Mathematics Culture Clash: Negotiating New Classroom Norms with Prospective Teachers

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JOURNAL OF MATHEMATICAL BEHAVIOR, 18 (4), 475 ± 509 ISSN 0732-3123. Copyright C 2000 Elsevier Science Inc. All rights of reproduction in any form reserved.

Mathematics Culture Clash: Negotiating New Classroom Norms with Prospective Teachers Betsy McNeal Ohio State University, Columbus, OH, USA

Martin A. Simon Pennsylvania State University, University Park, PA, USA

As part of an investigation of the mathematical and pedagogical development of prospective teachers, the second author taught mathematics to a group of undergraduate teacher candidates in a way that is compatible with current mathematics education reform principles. Initially, the lack of a shared basis for communication was evident when these students, acculturated to the practices of school mathematics, interacted with a teacher who was trying to promote inquiry mathematics. Our analysis of the data indicates that by the latter stages of the course, a classroom microculture characterized by inquiry mathematics had evolved. In this article, we examine the processes by which the participants in this classroom community negotiated norms and practices. The result of this analysis was an identification and elaboration of four categories of interaction central to the ongoing negotiation. This article illustrates how each of these categories of interaction contributed to the negotiation of new norms and practices.

Current efforts to reform mathematics education (e.g., National Council of Teachers of Mathematics, 1989; 1991) focus on teaching and learning mathematics as a process of exploring ideas and relationships, of making and testing conjectures, and of creating and evaluating arguments to justify those conjectures. This vision of mathematics education would imply a change from school mathematics to inquiry mathematics, the mathematics ``used by mathematically literate adults'' (Richards, 1991, p. 15). This vision not only challenges teachers' assumptions about mathematics and mathematics teaching and learning, but also asks them to teach a mathematics that they may never have experienced. Teachers are likely to continue to teach the way they were taught if teacher educators do not find a way to interrupt this self-perpetuating cycle. Our work is based on the premise that the practice of mathematics teaching is strongly influenced by teachers' theories of teaching, learning, and mathematics. Thus, in order for teachers to make significant changes in their practice, they need opportunities to reconstruct these personal theories. Such opportunities can result from participation in a mathematics (classroom) community characterized by mathematical and social practices Direct all correspondence to: Betsy McNeal, Ohio State University, 333 Arps Hall, 1945 N. High St., Columbus, OH 43210-1172, USA; E-mail: [email protected] 475

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supporting inquiry. For prospective teachers, mathematics classes in teacher preparation programs can provide opportunities to alter their relationships to mathematics as well as to mathematics learning and teaching (Simon, 1994; Simon & Brobeck, 1993; Wilcox, Schram, Lappan, & Lanier, 1991). We describe our analysis of the co-construction of such a classroom community by one teacher educator and his students ( preservice teachers). Recent theoretical and empirical work has emphasized an important aspect of a mathematics teacher's role, guiding the negotiation of classroom norms and practices (Yackel & Cobb, 1996). Although this is important in any classroom mathematics teaching, it is particularly critical as teachers who wish to establish a reform-oriented classroom work with students whose past mathematics experience has been traditional. We report a fine-grained analysis of such a negotiation process. As part of a 3-year study of the mathematical and pedagogical development of prospective elementary teachers, Simon taught a mathematics course designed as a whole class constructivist teaching experiment (Cobb, 2000; Simon, 2000). The vision of teaching and learning that guided this course is fundamentally compatible with current reform efforts and is described in Simon (1995). The course was organized around tasks that Simon created to promote students' construction of new mathematical ideas and reorganization of their prior understandings. Rather than lecturing, he posed problems, probed students' thinking, requested paraphrases of their ideas, managed and focused discussions, and avoided indicating the correctness of particular ideas. Students worked on problems in small groups before discussing them with the whole class. At the beginning of the course, most of the 26 undergraduate students were uncomfortable with mathematics both as learners and as future teachers. The students' responses to a focused writing assignment for the first week of class, as well as their actions and reactions in the earliest mathematical discussions, suggested that they expected to be passive recipients of procedures to be applied to problems posed by the teacher. They expected to be told how to solve particular types of mathematical problems and told when they had obtained the correct answer. Early in the course, encounters of these students acculturated to school mathematics with a teacher from a mathematics culture characterized by inquiry resulted in a culture clash. As these students interacted with a teacher whose expectations conflicted with those that they had come to anticipate in mathematics class, it was evident that the teacher and students lacked a shared basis of understanding. This showed in moments of surprise, misinterpretation, misunderstanding, or when community members simply talked past each other. Our preliminary analyses of the data indicated that by the latter stages of the course, a classroom microculture characterized by inquiry mathematics had evolved. Our subsequent analyses concentrated on identifying the processes by which this teacher and his students ( prospective teachers) negotiated norms and practices that supported an inquiry mathematics culture.1 This is the focus of this article. In the next section, we define mathematics culture as we use the term, then explicate the interactionist theory upon which our analysis is based. 1.

THEORETICAL FRAMEWORK

For purposes of this investigation, we define culture as encompassing both common knowledge (Edwards & Mercer, 1987) and common patterns of behavior, including those

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ways of perceiving, acting toward, interpreting, evaluating, and defining the objects, actions, and events of everyday life (Collins & Green, 1992). Membership in a cultural group requires knowledge of what constitutes acceptable participation in that group. A ``cultural group'' then can be characterized by the particular roles and attendant obligations and expectations that constitute normal social interaction, and by the participants' responses to divergent behavior. By mathematics culture, we refer to the common knowledge and patterns of behavior that characterize the activity of mathematics. Participants in a particular mathematics culture can therefore be distinguished by the ways they perceive, act on, and interpret mathematical objects, as well as how they evaluate and define mathematical problems, objects, justifications, and explanations. Following Cobb, Wood, Yackel, and McNeal (1992), we consider a mathematical explanation to be a statement that is offered in an attempt to clarify a communication, and a justification to be a statement intended to convince someone of the validity of a mathematical idea. The kinds of statements offered, and the situations in which explanations or justifications are offered, are thus relative to the mathematics culture. Normative aspects of explanations and justifications are further discussed in Cobb et al. (1992). Yackel and Cobb (1996) also distinguish between social norms (mutually constructed expectations for the appropriate behavior of participants in a particular cultural group) and sociomathematical norms. Whereas social norms might be enacted in any subject area, sociomathematical norms refer to those normative understandings that characterize the mathematical activity of a community. These might include understandings of what counts as an acceptable explanation or justification and what counts as a different, efficient, or sophisticated solution. Our focus is on the process by which both types of norms are established in a particular classroom microculture; distinguishing between them is not central to the work we report here. Our work is informed by the theory of symbolic interactionism (Blumer, 1969) as applied to the teaching and learning of mathematics by Bauersfeld, Krummheuer, and Voigt (1985). From this perspective, classroom cultures are jointly constructed through the interactions of teacher and students over time. Participants interpret situations on the basis of their prior experiences, and reorganize their interpretations to account for unexpected events, so human interaction is a process of ongoing and evolving interpretations of each other's actions and responses to those actions. We use the term negotiation (Voigt, 1985) to describe this process. Negotiation of meaning often occurs implicitly through the subtle adaptation of participants' actions to fit with their ongoing interpretations. This negotiation process can also be explicit: A conversation can have the give and take of negotiation where it is considered complete when the participants feel they have successfully communicated. From an interactionist perspective, individual sense-making and the classroom microculture are taken to be reflexively related: Individuals contribute to shaping the microculture while the microculture enables and constrains the sense-making of the individuals (Cobb & Bauersfeld, 1995). As they gradually come to understand each other, participants' interpretations become coordinated into routine responses, and patterns of interaction emerge. In this way, a classroom microculture is neither static nor created by the teacher, but is continually reconstituted through the mutual adaptation of its participants' interpretations of their activity. Voigt (1995, p. 164) suggested that ``We should conceptualize the change of a microculture as an evolution rather than as a

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rearrangement. In order to influence and direct that evolution, it is helpful to understand the regularities and dynamics of the processes within the classroom life.'' Where teacher and students bring compatible expectations and interpretations to classroom events, the microculture may be constituted and reconstituted quite smoothly. It might indeed appear to the participants as well as to an outsider that no negotiation occurs at all. However, when participants bring divergent expectations and interpretations, there are not only instances of misunderstanding and miscommunication, but moments where previously implicit understandings must be made explicit. It is this situation, in which teacher and students differ radically in their understandings and expectations, that we characterize as a culture clash. Cobb, Yackel, and Wood (1989) provide examples of the work one teacher did to engage her second grade students in renegotiating their classroom practices in response to situations in their mathematics class. This analysis revealed two levels of discourse in the classroom, talking about and doing mathematics in which norms are implicitly negotiated, and talking about talking about mathematics in which norms are explicitly negotiated.2 We extend their work by identifying and elaborating examples of implicit and explicit negotiation of norms in a mathematics class for prospective elementary teachers. We highlight tensions between the mathematics cultures of teacher and students and the adaptive process of constituting a classroom mathematics culture.

2.

