Teachers College Record, mathematics, rehearsals, New York, teacher communities, American Educational Research Journal, teacher community, teaching, Journal of Curriculum Studies, Teachers College Press, Open University Press, math teachers, Colleagues, elaboration, Guillermo, professional community, Educational Research, Teaching and Teacher Education, Cambridge University Press, teachers, classroom practice, Belinda, communities, University Press, The teachers, rehearsal, teaching rehearsals, teacher, teacher group, classroom events, professional conversations, Professional communities, American Educational Research Association, Teacher Education, student learning, Judith Warren Little, Corwin Press, professional learning communities, traditional practice, University of Chicago Press
Teaching Replays, Teaching Rehearsals, and Re-Visions of Practice: Learning From Colleagues in a Mathematics Teacher Community ILANA SEIDEL HORN Vanderbilt University's Peabody College Background/Context: Research shows that teachers' understandings of students, subject, and teaching influence their classroom practice. Additionally, teachers' colleagues have a role in shaping individuals' approaches to teaching and their responses to reform. Focus of Study: To understand how interactions with colleagues support teachers' informal learning, I examined teachers' collegial conversations in a highly collaborative teacher community. Setting: This research took place within a collaborative, improvement-oriented urban high school mathematics department that showed evidence of increasing access to and rigor in its curriculum. Participants: Six high school mathematics teachers
and the researcher as a participant observer worked together in a group focusing on detracking ninth-grade algebra classes. Research Design: This 2-year ethnographic study went inside the teacher community, with the researcher teaching alongside the teachers to gain access to their interactions and workings. data collection
and Analysis: Approximately 100 hours of teachers' collaborative conversations were observed and recorded through audio- and videotape and field notes. Using a situated learning framework and sociolinguistic analyses, I examined teacher-to-teacher talk that constituted episodes of pedagogical reasoning (EPRs) to understand how collegial conversations provide resources for teacher learning. Findings: Across EPRs, two forms of discourse were important sites for representing, and sometimes learning about, teaching practice, which I call teaching replays and teaching Teachers College Record
Volume 112, Number 1, January 2010, pp. 225259 Copyright © by Teachers College, Columbia University 0161-4681
226 Teachers College Record rehearsals. Extended replays or rehearsals supported a re-visioning routine, interactions in which teachers elaborated, reconsidered, or revised their understanding of complex teaching situations while providing particular, emotionally involving accounts of the classroom. The examination of these interactions further specifies conditions that support teachers' collegial learning. Conclusions: Theoretically, this article illustrates a process of learning as recontextualization, as the teachers work between general teaching principles and specific occurrences in their classrooms. Practically, by highlighting the work that teachers do to make sense of innovative practices, this analysis provides a description of how collegial conversations can support teachers' informal learning, supporting the development of professional communities
. ENTERING INTO A COLLABORATIVE COMMUNITY It is the first day I visit East High School.1 I arrive a little early since I was concerned about getting lost and gave myself extra time to get there. Since I am no longer hurried, I walk around, trying to shake myself out of a Friday afternoon daze. I slowly make my way to Guillermo Reyes' classroom. It is after school, and only a few students remain in the hall, talking loudly and laughing by an open metal locker. I find Guillermo's classroom at the end of the corridor. The room is large and spacious. Colorful math posters cover the walls: published ones describing the mathematical contributions of different cultures alongside student-made ones illustrating key mathematical ideas. Guillermo's desk is separated from the student desks by freestanding partitions. After he has settled some students to work on a make-up test in the back of the room, he goes behind the partitions and offers me a drink of juice or coffee. I smile, declining his offer as I set up my computer to take field notes. Shortly thereafter, several other teachers arrive. They have the weary look of teachers in the waning days of the school year, which I suspect is only intensified by the fact that it is a Friday afternoon. They spend the better part of the meeting going over a curriculum unit on probability that one teacher, Jill, has reworked to keep their students engaged in learning during the last days before summer. (Field notes, East High Algebra Meeting, June 6, 1999)
Learning from Colleagues 227 This field note excerpt describes my first encounter with the East High mathematics teacher community. Before I attended this meeting, my initial conversations with the department chair, Guillermo Reyes, made me hopeful that I had, after an almost yearlong search, actually found a collaborative mathematics teacher community. The meeting that followed confirmed my hopes. My field notes from that first observation are filled with side comments about the distinctive way the teachers talked and interacted. It was so strikingly different from the other math teacher groups I had observed during our research team's site selection process. My comments note the humor, teasing, and playfulness in the teachers' interactions, despite their obvious fatigue. I was struck by the extent to which the teachers pushed Jill to explain the mathematical goals and meaning of the different activities in the unit. When a tired Jill was pushed too far and protested the relentless stream of questions, Annie stepped in with a motherly tone to reassure her, saying, "It's not personal, Jill. We're just trying to understand it." But most of all, I was struck by the attention to mathematical concepts and students' understanding of ideas. Teachers' conceptions of students, mathematics, and pedagogy shape the way they teach. As a result, research on teacher learning and the relationship of that learning to teaching practice has become of central importance in our field (National Academy of Education, 1999). Understanding how teachers come to know what they know is a complex, multifaceted endeavor. To date, most studies of teacher learning have focused on the influence of formally organized educational activities on teacher knowledge (Franke & Kazemi, 2001; Grossman et al., 2000; Sherin & Han, 2004). Although formal professional development is an important site of inquiry, it is increasingly clear that teachers' school-level colleagues have an important role in shaping their approaches to classroom practice and their responses to reform. For this reason, the present study focuses on teachers' informal workplace learning, examining the mutually constitutive nature of teachers' knowledge and workplace practices. This analysis contributes to a relatively recent body of work that seeks to understand teacher learning through interactions with colleagues (Horn, 2005, 2007; Horn & Little, In press; Little, 2002, 2003; Little & Horn, 2007). How do teachers learn in and from interactions with their colleagues? This article examines two related discourse structures that, when inserted into a specific conversational routine, position teachers to learn about teaching as they talk and work together. These discourse structures, which I call teaching replays and teaching rehearsals, focus teachers on the specificity of the classroom and, when coupled with the re-visioning par-
228 Teachers College Record ticipation framework (Goffman, 1981), help them develop knowledge for teaching that is at once rooted in general principles
and deeply situated in practice. Replays and rehearsals are an important resource for this learning because they create scenes for listeners that involve them emotionally and cognitively in the discussion. Tannen (1989) claimed that "scenes are crucial in both thinking and feeling because they are composed of people in relation to each other, doing things that are culturally and personally recognizable and meaningful" (p. 16). During replays and rehearsals, teachers paint detailed and involving classroom scenes to engage one another in sense-making. Writers often use this strategy, as I did in the introduction of this article. By painting a detailed scene of my first encounter with the East High math teachers--their weariness at the end of the school year, their determination to understand the curriculum in a way that serves their students--I sought to engage readers in the intellectual and emotional experience I had entering into this collaborative group of teachers. These kinds of scenes play a role in the development of understanding. In this analysis, I describe the way that the scenes constructed through teaching replays and rehearsals provide a means for teachers to reimagine their classroom practice in interactions structured by a participation framework, the re-visioning routine. The remainder of this article is organized in the following manner. First, I provide the conceptual framework for this study, describing situated learning theory and its relationship to the analysis of discourse strategies. After reviewing relevant educational research
that informs this article, I introduce the present study and the analytic methods used. This is followed by an explanation of teaching replays and teaching rehearsals. These frame three extended examples of the conversational re-vision routine in action, which I claim provide important opportunities for professional learning. I conclude with a discussion of the implications of this work for teachers' professional development and future research
