horse for $50. He sold it for $60. Then he bought the horse for $70. He sold it again for $80. What is the financial outcome of these transactions?” This deceptively simple-looking problem never fails to cause uncertainty and confusion as to how it might be interpreted and solved. Usually students generate alternative ways of solving it and come up with reasonable-sounding explanations for right and wrong answers. In response to the problem, common answers students offer and defend 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 easily discern why or where the competing solution may be flawed. Typically these solutions read:

Earned $20: The man bought the horse twice, which means he spent $50 and $70, or $120 buying the horse. He also sold the horse twice, which means he made $140 from selling the horse.

Subtracting what he made from what he spent, 140

−

120, the result is that the man made $20.

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


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

The problem generated a lot of interest and a lively discussion about the different answers and solution methods. In their journals, preservice teachers wrote about being intrigued by “the number of ways in which people went about solving the horse problem,” and “how others explained how they got their answers.” Some seemed excited about the prospect of allowing students to discuss their solution strategies and explanations in a mathematics class, whereas others thought that “kids should ultimately be shown the right answer and how it is arrived at.” This and similar experiences seemed to make preservice teachers curious about students’ thinking and how they would justify their answers. It was therefore with great anticipation that the preservice teachers awaited their students’ responses to their first letters. The first letter exchange between Sally (preservice teacher) 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 you to try. When you write back, will you solve it in your letter, and write everything you think down, even the mistakes? That will help me see how you think in math. Here is the question:

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 you try 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 horse problem I came up with 70. This is how I got 70, start with 80

−

10. The second time I tride [tried] the horse problem I got 20 because you start with $80

−

$10 and look at the differents [difference] between $70 and $50 witch [which] = $20. [ml2]

Sally:

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

Bought – $50} He made

Sold – $60}

$10

$10 (60

−

50) +$10

Bought – $70}

Sold – $80}

He made

$10 (80

−

70)

$20

Or try using a number line.

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SANDRA CRESPO

Samantha:

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

After each letter writing session preservice teachers were required to write in their journals about the insights, surprises, and questions that their students’ 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 be honest, I was so pressed for time in reading and responding to 2 letters, that I didn’t think to ask her. From there, she must have looked at the answer, and seen that she didn’t have the 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 2 ways 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 teachers were inspired by the idea of listening to students’ mathematical thinking, once faced with their students’ work, they tended to focus on the correctness of their students’ answers. “Yes! They got it,” “This kid screwed up,” “Wow, was she ever off,” were some of the comments they made aloud while reading their students’ responses. Such comments highlight preservice teachers’ tendency to accept students’ right answers as evidence of understanding and to see students’ wrong answers as signs of confusion or carelessness. These tendencies were also apparent in the preservice teachers’ initial journal entries in which they wrote little about the meaning of students’ work. They noted, for example, whether students were successful or unsuccessful answering their questions and made no specific inferences about what or how the students were or were not understanding.

Sally’s written exchanges and later journal entry illustrate the interpretive challenge that this form of interaction with students presented to the preservice teachers. Quite often preservice teachers found themselves having to formulate a response letter with little idea as to what the student did or said. It is also important to note that Sally’s journal entry is quite


163 elaborate in comparison with the beginning journal entries from most preservice teachers in the study. Yet, Sally’s analysis focused mostly on the correctness of the student’s work. Consider for instance that rather than explore 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 in her 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 she did not quite understand the student’s solution. She seemed genuinely puzzled by the students’ work. Yet, her subsequent analysis focused on explaining how the student could have come up with the right answer despite producing an explanation for her work that, in Sally’s eyes, was confused and possibly made up. Rather than exploring and speculating what the student’s work might mean or suggest, Sally closed other possible interpretations when she conclusively stated that the student “must have” looked at the answer.

Samantha’s work, however, need not be so easily dismissed. A second look at her work could suggest an alternative interpretation. One may consider, for instance, that her solution is counter-intuitive, that is, it uses numbers that are not included in the problem (e.g., $10). This suggests that she is not mindlessly plugging in numbers as Sally insinuated. Taking this perspective makes the student’s work seem more sensible and reasonable even 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 the change between $70 and $80? Also, what is her $70 referring to – the $70 in the problem or the $70 that resulted from the first subtraction (80

−

10)?