METHODOLOGY

All class meetings were videotaped and transcribed, and field notes were taken by project researchers. Copies of the students' focused writing assignments (at the beginning and end of the course), weekly journal reflections, tests, and written work were collected. Simon kept a reflective notebook of his thinking about each class and of his planning for the next class. The videotapes and transcriptions became our primary data source because of our interest in the development of communal meanings. McNeal analyzed the data independently, and then she and Simon, the course instructor, worked to develop shared interpretations of classroom interactions. The analysis was thus separately and jointly constructed. Using techniques developed by Bauersfeld et al. (1985), we examined the transcripts line by line for meanings the participants seemed to attribute to each other's words and actions. We worked our way through the entire semester (Classes #1±25) documenting what we saw as the community's mathematical and social practices at each point in time. Specifically, we looked at how teacher and student roles were defined and tried to understand mathematical activity as practiced in the classroom community. We compared our understanding of the students' perceptions of their role to the teacher's expectations of them and, reciprocally, our understanding of the students' expectations of the teacher to his view of his role. We analyzed students' actions in and responses to class events in order to infer the students' interpretations of class activities, norms, and roles. When self-report data (in the form of classroom talk, journals, and focused writing assignments) conflicted with teacher intentions such as when students expressed disagreement with, or a lack of understanding of, the way class was being conducted, we considered these data to be reliable and relevant

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to our analyses. However, when students' self-report data suggested alignment with the intentions of the teacher, we used these data only as secondary corroborating evidence. This cautious approach to self-report data was based on an awareness that such data could have been biased by what students perceived to be the ``party line.'' In our initial analysis of the full set of videotapes and transcripts, we identified numerous situations in which students engaged in discussions that were remarkable for their mathematical substance and for the non-traditional teacher and student roles. By Class #15, such interactions seemed to characterize the classroom interactions. We thus took Class #15 to be indicative of changes that had developed since the first day of class, and analyzed the transcript of that class in terms of the specific norms and practices that it reflects. With this evidence that an inquiry mathematics culture had evolved, we reexamined Classes #1±15 to see how this evolution had taken place, that is how the norms and practices of inquiry mathematics had been constituted. Following Erickson (1986), we compared our interpretations of each class meeting to our interpretations of prior class meetings to check whether our emerging theories represented ideas previously overlooked or new developments in the data. In this way, we not only examined each class meeting individually, but also looked across Classes #1±15 for events and interactions that appeared to contribute to the development of the social and mathematical practices observed. We also looked for events that uncovered participants' differing assumptions, such as instances of overt disagreement or miscommunication. Such events further elaborated our understanding of the students' perspective. The resulting analysis explains discrepancies as well as patterns in the data. The data presented in this article, whether excerpted from transcripts or quoted from written artifacts, are indented. Summaries of transcript data are set off from quoted data using brackets. Extraneous words have been deleted in order to make the transcripts more readable; all other omissions are indicated by ellipses. ``T'' designates the teacher, Simon, and abbreviations are used for students' pseudonyms. Analytic commentary is interspersed with the data in normal format. Present tense is used to narrate an event or to indicate what comes next in the transcripts. Past tense is used to refer to data already presented. After each excerpt, we provide an analytic summary. Presenting analyses of both small and large data segments is consistent with our method of examining both the moment by moment meanings of words and actions and their relation to the development of the classroom microculture. We first present evidence from Class #15 that students and teacher had negotiated norms consistent with inquiry mathematics. In subsequent sections, we will trace the evolution of these norms chronologically.

3.

EVIDENCE OF NEW CLASSROOM NORMS: CLASS #15

The students began this course with a high level of discomfort about their ability to do and learn mathematics. They expected the teacher to pose problems, provide the solution methods, and evaluate students' answers. By Class #15, a pattern appeared to have emerged in these interactions: A problem is posed by teacher or a student, students work in small groups, the teacher asks for persuasive arguments, and the students discuss

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FIGURE 1. A ski ramp in Kansas.

them. The teacher participates by asking for a paraphrase, asking for clarification or demonstration, polling the group, or posing a new problem. Students participate by presenting ideas, asking each other questions, and challenging or refining ideas under discussion. There is evidence that students have shifted from attempting to comply with the demands of the teacher, to initiating and maintaining self-directed inquiry and exploration of mathematical ideas. Prior to the discussion in Class #15 presented below, the students began working in Class #11 on the following problem: In Kansas, there are no mountains for skiing. An enterprising group built a series of ski ramps and covered them with a plastic fiber that permitted downhill skiing. It is your job to rate them in terms of most steep to least steep. You have available to you the following measurements for each hill: the length and width of the base (measured along the ground) and the height (see Fig. 1). How would you determine the relative steepness using the information that you have?

Students worked on this and related problems in their small groups and whole class discussions over several class sessions. They perceived early on that steepness is increased by greater height and/or decreased length-of-base. The students' inquiry then shifted to whether the ratio of these quantities or their (subtracted) difference is an effective indicator of steepness. These came to be known as the ``ratio method'' and the ``difference method,'' respectively. The students struggled with what it would mean for a method to ``work,'' and had difficulty structuring their investigation so as to be able to draw conclusions (e.g., they seemed to choose random lengths for the height and base of their triangles, rather than comparing two triangles of same ratio, same difference, or identified as equally steep). Students also had difficulty determining what constituted evidence for or against either of the proposed methods (Simon & Blume, 1994). 3.1.

Class #15

[Based on their prior work with the ski-slope problem, the teacher poses another problem:] Order the following slopes [height, length of base, respectively] in terms of their steepness. Use any means about which you are sure it works. Then go back and explore our ratio and difference methods using these data: 3, 1 4, 6 11, 9 2, 6 9, 3 7, 11 12, 4.

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[After students have worked in small groups for about 40 min, the teacher opens the discussion, calling for arguments for the validity of either method.] T: So who would like to present us with some persuasive arguments? (pause) Georgia. G: We all drew these slopes and then we used the ratio method and what we found was three of the slopes were equivalent meaning that three of the slopes using the ratio method came out to be 3 . . . So what that told me was that . . . using the ratio, they could all be reduced down to 3, and they all had the same angle of steepness. But in the difference method, . . . they would have all been different. And with the rest of the slopes, we would have found negative numbers, if we would have used the difference method . . . . We just can't have negative slopes. I don't know, maybe in Kansas they have negative slopes . . . . It just doesn't make sense to me to have a negative slope. The teacher turns to the class for their consideration of Georgia's argument. T: Judy, do you have a question about what Georgia said? Ju: . . . I agree with what she said, but I drew them all out, and you can see in those she used there, where you get the same slope for them, that they look the same. Like the shape of them . . . but when you use the subtraction method, you get all different numbers. That's why I don't think that's accurate. Judy seemed to think that Georgia had not connected the information she had obtained by using the ratio method with the pictures her group had drawn, and so Judy added this point to the discussion. This suggests that she believed her role was to make sense of Georgia's remarks and to continue looking for clarification of the mathematics involved by adding to Georgia's idea. Sara, Molly, and Bobbie then offer a series of contributions that do not focus on which slopes are equally steep, but focus on finding still more persuasive methods of showing equal steepness. [Sara shows with Fig. 2 that embedding the triangles by superimposing their right angles allows one to see equal steepness by observing the parallel hypotenuses. Molly then claims that she and Lois are ``taking [Sara's] idea and then doing it better.'' They demonstrate in Fig. 3 that embedding the triangles by superimposing the top vertices and the vertical sides allows the observer to see equal steepness when the hypotenuses coincide. Bobbie adds that Sara and Molly have offered two of three possibilities; the triangles can also be embedded at the lower right vertex as in Fig. 4.]

FIGURE 2. Sara's drawing.

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FIGURE 3. Molly and Lois' drawing.

FIGURE 4. Bobbie's drawing.

These contributions indicate that, despite the group's understanding of Georgia's idea and agreement with Judy's point, the students did not believe discussion of the problem was complete. Rather, they seemed to believe that it was appropriate to generate more powerful justifications and to learn as much as possible from the mathematical context. As consensus builds on the validity of the ``ratio method'' and on the lack of validity of the ``difference method,'' Eve, Lilly, and Toni raise questions that delve further into the mathematics. Eve seeks a relationship between the physical model and the calculations: E: If we're doing the ratio method and you divide 9 by 3, OK, and you get your 3, what is that 3? . . . Is it the angle of degree or, what is it? Then Lilly, looking for greater understanding, suggests a possible connection between their mathematics problem and her own experience outside of class: Li: I was just wondering if that answer tells you, you know, when you go over the mountain and there's a truck stop and a sign that tells the trucks to gear down because it's a 7 percent slope or whatever, or I mean 7 percent grade. Do you allÐhave you seen those signs? As the discussion continues, Lois explains the meaning of the ratio in the problem context, encompassing both Eve's question and Lilly's speculation. Lo: [The 3 is] not . . . giving you a value for that side, or telling you that this angle is this much, or this side is this long. What it's sayingÐit's sort of like a rate . . . . It's like a rate of steepness. It's sort of almost like a comparison.