. CONCEPTUAL FRAMEWORK: SITUATING TEACHERS' LEARNING To account for teachers' opportunities to learn within their interactions with colleagues, I employed a situated learning framework (Lave & Wenger, 1991; Wenger, 1998). Situated learning provides a promising yet underused perspective for investigating teacher learning (Putnam & Borko, 2000). Instead of viewing teaching knowledge as residing in the head of individual teachers, a situated view highlights the ways in which teachers' knowledge is socially, culturally, and historically constructed. To capture these dimensions of teacher knowledge, in the design and
Learning from Colleagues 229 analysis of this study, I broaden my lens to investigate how knowledge and practice are negotiated in context, specifically through interactions with colleagues and experiences in the workplace. In keeping with this framework, I define learning as a change in participation in a community of practice. The teachers in this study constitute a community of practice because they meet Wenger's three criteria (1998): they were mutually engaged in the joint enterprise of detracking their mathematics classes, and they had a shared repertoire for doing this work, which included teaching from a common curriculum and meeting weekly to discuss their teaching. It is the nature of their mutual engagement that I highlight in this study, which, during the interactions that are at the center of this analysis, exhibited a high degree of conversational involvement (Tannen, 1989). PARTICIPATION FRAMEWORKS AND POSITIONING WITHIN COMMUNITIES To detail the character of the teachers' collegial interactions, I linked the notion of learning in communities of practice to another analytic construct, participation frameworks. From moment to moment in conversations, speech positions both speakers and listeners in a variety of ways, endowing them with a particular participation status relative to a given utterance. The codification of these various positions and expected conduct in relationship to such speech is called a participation framework (Goffman, 1981). By organizing the context of interactions, participation frameworks signal meanings and provide resources for sense-making. They are therefore important to understanding how teachers learn in interaction with their colleagues. By identifying participation frameworks within a teacher community, I can analyze how teachers are able to engage with topics under discussion, the specificity with which they are able to reveal and work through problems of practice, and the positions that are available to them during these conversations. These positions are made visible through the participation status that speakers take on and assign each other. Relevant to this analysis, interactional positioning can put somebody in a role of knowing more or knowing less, irrespective of their actual knowledge. For example, when a doctor examines a patient, she might do so using a consultation framework, aligning herself asymmetrically with the patient and using her role as doctor to position herself as the expert who holds the exclusive information relevant to the diagnosis. Alternatively, the doctor might take on an examination framework, positioning herself more symmetrically with
230 Teachers College Record the patient as they pool their respective information to reach a diagnosis. And, of course, there can be competing frameworks: The doctor may bring one framework to an interaction in which a patient seeks or assumes the other (Tannen & Wallat, 1993). The positioning that occurs within a given framework not only differently positions speakers as knowers; it differentially orients participants to the problem at hand. These positions may, for example, scaffold teachers' pursuit of a solution to a problem; alternatively, they may render a problem insoluble and therefore encourage the abandonment of such pursuit. In this way, frameworks carry with them metamessages about the nature of the problems being solved. In educational research, prior studies have shown how the framing of teaching problems within collegial communities differently positions teachers to learn about them (Horn, 2007) and act on them in practice (Coburn, 2006). INVOLVEMENT IN COLLEGIAL CONVERSATIONS In addition to codifying the roles and alignments that emerge from participation frameworks, interactions can be characterized by the degree to which they elicit involvement on the part of speakers and listeners. In her extensive examination of involvement strategies in interactions, Tannen (1989) described involvement as observable, active participation in conversation that is signaled by mutual engagement on the part of speakers and listeners, signaled by the way in which "speaking and listening include elements and traces of the other" (p. 12). Tannen argued that conversational involvement has both cognitive and emotional consequences. When speakers and listeners are mutually engaged in the construction of meaning, different kinds of meanings converge, creating coherence and insight. On the surface, it is easy to see how coherence and insight have cognitive implications, but they have emotional ones as well. Coherence and insight via conversation create an experience of connectedness and a metamessage of rapport between communicators. Tannen's (1989) work examined a number of involvement strategies that speakers use in conversation, comparing them to literary devices used by poets and writers, such as various forms of repetition and particularistic imagery. One involvement strategy that is especially salient to the present analysis is the use of constructed dialogue. Tannen wrote, The creation of voices occasions the imagination of a scene in which characters speak in those voices, and that these scenes occasion the imagination of alternative, distant, or familiar worlds, much as does artistic creation. . . . [T]he casting of ideas
Learning from Colleagues 231 as the speech of others is an important source of emotion in discourse. Recent work by ethnographers of communication on affect has come hand-in-hand with studies of evidentiality: How speakers frame the information they express, what authority they claim for it. (pp. 2526) In summary, using constructed dialogue as a storytelling device in conversations serves several purposes. First, constructed dialogue connects speakers and listeners in relation to the parties being represented, providing a means for involvement. Second, this discourse strategy paints a scene that brings the images and sounds of one world into another, allowing for commentary or consultation on events in other times and places. Finally, constructed dialogue provides an evidence base within discussions that is emotionally rich while maintaining the resonance of an empirical example. CONSTRUCTED DIALOGUE AS A RESOURCE FOR LEARNING IN COLLEGIAL CONVERSATIONS In this analysis, I focus on forms of discourse that I call teaching replays and teaching rehearsals; they use constructed dialogue to create classroom scenes during collegial conversations. Briefly, teaching replays provide blow-by-blow accounts of actual and sometimes ongoing classroom events, often with teachers narrating or acting out their part as teacher. In teaching rehearsals, teachers also narrate or act out classroom interaction, but they do so in either an imagined or an anticipatory fashion. When embedded in the re-visioning routine, these structures organize interaction, create alignments, and position teachers in relationship to each other and to the students they teach. The alignments created by replays and rehearsals themselves vary from setting to setting,2 but when the East High math teachers coupled them with the re-vision routine, they were an important resource for consultation and learning. Because I have introduced a number of technical terms in this conceptual framework, I provide Figure 1 to illustrate the relationships among these terms. By using constructed dialogue from the classroom, replays and rehearsals in the re-vision routine organized the teachers' learning from practice in several other ways that Tannen's (1989) analysis might anticipate. The stories told in this vivid fashion provided a sense of connectedness to the teachers involved in their telling. Such connectedness is a resource for learning because it means that listeners are involved in the construction of meaning in the conversation. Relatedly, these replays and rehearsals served as an emotionally compelling source of evidence in
232 Teachers College Record Figure 1. A schematic diagram illustrating the relationships among the terms used in the conceptual framework and the analysis the conversations. Finally, replays and rehearsals laminated the world of the classroom onto the world of the teacher community, positioning the teachers toward students and classroom situations
in ways that were consequential for their learning. I venture that this last feature of replays and rehearsals provides a means for teachers to recontextualize activities from one setting to another by allowing for a continual process of embedding one context (e.g., a classroom event) in another context (e.g., a collegial conversation) and back again. Such recontextualization has been posited as critical for the development of meaningful abstract thinking (van Oers, 1998). That is, when new problems arise in the midst of a familiar activity, people reorganize their actions to respond to these problems while attempting to fulfill the same goals. Such reorganization of activity requires an abstract understanding of what the activity is about. The teachers' conversations provided a place to negotiate that abstract understanding of their teaching, with the replays and rehearsals serving as a critical resource for reflecting on how they responded to the various problems that arose during the course of their teaching. Given the poten-
