THE JOURNAL OF THE LEARNING SCIENCES, 16(1), 37–79 Copyright © 2007, Lawrence Erlbaum Associates, Inc.

Fast Kids, Slow Kids, Lazy Kids: Framing the Mismatch Problem in Mathematics Teachers’ Conversations Ilana Seidel Horn College of Education University of Washington

This article examines the social nature of teachers’ conceptions by showing how teachers frame the “mismatch” of students’ perceived abilities and the intended school curriculum through conversational category systems. This study compares the conversations of 2 groups of high school mathematics teachers addressing the Mismatch Problem when implementing equity-geared reforms. Although East High teachers challenged conceptions that were not aligned with a reform, South High teachers reworked a reform mandate to align with their existing conceptions. This research found that the teachers’ conversational category systems modeled problems of practice; communicated assumptions about students, subject, and teaching; and were ultimately reflected in the curriculum. Because East High teachers supported greater numbers of students’ success in advanced mathematics, this study considers the relation between teachers’ understandings of student learning and the success of equity-geared math reforms. In addition, this study contributes to the understanding of how teacher conceptions of students are negotiated and reified in context, specifically through interactions with colleagues and experiences with school reform.

Several years ago, I was invited to conduct a workshop on mathematics reform for a group of school leaders. I asked a high school mathematics teacher to present with me. I had met Guillermo Reyes1 while conducting research on teacher learning in high school departments. As a co-chair of East High School’s mathematics department, one of the two in the study, Guillermo impressed me not only with his skill as a Correspondence should be addressed to Ilana Seidel Horn, College of Education, University of Washington, 115 Miller Hall, Box 363600, Seattle, WA 98195–3600. E-mail: [email protected] 1All

proper names of schools and participants are pseudonyms.

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teacher, but also as a mentor to other teachers working to reconceptualize their practice. During the workshop, we demonstrated a typical East High mathematics lesson, modeling a groupwork structure that involved a complex, open-ended task and emphasized strategies to support more equitable student participation in the mathematics. Afterward, one of the participants approached me. I knew him from my research as well. He was the Reform Coordinator of South High School, the second school in the study. He complimented me on the session, telling me how much he had learned from our presentation. Then he said almost desperately, “This is exactly what our math teachers need to be doing! Why aren’t we doing this?” It was an interesting moment, and one in which my dual roles of ethnographer and educator were in conflict. He voiced a question that our research team had puzzled over during the entire course of our 2-year study, and yet my obligations to my participants’ confidentiality disallowed me from pursuing the question with him. Nonetheless, I shared his bewilderment. How was it that East’s mathematics teachers, housed in a fairly typical urban comprehensive high school, had managed to make such impressive strides in improving their instruction, whereas their counterparts at South, a school widely recognized for its progressive orientation, remained committed to less effective teaching methods? Although there are certainly multiple answers to this question, the one I take up in this article focuses on the conceptual resources high school mathematics teachers bring to their encounters with equity-geared reforms. In educational research, such conceptual resources have often been investigated through the framework of teacher beliefs (Nespor, 1987; Stodolsky & Grossman, 2000; Thompson, 1992). This literature provides evidence that teachers’ beliefs influence classroom practice. To use an example relevant to this analysis, a common belief among teachers is that higher order thinking is not appropriate in the instruction of low-achieving students (Zohar, Degani, & Vaaknin, 2001). Such a belief may have enormous instructional consequences, as it constrains the field of possible actions that teachers might take while planning or enacting their lessons. Because beliefs seem to be a critical in shaping teacher practice, as a field, we need to better understand how teachers construct their beliefs (Thompson, 1992). In this analysis, I take up this question using a sociocultural framework of teacher learning (Lave & Wenger, 1991; Wenger, 1998). This framework requires an analytic shift away from beliefs as they are held by individuals and a move toward conceptions as they are distributed across individuals and settings, leading me to the investigation of teacher communities. Specifically, I examine the ways in which conceptions of students, subject, and teaching are embedded in teachers’ daily work, particularly as they encounter problems of practice and work to solve these problems in consultation with colleagues. This article begins with a description of the conceptual framework that informs this study. I then provide an account of the relevant educational context by describing a commonly felt teaching problem: the Mismatch Problem. This is

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followed by a report of the study context, research design, and analytic methods. Two cases of professional conversations are then presented and closely analyzed. The article concludes with a summary of the findings and discussion of the educational implications. CONCEPTUAL FRAMEWORK A situated perspective on learning assumes that people learn as they participate in the socially and culturally organized practices of a community (Lave & Wenger, 1991). I focus on the context of teacher learning through collegial conversations by bringing it together with two other strands of research. The first is research on teacher communities. This line of inquiry has sought to understand the location, nature, and dynamics of collegial interactions within school contexts. Second, because talk is one of the primary activities teachers engage in while working with their colleagues, I draw on studies about the resources for learning embedded in everyday talk, particular the conversational category systems that shape what people perceive in the world. Teacher Communities as Communities of Practice An abundance of research exists on teacher community, and yet it must be reconceived somewhat to align with the notion of community of practice. For more than two decades, teacher communities at the school level have become important sites of investigation in studies of teachers’ work. Increasingly, researchers have grown to appreciate the way that such subgroups can create fundamentally different settings, even within the same school (Ball & Lacey, 1980; Little, 1995; McLaughlin & Talbert, 2001; Siskin, 1994). In the high school, the most salient communities tend to reside at the level of subject-matter departments (Little, 1995; McLaughlin & Talbert, 2001; Siskin, 1994). Prior work on teacher communities suggests that teachers’ colleagues can influence teachers’ approaches to classroom practice (Coburn, 2001; Gutiérrez, 1996; Johnson, 1990) as well as shape their responses to reform (Ball & Bowe, 1992; Coburn, 2001; Little, 1995). However, we are only beginning to understand precisely how these responses get negotiated within teacher communities (Little, 2003). In my research, I have located resources for learning within teacher communities that provide teachers with a conceptual infrastructure—that is, linguistic and semiotic resources for learning and sense making about their work (Horn, 2005). In this analysis, I examine the ways in which conversational category systems constitute an important component of that infrastructure. Turning to Wenger (1998), I adopt a community of practice lens through which to view the mathematics teacher groups in this study.

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This perspective provides a more inclusive notion of community than those developed in the teacher community literature, as it remains more neutral about what constitutes a teacher community. In the teacher community literature, community is a term typically reserved for rarefied places in which teachers collaborate productively toward a common goal—be it the bolstering of conventional practice in “strong traditional communities” (McLaughlin & Talbert, 2001) or learning from practice in “inquiry communities” (Cochran-Smith & Lytle, 1999). Contexts in which teachers do not strive together toward a common vision of teaching but, for example, only “behave as if [they] agree,” have been distinguished as “pseudocommunities” (Grossman, Wineburg, & Woolworth, 2000, p. 13). According to Wenger (1998, p. 73) the three dimensions that characterize a community of practice are (a) mutual engagement, (b) a joint enterprise, and (c) a shared repertoire. This is a broader conception of community than the one put forth in the teacher community literature because it could, for example, potentially encompass the pseudocommunities that have been set apart in the teacher community literature. That is, the imperative to “behave as if we all agree” would not disqualify a group from being viewed as a community of practice. Instead, such a norm of interaction could be taken as part of the shared repertoire that competent members develop to mutually engage in the group’s joint enterprise. Wenger (1998) argued that communities of practice are an important unit of analysis in investigating learning, as such communities play an important role in meaning making. In particular, he described how meaning is negotiated in communities of practice through the complementary processes of participation and reification. Participation refers to “a process of taking part and also to the relations with others that reflect this process” (p. 55). Participation encompasses conflictual, harmonious, intimate, political, competitive, and cooperative relations among people. The complement to participation is reification. By reification, Wenger referred to the ways that we treat abstractions as if they substantially exist in the world. Such reifications represent aspects of a community’s joint enterprise, as “any community of practice produces abstractions, tools, symbols, stories, terms, and concepts that reify something of that practice in a congealed form” (p. 59). Together, these processes contribute to the making of meaning. Whereas participation facilitates local interpretation of reified ideas among members of a community, reification helps compensate for the potentially confusing fluidity and local specificity that arise out of participation. Between Participation and Reification: Resources for Learning in Everyday Talk How might teachers’ communities of practice support their learning? To pursue this question, I turned to studies of learning resources in everyday talk. Conversational category systems, the ways in which everyday language classifies things in

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the world, are one such resource. These category systems demand both participation and reification in their use. On the one hand, they are vital tools for participation, because collegial interactions are largely constituted by conversations. Thus, any event in which meaning is being constructed through participation will necessarily require the deployment of conversational categories. On the other hand, the categories themselves are embedded in particular terms, which congeal the ideas that they represent. The negotiation of these category systems, then, requires both critical meaning-making processes. Charles Goodwin (1994) explored this negotiation in some depth in his study of how professionals develop shared understandings of important aspects of their work by communicating a more or less standardized professional vision. He described the discursive practices that permit professionals to do so, helping them organize, notice, and interpret complex perceptual fields. Among the practices he described are coding schemes, or a “systematic practice used to transform the world into the categories and events that are relevant to the work of the profession” (p. 608). Some aspects of teachers’ work require the use of formal coding schemes, or those that are an official part of the functioning of the workplace. For example, in most U.S. schools, teachers are required to distinguish between A work and B work when assigning grades. From an ethnographic perspective, the local meanings of any such classification system remain an open question (Emerson, Fretz, & Shaw, 1995). In pursuing group members’ local knowledge of their setting (Geertz, 1983), ethnographers attend not only to the indigenous meanings of formal category systems (A vs. B work), but to the meanings of the informal category systems that arise in interaction as well. Informal category systems are a part of any workplace (Bowker & Star, 1999) and mark membership in a community. Following Goodwin (1994), I argue that these category systems also provide resources for professional learning. Here I explore one such set of informal categories: namely the categorization of students and its relation to teachers’ conceptions of their subject and teaching practice.2 Conversational category systems are an interesting focal point within a community of practice framework for learning. They provide heuristics for modeling the world, allowing people to represent problems of practice and work out possible solutions. But classifications necessarily simplify complex phenomena, and thus risk providing precision over validity (Bowker & Star, 1999). When teachers consult each other about problems of practice, they sometimes confront the limitations of these models, opening up the possibility for shifts in meaning as categories are negotiated while teachers alternately invoke, reinstate, and challenge the assump2The social organization of schools as it relates to categories of students has been explored previously. For example, McDermott (1993) argued that schools are organized to help categories such as learning disabled “acquire” students; Eckert (1989) examined the social spaces that separate jocks from burnouts in high schools.

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tions that underlie them. Language, patterns of interaction, and modes of participation become resources for modeling the world (Goodwin, 1994) and are intimately tied to teachers’ local professional cultures. By analyzing the conversational category systems that emerge within teacher communities, I uncover a sociocultural basis for teachers’ conceptions of their students, subject, and teaching practices and thus open a place to explore professional learning. I argue that conversational category systems shape teaching practice in three ways. Category systems deployed in teachers’ conversation (a) model problems of practice while simultaneously (b) communicating beliefs about students, subject matter, and teaching. As problems of practice are solved, these conceptions of students get (c) built into the organization of curriculum, naturalizing and reinforcing the beliefs they represent.

EDUCATIONAL CONTEXT Mathematics Education and the Mismatch Problem In this section, I describe a common framing of a problem, which I have come to call the Mismatch Problem, encountered by high school mathematics teachers in the current U.S. educational context. Over the past two decades, educators have gradually redefined the goals of, and audience for, mathematics instruction. They urge teachers to focus on processes such as problem solving, reasoning, mathematical connections, and communication in their classrooms—and to include all students (not just the “college intending”) in those activities (National Council of Teachers of Mathematics, 1989, 2000; RAND Mathematics Study Panel, 2003). Because of the disproportionately low numbers of poor students and students of color who participate in advanced mathematics classes (National Action Committee for Minorities in Engineering, 1997; National Commission on Mathematics and Science Teaching for the 21st Century, 2000; National Science Foundation, 2000; Oakes, 1990), equity-focused high school math reforms question the differential curricular opportunities made available to students. In response, mandates have emerged from policymakers to reconfigure the curriculum through strategies such as eliminating remedial courses, instituting heterogeneous classes, or whole-scale detracking. Many high school teachers experience these mandates as a challenge to their work. In my interactions with them as a teacher educator and researcher, I have found that they express the pressures of the current educational climate by invoking the Mismatch Problem. That is, teachers often see their students’prior achievements as incommensurate with a rigorous mathematical curriculum—and, in many cases, students themselves concur, only adding to teachers’challenge. At the same time, increased pressure from accountability structures such as curriculum standards and large-scale testing pushes teachers toward increased coverage of difficult content.

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Faced with either watering down the content of a college preparatory curriculum or failing large numbers of students, teachers experience a keenly felt dilemma: Do they remain loyal to the rigor of the curriculum, or do they teach to the level of their students? Most teachers end up feeling torn between the subject and their students, struggling and compromising as they try to reconcile their commitments to both. Conceptual Underpinnings of the Mismatch Problem Although this dilemma is genuinely felt, it reveals a particular conception of mathematics and students. I argue that this framing of the Mismatch Problem is rooted in a sequential view of the subject (Grossman & Stodolsky, 1995; McLaughlin & Talbert, 2001; Ruthven, 1987; Stodolsky & Grossman, 2000). That is, teachers frequently regard mathematics as a well-defined body of knowledge that is somewhat static and beholden to a particular order of topics. This was the collective view expressed by the South High mathematics teachers and made it seem impossible to participate meaningfully in some of the broader reforms taking place at their school. This perspective has logical consequences for both instruction and student learning. In light of this view, the main goal of teaching is to cover the curriculum in sequence to prepare students for subsequent coursework. Relatedly, students must learn prior topics in the sequence to move forward in the curriculum. Absent another model of mathematics, the teachers’ felt dilemma is understandable. The view of mathematics as a system of important and deeply connected ideas put forth by math educators (Boaler & Humphreys, 2005) and cognitive scientists (Resnick, 1988) has not found its way into many high schools. East’s mathematics teachers were a notable exception to this general rule. As shown subsequently, their collective work is governed by a connected and conceptual view of mathematics. The Mismatch Problem is built on a second common assumption: namely, that low-achieving students cannot learn mathematics through inquiry-type instruction (Raudenbush, Rowan, & Cheong, 1993; Zohar, Degani, & Vaaknin, 2001). Teachers often view cognitively demanding modes of instruction as inappropriate for low-achieving students. The department chair of South High frequently expressed this perspective, telling a novice teacher that collaborative learning, for example, only works with “the cream” (i.e., high-performing students). In fact, ample evidence has suggested that the metacognitive aspects of inquiry-based approaches greatly benefit all students, including those who have encountered past academic difficulties (Boaler, 1997; Boaler & Staples, in press; Schoenfeld, 2002; White & Frederiksen, 1998; Zohar & Dori, 2003). In this vein, the East High teachers worked to involve all of their students in a demanding and conceptually rich mathematics curriculum. In this article, I analyze conversations of the East and South High mathematics teachers as they work to address the Mismatch Problem. I argue that the conceptions teachers invoke to frame the Mismatch Problem reside at the heart of equita-

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ble teaching. As long as teachers believe that some kinds of mathematical activity are not viable for certain groups of students—whether because of students’ prior preparation or innate abilities—teachers will have little impetus to seriously engage in developing their pedagogy in ways that will reach all students with richer, more challenging content.

RESEARCH DESIGN Study Goals, Settings, and Participants The research for this article took place in the context of a larger project, Teachers’ Professional Development in the Context of Secondary School Reform,3 focusing on teachers’ work in the Mathematics and English departments of two California high schools. Our research team investigated the way teachers’ professional communities influenced both their professional learning and commitments to teaching using a multilevel, comparative case study design. Both South and East High Schools have strong local reputations for their involvement in school improvement efforts. Both schools are engaged in reforms aimed at reducing disparities in student achievement. They both enroll a middleand working-class population and have ethnically diverse student bodies. Notably, a greater number of East High’s students come from groups that have been traditionally underrepresented in higher education: 27% of South’s 750 students and 59% of East’s 1,500 students identify as African American or Latino/a.4 At the same time, the schools approach equity-geared reforms in different ways, providing fodder for analytic comparison. South High emphasizes school-wide reform. For about a decade, the school has restructured to support reform efforts. School-wide activities target change: teacher research groups, integrated student projects, and participation in whole-school reform networks. Although South High’s decade of reform overlaps with an era of activism from both the National Council of Teachers of Mathematics and its state level affiliates, these ideas—available to the South High mathematics teachers—have not taken hold in this school, which otherwise prides itself on innovation. East High, from the outside, looks more like a typical comprehensive high school. Grassroots efforts from the departments have driven some of the school-wide reforms, such as block scheduling, but the leadership in equity-geared reforms remains at the level of the subject areas. The mathematics department has led the way on many reforms—creating time for teachers to work together, plan collaboratively, 3Judith Warren Little, Principal Investigator. Funded by the Office of Educational Research and Improvement, and the Spencer and MacArthur Foundations. 4Student population sizes are rounded to the nearest multiple of 50.

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and try new pedagogies—establishing a culture of collaboration, risk, and improvement. An accreditation review in the mid-1980s spurred this effort when the review committee reported to the teachers that students felt that they “could not learn math” in the school’s program. They also advised the mathematics department to develop “an aggressive posture toward reform.” Taking this charge seriously, the department chair sought out all kinds of professional development with her colleagues. For example, they attended workshops on problem solving and were “surprised” at how much their students could do. This spurred a pursuit of ongoing instructional improvement that continued through and beyond the time of this study. In this research, I focus on the mathematics teachers at the two high schools and examine the ways their participation in various professional communities influences their informal learning. Professional communities are identified as groups that (a) have some kind of collective identity (e.g., “the Algebra Group” and “the Math Department”) and (b) are in some way oriented toward a collective conception of their work. Relating this construct to Wenger (1998), these criteria signal the existence of communities of practice—that is, groups with a shared repertoire of practices who are mutually engaged in some kind of joint enterprise. Teachers in self-identified groups marked their mutual engagement by giving themselves a name. The joint enterprise, in these cases, was their charge to teach their students mathematics. At the very least, these groups’ shared repertoire included regularly scheduled meetings. The more subtle aspects of the repertoire were for us to uncover empirically. It was clear from the outset that the two groups differed in both size (5 teachers at South vs. over 10 teachers at East High) and orientation toward mathematics teaching. Both groups felt strong commitments toward their students’ learning, although, because of the differences in orientation toward the Mismatch Problem, these manifest themselves differently in each site. Data-Collection Strategy Because of the focus on in situ teacher learning, the research relies on a variety of ethnographic data. These data were collected at the two case study schools over an 18-month period and include observational data of collegial encounters and classroom teaching, interviews with individual teachers and administrators, and a variety of relevant school and classroom documents. Although the larger study includes data collection at the individual, community, and policy context levels, in this analysis the focus is on the level of teacher community. Central to the research team’s data-collection strategy was the accumulation of records of collegial conversations and related artifacts. The majority of these conversations were audiotaped and transcribed, some were observed and recorded through running fieldnote records, and some were videotaped. We assumed that, among other things, these records would help us understand how teachers’ collectively conceptualized critical aspects of their work.

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Analyzing Pedagogical Reasoning This analysis started with a question: “What are the conceptual resources high school mathematics teachers bring to their encounters with equity-geared reforms?” To approach this question, I needed to find instances in which teachers’ encounters with these reforms were made visible. I used the heuristic of looking for moments of perturbation, such as when a teaching practice was being rendered as problematic in conversation. Such instances require teachers to engage in pedagogical problem solving, thus making available for analysis the sense-making resources they invoked within their communities. In line with the interpretive stance of ethnographic inquiry, I drew on analytic techniques arising out of the ethnomethodological tradition of seeking to understand people’s actions within their contexts and webs of meaning (Garfinkel, 1967; Geertz, 1973; Erickson, 1986). For fine-grained analysis of talk-in-interaction, I adopted complementary techniques from the ethnography of communication (Hymes, 1974; Saville-Troike, 1989). Guided by my question, I developed a unit of analysis called episodes of pedagogical reasoning (EPRs) to make sense of the data.5 I define the EPRs to be units of teacher-to-teacher talk in which teachers exhibit their understanding of an issue in their practice. Specifically, EPRs are moments in teachers’interaction in which they describe issues in or raise questions about teaching practice that are accompanied by some elaboration of reasons, explanations, or justifications. These episodes can be individual, single-turn utterances, such as “I’m not using that worksheet because it bores the kids.” Alternatively, these can be multiparty coconstructions over many turns of talk. My decision–rules for locating EPRs were mostly topical. Frequently, these longer episodes are signaled by a teacher’s question or the raising of a problematic issue, such as “I have a handful of kids who are not successful. How is this going to impact our classes next semester?” I then coded the EPRs by topical content so that I could compare thematically similar conversations across time and settings. For example, the codes student engagement, student failure, assessment, and alignment of students and curriculum were all topics of multiple EPRs. The last coding category yields the EPRs I focus on in this article. EPRs provide a means for comparing the ways teachers collectively reasoned about their work, providing insight into the social resources for teachers’ learning. Notably, discussions of curriculum at the two schools occurred at different levels of description. At South High, norms of privacy (Lortie, 1975) prevailed around classroom practice and little was discussed about actual classroom activities. The few times these were discussed, it was in an expository rather than a collaborative 5The concept of pedagogical reasoning has been discussed by others who studied teacher knowledge (McDiarmid & Ball, 1989; Wilson, Shulman, & Richert, 1987). What distinguishes my use of this construct is that I am investigating pedagogical reasoning as it emerges in teachers’everyday conversations.