This sort of exploration may lead to other plausible interpretations of the student’s solution. For instance, one could reasonably say that her

“made $70” solution addresses Sally’s first question of “how much money does he have in the end?” – a question about the resulting balance. A reasonable answer to this question is that in the end the man has $80 in his pocket. The student’s “made $70” answer, however, may have considered that 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. In the second attempt, the student may be addressing Sally’s second question

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SANDRA CRESPO

“did he make or lose money?” – a question about the resulting profit.

The student may have figured that she also needed to subtract the $50 the man started with from the $70 he had in the end. Though this is only a conjectured explanation for how the student may have worked out the problem, considering this as a possibility serves to open the avenues of interpretation, which in turn can suggest other ways of responding to and probing the student’s solution to the horse problem.

Similar to her students’ mathematical work, Sally’s work is not as clearcut or single-minded as my early discussion might suggest. Her work is also open to multiple interpretations and raises important interpretive questions about what she intended to see and was able to see in her student’s work. For instance, there are many indications that she was intending to attend 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 in math.” In her response letter she offered not one but two different solution methods to the horse problem. This suggests a concern for encouraging students to see multiple solutions to problems. In light of this, one also has to wonder why Sally chose to send her students the answer key (with only the answer) in the first place.

It is also interesting to note that in her analysis Sally could re-state her student’s solution word-by-word. She realized that she did not really know why the student subtracted 10 from 80. Did she mean that she had no real data or proof as to why she did this, or that she did not see why the 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 at the same time, Sally shared her conjecture or perhaps suspicion of how the student came up with the right answer. With this I wish to point out that I am not claiming that correctness is the only issue to which Sally was attending in her beginning interpretations. I do claim that correctness was a prevalent focus of attention when analyzing the student’s work at the beginning of the course. This claim is based on the examination of not only Sally’s work but also of other preservice teachers’ writings. This focus on correctness is also more apparent when contrasted with preservice teachers’ later interpretations.

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


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

Miriam responded as follows:

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

Buy 50

+70

120

140

−

120 = $20

Sell 60

+80

140

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 was wrong. I was at a total loss because I did not want to come out and say that she is wrong because she mentioned how easy the question was. Even in our class there were a number of people who had the same answer as her ($10) when we first got the question. I told Beth this and explained that it is not the correct answer, then I told her how to use one way to solve 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 the wrongness of the student’s work without delving into what the student may or may not have understood. Both preservice teachers responded by telling their students how to find the correct answer. Although, in contrast to Sally, Miriam claimed she “can understand” her student’s solution, her lack of elaboration, along with the response she made to correct her student’s thinking, suggest otherwise. Sally and Miriam’s inattention to the details in their students’ work were not isolated cases nor confined to students’ wrong answers. Preservice teachers whose students had reached the correct answers also seemed unaware of and inattentive to what they could learn from analyzing their students’ work. Some simply commented that their student: “did a good job at explaining how she got the answer”

(Linda), was “successful” answering their question (Mitch), or “seems advanced in math and writing compared to others’ letters” (Terry). Others, like Thea, simply noticed that their students’ letters “were short and contained little information.”

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

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SANDRA CRESPO understanding based on the correctness of the work they received. Very few preservice teachers raised questions about their students’ understanding of the specific mathematics. Those who did, as in the case of Sally and

Megan, did so out of suspicion that the student “must have looked at the answer” before solving the problem, (Sally) or that the student “did not work out the problem herself” (Megan).

In addition to making claims about the students’ mathematical understanding or lack thereof, preservice teachers’ interpretations also focused on their students’ mathematical abilities and attitudes. Based on the students’ questions, comments, and writing style, preservice teachers made claims about their students’ personal characteristics and dispositions as learners of mathematics. Consider Rosa’s interpretive comments after reading her student’s first letter.

After receiving the first letter I can tell that Paul is having problems in math. He enjoys aspects 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 help

Paul with his difficulties. With the help of the readings, I have come up with a number of 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 Paul through 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 to work through a problem. [Rosa, MJ2]

Rosa’s analysis of her students’ comments is both plausible and contestable. Rosa, however, had already begun to devise strategies to alleviate her student’s alleged difficulties with mathematics. Students’ comments, such as the statement from Rosa’s student that “as for math I need more practice,” often caught the preservice teachers’ attention. Interestingly, the preservice teachers seemed to relate to these types of comments more easily than to the actual mathematical work the students were providing.