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G: If we had to look at it on a triangle on the board, where would you say that we're findingÐwhere would that 3 go in the triangle? Lo: Think of like when G: (interrupts) No, you tell me where you think it would go. Lo: It's not like that though . . . It's not measuring any part of the triangle. It's measuring how much that triangle is tilting this way or that way sort of. G: What determines that triangle to tilt this way or that way? Lo: The relationship of the sides. How you draw them. Like, if one side is 3 and, if it goes up 3 and over 1, then that is going to determine those sides. Toni continues the inquiry into the meaning of the ratio: To: Is the number we come up with when we use the ratio method relevant unless we're comparing it to other triangles? Is that number that we get when we use the ratio method, is that even a relevant number unless we're comparing it with other slopes? [After some additional discussion, Lois speaks up.] Lo: I can clear this whole thing up right here. (goes to the blackboard and draws as she speaks) . . . The ratio method works, because it's like a rate of 3 to 1. So as you're going down, for every 3 you go down, you're also going over 1 . . . So when you said, ``I went down a mountain of 3,'' so I went down 3 and over 1, down 3, over 1, and that will eventually lead you to this point [the bottom of the slope] and that's why you have that line. Know what I mean? . . . And that's your rate, and that's what the down grade is. So it's not an arbitrary number. Although she used the mathematical term ``rate,'' Lois tried to relate her explanation to the questions and comments of her fellow students. 3.2.

Analytic Summary

For more than 30 minutes of this class and for two previous classes, the talk centered around a single problem: Decide whether or not the ratio and/or difference methods will allow one to order ski ramps by steepness from only the measurements of their height and length of base. Students decided that the ratio method would accomplish this goal through analysis of self-constructed drawings, and then continued the discussion, building off each other's contributions toward increasingly convincing justifications of these ideas. They determined that the difference method would not provide a valid description of slope because the numerical results did not reflect what they saw in their drawings. Students spontaneously generated and discussed questions about the meaning of the mathematical objects and relationships they were studying and made connections among each other's ideas, and between these mathematical ideas and their outside experiences. This interaction provides evidence that the teacher and his students had become a mathematics community in which it was the students' job to pursue understanding of mathematical ideas generated in the process of solving problems. Students viewed themselves as responsible for presenting and justifying mathematical ideas, listening to

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and understanding each other's ideas, evaluating each other's mathematical arguments, and finding connections among the different explanations offered. All of this seemed to occur in pursuit of the fullest possible understanding for the group, suggesting a belief that ideas are built as a community. In the next section, we will trace the patterns of action and interpretation in classroom interactions from which these changes developed.

4.

ANALYSIS OF THE NEGOTIATION PROCESS

Both non-traditional routines of the teacher (such as his non-evaluative stance) and discrete classroom events (such as the announcement of the first test) provided opportunities for implicit and explicit negotiation of the different perspectives that characterized the two mathematics cultures represented in this classroom.3 Although the teacher had a vision of what would constitute an inquiry mathematics community, changes in classroom norms and practices were jointly constituted in the course of classroom interactions. In this first subsection, we illustrate the kind of work the teacher did to initiate development of new roles, rights, and obligations for himself and his students by explicitly stating his expectations. In subsequent sections, we illustrate the mutual construction of norms by examining those events that prompted both explicit and implicit negotiations, and hence, shifts in behavior and interpretations. In this particular classroom, we found that negotiation was prompted by four types of interactions: those involving new experiences for students, the students' reflective journal, paradigm cases of mathematical activity, and discrepant events. Changes in communal meanings occurred as a result of explicit discussion at specific points in time and gradual adjustment of each participant's activity to the others'. 4.1.

Introducing the Course and Defining New Practices

One of the teacher's goals in the early class meetings was to begin to orient students toward particular social and mathematical practices that he believed would support mathematical inquiry. Thus, he talked more about classroom processes in these classes than in subsequent classes. In Class #1, the teacher begins preparing the students for a course that will emphasize the articulation of mathematical ideas and reflection on ideas and on one's own learning. Although, from the teacher's perspective, this class is the first step on the road to a new way of thinking about mathematics, the students probably have no way (at this point in time) to anticipate how different this classroom experience will be. Class #1. [The teacher makes general comments about the course. He points out that this course will probably be different from their prior experiences in math classes because he will not lecture and they will be expected to participate actively in class.] T: And just notice that, you know, you feel a little resistant to it, and pay attention to it and see if you can understand what's going on for you. That happens when people get involved in new things.

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[He explains that the topics listed on the syllabus are not fixed, but only what he anticipates they are going to do. He explains that he will weigh the final exam more than the first two if students show that they have learned the mathematics and reminds them to bring calculators and three different colored pens to each class, but leaves them to read for themselves the statement that: ``All mathematics work will be done in pen. Instead of erasing draw a single line through an error. This leaves a clear record of your thought process.'' Finally, he calls their attention to the weekly journal assignment detailed on a separate handout. Students are asked to respond to one of the two classes each week by addressing the headings: ``what I learned of and about mathematics,''4 ``how I learned,'' and ``affect.'' The handout states several purposes for this assignment, including:] 1. 2. 3. 4.

to enhance your learning of mathematics (by articulating what you have learned, what your questions are, and your confusion), to provide you an outlet for expressing feelings connected with the mathematics and/ or class experiences (e.g. elation, frustration, fear, confidence, etc.), to help you to learn more about how you learn mathematics . . . , and to provide additional opportunities for Dr. Simon to stay/become aware of your thoughts, learnings, and feelings in relation to the mathematics and the course and to respond to them. [At the end of this handout, students are given two pointers.]

1. 2.

The only way you will benefit from this activity is to be completely honest. [There are no] correct or preferred responses. Nothing that you write will help or hurt your grade. You should learn while you are writing if you are reflecting on important issues. You will not learn if you are only putting down what is very obvious to you. [The remainder of the class is spent learning each other's names by playing a game.]

Analytic Summary. In this first class, the teacher's description of the course was without context for the students, so his comments probably did not seem remarkable. It is likely that students had heard other teachers exhort them to focus on learning rather than on grades, encourage them to see him with problems, and even talk about weighting grades more at the close of the marking period. It was a little more unusual to have a mathematics teacher claim that he would not be lecturing, describe the topics of the course as ``not set in stone,'' and request that they leave errors visible to the reader of their assignments. Most unusual in a mathematics class was the journal assignment.5 Although the teacher intended his comments to prepare the students for a unique learning experience, this was only the first step in a lengthy process of negotiating new classroom norms. We can see that he attempted to indicate, from the first day, his intention to focus the students on learning how to learn mathematics (journal section on ``how I learned''); to treat students' concerns, frustrations, and confidence level as legitimate parts of the course (his anticipation of their resistance; affect section of the journal); to value students' thought processes (description of journal; instruction to cross out rather than erase); and to encourage articulation of, and reflection on, ideas as crucial learning tools (emphasis on journals and class participation). However, students did not anticipate the

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extent to which their ideas would be a focus of the course, and their understandings would drive the topics covered. The teacher's statements on this first day became recurring themes over the semester, not because of what he said, but because his subsequent actions were consistent with them and repeated over time. The students thus constructed meanings that were more compatible with the teacher's meanings through hearing and interpreting his stated expectations, but also through interpreting actions consistent with those statements, and hearing them restated in particular contexts. In Class #2, Simon explicitly defines some of the new practices that he wants students to adopt. He states his expectations explicitly because he anticipates that they will differ significantly from the students' prior experiences in mathematics class. Class #2. [Before setting the specifics of a mathematical task that students are to complete in small groups, the teacher describes what he means by working in a group.] T: What I don't mean is a whole bunch of people working independently on the task and checking their answer at the end to see if they got the same answer . . . When I say working as a group, I'm talking about cooperating all through the activity and discussing what you want to do and why you're doing it and making sure that every member of the group understands what's going on. The teacher's statement indicated that he considered individual engagement in group work to be important, and that he wanted the group to share responsibility for the understanding of its members. [The teacher explains the first mathematical task of the course, then describes what he will expect after the students have completed it.] T: I'm interested that you can come back and tell the group how you went about doing this task. Not just a number that you came up with, OK? So that's what you're gonna be ready to report, is how you went about doing it. [The students work together for about 10 min, talking out their solutions. The teacher moves around the room observing their actions and listening to their talk. He calls them back together when he judges that all groups have completed the task. Occasionally within a group, someone asks if everyone understands.] The students' attempts to monitor classmates' understanding indicated that they had heard and were complying with the teacher's instructions. Before beginning discussion of the mathematics, the teacher again specifies certain practices that he would like them to adopt, this time for whole class discussion. T: It doesn't matter who says it, [if] somebody says something in here, it's all of our jobs to understand what that person said. That has nothing to do [with] whether it was right or wrong. We can't even respond to it until we have understood it. [He tells them to ``take responsibility for understanding'' by asking to hear the idea again, or asking about the meaning of what was said.]