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way. At East High, classroom practice and activities were an important part of ongoing discussions about curriculum, as were issues of pacing, assessment, and student response. Understanding this particular difference in the communities is beyond the scope of this article. Although the intensive discussions about curriculum took place at different levels, they represent what these groups had come to recognize as discussable instantiations of curriculum and are therefore useful in a comparative analysis. Focal Data Selection In this article, I compare two thematically related EPRs to illustrate the differences in the reasoning resources and practices of the teachers at East and South High Schools. Neither EPR was typical within the respective teacher community—both were longer and more in-depth than the average—but the conceptions of students and subject represented within each EPR were representative of the conceptions of students and subject found elsewhere in their communities, albeit more clearly elaborated on, making them richer for analysis. Both of these EPRs were spurred by a confrontation with the Mismatch Problem, so the teachers’ reasoning about issues of alignment of students and curricula is made apparent. The EPR from East High comes from a transcript of a videotaped meeting in September 21, 1999. In it, many details of interaction are available for interpretation, including direct quotes, gaze, and body language. The EPR from South High comes from qualitatively different data: fieldnotes from a daylong meeting on January 31, 2000. I interpret the interactions at each site appropriately for the kind of data available. Details, such as gesture and intonation, were not as available in the fieldnote data. However, the ways the teachers framed the Mismatch Problem and used their collectively developed understanding to work on a solution was available in both the fieldnote and the video data and therefore serves as a basis for this comparison. Limitations of the Analysis Although this analysis reveals important dimensions and consequences of category construction in teachers’ work, it is limited in its access to the full complexity of the social processes under examination. The analysis tracks the making (and remaking) of the category systems revealed in teachers’ talk and activity, but this talk and activity is not fully transparent. An important dimension of the social practice surrounding the interactions is therefore less available for analysis. The category systems exhibited in teachers’ talk may reveal schematic aspects of the work culture—the “relatively stable, prefabricated aspects of practice that actors have access to as they enter into engagement” (Hanks, 1996, p. 233)—but they often obscure the emergent ones. In other words, participants may utilize the category

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systems strategically in interaction for their own emergent purposes. The exact nature and multiplicity of those purposes are often in process and negotiation and are not always available for analysis, even with rich supporting ethnographic data. In my analysis, I try to be mindful of this complexity and provide interpretations that reflect the data and information to which I have access. Despite these social complexities, this analysis reveals important resources for teachers’ learning in the workplace. The decisions that the teachers made as a result of these conversations have a very real impact on students’ opportunities for learning mathematics. The discursive resources that went into the deliberation and construction of these decisions are available for examination and are at the heart of this analysis. I argue that the consequences of these categories are great enough to merit examination in themselves, even when the social complexity is considered.

CASE 1: THE “FAST KIDS” PROBLEM AT EAST HIGH SCHOOL Overview In this section, I present the coconstruction of students, mathematics, and teaching practices at East High School through the conversational categories that model problems of practice, communicate assumptions, and are ultimately reflected in the curriculum. I provide some background to the event, contextualizing the teachers’ conversations and activities. A close analysis of four sub-EPRs follows. I pay particular attention to the ways in which students, mathematics curriculum, and teaching practices are coconstructed in interaction. Finally, I synthesize these strands of analysis and reflect on the implications for understanding the teachers’ implicit models of students, mathematics, and teaching practice. Background of the Event The teachers’ at East High were in their first year of full-scale detracking in their mathematics courses. The previous year, two teachers, Guillermo Reyes and Carrie Edwards, piloted the group’s detracking efforts by teaching Algebra 1 content to their Math A (pre-algebra) students. They reported that the students who failed this experimental course did so because of attendance and homework issues and not because of an inability to handle the material. At East High, the school year is organized into two terms, with students taking four 90-min courses per term. A term-long class carries the same number of credits as a traditional 1-year course. To accommodate their reworking of the curriculum, the algebra course was taught over two terms, or a full academic year. All students took the two-term algebra course. To sustain their ongoing in-

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TABLE 1 A Summary of Transcript Conventions Symbol / = (.), (5s) [, // , ? :: . italics CAPITAL LETTERS (??), (cow) (( laughs )) (1), (2)

Description Self-interruption No gap between utterances Very slight pause, 5-sec pause Beginning of overlapping utterances, end of overlapping utterances Low rise in intonation High rise in intonation Marks lengthened syllable, each “:” equals one “beat” Low fall in intonation Marks stress Increased volume Unclear reading, tentative reading Marks other voice qualities or actions Links talk to gestures described at end of transcribed turn

vestigation into curriculum and activities that support the heterogeneity of their classes, East High’s algebra teachers met weekly, donating 1.5 hr of their time after school. The analysis that follows examines an extended EPR that took place during one of these after-school meetings. All of the members of the Algebra Group were present at this September 21, 1999 meeting. The primary participants in this episode sequence were Guillermo Reyes, Carrie Edwards, and Tina Lee, an intern. Also present were Jill Larimer, Howard Silver, Charlie García, Christy Roleri, Annie Stolz, Alice March, Judy Stone, and myself. All of the teachers had at least one section of algebra in their schedule. We gathered after school in Charlie’s portable classroom. We started the meeting by working in small groups on a couple of conceptual fraction activities Carrie had worked on with her classes. We then reviewed an activity with an algebra manipulative, with experienced teachers sharing what they had learned from using it before with their classes. The meeting was to go from 3:30 p.m. to 5:00 p.m., and with about 11 minutes remaining, Guillermo suggested that we do a “quick check-in.” The teachers organized their desks in a circle and began the check-in. This episode sequence is excerpted from Tina’s check-in.6 The following video transcript includes notes about movement and gaze in the form of numbers inserted in the text. For an explanation of other transcript conventions, see Table 1.

6“Check-in” has a certain routine structure in the Algebra Group. In this routine, members would state where they were in the common curriculum and what activities they had been doing with their classes. Any questions or issues that arose since the last meeting then followed.

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Analysis of Case 1

Episode 1.1: Tina’s Problem: Closing the Gap Between Fast and Slow Kids 01 Tina: 02 03 04 05 06 07 08 09 10 11 12 13 Christy: 14 Carrie: 15 Tina: 16 Carrie: 17 Tina: 18 19 20 21 22 23 24 25 26 27 28 Christy: 29 Tina: 30 31 32 33 34 35

OK for me, I’m going over perimeters and I’m introducing a little bit of surface area. I’m doing graphical interpretation, finding patterns. I just gave them their second quiz, class average was a B. I give a weekly quiz and I think it’s been helpful. My students, I don’t know where they’re from, but they’re doing so well, I mean they know the difference between a linear graph versus exponential. But the thing about my students (1) is that there’s kids that know a lot (2) and then there’s kids that (.) you know, feel like they’re slow learners (3). (1) Right and left hand alternate drum beat gesture, flexing at wrist (2) Gestures two hands parallel, on her left side, indicating “kids who know a lot” (3) Uses same gesture to her right side, indicating “kids who feel like they’re slow learners” (Nodding vigorously from “a lot” to “slow learners”) (Leans back in desk, props self up on elbows.) And I’m trying to find group-worthy activities (Nods) where the kids who are fast learners (1) and the kids who are slow learners (2), that it can close the gap (3). So that the kids that are slow learners can contribute and can you know feel (4) smart (5), but I don’t know if I can find activities that are group-worthy activities like that. Because I can feel the um frustration of the fast learners, like (1) Puts out left hand, where she had indicated “kids who know a lot” (2) Puts out right hand, where she had indicated “kids who feel like they’re slow learners” (3) Waves hands together, to close “gap” (4) Holds overlapping hands to chest (5) Looks at Guillermo (quietly; nodding) Yeah. “This is easy! (1) I already know the answer!” And then there’s kids that are slow learners (2) that are like, “Give me a chance to find the answer!” and it’s almost like they kind of give up (3) because they feel like it’s a speed competition, like who can get the answer the fastest kind of thing. And I’m trying to close the gap (4) between that and that’s been one of my frustrations (5) I think (6). Otherwise, they’re doing well and just trying to find activities (7) group-worthy activities (8). (1) Plops hand on desk in mock frustration

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(2) Gestures with right hand (3) Leans to the right, as if moving away from the “fast kids” on the left (4) Pushing left and right hands together (5) Bouncing hands in prayer position in front of her chest (6) Opens hands, palms up (7) Slaps right knuckles in left palm (8) Looks to Guillermo

Kinds of kids. Tina opens by saying something positive about her kids (“they’re doing so well,” line 5). She starts by contrasting “kids that know a lot” to “kids that […] feel like they’re slow learners” (7–8). She is reluctant to construct this last category, searching for a way to describe what she has seen in her class. Her description is supported by Christy, also an intern, who nods vigorously as she speaks (13). Carrie, a more experienced teacher who helped pilot the detracking effort, retracts slightly at Tina’s words, leaning back in her seat (14). As Tina explains her problem, several things happen to these initial categories. First, “kids that know a lot” becomes “kids who are fast learners.” This suggests that, for Tina, knowing a lot of mathematics is related to being a fast learner. Also, Tina reenacts the frustrations felt by what she sees as two groups of students (29–31). In her framing of the Mismatch Problem, she suggests that the problem lies in her activities, not simply in the kids themselves. Mathematics curriculum. Tina sees the problem in her class, in part, as a result of her curriculum. In introducing her problem, she reports her search for group-worthy activities. She wants to find activities that will allow the “slow learners” to “contribute” and “feel smart” (18–19), as opposed to the “speed competition” that she sees now (32). Tina thus reveals her understanding of the Algebra Group’s notion of “group-worthy activities” to be ones that allow “slow learners” to participate positively. Notably, when she is enacting the frustrations of the fast students, she voices their complaint as “I already know the answer!” (29). This suggests that, in Tina’s classroom, students may perceive the answer as an endpoint for mathematical activity. Teaching practices. By posing this problem at all, Tina as a teacher, expresses a sense of responsibility for the dynamics in her classroom. Indeed, after gesturing to her right and left to indicate the groups of students she sees in her classroom, she indicates her frustration with a gesture to her heart, placing herself in the physical and emotional center of this mismatch dilemma (39). Although she has not been able to close the gap she sees between her students, she

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feels obligated to address it. Her strategy has been to search out group-worthy activities. This response reveals a particular conception of teaching: A teacher is responsible for classroom dynamics, and a teacher has the right (or perhaps even an obligation) to vary the curricular activities as a way of addressing issues in his or her classroom. Both of these conceptions suggest an active role for the teacher: the teacher as a problem solver and the teacher as a curriculum “tinkerer.”

Episode 1.2: Carrie’s Brainstorm: Activity or Status? 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

Carrie: Tina: Carrie: Tina: Carrie: Tina:

Have they done the perimeter lab gear7 yet? Yeah, they did it today. (Nods.) And that still happened? Yeah, yeah. (Nods.) It still happened. Because, um. Let’s see what happened, I mean, (1) you should read the problem some of them wrote. This is a “guess my number” problem and I took this/ each of them wrote their own problems and I took it and I put it in the quiz (2). This guy wrote: “The number multiplied by 6, subtracting 50 and multiplying it by itself is 4,096.” This (3) is a very sophisticated problem to write. And (4) I also took one that’s more simpler. And I just don’t know, I don’t know (5) how to close the gap I guess. (3s) (6) (1) Picks up quiz (2) Raises quiz slightly (3) Flips quiz toward group (4) Points to quiz again (5) Gestures with right hand and left hand far apart, showing the gap, then pushing it closed (6) Looks at Guillermo Carrie: I wonder if it’s: (.) not just (.) the activities you’re doing but also just status. Tina: Mhm. Carrie: You know? I mean even if you did give them a group-worthy task, those kids who: (.) feel like they have low status will just continue to play that

7“Perimeter lab gear” refers to an activity that involves finding the perimeters of two-dimensional rectilinear polygons built with algebra tiles (Piccioto & Wah, 1994). The tiles provide a geometric representation of variable expressions. The perimeter activity provides a geometric representation for combining like terms.

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70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

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role Tina: (Nodding.) Carrie: (Nodding) because (.) they think that that’s what they’re supposed to do. Tina: Mhm. So that’s what my struggle is, what can I do to (.) make them feel motivated? make them feel like (.) I’m a part of this group, make them feel like I’m smart too. Carrie: (Nodding) Yeah. Tina: I’ve been going to the bookstore looking for, you know, (turns quiz paper over) all kinds of activities, but it’s really hard, Carrie: (Nodding) Tina: picking what works (Looks at Guillermo) and I’m not really good at reading directions either so Christy, others: (Laugh) Tina: (Laughs) so I’ve got to see (1) and learn (2). And (3) I’ve been looking at a lot of stuff that you’ve been doing and I’ve done a lot of it in class, but um/ I mean, a lot of it works, but I still need a lot more (4) I guess. (1) Makes vertical karate chop gesture with right hand (2) Waves hand back toward head (3) Gestures toward Carrie with right hand (4) Digging gesture with right hand Carrie: What block do you teach? Tina: Huh? Carrie: What block. Do you teach? Tina: Oh. First block. (Looks to Guillermo) Carrie: I mean, it could be (.) something where some of us could go in and watch and, you know, give you feedback.

Kinds of curriculum or kinds of kids? In this episode, Carrie offers an alternative analysis to Tina’s problem. She asks and learns that Tina found this “gap” between her students with the lab gear activity—one that she seems to believe is group-worthy. She cues Tina to explain what happened with that activity in her class (4) by nodding and not speaking again, giving Tina another turn in their dialogue. Although Tina prompts herself to recall “what happened,” she goes on to describe a kid’s response on the quiz and does not elaborate on her observations about the kids and the lab gear activity. It is not clear why their communication does not line up: Carrie’s question about lab gear was answered by a report of the answer on a quiz. Perhaps Tina’s distress overrides her ability to hear or respond to Carrie’s question, and she instead needs to share more evidence of “the gap.” Alternatively, her frequent glances at Guillermo may indicate that she is distracted, awaiting his response.

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Kinds of kids and teaching practices. Carrie offers another explanation to what Tina is seeing in her classroom. Instead of the “fast learners” and “slow learners” Tina has described, Carrie proposes that the issue might be one of status (65–66).8 She describes for Tina how low-status students “will continue to play that role […] because they think that’s what they’re supposed to do.” Carrie’s analysis positions Tina differently as a teacher in relation to her class. Now, her work is not one of finding the right activity; it has become addressing the academic status issues among the students in her classroom. Notably, Tina again misses Carrie’s communication. She acknowledges her analysis (“Mhm”) and then says that her struggle is to make them feel “motivated” (74). Tina shifts the teaching work away from addressing status issues to motivating her students—work that perhaps makes more sense to her. She continues by returning to her description of the search for better curricular activities. Carrie then proposes something more concrete by offering the group’s situated assistance with her problem through peer observation. Even though Carrie has not managed to communicate her ideas effectively to Tina, the group’s practices of peer observation and debriefing provide her with another opportunity to help Tina reconceptualize her problem. Episode 2.3: Guillermo’s Prediction: Modeling Thinking About Kids 97 Guillerm: (Shifts in desk). Um. OK, I have a prediction. (.) My prediction is that you won’t be able to do anything about it (1). But that, I think that’s 98 from (.) from thinking about a group of kids (2) as slow learners and 99 that’s, that’s (3) how we’re, we’re acclimatized to, to (.) think about 100 learning (4). And, and I don’t really (5)/ what I find is that when I 101 have mindsets (.) like that that they get in my way (6) in terms of 102 thinking about (7) the curriculum. So (8) it wasn’t a real predic103 tion but I was just trying to be shocking. Um= 104 (1) Waves pen toward Tina 105 (2) Uses pen to gesture a circle on the desk, indicating a “group” 106 of kids 107 (3) Shrugs with right shoulder 108 (4) Alternates left and right hand in shaking gesture in front of face 109 (5) Scratching back of head 110 (6) Uses right hand to point to himself 8Carrie is using language and ideas from complex instruction (Cohen, 1994; Cohen & Lotan, 1997). Complex instruction is a key component of East High’s detracking work. See Horn (2006) for more details.

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111 112 113 Tina: 114 Judy, others: 115 Guillerm: 116 Tina: 117 Guillerm: 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 Carrie: 136 Guillerm: 137 138 139 140 141 142 143 144 145 146 147 148

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(7) Uses pen to indicate a small circle (8) Leans toward Tina a little =It worked.

(Laugh) (laughs) I wasn’t trying to be mean. (Shakes head) You weren’t. (Leaning chin on fist.) Um. One thing I’m, I’m thinking about is (.) the ones that are moving through things really quickly often they’re not stopping to think about what they’re doing, what there is to learn from this activity. So one thing to think about is (.) uh, helping/helping kids that are not stopping (1) to set their own learning agenda, like what should they be working on. (.) What are they learning. And they have to be learning something. (2) (1s) Because even if you have great activities, if if the perception in the class is, um, (3) I’m fast (4), I’m not as fast (5), it’s not going to help with the status issues I don’t think. (2s) But but a kid knowing OK, I can get through this quickly (6) but I’m working on X, being a better group member, because it’s going to help me in my future classes. = (1) Shakes head (2) Taps pen on desk (3) Puts right and left hands, palms down, in front of him (4) Indicates to the left, where Tina had indicated fast kids (5) Indicates to the right, where Tina had indicated the other kids (6) Rapidly moves pen down notepad (Nodding) = Um. Showing off math tools, um, because I know how to do it with a t-table but I don’t know how that relates to a graph yet. Um. (1) I’m making stuff up because I don’t know your kids. But, but like find/think of the ones that you think of as fast learners (2) and figure out what they’re slow at (3). Like what are they slow learners about (.) because that’s their agenda (4) then make that, this is totally simplistic, but (laughing) like, help, help them pick an agenda for their work. (1) Shrugs right shoulder (2) Shakes fist with pen (3) Fist hits desk (4) Waves pen across notepad, writing an “agenda” Tina: (Nods) Guillerm: And one vehicle for doing that is a, is a problem of the week, just one bigger problem (1), that they actually work on independently. Because

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149 150 151

i:f, one thing that I learned in the eighth grade9 is that kids are workthe same complex problems (2) but independently they can see (3) other kids’ strategies for how to/ (3s) for how they approach the problem and how they show their reasoning. That maybe they 152 wouldn’t have thought of because they’re too busy like playing 153 school? (4) I’ve got the answer? I’ve got it done quickly. So (.) 154 they’re slow at something. There’s something to get smarter at. 155 (1) Gestures a circle on notepad 156 (2) Points to different points in gestured circle 157 (3) Gestures out from his eyes toward group 158 (4) Uses pen to make quick writing gestures in the air 159 Tina: (??) (Turns toward Carrie) Guillerm: So fi/ fi/ give them a problem of the week. 160 Tina: Do, do you give problem of the week, Carrie? (Points to Carrie) Carrie: (Nodding) Sure (I can make a note of that). 161 Tina: OK, so I’ll just follow after you. Carrie: Hee hee! Ah!