Their previous experiences with mathematics seemed more readily available for aiding interpretations of such comments. Megan, for example, concluded: “I can tell that the students are trying to impress me,” and the reason she provided was “because they both stated in different words that math is great.”

As Rosa’s and Megan’s interpretive comments illustrate, preservice teachers were noticing clues not only in students’ mathematical work but also in the comments, questions, and suggestions the students were providing. From the letter’s length to its decorations to its personal undertones, preservice teachers were often drawing and expressing inferences and conclusions about their students’ personal characteristics. Comparisons among students were often referenced in such interpretations. These comparisons were more prominent, however, in those preservice teachers


167 who corresponded with two students. Lesley, for example, concluded the following about the differences she observed in her two students’ first letters.

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

Students’ brief responses to the preservice teachers’ questions and their lack of elaboration on their answers were often interpreted as a lack of interest 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 in math.” Likewise, Linda derived a similar interpretation: “If I wanted to generalize, I could make the assumption that, based on his writing and spelling abilities (which are dreadful), he’s not good at math, but I think

I will wait a bit on that.” Not much later, however, Linda stated: “I don’t think he has much of an understanding of how to work with numbers either in 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 with math, it was her spelling – it was atrocious!” said Nilsa of her student’s first two letters. Similarly, Linda commented about her students’ communication skills. “I’m not sure how things are going to go with David. He doesn’t write as much as Shelley, nor does he spell very well (that’s the English teacher in me coming out!).” Students’ misspellings coupled with their miscommunication exerted a powerful, though often inadvertent, influence on the preservice teachers’ conclusions about their students’ mathematical abilities and attitudes.

A Later Exchange

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

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SANDRA CRESPO

I must admit that I went to our text to get a more complete understanding of what area and perimeter are. I tried doing some of the activities suggested in the book to apply what I knew about these concepts. I then went back to the handout to see if this information could help me in answering the questions: e.g.: “is it possible to have two figures that have the same perimeter but different areas” and “what about two figures that have the same area but different perimeters?” I found myself doodling on scrap paper to try to answer these questions. I drew a square and a long, skinny rectangle, both of which had a perimeter of

16; thus 2 shapes can be different yet have the same perimeter. I then drew a grid to answer the 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 understandings 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 openended design similar to that of the journal prompt I had provided. Her area-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 them to make different size rectangles? Do you think they will all have the same perimeter? Why or why not? Do you think they will all have the same area? Why or why not? Now find the area 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 notice anything about the rectangles’ area? If so what? [ml3]

Student 1 (Samantha):

Rgle 1

Rgle 2

Rgle 3 etc. . . .


6 4 20 24

8

4

3

6

18

16

24

24

24

12

1

2

50

28

24

24

I don’t agree that they all have the same perimeter. Because I tride [tried] it and they don’t have 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):


169

Rgle 1

Rgle 2

Rgle 3 etc. . . .


6 4 20 24

8

4

3

6

18

16

24

24

24

12

1

2

24

28

24

24

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

Afterwards Sally wrote in her journal:

Today we got responses back from our students on the area/perimeter questions. Samantha and Jordan worked on the question together and did a good job, though a couple of times they seemed to have forgotten a side when adding up the perimeters of some rectangles. I thought 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 rectangle of a length of 4 and width of 6 to be 16. It seems to me that they forgot to include one of the sides of 4 in the addition of sides, but it’s very interesting that they didn’t initially make 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 preservice teachers’ journal entries were noticeably more analytical of the mathematics involved in the students’ responses. One example is found in Sally’s comments, for example, which show that – differently from her horse problem journal entry – her focus was not solely on the correctness but also on the meaning of the students’ work. Notice that even though

Sally thought that the students “did a good job” solving her problem she continued to investigate her students’ work. She highlighted some of the interesting features she noticed. She speculated (rather than concluded) that the students “seemed” to be forgetting a side when adding up the lengths of the sides, a common error students make when they deal with perimeters of regular shapes. She also noticed that her students did not recognize the identical properties of the 6x4 and 4x6 rectangles that relate to the commutative property of multiplication and the conceptual relationship between the dimensions and the linear and surface measurements of rectangular shapes.

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

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SANDRA CRESPO

Sally, Thea had become curious about her student’s understanding of area and perimeter. She said, “I’m really interested to hear what the students are 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 I was 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. How is it going? Are you having trouble? Pretend that I have no clue what area and perimeter means. Can you explain what they mean in your own words? What is area? What is peri-

meter? Try to explain these words to me. [ml4].

