Educational Researcher

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Paul Cobb, Kay McClain, Teruni de Silva Lamberg and Chrystal Dean. Situating Teachers' .... practice as developed by Lave and Wenger (1991), Rogoff (1995),.
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Situating Teachers' Instructional Practices in the Institutional Setting of the School and District Paul Cobb, Kay McClain, Teruni de Silva Lamberg and Chrystal Dean EDUCATIONAL RESEARCHER 2003 32: 13 DOI: 10.3102/0013189X032006013 The online version of this article can be found at: http://edr.sagepub.com/content/32/6/13

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Situating Teachers’ Instructional Practices in the Institutional Setting of the School and District by Paul Cobb, Kay McClain, Teruni de Silva Lamberg, and Chrystal Dean In this article, we describe an analytic approach for situating teachers’ instructional practices within the institutional settings of the schools and school districts in which they work. In doing so, we draw on an ongoing collaboration with a group of teachers in an urban school district to illustrate both the approach and its usefulness in guiding the development of analyses that feedback to inform such collaborations. The approach involves delineating communities of practice within a school or district and analyzing three types of interconnections between them that are based on boundary encounters, brokers, and boundary objects.

Engestrom’s (1998) analysis of the relatively few lasting effects of U.S. school reform efforts highlights the bifurcation in the research literature. He observes that, on the one hand, school reforms have remained at the level of systems and structures, not reaching the daily practices of teaching and learning in classrooms. On the other hand, attempts to change the daily instructional practices have themselves not been particularly effective in the long run either. The dichotomy of systems and structures, on the one hand, and daily practices, on the other hand, may be an important reason for the difficulties. (p. 76)

Engestrom goes on to discuss what he calls a middle level between the formal structures of schools and the content and methods of instruction.

ur purpose in this article is to outline an analytic approach that enables us to view teaching as a distributed activity and to situate teachers’ instructional practices within the institutional settings of the schools and school districts in which they work. We know from both first-hand experience and from a number of more formal investigations that teachers’ instructional practices are profoundly influenced by the institutional constraints that they attempt to satisfy, the formal and informal sources of assistance on which they draw, and the materials and resources that they use in their classroom practice (Ball & Cohen, 1996; Brown, Stein, & Forman, 1996; FeimanNemser & Remillard, 1996; Nelson, 1999; Senger, 1999; Stein & Brown, 1997). However, the development of an interpretive perspective that seeks to situate teachers’ instructional practices with respect to the affordances and constraints of the schools and districts in which they work gives rise to a theoretical challenge that is evident in a bifurcation in the research literature on teacher change. As Franke, Carpenter, Levi, and Fennema (2001) observe, one body of scholarship on teacher change focuses on the role of professional development in supporting teachers’ reorganization of their instructional practices and their views of themselves as learners. A second, largely independent body of scholarship is concerned with the structural or organizational features of schools and with how changes in these conditions can result in changes in classroom instructional practices. The analytic approach that we propose moves beyond these separate lines of work by focusing squarely on teachers’ interpretations and understandings while simultaneously treating those interpretations and understandings as situated in and at least partially constituted by the institutional settings in which they work.

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Educational Researcher, Vol. 32, No. 6, pp. 13–24

The middle level consists of relatively inconspicuous, recurrent, and taken-for-granted aspects of school life. These include grading and testing practices, patterning and punctuation of time, uses (not contents) of textbooks, bounding and uses of the physical space, grouping of students, patterns of discipline and control, connections to the world outside the school, and interactions among teachers as well as between teachers and parents [and administrators]. (p. 76)

Engestrom characterizes these middle-level features as senseand identity-building processes and argues that they largely determine the sense of schoolwork, and thus the experience of what it means to be a teacher or a student within the institutional setting of a particular school and district. His arguments provide an initial orientation by steering us away from a structural perspective on the school as an institution and toward a focus on teachers’ activities as they participate in what he terms the taken-for-granted aspects of school life. This latter orientation is concerned with the school and school district as lived organizations rather than as formal structural systems that have been abstracted from the activities of the persons who constitute them. Throughout the article, we draw on our ongoing collaboration with a group of middle school mathematics teachers to ground our discussion of the theoretical constructs that we use. We first clarify that when we focus on the functions of teaching and the activities of the people who contribute to the accomplishment of these functions, it becomes reasonable to characterize teaching as a distributed activity. We then follow Wenger (1998) in developing a perspective on schools and school districts as lived organizations that can be viewed as configurations of communities of practice. We next describe the approach that we take when analyzing the interconnections between various communities of practice within a configuration by focusing on (a) boundary encounters in which members of different communities engage in activities together, (b) the role of brokers who are at least peripheral AUGUST/SEPTEMBER 2003

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members of two or more communities, and (c) the role of boundary objects that have been incorporated into the practices of two or more communities. In the course of this discussion, we pay particular attention to the last of these three types of interconnections as the use of tools and artifacts is a relatively inconspicuous, recurrent, and taken-for-granted aspect of school life that is underdeveloped in the research literature both on teacher professional development (Marx, Blumenfeld, Krajcik, & Soloway 1998; Putnam & Borko, 2000) and on policy and educational leadership (Spillane, Halverson, & Diamond, 1999). Against this background, we illustrate the usefulness of the analytic approach by clarifying how ongoing analyses of the institutional setting in which the group of collaborating middle school teachers developed their instructional practices has fed back to inform our work with them.

Interviewer: How does the [reform series] relate to the Prescribed Instructional Program?