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Finally, the teacher spells out what a student should do when requested to paraphrase another students' idea, indicating simultaneously how such requests should be interpreted. T: Now, when I ask for a paraphrase, it doesn't mean the answer was right or wrong. It only means that I judge it's worthwhile stopping to make sure people understand. So you would get a chance to hear it explained in a different way. I would get a sense of whether people are understanding and so on. It will improve our communication and making sure people understand. Question on that? (pause) So, who can paraphrase for me why I might ask you to paraphrase? Or what it means? Nadine? N: Maybe somebody didn't understand the way you said it, but maybe they'd understand it a different way than how you explained yourself.6 T: OK. Lilly? Li: Also, to you as a teacher to have an understanding to make sure everybody else is understanding. So even if that person is giving you the right answer, you're gonna make sure everybody else agrees with it, or they also understand that they know that that's the right answer. T: Um hum [agrees]. But it will not be any indication from me as to whether it was right or wrong, because we can't discuss it either way until everybody has understood it. Nadine and Lilly understood that hearing students paraphrase each other would both help students understand and offer the teacher an opportunity to monitor students' understanding. Lilly also seemed to associate paraphrasing with highlighting the right answer. She probably had no context from the culture of school mathematics for understanding why a teacher would want to make sure that everyone understood something that was incorrect. Analytic Summary. Students and teacher seemed to be talking past each other on this point because of the different views in their respective mathematics cultures on how mathematics is learned. The students believe that mathematics is learned by the teacher orienting them to correct mathematics; the teacher believes that learning requires that students generate and evaluate mathematical ideas. Once students become engaged in the activities of Class #2, the teacher's prefacing statements begin to take on new significance. 4.2.

Interactions Involving New Experiences for Students

New experiences with mathematics played a significant role in the negotiation of norms and practices for doing mathematics in this classroom community. Contrasts between these and their prior experiences pushed students to reinterpret their prior and current experiences. For example, in Class #2, new experiences included being asked for justifications of their answers (even after they have correctly stated the steps of a procedure), and receiving no evaluation of their responses. These experiences prompted the students to reconsider their roles and the teacher's role, to recognize their lack of conceptual understanding, and to reconsider the role of authority in their mathematical learning. Although students experienced the events as new and different, changes in their understanding of classroom mathematics activity only came with the consistency of experiences over time.

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Class #2. [The teacher gives each group a different non-square cardboard rectangle and asks them to find the maximum number of copies of the rectangle that can be placed on their rectangular table without overlapping or cutting. After all the groups have completed the task, the teacher brings the students back together and initiates discussion.] T: Remember, your job is to understand, and I would like you to think about, as you are listening to people's methods, whether that method works. Who would like to speak for their group? Georgia? G: [Georgia refers to ``squares'' although each of the groups worked with a non-square rectangle.] With our group, we took the width of our square which was this little ittybitty square. And we took it, and we measured first the width of [the table], and we started up in the top left hand corner and measured all the way down how many widths of that square there were, and then we went across the table, and we timesed [sic] that, and we got 290 squares. [The teacher asks Georgia to physically demonstrate her group's measuring actions.] The teacher's response served to focus attention on the process by which the group had reached its answer, rather than on the answer itself, and implied his desire that everyone understand what Georgia's group had done. [After hearing the same method from another group, the teacher asks the class whether they all measured along the length and the width and then multiplied.] When each group confirms this, he challenges the students to justify their strategy. T: The job was to cover this whole table. Why did you choose to measure this edge and that edge and multiply? B: It seemed like the easiest way. T: . . . Is it an easy way to get a correct answer? Now why measure along an edge and another edge. How is doing that, and multiplying two numbers, related to covering this whole table with rectangles? You seemed to all think that was a good way to go about the problem. Why did you think that was? Deb. D: Because, in previous math classes, you learned the formula for area is length times width so probably everybody has the idea. T: And all those evil math teachers you were talking about [in an earlier discussion] (laughter) and you're gonna now take their word for it? M: They showed us that. T: How do we know if they are right? M: Because they showed us. (laughter) T: Blind faith? M: The teachers, you know, they showed us how it worked. In this interaction, Bobbie (B) and the teacher negotiated the meaning of the question ``why multiply.'' Bobbie seemed to interpret this as asking for her motivation. However, Simon wanted the group to think about why multiplication is an appropriate operation, a

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FIGURE 5. Judy's 3  3 grid.

conceptual, rather than a motivational issue, so he reformulated his question. Deb (D) and Molly (M) resorted to teachers as authorities in their first attempts to justify the use of multiplication. Simon responded by teasing Molly, challenging her to inquire beyond the word of previous teachers. Jonnie tries to elaborate on their idea. Jo: If you put down on this table . . . all the rectangles, and then you timesed [sic] like the width and the length together, and you got an answer and then you added all of them together and the answer was the same. Both ways. So you knew that that method, the length times the width, would work because they had us add them up after we timesed [sic] them. [In response to a request from the teacher, Judy paraphrases Jonnie's idea by using the rows and columns of a 3  3 grid as in Fig. 5.] After acknowledging that multiplication gives the same answer as counting in the examples students have previously encountered, the teacher challenges them to extend their thinking. T: So it seems to work. And it worked on the ones you had in school before. Does that always work? (pause) All the time, most of the time, some of the time? M: In rectangles or squares. T: It always works for rectangles or squares? M: From what I've seen. T: From what you've seen. There might have been some rectangles you haven't met yet? In an effort to ascertain whether the students are having difficulty creating a justification, or whether they simply do not believe the method will work in all cases, the teacher polls the students: T: Will it always work, if we count the rectangles along [the top row of the grid], and the rectangles along [the side of the grid], and multiply that? We'll get the same answer as we would if we counted them all up? (pause) How many believe that will always work? Raise your hand. (several) How many are sure it won't always work? (none) How many are not sure? (a few) That's a fine place to be. This poll explicitly permitted students to take a position of uncertainty and encouraged them to monitor their understanding. His acceptance of this position may have contributed

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to making this classroom a safe place to be uncertain7 and a place where understanding is expected to develop over time. The discussion continues. M: Well, it would work because multiplying and adding are related in that multiplying is like adding groups and so it would always work because you add them up to see how many is in the square, and to multiply the groups that go like that, that'll always work. You would get the same number . . . if you added them, or if you multiplied that side times that side. Because you're adding, I mean, you're multiplying the number of groups by the number in the groups which is the same as adding them all up. T: You multiply the number of groups times the number in the group. (writes ``# groups  # in groups'' on the board) Does anybody understand what she means by that? Eve? [Eve paraphrases Molly's point using an example of five circles with five stars in each.] Molly's response directly addressed the use of multiplication and offered a deductive justification of why it would always work. The teacher recognized her contribution as different from what other students had offered. He wrote the idea on the board, using her words, in order to focus discussion on it. Without evaluating it, he turned to see if the class understood it. Eve explained Molly's idea by giving the canonical school definition of multiplication. The teacher does not assume that others understand Molly's point or share his perception of its importance, so he asks how this idea is ``related to our problem of counting these rectangles.'' Sara states a basic disagreement with Molly's idea, initiating the first student±student conversation in a class discussion. In subsequent classes, these become an increasingly integral part of class. Sa: I don't think we're multiplying by the number of groups because the number of groups is two. We're not multiplying by 2. T: So, you say it doesn't apply here? M: What do you mean by the number of groups is 2? Sa: The number. If we're gonna have a group from up top and a [group] running down. So, you have two groups where you multiply those. [Molly and Sara debate what a ``group'' is.] T: (to Molly) Show us what you mean up there [in Fig. 5]. M: There's three groups (pointing to each square in the first column) . . . and then these [three squares in each column] are the numbers in the groups. So, it's three groups times the number in the groups. [The teacher asks Molly again what a group is and asks questions to prompt clarification of her thinking that each column is a group of three.] When Molly responded to Sara with a question, the teacher did not interfere until he saw an opportunity to help Molly clarify her idea. His response suggested that this

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student±student interaction was appropriate in this class. He neither evaluated Sara's idea, nor tried to clarify Molly's idea himself, although he did assist her in restating it systematically to the rest of the class. The teacher continues to probe the class' understanding of Molly's idea and to assess whether they are connecting Molly's justification with the original task. He asks for a restatement of the idea that omits the word ``group.'' When Georgia's paraphrase suggests that she did not interpret the teacher's question and Molly's idea as being about justification, the teacher renegotiates his request for justification. T: G: T: G: T: G: T: G: T: G:

I lost track of the ``why multiply?'' Why multiply? I didn't say, ``why multiply?'' OK. That was, maybe my question wasn't clear. Why multiply? Molly was giving us a reason why we multiply, right? We want to find the area . . . that's what you do when you find the area. You multiply. Why? 'Cause that's the way we've been taught. And it's an easy way to do it. I'm one of those people who doesn't believe anything unless he is persuaded. If we had to sit here and measure, go across, and do every one of these, we would miss our next class. (laughter)

This interaction and others suggested that not all members of the class understood the justification issue that Simon was putting before them and the progress on that issue represented by Molly's contribution. This was also evident in the students' journals. [Lois offers another explanation of why multiplication will always work, one compatible with Molly's. After trying to explain how each square in the left-hand column of Fig. 5 could be thought of as representing the three squares in each row, she turns to her classmates.] Lo: T: Ss: T: Lo: T:

I'm not making very much sense to you guys, but I know what I mean. (laughter) I think I understand what you're saying. I do too. Why don't you find out if other people do? Would anyone like to paraphrase that for me? (laughter) All right! (raises his arm in a cheer as students laugh)

While at the board, Lois' objective was to communicate with her classmates (``I'm not making very much sense to you guys''). By encouraging her to find out whether they understood, the teacher supported her interpretation of this as a discussion for the benefit of the whole class, rather than just for the teacher's approval. Her joking imitation of the teacher's language in requesting a paraphrase indicated her understanding of this tool. Analytic Summary. This discussion of their first mathematical work can be seen as the awkward meeting of members of different mathematics cultures. The teacher's questioning