June:

I think that area means space inside the shape and perimeter means distance around the shape. 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 they mean 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. (Then she provided a picture of a square divided up into 42 smaller squares, each square marked from 1–42 on the inside, on the outside the squares were counted from 1–28). So she wrote the perimeter is 28 and the area is 42.

I was quite pleased with her response because she made the effort to explain it in her own way rather than copying something from a text. From her response I think that she understands what area and perimeter means. I was looking forward to how she would define area (perimeter seems to be less of a problem for most people), as I myself am not quite sure how I would define it for the students. June said area means the space inside the shape. This seemed to me to be a good response but that lead me to wonder how she would define volume, or distinguish between the two. I’m also puzzled about why she did not put cm

2 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


171 and perimeter as we would like our students to have. Squaring whatever unit you use to measure 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, as other preservice teachers did, on her student’s work as being “interesting,” and provided a much more elaborate and detailed analysis of her student’s mathematical work. Notice that, although Thea was impressed with her student’s response and thought it showed that the student understood the concepts, she continued to look beyond the surface of the student’s answer.

As a result, Thea began to raise questions about the completeness and interconnectedness of her student’s ideas about area: “[I] wonder how she would define volume or distinguish between the two?” She also noticed that the student did not indicate a unit of measurement in her definition of area, and she found this puzzling. It is interesting to note that, rather than making a quick evaluation, Thea considered alternative reasons that could explain the student’s omission of the unit of measurement in her definition of area: “Does she not understand this part of it yet, or did she just forget?”

Overall, preservice teachers’ later interpretations became more detailed, more exploratory, and less conclusive. Analytical comments became more prominent and frequent in the preservice teachers’ discourse.

The comments revealed greater attention towards the meaning of student’s mathematical thinking rather than surface features. Examples of such comments included: “Something that I found interesting in Jess’ answer is that she knew that she wanted the largest area of land but she did not state this fact” (Carly); “I noticed that he changed all the units from meters to centimeters” (Megan); “How did she get 90 cm 2 ? All her perimeters were in meters, and all her areas were in centimeters

2

” (Nilsa).

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

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

Although the descriptive/interpretive tool helped focus preservice teachers’ attention onto their students’ mathematical thinking, it was the contradictions and surprises preservice teachers found in students’ work that challenged their conclusive and evaluative claims. Miriam, for example, who initially doubted her students’ mathematical abilities, soon found 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’s letter) for after,” could not contain her amazement when her student’s fourth 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 attitudes figured out, received contradictory evidence in their working on the “staking your claim” (area-perimeter) problem she had sent to both students.

I thought this problem would be challenging, but I never thought Jake would find it so difficult. . . . I was quite shocked that Jake had such difficulties with this problem because he has mastered [the] majority of problems I have given him in the past. None of my problems in the past have dealt with perimeter and area. I am not sure how much work

Jake has done in geometry in the past, therefore I may have been expecting too much from him based on past letters. Area and Perimeter may be Jake’s weak area in Math. I am not sure though. As far as I know he may have not been feeling well on the day he answered my question. . . . I must admit that I thought Mary would have more difficulties with this problem than Jake, because of past responses to letters. [Our instructor] stated in class today that people are weaker and stronger in different areas. This has proven to be true in the letters I received this week. [Megan, MJ5].

The face-to-face meetings with students provided many of the preservice teachers with contradictory data, which led them to question and revise their earlier convictions and claims about their students’ mathematical attitudes and abilities. Some preservice teachers, for example, were able to see that students who seemed very talkative and outgoing in their letters became shy and uncooperative during group work sessions.

Others saw their students, who in their letters seemed uninterested and unmotivated to do mathematics, show a different side when interacting in person.

Meeting and working with their students in person helped some preservice teachers look at their students from a different perspective. “We went over some of the questions I had asked her in previous letters,” commented

Nilsa, only to find that “she had no trouble with them.” “I think she is more of a talker than a writer,” Nilsa rationalized afterwards. Like Carly and


173

Nilsa, other preservice teachers began to formulate alternative explanations for their students’ previous “lackings.” A few examples follow: “He’s an enthusiastic and intelligent student but his organization skills are a bit weak” (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 I gave her” (Miriam).