Learning from Colleagues 233 tial for these conversations to support the teachers' learning, the identification of interactive collegial structures that allow for such interpenetration of contexts and the communication of abstract goals will help uncover the relationship between teacher communities and teacher learning. STUDYING LEARNING WITHIN TEACHER COMMUNITIES The approach I use to study the social nature of teachers' learning has its roots in earlier work on teacher communities. In this literature, we find that despite the isolating egg-crate organization of schooling (Lortie, 1975/2001), explanations for the persistence of traditional practice go beyond the individual. For several decades, researchers have recognized the way that teacher subgroups can create fundamentally different settings within the same school (Ball & Lacey, 1980; Little, 1993; McLaughlin & Talbert, 2001; Siskin, 1994). In the high school, the most salient communities tend to reside at the level of subject-matter departments (Little, 1993; McLaughlin & Talbert; Siskin & Little, 1995). There is evidence that these teacher communities can influence approaches to classroom practice (Gutiйrrez, 1996, 1999; Johnson, 1990) and shape responses to reform (Ball & Bowe, 1992; Coburn, 2001; Little, 1995; Spillane, 1999). Within this literature, teacher learning has been identified as a feature of certain kinds of teacher groups. For instance, in "strong teacher learning communities" (McLaughlin & Talbert, 2001), teachers collaborate to reinvent practice. Relatedly, inquiry communities are places that support teachers' "inquiry stance on practice," helping them "generate local knowledge, envision and theorize their practice, and interpret and interrogate the theory and research of others" (Cochran-Smith & Lytle, 1999, p. 289). Elsewhere, "professional learning communities" have been identified within schools as groups of teachers who share values, focus on student learning, collaborate, deprivatize their practice, and use reflective dialogue (Bryk, Camburn, & Louis, 1997; Kruse, Louis, & Bryk, 1995; Louis, Marks, & Kruse, 1996). But teachers can sometimes come together without this kind of community--and, presumably, with a different quality of learning. Achinstein (2002) described the almost inevitable conflict that arises when teachers come together to solve problems of practice. Some teachers find this kind of conflict intolerable and avoid it altogether. As Grossman, Wineburg, and Woolworth (2001) have described, there is a difference between "community" and "pseudocommunity," in which the latter is characterized by the imperative to "behave as if we all agree" (p. 955).
234 Teachers College Record Pseudocommunities lack the communities' capacity to tackle problems of practice, thereby disallowing teachers the same kinds of opportunities to examine taken-for-granted assumptions and move toward change. The norms of pseudocommunities can be counterproductive to inquiry, because groups increase their ability to investigate practice when they develop ways of publicly disagreeing (Loucks-Horsley, Love, Stiles, Mundy, & Hewson, 2003; Pfieffer & Featherstone, 1996). From this literature, we see that collective innovation, inquiry, and capacity to disagree may signal the presence of teachers who are poised to learn from their colleagues. In fact, the productive conversational re-vision routine identified in this analysis provides a means for questioning or challenging colleagues. Although these studies provide insight into the location and dynamics of teacher communities, they do not detail the kinds of resources for learning that exist inside of them. In line with Little's (2002, 2003) call to investigate the development opportunities that exist in teachers' ordinary daily work, I have located sites for learning within teacher communities by identifying elements that constitute a conceptual infrastructure for making sense of their work. By conceptual infrastructure, I refer to a variety of linguistic and semiotic resources for learning about and interpreting the work of teaching (Horn, 2005). More specifically, I found three different kinds of community-based resources for teachers' learning. First, collegial interactions (or a lack thereof) shaped teachers' interpretations and specified the local meanings of artifacts for teaching (e.g., curriculum, slogans). Second, informal category systems in teachers' conversations ("fast kids," "college-bound kids") provided a basis for modeling problems of practice while communicating assumptions about students, subject, and teaching that would get built into the solutions. Finally, through the rendering of practice in conversation, teachers defined what was locally "sharable" and made different aspects of classroom events available for consultation. Over time, these renderings represented prototypical events that could be linked to appropriate pedagogical responses. It is this last feature of the conceptual resources for learning about teaching that I probe in this analysis by examining the participation framework of re-visioning that makes use of teaching replays and teaching rehearsals. Recall that teaching replays provide blow-by-blow accounts of actual and sometimes ongoing classroom events, with teachers often acting out their part as teacher. In teaching rehearsals, teachers also portray and often act out classroom interaction, but they do so in either an imagined or an anticipatory fashion. The close rendering of the classroom created multiple opportunities for the teachers' collaborative
Learning from Colleagues 235 pedagogical problem solving. First, by locating problems in the specific interactions of the classroom, the teachers often faced the ambiguity and complexity of their teaching choices. Second, by sharing the normally private events of the classroom with their peers in detail, they coordinated expectations and teaching strategies, creating a more consistent environment for their students. In addition, by taking on both the student and teacher voices in these replay and rehearsal-laden conversations, they laminated student identities onto themselves as teachers, intertwining their voices in the roles of teacher-as-teacher and teacher-asstudent. The involvement that these constructed replay- and rehearsalrich conversations created afforded both emotional support for the teaching problems and a set of empirical examples that became archetypical of certain teaching problems shared among the participants. Coupled with their re-vision routine in which they reconsidered aspects of the replay and rehearsal scenes, the teachers were thus poised to learn from these conversations. By extensively taking on the student voice and perspective in their considerations of practice and by pushing each other to reflect on their significance, the teachers constantly considered their students' intellectual and emotional responses to their teaching, positioning one another as capable of solving the problems of teaching that inevitably arose in their classrooms. BACKGROUND OF STUDY In this study, I analyze the conversations of teachers in a U.S. urban high school who have worked to increase equity in their mathematics classrooms. East High School is located in a working-class neighborhood and serves a diverse student population. Over the course of 15 years, the teachers have worked together to increase access and rigor in their mathematics classes. About 15% of East's students are identified as English language learners, and approximately 40% are classified as Latina/Latino, 25% are classified as African American, 20% are classified as European American, and the remaining students are from a mix of other ethnic groups. Despite the large number of students who come from groups historically disenfranchised from education, students enroll in advanced mathematics courses at higher than average rates, indicating that the teachers have achieved some success in meeting their goal of increasing access to mathematics. In addition, a comparative study of student mathematics performance showed that although East's students enter high school with weaker mathematical preparation than their college preparatory peers in more affluent high schools, they outperformed them after 2 years of instruction, indicating some success toward the goal of rigor
236 Teachers College Record (Boaler & Staples, 2008).3 Understanding how these teachers have effectively implemented math- ematics reforms would therefore greatly contribute to our larger efforts at improving practice and creating more equitable classrooms. To do so, I went inside this reform-oriented, highly collaborative teacher community to uncover the important features of collegial environments that support innovation and learning. By and large, the teachers worked toward a goal of ambitious teaching--that is, teaching that is "intellectually ambitious in that it [focuses] on students learning to perform complex tasks and undertake sophisticated analyses of ideas, and socially ambitious in that it [attends] to the success of students not currently well-served by education systems" (Lampert & Graziani, 2005, p. 1). The East algebra teachers were highly collaborative: They met weekly to work on their collective project of detracking the math curriculum, reporting in a survey that they spent approximately a third of their planning time working with colleagues. Table 1 is a summary of the participants in this study. Table 1. East High Algebra Teachers Included in This Analysis * UNDERSTANDING THE EAST HIGH ALGEBRA GROUP AS A COMMUNITY OF PRACTICE Research design and data collection. During a 2-year ethnographic study in which I taught and worked alongside the East algebra teachers, our
Learning from Colleagues 237
research team sought to go inside teacher communities as a way to better understand their contribution to learning and improvement of classroom practice. I collected a variety of data targeted to capture the East teachers' formal and informal collegial interactions. The Primary Data
for this analysis are the records of the teachers' collaborative meetings and conversations, which I observed for approximately 100 hours over the course of the study. The majority of these meetings were video- or audiotaped and transcribed (55 hours), and the remainder were recorded via field notes. Although other data are not the focus of this analysis, I also observed in teachers' classrooms and interviewed focal teachers about their experiences working and teaching in this department. My own experiences as a teacher-learner in this community serve as a validity check for the claims I make in this article. Table 2 shows a summary of the data collected.