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indicated that he expects students to justify the mathematical statements they offer. However, the students' responses showed that they were surprised by this and that their standards for justification differed from the teacher's. Where the school mathematics culture had accustomed them to expect ideas to be provided by the teacher, this teacher asked students to generate justifications based on their own reasoning. Although there was a range of responses, 9 of the 16 journal entries submitted indicated that the experiences of Class #2 differed significantly from the students' prior experiences in mathematics class. Not only do their comments offer a glimpse of the emotions around these new experiences, but they also provide evidence of which aspects of Class #2 were new to the students. One new experience was the lack of evaluative feedback from the teacher. This ran counter to the students' prior experiences of validation in a school mathematics culture, in which evaluation by the teacher is a regular and predictable part of classroom interactions (Mehan, 1979). Several students comment in their journals on the unusual role the teacher took in Class #2. Some enjoyed the new challenges they faced as a result. K: The factor that most influenced my learning was the kind of questioning by [the teacher] that allowed all thoughts to be acceptable, in his eyes, and allowed the group to examine them . . . An idea was never evaluated as good or bad but all ideas had an equal chance to be looked at. [This] made it tremendously easy to share any thoughts. A: (By) everyone being allowed to give input and ideas toward solving the problem . . . we are not just listening to a teacher and copying what he/she says or writes on the board. We are actually thinking throughout the entire class whether we answer out loud or not. The students are actually doing all of the figuring. Jo: In other math classes we were given lectures and had to try to teach ourselves the reasoning behind what was being asked of us to do. In our classroom now we are already allowed to show our opinions and . . . I do not have to worry about being thought of as stupid, [nor] worry about whether we are right or wrong. Others found this challenge unsettling as it could require some changes on their part. D: [Knowing] what everyone else's ideas on how to solve the problem were . . . expanded my outlook on the problem and it made me think about it in different ways. [The teacher] did a good job of making us think about what we had to do. It was very strange not to have him teach like other teachers but I think it is a good thing although it may take me a little while to get used to this method of teaching. Ln: I am not used to the class format where anybody can challenge your answer and speak out against you in a debate format. For still others, this change in the teacher's role evoked some concern. G: I felt a sense of frustration when there was no real answer given to the reason for why it was wrong or right to do the problem in the manner that we did. Ju: We never received a clear explanation of ``why'' that rule is true.

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The phrases ``there was no real answer given'' and ``we never received'' also indicate their conceptions of teacher's and students' roles: Teachers provide answers, students wait to receive them. These conceptions were echoed in several of the journals, although not so negatively. Dissatisfaction with and confusion about the teacher's refusal to identify correct and incorrect responses increased over time for some students and was explicitly discussed in later classes. These students had not been in the habit of offering justifications, so this was another new experience. Although the issue was not identified explicitly, the events of Class #2 began negotiation of the practice of justification. E: I realized today that when I was in grade school I was taught only by laws. Nothing had reasoning behind itÐit was just the way it had to be done. I felt as if Dr. Simon was dragging information out of our brains. He asked us that dreaded questionÐWhy? Br: I liked it better when we were told what was true and memorized it, it avoided a lot of confusion . . . I assume from the discussion that we are expected to be able to prove what we say is true. That would seem logical except for the fact that I don't know how to do it. As a foundation for mathematical inquiry, the teacher wanted to establish as norms that both teacher and students ask ``Why,'' and students expect to justify their claims. Rather than telling them that he would only accept ideas they could justify, the teacher attempted to have the students experience a need for justification. Ensuing small group talk and journals suggested that students had formed the impression that he might ask ``why'' in response to any statement made in class. Connected with negotiation of the practice of justification is negotiation of what counts as an acceptable justification. To convince themselves, the students may not have needed to consider whether their method ``always works.'' Instead, they may have responded to the teacher's question by attempting to figure out what he wanted. R: To me it seems like there is only one way to do [the rectangles problem] . . . I do not know what other people learned when they were in grade school but I learned you measure the length and the width and then you multiply the measurements . . . I still do not understand all the confusion and questions brought [up] in class about the solution. Several students expressed frustration with their inability to explain their thinking. Over time, students' justifications became more compatible with the teacher's expectation of deductive arguments (Simon & Blume, 1996). This will become evident as we continue the story through Class #15. The responses of Karen (K), Agnes (A), Deb, and especially Jonnie quoted above suggest that student input was also uncommon in their prior mathematics classes. Finally, some students seemed to recognize a lack of understanding in the mathematics that they repeated (e.g., Eve, Bridget, abbreviated by ``Br''), and their lack of authority in deciding what constitutes correct mathematics (e.g., Agnes, Eve). While these new experiences seem to have begun making past experiences problematic for some, other

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students' responses (e.g., Rebecca abbreviated by ``R'') suggested that they and the teacher had not communicated. Looking again at the mathematical discussion in Class #2, we see that the teacher initiated a process of negotiating new norms for talking about and doing mathematics. He did this by explicitly stating his expectations for group work and whole class discussion in terms of what he wanted them to be doing (e.g., explaining their thinking) and what he would be doing (e.g., asking for a paraphrase). New meanings were further developed as students encountered experiences that served both to uncover expectations they brought to class and to provide examples of the teacher's new expectations. We see this negotiation of classroom norms as a mutually adaptive process that may at first be masked by the asymmetrical power relation between teacher and students. Certainly, the teacher's and students' adaptations to each other were motivated by different intentions and took different forms. Students attempted to respond effectively to the teacher's questions, adapting their responses on the basis of the teacher's subsequent communications (explicit and implicit) and their evolving understanding of their respective roles and of classroom norms. In Class #2, the students seemed to identify early on that the teacher's expectations differed from what they were used to in previous mathematics classes. They were not sure what those expectations were and how to meet them. Thus, each cycle of teacher question, student response, teacher question led the students to further attempt to adapt both the nature and content of their responses. The teacher's adaptations were motivated by his commitment to negotiate classroom norms regarding the role of deductive justification, conceptual understanding, and student validation of ideas. Although he made a conscious attempt to influence classroom norms, he was nonetheless involved in a process of ongoing adaptation to the students' responses. His adjustments involved working to understand their mathematical understandings and the meaning that they made of his questions, finding ways to acknowledge their current ideas, and generating subsequent questions based on his understanding of their current ideas and meanings. Students participated in the definition of their roles through their attempts to interpret and meet the teacher's expectations. As they engaged in behaviors that the teacher found appropriate, his verbal or non-verbal support served to legitimize these actions and their underlying interpretations. For example, the teacher told the students that they should make sure they understood each other's ideas. That he did not interfere when Sara questioned Molly's definition of a group supported his stated expectation. In an instance not included here, Penny described her group's work and mentioned an answer of 13.88 rectangles. Bridget asked her how they had gotten such a precise answer with only a cardboard rectangle with which to measure. Afterwards, when she explained her question saying she was ``just curious,'' the teacher responded, ``It's a good question.'' The teacher thus supported Bridget's action first by not interfering in the discussion and then by praising the question. This example contrasts with the practice of school mathematics in which students are often encouraged to ask questions (``There's no such thing as a dumb question''), but subsequently choose not to do so when the teacher responds to their questions with, ``Who remembers what we said about that?'' Just as the students' role was defined both directly and indirectly, aspects of the teacher's role were also defined in the discussions. For example, the teacher stated his intentions to monitor students' understandings during class discussions by requesting

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paraphrases. He followed through on this and showed a belief that his role included assisting students in their efforts to communicate with each other. This was evident when he assisted Molly in revealing those aspects of her thinking that he suspected were unclear to some of her classmates (e.g., what she meant by a ``group'') and encouraging students to seek better understanding for themselves as shown by his praise of Bridget's question. We note that this was the only praise offered in the entire class period. Rather than using praise to evaluate students' responses, this teacher used praise to indicate particular types of mathematical activity that he valued, hoping to guide students to redefine what would constitute effective participation in this classroom. 4.3.