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

In my journal, I wrote several times regarding how it seemed that “Lynn” had more difficulty explaining and clarifying her math thinking on paper than “Parker.” . . . When

I compared my journal reflections with the letters, I was very surprised to discover that

Parker 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 practices when she reviewed her journal entries and her responses to the student she had perceived to be of lowest ability. In her case report, Linda discussed how David’s messy work and misspellings played a role in her developing perception of him as a less capable student. Her perceptions,

Linda was dismayed to find out, affected her communication with her students. She reported that she found substantial differences in her “writing style, questioning, length of letters, and problems posed.” She wondered whether and how these differential communications with her students in turn affected her students’ mathematical work. This and the preceding examples show that preservice teachers had begun to see contradictions in their initial claims and to look for alternative explanations, which often incorporated considerations for the content, context, and format affecting their students’ mathematical performance.

EXAMINING INFLUENTIAL FACTORS

I have highlighted the main features of preservice teachers’ beginning interpretations of their students’ mathematical work and the ways in which their interpretations changed over the course of their interactive experiences with students. In this section, I examine what might have helped the preservice teachers begin to see more than right and wrong answers in their students’ mathematical work and try to illuminate the reasons behind the interpretive turns.

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SANDRA CRESPO

Introduction of Challenging yet Accessible Problems

Ball (1988) shared her deliberations on the kinds of tasks she typically chose for prospective teachers in an initial teacher education course. She discussed the importance of choosing a task that was unfamiliar, intriguing though inviting, in order to ensure prospective teachers’ genuine engagement with the mathematical content they were studying. The NCTM

Professional Standards (1991) also highlighted the role that a carefully selected task can play in promoting students’ learning of mathematics.

Similarly, in this study, the introduction of challenging yet manageable mathematical problems seemed to have played a role in preservice teachers’ patterns of interpretations.

The introduction of unfamiliar mathematical tasks, that is, tasks that challenged and extended preservice teachers’ own understanding of mathematics, helped them move away from their evaluative interpretations.

Notice that in the case of the horse problem, the familiarity of the mathematical content and format of the problem may have played a role in preservice teachers’ superficial and unproblematic interpretations of their students’ work. It is interesting to note that once the mathematical tasks became less familiar, such as the area/perimeter problem, preservice teachers’ interpretations became less certain, more exploratory, and increasingly respectful of their students’ mathematical sense making. To use Duckworth (1987) and Schön’s (1983) notion, preservice teachers began to challenge themselves to “give reason” to students’ mathematical work. That is, they began to raise questions and attempt to understand their students’ seemingly impenetrable and indecipherable work.

The relationship between the type of mathematical task and preservice teachers’ interpretations makes, in retrospect, a lot of sense. Consider, for instance, that familiar problems are more likely to yield familiar responses, which preservice teachers are likely to recognize as similar to their own mathematical work and therefore be less inclined to question and explore. Unfamiliar problems, on the other hand, are likely to yield unfamiliar responses from students, therefore increasing the likelihood that preservice teachers inquire into students’ mathematical thinking.

This is not to say that this is an unproblematic relationship. Recognizing their own ways of thinking in students’ work helped the preservice teachers recognize meaning and analyze students’ work, whereas receiving unfamiliar work from students could also make it difficult for preservice teachers to recognize meaning in such work. Nevertheless, is important for teacher educators to consider the careful selection and introduction of mathematical tasks, not only to ensure prospective teachers genuine mathe-


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

Content and Form of Reflective Writing

Journal writing is a popular activity promoted in teacher preparation programs as a means to engage preservice teachers in critical reflection and analysis of their own teaching and learning of subject matter. Teacher educators and researchers, however, have found it difficult to move preservice teachers’ writing from a focus on personal and emotional aspects of learning-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 preservice teachers’ writing takes the form of reporting and summaries of their experiences (Anderson, 1992). The content and form of preservice teachers’ writing, therefore, is an important factor to consider and examine, particularly if writing plays such a prominent communicative and reflective role as in this study.

The introduction of the descriptive/interpretive journal writing tool during the third week was an intervention that seemed to have aided preservice teachers’ interpretive turns. Although not all preservice teachers used the double-column format to the same extent, its introduction provided a model for reporting and reflecting on students’ mathematical work. In addition, this tool seemed to have helped focus preservice teachers’ writings onto the examination of students’ work. Such a focus was important if we consider that the value of reflecting on students’ work may initially not have been apparent to preservice teachers. Notice that by the time preservice teachers made their journal entries, they had already responded to their students’ work. These after-the-fact interpretations may have seemed too late to inform their already-sent responses. In turn, the value of focusing their reflective writing on further interpreting the students’ work may not have been readily apparent to preservice teachers until they were asked to do so.