Table 2. A Summary of the Data Collected for This Study
East High Mathematics Teacher Community Data
Algebra group meetings
20 (1.52 hours each)
Other professional activities
1 meeting + 3-day conference
Summer "Algebra Week"
5 days (about 8 hours/day)
Analytic methods. Given the amount of meeting data available, I developed a unit of analysis that allowed me to make sense of how the teachers were learning from each other in their conversations. To do so, I looked at what I have called episodes of pedagogical reasoning (EPRs). I define the EPRs as units of teacher-to-teacher talk in which teachers exhibit their reasoning about an issue in their practice. Specifically, EPRs are moments in teachers' interaction when they describe issues in, or raise questions about, teaching practice, and these descriptions 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, they can be multiparty co-constructions over many turns of talk. My decision-rule for locating EPRs is based on topical shifts. That is, the teachers switching conversational topics signaled the boundary of an EPR. Some longer EPRs were decomposed into sub-EPRs, as teachers explored different facets of a single problem. Frequently, these longer episodes are signaled by a teacher's question or the raising of a problematic issue,
238 Teachers College Record
such as, "I have a handful of kids who are not successful. How is this going to impact our classes next semester?" Although the content of the EPRs provides an interesting portrait of the types of problems the teachers' addressed over the course of the year, in this analysis, I am more interested in the kinds of resources for learning that were mobilized during EPRs. I thus look at the in-depth conversations about problems of practice and examine the participant frameworks and conversational routines that supported collaborative pedagogical reasoning. The following transcript conventions (Figure 2) were used in the excerpts that follow.
Figure 2. A summary of transcript conventions used in the replay and rehearsal excerpts
/ = . ? Italics (??), (cow) ((gesture)) [comment]
Self-interruption No gap between utterances Low fall in intonation Rise in intonation Speaker emphasis Unclear reading, tentative reading Describes voice qualities, actions, or pauses Researcher's comment added to clarify conversation
TEACHING REPLAYS AND TEACHING REHEARSALS In this first section of findings, I provide a more detailed description of teaching replays and teaching rehearsals. I argue that they served four main purposes in the extended episodes of pedagogical reasoning. At their most basic level, they provided a way to garner support for the often emotionally intense work of teaching. By relaying episodes of frustration, confusion, or humor, the teachers could recruit sympathetic (if secondhand) witnesses to the events in their otherwise closed-off classrooms. Second, replays and rehearsals provided an evidence base for consultation by providing specific images of classroom practice, laminating the world of the classroom onto the world of the teacher community. Because the scenes portrayed were emotionally involving, participants frequently engaged with their details in considering the problem being represented. In addition, replays and rehearsals provided a means of developing general knowledge for teaching when seemingly disparate events were linked via a unifying principle or, alternatively, when gener-
Learning from Colleagues 239 alized accounts of classroom practice were conveyed. Finally, the constructed dialogue within the replays and rehearsals not only involved the teachers emotionally but also created an alliance among them. As Tannen (1989) anticipated, and as I illustrate in the following examples, in addition to whatever content is conveyed through replays and rehearsals, such constructed dialogue "expresses the relationship not between the quoted party and the topic of talk but rather the quoting party and the audience to whom the quotation is delivered" (p. 109).4 TEACHING REPLAYS Teaching replays are accounts of specific classroom events and often include blow-by-blow renderings of interactions with students. As Goffman (1974) described, replays recount "a personal experience, not merely reports on an event" (p. 504). They are often characterized by second-person ("you") address of students and constructed dialogue between teachers and students. Embedded in collegial conversations, replays allow teachers to troubleshoot problems with colleagues by providing an emotionally compelling snapshot of a specific or ongoing situation. At the same time, replays often help teachers garner support because they provide interactional openings for listeners to empathetically insert themselves while allowing speakers to express their feelings about a situation. The replay underlined next takes place during a mentor teacher/student teacher meeting. Belinda has arrived at the meeting with a list of questions she has for her mentor, Guillermo, and she solicits his advice on a particular rut she is in with a student who is taking advantage of her homework policy. Note how she starts the replay with a specific image of the student's expression. There is slight (and understandable) sarcasm in her voice as she imitates the student, providing an outlet for her ire: That sheepish grin I get in the morning when I'm checking homework and like, "It's not here?" ((quiet laugh)). ((imitating student's voice)) "Nooo, I'll come in at lunch!" ((Returns to sweet teacher voice)) "Yes, you're doing it [your homework], that's really good." (Audio transcript, student-teacher/mentor-teacher meeting, September 24, 1999) In this brief retelling of an ongoing interaction, Belinda creates a scene
240 Teachers College Record for her mentor teacher. She provides the instructional context for this recurring interaction ("when I'm checking homework") and provides an image that suggests the student's positioning of himself within the interaction ("that sheepish grin.") The present tense of this account signals that this exchange over incomplete homework has happened repeatedly with this student (and that it has an implied future as well). The dialogue may or may not be a literal account of what has transpired in her classroom, but it clearly instantiates the nature of the ongoing interaction. Her replay has both emotional and cognitive dimensions. In her recounting, Belinda provides an account of her frustration while also illustrating her attempt to address the situation. By portraying both her version of the student and her own response, she has opened up a place in the conversation for her mentor teacher to give her specific advice on the problem she is experiencing, which he does in great detail. She thus positions herself as an advice seeker, because her mentor is positioned as a teacher with expertise to share. LINKING REPLAYS TO CREATE A CLASS OF EVENTS When replays are linked, they support the development of general knowledge for teaching. The excerpt that follows was taken from a 4-hour retreat between Jill and her mentee, Alice. They met at a coffee shop one morning, with Alice bringing a range of concerns and questions to consult with Jill about. In the course of the following answer, Jill replays to Alice her students' response to problems involving multiplying fractions. In the context of Jill's role as a mentor, the replay provides a way to model a routine response to a common anxiety among students: "They're really upset about multiplying fractions, `No one ever taught me this stuff!' Just like the conversation we had this morning, `I've never learned this before.' So I said `Okay, I'll teach it to you. Here we go. If you want to learn it, no problem.'" (Audio transcript, mentor/mentee meeting, April 7, 2000) In this excerpt, the first replay ("No one ever taught me this stuff!") gets linked to a second replay ("I've never learned this before.") via the sentence "just like the conversation we had this morning." By likening two different student responses to challenging mathematics, Jill begins to establish a general class of students' anxious protests about their prior learning. In this sense, Jill's reported response ("Okay, I'll teach it to you") applies to the situation at hand and to the situation in general in
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language that is directly importable to the classroom. Whereas in the earlier example, Belinda used a replay to seek advice, here, Jill uses a replay to share advice and wisdom about teaching.