Interactions Involving the Journal

Between classes, throughout the semester, the teacher read the students' journal reflections on the previous class and commented in writing on individual entries. With Class #3, he established a routine of opening each class with comments about issues relevant to the whole group that arose in the journals. We came to view this teacher talk about journal themes as part of a process of negotiating mathematical and social practices appropriate for competent participation in this community: Students participate in a class activity; students write a journal entry; the teacher responds individually and to the group based on the journals; students adapt their classroom activities and journal writing in response to his comments; the teacher responds again to students' activities and writings. The teacher's public responses addressed students' and teacher's roles, students' understanding of mathematical content, students' feelings about this mathematics course, the goals of the course, and the importance and meaning of articulating mathematical ideas. Sometimes the teacher responded in action rather than words. We illustrate the function of interactions involving the journal in the following example in which the teacher and students negotiate their respective roles. Discussion of the mathematical ideas raised in Class #2 continued in Class #3. Afterwards, many students wrote that they were frustrated and confused. Journals after Class #3. G: One thing I find very frustrating is that we were never really given an appropriate answer. I feel we need some kind of reinforcement. B: This is when the frustration began . . . When you asked the students how sure they were with the answers presented, everyone raised their hand, including me, but when called upon to give an explanation of the problem no one was confident with the answers. Jo: I was pretty confident I knew how to explain it, but once everyone began to give their explanations, I became confused and a little frustrated . . . the students' ideas and their way of wording the ideas confused me . . . and began to frustrate me. In response to this expression of frustration over the teacher not giving indications of the correct mathematics, the teacher initiates a game and discussion. Class #4. [The teacher introduces ``What's in the Bag?'' In this game, students figure out what the teacher has hidden in his bag by asking yes/no questions about the object, but not

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questions that name what is in the bag (e.g., ``Can you measure with it?'' but not ``Is it a ruler?''). When some students claim that they know what is in the bag, yet want him to show it to them, the teacher facilitates a discussion as to whether it is necessary and constructive to do so. In this context, he discusses his refusal to evaluate their mathematical ideas.] T: I want you to begin to know things in mathematics not because I tell you they're so . . . but because you understand it so well that you're sure this is the way it works. . . . Now, there's a problem with this, which is that most of you have never had this experience before in mathematics . . . I mean if I said, ``This is the way you add these numbers . . . everybody got it? OK, do 15 of those,'' . . . you would know what to expect. But I ask you to ask yourself the question of how well that worked for you. Did you come away with a solid understanding of the mathematics that you studied up 'til now? Journals after Class #4. Sa: Today's class left me confused and frustrated . . . I feel as if I'm being pushed too far and that I can't go any further . . . In the last journal I wrote, I was feeling confident in what I knew and my ability to do math. Now all my feelings have gone to the other extreme. In the next class, I hope to get a grip on my ideas and the ideas of the class. I hope to begin to understand my thoughts and my classmates' examples a lot more clearly. E: I felt more comfortable in class today because I realized that I am not the only one who leaves the class confused and still thinking about the problems. I got frustrated because there seemed to be no right or wrong answers, but I guess that's what makes us keep thinking about the questions and trying to come up with some reasoning. B: At this point, I really don't think I'm learning anything, at least in the form of mathematical concepts. Explanations from other students give me somewhat of a vague understanding on why a problem is done the way it is. At your request, I'm going to try and participate more to see if my level of understanding rises. In response to these journals, the teacher initiates another discussion of the game. Class #5. [Simon asks students to consider what would happen if he played this game regularly with his students with one change: What would happen if each time they played the game he told them or showed them whether or not they were right? Students offer a variety of opinions from ``students will cease to ask questions,'' to ``students will learn how to ask better questions.'' The teacher temporarily closes the discussion by giving examples of occasions in the real world where no authority can judge whether an answer is correct. He then makes some suggestions about how students can learn in this situation where the correctness of their ideas is not evaluated by the teacher.] T: Your job, and the way you can be successful in this class and successful in learning mathematics, is to make sure that you understand whatever we're doing. And if you don't, be clear with what you don't understand. Most times we're going to come back to it in future classes, but don't let us get off that topic until you're satisfied you really understand it thoroughly.

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Analytic Summary. In the cycle we described here (Classes #3±5), the teacher responded (in Class #4) to communications in the students' journals with a game followed by explicit discussion of his reasons for taking a non-evaluative stance, one that he acknowledged was discomfiting. In their next journals, some students responded directly to this by articulating their understandings of the reasons he gave, while others continued to express dissatisfaction. In Class #5, the teacher responded again to the journals, pushing a little harder on his point and becoming still more explicit about his expectations of them as learners of mathematics and as participants in a clash of mathematics cultures. This negotiation of whose role it was to determine mathematical validity continued over much of the semester, albeit less explicitly. The teacher's efforts to respond to students' concerns as they arose played a significant part in the process of negotiating new classroom practices because these concerns indicated incompatibility between the students' and teacher's perspectives. In school mathematics, confusion is taken as an indication of failure on the part of either teacher or students. By attending to the students' frustration, the teacher contributed to the negotiation of new meanings for such experiences. The teacher also showed he was willing to adapt class sessions to address students' ideas, confusion, and emotions. Students seemed to cooperate tentatively with his expectations as they tried to make sense of events in this class. In summary, interactions involving the journal contributed to the negotiation process in a number of ways, including raising issues of affect, uncovering differing interpretations of class activities, and clarifying the purpose of journal-writing itself. The journal served as an important tool in the overall process of negotiation in this classroom community for several reasons: (1) Every student had a voice in this part of the negotiation. (2) The teacher's comments were explicitly identified as responses to students' communications. (3) The journal provided the teacher with student responses to classroom activity beyond what he gleaned during class sessions. (4) The journal had the effect of uncovering students' differing interpretations of classroom events. (5) The teacher used this tool to respond to individuals and to the class as a whole. An implication of this last point is that the private negotiation between teacher and individual students can contribute indirectly to the negotiation of classroom norms and practices. These private negotiations had an effect because the teacher and student reoriented their activity (to each other) in these negotiations, and because there is a reflexive relationship between individuals' conceptions and patterns of classroom interaction (Voigt, 1995). The journal and teacher responses played a key role in the initial stages of this negotiation. 4.4.

Interactions Involving a Paradigm Case

As the negotiation of classroom norms and practices proceeded, the teacher recognized particular opportunities to influence this process, situations that he identified as demonstrating a shift toward the types of mathematical activity that he sought to promote. By calling the students' attention to such situations, what Cobb et al. (1989) referred to as ``framing paradigm cases,'' he was able to clarify and provide a concrete example of productive mathematical interaction. One such instance occurs in Class #4, when for the first time the students spontaneously initiate a mathematical exploration.8 We describe the mathematical activity of Class #4 and

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FIGURE 6. The ``blob.''

then illustrate how the teacher frames these events as a paradigm case of desirable mathematical and social practice in Class #5. Substantial analysis of these events is included in order to illustrate aspects of the constitution of several different classroom norms. Class #4. [In a discussion about the meaning of area, the teacher spontaneously poses this problem:] T: How many different ways can you generate to find the area of Fig. 6? [The students talk about this in their groups, becoming very engaged in their discussions. When the teacher calls the class together, he expresses pleasure in what he has observed.] T: There's some wonderful mathematical discussions going on. There are enough interesting questions being brought up to keep us going for a long time . . . [He polls the groups, asking how many have at least one, at least three, and at least five methods for finding the area of Fig. 6. He proposes a brainstorming approach.] Let's hear some of the ideas. Let's just get them up [on the board]. Now, some of these may work and some of these may not work, right? Let's just get the ideas up there, we won't decide yet whether they work or not, OK? Lilly? Li: OK. We had a big debate about whether or notÐit's not a consensus in our group whether this works. But, if you took some kind of rope or something, and measured the whole outside of the area, and then pulled that out into a shape like a rectangle or a square, you'd get the area. But what that led to was [Students start talking in their groups.] T: (to Lilly) Hang on a second, it seems like people want to discuss that. (to the class) Go ahead, discuss it for a minute. [The students request string, measuring tapes, rulers, and other materials, and the teacher searches for and distributes these materials to the students as they are requested.] Although the teacher intended the class to hear all methods before evaluating any, the students launched an investigation of the ``string method.'' The energy level in the class

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was very high, and the teacher's support of this student initiative served to legitimize their spontaneous investigation. [When the teacher reconvenes the class discussion, Lilly shares with the class a new question that has emerged in her group regarding the string method.] Li: It seems like it's a difference in what area is and what volume is to do it like this. T: What area is and what volume is? Li: Because (inaudible) it seems like it does change, [pulling the string into another shape] does change the amount of area that's in there. T: So [the string method] wouldn't work? Li: So I think we're feeling like it wouldn't work, but we're still questioning why because of the whole idea of conservation. Well, Molly, show your example of this. This is great. Here, Molly swayed us here with her M: (using three soda cans) They were saying, ``Well, it has to be the same because if you have three cans stacked, if you put three cans [next to each other], it's still the same thing.'' But I said, ``Well, you really aren't changing the outline of this can when you stack three cans up this way.'' Or, I mean, it is changing a lot when you pull this [rope] up, well, it's not going to hold as much if you squeeze it closed (holds up a crushed can). I mean, that's the difference between volume and area. This, you know, we were trying T: Does anybody understand what they're saying? This question reinforced the teacher's commitment that everyone understand the contributions of others. Ss: No. (Bridget indicates that she understands.) T: Bridget, can you say it in your own words? Br: She's saying that if you stack them up three ways, up or to the side, it's going to be the same length, but, you know, if you squeeze T: Same amount of soda. Br: Right, but if you squeeze the can, it's not going to hold as much, whereas this rope, it's just much more flexible. There's more flexibility, you can change it around. I understand though. M: I don't think so. I mean, I'll try to say what I meant . . . it's just the opposite of the rope, it's not going to hold as much. It's not going to have the same area, I think, if you pull it apart, as if you, all right . . . imagine this is the whole [soda] can, OK? When you scrunch it up, it's not going to hold as much. You know what I mean? It's not going to be the same area, but if you, you know, stack three high . . . it's a different thing you're talking about. It's not this. (pause) That's basically what I think. It is a difference. You can't just stretch something out and say it's the same thing as what it was when it was scrunched up. [Deb comes to the overhead projector to address the validity of the strategy using a piece of rope as the perimeter of the blob in Fig. 6. She tries to illustrate how rearranging the rope would change the area.]