Teaching As a Distributed Activity The contention that teaching can be characterized as a distributed activity might initially seem highly questionable, given that most American teachers work in relative isolation and have limited informal professional networks (Lortie, 1975; Meyer & Rowan, 1977; Weick, 1976). However, this contention becomes more plausible when we shift our focus from the competencies and actions of individual teachers working alone in their classrooms to the functions of teaching as they are accomplished in schools and school districts.1 In the case of mathematics, these functions are not restricted to interacting with students in the classroom to support their mathematical learning but also include: • Organizing for mathematics teaching and learning by, for example, delineating instructional goals and by selecting and adapting instructional activities and other resources. • Making mathematics learning and teaching visible by, for example, interpreting test scores or posing tasks designed to generate a record of students’ mathematical reasoning. When we analyze how these latter two functions are actually accomplished, it almost invariably proves to be the case that a number of persons in various designated positions within the school and district are involved in accomplishing them. As an illustration, the district in which the collaborating teachers work, which we call Jackson Heights, is located in a state with a high-stakes accountability program in which students are tested in mathematics at each grade level. As part of this accountability program, the State Department of Education produced a document that we will call the Prescribed Instructional Program2 that specifies the mathematics objectives for each grade level. As part of compliance and in an attempt to reform efforts in mathematics, the district adopted two different textbook series for middle school mathematics. One textbook is compatible with current mathematics education reform recommendations (e.g., National Council of Teachers of Mathematics, 1991, 2000) and the other is a traditional textbook series. In conjunction with the current reform efforts in the district, the mathematics specialists produced a pacing guide to assist the teachers in coordinating the two textbook series with the Prescribed Instructional Program. Their intent in doing so was to ensure that their vision for mathematics instruction was realized in classrooms, namely that the teachers would use the reform series as the primary basis for mathematics instruction. In the following interview excerpt, one of the district’s mathematics specialists clarifies this intent.

As the above exchange indicates, in producing and revising the pacing guide, the specialist and her colleagues were organized for mathematics teaching and learning. In addition to disclosing that a number of persons in a school or district are involved in accomplishing the functions of teaching, analyses of these functions also reveal that the contributors typically use a range of tools (e.g., documents listing state-mandated curriculum objectives, pacing guides, textbooks, classroom observation forms, reports of test scores, copies of students’ written work). It is in this sense that teaching can be viewed as distributed activity that is accomplished collectively by a number of persons using a variety of tools. It is important to stress that this distributed perspective on teaching does not imply that people within a school or district necessarily coordinate their activities seamlessly or smoothly. As will become apparent when we describe the institutional setting in which the collaborating teachers worked, teaching is frequently a site of tension in that people within a school or district are frequently pursuing conflicting agendas. It is these efforts of the members of different communities of practice to pursue sometimes-conflicting instructional visions and to gauge the extent to which their visions have been realized in classrooms that constitute the immediate institutional setting within which teachers develop and refine their instructional practices.

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Specialist:

We felt that it, the [reform series], was aligned fairly well with the Prescribed Instructional Program. Other than geometry . . . We do go back and revise the pacing guide [each year]. There are some gaps in the [reform series] that we have to fill in with material from the Prescribed Instructional Program from the [traditional series].

Interviewer: So the pacing guide gives you an idea of how you could work through the [reform series] and supplement it with the [traditional series]. Specialist:

Exactly.

Communities of Practice The approach that we propose for analyzing the institutional settings in which particular groups of teachers work involves identifying the communities of practice within a school or district whose missions or enterprises are concerned with the teaching and learning of mathematics. The notion of a community of practice as developed by Lave and Wenger (1991), Rogoff (1995), and Wenger (1998) has been used relatively widely to characterize professional teaching communities, particularly those that have been established in collaboration with researchers (Franke & Kazemi, in press; Grossman, Wineburg, & Woolworth, 2000; Lehrer & Schauble, 1998; Nelson & Hammerman, 1996; Stein, Silver, & Smith, 1998; Warren & Rosebery, 1995). As the authors of these analyses make clear, a group of teachers who are working together on instructional issues do not necessarily constitute a community of practice.3 Wenger (1998) discussed three interrelated dimensions that serve to characterize a community of practice:

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• A joint enterprise. In the case of a professional teaching community, the joint enterprise might be that of ensuring that students come to understand central mathematical ideas while simultaneously performing more than adequately on high-stakes assessments of mathematics achievement. • Mutual relationships. In the case of a professional teaching community, these relationships encompass general norms of participation as well as norms that are specific to mathematics teaching such as the standards to which the members of the community hold each other accountable when they justify pedagogical decisions and judgments. • A well-honed repertoire of ways of reasoning with tools and artifacts. In the case of a professional teaching community, this repertoire includes normative ways of reasoning with instructional materials and other resources when planning for instruction. It also includes normative ways of using tasks and other resources to make students’ mathematical reasoning visible. The potential value of this construct to the issue of locating teachers’ instructional practices in institutional context stems from the manner in which it brings together: (a) theories of social structure that give primacy to institutions, norms and rules, and (b) theories of situated experience that give primacy to the dynamics of everyday existence and the local construction of interpersonal events (Wenger, 1998, pp. 12–13). These two types of theories correspond to the dichotomy we discussed in the literature on teacher change between analyses that focus on the structural or organizational features of schools and analyses that focus on the role of professional development in supporting teachers’ reorganization of their instructional practices. Wenger is explicit in characterizing a community of practice as a midlevel unit that does not go as far as either social structures in the abstract or the detailed choreography of interactions. Instead, it captures social structures that are within the scope of people’s engagement with the world. In the case of mathematics teaching, these familiar structures include the relatively inconspicuous, recurrent, and taken-for-granted features of school life to which Engestrom (1998) referred. To ground the discussion, we draw on our ongoing collaboration with a group of six middle school teachers who work in four schools in the Jackson Heights district (JHD). As we have noted, this urban school district that serves a 60% minority student population is located in a state with a high-stakes accountability program. The district had received an external grant to support its reform efforts prior to our collaboration with the teachers. We began working in the district to provide teacher development at the invitation of the district’s mathematics coordinator who then selected the teachers with whom we have collaborated. She was interested in efforts that focused on the middle grades and reported that a significant proportion of the teachers at this level continued to use the traditional textbook series as the primary basis for their instruction. During the 2 years in which we have worked with the teachers, we have conducted 3-day work sessions each summer, three 1-day sessions during the first school year, and six 1-day sessions during the second school year. Our long-term goal is to support the teachers’ development of instructional practices in which they place their students’ reason-