BETWEEN REPLAY AND REHEARSAL
Like teaching replays, teaching rehearsals also portray vivid scenes of classroom events, often in a blow-by-blow manner, reporting both teacher and student talk. The main difference is that teaching rehearsals are more generalized and are not necessarily linked to a single event. In fact, rehearsals can represent either general, ongoing interactions or ones that teachers might anticipate. In these data, teaching rehearsals were often scenes portrayed by experienced teachers to provide an example or illustrate a point, with the benefit of being, in Goffman's (1974) terms, "decoupled from consequentiality" (p. 59). Linguistically, they are usually signaled by generalized pronouns, "they" representing the students, and "you" representing the teacher. They are often marked by the present tense to indicate their ongoing nature, whereas teaching replays are more often in past tense. Replays and rehearsals are sometimes interwoven in conversations as the teachers move between the particular and the general in making sense of the events in their classroom. Teaching rehearsals provide opportunities to link notions about teaching to generalized examples of practice, thus supporting the generation of principled teaching knowledge. In the following example, the participation frameworks of replay and rehearsal actually overlap in interaction. A number of linguistic cues signal the rehearsal aspect of the account: Students are represented by a generalized "they" and teachers by a generalized "you," and the present tense flags an ongoing occurrence. Although a specific instance of teaching is described, the story does not appear to depict a literal conversation. During an after-school team meeting, Annie uses a rehearsal to describe a common error that students make when they are first constructing linear graphs: that of disregarding scale:
Well you know it (what I love) is when they give you this graph, when they just kind of throw numbers on a piece of paper
. So they have this gorgeous graph hung up but it's got like the ((traces out jagged line with her finger)) hiccups in it. So I look at it and I see that they've not spaced things correctly. So they go, "Well, I just
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hate doing that. I don't want to do that." "By all
means! Don't do it!" I say. "Then why are we signed up
for the course?" "This is, like, my job."
2. Teachers: ((Laughing))
You know? "Well I never do it like that." And of course,
you always say, "And how old are you? You're gonna
have a lot of `nevers' in your life and it's gonna get
started right here." So, you know, it's just funny.
This girl had this gorgeous graph and it was like ((ges-
turing again with her finger)) didee, didee, and it was
because you could just see, right at the graph, where
she's decided to jump three [on the scale] instead of
"Oh, I don't need that number!"
5. Guillermo: Right. I like when they blame it on their last year's
6. (Audio transcript, East High algebra meeting, October 5, 1999)
After ironically expressing her "love" for this common event, Annie starts with a general, rehearsal-like description of the student scaling error and the interactions that surround it. The rehearsal is signaled by linguistic cues such as describing the students as a unified "they" and the teachers in the second-person "you." Her generic description represents the universality of the dialogue she is replaying. The reported dialogue is a bit fanciful ("Then why are we signed up for this course?" may or may not be a direct quote from a student) and her tone mirthful. As is typical of constructed dialogue, this "quote" serves more to represent a prototypical student stance, requiring teachers to justify classroom activities while also providing humor to a sympathetic teacher audience. The general nature of the account is underscored in Turn 3, when Annie says "and of course you always say," emphasizing the commonness of the exchange. In the same turn, she segues into an actual replay of a girl in her class. By embedding the replay in the rehearsal, she locates the particular classroom incident in a general class of incidents that she assumes her colleagues have shared, an assumption that is ratified by the utterances that follow her account. The listener-supplied dialogue I provide in Turn 4 signals my involvement with the scene that Annie is portraying, as does Guillermo's expansion in Turn 5. Collectively, we are elaborating on the phenomenon described, providing ourselves with a sense of rapport and identification as we align with Annie's laughter about a common teaching experience.
Learning from Colleagues 243 TYPES OF TEACHING REHEARSALS In a certain sense, teaching rehearsals are a common way for teachers in many places to communicate with each other about practice. A teacher will, for example, share a particular mode of exposition with colleagues by going up to the board and running through a sample presentation. Student teachers will often rehearse lessons with their classmates as a dry run for real teaching. Or, a professional development provider will sometimes run through an activity with teachers as a model lesson and have the teachers role-play the part of the students. Unlike the lessons of formal professional development, the East teachers' model lessons are (literally) situated in their classrooms, with colleagues who can provide collective knowledge and understanding of the school, student, and teaching contexts. If the transfer of knowledge and practices is really an issue of recontextualization, requiring learners to map the work of one context onto another, then this condition minimizes that burden. The situated aspect of the algebra teachers' model lesson rehearsals helped address the problems involved in mapping an idealized lesson onto the teachers' complex classroom reality, because many of the highly particular, multilevel issues that are always a part of teaching are held in common by the participants. Lampert (2001) described the complexity of teaching practice by locating problems within three overlapping and interacting "problem spaces": the teacher's connection to the student; the content; and the studentcontent relationship. These connections are often in tension, or even conflict, with one another. The collaboration among East's math teachers supported the sustained investigation of teaching practice by stabilizing certain dimensions of this problem space by, for instance, working from a common curriculum. Teaching rehearsals pressed on the inclusion of the teacher-student and teacher-(student-content) dimensions of this problem space by constantly invoking students' responses during mathematics class in the teachers' discussions. In my analysis of the East teachers' conversations, I found two types of teaching rehearsals: anticipatory rehearsals and practice teaching rehearsals. Anticipatory rehearsals. Anticipatory rehearsals help teachers communicate typical or expected events in their mathematics classes. In a previous example, Annie uses an anticipatory rehearsal to frame her story about the girl with the hiccupping graph. She conveys knowledge about the kinds of mistakes students tend to make when they are first learning to graph. In doing so, she positions herself as somebody with enough experience to make the generalizations about the work of teaching that she represents in the rehearsal scene. Teachers who ratify her account align
244 Teachers College Record themselves with her generalization. Others may question her for more detail, positioning themselves either as neophytes or as teachers who have not encountered the phenomenon portrayed. Alternatively, they could disagree with her framing altogether. All these positions are available within the anticipatory rehearsal framework. Elsewhere, when teachers are discussing curriculum or activities, they might use an anticipatory rehearsal as a way of trying them out. The following comes from a field note excerpt, so it is formatted differently from the audio transcripts shown elsewhere in this article. It comes from the first meeting I observed at East High and describes Guillermo using a teaching rehearsal to figure out how to use a proposed activity to explain standard deviation to his students. Notice how he rehearses classroom language as he reviews the activity. Guillermo looks over the worksheet Jill has distributed. He says aloud, "So standard deviation helps decide what does `in here' mean. Whether they learn/ go through an algorithm or this messy thing that doesn't make much sense, or whether they just get it from their calculators, either way the connection to something real. . . Is it in this area?" He points to the area underneath the normal curve that would be within two standard deviations of the mean. He pauses and then continues. "What's the difference between the data point and the mean? What's the square root all about? To get rid of the negative. . . " He looks up at his colleagues around the table and says: "Try out this algorithm." He points to the algorithm for finding standard deviation. "Try explaining each step." (Field notes, East High Algebra Meeting, June 6, 1999) Guillermo uses the anticipatory rehearsal to make sense of an activity. In the first rehearsal segment, he tries out language that might convey the idea of standard deviation to his students (it "helps decide what does `in here' mean"). In between the first and second rehearsal segments, he reveals his purpose, stating that he is seeking a way for students to connect the idea of standard deviation to "something real." In the last rehearsal segment, he anticipates the kind of clarifying questions he might need to ask to uncover its meaning (e.g. "What is the difference between the data point and the mean?"). In this way, Guillermo evaluates the activity based on how well he thinks it can support the development of student understanding. Within this anticipatory rehearsal framework, he positions himself as a teacher in relationship to the content. He is trying out language that will make the main ideas of the activity accessible