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D: The area is what's inside the rope, OK? So if you're changing, like this is narrow here, and if you're changing that to move it out here to get something straight to find out what the area is, you're going to make this part in here bigger. K: But would another part be smaller compensating for it somewhere else? [Deb is unable to respond convincingly and the discussion continues.] The issues of justification that arose here are discussed in detail in Simon and Blume (1996). In Class #5, the teacher highlights the events of Class #4 as a paradigm case of appropriate mathematical activity. His comments encourage students to interpret their initiative and independent action as appropriate. Class #5. [The teacher opens class with comments on the previous exploration.] T: I find myself really looking forward to this class, and particularly because of the neat things that you all are doing each class, I look forward to seeing what's going to happen. My favorite class was the last one . . . because you all just kind of took over with something that you wanted to explore. In fact, if you remember, when Lilly made her suggestion about putting the rope around the pond or the blob or whatever you want to call it, I was saying at the time, ``We just want to brainstorm . . . We're not going to discuss whether they work or not.'' You all were so eager to figure out whether that worked or not, that I just had to give in and say, ``Go do it.'' And . . . there was really a feeling that . . . ``we can figure out whether this works or not.'' . . . That was exciting because you were really involved in a mathematical exploration. You wanted to know the answer and you knew you could figure out how that worked. And you may still have some questions about it; that's fine. It's not a matter of whether it's resolved instantaneously, but that's the kind of activity in here that makes me excited. Analytic Summary. Negotiation of norms and practices is ongoing and often implicit in everyday classroom mathematical activity. The community develops expectations and modes of operating, often without the conscious awareness of participants. Meanings are established as a result of community members' interpretations of classroom interaction. Brousseau (1983) calls this largely implicit process the establishment of the contrat didactique (didactical contract). Implicit negotiations can result in definition of what is valued, what particular behaviors mean, what it means to be effective, and what constitute the various roles of classroom community members, in short, the characteristics of a classroom microculture. The activities of Class #4 supported development of key norms in several ways. The teacher not only allowed, but also encouraged the students' disruption of his plan to generate a list of ideas. When he began gathering materials at their request, he encouraged students to pursue their investigations. These actions showed that he valued discussion of mathematics, curiosity, and initiative. Taking a poll of the number of ideas before hearing any of them had the effect of focusing attention on the creativity and multiplicity of solutions. There was also development along some of the themes raised in Classes #1±3.

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Students were beginning to evaluate and value each other's contributions: When Lilly called on Molly to share her idea with the whole class, she recognized a powerful argument and her own role in the public airing of such arguments. Similarly, some students were beginning to see themselves as determining whether an argument is valid: Molly and Bridget, then Karen and Deb, engaged each other in public debate, indicating a belief that this was appropriate behavior. The excerpts from Classes #4 and 5 illustrate the interplay of implicit and explicit negotiation of classroom norms and practices, as well as the specific role of framing paradigm cases. Although experience itself can have a notable effect on community members, explicit highlighting of a paradigm case of mathematical activity has potential to be a still more powerful means of affecting classroom practice because it brings together implicit understandings with officially sanctioned interpretations. The teacher framed the experiences of Class #4 in a particular wayÐpointing to the students' changing relationship to mathematics (``We can figure out . . .''). In this way, he prompted and guided reflection on and interpretation of their prior activity and of his response to it (Cobb et al., 1989). 4.5.

Interactions Involving a Discrepant Event

Our analysis of negotiations prompted by interactions involving new experiences, journals, and paradigmatic cases has been structured chronologically as well as thematically. Looking across episodes, there appears to be a growing understanding among the students that the main source of mathematical information is students' ideas and that all ideas are subject to possible refinement or disagreement. However, ``because of the ambiguity and different background understandings in the classroom, the negotiation of meaning in the microsituation is fragile . . . there is a permanent risk of a collapse and disorganization of the interactive process.'' (Voigt, 1995, p. 178) Such instances of communicative breakdown are important sites for analysis precisely because of their power to reveal ``different background understandings.'' As long as interaction proceeds smoothly, participants tend to assume shared meanings, norms, and practices. Events that expose a lack of fit among participants' perspectives play a significant part in the ongoing negotiation of classroom mathematical activity and of the roles of teacher and students. Discussion of the first exam was such an event, providing evidence of different perspectives that were discussed explicitly, as well as aspects of those perspectives that remained implicit. Class #7. [The teacher opens class by announcing the first exam. Although the class gasps, they discuss the day's mathematical task until Georgia interrupts her own contribution to the mathematical discussion with the following comment.] G: I know you won't do this but I kind of wish you would just like tell us. I mean because, can we share concerns? . . . this is a 3 credit course for me, and it's a grade, and . . . right now I am dumbfounded about this . . . I don't see usÐmaybe I have a mental block, I don't knowÐbut I just don't see us going forward. I kind of just see us stopped . . . I am so frustrated! T: Do you know what it is you'd like me to tell you?

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G: I don't know, I mean I just wish T: I'll tell you something. G: that it would be normal. I mean, (laughter) I realized that when I signed up for this class I had no idea. T: No idea. (to the class) Did anybody have any idea? Just be honest. Did you have an idea what you were getting into? Georgia's outburst was tacitly accepted through the teacher's responses to her. First, he asked if she knew what she wanted, then he accepted her comments by drawing in the rest of the class. Lois raises her hand to offer a comment that has ``to do with the rectangle,'' assuming the teacher will brush off Georgia's comments and resume the mathematical discussion, but he does not. His reply, ``OK, hang on to what you want to say,'' tacitly affirms Georgia's right to bring up her feelings in the class discussion. [Sara announces that she is ``going to help'' Georgia express her concerns.] Sa: I think her concern is coming from you told us we were having the exam in a week. G: . . . even though I'm the onlyÐwe're the only brave ones to say . . . well, what's that exam going to be on. And then we look at the grading system and it's like 46 is still a D [according to the grading scheme on the syllabus] . . . , and we're like, ``Oh, my God!'' (laughter) T: OK, for how many of you was there a shift in your experience here [with announcement of an exam]? (many) OK. I was afraid of that . . . Let's quit 10 minutes before the end of the period and talk about how one might think about this exam in this abnormal situation. Let me just say a couple of things about it. Number one is, I anticipated that as soon as I mentioned exam that was going to change the comfort level in here. And I don't know how to not do that . . . I can help you be more relaxed about thinking about the exam, but if I had my druthers there would be no exams. There would be no grades, and we could just be involved in the experience of learning and teaching. But that's not the real world we live in. It's not the one I teach in. It's not the one you're likely to teach in. So we're going to have to bring those worlds together. So 10 min before the end of the period, you make sure that I stop, and we will talk about what you might expect from the unexpected. [When they return to discussion of the exam, the teacher describes the form of the exam by comparing it to the students' past experiences with mathematics. For example, they will be called on to explain their thinking as they have done in class, rather than to calculate. A student asks if the exam will be like the ``worksheets'' of problems done in class.] T: Right. It's going to ask you to write something that's going to show your understanding or lack of. (laughter) (something inaudible) G: How can you grade our understanding? T: How can I grade your understanding?

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G: Yeah. That's kind of like our opinion. T: That's kind of like your opinion? I don't think so. Let me see if I can think of a colorful way to answer that question. When I was a kid, Georgia, I believed that trees caused wind and moved back and forth and that's what made us feel wind. I mean, do you think that I understood wind very well? Ss: (laughter) No. T: Yeah. So you have some basis of evaluating my understanding of wind. You might have understood why I thought that. Toni tries to fit the teacher's comments into her experience of the course to date. To: So if we generate things like we did in the classroom, but not necessarily come up with a correct answer then we're kind of on the right track for what you want? T: Well, the problems are not going to be arbitrary. The problems are going to have to do with the specific understanding that you dealt with in the problems that we have dealt with so far. Now I'm not going to ask you the exact same problem. Don't expect to meet with Luisa and Ruiz [from a previous problem]. (laughter) OK? To: But I don't think T: But Hector and Molly may have a problem. And they will have a problem which will have the same basic understanding as understandings that we dealt with in the original tiles problem, one of the . . . five or six questions that I gave you [and] the question on the transparency. OK? It's not that I'm going to bring in . . . new stuff to test you on. [After commenting on its form, he describes some ways to study for the exam.] I would review each of those problems and ask myself, ``do I thoroughly understand the mathematics of this problem?'' and if I don't, then spend some more time with it, maybe talk to colleagues. I would pose questions in class. Eve? E: This is going to sound really stupid. But you said, ``Ask yourself if you thoroughly understand it.'' But I don't know if I thoroughly understand because I never was told if that was the correct understanding, if my understanding was a correct one. Does that make sense? I mean, I think I'm right, but I don't know if you think I'm right. The teacher comments on the complexity of the assessment situation in this atypical class. T: Let's try to make this more concrete. These are good questions that you're asking. And they're difficult questions that kind of get skimmed over when students and teachers play the game of . . .`` I'll tell you what you should know, you memorize it, and you give it back to me on the test.'' It makes it all simple. [When the teacher asks Eve for an example, she replies that she thinks she understands ``why multiply,'' but with a qualification that refers to an earlier discussion not included here.] E: Remember how we were talking about the overlapping [of rectangles] and people were agreeing, and people were disagreeing, and all this stuff? But what if