ing at the center of their instructional decision making (Cobb and McClain, 2001). Methodologically, we have used what Spillane (2000) refers to as a snowballing methodology and what Talbert and McLaughlin (1999) term a bottom-up strategy to delineate the communities of practice within the JHD whose missions or enterprises are concerned with the teaching and learning of mathematics. The first step in this type of strategy involved conducting audio-recorded semistructured interviews with the collaborating teachers to identify people within the district who influenced, in some significant way, how they teach mathematics. The issues addressed in these interviews included the professional development activities in which the teachers have participated, their understanding of the district’s policies for mathematics instruction, the people to whom they are accountable, their informal professional networks, and the official sources of assistance on which they can draw. In order to corroborate these interview data, we also administered a survey that addressed these same issues to all the mathematics teachers who work in the same schools as the collaborating teachers. The second step involved interviewing the people identified in the teacher interviews and surveys, in order to understand both their agendas as they relate to mathematics instruction and the means by which they attempt to achieve those agendas. We then continued this process as we identified additional people in this second round of interviews who actively attempt to influence how mathematics is taught in the district. The communities of practice that we identified by analyzing these data are the districtwide mathematics leadership community, the school leadership communities in the four schools in which the teachers work, and the professional teaching community.

The Mathematics Leadership Community The mathematics leadership community includes the mathematics coordinator and three mathematics specialists as core members, and a number of teachers as more peripheral members. In addition to the semistructured interviews conducted with the core members, the data generated to document the activities of this community included follow-up interviews with the core members, observations of professional development sessions conducted in the district by the mathematics leaders, discussions with the mathematics leaders, and an ongoing e-mail exchange. These data consistently indicate that the joint enterprise of this community was to improve mathematics performance of all students, and of minority students in particular, by helping teachers use the reform textbook series as the basis for their mathematics instruction. For example, one of the mathematics specialists explained the rationale for adopting the reform series by first observing: Specialist:

The state requires Algebra One for graduation now. That has not always been true. So, it is about the same time that it was a part of the landscape. I guess we saw as one of the goals, to narrow the gap between minority students’ and White students’ performance.

Interviewer: When you a saw gap, what was the main reason for that gap? Specialist:

We were not sure if it was the materials we were using, or the way they were presented or whether AUGUST/SEPTEMBER 2003

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it was socioeconomic. We thought it was probably some or all of those things. We looked at all the [available instructional] materials. And we thought that using standards-based materials, presenting it [mathematics] in a different way [would narrow the performance gap]. She then went on to explain the rationale for this conjecture: Specialist:

Because standards-based materials incorporate a different way or method of presenting the materials. We looked at our past performance and looked at what we have been doing that does not work. We have this [achievement] gap. We have lowperforming students. So, those kids who are going to get it are going to get it anyway. So, it was those students we worry about, those students who aren’t going to get it right away. What can we do to help them do a better job?

It was also apparent from the initial interviews that the core members of the Mathematics Leadership Community viewed themselves as participants in a broader community of mathematics education reformers4 and had a relatively deep understanding of and a commitment to the general intent of reform proposals for mathematics teaching. The tools with which the members of this community reasoned, as they organized for mathematics instruction, included the reform and traditional textbook series, the Prescribed Instructional Program, and the pacing guide that they produced and revised each year. However, the mathematics leaders all indicated that they relied almost exclusively on scores on the state-mandated test to make students’ learning visible. A division of labor was evident in this community in that the mathematics coordinator spent the bulk of her time completing administrative tasks and coordinating with other groups both within and outside the district. For their part, the three mathematics specialists visited teachers’ classrooms to assist them and also organized professional development sessions on the use of the reform textbook series that they conducted in collaboration with the teachers who were peripheral members of this leadership community.

School Leadership Communities The leadership communities in each of the four schools in which the collaborating teachers worked consisted of the principal and two or more assistant principals. In addition, one or more mathematics teachers in each school was a peripheral member and, for example, occasionally conducted observations of other teachers at the request of a school administrator. We have relied on the semistructured interviews conducted with leaders in each school to document the activities of these communities, and have triangulated these interviews with the collaborating teachers’ descriptions of the settings of their work. These data document that the enterprise of each of the school leadership communities is to raise students’ scores on the state-mandated achievement test. For example, one principal defined a good teacher as someone who has “good test scores” and another stated that “in mathematics an exceptional teacher has kids rank well in academics and behavior.” When asked to whom she was accountable, the latter 16

principal went on to clarify that “the district office focuses on test scores.” The primary tool that members of the school leadership communities used to organize for mathematics teaching and learning was the Prescribed Instructional Program produced by the State Department of Education. To make mathematics teaching and learning visible, they regularly conducted drop-in classroom visits during which they focused on the match between lesson objectives that teachers are required to write on their whiteboard and the objectives specified by the Prescribed Instructional Program as well as students’ behavior and level of engagement. For example, one principal explained: When I walk into a classroom, I look at the objectives on the board, if the students are misbehaving or working in groups. I ask students what they are doing, if they are enjoying the lesson. I look to see if they [teachers] are following the objectives on the board and the lesson plan. I also look to see how the lesson plan is organized.

The school leaders were aware of the mathematics leaders’ efforts to reform mathematics instruction in the district. However, the school leaders indicated that they were open to a range of instructional techniques and to the use of the traditional textbook series as the primary basis for instruction, provided the teachers met the school leaders expectations with respect to instructional objectives and student engagement. Consistent with the limited attention that they gave to the substance of teachers’ instructional practices, each of the school leaders described current reform recommendations in mathematics education in terms of the demathematized generalities reported by Spillane (2000) (e.g., using small group work, manipulatives, and real-world problems to achieve traditional instructional goals). The following comment by one of the principals is representative in this regard: Some teachers completely bought into it [the reform effort in the district] because it is more hands on, involves divergent thinking, and more than one way to solve a problem. Some good math teachers who have been teaching for 15 years or more don’t want to do it. We have different ways of delivering math instruction [in this school]. Our focus is on the Prescribed Instructional Program and preparing for the next level.