Learning from Colleagues 245 to imagined students. His colleagues are an audience for his rehearsal, and he positions them both as imaginary students when he looks up at them to practice his instructional language and as experienced colleagues who might help him judge the discourse he rehearsed. Teaching rehearsals as "practice lessons." The East teachers would occasionally model lessons to share expertise and training. One teacher would play the role of teacher while the others would play the role of students. This allowed the "teacher" to model the discourse around the activities, while the "students" could understand the activity's relationship to the mathematics they were trying to teach. "Practice lesson" rehearsals may not be unusual in other teacher communities, but the East teachers' rehearsals were striking because of the amount of time they occupied and the ways in which they focused teachers' attention on the students' experience. This type of rehearsal is obviously lengthier; for reasons of space, I direct readers to Example 3 in the next section for an illustration. Summary of teaching replays and teaching rehearsals. Teaching replays provide specific images of classroom interactions that serve two main purposes. First, they represent the classroom in a way that allows for specific consultation on problems of practice. Second, when replayed events are linked, they become classes of events that support generalizations about appropriate pedagogical responses. Likewise, teaching rehearsals can frame replays to locate them within a class of common classroom events. In addition, rehearsals can provide a means for teachers to project activities in their classrooms and imagine students' responses. They are what Goffman (1974) called spaces of "utilitarian make-believe," allowing for the development of classroom interactional skills in a less consequential setting. In this next section, I illustrate how teaching replays and teaching rehearsals can support opportunities for teacher learning when embedded in a conversational re-vision routine. USING TEACHING REPLAY AND TEACHING REHEARSALS TO SUPPORT TEACHER LEARNING: RE-VISIONS OF CLASSROOM PRACTICE5 As the preceding excerpts illustrate, a range of knowledge for teaching was displayed through the teaching replays and rehearsals. But how did these classroom-specific, multiply-perspectived accounts of practice support teacher learning? In this section, I propose one mechanism for supporting teacher learning through a conversational routine that supports re-visions of practice.
246 Teachers College Record When I analyzed the more elaborated episodes of pedagogical reasoning, I noticed that accounts of practice (frequently represented through replays and rehearsals) would become transformed over the course of interaction. Generally speaking, the pattern went as follows. A teacher would render a classroom event, usually in replay or sometimes in rehearsal form. Eventually, a colleague would prompt for elaboration, either by posing a question or interjecting an interpretive principle about teaching. This would sometimes lead to a re-vision of the initial account, in the form of either an elaboration or a reinterpretation of events. This sequence of interactions is represented in Figure 3. This routine emerges out of the conversational involvement that the East math teachers exhibited in their meetings. The re-vision routine becomes a mechanism for a collaborative diagnosis of teaching problems, positioning both the speaker and listeners as having important information to share in the conversations. It is my claim that these re-visions signal teachers' learning from their colleagues. In a situated learning framework, learning is marked by changing participation within a community (Lave & Wenger, 1991). As teachers interact through the frameworks of replays and rehearsals, their participation changes on two different levels. First, as they talk and work together, their participation changes within the conversation as they specify and revise classroom stories to consider features of the classroom that had previously been overlooked in their description or enactment of an event. Second, through teaching rehearsals, their projected participation as teachers in an imagined classroom changes as they reconsider interpretations and pedagogical choices. As replays are elaborated or rehearsals brought to bear on problems of practice, participants in this discourse community are repositioned to address the problems represented within the collective accounts of teaching. Figure 3. Sequence of the re-visioning participation framework Next, I illustrate how replays and rehearsals supported re-visions of practice, taking three examples in which a prompt for elaboration was taken up and integrated into a revised account of the teaching event. This analysis will demonstrate the various ways that this process unfolds among colleagues.
Learning from Colleagues 247
EXAMPLE 1: REPLAY BEGETS REHEARSAL
The following interaction took place between Guillermo and Belinda during a mentorteacher/studentteacher meeting. Guillermo had asked Belinda to write up a list of questions to bring to the meeting, and she is elaborating on one in particular. As in previous sections, underlined text signals a replay or rehearsal. These excerpts come from an audio transcript of a meeting that took place on September 24, 1999.
Part 1: Rehearsal
1. Belinda: 2. Guillermo: 3. Belinda: 4. Guillermo: 5. Belinda:
So that was part of this question. ((reading)) Do they know how to answer each other in their groups when they ask each other questions in their groups? `Cause like I was watching Angel ask David a question. And David kind of tossed an answer, he's like, "You do it this way." And it clearly didn't satisfy her, but she didn't come back with another question. And he didn't register that she hadn't processed, understood. Mm. Right. what he said. Right. So
Belinda has supported her question about whether students know how to answer each other when working collaboratively by replaying a specific instance in which she saw communication between two students break down. Although Belinda is calm in her retelling, there is emotional urgency in the scene; she seems to convey empathy for Angel because she did not get the help from David that she needed.
Part II: Elaboration
The specificity of her account allows Guillermo to prompt her for an elaboration
6. Guillermo: 7. Belinda: 8. Guillermo:
Did you intervene? Did you ((trails off)) It was um during the warm-up. Oh, when you said you weren't gonna, oh. That's too bad.
248 Teachers College Record
And I saw her thinking and I wasn't quite sure if she was going to go with it. So.
Guillermo prompts for an elaboration from Belinda in Turn 6. Because Guillermo shares a classroom with Belinda, he understands her telegraphic comment in Turn 7. He recalls that she had told the students that she would not answer any questions while they worked on the warmup. Although his prompt about intervention does not support further elaboration, she manages to fill in more detail in Turn 9.
Part III: Re-vision
After a couple intervening turns, Guillermo helps Belinda imagine an intervention through the use of a rehearsal:
`Cause `cause there's/ I mean, "Angel, it was great that you asked a question. David you tried to answer it, but did you see what happened? Um, you gave an answer but you weren't checking to see whether you were really being helpful to Angel." And then. "Angel, was he being helpful? No, then can you think of, so now think of some follow-up questions, like if he just gives you an answer, say `I don't want just an answer, I want (this that).'"