Kinds of kids. Guillermo’s turn changes the way kids are talked about (97–112). Although Carrie’s response in Episode 1.2 proposes a different dimension (status) in Tina’s category system for kids, Guillermo explodes the scheme altogether by complicating the single dimension of speed (fast–slow), transforming it into a multiplicitous, relational category. He does this in several ways. First, he changes “fast” into “the ones that are moving through things really quickly.” In Tina’s original statement, “fast” seems to be a desirable quality in a student; but here, Guillermo claims that speed may also be a liability, as fast kids “often [are] not stopping to think about what they’re doing.” He reiterates this idea in his final turn, when he describes kids who are “too busy playing school” to consider multiple strategies for a complex problem. He implies that their focus on task completion may be adaptive to schooling, but not necessarily to complex thinking. Second, Guillermo makes “fast” a relative category when he says “think of the ones that you think of as fast learners and figure out what they’re slow at.” His unpacking of the category “fast” provides the possibility for a more complex and particular view on the students in Tina’s classroom. He underscores this in saying, “I’m making stuff up because I don’t know your kids,” revealing a perspective that his generalizations are limited because they might not map directly onto Tina’s students. 9The group makes frequent references to the 8th grade, which constitutes a kind of existence proof for their detracking work. Up until 2 years prior to this meeting, East High School was an 8 though 12 grade school. The math teachers, over the course of a decade, created a detracked 8th grade curriculum.

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Kids are also animated (M. H. Goodwin, 1990) through Guillermo’s talk, as when he uses the first person to voice their imagined perceptions as they sit in Tina’s classroom (“I’m fast, I’m not as fast”) and their ownership of a learning agenda (“I know how to do it with a t-table but I don’t know how that relates to a graph yet”). These imagined utterances impart a view that students themselves have active roles in the classroom—both in negotiating their status and in participating in learning.10

Mathematics curriculum. Although mathematics is not discussed at length, a distinctly nonsequential view of mathematics undergirds Guillermo’s statements. Where it is represented directly in Guillermo’s talk, mathematics is a subject with connections: Guillermo imagines examples of student-learning agendas and picks one that involves connecting tables and graphs. More subtly, Guillermo’s reworking of Tina’s categories of students ties in with notions of mathematical competence. Because students, in his terms, are not simply “fast” or “slow” learners of mathematics, the subject itself takes on more texture. Mathematical competence is not an accumulation of mastery over procedures—something that students are more or less facile with—but a connected web of ideas that requires careful consideration. He reveals this last view of mathematical competence when he expresses concern about “the ones who move things really quickly […] not stopping to think about […] what there is to learn from this activity.” Guillermo proposes that Tina introduce a specific kind of activity, a problem of the week, thus addressing her original question about activities. In doing so, he validates her concern about curriculum while tying it back to the reworked student classification system. His description of problems of the week (complex problem with multiple strategies that require students to demonstrate their reasoning, 150–151) suggests a problem-solving perspective on mathematics. Teaching practices. Teaching, likewise, is made more subtle and complex. Teachers do have to choose good activities: Guillermo ends the episode by suggesting that Tina give her students a problem of the week. But teachers also have to address status issues in their classrooms, as Carrie proposed earlier. Guillermo makes Carrie’s suggestion more specific by describing student-learning agendas. Learning agendas are “what [students] should be working on, what they are learning” (121–122) or “what they are slow learners about” (139–140). Teachers have an active role in mediating curriculum in this representation of teaching. Teachers’ and students’ agency is highly engaged in this notion. Classroom dy10This

practice of inserting imagined classroom dialogue into conversations is explored more in Horn (2002).

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namics are not simply consequences of curriculum and static student ability; teachers help negotiate students’ roles in the classroom by helping them to “pick an agenda for their work,” coordinating their academic goals with those of their students. Guillermo’s notion of learning agendas renders teaching as particular work that requires close attention to specific kids and their purposes in the classroom. Tina is not left on her own, however, to integrate these complex practices in her classroom. Carrie has already offered Tina the support of the algebra teachers through in-class observation of Tina’s teaching. Tina then garners even more support by proposing that she “follow after” Carrie—a practice where a newer teacher stays a day behind a more experienced teacher in the progress of the curriculum, thus allowing the newer teacher to observe, synthesize, and adapt a lesson before bringing it to her classroom.

Episode 1.4: “The One Kid, I Forgot his Name”: Getting Specific 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185

Guillerm: The one kid, I forgot his name, the one that we talked to? Tina: [student’s name]=? Guillerm: =Yeah. (2s) And he is/ he does have a (.) ton of background and lots of desire to learn and lots of intellectual curiosity. He probably is going to get most things quickly so (.) mathematically, you could really/ you’d be looking in bookstores forever trying= Tina: =Yeah= Guillerm: =to find problems. But he/ he’s also impetuous? and he doesn’t, he gets locked (1) into one thought pattern and doesn’t break out (2) very easily. So you may not be able to really challenge him mathematically but I think in terms of (.) the complexity of his thinking you can. (1) Cupped right hand arcs and stops at point in front of chest (2) Open, loose fists waving back and forth Tina: (Nods)Complexityofhisthinkingandumalsobeingabletoworkwith other group members. (Sits up straighter and indicates a circle with right hand.) Guillerm: Yeah. Tina: A lot of, um, the kids in my class, they’re very individualistic Carrie: (Quietly) Mhm. (Nodding)

Kinds of kids. Because he has observed her class, Guillermo is able to get even more specific in his response to Tina. He effectively models alternative language for talking about a kid who might be considered “fast”: This kid has a “ton of background and lots of desire to learn and lots of intellectual curiosity” (169–170).

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At the same time, the kid is represented as “impetuous” and as getting “locked into one thought pattern” (174–175). In these remarks, Guillermo creates a multidimensional classification system of students that grows ever more complex, now encompassing qualities such as “carefulness” and “ability to see multiple perspectives.”

Mathematics curriculum and teaching practices. Guillermo represents mathematics curriculum as a dynamic part of the classroom. Students’ engagement or disengagement is not inherent to a given activity. This is because the teacher can work to bring students into mathematics by elaborating on what is written to push on the complexity of the student’s thinking (177). In this reframing, the “fast kids” problem is not solely about finding the right activity, but also about an appropriate pedagogical response. Discussion of Case 1: A Nonsequential View of Mathematics and Its Consequences In these episodes, the teachers discussed Tina’s rendition of the Mismatch Problem—the gap between fast and slow kids in her classroom. Her problem is addressed by examining issues of students (learning agendas, categories of kids), mathematics curriculum (group-worthy problems, problems of the week), and teaching practices (setting learning agendas, incorporating problems of the week, following an experienced teacher). Tina described her efforts to remedy the problem by finding group-worthy problems and was seeking assistance in finding curricular activities that would allow her to better close the gap. Carrie and Guillermo responded to her by working to reframe her problem, challenging her initial categories of “fast” and “slow” learners. In doing so, they were constructing a more complex model for mathematics and student learning than the one Tina introduced. Guillermo, in particular, changed the talk about students by using specific language and talking about actual students. Tina’s initial framing posed a problem of matching students and curriculum by finding group-worthy problems. Carrie and Guillermo’s reframing legitimated Tina’s impulse to find better curriculum and activities (as evidenced by his suggestion that she do a problem of the week) while also imploring her to take action in her classroom to address student status issues and set learning agendas. The teachers’ talk about students, shaped by their frame for problems of schooling, tied into conceptions of both mathematics curriculum and teaching. Guillermo’s talk in Episode 1.3 questioned the notion that “fast kids” were necessarily better at math. The model he deployed represented mathematics not as a subject where only the quickest survive (Boaler, 1997), but rather one in which students might bring a variety of abilities to be drawn on. In addition, both he

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and Carrie extended Tina’s portrayal of her work as teacher. Tina characterized her work as one of seeking the right curriculum (“I’ve been going to the bookstore looking for […] all kinds of activities, but it’s really hard”). Carrie pointed out the ways that students’ status issues could undo her best efforts toward curriculum matching (“even if you did give them a group-worthy task, those kids who feel like they have low status will just continue to play that role”). Guillermo extended Carrie’s point about status, turning Tina’s whole framing of the problem on its head by pointing out the problems that the fast kids often have (“often they’re not stopping to think about what they’re doing”) and the kinds of interventions Tina could do as a teacher (“help […] kids that are not stopping to set their own learning agenda”). Teachers have a complex role in the classroom in this conversation. Although Guillermo and Carrie pushed on Tina’s framing of her problem by expanding Tina’s role as teacher, Tina was not left alone to sort out their suggestions. In addition to modeling a possible learning agenda for one of Tina’s students, the group offered its support through peer observation and following. These structures would provide learning resources for Tina. Their existence made it easier for the group to challenge one another’s ideas and convey ideas that went beyond the available language through other kinds of support and follow-up.

CASE 2: REDESIGNING COURSE OFFERINGS AT SOUTH HIGH SCHOOL Overview In this section, I present the coconstruction of students, mathematics, and teaching practices at South High School through an examination of the teachers’ conversational category systems for students used to model the mismatch problem. I provide some background to the event, contextualizing the teachers’ conversation and activity. A close analysis of five sub-EPRs follows, with particular attention to the ways in which students, mathematics curriculum, and teaching practices are coconstructed through the teachers’ interactions. Finally, I synthesize the close analysis by reviewing the category system that developed across the episodes and the underlying conceptions represented. Background of Event The South High mathematics teachers were asked by their district to eliminate remedial courses from their curriculum. Before the day’s work began, the teachers had de-

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cided to replace Math A and Math B (pre-algebra courses) with a 2-year version of the introductory algebra course. The new course would have the same content as a traditional Algebra 1 course, but it would be spread out over 2 years. However, the teachers had to provide 4 years of mathematics courses for these students, as per the district’s mandate. The mathematics teachers were given a paid workday to finalize their response to the mandate. During their 6 hours together, they were also asked to determine the alignment of their subject standards to the state standards, make sure that all courses have identifiable standards, and propose a new curriculum structure without any remedial courses. The analysis that follows focuses on the work and interactions of this last task. The teachers had to eliminate the remedial courses—anything below the college preparatory sequence—while providing 4 years of mathematics coursework for all ninth graders who did not attain a certain grade level certification on a district standardized test. On January 19, 2000, all five members of South’s math department were present for this meeting. They were Barbara Moore, Dan Marcus, Rosa Feliciano, Noah Banes, and Lucy Chan. We met together in Noah’s classroom and sat in a small circle at the front of the room. Barbara, the department head, led the meeting as she typically did. As was common, she had more frequent and longer turns of talk. Dan’s participation was the second most intensive in this meeting. Although Barbara had more years of teaching experience, Dan had been at the school longer and could speak to the history of current structures. Noah, a second-year teacher, was often sought out for input, as he had the strongest mathematics background in the group. Lucy was an overwhelmed first-year teacher whose opinion was seldom offered or solicited, and Rosa’s participation in group discussions was infrequent, although she seemed to know the most about union regulations and contractual rules. They started out the day by finalizing their selection of a textbook for the course before they went on to write the course descriptions required by the district. As the teachers worked together on these projects, they had a number of drop-in visitors, including a district representative answering questions. As a result, the meeting had a public feeling, and there were constant reminders that the district was both underwriting and overseeing it. I acted as a “note taker” in that meeting, allowing me to create in-the-moment field jottings on my laptop computer, which I then refined into full fieldnotes on that same day, filling in missing shorthand and adding remembered details to tell a coherent story of what I observed. Only those words actually taken down at the time of my observation are placed in direct quotes in the fieldnotes. Speech not taken down directly at the time is either presented as an indirect quotation or paraphrased (Emerson, Fretz, & Shaw, 1995).

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Episode 2.1: “I Don’t Think We Can Fly Geometry”: The Case for New Courses 1 2 3 4 5 6 7 8 9 10 11 12 13

Dan says, “I don’t think we can fly geometry.” Barbara agrees. “You brought it up, and you’re telling the truth” that “our kids cannot get through our geometry as we teach it now.” They discuss whether the kids can get through their geometry course. They talk about the foundations for geometry in the algebra textbook. Barbara says: “In this book that we have, they do formulas for circumference of a circle, similar triangles.” She continues, “What they won’t be able to handle is the logic in the two-column proofs.” “Our regular kids can’t handle that.” “One of the problems with the kids we’re putting in [the 2-year algebra course] is they don’t have the logic component.” We’ll still have “kids” who “can’t pass the SAT–9 geometry.” “What you’re doing is figuring out our hierarchy and that’s good.”

Mathematics curriculum and kinds of kids. Dan’s observation starts this EPR by putting forth a problem—namely, their collective capacity to “fly geometry” (1). This establishes the necessity for other alternative college preparatory courses beyond the new 2-year algebra sequence. In addressing the problem, Barbara introduces the first category of students when she restates the problem as “our kids cannot get through our geometry as we teach it now” (3). The meaning of “our kids” has not been made clear in this utterance, but in other conversations “our kids” are contrasted with students at other, higher performing schools. Barbara then specifies which subset of “our kids” is of concern: She hones in on the kids in the 2-year algebra course (10). She then fleshes out some characteristics of these students, providing a resource for those around her to learn her conceptions of them. Similar to the regular kids (with whom they are contrasted) these students “won’t be able to handle the logic in the two-column proofs” (9). When she states that they “don’t have the logic component” (11), she is broadcasting a common belief about low-achieving students (Raudenbush, Rowan, & Cheong, 1993; Zohar, Degani, & Vaaknin, 2001). She also explains that these students “can’t pass the SAT–9 geometry” (11-–12). Although the 2-year algebra class does not yet exist, Barbara appears to be applying her understandings about the type of kids in the school’s current remedial courses. Metaphorically speaking, the remedial courses can be thought of as bins that hold certain kinds of students, and although the labels and course content of those bins might be changing, the types of students they contain are not. In her talk, Barbara both elaborates on the collective understanding of the problem at hand and communicates some conceptions of our kids, the kids in the 2-year

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algebra course, and regular kids. She elaborates on the Mismatch Problem that Dan has put forth by identifying a particular part of the curriculum, the two-column proofs, which even the regular kids “can’t handle.” Because these students “don’t have the logic component” (11), the standard geometry curriculum is an untenable option for them. Barbara refers to the state’s standardized test, the SAT–9, as further evidence that the students in this course will not be adequately prepared for geometry (12). Her understanding of the state’s testing system as being tied to the curriculum also becomes a resource for maintaining these students in a separate course sequence. The Mismatch Problem between the district’s mandate to eliminate remedial courses and the teachers’ perceptions of the geometry curriculum and the students requires further deliberation (and expansion) on the part of the teachers.

Episode 2.2: Not “the Quickest Kids”: A 2-Year Geometry Class 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Dan says, “Until they really enforce keeping the eighth graders back” until they are ready for 9th grade work, we know “we’re not going to be getting the quickest kids.” “Where will we send the kids out of [the second half of algebra] who aren’t ready for geometry?” “Should we keep Math C?” Barbara proposes, “We can call it Integrated Math Concepts?” Dan says, “Do we want to still have another 2 years of something besides geometry?” Barbara says, “The district will not fund any remedial class, so as long as it isn’t remedial.” “If you taught a hands-on geometry, if you took Discovering Geometry [a hands-on geometry textbook] and spread it out over 2 years, the district would buy that. That’s not a remedial course.” Dan says, “Should we be thinking about a 2-year geometry class?” Noah: “A 1-year hands-on geometry class would not be a pathway to advanced algebra.” Barbara leans back in her chair and crosses her arms. She raises her eyebrows and says, “Serra [the author of Discovering Geometry] claims it would.” Noah shrugs and says, “OK.” Barbara says, “I say that the kid is ready for advanced algebra at a community college level, but we don’t have to do it.”

Kinds of kids. Dan’s model of the Mismatch Problem is elaborated on further: Because of inadequate filtering in the eighth grade, the teachers will not get the quickest kids (15–16). This provides an amplified understanding of who the kids in the 2-year algebra course are: They include some not-quick kids who will not be ready for geometry at the end of their time in that course.

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Mathematics curriculum. The model of the Mismatch Problem opens itself up to a particular range of solutions. In this case, it seems to be understood that the kids in the 2-year algebra course have enough not-quick kids among them to warrant a separate follow-up course in the curriculum. Barbara proposes offering a “hand-on” class spread out over 2 years (23–24). As they deliberate the nature of this follow-up course, the teachers reveal their sense of mathematically legitimate (and less legitimate) curriculum, as Barbara and Noah compare a formal geometry course to a hands-on geometry course. Noah’s concern about preparation (discussed next) suggests that such a course, in his view, would be more of a stopgap than a bona fide mathematics course. The presentation of mathematics, then, determines its validity for the teachers and not its accessibility to students. The teachers face the challenge of creating a mathematics class that accommodates the not-quick kids but, at the same time, per district mandate, is not remedial (21–22). A hands-on geometry over 2 years satisfies these constraints (23–24). Mathematics curriculum and teaching practices. Embedded in this episode is another elaboration on the problem the teachers face reconciling the district mandate with their own commitments. When Noah proposes that even a 1-year, hands-on geometry course would not be a pathway to advanced algebra (26–27), he suggests that this alternative only pushes the problem of adequate student preparation further up the curricular ladder. Although Barbara invokes the author of the inductive geometry text as an outside authority to support the course’s legitimacy (28–29), the audience for her comment seems to be a district-level judge of their plan rather than the teachers in the room who know better. She lets Noah know that the preparation problem that he anticipates will not be theirs (“we don’t have to do it”), and that the students will be adequately prepared for a community college advanced algebra class (31–32). Kinds of kids and mathematics curriculum. Because of the irreconcilability of the not-quick kids and the highly valued two-column proofs, the teachers build a separate (albeit less mathematically legitimate) geometry course into their curriculum. The two-year course will provide a place for the not-quick kids to learn hands-on geometry and avoid the rigors of two-column proofs. Episode 2.3: “These Are the Kids Who Forget if They’re Not Reminded Every Week, Every Day”: Specifying Curriculum 33 Barbara says: “Integrated Math/Geometry Concepts.” “Mike Serra’s 34 book with a lot of algebra support.” These are the kids who “forget if

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they’re not reminded every week, every day.” Dan says, “So we can still call it … ” Barbara continues, “We have to fancy up the name.” These kids will “not be ready for two-column proofs.” She’s trying out names, things like “inductive geometry.” Noah: “As far as the names of math classes, I don’t think we should change it so colleges can recognize it on transcripts.” Dan: “We’ll still call it geometry […]” [gives it a label parallel to that of the two-year algebra course].

Kinds of kids and teaching practices. In this episode, Barbara imagines some of the details of the course: the name and the instructional strategies. These details communicate more about the kids in the two-year geometry course (who are the not-quick kids exiting from the two-year algebra course). Barbara describes the content of the course: hands-on geometry with “a lot of algebra support” (34). She goes on to describe the teaching needed for this kind of course by appealing to another characteristic of the kids in the two-year algebra course. The kinds of kids enrolled in two-year geometry need repetitive content, as they “forget” what they’ve learned “if they’re not reminded every week, every day” (34–35). Thus, she is elaborating her understanding of these students while articulating the kind of instruction they require. Mathematics curriculum and kinds of kids. The district’s mandate to eliminate remedial courses has compelled the teachers to create a four-year sequence of slowed-down, yet not remedial, courses. Although the teachers have expressed doubt about the mathematics in a two-year, hands-on geometry course, they recognize that they will have to pass muster with the district. Thus they must arrive at a name for their newly formed courses. Barbara’s inclination is to give it a label that distinguishes it from the regular geometry class (37–39), referring to the omission of two-column proofs, a topic for which, she reminds the group, these students will “not be ready” (38). Noah argues for a more standard name so that “colleges can recognize” the course (40–41). Here, and again in Episode 2.5, Noah, a second-year teacher, appears either uninitiated—or perhaps resistant11—to the group’s conceptions of the kids in the 2-year algebra course. In that episode, Dan and Barbara seek to distin11Noah’s own biography belies deterministic classification of students’ abilities. He was a student who had been routed out of the mainstream into an alternative high school where, according to his own account, he did not learn much math. He eventually went to a community college, discovered a love for math, and graduated as a Phi Beta Kappa math major from a prestigious university.

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guish the 2-year geometry kids (a subset of the 2-year algebra kids) from the college-bound kids, but Noah still imagines these kids applying to colleges who will want to see a recognizable course name.

Episode 2.4: College-bound, Lazy Kids, and Screw-ups: Figuring Out Who Belongs in 2-Year Geometry 43 44 45 46 47 48 49 50 51 52

Dan says, “The college-bound kids won’t take it.” Barbara adds, “Our lazy kids will sign onto it. If the way it’s written says the [state universities] accept it.” Dan demurs, “Even a lazy kid who wants to go to college won’t take it.” Barbara continues, “Our lazy kids will say, ‘That teacher’s too hard, and that teacher’s too hard.’” The “counselors say, ‘OK then take this class.’” Dan says, “If they want to be a screw-up and take two years to get through one year’s worth of work they have the right to screw around.” Noah asks, “Would we let freshmen sign up for them?” Barbara says, “No, it’s for 10th through 12th.”