The school leaders, like the mathematics leaders, relied on scores on the state-mandated test to make students’ learning visible. It was apparent from both their interview responses and the collaborating teachers’ comments that they also used these scores to make mathematics teaching visible by inferring the extent to which teachers had addressed the objectives specified by the Prescribed Instructional Program. The primary function of instructional leadership in mathematics appeared to be that of monitoring and assessing teachers’ instructional practices with respect to content coverage and student engagement. It is apparent from these cursory accounts of the mathematics leadership and school leadership communities that the visions for mathematics teaching and learning that they attempted to realize in classrooms differed significantly and were in partial conflict. For the school leaders, mathematics teaching was a relatively routine activity whereas for the mathematics leaders it was a complex and demanding activity that requires a deep understanding of both students’ mathematical reasoning and the mathematical

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ideas that are the focus of instruction. Whereas the school leaders viewed different instructional techniques as alternative ways to address traditional goals, the mathematics leaders conceptualized instructional goals in terms of central mathematical ideas and viewed mathematical communication and argumentation not as a possible instructional strategy but as an important goal in its own right. More generally, the school leaders appeared to participate in the discourse of high-stakes testing and accountability (Confrey, Bell, & Carrejo, 2001) whereas the mathematics leaders participated in the discourse of reform in mathematics education.

The Professional Teaching Community It was not until we had worked with the teachers for 18 months that we and the teachers could be said to constitute a community of practice with a joint enterprise. Prior to this, interactions among the teachers during the work sessions that we conducted with them frequently involved what Grossman, Wineburg, and Woolworth (2000) term pseudo-agreements that serve to mask differences in viewpoints. At about the same time that the teachers began to challenge each other’s proposals and interpretations, they also voiced a desire to engage in joint planning and to observe each other’s teaching. This was a significant development as the initial interviews that we had conducted with the collaborating teachers indicated that their informal professional networks were extremely limited and that their mathematics instruction was highly privatized. These findings were corroborated by the survey responses of the other mathematics teachers in their schools. This deprivatization of practice has continued in that the teachers subsequently agreed to circulate video recordings of lessons that they planned and conducted in pairs for analysis and discussion during work sessions. Shortly after these developments, the mathematics specialist who helped organize the work sessions began to participate as a full member of the professional teaching community and the district’s mathematics coordinator began to attend most of the work sessions and to participate as a peripheral member. As will become apparent, when we describe the nature of their participation, they now viewed the teachers as potential contributors to their enterprise of reforming mathematics instruction in the district. The data we have generated to document the learning of the professional teaching community include semistructured interviews conducted with the teachers each year, videotapes of all work sessions, and copies of all material artifacts produced by the teachers.5 To document the teachers’ instructional practices, we have generated three modified teaching sets (Simon & Tzur, 1999) each year for each teacher. A teaching set consists of a series of classroom observations followed by an audio-recorded semistructured interview with the teacher that focuses on instructional planning and on reflections on lessons. As the teachers only allowed us to videotape their teaching after we had worked with them for 6 months, we initially had to record our classroom observations as field notes. There were identifiable regularities in the teachers’ instructional practices when we first began collaborating with them, even though they worked in relative isolation. In general, their instruction focused on students’ acquisition and application of procedures for operating on mathematical symbols, and on the learning of definitions for mathematical terms. Almost invari-

ably, the teachers first demonstrated the procedure for solving a particular type of problem in a step-by-step manner, and then assigned similar problems for the students to solve. Their assessments of students’ reasoning were limited to the correctness of answers. The adjustments they made when students did not produce correct answers typically involved either explaining the procedure for a second time or asking students to check whether they had performed the steps correctly. The teachers’ instructional practices were more heterogeneous after they had collaborated with us for 2 years. Differences were apparent in the extent to which the teachers expected students’ to justify their reasoning, posed questions to understand students’ solution processes, expected students to listen to and make judgments about others’ explanations, and encouraged multiple solutions. All but one of the teachers had previously used the traditional textbook series as the primary basis for their instruction. Three now relied primarily on the reform textbook series and the other three used the two series in combination. It was also noticeable that only two of the teachers continued to use textbooks as blueprints for instruction. The remaining four teachers modified and adapted textbook lessons based on their understanding of their students’ reasoning. Despite these differences, we were able to identify some general patterns in their teaching. For example, there was an overall shift away from demonstrating procedures and toward leading whole class discussions that focused on students’ solutions. This summary description of the JHD serves to illustrate the types of communities of practice that might be delineated and characterized in terms of the joint enterprises that are established, the forms of engagement that are involved, and technical repertoires that they entail. The decision as to how far to continue with an analysis is a pragmatic one. For example, we could have extended the analyses of the JHD to include other communities and groups such as parents and local business leaders. However, there was no indication from the interviews we conducted and from other data sources that members of either of these groups influenced how the teachers, mathematics leaders, and school leaders organized for mathematics teaching and learning or made mathematics teaching and learning visible. We will consider the relations between the teachers’ evolving instructional practices and the institutional setting in which they worked when we illustrate the process of documenting interconnections between communities of practice. First, however, we place the analytical approach of delineating communities of practice in broader theoretical context by contrasting a school or district viewed as a designed organization and as a lived organization. Schools and Districts Viewed As Designed Organizations and As Lived Organizations A school or school district viewed as a designed organization consists of formally designated roles and divisions of labor together with official policies, procedures, management systems, organizational units, and the like. Wenger (1998) uses the term designed organization to indicate that its various elements were designed to carry out specific tasks or to perform particular functions. In contrast, a school or school district viewed as a lived organization consists of a configuration of interconnected communities of practice. This latter orientation is apparent in our summary description of the JHD. AUGUST/SEPTEMBER 2003