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somebody wrote, ``Well I think you could overlap these rectangles,'' and articulates it in a way that they really thought it would happen. Then would you mark it wrong or, you know what I mean? T: OK. First of all, the issue about the overlapping when I rotated the rectangle . . . I contend that we haven't gotten to the bottom of that yet. OK? But I think we're making progress on it. You may not have noticed, but we're doing some work that's going to allow us to get back to this, I think, with some new insights . . . . Anything that we have not resolved in this class, I will not ask you about. E: Oh, OK. T: So I'm not going to ask you about the rotated rectangle unless we get to the bottom of that. I think we've gotten to the bottom of the string around the blob. Eve's question about assessment of understandings that are well supported and clearly argued showed that she perceived the emphasis in this class to be on providing sufficient evidence and convincing argument. She asked if a good argument would be good enough to be judged correct, or if a correct answer would also be necessary. Eve seemed satisfied with the teacher's reply that only resolved issues would be included on the exam. It is not clear whether she found this approach acceptable, was satisfied to know that the particular topic that she brought up was not going to be on the exam without more resolution, or simply did not know how to advance the discussion. Analytic Summary. The students seemed to view the announcement of an exam as an event that did not fit with the classroom microculture as it had evolved to that point. The ensuing discussion uncovered discrepancies between teacher's and students' views of their roles. Discrepant events such as this are significant to the development of a classroom community precisely because they raise issues for discussion, and thus for negotiation of meaning. The issue of grading caused teacher and students to struggle to connect this vestige of traditional teaching with their respective current views of interaction in this classroom. During the first few weeks of the semester, the students' notions of teacher had shifted from someone who imparts knowledge and arbitrates validity toward someone who poses problems and facilitates discussion. Based on their interpretations of this teacher's participation in class discussions, the students had come to believe that he would only evaluate the appropriateness of their activity and not the content of their contributions. Having experienced non-judgmental responses to their classroom contributions, students had come to believe that effective participation consisted of honest expression of ideas and their rationale. There was no stigma attached to ideas that were shown not to work, and the teacher did not get upset with students who did not understand. Thus, the mathematical content of students' ideas was not seen as a key factor in effective participation. The notion of an exam that would be used to evaluate and grade their understandings, therefore was seen as counter to the evolving microculture. Students wondered how the teacher, whose (redefined) role it was to accept all answers without judgment, could now judge their answers. Differences in prior experience, knowledge, and motivations between teacher and students make communication problematic. At this point in time, students were aware of their own learning about mathematics, but not aware of their learning of mathematics.

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When the teacher asked the group in Class #6 (not included here), what ``big ideas'' they had learned from the blob investigation (Class #4), their responses did not address the mathematical concepts underlying their activity (e.g., conditions for invariance of area), but concentrated on problem-solving strategies they had learned and a recognition that mathematics can be active and creative. The teacher's intention, however, had been to engage them in understanding particular concepts related to area. Much of the incompatibility of interpretations apparent in discussion of the exam can be traced to this point: Students did not recognize that they were developing significant mathematical understandings, and because they did not focus on their conceptual growth, the exam as described by the teacher seemed inappropriate to them. Though he attempted to respond to students' concerns and to guide their interpretation of his decisions about the exam, the teacher was not able to span this gulf in perspective about what was being learned in this mathematics class. Because he did not provide explicit validation of ideas, there was no one to say, ``Here are the big ideas'' and ``This is the correct version.'' Furthermore, as in the example of the wind, the adult can see that the child's idea of wind was a poor understanding, but the child cannot see it for himself. In fact, one can only become aware of the flaws in one's understanding after one learns more. Eve voiced a thought probably held by many students, ``I think I'm right, but I don't know if you think I'm right.'' Although she had some confidence in her understanding of some of the ideas they had discussed in class, once the situation involved grading, she found her own judgment insufficient. Her comment is quite reasonable in that she knew that it was the teacher's judgment, rather than her own, that would determine her grade on the exam. In fact, she had put her finger on a fundamental paradox in this kind of teaching: While the teacher's non-evaluative stance was intended to encourage students to develop confidence in their own ability to validate their ideas, he was also engaged in evaluating student understandings to guide his pedagogical decisions and to fulfill his responsibility for grading. Although other important events took place between Classes #7 and 15, our analysis suggests that they are well represented by the types of examples that we have presented. To accommodate space limitations, we have chosen examples from the first seven classes because these early experiences seemed to be the most formative.

5.

CONCLUDING DISCUSSION

We see the constitution of a classroom microculture that supports inquiry as fundamental to realizing the goals of the current mathematics education reform. Through participation in such a classroom community, students (and prospective teachers) can develop a new relationship with mathematics. Cognitive development of the type envisioned requires a social context in which everyday activities support such development and in which productive mathematical activity is self-perpetuating. In contrast to studies that focus on the learning of individuals to think differently about mathematics (e.g., Schifter, 1994; Schram & Wilcox, 1988; Simon & Brobeck, 1993), this work emphasized the evolution of a group's interpretation of words and actions in mathematics class and the establishment of patterns of interaction that support inquiry mathematics. For prospective teachers, participation in an inquiry mathematics culture is important because such participation can provide them a foundation for promoting mathematical inquiry in their own future

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classrooms as well as assisting their individual mathematical and pedagogical development. Can teachers' participation in a culture of inquiry mathematics result in the same degree of impact on their beliefs about and understandings of mathematics as their past experiences in school mathematics? In our analysis, we came to see the teacher's and students' differing perspectives as representative of differences in the cultures of inquiry mathematics and school mathematics. The analogy of a culture clash helped us to make sense of the tensions, misunderstandings, and interpretations of the other that were characteristic particularly of the early class sessions. This provided an organizing structure for thinking about the mutually adaptive process by which the group constructed norms and practices that support mathematical inquiry. Furthermore, this analogy served to highlight the interrelationship of mathematical and social understandings. Challenging students' ideas and justifications not only provoked cognitive reorganization of mathematical concepts in many students, but also prompted reorganization of their perceived roles in mathematics class. By characterizing the different types of interactions that figured prominently in the negotiation of classroom norms and practices (interactions involving new experiences, the journal, paradigm cases, and discrepant events), we were able to organize an instantiation of this process and provide images of how a teacher (teacher educator) consciously and intentionally participates in this key aspect of promoting inquiry mathematics. It is in so doing that we have attempted to extend the work of Cobb et al. (1989) and Cobb and Bauersfeld (1995). Finally, our analysis demonstrates the usefulness of the interactionist perspective in accounting for the transformation of the mathematics classroom experience. Our analysis of classroom interactions provides numerous examples of how norms and practices do not change simply by virtue of the teacher using his authority to assert a new set of rules accompanied by student compliance. By examining interactions that occurred over time, we were able to illustrate the role played by non-traditional teacher routines and by ongoing negotiation (mutual adaptation). In this era of reform, teacher educators regularly attempt to engage traditionally prepared student teachers in mathematical inquiry. As they do so, the potential for culture clashes exists. This culture clash is echoed in schools where elementary or secondary teachers have one perspective on mathematics, and students have another. Skemp (1976) describes this as a ``mismatch'' of relational teaching with instrumental understanding (or vice versa), but a cultural perspective takes more than the goals and motivations of the participants into account, it encompasses differences in roles, and meanings of interactions, actions, and values. Because we believe that the negotiation of new classroom norms and practices for school math will play a key role in the ultimate success of these reform efforts, it is imperative that we understand these processes and ultimately prepare teachers to participate in them effectively. Acknowledgment: Excerpts of this work appeared in the Proceedings of the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, edited by David Kirshner and in the Proceedings of the Nineteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, edited by Jane Swafford and John Dossey. The CEM project was supported by the National Science Foundation under Grant

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No. TPE-9050032. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We would like to thank Hilda Borko, Ron Tzur, Michael Beeth, and Paul Vellom for their comments on earlier drafts of this article.

NOTES 1. Simon and McNeal (in preparation (a),(b)) provide detailed analyses of the development of particular norms that emerged in this classroom community. 2. Pimm (1994) also describes teachers' meta-comments as they attempt to initiate their students into new ways of doing classroom mathematics. 3. Our reference to just two cultures of mathematics reflects the coherence we found in students' expectations and interpretations of classroom activities (despite possible variations in their prior classroom cultures) and the clear distinction between these and the teacher's expectations and interpretations. 4. With Ball (1991), we distinguish what is learned of mathematics, meaning particular mathematical content, from what is learned about mathematics, meaning the nature of mathematics as a specialized human activity. 5. The journal assignment may have made sense to those students who expected the course to focus on teaching, rather than on mathematics. Confusion about the goals of the course was explicitly discussed in Class #3. 6. From the videotape, we interpreted ``you'' to mean any member of the class. 7. Six of 16 journals after this class commented on the relaxed classroom atmosphere. 8. This was not the only occasion on which the teacher framed particular actions as examples of desirable mathematical activity. For instance, the class was reviewing solutions to the test problems (in Class #10), the teacher called attention to one student's solution saying, ``Notice the problem solving that's gone into that.''

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