Mathematics curriculum and kinds of kids. In this episode more than the others, the teachers show the ways that the old course structure served as a classificatory infrastructure, binding kinds of kids with kinds of courses and separating college-bound kids from noncollege-bound kids. In other words, the curricular bins maintained these student categories, helping the teachers’ make sense of the kinds of students associated with kinds of courses, as they have done in the conversation so far. At the same time, the mandate has forced the teachers to rework the course structure and eliminate remedial classes. As a consequence, the new classes that they created have names that blur the crucial college-bound and noncollege-bound division. In Episode 2.4, Barbara and Dan introduce new kinds of kids into the course of this discussion as they try to reconstruct this boundary. College-bound kids will not put themselves in classes with the two-year geometry kids (43), but lazy kids might be drawn in (44–45). A lazy kid—or a college-bound kid who wants to avoid hard teachers (47–48)—who ends up in this new course then becomes a screw-up (49). Noah, still not fully on board with the student-curriculum classification system being reified, asks if freshmen can enroll in the two-year geometry course (51). His question is quickly answered by Barbara (52), who later in the meeting explains that freshmen in geometry are a special kind of college-bound kid: “That’s the seventeen kids I have right now [in freshmen geometry]. They’ll become the thirteen kids who will be in Noah’s calculus course.” This comment works to justify the separation of these students as they are being readied for a

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special place in the curriculum. At the same time, it naturalizes the filtering function of mathematics, as seventeen students will eventually be winnowed down to thirteen. The courses here are not a reflection of content alone. They are also a reflection of the students they enroll. The new two-year geometry course can again be likened to a sorting bin, and in their discussion, the teachers decide who belongs and who does not. By keeping college-bound kids out, they are, in a sense, working around the mandate to eliminate remedial courses by recreating a remedial space with a “fancier,” college-bound sounding name. Although the labels of the twoyear algebra and two-year geometry courses sound more similar to mathematically legitimate courses, operationally, they will contain the same students that the previous courses contained. Such “workarounds” commonly occur when imposed standards do not fit users’ needs (Bowker & Star, 1999). That is, the district has required the elimination of remedial courses, which demands that the teachers reconfigure their course structure. Truly reworking their course structure would require them to take up curricular issues in a serious and sustained manner. They have not developed work practices to support such an endeavor, nor do the six hours the district allotted for this task allow the math teachers to explore such ways of working.

Episode 2.5: Maturation, Innovation, and Other Modes of Recovery: Constructing Kids’ Trajectories 53 54 55 56 57

Noah is looking at the diagrams of the math pathways. Barbara says, “[…] A freshman who fails algebra this year can jump into that [the lower track] pathway.” Noah goes to the white board to write up pathways. He draws: 9th [algebra] [1st half of Alg] [geometry]

58 59 10th Geo [1st half Geo][2nd half Alg] [Alg] [Advanced Alg] 60 61 62 63 64 65 66 67 68

As he’s writing, the teachers begin to talk rapidly, overlapping one another’s talk. I catch these bits of discussion: The kids might want to go at a “faster pace.” Sophomore year some of those “kids really mature.” Or, Barbara says, “They might get turned on by an innovative teacher.” A question is raised: Can kids go backwards in the pathways? The conversation slows. Dan says, “Don’t go backwards, it looks bad.” Barbara cautions, “Don’t assume the counselors will know that going backwards is possible.” Noah asks, “Well, do we have it backwards the way its written now?” It’s decided that although kids can go backwards, it will not be an official part

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69 70 71 72 73 74 75 76

of the pathways representation that they need to hand in. […] Dan says, “From [the first half of geometry], they can go to [the second half] or [geometry]. So in [the second half of algebra], they can go to either of these (he’s pointing to [geometry] or [the first half of geometry]), and the same thing for [algebra].” […] Dan continues, “[geometry] goes to [advanced algebra], math analysis, and calculus.” Noah writes a happy face next to Calculus—a course he teaches. They fill in the rest of the table.

Mathematics curriculum and kinds of kids. The district requested that the teachers represent the solution to their course elimination problem by creating a curricular pathways chart. In this episode, the teachers talk as they represent their solution to the Mismatch Problem that the district’s mandate brought with it. What is interesting here is the way that the talk about students shifts. Up until now, most of the talk has presumed highly determined student abilities: That is, once teachers know a category that students fit in, they freely make certain assumptions about their mathematical abilities and judgments about appropriate instructional strategies. In this episode, these classifications open up to greater possibility. This can be seen as a shift that comes with the shift in microcontext. Prior to this, the discussion centered on imagined classrooms with students and particular curricula. Now, the teachers are literally sketching out the trajectories for students who might start in different places in the curriculum. Their focus changes from concern about their overall curricular organization to the possibilities they are creating for their students. Their talk in this episode reveals the ways for students to change the trajectories that their freshmen course placement might set out for them. The teachers represent the curricular pathways by starting at the possible entry points for entering 9th graders (57). The opening comment acknowledges that some path hopping might be appropriate in cases of failure, as in the example of the 9th grader failing the one-year algebra course that could then move to the two-year course during 10th and 11th grades (54–55). The teachers then discuss ways that students might recover into a higher track through a desire for a faster pace, maturation, or by being turned on by an innovative teacher (62–63). Thus new “recovery” arrows are added to the curricular pathways. Perhaps because the kids in the two-year algebra course need to be reminded “every week, every day” (Episode 2.3), the seeming redundancy of going from the first half of algebra to a full year of algebra does not merit comment. This suggests again that the tracking of college-bound and noncollege-bound kids is more important than the apparent content of the course. The academic status represented by the curricular paths matters more than the content they supposedly cover.

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Discussion of Case 2: Fragmented Professional Vision of Students, Stable Vision of Mathematics As these teachers work to resolve the Mismatch Problem, they reveal a number of conceptions about students, mathematics, and teaching. These conceptions may have been broadcasted more clearly by particular individuals; however, they are distributed beyond individuals, stretching across the curricular structure to such an extent that its underlying conceptual organization needs to be preserved, despite a mandate to eliminate remedial courses. The South High mathematics teachers eliminate remedial courses by replacing them with a sequence of slowed-down college preparatory classes, the two-year algebra sequence followed by a two-year geometry sequence. If the titles of these courses have some meaning, students who take them would have a chance to encounter a more challenging curriculum than that of the remedial courses. At the same time, students who successfully complete all four courses still do not qualify to apply to the state’s university system, thus bypassing the concerns that may have shaped the district’s mandate to eliminate remedial math courses in the first place. The South High mathematics teachers’ understanding of their students appears to be strongly linked to their understanding of their school’s curricular organization. The mandate they received to eliminate remedial courses does not mesh with their understanding of their students and subject. As they work to resolve this mismatch, they deploy and elaborate on their understanding of the students in their courses. As I summarize, I use the categories put forth by the teachers and elaborate on some of the attributes of these students to illustrate what can be learned about the relations among students, mathematics, and teaching. The kids in the two-year algebra course require instruction that relies on a lot of repetition. Some of them are not quick and therefore may become kids in the two-year geometry course. These students lack the logic component, making it difficult to teach them two-column proofs—a necessary part of a legitimate geometry curriculum, which would leave them insufficiently prepared for advanced algebra, except perhaps at the community college. Because the new curricular infrastructure blurs the boundary between college-prep and noncollege-prep classes, college-bound kids could become confused or the lazy kids among them enticed into become screw-ups by taking two years to do a one-year class. The teachers’ talk about kids ties into how they conceptualize mathematics as a subject and, relatedly, possibilities for the mathematics curriculum. The categories for kids appear to lie on the two dimensions of speed and motivation, as represented in Figure 1.

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FIGURE 1 The categories of kids presented in the South High math teachers’ conversations. Here the categories are shown sorted along two dimensions—speed and motivation— and color coded accordingly. This representation is not meant to be literal, but rather to argue that these two dimensions can explain much of what the categories are supposed to represent about kids.

The conceptions of students get built into the curriculum, but for the teachers, the categories of kids are not static. Although certain categories may be viable in South High’s mathematics department, the category system shifts in subtle but important ways to suit the teachers’ different purposes. Once it was clear that a parallel noncollege-prep, non-remedial track was in place, the discussion shifted to possibilities for individual students in the school’s curriculum—quite likely the exceptional cases, because the larger course structure did not imagine these shifts to be common enough to create the curriculum around them—and some of the kids in the two-year algebra course suddenly had new options. These teachers deployed a fragmented professional vision of students in changing and newly reifying their work. Their categories for students shifted along with the conversational referents (course organization vs. students’ opportunities) from a set of highly determined categories to more open categories that allow for more possibilities. Admittedly, these possibilities were vaguely delimited, and during the day’s meeting, actual tales of recovery did not make it to the floor. The teachers also exhibited a professional vision about mathematics curriculum as an entailment of their vision of kids. This vision seemed imperturbably stable, show-

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ing no shifts with the changing contexts of the conversation. Most notably, the quality of the mathematics curriculum was linked with the qualities of students. Two-column proofs, for example, were considered an important part of the legitimate geometry curriculum, whereas hands-on geometry was not a pathway into advanced algebra (Episode 2.2). Students’ possibilities for recovery, as outlined in Episode 2.5, require success in the high-caliber math courses, even if it means redundant content. Although some students might have a better understanding of, for example, the isosceles triangle theorem through a (less legitimate) hands-on investigation than by working on a (more legitimate) two-column proof, that understanding was not deemed sufficient by the course “hierarchy” the teachers developed. To move forward, students have to succeed in the traditional geometry course. This decision values the mastery of mathematical formalisms over other modes of understanding, despite research that has shown that students often master mathematical formalisms with little understanding (Boaler, 1997; Schoenfeld, 1985. The vision of mathematics, then, is one in which formalisms are more mathematically legitimate than other kinds of understanding. Perhaps not coincidentally, this vision is in good keeping with the external accountability pressures that are linked to assessments that reify this same model of mathematics.

DISCUSSION Two Responses to Mismatch: Group-Worthy or Worthy Groups? In this section I summarize common themes found at both sites by comparing and contrasting the modeling of the Mismatch Problem as portrayed in the two conversations. Although the two teacher groups differed in size, history, and orientation to teaching, there were nonetheless commonalities in what unfolded in these two events.

Equity-Geared Reform Sparked Conversations About the Mismatch Problem In these events, two groups of teachers worked together to change their mathematics classes to address inequalities in students’ educational opportunities. In the course of doing so, they had to address perceived mismatches between their students and (proposed and actual) mathematics curriculum. To model the Mismatch Problem, they invoked categories of students that carried with them certain assumptions about subject matter and teaching. In this analysis, I argue that the kinds of categories in use and the entailments of those categories implicate teachers’ de-

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cision making about curriculum and ultimately impact students’ educational opportunities as they get naturalized in the curriculum.

The Mismatch Problem Was Examined at Length by Both Groups of Teachers Despite a general consensus among the teachers at both schools about the importance of increasing educational opportunities of their students, the nuts and bolts of how to do this demanded intensive discussion of the Mismatch Problem. Both Tina (Case 1) and Dan (Case 2) presented a version of the Mismatch Problem to their colleagues. The particular contexts of the reported mismatches differ significantly. Whereas Tina reported an experienced mismatch between the detracked algebra curriculum and the fast and slow kids in her classroom, Dan anticipated a mismatch between the two-year algebra students and South High’s geometry curriculum. Dan reported his problem tersely and collectively (“I don’t think we can fly geometry”), whereas Tina’s account was lengthy and personal (Episode 1.1). This difference provides one possible reason why the South High teachers might have been operating with a more closed classification system for students: They were discussing hypothetical scenarios, whereas Tina was discussing the actual students in her classes. The particularity of her problem became an important resource for both Carrie and Guillermo to challenge her characterization of her students as “fast” and “slow.”

Both Conversations Imply a Source for the Mismatch Problem, Although the Roots Are Found in Different Places The ownership of the Mismatch Problem moved closer to the teacher through the course of the conversation at East High, whereas it remained largely on the students throughout the episode sequence at South High. In the discussions, the teachers’ proposed framings and solutions indicate their understandings of the problem’s cause. At East High, Tina reported searching for “group-worthy problems” while expressing doubt in her ability to “find problems like that.” She placed the responsibility for the mismatch on herself, the teacher, as the organizer of the curriculum. Carrie and Guillermo expanded on her responsibility by including her role as mediator of student status (Episode 1.2) and setter of student-learning agendas (Episodes 1.3 and 1.4). At South High, on the other hand, the students appear to be at the root of the Mismatch Problem: The anticipated students “won’t be able to handle […] the logic in the two-column proofs” (Episode 2.1). Even in Episode 2.4, when the student classification system opened up, except for “being turned on by an innovative teacher,” the modes of recovery to the college-preparatory

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track—maturation, a desire for a speedier curriculum—depend on changes in the students. The burden of the Mismatch Problem remains largely with them.

At Both Schools, the Prevailing Conceptions of Students, Mathematics, and Teaching Were Reinstated Through These Conversations About the Mismatch Problem Tina’s framing of the problem was quietly supported by Christy, another novice in the East High group, but it was vocally challenged by two of the more experienced teachers, including one of the department chairs, thereby preserving the detracked curriculum. Through their interactions with Tina, the senior teachers maintained a curriculum and promoted teaching practices reflective of their understanding of students and mathematics. In contrast, Barbara, the department chair and the most experienced teacher in the department at South High, ratified Dan’s problem quickly. Because of this, Dan’s framing of the Mismatch Problem was elaborated on in the conversation and eventually built into the tracked curricular pathways.

The Complex Classification of Students in Use at East High School Implicates a Different Classroom Role for the Teachers East High’s teachers have developed a variety of work practices to support their collective learning about their expanded role. These teachers’ nonsequential view of mathematics is clearly not typical (Grossman & Stodolsky, 1995). The East High teachers’ model of students places more responsibility on the teachers for addressing potential “mismatches” between curriculum and kids. The onus is not on the students to “mature” or “get turned on”; instead, the teachers must work actively in their classrooms to find group-worthy curriculum, address status issues, and set learning agendas for individual—even “fast”— students.

The East High Teachers Had Developed a Variety of Participatory Structures That Allow Them to Collectively Make Sense of and Question the Open Categories Reified by the Curriculum The East High teachers observe each other regularly, both to model lessons and solicit feedback. They plan and consult with each other on a weekly basis. These participation structures allow them to create a more coherent and coordinated model to guide their work. At South High, the mathematics teachers do not seem to have the kinds of work practices that could support them in questioning their un-

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derstanding of students and mathematics. Their conversation reinscribes their more deterministic classifications for students into the curriculum. CONCLUSION As numerous research studies have documented, many equity-geared mathematics reforms are transformed in the process of implementation, often due to teachers’ apparent misunderstandings of, or outright resistance to, the reform’s intent. The demands of these reforms challenge teachers on numerous fronts: They must teach more and harder content to increasingly heterogeneous classes, engage struggling or apathetic students, and respond to high-stakes accountability systems. In this article, I highlight a conceptual basis for the transformation of these reforms by pointing to the ways extant understandings of students, subject, and teaching can pose keenly felt dilemmas for teachers as they try to reconcile perceived mismatches between the students and curriculum. These conceptions are encapsulated in the category systems embedded in teachers’ everyday conversations with their colleagues and built into the curricular structures through which they view them. Because these category systems provide a vocabulary for fleshing out problems of practice, any solutions that emerge are informed by these underlying conceptions, providing a resource for teacher learning and for making choices about practice. This finding has implications for educational research and practice, particularly for the areas of reform implementation and teachers’ professional development. Because of the role category systems play in modeling problems of practice, they ultimately define an important part of teachers’ zones of enactment (Spillane, 1999), delineating what seems possible when policies face the complex realities of schools and classrooms. For example, if a category system explains students’ success or failure by ascribing them varying degrees of ability and motivation, it delimits a range of reasonable pedagogical responses. Teachers can do little to change students’ innate abilities—their best chance is to be engaging, perhaps overcoming low levels of motivation. Even if they manage to engage students, they may not be able to overcome the perceived deficiencies in their abilities to learn (e.g., students who “forget if they’re not reminded every day.”) However, if teachers are given conceptual support to view mathematics in a more connected way, if they are provided with more complex categories of students in discussions issues in practice, the problem space for teaching shifts accordingly. By introducing the dimension of academic status, for example, to explain students’ reluctance to participate in classroom activities, a new set of pedagogical responses becomes possible. Teachers can find ways to intervene by, for instance, “assigning competence” to these students (Cohen, 1994; Cohen & Lotan, 1997), using their authority to publicly and specifically acknowledge

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intellectual contributions of low-status students, thereby increasing their status in the eyes of their peers. Likewise, if mathematical alacrity is not simply taken as a virtue, “fast kids” can engage in more complex mathematical thinking with the guidance of an attentive teacher, thereby deepening their own understanding of the subject. For equity-geared mathematics reforms to succeed, it appears that related professional development needs to do more than supply teachers with useful mathematical knowledge and pedagogical practices. The sequential view of mathematics needs to be taken on, and with it, a more adequate technical language to represent the complexity of student learning incorporated into teachers’ models of practice. By engaging teachers in careful examination of student work (Franke & Kazemi, 2001) or videotaped episodes of classroom practice (Sherin, 2003), some of these more nuanced descriptions may emerge as teachers’ informal reasoning shifts from explanatory to evidence-based (Brem & Rips, 2000). Category systems form the basis for conversational models of problems of practice. The conceptions of subject, students, and teaching underlying such category systems then get built into whatever solutions emerge. These conceptions may get built into the infrastructure of schooling through curricular pathways or, more subtly, classroom-level teaching practices. Empirically, this study provides researchers with a means to examine prevalent conceptions operating within a teacher community. I do not claim that all individual members of the two groups hold the conceptions described in these conversations, or hold them in the same ways. In fact, there is evidence even within these conversations that a range of conceptions exist (as in Christy’s head nodding with Tina’s description of her problem at East High, or Noah’s questions about ninth graders in the two-year geometry course at South High). Instead, I claim that these were the conceptions that were given voice and operationalized in the teachers’ collective decision-making process. Future research can examine the nature of teachers’ group roles making some conceptions more viable than others. It is not coincidental that the department chairs (Barbara at South High and Guillermo at East High) most clearly broadcasted the dominant conceptions in their respective groups. Various roles seem to emerge when teachers discuss problems of practice, and they may be linked to the acceptance or rejection of category systems and their underlying conceptions in collegial conversations. In this study, the conversational category systems and their affiliated conceptions of subject and practice reflect the range of values voiced in interviews and the observed classroom practices. The teachers’ informal category systems may not fully explain why progressive mathematical pedagogy flourished in the unlikely terrain of East High, whereas it never took root in the seemingly fertile ground of South High. Future work should examine if this confluence can be generalized to other teacher communities and other settings.

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ACKNOWLEDGMENTS The study was supported by a Spencer Dissertation Fellowship. The dissertation received the 2003 AERA Division K Outstanding Dissertation Award and the Outstanding Dissertation Award from University of California–Berkeley’s Graduate School of Education. I thank Rogers Hall, Judith Warren Little, Alan Schoenfeld, Anne Haas Dyson, and Susan Jurow for their assistance and support in the early stages of this analysis. More recently, I have benefited from conversations and correspondence with Manka Varghese, Jennifer Stone, Bret Norris, Yasmin Kafai, Janet Kolodner, and several thoughtful anonymous reviewers of this journal. Any errors that remain are my own.

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McLaughlin, M. W., & Talbert, J. E. (2001). Professional communities and the work of high school teaching. Chicago: University of Chicago Press. National Action Committee for Minorities in Engineering. (1997). Engineering and affirmative action: Crisis in the making. New York: Author. National Commission on Mathematics and Science Teaching for the 21st Century. (2000). Before it’s too late: A report to the nation. Washington, DC: U.S. Department of Education. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Science Foundation. (2000). Science and engineering indicators 2000. Washington, DC: Author. Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of Curriculum Studies, 19(4), 317–328. Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and tracking on opportunities to learn mathematics and science. Santa Monica, CA: RAND. Piccioto, H., & Wah, A. (1994). Algebra: Themes, concepts, and tools. Aslip, IL: Creative Publications. RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education. Santa Monica, CA: Author. Raudenbush, S. W., Rowan, B., & Cheong, Y. F. (1993, Fall). High order instructional goals in secondary schools: Class, teacher, and school influences. American Educational Research Journal, 30(3), 523–553. Resnick, L. B. (1988). Treating mathematics as an ill-structure discipline. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32–60). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Ruthven, K. (1987). Ability stereotyping in mathematics. Educational Studies in Mathematics, 18, 243–253. Saville-Troike, M. (1989). The ethnography of communication: An introduction. New York: Basil Blackwell. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic. Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13–25. Sherin, M. G. (2003). Using video clubs to support conversations among teachers and researchers. Action in Teacher Education, 24(4), 33–45. Siskin, L. S. (1994). Realms of knowledge: Academic departments in secondary schools. London: Falmer. Spillane, J. P. (1999). External reform initiatives and teachers’ efforts to reconstruct their practice: The mediating role of teachers’ zones of enactment. Journal of Curriculum Studies, 31(2), 143–175. Stodolsky, S., & Grossman, P. L. (2000). Changing students, changing teaching. Teachers College Record, 102(1), 125–172. Thompson, A. (1992). Teachers’ beliefs and conceptions. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). New York: MacMillan. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. New York: Cambridge University Press. White, B., & Frederiksen, J. (1998). Inquiry, modeling, and metacognition: Making science accessible to all students. Cognition and Instruction, 16(1), 3–118.