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Brown and Duguid (1991) note that the designed organization reflects the dominant assumptions of what they refer to as the organizational core. In the case of a school district, this is the view of the organization that is inscribed in official organizational charts, policy statements, and manuals. They go on to argue that analyzing an organization as complex as a school district only as a designed organization will not necessarily provide an adequate account of how work is actually organized and accomplished (see also Brown & Duguid, 1991; Kawatoko, 2000; Ueno, 2000; Wenger, 1998). This is because people frequently square prescribed procedures and activities with the exigencies of their circumstances. Brown and Duguid also clarify that actual collaborations often do not correspond to formally appointed groups, committees, and task forces. Instead, the communities of practice within which work is actually organized are frequently noncanonical and not officially recognized (cf. Brown & Duguid, 2000). The professional teaching community that had come to involve a collaboration between teachers, mathematics leaders, and researchers is a case in point in that it was yet to be formally recognized by most administrators in designated leadership positions within the district. It is important to note that in analyzing a school or district as a lived organization, we do not presuppose the functional organizational units but instead treat the identification of the communities of practice in which work is actually accomplished as an empirical question. This analytic perspective also orients us to extend our analysis of the sites of teachers’ learning beyond visible professional development activities such as workshops, seminars, and study groups organized by the school or district. It is for this reason that we have documented both the collaborating teachers’ nascent professional networks and the extent to which their classrooms are sites for their learning. Furthermore, in adopting this perspective, we do not assume that school and district leadership resides exclusively with the individuals who occupy designated leadership positions in the designed organization. Instead, we follow Spillane, Halverson, and Diamond (1999, 2001) by discerning how various leadership functions are actually accomplished, with the expectation that we will find that many are, in fact, distributed across several people.

during the school year that focused on the use of the reform textbook series. As the collaborating teachers all gave positive accounts of these sessions and indicated that, as a result, they were better prepared to discern the mathematical intent of the nonstandard problems in these textbooks, it would appear that these boundary encounters contributed to the human resources within the district (i.e., the knowledge, skills, and commitments of individuals). As a second example, we have noted that the school leaders participated in the activities of their communities by conducting classroom visits to monitor and assess teachers’ instructional practices. As a consequence of these boundary encounters, the teachers viewed classroom observations as situations for assessment rather than assistance and initially attempted to delimit the access that we and other teachers had to their classrooms.6 When we asked, during a meeting of the professional teaching community, why visitors to their classrooms made them uncomfortable, one simply stated to the others’ agreement, “We don’t want to be wrong.” This privatization of instructional practices within the district was evident in the teachers’ initial view of themselves as independent professionals who worked within perceived institutional constraints by relying almost exclusively on their own resources. For example, during the initial interview, one of the collaborating teachers used the metaphor of constructing a building to explain what he viewed as distinctive about his instructional practices.

Interconnections Between Communities of Practice

Teacher:

To this point, we have illustrated the potential value of analyzing schools and school districts as configurations of communities of practice and have focused on the process of identifying relevant communities. The issue to which we now turn is that of how the relations between communities might be analyzed. In doing so, we distinguish between three types of interconnections: boundary encounters, brokers, and boundary objects.

Boundary Encounters Participation in the activities of a community of practice constitutes the immediate social context in which members learn. The first type of interconnection arises when members’ participation in the practices of a community involves boundary encounters in which they engage in activities with members of another community. For example, the mathematics leaders in the JHD participated in the activities of their community by conducting both summer work sessions and study-group meetings with teachers 18

Interviewer: What comes from outside the classroom that you actually attend to or has some influence on how you teach? You talked about the Prescribed Instructional Program and the pacing guide. Teacher:

Those are all things that create my universe. They create my framework, what I hang on that framework, and what gives my stuff style. You know, it’s like OK, I’ve got four posts and beams and I am going to make a house out of that. After that, whether I’ve got a Dutch or a French colonial or whatever.

Interviewer: Are we missing any beams? Are there any other beams that we should focus on? The structure, no. It’s the State Prescribed Instructional Program and the textbooks that I have in my classroom. Beyond that, it is the way I teach and why I teach the way I do, and how I am evolving.

Gamoran, Anderson, Quiroz, Secada, Williams, and Ashman’s (2003) analysis of reform efforts in five school districts indicates that the institutionalization of teaching as a private activity in the JHD reduced the extent to which the teachers’ classrooms could be sites for their learning, thereby curbing the generation of both human and social resources within the district (i.e., the relationships and methods of communication of groups of people engaged in joint activities).

Brokers The second type of interconnection that we document when analyzing the institutional settings in which teachers work in a particular school or district concerns the activities of brokers, who are at least peripheral members of two or more communities of

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practice. Brokers can bridge between the activities of different communities by facilitating the translation, coordination, and alignment of perspectives and meanings (Wenger, 1998). Their role can be important in developing alignment between the enterprises of different communities of practice. We therefore regarded it as highly significant that we were unable to identify a single broker between the professional teaching community, the mathematics leadership community, and school leadership communities when we first began working in the JHD. This absence of brokers accounts to some extent for the lack of alignment of the agendas of the mathematics leadership and school leadership communities, and thus for the tensions that the teachers reported experiencing. The major development concerning brokers that has occurred during our collaboration with the teachers is that two mathematics leaders have become members of the professional teaching community. In the course of their participation, these two leaders have frequently related activities in the professional teaching community to ongoing initiatives within the district and have recruited the teachers to assist in revising the pacing guide, to pilot a textbook series being considered for adoption by the district, and to serve on textbook adoption committees. These interventions have encouraged the teachers to consider how they can contribute to the improvement of mathematics teaching in the district, thereby broadening their purview beyond their individual classrooms. Our observations of professional development sessions conducted in the district by the mathematics leaders indicate that these sessions have shifted away from a conventional workshop format and toward joint inquiry into problems of mathematics teaching and learning that characterize meetings of the professional teaching community. Taken together, these developments indicate that as a consequence of the brokering activities of the mathematics leaders, the enterprises of the professional teaching community and the mathematics teaching community have become relatively closely aligned.