While acknowledging that the parameters Belinda had set up disallowed this kind of intervention in the class period
that she reported, Guillermo's anticipatory rehearsal helps provide her with a specific image of how she might intervene in a similar situation in the future.
EXAMPLE 2: A REPLAY REVISED
In the following example taken from an after-school algebra meeting, Alice, a first-year teacher, uses a replay to report on the "mayhem" she experienced in her class. These excerpts are from an audio transcript of the October 12, 1999, algebra meeting.
Part I: Replay
Uh, well my frustration, I think, was just, I started the geoboards today and it, it felt like mayhem? Like, it felt like no one kind of understood--I just had a vision of
Learning from Colleagues 249
what it--I thought it should look like and it didn't look anything like that and then. . . I was trying to keep students together in their groups, but they, they weren't staying together. And then. . . What was happening? So then I wanted to communicate the whole putting the rectangle around the triangle? but it's like, if I do it in front of class, no one's paying attention but if I go around to groups, I felt like I wasn't communicating it to all the students? So I think that--and after processing it with Jill, I think they were getting stuff done? It's just that I have a vision of what group work should look like, and it's not looking anything like that.
Alice sets the scene by interweaving emotional language ("frustration," "mayhem") with a description of the instructional activity ("geoboards"), creating an involving and particular account for her colleagues. Alice's replay fills out her experience of the mayhem that arose when her lesson went awry. Her replay includes her own internal monologue as a teacher as she explicates the dilemma she felt between clarifying an activity in front of the whole class and communicating to individual groups.
Part II: Prompt for elaboration
The teachers signaled their involvement with Alice's tale, empathizing with her and the chaos she experienced. Then the teachers prompt her to elaborate her account. Howard suggests, "I mean, you put something like that in their hands for the first time and there's a certain level of play with it," while Judy wonders if the mayhem came about because of "the fact that it involved rubber bands?" In a series of turns, Guillermo asks her if she can "identify the source of the squirreliness," keeping Alice in the position of primary interpreter of the classroom events.
III. Revised Replay
After considering her colleagues' questions, Alice arrives at the following re-vision, providing a different set of details to her scene:
I'm not sure, I think it even started with the warm-up. Like they weren't . . . I don't know. Maybe it was a sense that they don't have a concept of area at all. So maybe they were afraid of not being able to do these ideas. It was like they were just counting the squares the whole
250 Teachers College Record
time. I kept saying, "Is there a rectangle there?" and that was like going beyond for them. So maybe it's just that the concepts are challenging for them. Linguistic cues signal Alice's uncertainty. The hedging and false starts in the revised account indicate that this is an emerging version of what happened in the classroom that has not yet been closely considered. Guillermo's, Judy's, and Howard's questions position Alice as having important additional information, inviting just such a reformulation. From a learning perspective, this positioning provides an interactional space for Alice to reflect on the source of the mayhem. Her new account considers the students' mathematical understanding: They were just counting the squares to figure out the area of the figures they were making on the geoboards, and she wanted them to find rectangles that would allow them to calculate the area more efficiently. This re-vision shifts the problem space for her as a teacher, opening up new issues of practice. EXAMPLE 3: REHEARSING EXPLANATIONS TO PROVIDE CONCEPTUALLY CLEAR LANGUAGE The activity described next took place during a practice lesson-type teaching rehearsal. These excerpts come from a video transcript of the September 21, 1999, algebra meeting. Carrie, a third-year teacher, is modeling a lesson using algebra tiles. In a corner of the classroom, three teachers are working together on a particularly challenging problem. In the problem, the teachers must construct a polygon, using specific algebra tiles, that has a certain perimeter. The challenge is that one of the tiles has a length measuring x, but the final perimeter does not have an x term in it (see Figure 4). Figure 4. The problem the teachers are working on during the rehearsal in Example 3. They are asked to arrange these tiles into a polygon with perimeter 4y + 4.
y y block 1
In the context of this practice lesson teaching rehearsal, Alice has been working through this problem. She has figured out a solution, the one
Learning from Colleagues 251 represented in Figure 5. Annie and Carrie, acting as teachers, have been attending to her explanation: Figure 5. The polygon built by the teacher group. The measures of the sides are indicated. The sides marked A and B are discussed in the article. 2 Part I: Reflecting on a Rehearsal Because the teachers have come to an agreed-upon solution, Carrie reminds them that the objective for them, as teachers, is to reflect on the variety of strategies they used in building the polygons. 1. Carrie: But um we were talking about a strategy they might use. You really should write that down. 2. Alice: So the strategy that I'm saying if they're, again, so, if the x's cancel out? 3. Carrie: ((Writing on a paper)) (6-second pause) Canceling out ... Part IIa: Introducing an alternative perspective At this point, Carrie writes down Alice's strategy as stated. However, Annie objects to the language of "canceling": 4. Annie: "Making zeroes." That's the way I like to put it.
252 Teachers College Record 5. Carrie: Making zeroes. ((writing on paper)) 6. Annie: Right. Because then they're going to do the zero mon- ster?6 ((Pats one hand on top of the other.)) You know, and then they're going to have negative and positive ((index fingers out in front, parallel)) when they start using the mat?7 So you want to start "making zeroes." Because canceling, you know, "canceling," they think that they just throw it away. 7. Carrie: ((Nods, writing.)) In Turn 4, Annie first brings up the language of "making zeroes" as a matter of preference ("That's the way I like to put it"). Carrie transcribes it alongside Alice's strategy of "canceling" (Turn 5). She does not appear to have understood "making zeroes" as a renaming of "canceling." Annie takes her reasoning further, giving a clearer explanation of her objection. The first two reasons are curricular as she makes references to specific activities ("zero monster," representing negative and positive values on "the mat"). Her final reason points to a problem with student thinking: "canceling, they think that they just throw it away." She then likens this last reason to another situation in which the language of canceling is problematic: 8. Annie: So same, thing (becomes one). Division? Annie raises the same objection about the term canceling in another situationthe simplification of fractions with like factors in the numerator and denominator. By bringing up this second example, Annie is making a more general point about the conceptual opacity of the term cancel. Her preference for "making zeroes" has thus become a principled stance, one linked to student understanding. At this point, Carrie is paying close attention to Annie's objection: 9. Carrie: Uh-huh. That's something I really made a mess of the first--last year when I taught it? I used "balance" and "cancel" and, and "make zeroes" and I used them all all over the place Somewhere in the midst of Annie's elaboration, Carrie has recognized the confusion of using multiple metaphors for the same idea of combining like terms with opposite signs (positive and negative). She links it to the "mess" she made when she taught it, as she rattles off several terms she used interchangeably for the same mathematical idea.