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Wilson, S., Shulman, L. S., & Richert, A. E. (1987). “150 different ways” of knowing: Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring teachers’ thinking (104–124). London: Cassell Educational Limited. Zohar, A. & Dori, Y. (2003). Higher order thinking skills and low-achieving students: Are they mutually exclusive? Journal of the Learning Sciences, 12(2), 145–181. Zohar, A., Vaaknin, E., & Degani, A. (2001). Teachers’ beliefs about low achieving students and higher order thinking. Teaching and Teacher Education, 17, 469–485.

Fast Kids, Slow Kids, Lazy Kids: Framing the Mismatch Problem in Mathematics Teachers’ Conversations Ilana Seidel Horn College of Education University of Washington

This article examines the social nature of teachers’ conceptions by showing how teachers frame the “mismatch” of students’ perceived abilities and the intended school curriculum through conversational category systems. This study compares the conversations of 2 groups of high school mathematics teachers addressing the Mismatch Problem when implementing equity-geared reforms. Although East High teachers challenged conceptions that were not aligned with a reform, South High teachers reworked a reform mandate to align with their existing conceptions. This research found that the teachers’ conversational category systems modeled problems of practice; communicated assumptions about students, subject, and teaching; and were ultimately reflected in the curriculum. Because East High teachers supported greater numbers of students’ success in advanced mathematics, this study considers the relation between teachers’ understandings of student learning and the success of equity-geared math reforms. In addition, this study contributes to the understanding of how teacher conceptions of students are negotiated and reified in context, specifically through interactions with colleagues and experiences with school reform.

Several years ago, I was invited to conduct a workshop on mathematics reform for a group of school leaders. I asked a high school mathematics teacher to present with me. I had met Guillermo Reyes1 while conducting research on teacher learning in high school departments. As a co-chair of East High School’s mathematics department, one of the two in the study, Guillermo impressed me not only with his skill as a Correspondence should be addressed to Ilana Seidel Horn, College of Education, University of Washington, 115 Miller Hall, Box 363600, Seattle, WA 98195–3600. E-mail: [email protected] 1All

proper names of schools and participants are pseudonyms.

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teacher, but also as a mentor to other teachers working to reconceptualize their practice. During the workshop, we demonstrated a typical East High mathematics lesson, modeling a groupwork structure that involved a complex, open-ended task and emphasized strategies to support more equitable student participation in the mathematics. Afterward, one of the participants approached me. I knew him from my research as well. He was the Reform Coordinator of South High School, the second school in the study. He complimented me on the session, telling me how much he had learned from our presentation. Then he said almost desperately, “This is exactly what our math teachers need to be doing! Why aren’t we doing this?” It was an interesting moment, and one in which my dual roles of ethnographer and educator were in conflict. He voiced a question that our research team had puzzled over during the entire course of our 2-year study, and yet my obligations to my participants’ confidentiality disallowed me from pursuing the question with him. Nonetheless, I shared his bewilderment. How was it that East’s mathematics teachers, housed in a fairly typical urban comprehensive high school, had managed to make such impressive strides in improving their instruction, whereas their counterparts at South, a school widely recognized for its progressive orientation, remained committed to less effective teaching methods? Although there are certainly multiple answers to this question, the one I take up in this article focuses on the conceptual resources high school mathematics teachers bring to their encounters with equity-geared reforms. In educational research, such conceptual resources have often been investigated through the framework of teacher beliefs (Nespor, 1987; Stodolsky & Grossman, 2000; Thompson, 1992). This literature provides evidence that teachers’ beliefs influence classroom practice. To use an example relevant to this analysis, a common belief among teachers is that higher order thinking is not appropriate in the instruction of low-achieving students (Zohar, Degani, & Vaaknin, 2001). Such a belief may have enormous instructional consequences, as it constrains the field of possible actions that teachers might take while planning or enacting their lessons. Because beliefs seem to be a critical in shaping teacher practice, as a field, we need to better understand how teachers construct their beliefs (Thompson, 1992). In this analysis, I take up this question using a sociocultural framework of teacher learning (Lave & Wenger, 1991; Wenger, 1998). This framework requires an analytic shift away from beliefs as they are held by individuals and a move toward conceptions as they are distributed across individuals and settings, leading me to the investigation of teacher communities. Specifically, I examine the ways in which conceptions of students, subject, and teaching are embedded in teachers’ daily work, particularly as they encounter problems of practice and work to solve these problems in consultation with colleagues. This article begins with a description of the conceptual framework that informs this study. I then provide an account of the relevant educational context by describing a commonly felt teaching problem: the Mismatch Problem. This is

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followed by a report of the study context, research design, and analytic methods. Two cases of professional conversations are then presented and closely analyzed. The article concludes with a summary of the findings and discussion of the educational implications. CONCEPTUAL FRAMEWORK A situated perspective on learning assumes that people learn as they participate in the socially and culturally organized practices of a community (Lave & Wenger, 1991). I focus on the context of teacher learning through collegial conversations by bringing it together with two other strands of research. The first is research on teacher communities. This line of inquiry has sought to understand the location, nature, and dynamics of collegial interactions within school contexts. Second, because talk is one of the primary activities teachers engage in while working with their colleagues, I draw on studies about the resources for learning embedded in everyday talk, particular the conversational category systems that shape what people perceive in the world. Teacher Communities as Communities of Practice An abundance of research exists on teacher community, and yet it must be reconceived somewhat to align with the notion of community of practice. For more than two decades, teacher communities at the school level have become important sites of investigation in studies of teachers’ work. Increasingly, researchers have grown to appreciate the way that such subgroups can create fundamentally different settings, even within the same school (Ball & Lacey, 1980; Little, 1995; McLaughlin & Talbert, 2001; Siskin, 1994). In the high school, the most salient communities tend to reside at the level of subject-matter departments (Little, 1995; McLaughlin & Talbert, 2001; Siskin, 1994). Prior work on teacher communities suggests that teachers’ colleagues can influence teachers’ approaches to classroom practice (Coburn, 2001; Gutiérrez, 1996; Johnson, 1990) as well as shape their responses to reform (Ball & Bowe, 1992; Coburn, 2001; Little, 1995). However, we are only beginning to understand precisely how these responses get negotiated within teacher communities (Little, 2003). In my research, I have located resources for learning within teacher communities that provide teachers with a conceptual infrastructure—that is, linguistic and semiotic resources for learning and sense making about their work (Horn, 2005). In this analysis, I examine the ways in which conversational category systems constitute an important component of that infrastructure. Turning to Wenger (1998), I adopt a community of practice lens through which to view the mathematics teacher groups in this study.

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This perspective provides a more inclusive notion of community than those developed in the teacher community literature, as it remains more neutral about what constitutes a teacher community. In the teacher community literature, community is a term typically reserved for rarefied places in which teachers collaborate productively toward a common goal—be it the bolstering of conventional practice in “strong traditional communities” (McLaughlin & Talbert, 2001) or learning from practice in “inquiry communities” (Cochran-Smith & Lytle, 1999). Contexts in which teachers do not strive together toward a common vision of teaching but, for example, only “behave as if [they] agree,” have been distinguished as “pseudocommunities” (Grossman, Wineburg, & Woolworth, 2000, p. 13). According to Wenger (1998, p. 73) the three dimensions that characterize a community of practice are (a) mutual engagement, (b) a joint enterprise, and (c) a shared repertoire. This is a broader conception of community than the one put forth in the teacher community literature because it could, for example, potentially encompass the pseudocommunities that have been set apart in the teacher community literature. That is, the imperative to “behave as if we all agree” would not disqualify a group from being viewed as a community of practice. Instead, such a norm of interaction could be taken as part of the shared repertoire that competent members develop to mutually engage in the group’s joint enterprise. Wenger (1998) argued that communities of practice are an important unit of analysis in investigating learning, as such communities play an important role in meaning making. In particular, he described how meaning is negotiated in communities of practice through the complementary processes of participation and reification. Participation refers to “a process of taking part and also to the relations with others that reflect this process” (p. 55). Participation encompasses conflictual, harmonious, intimate, political, competitive, and cooperative relations among people. The complement to participation is reification. By reification, Wenger referred to the ways that we treat abstractions as if they substantially exist in the world. Such reifications represent aspects of a community’s joint enterprise, as “any community of practice produces abstractions, tools, symbols, stories, terms, and concepts that reify something of that practice in a congealed form” (p. 59). Together, these processes contribute to the making of meaning. Whereas participation facilitates local interpretation of reified ideas among members of a community, reification helps compensate for the potentially confusing fluidity and local specificity that arise out of participation. Between Participation and Reification: Resources for Learning in Everyday Talk How might teachers’ communities of practice support their learning? To pursue this question, I turned to studies of learning resources in everyday talk. Conversational category systems, the ways in which everyday language classifies things in

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the world, are one such resource. These category systems demand both participation and reification in their use. On the one hand, they are vital tools for participation, because collegial interactions are largely constituted by conversations. Thus, any event in which meaning is being constructed through participation will necessarily require the deployment of conversational categories. On the other hand, the categories themselves are embedded in particular terms, which congeal the ideas that they represent. The negotiation of these category systems, then, requires both critical meaning-making processes. Charles Goodwin (1994) explored this negotiation in some depth in his study of how professionals develop shared understandings of important aspects of their work by communicating a more or less standardized professional vision. He described the discursive practices that permit professionals to do so, helping them organize, notice, and interpret complex perceptual fields. Among the practices he described are coding schemes, or a “systematic practice used to transform the world into the categories and events that are relevant to the work of the profession” (p. 608). Some aspects of teachers’ work require the use of formal coding schemes, or those that are an official part of the functioning of the workplace. For example, in most U.S. schools, teachers are required to distinguish between A work and B work when assigning grades. From an ethnographic perspective, the local meanings of any such classification system remain an open question (Emerson, Fretz, & Shaw, 1995). In pursuing group members’ local knowledge of their setting (Geertz, 1983), ethnographers attend not only to the indigenous meanings of formal category systems (A vs. B work), but to the meanings of the informal category systems that arise in interaction as well. Informal category systems are a part of any workplace (Bowker & Star, 1999) and mark membership in a community. Following Goodwin (1994), I argue that these category systems also provide resources for professional learning. Here I explore one such set of informal categories: namely the categorization of students and its relation to teachers’ conceptions of their subject and teaching practice.2 Conversational category systems are an interesting focal point within a community of practice framework for learning. They provide heuristics for modeling the world, allowing people to represent problems of practice and work out possible solutions. But classifications necessarily simplify complex phenomena, and thus risk providing precision over validity (Bowker & Star, 1999). When teachers consult each other about problems of practice, they sometimes confront the limitations of these models, opening up the possibility for shifts in meaning as categories are negotiated while teachers alternately invoke, reinstate, and challenge the assump2The social organization of schools as it relates to categories of students has been explored previously. For example, McDermott (1993) argued that schools are organized to help categories such as learning disabled “acquire” students; Eckert (1989) examined the social spaces that separate jocks from burnouts in high schools.

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tions that underlie them. Language, patterns of interaction, and modes of participation become resources for modeling the world (Goodwin, 1994) and are intimately tied to teachers’ local professional cultures. By analyzing the conversational category systems that emerge within teacher communities, I uncover a sociocultural basis for teachers’ conceptions of their students, subject, and teaching practices and thus open a place to explore professional learning. I argue that conversational category systems shape teaching practice in three ways. Category systems deployed in teachers’ conversation (a) model problems of practice while simultaneously (b) communicating beliefs about students, subject matter, and teaching. As problems of practice are solved, these conceptions of students get (c) built into the organization of curriculum, naturalizing and reinforcing the beliefs they represent.

EDUCATIONAL CONTEXT Mathematics Education and the Mismatch Problem In this section, I describe a common framing of a problem, which I have come to call the Mismatch Problem, encountered by high school mathematics teachers in the current U.S. educational context. Over the past two decades, educators have gradually redefined the goals of, and audience for, mathematics instruction. They urge teachers to focus on processes such as problem solving, reasoning, mathematical connections, and communication in their classrooms—and to include all students (not just the “college intending”) in those activities (National Council of Teachers of Mathematics, 1989, 2000; RAND Mathematics Study Panel, 2003). Because of the disproportionately low numbers of poor students and students of color who participate in advanced mathematics classes (National Action Committee for Minorities in Engineering, 1997; National Commission on Mathematics and Science Teaching for the 21st Century, 2000; National Science Foundation, 2000; Oakes, 1990), equity-focused high school math reforms question the differential curricular opportunities made available to students. In response, mandates have emerged from policymakers to reconfigure the curriculum through strategies such as eliminating remedial courses, instituting heterogeneous classes, or whole-scale detracking. Many high school teachers experience these mandates as a challenge to their work. In my interactions with them as a teacher educator and researcher, I have found that they express the pressures of the current educational climate by invoking the Mismatch Problem. That is, teachers often see their students’prior achievements as incommensurate with a rigorous mathematical curriculum—and, in many cases, students themselves concur, only adding to teachers’challenge. At the same time, increased pressure from accountability structures such as curriculum standards and large-scale testing pushes teachers toward increased coverage of difficult content.

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Faced with either watering down the content of a college preparatory curriculum or failing large numbers of students, teachers experience a keenly felt dilemma: Do they remain loyal to the rigor of the curriculum, or do they teach to the level of their students? Most teachers end up feeling torn between the subject and their students, struggling and compromising as they try to reconcile their commitments to both. Conceptual Underpinnings of the Mismatch Problem Although this dilemma is genuinely felt, it reveals a particular conception of mathematics and students. I argue that this framing of the Mismatch Problem is rooted in a sequential view of the subject (Grossman & Stodolsky, 1995; McLaughlin & Talbert, 2001; Ruthven, 1987; Stodolsky & Grossman, 2000). That is, teachers frequently regard mathematics as a well-defined body of knowledge that is somewhat static and beholden to a particular order of topics. This was the collective view expressed by the South High mathematics teachers and made it seem impossible to participate meaningfully in some of the broader reforms taking place at their school. This perspective has logical consequences for both instruction and student learning. In light of this view, the main goal of teaching is to cover the curriculum in sequence to prepare students for subsequent coursework. Relatedly, students must learn prior topics in the sequence to move forward in the curriculum. Absent another model of mathematics, the teachers’ felt dilemma is understandable. The view of mathematics as a system of important and deeply connected ideas put forth by math educators (Boaler & Humphreys, 2005) and cognitive scientists (Resnick, 1988) has not found its way into many high schools. East’s mathematics teachers were a notable exception to this general rule. As shown subsequently, their collective work is governed by a connected and conceptual view of mathematics. The Mismatch Problem is built on a second common assumption: namely, that low-achieving students cannot learn mathematics through inquiry-type instruction (Raudenbush, Rowan, & Cheong, 1993; Zohar, Degani, & Vaaknin, 2001). Teachers often view cognitively demanding modes of instruction as inappropriate for low-achieving students. The department chair of South High frequently expressed this perspective, telling a novice teacher that collaborative learning, for example, only works with “the cream” (i.e., high-performing students). In fact, ample evidence has suggested that the metacognitive aspects of inquiry-based approaches greatly benefit all students, including those who have encountered past academic difficulties (Boaler, 1997; Boaler & Staples, in press; Schoenfeld, 2002; White & Frederiksen, 1998; Zohar & Dori, 2003). In this vein, the East High teachers worked to involve all of their students in a demanding and conceptually rich mathematics curriculum. In this article, I analyze conversations of the East and South High mathematics teachers as they work to address the Mismatch Problem. I argue that the conceptions teachers invoke to frame the Mismatch Problem reside at the heart of equita-

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ble teaching. As long as teachers believe that some kinds of mathematical activity are not viable for certain groups of students—whether because of students’ prior preparation or innate abilities—teachers will have little impetus to seriously engage in developing their pedagogy in ways that will reach all students with richer, more challenging content.

RESEARCH DESIGN Study Goals, Settings, and Participants The research for this article took place in the context of a larger project, Teachers’ Professional Development in the Context of Secondary School Reform,3 focusing on teachers’ work in the Mathematics and English departments of two California high schools. Our research team investigated the way teachers’ professional communities influenced both their professional learning and commitments to teaching using a multilevel, comparative case study design. Both South and East High Schools have strong local reputations for their involvement in school improvement efforts. Both schools are engaged in reforms aimed at reducing disparities in student achievement. They both enroll a middleand working-class population and have ethnically diverse student bodies. Notably, a greater number of East High’s students come from groups that have been traditionally underrepresented in higher education: 27% of South’s 750 students and 59% of East’s 1,500 students identify as African American or Latino/a.4 At the same time, the schools approach equity-geared reforms in different ways, providing fodder for analytic comparison. South High emphasizes school-wide reform. For about a decade, the school has restructured to support reform efforts. School-wide activities target change: teacher research groups, integrated student projects, and participation in whole-school reform networks. Although South High’s decade of reform overlaps with an era of activism from both the National Council of Teachers of Mathematics and its state level affiliates, these ideas—available to the South High mathematics teachers—have not taken hold in this school, which otherwise prides itself on innovation. East High, from the outside, looks more like a typical comprehensive high school. Grassroots efforts from the departments have driven some of the school-wide reforms, such as block scheduling, but the leadership in equity-geared reforms remains at the level of the subject areas. The mathematics department has led the way on many reforms—creating time for teachers to work together, plan collaboratively, 3Judith Warren Little, Principal Investigator. Funded by the Office of Educational Research and Improvement, and the Spencer and MacArthur Foundations. 4Student population sizes are rounded to the nearest multiple of 50.

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and try new pedagogies—establishing a culture of collaboration, risk, and improvement. An accreditation review in the mid-1980s spurred this effort when the review committee reported to the teachers that students felt that they “could not learn math” in the school’s program. They also advised the mathematics department to develop “an aggressive posture toward reform.” Taking this charge seriously, the department chair sought out all kinds of professional development with her colleagues. For example, they attended workshops on problem solving and were “surprised” at how much their students could do. This spurred a pursuit of ongoing instructional improvement that continued through and beyond the time of this study. In this research, I focus on the mathematics teachers at the two high schools and examine the ways their participation in various professional communities influences their informal learning. Professional communities are identified as groups that (a) have some kind of collective identity (e.g., “the Algebra Group” and “the Math Department”) and (b) are in some way oriented toward a collective conception of their work. Relating this construct to Wenger (1998), these criteria signal the existence of communities of practice—that is, groups with a shared repertoire of practices who are mutually engaged in some kind of joint enterprise. Teachers in self-identified groups marked their mutual engagement by giving themselves a name. The joint enterprise, in these cases, was their charge to teach their students mathematics. At the very least, these groups’ shared repertoire included regularly scheduled meetings. The more subtle aspects of the repertoire were for us to uncover empirically. It was clear from the outset that the two groups differed in both size (5 teachers at South vs. over 10 teachers at East High) and orientation toward mathematics teaching. Both groups felt strong commitments toward their students’ learning, although, because of the differences in orientation toward the Mismatch Problem, these manifest themselves differently in each site. Data-Collection Strategy Because of the focus on in situ teacher learning, the research relies on a variety of ethnographic data. These data were collected at the two case study schools over an 18-month period and include observational data of collegial encounters and classroom teaching, interviews with individual teachers and administrators, and a variety of relevant school and classroom documents. Although the larger study includes data collection at the individual, community, and policy context levels, in this analysis the focus is on the level of teacher community. Central to the research team’s data-collection strategy was the accumulation of records of collegial conversations and related artifacts. The majority of these conversations were audiotaped and transcribed, some were observed and recorded through running fieldnote records, and some were videotaped. We assumed that, among other things, these records would help us understand how teachers’ collectively conceptualized critical aspects of their work.