Boundary Objects The first two types of interconnections that we have discussed are based on participation and are usually visible both to observers and to participants. The third type of interconnection between the communities of practice is typically less visible and is based on what Wenger (1998) terms reification7 rather than participation. Wenger defines reification as “the process of giving form to our experience by producing objects that congeal this experience into ‘thingness’” (p. 58). He argues that in creating reifications, “we project our meanings into the world and then we perceive them as existing in the world, as having a reality of their own” (p. 58). However, as Wenger emphasizes, reification cannot capture the richness of lived experience precisely because it is frozen into a concrete form such as a text. What is important about all these objects is that they are only the tip of an iceberg, which indicates larger contexts of significance realized in human practices. Their character as reifications is not only in their form but also in the processes by which they are integrated into these practices. Properly speaking, the products of reification are not simply concrete, material objects. Rather, they are reflections of these practices, tokens of vast expanses of human meanings. (p. 61)

The reifying object is, therefore, a relatively transparent carrier of meaning for members of the community in which it was created. In contrast, there is the very real possibility that these objects will be used differently and come to have different meanings if they are incorporated into the practices of other communities. Star and Griesemer (1989) argue that reifying objects can play a significant role in enabling the members of different communities to coordinate their activities even when they are used differently and have different meanings. As they demonstrate, successful coordination does not require that members of different communities achieve consensus. Instead, the use of the objects in different communities makes it possible for them to function as common boundary objects around which the members of the different communities can organize their activity. Consequently, as Star and Griesemer emphasize, boundary objects do not carry meanings across boundaries but instead constitute focal points around which interconnections between communities emerge. In this respect, boundary objects can serve as tools for communication between the members of different communities even though they do not provide a ready-made bridge between perspectives and meanings. As an illustration, the mathematics leaders in the JHD reified their vision of teachers building on students’ reasoning to support their understanding of central mathematical ideas when they developed and revised the pacing guide. This guide, it will be recalled, mapped the two textbook series onto the Prescribed Instructional Plan that lists the objectives assessed by the statemandated test. As we have seen, when we first began working with the teachers, they used the pacing guide by developing lessons that tended to focus on performing and applying mathematical procedures. Thus, the pacing guide functioned as a boundary object even though it did not carry the mathematics leaders’ instructional vision to the teachers. Although the teachers continue to use the pacing guide, there has been an overall shift in their instructional practices from focusing on mathematical procedures toward supporting students’ understanding of mathematical ideas. We view this development as significant, given that the pacing guide affords the formulation of teaching trajectories that are concerned with the coverage of content objectives. Although it does not preclude the formulation of learning trajectories that are concerned with students’ reasoning and the means of supporting its development, such trajectories in effect have to be read into the guide. In this regard, the pacing guide can be contrasted with a planning tool used by Japanese teachers that specifies both the most frequent student solutions to particular types of problems and the ways in which teachers can capitalize on those solutions to achieve their instructional agendas (Stigler & Hiebert, 1999). We noted that the teachers’ instructional practices were relatively homogeneous when we first began working with them, even though their informal professional contacts were limited. We also saw that the school leaders used the Prescribed Instructional Program to establish their agendas for mathematics instruction and explicitly communicated to the teachers in their schools that they expected to observe instruction directly related to objectives assessed by the state-mandated test. The Prescribed Instructional Program served as a boundary object in the context of a relationship characterized by assessment rather than assistance. Most of AUGUST/SEPTEMBER 2003

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the teachers, in fact, indicated that they were so familiar with the Prescribed Instructional Program that they rarely had to refer to it directly when they organized for mathematics teaching and learning. Importantly, the Prescribed Instructional Program affords the formulation of teaching trajectories that focus on a large number of relatively narrow goals rather than learning trajectories that focus on central mathematical ideas. The teachers had few social resources on which they could draw and apparently did not have the personal resources to read either more encompassing goals or a learning trajectory into the Prescribed Instructional Program as they attempted to meet school leaders’ expectations. In our view, these aspects of the institutional setting in which the teachers worked account for the homogeneity that we observed in their instructional practices when we first began working with them, even though the school leaders were open to a range of instructional techniques. It is important to note that the characterization we have given of this institutional setting is not cast in terms of systems and structures that fail to make contact with the teachers’ classroom instructional practices. Instead, we have developed descriptions of the institutional setting that had a reality not only in the structure of the school district as a lived organization but also in the collaborating teachers’ personal experience. For example, the teachers experienced and continually had to cope with a tension between the agendas of the school leadership communities and mathematics leadership community as they developed their instructional practices. It should also be apparent that although the perspective we take on a school district as a lived organization is not based on economic analogies that involve the flow of resources, it does enable us to account for access to material resources (e.g., instructional materials, release time and joint planning time, and outside consultants) and for the generation of human and social resources. As a final observation, it is worth noting that the JHD would have appeared to be relatively static had we analyzed it as a designed organization. In contrast, the various developments that we documented when we analyzed the district as a lived organization indicate that the institutional setting in which the teachers worked was dynamic and evolving. From this analytic perspective, the task of accounting for teachers’ learning, therefore, involves analyzing the coevolution of their instructional practices and the institutional setting in which they develop and refine those practices.

Usefulness To this point, we have clarified several constructs that are key to the analytic approach that we propose for locating teachers’ instructional practices within the institutional settings of their schools and districts. We now consider the usefulness of this approach by illustrating how our ongoing analyses of the JHD have informed our collaboration with the teachers. As we have seen, the school leaders considered instructional leadership in mathematics to be an important part of their work. However, the collaborating teachers and the mathematics leaders both explicitly rejected the suggestion that the school leaders gave a high priority to instructional issues and indicated that they viewed the school leaders solely as managerial or administrative leaders. We inferred from this discrepancy in perspectives that the teachers 20