Being directed to focus the writing upon the students’ work helped preservice teachers become more self-conscious and analytical. They reported that spelling out their analysis of their students’ work: helped them to “move beyond superficial considerations to a deeper more critical analysis” (Sally) and become “more insightful” (Terry); and allowed them

“in time to (estimate) guess a little better why students answered questions in a certain way” (Megan). Writing and reflecting on students’ work with an explicit focus and format played an important role in developing preservice teachers’ interpretations.

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The case study at the end of the term provided another context for preservice teachers to engage in reflective writing and to revisit and reinterpret their students’ mathematical work. This assignment allowed preservice teachers in this study to gain further insights that they might not have had otherwise. Revisiting the students’ data and their own journal reflections on the data provided an occasion for preservice teachers to reconsider and revise their convictions about their students’ attitudes and abilities as well as to uncover flaws in their own interpretations. Access to tangible records and the process of revisiting such records was an important aid to the process of meaning making and self-examination involved in reflective writing (Wassermann, 1993).

Interactive Experiences With Students

Field experiences and field-related experiences are common activities in teacher preparation programs. Although the practice setting has long been considered the authentic place for learning to teach, it has become increasingly clear that the practice setting may not necessarily be the best nor the only place for learning to teach (Feiman-Nemser & Buchmann, 1986).

Finding ways to help prospective teachers make problematic (and an object of investigation) what they experience in the practice setting has been the focus of much of the recent research and practice in teacher education. The incorporation of an unconventional field-related experience can therefore provide insights into important features to consider when designing field experiences for prospective teachers.

Different from traditional forms of interactions, written interactions provided the opportunity to disregard managerial or disciplinary issues and school and curricular pressures which are often the focus of preservice teachers’ attention. In addition, delayed interactions with students made it possible for preservice teachers to engage in the collaborative analysis of students’ mathematical thinking with other preservice teachers. At the same time, this form of interaction brought to the foreground concerns about students’ abilities to communicate their ideas in writing. This, in turn, helped many preservice teachers begin to make problematic the act of interpreting and making sense of students’ written work.

The delayed interaction with students also provided the time and opportunity for preservice teachers to question and extend their own mathematical ideas. Examining their students’ responses sometimes served to engage preservice teachers in further mathematical explorations of the problems they had posed (as was shown earlier in Thea’s case). There were also occasions when their attempts to make sense of a student’s response led preservice teachers to make some mathematical discoveries of their


177 own. For example, the analysis of her student’s work led Miriam to revise her own solution to a problem she had sent to her student. Her problem asked to find a rectangular shape that would give the maximum area for a given perimeter. After reading her students’ explanation for why the square would give the most amount of area, Miriam began to delve more deeply into the definitions and classifications of shapes, particularly whether and why 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 have so readily available. These preservice teachers often gave their students the same problem and therefore were able to obtain more data about how two different students responded to their mathematical problems and questions. Writing to two students also raised pedagogical challenges, which corresponding with one student did not. Reading and writing two letters in the same amount of time that others had for reading and writing one was a source of difficulty and concern for these preservice teachers. Sally, for example, worried about being too rushed to carefully think and reflect during the writing of her letters. Megan, in turn, worried about providing equal time and quality of attention to each student. As a result, some of these preservice teachers focused their investigations primarily on the differences between the two students’ mathematical work.

The face-to-face and more immediate interactions provided opportunities 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 structure and setting played in their students’ mathematical work. This, as it turned out, became an important source of tension and deliberation in preservice teachers’ journals. They had constructed conjectures and had made assumptions about their students as learners of mathematics based on the students’ written work. Meeting their students in person provided another source of data to confirm, elaborate, and challenge those assumptions.

CONCLUDING NOTE

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

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SANDRA CRESPO only a few preservice teachers who initially wondered whether students would be able and willing to communicate their thinking and whether they themselves would be able to understand it.