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Analyzing Pedagogical Reasoning This analysis started with a question: “What are the conceptual resources high school mathematics teachers bring to their encounters with equity-geared reforms?” To approach this question, I needed to find instances in which teachers’ encounters with these reforms were made visible. I used the heuristic of looking for moments of perturbation, such as when a teaching practice was being rendered as problematic in conversation. Such instances require teachers to engage in pedagogical problem solving, thus making available for analysis the sense-making resources they invoked within their communities. In line with the interpretive stance of ethnographic inquiry, I drew on analytic techniques arising out of the ethnomethodological tradition of seeking to understand people’s actions within their contexts and webs of meaning (Garfinkel, 1967; Geertz, 1973; Erickson, 1986). For fine-grained analysis of talk-in-interaction, I adopted complementary techniques from the ethnography of communication (Hymes, 1974; Saville-Troike, 1989). Guided by my question, I developed a unit of analysis called episodes of pedagogical reasoning (EPRs) to make sense of the data.5 I define the EPRs to be units of teacher-to-teacher talk in which teachers exhibit their understanding of an issue in their practice. Specifically, EPRs are moments in teachers’interaction in which they describe issues in or raise questions about teaching practice that are accompanied by some elaboration of reasons, explanations, or justifications. These episodes can be individual, single-turn utterances, such as “I’m not using that worksheet because it bores the kids.” Alternatively, these can be multiparty coconstructions over many turns of talk. My decision–rules for locating EPRs were mostly topical. Frequently, these longer episodes are signaled by a teacher’s question or the raising of a problematic issue, such as “I have a handful of kids who are not successful. How is this going to impact our classes next semester?” I then coded the EPRs by topical content so that I could compare thematically similar conversations across time and settings. For example, the codes student engagement, student failure, assessment, and alignment of students and curriculum were all topics of multiple EPRs. The last coding category yields the EPRs I focus on in this article. EPRs provide a means for comparing the ways teachers collectively reasoned about their work, providing insight into the social resources for teachers’ learning. Notably, discussions of curriculum at the two schools occurred at different levels of description. At South High, norms of privacy (Lortie, 1975) prevailed around classroom practice and little was discussed about actual classroom activities. The few times these were discussed, it was in an expository rather than a collaborative 5The concept of pedagogical reasoning has been discussed by others who studied teacher knowledge (McDiarmid & Ball, 1989; Wilson, Shulman, & Richert, 1987). What distinguishes my use of this construct is that I am investigating pedagogical reasoning as it emerges in teachers’everyday conversations.

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way. At East High, classroom practice and activities were an important part of ongoing discussions about curriculum, as were issues of pacing, assessment, and student response. Understanding this particular difference in the communities is beyond the scope of this article. Although the intensive discussions about curriculum took place at different levels, they represent what these groups had come to recognize as discussable instantiations of curriculum and are therefore useful in a comparative analysis. Focal Data Selection In this article, I compare two thematically related EPRs to illustrate the differences in the reasoning resources and practices of the teachers at East and South High Schools. Neither EPR was typical within the respective teacher community—both were longer and more in-depth than the average—but the conceptions of students and subject represented within each EPR were representative of the conceptions of students and subject found elsewhere in their communities, albeit more clearly elaborated on, making them richer for analysis. Both of these EPRs were spurred by a confrontation with the Mismatch Problem, so the teachers’ reasoning about issues of alignment of students and curricula is made apparent. The EPR from East High comes from a transcript of a videotaped meeting in September 21, 1999. In it, many details of interaction are available for interpretation, including direct quotes, gaze, and body language. The EPR from South High comes from qualitatively different data: fieldnotes from a daylong meeting on January 31, 2000. I interpret the interactions at each site appropriately for the kind of data available. Details, such as gesture and intonation, were not as available in the fieldnote data. However, the ways the teachers framed the Mismatch Problem and used their collectively developed understanding to work on a solution was available in both the fieldnote and the video data and therefore serves as a basis for this comparison. Limitations of the Analysis Although this analysis reveals important dimensions and consequences of category construction in teachers’ work, it is limited in its access to the full complexity of the social processes under examination. The analysis tracks the making (and remaking) of the category systems revealed in teachers’ talk and activity, but this talk and activity is not fully transparent. An important dimension of the social practice surrounding the interactions is therefore less available for analysis. The category systems exhibited in teachers’ talk may reveal schematic aspects of the work culture—the “relatively stable, prefabricated aspects of practice that actors have access to as they enter into engagement” (Hanks, 1996, p. 233)—but they often obscure the emergent ones. In other words, participants may utilize the category

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systems strategically in interaction for their own emergent purposes. The exact nature and multiplicity of those purposes are often in process and negotiation and are not always available for analysis, even with rich supporting ethnographic data. In my analysis, I try to be mindful of this complexity and provide interpretations that reflect the data and information to which I have access. Despite these social complexities, this analysis reveals important resources for teachers’ learning in the workplace. The decisions that the teachers made as a result of these conversations have a very real impact on students’ opportunities for learning mathematics. The discursive resources that went into the deliberation and construction of these decisions are available for examination and are at the heart of this analysis. I argue that the consequences of these categories are great enough to merit examination in themselves, even when the social complexity is considered.

CASE 1: THE “FAST KIDS” PROBLEM AT EAST HIGH SCHOOL Overview In this section, I present the coconstruction of students, mathematics, and teaching practices at East High School through the conversational categories that model problems of practice, communicate assumptions, and are ultimately reflected in the curriculum. I provide some background to the event, contextualizing the teachers’ conversations and activities. A close analysis of four sub-EPRs follows. I pay particular attention to the ways in which students, mathematics curriculum, and teaching practices are coconstructed in interaction. Finally, I synthesize these strands of analysis and reflect on the implications for understanding the teachers’ implicit models of students, mathematics, and teaching practice. Background of the Event The teachers’ at East High were in their first year of full-scale detracking in their mathematics courses. The previous year, two teachers, Guillermo Reyes and Carrie Edwards, piloted the group’s detracking efforts by teaching Algebra 1 content to their Math A (pre-algebra) students. They reported that the students who failed this experimental course did so because of attendance and homework issues and not because of an inability to handle the material. At East High, the school year is organized into two terms, with students taking four 90-min courses per term. A term-long class carries the same number of credits as a traditional 1-year course. To accommodate their reworking of the curriculum, the algebra course was taught over two terms, or a full academic year. All students took the two-term algebra course. To sustain their ongoing in-

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TABLE 1 A Summary of Transcript Conventions Symbol / = (.), (5s) [, // , ? :: . italics CAPITAL LETTERS (??), (cow) (( laughs )) (1), (2)

Description Self-interruption No gap between utterances Very slight pause, 5-sec pause Beginning of overlapping utterances, end of overlapping utterances Low rise in intonation High rise in intonation Marks lengthened syllable, each “:” equals one “beat” Low fall in intonation Marks stress Increased volume Unclear reading, tentative reading Marks other voice qualities or actions Links talk to gestures described at end of transcribed turn

vestigation into curriculum and activities that support the heterogeneity of their classes, East High’s algebra teachers met weekly, donating 1.5 hr of their time after school. The analysis that follows examines an extended EPR that took place during one of these after-school meetings. All of the members of the Algebra Group were present at this September 21, 1999 meeting. The primary participants in this episode sequence were Guillermo Reyes, Carrie Edwards, and Tina Lee, an intern. Also present were Jill Larimer, Howard Silver, Charlie García, Christy Roleri, Annie Stolz, Alice March, Judy Stone, and myself. All of the teachers had at least one section of algebra in their schedule. We gathered after school in Charlie’s portable classroom. We started the meeting by working in small groups on a couple of conceptual fraction activities Carrie had worked on with her classes. We then reviewed an activity with an algebra manipulative, with experienced teachers sharing what they had learned from using it before with their classes. The meeting was to go from 3:30 p.m. to 5:00 p.m., and with about 11 minutes remaining, Guillermo suggested that we do a “quick check-in.” The teachers organized their desks in a circle and began the check-in. This episode sequence is excerpted from Tina’s check-in.6 The following video transcript includes notes about movement and gaze in the form of numbers inserted in the text. For an explanation of other transcript conventions, see Table 1.

6“Check-in” has a certain routine structure in the Algebra Group. In this routine, members would state where they were in the common curriculum and what activities they had been doing with their classes. Any questions or issues that arose since the last meeting then followed.

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Analysis of Case 1

Episode 1.1: Tina’s Problem: Closing the Gap Between Fast and Slow Kids 01 Tina: 02 03 04 05 06 07 08 09 10 11 12 13 Christy: 14 Carrie: 15 Tina: 16 Carrie: 17 Tina: 18 19 20 21 22 23 24 25 26 27 28 Christy: 29 Tina: 30 31 32 33 34 35

OK for me, I’m going over perimeters and I’m introducing a little bit of surface area. I’m doing graphical interpretation, finding patterns. I just gave them their second quiz, class average was a B. I give a weekly quiz and I think it’s been helpful. My students, I don’t know where they’re from, but they’re doing so well, I mean they know the difference between a linear graph versus exponential. But the thing about my students (1) is that there’s kids that know a lot (2) and then there’s kids that (.) you know, feel like they’re slow learners (3). (1) Right and left hand alternate drum beat gesture, flexing at wrist (2) Gestures two hands parallel, on her left side, indicating “kids who know a lot” (3) Uses same gesture to her right side, indicating “kids who feel like they’re slow learners” (Nodding vigorously from “a lot” to “slow learners”) (Leans back in desk, props self up on elbows.) And I’m trying to find group-worthy activities (Nods) where the kids who are fast learners (1) and the kids who are slow learners (2), that it can close the gap (3). So that the kids that are slow learners can contribute and can you know feel (4) smart (5), but I don’t know if I can find activities that are group-worthy activities like that. Because I can feel the um frustration of the fast learners, like (1) Puts out left hand, where she had indicated “kids who know a lot” (2) Puts out right hand, where she had indicated “kids who feel like they’re slow learners” (3) Waves hands together, to close “gap” (4) Holds overlapping hands to chest (5) Looks at Guillermo (quietly; nodding) Yeah. “This is easy! (1) I already know the answer!” And then there’s kids that are slow learners (2) that are like, “Give me a chance to find the answer!” and it’s almost like they kind of give up (3) because they feel like it’s a speed competition, like who can get the answer the fastest kind of thing. And I’m trying to close the gap (4) between that and that’s been one of my frustrations (5) I think (6). Otherwise, they’re doing well and just trying to find activities (7) group-worthy activities (8). (1) Plops hand on desk in mock frustration

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36 37 38 39 40 41 42

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(2) Gestures with right hand (3) Leans to the right, as if moving away from the “fast kids” on the left (4) Pushing left and right hands together (5) Bouncing hands in prayer position in front of her chest (6) Opens hands, palms up (7) Slaps right knuckles in left palm (8) Looks to Guillermo

Kinds of kids. Tina opens by saying something positive about her kids (“they’re doing so well,” line 5). She starts by contrasting “kids that know a lot” to “kids that […] feel like they’re slow learners” (7–8). She is reluctant to construct this last category, searching for a way to describe what she has seen in her class. Her description is supported by Christy, also an intern, who nods vigorously as she speaks (13). Carrie, a more experienced teacher who helped pilot the detracking effort, retracts slightly at Tina’s words, leaning back in her seat (14). As Tina explains her problem, several things happen to these initial categories. First, “kids that know a lot” becomes “kids who are fast learners.” This suggests that, for Tina, knowing a lot of mathematics is related to being a fast learner. Also, Tina reenacts the frustrations felt by what she sees as two groups of students (29–31). In her framing of the Mismatch Problem, she suggests that the problem lies in her activities, not simply in the kids themselves. Mathematics curriculum. Tina sees the problem in her class, in part, as a result of her curriculum. In introducing her problem, she reports her search for group-worthy activities. She wants to find activities that will allow the “slow learners” to “contribute” and “feel smart” (18–19), as opposed to the “speed competition” that she sees now (32). Tina thus reveals her understanding of the Algebra Group’s notion of “group-worthy activities” to be ones that allow “slow learners” to participate positively. Notably, when she is enacting the frustrations of the fast students, she voices their complaint as “I already know the answer!” (29). This suggests that, in Tina’s classroom, students may perceive the answer as an endpoint for mathematical activity. Teaching practices. By posing this problem at all, Tina as a teacher, expresses a sense of responsibility for the dynamics in her classroom. Indeed, after gesturing to her right and left to indicate the groups of students she sees in her classroom, she indicates her frustration with a gesture to her heart, placing herself in the physical and emotional center of this mismatch dilemma (39). Although she has not been able to close the gap she sees between her students, she

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feels obligated to address it. Her strategy has been to search out group-worthy activities. This response reveals a particular conception of teaching: A teacher is responsible for classroom dynamics, and a teacher has the right (or perhaps even an obligation) to vary the curricular activities as a way of addressing issues in his or her classroom. Both of these conceptions suggest an active role for the teacher: the teacher as a problem solver and the teacher as a curriculum “tinkerer.”

Episode 1.2: Carrie’s Brainstorm: Activity or Status? 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

Carrie: Tina: Carrie: Tina: Carrie: Tina:

Have they done the perimeter lab gear7 yet? Yeah, they did it today. (Nods.) And that still happened? Yeah, yeah. (Nods.) It still happened. Because, um. Let’s see what happened, I mean, (1) you should read the problem some of them wrote. This is a “guess my number” problem and I took this/ each of them wrote their own problems and I took it and I put it in the quiz (2). This guy wrote: “The number multiplied by 6, subtracting 50 and multiplying it by itself is 4,096.” This (3) is a very sophisticated problem to write. And (4) I also took one that’s more simpler. And I just don’t know, I don’t know (5) how to close the gap I guess. (3s) (6) (1) Picks up quiz (2) Raises quiz slightly (3) Flips quiz toward group (4) Points to quiz again (5) Gestures with right hand and left hand far apart, showing the gap, then pushing it closed (6) Looks at Guillermo Carrie: I wonder if it’s: (.) not just (.) the activities you’re doing but also just status. Tina: Mhm. Carrie: You know? I mean even if you did give them a group-worthy task, those kids who: (.) feel like they have low status will just continue to play that

7“Perimeter lab gear” refers to an activity that involves finding the perimeters of two-dimensional rectilinear polygons built with algebra tiles (Piccioto & Wah, 1994). The tiles provide a geometric representation of variable expressions. The perimeter activity provides a geometric representation for combining like terms.

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70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

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role Tina: (Nodding.) Carrie: (Nodding) because (.) they think that that’s what they’re supposed to do. Tina: Mhm. So that’s what my struggle is, what can I do to (.) make them feel motivated? make them feel like (.) I’m a part of this group, make them feel like I’m smart too. Carrie: (Nodding) Yeah. Tina: I’ve been going to the bookstore looking for, you know, (turns quiz paper over) all kinds of activities, but it’s really hard, Carrie: (Nodding) Tina: picking what works (Looks at Guillermo) and I’m not really good at reading directions either so Christy, others: (Laugh) Tina: (Laughs) so I’ve got to see (1) and learn (2). And (3) I’ve been looking at a lot of stuff that you’ve been doing and I’ve done a lot of it in class, but um/ I mean, a lot of it works, but I still need a lot more (4) I guess. (1) Makes vertical karate chop gesture with right hand (2) Waves hand back toward head (3) Gestures toward Carrie with right hand (4) Digging gesture with right hand Carrie: What block do you teach? Tina: Huh? Carrie: What block. Do you teach? Tina: Oh. First block. (Looks to Guillermo) Carrie: I mean, it could be (.) something where some of us could go in and watch and, you know, give you feedback.

Kinds of curriculum or kinds of kids? In this episode, Carrie offers an alternative analysis to Tina’s problem. She asks and learns that Tina found this “gap” between her students with the lab gear activity—one that she seems to believe is group-worthy. She cues Tina to explain what happened with that activity in her class (4) by nodding and not speaking again, giving Tina another turn in their dialogue. Although Tina prompts herself to recall “what happened,” she goes on to describe a kid’s response on the quiz and does not elaborate on her observations about the kids and the lab gear activity. It is not clear why their communication does not line up: Carrie’s question about lab gear was answered by a report of the answer on a quiz. Perhaps Tina’s distress overrides her ability to hear or respond to Carrie’s question, and she instead needs to share more evidence of “the gap.” Alternatively, her frequent glances at Guillermo may indicate that she is distracted, awaiting his response.

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Kinds of kids and teaching practices. Carrie offers another explanation to what Tina is seeing in her classroom. Instead of the “fast learners” and “slow learners” Tina has described, Carrie proposes that the issue might be one of status (65–66).8 She describes for Tina how low-status students “will continue to play that role […] because they think that’s what they’re supposed to do.” Carrie’s analysis positions Tina differently as a teacher in relation to her class. Now, her work is not one of finding the right activity; it has become addressing the academic status issues among the students in her classroom. Notably, Tina again misses Carrie’s communication. She acknowledges her analysis (“Mhm”) and then says that her struggle is to make them feel “motivated” (74). Tina shifts the teaching work away from addressing status issues to motivating her students—work that perhaps makes more sense to her. She continues by returning to her description of the search for better curricular activities. Carrie then proposes something more concrete by offering the group’s situated assistance with her problem through peer observation. Even though Carrie has not managed to communicate her ideas effectively to Tina, the group’s practices of peer observation and debriefing provide her with another opportunity to help Tina reconceptualize her problem. Episode 2.3: Guillermo’s Prediction: Modeling Thinking About Kids 97 Guillerm: (Shifts in desk). Um. OK, I have a prediction. (.) My prediction is that you won’t be able to do anything about it (1). But that, I think that’s 98 from (.) from thinking about a group of kids (2) as slow learners and 99 that’s, that’s (3) how we’re, we’re acclimatized to, to (.) think about 100 learning (4). And, and I don’t really (5)/ what I find is that when I 101 have mindsets (.) like that that they get in my way (6) in terms of 102 thinking about (7) the curriculum. So (8) it wasn’t a real predic103 tion but I was just trying to be shocking. Um= 104 (1) Waves pen toward Tina 105 (2) Uses pen to gesture a circle on the desk, indicating a “group” 106 of kids 107 (3) Shrugs with right shoulder 108 (4) Alternates left and right hand in shaking gesture in front of face 109 (5) Scratching back of head 110 (6) Uses right hand to point to himself 8Carrie is using language and ideas from complex instruction (Cohen, 1994; Cohen & Lotan, 1997). Complex instruction is a key component of East High’s detracking work. See Horn (2006) for more details.

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111 112 113 Tina: 114 Judy, others: 115 Guillerm: 116 Tina: 117 Guillerm: 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 Carrie: 136 Guillerm: 137 138 139 140 141 142 143 144 145 146 147 148

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(7) Uses pen to indicate a small circle (8) Leans toward Tina a little =It worked.

(Laugh) (laughs) I wasn’t trying to be mean. (Shakes head) You weren’t. (Leaning chin on fist.) Um. One thing I’m, I’m thinking about is (.) the ones that are moving through things really quickly often they’re not stopping to think about what they’re doing, what there is to learn from this activity. So one thing to think about is (.) uh, helping/helping kids that are not stopping (1) to set their own learning agenda, like what should they be working on. (.) What are they learning. And they have to be learning something. (2) (1s) Because even if you have great activities, if if the perception in the class is, um, (3) I’m fast (4), I’m not as fast (5), it’s not going to help with the status issues I don’t think. (2s) But but a kid knowing OK, I can get through this quickly (6) but I’m working on X, being a better group member, because it’s going to help me in my future classes. = (1) Shakes head (2) Taps pen on desk (3) Puts right and left hands, palms down, in front of him (4) Indicates to the left, where Tina had indicated fast kids (5) Indicates to the right, where Tina had indicated the other kids (6) Rapidly moves pen down notepad (Nodding) = Um. Showing off math tools, um, because I know how to do it with a t-table but I don’t know how that relates to a graph yet. Um. (1) I’m making stuff up because I don’t know your kids. But, but like find/think of the ones that you think of as fast learners (2) and figure out what they’re slow at (3). Like what are they slow learners about (.) because that’s their agenda (4) then make that, this is totally simplistic, but (laughing) like, help, help them pick an agenda for their work. (1) Shrugs right shoulder (2) Shakes fist with pen (3) Fist hits desk (4) Waves pen across notepad, writing an “agenda” Tina: (Nods) Guillerm: And one vehicle for doing that is a, is a problem of the week, just one bigger problem (1), that they actually work on independently. Because

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149 150 151

i:f, one thing that I learned in the eighth grade9 is that kids are workthe same complex problems (2) but independently they can see (3) other kids’ strategies for how to/ (3s) for how they approach the problem and how they show their reasoning. That maybe they 152 wouldn’t have thought of because they’re too busy like playing 153 school? (4) I’ve got the answer? I’ve got it done quickly. So (.) 154 they’re slow at something. There’s something to get smarter at. 155 (1) Gestures a circle on notepad 156 (2) Points to different points in gestured circle 157 (3) Gestures out from his eyes toward group 158 (4) Uses pen to make quick writing gestures in the air 159 Tina: (??) (Turns toward Carrie) Guillerm: So fi/ fi/ give them a problem of the week. 160 Tina: Do, do you give problem of the week, Carrie? (Points to Carrie) Carrie: (Nodding) Sure (I can make a note of that). 161 Tina: OK, so I’ll just follow after you. Carrie: Hee hee! Ah!