and the mathematics leaders, on the one hand, and the school leaders, on the other, had very different understandings of what instructional leadership entails. Given the tension between the enterprises of the different communities as they relate to mathematics teaching and learning, we also conjectured that it would be important for some of the school leaders (ideally one from each school) to become at least peripheral members of the professional teaching community so that they could act as brokers. As a first step, we intervened relatively directly during a meeting of the professional teaching community by sharing the general findings of the interviews that we had conducted with school leaders. Our goal in doing so was to enable the members of this community to better appreciate the perspective of the school leaders. In the course of several protracted discussions, we were able to promote an alternative explanation for the school leaders’ actions, namely that they cared deeply about students’ intellectual welfare and intervened on the basis of their understandings of mathematics, the process of mathematics learning, and the means of supporting learning. Against this background, we conducted an activity during the second summer of our collaboration in which the members of the professional teaching community developed a conjectured learning trajectory for the school leaders. This involved specifying the school leaders’ current understandings, a potential endpoint for their activity as instructional leaders, and the possible means of supporting their development. The first step in this conjectured trajectory involves perturbing the school leaders’ current view that mathematics teaching is a relatively routine activity and that the student achievement will improve if teachers merely cover the prescribed content and maintain student engagement. With this initial goal in mind, the teachers then made plans to generate evidence of their students’ mathematical reasoning that, they conjectured, would challenge the school leaders’ assumptions about mathematics teaching and learning. The rationale for this course of action is that some of the school leaders might develop reason and motivation for participating in the activities of the professional teaching community. Planning ahead, we conjecture that the hoped for alignment of the three communities will be relatively fragile, even if it is achieved, as it will depend on the continuing participation of particular individuals in the activities of the professional teaching community. We therefore anticipate that it will also be necessary to develop tools that can support the alignment of communal enterprises on a continuing basis. These collaboratively developed tools might include both an instructional plan for middle school mathematics that builds on investigations of students’ mathematical learning and the means for making the types of mathematical learning and teaching inherent in this plan visible. If these tools are used by members of different communities and are constituted as boundary objects, they would support brokering and the bridging of perspectives, thereby making the alignments of communal enterprises potentially more sustainable. The examples that we have given of how our analyses have informed our collaboration with the teachers in the JHD illustrate several more general aspects of the analytic approach that make it pragmatically useful. First, because the types of analyses that we propose attempt to account for the interpretations that teachers and other persons make and the understandings that they

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develop, they are likely to have face validity for members of different communities in that they acknowledge and account for the frustrations and antagonism that they experience. Second, because analyses of this type situate people’s actions within the school or district as a lived organization, they constitute resources that might enable members of particular communities to move beyond viewing members of other communities merely as impediments to their agendas. As members of one community look at others through an analysis, they might instead begin to appreciate that their actions are rational (or at least understandable), given the constraints and affordances of their institutional niches within the school or district as lived. Third, analyses of this type support the formulation of strategies for institutional change that involve the creation of new tools as prospective boundary objects as well as the orchestration of boundary encounters and the development of brokers. This relatively broad orientation to the process of renegotiating the institutional settings in which teachers work is significant, given Wenger’s (1998) observation that mutual engagement and reification offer two complementary ways of attempting to shape the future, and that one is rarely effective without the other. Taken together, these three aspects of the analytic approach clarify that it constitutes a general method for understanding the specific settings in which particular groups of teachers work. We contend that this approach enables us to be more effective in collaborating with teachers in that it involves the development of testable conjectures about the constraints and affordances of the institutional settings in which they develop and revise their instructional practices. The resulting analyses orient us to consider whether our collaboration with a group of teachers should entail concerted attempts to bring about change in these settings and to formulate potentially revisable strategies for doing so with them as part of the process of supporting their learning. In the case of the JHD, the teachers’ engagement in a collaboration of this type has involved discernable shifts in their perspective on the institutional setting of their teaching. At the outset, they voiced a number of concerns and frustrations but appeared to view these as beyond their control. They subsequently developed an initial awareness that their instructional practices were partially constituted by the institutional setting in which they worked (e.g., their privatization of their instructional practices reflected their experience of classroom observations as situations for assessment). More recently, they were able to collectively identify a range of affordances and constraints for their learning and could distinguish the constraints that they conjectured they might be able to affect. We contend that in cases such as Jackson Heights, in which the professionalization of teaching involves the development of an institutional setting in which it is possible for teachers to act as professionals, teachers’ developing understanding of the relations between the institutional setting in which they work and their instructional practices is a crucial aspect of their learning. Discussion As our primary focus in this article has been on the institutional settings in which teachers develop and revise their instructional practices, we should emphasize that we also find it essential to focus on individual teachers’ instructional practices and peda-

gogical reasoning. In doing so, we attend to individual teachers’ personal resources such as their conceptualizations of particular mathematical domains, their understanding of the development of students’ reasoning in these domains, and the mathematical possibilities that they see in their students’ solutions and explanations. However, rather than analyzing these personal resources as attributes of individual teachers per se, we view them as characteristics of individual teachers’ activity in a particular institutional setting. A notion that comes to the fore when we adopt this latter perspective is that of teachers’ access to particular forms of pedagogical reasoning. As an illustration, our characterization of the institutional setting in which the Jackson Heights teachers worked when we began collaborating with them takes account of the affordances of the tools they used, their lack of social resources, and the manner in which they were held accountable for content coverage. We also saw that their instructional practices were initially relatively homogeneous and focused on the acquisition and application of mathematical procedures. On this basis, we inferred that the teachers did not have the personal resources to read either central mathematical ideas or learning trajectories into the tools they used as they organized for mathematical teaching and learning. However, we did not view this as a deficiency of the teachers but instead interpreted it as an indication that the setting in which they had developed their practices did not give them access to these forms of pedagogical reasoning. It was primarily for this reason that we concluded that our collaboration with them should include a sustained attempt to bring about changes in this setting so that they would have access to forms of pedagogical reasoning that involve explicitly focusing on, documenting, and adjusting to students’ developing mathematical understandings. We can further clarify the perspective that we have proposed on the institutional setting of the school and district by contrasting what Cole (1996) terms the concentric circles view of context with that of the metaphor of context as “to weave together.” In the concentric circles view, a professional teaching community is seen to be located within a school, which is located within a school district, which is located within a local community that includes students’ families, which is located within a broader sociopolitical setting that includes state and federal educational policies. The resulting image is that of the institutional setting of teaching as a series of nested layers, each of which surrounds a professional teaching community. As Cole notes, this image orients us to focus on how the more inclusive layers shape the lower levels. In taking this stance, we would assume that the relation between layers is unidirectional with outer layers influencing inner layers but not vice versa. The view of context as a weaving together challenges the assumption of unidirectional hierarchy inherent in the concentric circles view (Cole, 1996; Jacob, 1997). In the JHD, for example, a weaving together was apparent in the interconnections of the professional teaching community with the leadership communities in the teachers’ schools as well as with the mathematics leadership community that would be located in the putative district layer. The contrast between analyses of the institutional settings of teaching that take as their point of reference either the nesting of formal administrative units or the weaving together of the enterprises of distinct communities of practice approximates AUGUST/SEPTEMBER 2003