I have, however, documented the ways in which preservice teachers began to reflect on and change their interpretations of their students’ mathematical work. I have characterized preservice teachers’ interpretations as taking two important interpretive turns. The first turn refers to a change in the focus of interpretation, from correctness to meaning. The second turn can be thought of as a change in the interpretive approach itself, from quick and conclusive to more deliberate and tentative. The first interpretive turn saw changes in preservice teachers’ interpretations from a limited focus on the correctness of the students’ solutions towards additionally attending to, exploring, and recognizing mathematical meaning in students’ solutions to problems. The second interpretive turn saw preservice teachers who initially made quick and conclusive judgments about their students’ mathematical abilities and attitudes begin to make more thoughtful and provisional interpretive claims.

These interpretive changes became apparent both in preservice teachers’ written interactions with students and in their reflective writing.

Their weekly journal entries became lengthier, more elaborate, and more detailed concerning the students’ work. In addition, their interpretations became more analytical and exploratory of the mathematics in their students’ work. I offered three factors as important contributors to the reported interpretive changes: (a) the introduction of challenging but accessible mathematical problems, (b) the content and form of reflective writing, and (c) interactive experiences with students. Although I did not regard these as the sole contributing factors, I did recognize their prominent role in overturning the patterns of interpretations of the preservice teachers in this study.

The study brings to the forefront the importance of attending to the interpretive discourse and practices teacher candidates bring to their teacher preparation courses. It also shares some insights into the ways in which a course-based interactive experience with students provided an occasion for preservice teachers to make their interpretations explicit and the object of their investigations. Furthermore, the study builds upon

Davis (1996) conceptual categories of teachers’ orientations to listening in mathematics classrooms by using and elaborating the categories of evaluative and interpretive listening as useful ways to characterize and analyze preservice teachers’ interpretations. Noticeably absent from this study, however, was Davis’ third category of hermeneutic listening. I found no evidence of this form of interpretation in the data collected for this study.


179

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

The study also reports evidence of change in the focus and approach of preservice teachers’ interpretations. This is particularly salient when we consider that the participants had weak mathematical backgrounds.

This raises questions about the role that preservice teachers’ often incomplete understanding of mathematics plays in the kinds of interpretations they make. This is a question about the relationship between content and pedagogical content knowledge which in this study came to the fore in the discussion on how the introduction of open-ended and exploratory types of 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 say that this reversed relationship also warrants further study: How does the analysis of students’ mathematical work affect the growth of preservice teachers’ mathematical understanding?

In addition, the study underscores the importance that preservice teachers’ evaluative interpretations be challenged. The research on teachers’ expectations has long discussed the dangers of teachers’ overgeneralized and conclusive inferences about students’ mathematical abilities (e.g., Good, 1987). This work has shown that the expectations teachers have of students who they believe to be more capable tend to be more respectful and demanding than the expectations they hold for students who they perceive to be less capable. Brophy and Good (1974) found that teachers who form these unexamined ways of seeing and judging develop rigid and stereotyped perceptions of their students based on prior records or on first impressions. This is also true for preservice teachers.

Another danger of ignoring the conclusive and evaluative discourse that preservice teachers might bring to their mathematics teacher education courses is the message that it carries about the certainty and predictability of teaching practice and students’ thinking. This kind of discourse serves to limit teacher education students’ ability to conceptualize new and richer images for mathematics teaching and learning. Ball and Chazan

(1994), for example, discussed the pernicious effect of the evaluative and judgmental discourse in current descriptions and discussions of mathematics teaching practice. Such a discourse, closes rather than opens practitioners’ conversations about mathematics teaching practice. Furthermore, “the common syntax of shoulds and should haves distorts practice with a stance of implied clarity” (Ball & Chazan, 1994, p. 4). An alternative syntax “of could and might,” as they proposed, would not only

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SANDRA CRESPO better represent the uncertain nature of teaching practice, but also help widen 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 is also useful to consider what kind of experiences might invite them to reframe their interpretive lenses. This study offers a particular pedagogical intervention teacher educators might consider when they offer opportunities to investigate students’ thinking. Where and how such an intervention would fit within, and build upon, the range of experiences available in a teacher preparation program is a larger, though important, question to consider when designing such experiences, especially because one single encounter of this type of experience in an isolated course is unlikely to have the kind of impact that teacher education programs hope to have in the preparation of beginning teachers. We need a more concerted strategy in order to help prospective teachers begin to seriously consider Davis’

(1996) vision of “teaching as listening,” where teachers see an open-ended inquiry approach towards their students’ mathematical work as a viable and desirable teaching practice.

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513G Erickson Hall

Michigan State University

East Lansing, MI 48824-103

E-mail: crespo@msu.edu

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