Kinds of kids. Guillermo’s turn changes the way kids are talked about (97–112). Although Carrie’s response in Episode 1.2 proposes a different dimension (status) in Tina’s category system for kids, Guillermo explodes the scheme altogether by complicating the single dimension of speed (fast–slow), transforming it into a multiplicitous, relational category. He does this in several ways. First, he changes “fast” into “the ones that are moving through things really quickly.” In Tina’s original statement, “fast” seems to be a desirable quality in a student; but here, Guillermo claims that speed may also be a liability, as fast kids “often [are] not stopping to think about what they’re doing.” He reiterates this idea in his final turn, when he describes kids who are “too busy playing school” to consider multiple strategies for a complex problem. He implies that their focus on task completion may be adaptive to schooling, but not necessarily to complex thinking. Second, Guillermo makes “fast” a relative category when he says “think of the ones that you think of as fast learners and figure out what they’re slow at.” His unpacking of the category “fast” provides the possibility for a more complex and particular view on the students in Tina’s classroom. He underscores this in saying, “I’m making stuff up because I don’t know your kids,” revealing a perspective that his generalizations are limited because they might not map directly onto Tina’s students. 9The group makes frequent references to the 8th grade, which constitutes a kind of existence proof for their detracking work. Up until 2 years prior to this meeting, East High School was an 8 though 12 grade school. The math teachers, over the course of a decade, created a detracked 8th grade curriculum.

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Kids are also animated (M. H. Goodwin, 1990) through Guillermo’s talk, as when he uses the first person to voice their imagined perceptions as they sit in Tina’s classroom (“I’m fast, I’m not as fast”) and their ownership of a learning agenda (“I know how to do it with a t-table but I don’t know how that relates to a graph yet”). These imagined utterances impart a view that students themselves have active roles in the classroom—both in negotiating their status and in participating in learning.10

Mathematics curriculum. Although mathematics is not discussed at length, a distinctly nonsequential view of mathematics undergirds Guillermo’s statements. Where it is represented directly in Guillermo’s talk, mathematics is a subject with connections: Guillermo imagines examples of student-learning agendas and picks one that involves connecting tables and graphs. More subtly, Guillermo’s reworking of Tina’s categories of students ties in with notions of mathematical competence. Because students, in his terms, are not simply “fast” or “slow” learners of mathematics, the subject itself takes on more texture. Mathematical competence is not an accumulation of mastery over procedures—something that students are more or less facile with—but a connected web of ideas that requires careful consideration. He reveals this last view of mathematical competence when he expresses concern about “the ones who move things really quickly […] not stopping to think about […] what there is to learn from this activity.” Guillermo proposes that Tina introduce a specific kind of activity, a problem of the week, thus addressing her original question about activities. In doing so, he validates her concern about curriculum while tying it back to the reworked student classification system. His description of problems of the week (complex problem with multiple strategies that require students to demonstrate their reasoning, 150–151) suggests a problem-solving perspective on mathematics. Teaching practices. Teaching, likewise, is made more subtle and complex. Teachers do have to choose good activities: Guillermo ends the episode by suggesting that Tina give her students a problem of the week. But teachers also have to address status issues in their classrooms, as Carrie proposed earlier. Guillermo makes Carrie’s suggestion more specific by describing student-learning agendas. Learning agendas are “what [students] should be working on, what they are learning” (121–122) or “what they are slow learners about” (139–140). Teachers have an active role in mediating curriculum in this representation of teaching. Teachers’ and students’ agency is highly engaged in this notion. Classroom dy10This

practice of inserting imagined classroom dialogue into conversations is explored more in Horn (2002).

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namics are not simply consequences of curriculum and static student ability; teachers help negotiate students’ roles in the classroom by helping them to “pick an agenda for their work,” coordinating their academic goals with those of their students. Guillermo’s notion of learning agendas renders teaching as particular work that requires close attention to specific kids and their purposes in the classroom. Tina is not left on her own, however, to integrate these complex practices in her classroom. Carrie has already offered Tina the support of the algebra teachers through in-class observation of Tina’s teaching. Tina then garners even more support by proposing that she “follow after” Carrie—a practice where a newer teacher stays a day behind a more experienced teacher in the progress of the curriculum, thus allowing the newer teacher to observe, synthesize, and adapt a lesson before bringing it to her classroom.

Episode 1.4: “The One Kid, I Forgot his Name”: Getting Specific 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185

Guillerm: The one kid, I forgot his name, the one that we talked to? Tina: [student’s name]=? Guillerm: =Yeah. (2s) And he is/ he does have a (.) ton of background and lots of desire to learn and lots of intellectual curiosity. He probably is going to get most things quickly so (.) mathematically, you could really/ you’d be looking in bookstores forever trying= Tina: =Yeah= Guillerm: =to find problems. But he/ he’s also impetuous? and he doesn’t, he gets locked (1) into one thought pattern and doesn’t break out (2) very easily. So you may not be able to really challenge him mathematically but I think in terms of (.) the complexity of his thinking you can. (1) Cupped right hand arcs and stops at point in front of chest (2) Open, loose fists waving back and forth Tina: (Nods)Complexityofhisthinkingandumalsobeingabletoworkwith other group members. (Sits up straighter and indicates a circle with right hand.) Guillerm: Yeah. Tina: A lot of, um, the kids in my class, they’re very individualistic Carrie: (Quietly) Mhm. (Nodding)

Kinds of kids. Because he has observed her class, Guillermo is able to get even more specific in his response to Tina. He effectively models alternative language for talking about a kid who might be considered “fast”: This kid has a “ton of background and lots of desire to learn and lots of intellectual curiosity” (169–170).

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At the same time, the kid is represented as “impetuous” and as getting “locked into one thought pattern” (174–175). In these remarks, Guillermo creates a multidimensional classification system of students that grows ever more complex, now encompassing qualities such as “carefulness” and “ability to see multiple perspectives.”

Mathematics curriculum and teaching practices. Guillermo represents mathematics curriculum as a dynamic part of the classroom. Students’ engagement or disengagement is not inherent to a given activity. This is because the teacher can work to bring students into mathematics by elaborating on what is written to push on the complexity of the student’s thinking (177). In this reframing, the “fast kids” problem is not solely about finding the right activity, but also about an appropriate pedagogical response. Discussion of Case 1: A Nonsequential View of Mathematics and Its Consequences In these episodes, the teachers discussed Tina’s rendition of the Mismatch Problem—the gap between fast and slow kids in her classroom. Her problem is addressed by examining issues of students (learning agendas, categories of kids), mathematics curriculum (group-worthy problems, problems of the week), and teaching practices (setting learning agendas, incorporating problems of the week, following an experienced teacher). Tina described her efforts to remedy the problem by finding group-worthy problems and was seeking assistance in finding curricular activities that would allow her to better close the gap. Carrie and Guillermo responded to her by working to reframe her problem, challenging her initial categories of “fast” and “slow” learners. In doing so, they were constructing a more complex model for mathematics and student learning than the one Tina introduced. Guillermo, in particular, changed the talk about students by using specific language and talking about actual students. Tina’s initial framing posed a problem of matching students and curriculum by finding group-worthy problems. Carrie and Guillermo’s reframing legitimated Tina’s impulse to find better curriculum and activities (as evidenced by his suggestion that she do a problem of the week) while also imploring her to take action in her classroom to address student status issues and set learning agendas. The teachers’ talk about students, shaped by their frame for problems of schooling, tied into conceptions of both mathematics curriculum and teaching. Guillermo’s talk in Episode 1.3 questioned the notion that “fast kids” were necessarily better at math. The model he deployed represented mathematics not as a subject where only the quickest survive (Boaler, 1997), but rather one in which students might bring a variety of abilities to be drawn on. In addition, both he

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and Carrie extended Tina’s portrayal of her work as teacher. Tina characterized her work as one of seeking the right curriculum (“I’ve been going to the bookstore looking for […] all kinds of activities, but it’s really hard”). Carrie pointed out the ways that students’ status issues could undo her best efforts toward curriculum matching (“even if you did give them a group-worthy task, those kids who feel like they have low status will just continue to play that role”). Guillermo extended Carrie’s point about status, turning Tina’s whole framing of the problem on its head by pointing out the problems that the fast kids often have (“often they’re not stopping to think about what they’re doing”) and the kinds of interventions Tina could do as a teacher (“help […] kids that are not stopping to set their own learning agenda”). Teachers have a complex role in the classroom in this conversation. Although Guillermo and Carrie pushed on Tina’s framing of her problem by expanding Tina’s role as teacher, Tina was not left alone to sort out their suggestions. In addition to modeling a possible learning agenda for one of Tina’s students, the group offered its support through peer observation and following. These structures would provide learning resources for Tina. Their existence made it easier for the group to challenge one another’s ideas and convey ideas that went beyond the available language through other kinds of support and follow-up.

CASE 2: REDESIGNING COURSE OFFERINGS AT SOUTH HIGH SCHOOL Overview In this section, I present the coconstruction of students, mathematics, and teaching practices at South High School through an examination of the teachers’ conversational category systems for students used to model the mismatch problem. I provide some background to the event, contextualizing the teachers’ conversation and activity. A close analysis of five sub-EPRs follows, with particular attention to the ways in which students, mathematics curriculum, and teaching practices are coconstructed through the teachers’ interactions. Finally, I synthesize the close analysis by reviewing the category system that developed across the episodes and the underlying conceptions represented. Background of Event The South High mathematics teachers were asked by their district to eliminate remedial courses from their curriculum. Before the day’s work began, the teachers had de-

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cided to replace Math A and Math B (pre-algebra courses) with a 2-year version of the introductory algebra course. The new course would have the same content as a traditional Algebra 1 course, but it would be spread out over 2 years. However, the teachers had to provide 4 years of mathematics courses for these students, as per the district’s mandate. The mathematics teachers were given a paid workday to finalize their response to the mandate. During their 6 hours together, they were also asked to determine the alignment of their subject standards to the state standards, make sure that all courses have identifiable standards, and propose a new curriculum structure without any remedial courses. The analysis that follows focuses on the work and interactions of this last task. The teachers had to eliminate the remedial courses—anything below the college preparatory sequence—while providing 4 years of mathematics coursework for all ninth graders who did not attain a certain grade level certification on a district standardized test. On January 19, 2000, all five members of South’s math department were present for this meeting. They were Barbara Moore, Dan Marcus, Rosa Feliciano, Noah Banes, and Lucy Chan. We met together in Noah’s classroom and sat in a small circle at the front of the room. Barbara, the department head, led the meeting as she typically did. As was common, she had more frequent and longer turns of talk. Dan’s participation was the second most intensive in this meeting. Although Barbara had more years of teaching experience, Dan had been at the school longer and could speak to the history of current structures. Noah, a second-year teacher, was often sought out for input, as he had the strongest mathematics background in the group. Lucy was an overwhelmed first-year teacher whose opinion was seldom offered or solicited, and Rosa’s participation in group discussions was infrequent, although she seemed to know the most about union regulations and contractual rules. They started out the day by finalizing their selection of a textbook for the course before they went on to write the course descriptions required by the district. As the teachers worked together on these projects, they had a number of drop-in visitors, including a district representative answering questions. As a result, the meeting had a public feeling, and there were constant reminders that the district was both underwriting and overseeing it. I acted as a “note taker” in that meeting, allowing me to create in-the-moment field jottings on my laptop computer, which I then refined into full fieldnotes on that same day, filling in missing shorthand and adding remembered details to tell a coherent story of what I observed. Only those words actually taken down at the time of my observation are placed in direct quotes in the fieldnotes. Speech not taken down directly at the time is either presented as an indirect quotation or paraphrased (Emerson, Fretz, & Shaw, 1995).

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Episode 2.1: “I Don’t Think We Can Fly Geometry”: The Case for New Courses 1 2 3 4 5 6 7 8 9 10 11 12 13

Dan says, “I don’t think we can fly geometry.” Barbara agrees. “You brought it up, and you’re telling the truth” that “our kids cannot get through our geometry as we teach it now.” They discuss whether the kids can get through their geometry course. They talk about the foundations for geometry in the algebra textbook. Barbara says: “In this book that we have, they do formulas for circumference of a circle, similar triangles.” She continues, “What they won’t be able to handle is the logic in the two-column proofs.” “Our regular kids can’t handle that.” “One of the problems with the kids we’re putting in [the 2-year algebra course] is they don’t have the logic component.” We’ll still have “kids” who “can’t pass the SAT–9 geometry.” “What you’re doing is figuring out our hierarchy and that’s good.”

Mathematics curriculum and kinds of kids. Dan’s observation starts this EPR by putting forth a problem—namely, their collective capacity to “fly geometry” (1). This establishes the necessity for other alternative college preparatory courses beyond the new 2-year algebra sequence. In addressing the problem, Barbara introduces the first category of students when she restates the problem as “our kids cannot get through our geometry as we teach it now” (3). The meaning of “our kids” has not been made clear in this utterance, but in other conversations “our kids” are contrasted with students at other, higher performing schools. Barbara then specifies which subset of “our kids” is of concern: She hones in on the kids in the 2-year algebra course (10). She then fleshes out some characteristics of these students, providing a resource for those around her to learn her conceptions of them. Similar to the regular kids (with whom they are contrasted) these students “won’t be able to handle the logic in the two-column proofs” (9). When she states that they “don’t have the logic component” (11), she is broadcasting a common belief about low-achieving students (Raudenbush, Rowan, & Cheong, 1993; Zohar, Degani, & Vaaknin, 2001). She also explains that these students “can’t pass the SAT–9 geometry” (11-–12). Although the 2-year algebra class does not yet exist, Barbara appears to be applying her understandings about the type of kids in the school’s current remedial courses. Metaphorically speaking, the remedial courses can be thought of as bins that hold certain kinds of students, and although the labels and course content of those bins might be changing, the types of students they contain are not. In her talk, Barbara both elaborates on the collective understanding of the problem at hand and communicates some conceptions of our kids, the kids in the 2-year

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algebra course, and regular kids. She elaborates on the Mismatch Problem that Dan has put forth by identifying a particular part of the curriculum, the two-column proofs, which even the regular kids “can’t handle.” Because these students “don’t have the logic component” (11), the standard geometry curriculum is an untenable option for them. Barbara refers to the state’s standardized test, the SAT–9, as further evidence that the students in this course will not be adequately prepared for geometry (12). Her understanding of the state’s testing system as being tied to the curriculum also becomes a resource for maintaining these students in a separate course sequence. The Mismatch Problem between the district’s mandate to eliminate remedial courses and the teachers’ perceptions of the geometry curriculum and the students requires further deliberation (and expansion) on the part of the teachers.

Episode 2.2: Not “the Quickest Kids”: A 2-Year Geometry Class 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Dan says, “Until they really enforce keeping the eighth graders back” until they are ready for 9th grade work, we know “we’re not going to be getting the quickest kids.” “Where will we send the kids out of [the second half of algebra] who aren’t ready for geometry?” “Should we keep Math C?” Barbara proposes, “We can call it Integrated Math Concepts?” Dan says, “Do we want to still have another 2 years of something besides geometry?” Barbara says, “The district will not fund any remedial class, so as long as it isn’t remedial.” “If you taught a hands-on geometry, if you took Discovering Geometry [a hands-on geometry textbook] and spread it out over 2 years, the district would buy that. That’s not a remedial course.” Dan says, “Should we be thinking about a 2-year geometry class?” Noah: “A 1-year hands-on geometry class would not be a pathway to advanced algebra.” Barbara leans back in her chair and crosses her arms. She raises her eyebrows and says, “Serra [the author of Discovering Geometry] claims it would.” Noah shrugs and says, “OK.” Barbara says, “I say that the kid is ready for advanced algebra at a community college level, but we don’t have to do it.”

Kinds of kids. Dan’s model of the Mismatch Problem is elaborated on further: Because of inadequate filtering in the eighth grade, the teachers will not get the quickest kids (15–16). This provides an amplified understanding of who the kids in the 2-year algebra course are: They include some not-quick kids who will not be ready for geometry at the end of their time in that course.

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Mathematics curriculum. The model of the Mismatch Problem opens itself up to a particular range of solutions. In this case, it seems to be understood that the kids in the 2-year algebra course have enough not-quick kids among them to warrant a separate follow-up course in the curriculum. Barbara proposes offering a “hand-on” class spread out over 2 years (23–24). As they deliberate the nature of this follow-up course, the teachers reveal their sense of mathematically legitimate (and less legitimate) curriculum, as Barbara and Noah compare a formal geometry course to a hands-on geometry course. Noah’s concern about preparation (discussed next) suggests that such a course, in his view, would be more of a stopgap than a bona fide mathematics course. The presentation of mathematics, then, determines its validity for the teachers and not its accessibility to students. The teachers face the challenge of creating a mathematics class that accommodates the not-quick kids but, at the same time, per district mandate, is not remedial (21–22). A hands-on geometry over 2 years satisfies these constraints (23–24). Mathematics curriculum and teaching practices. Embedded in this episode is another elaboration on the problem the teachers face reconciling the district mandate with their own commitments. When Noah proposes that even a 1-year, hands-on geometry course would not be a pathway to advanced algebra (26–27), he suggests that this alternative only pushes the problem of adequate student preparation further up the curricular ladder. Although Barbara invokes the author of the inductive geometry text as an outside authority to support the course’s legitimacy (28–29), the audience for her comment seems to be a district-level judge of their plan rather than the teachers in the room who know better. She lets Noah know that the preparation problem that he anticipates will not be theirs (“we don’t have to do it”), and that the students will be adequately prepared for a community college advanced algebra class (31–32). Kinds of kids and mathematics curriculum. Because of the irreconcilability of the not-quick kids and the highly valued two-column proofs, the teachers build a separate (albeit less mathematically legitimate) geometry course into their curriculum. The two-year course will provide a place for the not-quick kids to learn hands-on geometry and avoid the rigors of two-column proofs. Episode 2.3: “These Are the Kids Who Forget if They’re Not Reminded Every Week, Every Day”: Specifying Curriculum 33 Barbara says: “Integrated Math/Geometry Concepts.” “Mike Serra’s 34 book with a lot of algebra support.” These are the kids who “forget if

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they’re not reminded every week, every day.” Dan says, “So we can still call it … ” Barbara continues, “We have to fancy up the name.” These kids will “not be ready for two-column proofs.” She’s trying out names, things like “inductive geometry.” Noah: “As far as the names of math classes, I don’t think we should change it so colleges can recognize it on transcripts.” Dan: “We’ll still call it geometry […]” [gives it a label parallel to that of the two-year algebra course].

Kinds of kids and teaching practices. In this episode, Barbara imagines some of the details of the course: the name and the instructional strategies. These details communicate more about the kids in the two-year geometry course (who are the not-quick kids exiting from the two-year algebra course). Barbara describes the content of the course: hands-on geometry with “a lot of algebra support” (34). She goes on to describe the teaching needed for this kind of course by appealing to another characteristic of the kids in the two-year algebra course. The kinds of kids enrolled in two-year geometry need repetitive content, as they “forget” what they’ve learned “if they’re not reminded every week, every day” (34–35). Thus, she is elaborating her understanding of these students while articulating the kind of instruction they require. Mathematics curriculum and kinds of kids. The district’s mandate to eliminate remedial courses has compelled the teachers to create a four-year sequence of slowed-down, yet not remedial, courses. Although the teachers have expressed doubt about the mathematics in a two-year, hands-on geometry course, they recognize that they will have to pass muster with the district. Thus they must arrive at a name for their newly formed courses. Barbara’s inclination is to give it a label that distinguishes it from the regular geometry class (37–39), referring to the omission of two-column proofs, a topic for which, she reminds the group, these students will “not be ready” (38). Noah argues for a more standard name so that “colleges can recognize” the course (40–41). Here, and again in Episode 2.5, Noah, a second-year teacher, appears either uninitiated—or perhaps resistant11—to the group’s conceptions of the kids in the 2-year algebra course. In that episode, Dan and Barbara seek to distin11Noah’s own biography belies deterministic classification of students’ abilities. He was a student who had been routed out of the mainstream into an alternative high school where, according to his own account, he did not learn much math. He eventually went to a community college, discovered a love for math, and graduated as a Phi Beta Kappa math major from a prestigious university.