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the contrast between a school or district viewed as a designed organization and as a lived organization. It is when we analyze how the functions of teaching and leadership are accomplished in a particular school or district that the interweaving of the enterprises of the communities that actually carry out these functions becomes apparent. As the case of the JHD illustrates, it is by documenting how the enterprises of various communities have become interwoven that we are able to analyze the institutional setting of teaching in a manner that accounts for the inconspicuous, recurrent, and taken-for-granted aspects of teachers’ professional lives. As we have seen, the processes by which the enterprises of different communities of practice become interwoven are not limited to participation but also include reification and the emergence of boundary objects around which the members of different communities organize their activities. The critical role of boundary objects becomes all the more apparent once we note that members of different communities look at each other through the tools that they use (McDermott, 1976; Scott, 1998; Ueno, 2000). For example, the school leaders in the JHD looked at the teachers through the Prescribed Instructional Program and students’ test scores and saw the extent to which they were managing their classrooms appropriately and addressing specified instructional objectives. The tools that members of different communities use to make mathematics teaching and learning visible are not solely resources for perception, but also serve as resources that they use to account for their decisions and judgments. For example, the mathematics leaders in Jackson Heights used students’ test scores as a resource to justify to the teachers the changes they made in the pacing guide. In addition to clarifying how the enterprises of different communities become interwoven as their members organized their activities around common boundary objects, our analysis of the function of boundary objects in the JHD underscores the tool-mediated nature of the institutional settings in which teachers develop and revise their instructional practices. As we have noted, the use of tools and artifacts is an underdeveloped theme in the research literature on both teacher professional development, and policy and educational leadership. Conclusion In this article, we have proposed an analytic approach that seeks to challenge the dichotomy between two bodies of scholarship, one concerned with the organizational features of schools, and the other with the role of professional development in supporting teachers’ learning. The analytic approach focuses on the functions of teaching and delineates the communities of practice whose members contribute to the accomplishment of these functions. As a consequence, teaching is characterized as an activity that is distributed across a configuration of communities of practice within a school or district viewed as a lived organization. Analyses of this type attend to interconnections between the communities of practice within a school or district that involve boundary encounters, the role of brokers, and the coordination of activity around common boundary objects. As we have seen, this latter focus brings the tool-mediated nature of both teaching and instructional leadership to the fore. Analyses of these various interconnections portray the institutional setting in which 22

teachers’ instructional practices are situated as a fabric in which the enterprises of different communities and the activities of their members are woven together. As we illustrated, this analytic perspective is pragmatically useful in that analyses can feed back to inform ongoing collaborations with teachers and administrators. NOTES

The analysis reported in this article was supported by the National Science Foundation (NSF) under grant No. REC9814898 and by the Office of Educational Research and Improvement (OERI) under grant number R305A60007. The opinions expressed do not necessarily reflect the views of NSF or of OERI. We are grateful to James Spillane, Adam Gamoran, and colleagues in the Jackson Heights school district for their constructive critiques of a previous draft of this article. 1 In focusing on the functions of teaching, we follow Spillane, Halverson, and Diamond (1999, 2001) who elaborate a distributed perspective on school leadership by focusing on the functions of leadership rather than on the activities of persons occupying formal leadership positions. 2 Here and elsewhere, we substitute the fictitious name we have coined for the document to maintain the anonymity of the school district. 3 In speaking of a professional teaching community, we adhere to the definitions and criteria given by Secada and Adajian (1997) and Gamoran et al. (2003). 4 Wenger (1998) makes it clear that participation, as he uses the term, is not restricted to engagement at the local level of a community of practice but can also involve participation in the practices of relatively broad social communities. 5 The process of documenting the learning of a professional teaching community involves identifying the successive norms that became established for (a) general participation, (b) mathematical reasoning, (c) pedagogical reasoning, and (d) strategic norms (i.e., the ways of understanding the institutional setting for mathematics teaching that have become normative within the professional teaching community). A discussion of the criteria that need to be satisfied when identifying communal norms can be found in Cobb, Stephan, McClain, & Gravemeijer (2001). 6 More generally, the JHD was primarily characterized by what Tharp and Gallimore (1988) term a chain of assessment rather than by a chain of assistance. 7 Reification, as Wenger (1998) defines it, should not be confused with Sfard’s (1991, 1994) use of this same term. For Sfard, reification is the process by which mathematical objects are constructed from operational mathematical processes. Wenger’s use of the term is less technical and refers to the process by which members of a community create objects that, for them, carry particular, practice-based meanings. As he makes clear, the process of reification complements participation in the sense that mutual engagement typically involves the use of artifacts that are the products of prior reifications. REFERENCES

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PAUL COBB is a professor of mathematics education in the Department of Teaching and Learning, Vanderbilt University, Box 330 Peabody College, 1930 South Drive, Nashville, TN 37203; paul.cobb@ vanderbilt.edu. His research interests include classroom instructional design and analysis, the development of professional teaching communities, the institutional setting of teaching, and issues of diversity and equity as they play out in the mathematics classroom.

Box 330 Peabody College, 1930 South Drive, Nashville, TN 37203; [email protected]. Her research interests include a focus on teacher change. In particular, she is researching the development and refinement of trajectories for teacher change in the context of professional teaching communities and the means of support for that change, including tools and artifacts. TERUNI DE SILVA LAMBERG is a research associate in the Department of Teaching and Learning, Vanderbilt University, Box 330 Peabody College, 1930 South Drive, Nashville, TN 37203; teruni.d.lamberg@ vanderbilt.edu. Her research interests include institutional context, teacher education, and children’s mathematical thinking. CHRYSTAL DEAN is research assistant in the Department of Teaching and Learning, Vanderbilt University, Box 330 Peabody College, 1930 South Drive, Nashville, TN 37203; [email protected]. Her research interests include supporting the development and learning of professional mathematics teaching communities.

KAY MCCLAIN is an assistant professor of mathematics education in the Department of Teaching and Learning, Vanderbilt University,

Manuscript received May 9, 2002 Final revision received January 22, 2003 Accepted February 5, 2003

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