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guish the 2-year geometry kids (a subset of the 2-year algebra kids) from the college-bound kids, but Noah still imagines these kids applying to colleges who will want to see a recognizable course name.

Episode 2.4: College-bound, Lazy Kids, and Screw-ups: Figuring Out Who Belongs in 2-Year Geometry 43 44 45 46 47 48 49 50 51 52

Dan says, “The college-bound kids won’t take it.” Barbara adds, “Our lazy kids will sign onto it. If the way it’s written says the [state universities] accept it.” Dan demurs, “Even a lazy kid who wants to go to college won’t take it.” Barbara continues, “Our lazy kids will say, ‘That teacher’s too hard, and that teacher’s too hard.’” The “counselors say, ‘OK then take this class.’” Dan says, “If they want to be a screw-up and take two years to get through one year’s worth of work they have the right to screw around.” Noah asks, “Would we let freshmen sign up for them?” Barbara says, “No, it’s for 10th through 12th.”

Mathematics curriculum and kinds of kids. In this episode more than the others, the teachers show the ways that the old course structure served as a classificatory infrastructure, binding kinds of kids with kinds of courses and separating college-bound kids from noncollege-bound kids. In other words, the curricular bins maintained these student categories, helping the teachers’ make sense of the kinds of students associated with kinds of courses, as they have done in the conversation so far. At the same time, the mandate has forced the teachers to rework the course structure and eliminate remedial classes. As a consequence, the new classes that they created have names that blur the crucial college-bound and noncollege-bound division. In Episode 2.4, Barbara and Dan introduce new kinds of kids into the course of this discussion as they try to reconstruct this boundary. College-bound kids will not put themselves in classes with the two-year geometry kids (43), but lazy kids might be drawn in (44–45). A lazy kid—or a college-bound kid who wants to avoid hard teachers (47–48)—who ends up in this new course then becomes a screw-up (49). Noah, still not fully on board with the student-curriculum classification system being reified, asks if freshmen can enroll in the two-year geometry course (51). His question is quickly answered by Barbara (52), who later in the meeting explains that freshmen in geometry are a special kind of college-bound kid: “That’s the seventeen kids I have right now [in freshmen geometry]. They’ll become the thirteen kids who will be in Noah’s calculus course.” This comment works to justify the separation of these students as they are being readied for a

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special place in the curriculum. At the same time, it naturalizes the filtering function of mathematics, as seventeen students will eventually be winnowed down to thirteen. The courses here are not a reflection of content alone. They are also a reflection of the students they enroll. The new two-year geometry course can again be likened to a sorting bin, and in their discussion, the teachers decide who belongs and who does not. By keeping college-bound kids out, they are, in a sense, working around the mandate to eliminate remedial courses by recreating a remedial space with a “fancier,” college-bound sounding name. Although the labels of the twoyear algebra and two-year geometry courses sound more similar to mathematically legitimate courses, operationally, they will contain the same students that the previous courses contained. Such “workarounds” commonly occur when imposed standards do not fit users’ needs (Bowker & Star, 1999). That is, the district has required the elimination of remedial courses, which demands that the teachers reconfigure their course structure. Truly reworking their course structure would require them to take up curricular issues in a serious and sustained manner. They have not developed work practices to support such an endeavor, nor do the six hours the district allotted for this task allow the math teachers to explore such ways of working.

Episode 2.5: Maturation, Innovation, and Other Modes of Recovery: Constructing Kids’ Trajectories 53 54 55 56 57

Noah is looking at the diagrams of the math pathways. Barbara says, “[…] A freshman who fails algebra this year can jump into that [the lower track] pathway.” Noah goes to the white board to write up pathways. He draws: 9th [algebra] [1st half of Alg] [geometry]

58 59 10th Geo [1st half Geo][2nd half Alg] [Alg] [Advanced Alg] 60 61 62 63 64 65 66 67 68

As he’s writing, the teachers begin to talk rapidly, overlapping one another’s talk. I catch these bits of discussion: The kids might want to go at a “faster pace.” Sophomore year some of those “kids really mature.” Or, Barbara says, “They might get turned on by an innovative teacher.” A question is raised: Can kids go backwards in the pathways? The conversation slows. Dan says, “Don’t go backwards, it looks bad.” Barbara cautions, “Don’t assume the counselors will know that going backwards is possible.” Noah asks, “Well, do we have it backwards the way its written now?” It’s decided that although kids can go backwards, it will not be an official part

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69 70 71 72 73 74 75 76

of the pathways representation that they need to hand in. […] Dan says, “From [the first half of geometry], they can go to [the second half] or [geometry]. So in [the second half of algebra], they can go to either of these (he’s pointing to [geometry] or [the first half of geometry]), and the same thing for [algebra].” […] Dan continues, “[geometry] goes to [advanced algebra], math analysis, and calculus.” Noah writes a happy face next to Calculus—a course he teaches. They fill in the rest of the table.

Mathematics curriculum and kinds of kids. The district requested that the teachers represent the solution to their course elimination problem by creating a curricular pathways chart. In this episode, the teachers talk as they represent their solution to the Mismatch Problem that the district’s mandate brought with it. What is interesting here is the way that the talk about students shifts. Up until now, most of the talk has presumed highly determined student abilities: That is, once teachers know a category that students fit in, they freely make certain assumptions about their mathematical abilities and judgments about appropriate instructional strategies. In this episode, these classifications open up to greater possibility. This can be seen as a shift that comes with the shift in microcontext. Prior to this, the discussion centered on imagined classrooms with students and particular curricula. Now, the teachers are literally sketching out the trajectories for students who might start in different places in the curriculum. Their focus changes from concern about their overall curricular organization to the possibilities they are creating for their students. Their talk in this episode reveals the ways for students to change the trajectories that their freshmen course placement might set out for them. The teachers represent the curricular pathways by starting at the possible entry points for entering 9th graders (57). The opening comment acknowledges that some path hopping might be appropriate in cases of failure, as in the example of the 9th grader failing the one-year algebra course that could then move to the two-year course during 10th and 11th grades (54–55). The teachers then discuss ways that students might recover into a higher track through a desire for a faster pace, maturation, or by being turned on by an innovative teacher (62–63). Thus new “recovery” arrows are added to the curricular pathways. Perhaps because the kids in the two-year algebra course need to be reminded “every week, every day” (Episode 2.3), the seeming redundancy of going from the first half of algebra to a full year of algebra does not merit comment. This suggests again that the tracking of college-bound and noncollege-bound kids is more important than the apparent content of the course. The academic status represented by the curricular paths matters more than the content they supposedly cover.

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Discussion of Case 2: Fragmented Professional Vision of Students, Stable Vision of Mathematics As these teachers work to resolve the Mismatch Problem, they reveal a number of conceptions about students, mathematics, and teaching. These conceptions may have been broadcasted more clearly by particular individuals; however, they are distributed beyond individuals, stretching across the curricular structure to such an extent that its underlying conceptual organization needs to be preserved, despite a mandate to eliminate remedial courses. The South High mathematics teachers eliminate remedial courses by replacing them with a sequence of slowed-down college preparatory classes, the two-year algebra sequence followed by a two-year geometry sequence. If the titles of these courses have some meaning, students who take them would have a chance to encounter a more challenging curriculum than that of the remedial courses. At the same time, students who successfully complete all four courses still do not qualify to apply to the state’s university system, thus bypassing the concerns that may have shaped the district’s mandate to eliminate remedial math courses in the first place. The South High mathematics teachers’ understanding of their students appears to be strongly linked to their understanding of their school’s curricular organization. The mandate they received to eliminate remedial courses does not mesh with their understanding of their students and subject. As they work to resolve this mismatch, they deploy and elaborate on their understanding of the students in their courses. As I summarize, I use the categories put forth by the teachers and elaborate on some of the attributes of these students to illustrate what can be learned about the relations among students, mathematics, and teaching. The kids in the two-year algebra course require instruction that relies on a lot of repetition. Some of them are not quick and therefore may become kids in the two-year geometry course. These students lack the logic component, making it difficult to teach them two-column proofs—a necessary part of a legitimate geometry curriculum, which would leave them insufficiently prepared for advanced algebra, except perhaps at the community college. Because the new curricular infrastructure blurs the boundary between college-prep and noncollege-prep classes, college-bound kids could become confused or the lazy kids among them enticed into become screw-ups by taking two years to do a one-year class. The teachers’ talk about kids ties into how they conceptualize mathematics as a subject and, relatedly, possibilities for the mathematics curriculum. The categories for kids appear to lie on the two dimensions of speed and motivation, as represented in Figure 1.

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FIGURE 1 The categories of kids presented in the South High math teachers’ conversations. Here the categories are shown sorted along two dimensions—speed and motivation— and color coded accordingly. This representation is not meant to be literal, but rather to argue that these two dimensions can explain much of what the categories are supposed to represent about kids.

The conceptions of students get built into the curriculum, but for the teachers, the categories of kids are not static. Although certain categories may be viable in South High’s mathematics department, the category system shifts in subtle but important ways to suit the teachers’ different purposes. Once it was clear that a parallel noncollege-prep, non-remedial track was in place, the discussion shifted to possibilities for individual students in the school’s curriculum—quite likely the exceptional cases, because the larger course structure did not imagine these shifts to be common enough to create the curriculum around them—and some of the kids in the two-year algebra course suddenly had new options. These teachers deployed a fragmented professional vision of students in changing and newly reifying their work. Their categories for students shifted along with the conversational referents (course organization vs. students’ opportunities) from a set of highly determined categories to more open categories that allow for more possibilities. Admittedly, these possibilities were vaguely delimited, and during the day’s meeting, actual tales of recovery did not make it to the floor. The teachers also exhibited a professional vision about mathematics curriculum as an entailment of their vision of kids. This vision seemed imperturbably stable, show-

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ing no shifts with the changing contexts of the conversation. Most notably, the quality of the mathematics curriculum was linked with the qualities of students. Two-column proofs, for example, were considered an important part of the legitimate geometry curriculum, whereas hands-on geometry was not a pathway into advanced algebra (Episode 2.2). Students’ possibilities for recovery, as outlined in Episode 2.5, require success in the high-caliber math courses, even if it means redundant content. Although some students might have a better understanding of, for example, the isosceles triangle theorem through a (less legitimate) hands-on investigation than by working on a (more legitimate) two-column proof, that understanding was not deemed sufficient by the course “hierarchy” the teachers developed. To move forward, students have to succeed in the traditional geometry course. This decision values the mastery of mathematical formalisms over other modes of understanding, despite research that has shown that students often master mathematical formalisms with little understanding (Boaler, 1997; Schoenfeld, 1985. The vision of mathematics, then, is one in which formalisms are more mathematically legitimate than other kinds of understanding. Perhaps not coincidentally, this vision is in good keeping with the external accountability pressures that are linked to assessments that reify this same model of mathematics.

DISCUSSION Two Responses to Mismatch: Group-Worthy or Worthy Groups? In this section I summarize common themes found at both sites by comparing and contrasting the modeling of the Mismatch Problem as portrayed in the two conversations. Although the two teacher groups differed in size, history, and orientation to teaching, there were nonetheless commonalities in what unfolded in these two events.

Equity-Geared Reform Sparked Conversations About the Mismatch Problem In these events, two groups of teachers worked together to change their mathematics classes to address inequalities in students’ educational opportunities. In the course of doing so, they had to address perceived mismatches between their students and (proposed and actual) mathematics curriculum. To model the Mismatch Problem, they invoked categories of students that carried with them certain assumptions about subject matter and teaching. In this analysis, I argue that the kinds of categories in use and the entailments of those categories implicate teachers’ de-

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cision making about curriculum and ultimately impact students’ educational opportunities as they get naturalized in the curriculum.

The Mismatch Problem Was Examined at Length by Both Groups of Teachers Despite a general consensus among the teachers at both schools about the importance of increasing educational opportunities of their students, the nuts and bolts of how to do this demanded intensive discussion of the Mismatch Problem. Both Tina (Case 1) and Dan (Case 2) presented a version of the Mismatch Problem to their colleagues. The particular contexts of the reported mismatches differ significantly. Whereas Tina reported an experienced mismatch between the detracked algebra curriculum and the fast and slow kids in her classroom, Dan anticipated a mismatch between the two-year algebra students and South High’s geometry curriculum. Dan reported his problem tersely and collectively (“I don’t think we can fly geometry”), whereas Tina’s account was lengthy and personal (Episode 1.1). This difference provides one possible reason why the South High teachers might have been operating with a more closed classification system for students: They were discussing hypothetical scenarios, whereas Tina was discussing the actual students in her classes. The particularity of her problem became an important resource for both Carrie and Guillermo to challenge her characterization of her students as “fast” and “slow.”

Both Conversations Imply a Source for the Mismatch Problem, Although the Roots Are Found in Different Places The ownership of the Mismatch Problem moved closer to the teacher through the course of the conversation at East High, whereas it remained largely on the students throughout the episode sequence at South High. In the discussions, the teachers’ proposed framings and solutions indicate their understandings of the problem’s cause. At East High, Tina reported searching for “group-worthy problems” while expressing doubt in her ability to “find problems like that.” She placed the responsibility for the mismatch on herself, the teacher, as the organizer of the curriculum. Carrie and Guillermo expanded on her responsibility by including her role as mediator of student status (Episode 1.2) and setter of student-learning agendas (Episodes 1.3 and 1.4). At South High, on the other hand, the students appear to be at the root of the Mismatch Problem: The anticipated students “won’t be able to handle […] the logic in the two-column proofs” (Episode 2.1). Even in Episode 2.4, when the student classification system opened up, except for “being turned on by an innovative teacher,” the modes of recovery to the college-preparatory

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track—maturation, a desire for a speedier curriculum—depend on changes in the students. The burden of the Mismatch Problem remains largely with them.

At Both Schools, the Prevailing Conceptions of Students, Mathematics, and Teaching Were Reinstated Through These Conversations About the Mismatch Problem Tina’s framing of the problem was quietly supported by Christy, another novice in the East High group, but it was vocally challenged by two of the more experienced teachers, including one of the department chairs, thereby preserving the detracked curriculum. Through their interactions with Tina, the senior teachers maintained a curriculum and promoted teaching practices reflective of their understanding of students and mathematics. In contrast, Barbara, the department chair and the most experienced teacher in the department at South High, ratified Dan’s problem quickly. Because of this, Dan’s framing of the Mismatch Problem was elaborated on in the conversation and eventually built into the tracked curricular pathways.

The Complex Classification of Students in Use at East High School Implicates a Different Classroom Role for the Teachers East High’s teachers have developed a variety of work practices to support their collective learning about their expanded role. These teachers’ nonsequential view of mathematics is clearly not typical (Grossman & Stodolsky, 1995). The East High teachers’ model of students places more responsibility on the teachers for addressing potential “mismatches” between curriculum and kids. The onus is not on the students to “mature” or “get turned on”; instead, the teachers must work actively in their classrooms to find group-worthy curriculum, address status issues, and set learning agendas for individual—even “fast”— students.

The East High Teachers Had Developed a Variety of Participatory Structures That Allow Them to Collectively Make Sense of and Question the Open Categories Reified by the Curriculum The East High teachers observe each other regularly, both to model lessons and solicit feedback. They plan and consult with each other on a weekly basis. These participation structures allow them to create a more coherent and coordinated model to guide their work. At South High, the mathematics teachers do not seem to have the kinds of work practices that could support them in questioning their un-

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derstanding of students and mathematics. Their conversation reinscribes their more deterministic classifications for students into the curriculum. CONCLUSION As numerous research studies have documented, many equity-geared mathematics reforms are transformed in the process of implementation, often due to teachers’ apparent misunderstandings of, or outright resistance to, the reform’s intent. The demands of these reforms challenge teachers on numerous fronts: They must teach more and harder content to increasingly heterogeneous classes, engage struggling or apathetic students, and respond to high-stakes accountability systems. In this article, I highlight a conceptual basis for the transformation of these reforms by pointing to the ways extant understandings of students, subject, and teaching can pose keenly felt dilemmas for teachers as they try to reconcile perceived mismatches between the students and curriculum. These conceptions are encapsulated in the category systems embedded in teachers’ everyday conversations with their colleagues and built into the curricular structures through which they view them. Because these category systems provide a vocabulary for fleshing out problems of practice, any solutions that emerge are informed by these underlying conceptions, providing a resource for teacher learning and for making choices about practice. This finding has implications for educational research and practice, particularly for the areas of reform implementation and teachers’ professional development. Because of the role category systems play in modeling problems of practice, they ultimately define an important part of teachers’ zones of enactment (Spillane, 1999), delineating what seems possible when policies face the complex realities of schools and classrooms. For example, if a category system explains students’ success or failure by ascribing them varying degrees of ability and motivation, it delimits a range of reasonable pedagogical responses. Teachers can do little to change students’ innate abilities—their best chance is to be engaging, perhaps overcoming low levels of motivation. Even if they manage to engage students, they may not be able to overcome the perceived deficiencies in their abilities to learn (e.g., students who “forget if they’re not reminded every day.”) However, if teachers are given conceptual support to view mathematics in a more connected way, if they are provided with more complex categories of students in discussions issues in practice, the problem space for teaching shifts accordingly. By introducing the dimension of academic status, for example, to explain students’ reluctance to participate in classroom activities, a new set of pedagogical responses becomes possible. Teachers can find ways to intervene by, for instance, “assigning competence” to these students (Cohen, 1994; Cohen & Lotan, 1997), using their authority to publicly and specifically acknowledge

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intellectual contributions of low-status students, thereby increasing their status in the eyes of their peers. Likewise, if mathematical alacrity is not simply taken as a virtue, “fast kids” can engage in more complex mathematical thinking with the guidance of an attentive teacher, thereby deepening their own understanding of the subject. For equity-geared mathematics reforms to succeed, it appears that related professional development needs to do more than supply teachers with useful mathematical knowledge and pedagogical practices. The sequential view of mathematics needs to be taken on, and with it, a more adequate technical language to represent the complexity of student learning incorporated into teachers’ models of practice. By engaging teachers in careful examination of student work (Franke & Kazemi, 2001) or videotaped episodes of classroom practice (Sherin, 2003), some of these more nuanced descriptions may emerge as teachers’ informal reasoning shifts from explanatory to evidence-based (Brem & Rips, 2000). Category systems form the basis for conversational models of problems of practice. The conceptions of subject, students, and teaching underlying such category systems then get built into whatever solutions emerge. These conceptions may get built into the infrastructure of schooling through curricular pathways or, more subtly, classroom-level teaching practices. Empirically, this study provides researchers with a means to examine prevalent conceptions operating within a teacher community. I do not claim that all individual members of the two groups hold the conceptions described in these conversations, or hold them in the same ways. In fact, there is evidence even within these conversations that a range of conceptions exist (as in Christy’s head nodding with Tina’s description of her problem at East High, or Noah’s questions about ninth graders in the two-year geometry course at South High). Instead, I claim that these were the conceptions that were given voice and operationalized in the teachers’ collective decision-making process. Future research can examine the nature of teachers’ group roles making some conceptions more viable than others. It is not coincidental that the department chairs (Barbara at South High and Guillermo at East High) most clearly broadcasted the dominant conceptions in their respective groups. Various roles seem to emerge when teachers discuss problems of practice, and they may be linked to the acceptance or rejection of category systems and their underlying conceptions in collegial conversations. In this study, the conversational category systems and their affiliated conceptions of subject and practice reflect the range of values voiced in interviews and the observed classroom practices. The teachers’ informal category systems may not fully explain why progressive mathematical pedagogy flourished in the unlikely terrain of East High, whereas it never took root in the seemingly fertile ground of South High. Future work should examine if this confluence can be generalized to other teacher communities and other settings.

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ACKNOWLEDGMENTS The study was supported by a Spencer Dissertation Fellowship. The dissertation received the 2003 AERA Division K Outstanding Dissertation Award and the Outstanding Dissertation Award from University of California–Berkeley’s Graduate School of Education. I thank Rogers Hall, Judith Warren Little, Alan Schoenfeld, Anne Haas Dyson, and Susan Jurow for their assistance and support in the early stages of this analysis. More recently, I have benefited from conversations and correspondence with Manka Varghese, Jennifer Stone, Bret Norris, Yasmin Kafai, Janet Kolodner, and several thoughtful anonymous reviewers of this journal. Any errors that remain are my own.

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