Learning from Colleagues 253 Part IIb: Prompting for elaboration Alice, the novice teacher, then poses a question. Carrie begins to formulate an answer, her voice strong and confident, but as she reaches the end of her explanation (end of Turn 11), she struggles for clarity: 10. Alice: What works better? 11. Carrie: Okay. So "making zeroes," you know, whenever you have a positive ((left hand in air)) and a negative ((right hand below)). And I would use the word "canceling out" for when you, yeah, when had something in the positive ((left hand up)) and something in the negative ((right hand below)) and taking both away ((pulls both hands toward herself)), so I was saying the word "canceling out." So canceling out should be used when. ((3 second pause)). No maybe not at all. I don't know. I have to teach it again. Alice's question positions Annie and Carrie as more experienced, opening an interactional slot for them to provide an authoritative answer. At first, Carrie is happy to oblige, formulating what seems to be a clear answer to the question. The previous discussion about the conceptual opacity of "canceling," however, cannot be ignored. Toward the end of Turn 11, she interrupts herself, doubting the assertive answer she has been constructing. After a 3-second pause, she tentatively concludes that perhaps the term should not be used "at all," then says that she has to "teach it again" to really sort out the answer to Alice's question. Part III: Revising Alice then turns to Annie, who does not appear to be as befuddled by this issue. 12. Alice: ((Looking at Annie)) Do you use canceling out? What words do you use? 13. Annie: Well, I wouldn't use canceling= 14. Carrie: =Yeah, at all= 15. Annie: =because then they take that into division. So it's a word that they use in division from way back ((left arm waves behind her)). 16. Carrie: Mhm. 17. Annie: So when I, what I try to say is "You're making one."
254 Teachers College Record Because then if it's something that's like this ((writing on paper)), like x over x-squared--If this goes away, and that's x over x. If this goes away, then they can see it's = 18. Carrie: =One.= 19. Annie: =1 over x. That's why I say, "You're making ones." And I say, "Can you ever divide by anything um and make a zero? Well, then, (???). I can divide a number into zero." 20. Carrie: So "making ones," "making zeroes"= 21. Annie: =Making ones, making zeroes. Although Annie's response is primarily directed at Alice, Carrie's interjections (Turns 14 and 18) signal that she is involved with Annie's reasoning. Annie enacts her rehearsal (Turns 17 and 19) around a type of problem that often causes confusion for students, simplifying x/x2. In doing so, she further explicates her mathematical knowledge for teaching, sharing with Alice a common student confusion that arises out of algebraic notation. Although Alice is ostensibly the less experienced teacher seeking advice, Carrie shifted positions from one of an experienced teacher sharing knowledge to one of a fellow learner. Indeed, she has arrived at a different conclusion about the clearest instructional language by the end of the conversation. DISCUSSION: CONVERSATIONAL RESOURCES FOR PROFESSIONAL LEARNING This article explores the interactions that take place within a collaborative teacher community, seeking to specify how conversations can support teacher learning through emotional and intellectual involvement in problems of practice. Whereas earlier research on teacher community signals the existence of reflective dialogue in improvement-oriented collaborations, this study identifies specific aspects of collaborative practice that help account for teacher learning. In particular, through teaching replays and teaching rehearsals, teachers create vivid and often emotionally compelling scenes to illustrate and examine problems of practice. By providing portraits of actual or anticipated events, replays and rehearsals allow teachers to substantiate claims about teaching, anticipate student responses, garner emotional support, and create general classes of like events to communicate knowledge about practice. When coupled with conversational routines that probe and question these representations of teaching, replays and rehearsals can support re-visions of classroom practice, creating productive opportunities for teacher learning. Like the doctor's examination routine described in the beginning of this article,
Learning from Colleagues 255 the re-vision routine used in this teacher community positioned both novice and experienced teachers as having important knowledge to contribute to making sense of problems of practice. By relating specific instances to larger principles of practice, these conversations provide a basis for recontextualization, given that abstract models of teaching are reflected against the particulars of the classroom. Because of the way these conversations position the East High math teachers as collaborative problem solvers, I propose that conversations such as the ones analyzed here support a stance of adaptive expertise in teaching. According to Crawford and her colleagues, teachers who take a stance of adaptive expertise share a number of characteristics (Crawford, Schlager, Toyama, Riel, & Vahey, 2005). These include an epistemic stance that views the world as complex; a willingness to reveal and work at the limits of one's knowledge; an inclination toward learning rather than merely applying knowledge; data-oriented reasoning; seeking and analyzing feedback; and monitoring results and performance. Mapping the teachers' activities onto this list, I see the East math teachers' conversations as embodying a stance of adaptive expertise in the following ways. Through their practice-laden conversations, the teachers take a view of teaching as complex and worthy of extensive examination. In prompting for elaboration in response to problem statements, they regularly reveal their own uncertainty about the enterprise. They seek out advice from one another to improve their practice. Although the majority of the advice seeking is done by newer teachers, more experienced teachers (such as Carrie in Example 3) also willingly revise their ideas when they recognize their limits. The replays and rehearsals provide "evidence" for the claims they are making, and the conversational routines that probe into these accounts seek disconfirming evidence for the claims teachers make about classroom events. The frequency of these types of conversations (and their frequent observations in each other's classrooms) suggests that the teachers are constantly seeking feedback from one another. Many of the conversations are opportunities to check in about the outcome of teaching activities across classrooms, providing a means of monitoring results. This analysis provides an existence proof, detailing the nature of teachers' mutual engagement in the collegial conversations of one collaborative community. Whether or not these kinds of conversations are found in similar communities is an empirical question yet to be explored. It may be that the vivid, emotionally involving conversations found within this teacher community and the routines for probing classroom accounts while positioning speakers and listeners as knowledgeable collaborators are characteristic of teacher learning communities. Alternatively, it may
256 Teachers College Record be that there are other mechanisms and structures through which teacher communities support adaptive expertise. In any case, teaching, especially ambitious teaching that seeks to include all students in rigorous versions of subject matter, is an expedition into complex and often uncharted terrain. The East teachers' notable success in engaging students in challenging mathematics may well be attributable to the problem-solving resources they made available to themselves within these professional conversations. Acknowledgments This analysis was supported by a University of Washington College of Education Junior Faculty grant. An earlier version of this analysis was presented at the European Association for Research on Learning SIG on Teaching and Teacher Education biennial conference, Garryvoe, Ireland. Notes 1. A pseudonym. All proper names are pseudonyms except my own when it appears in transcripts. I refer to myself as "Lani" because that is what the teachers called me. 2. The study of the East High math teachers comes out of a larger study examining four high school teacher communities (Judith Warren Little, PI). Although this group was the most collaborative of the four, we did see replays and rehearsals in some form in all the teacher groups. East High's math teachers were unique in the alignments they took on in relation to the replays and in the extent to which students' perspectives were represented and considered. 3. We have received participants' permission to link this study of the teachers' work to the subsequent study of student learning. 4. In this teacher group, the alliance was among the teachers against the challenges of teaching; one can easily imagine another context in which the alliance is created among the teachers against the students. 5. Parts of this analysis are an outgrowth of a collaboration with Judith Warren Little. 6. This refers to an activity about opposite integers "making zero." 7. This refers to a mat used for solving equations using the algebra tiles. References Achinstein, B. (2002). Community, diversity, and conflict among schoolteachers: The ties that blind. New York: Teachers College Press. Ball, S. J., & Bowe, R. (1992). Subject departments and the implementation of National Curriculum
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Learning from Colleagues 259 Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. New York: Cambridge University Press. ILANA SEIDEL HORN is an associate professor of mathematics education
and learning sciences at Vanderbilt University's Peabody College. Her research focuses on equitable mathematics teaching
in urban high schools, particularly as it relates to teachers' learning in professional communities and student learning through classroom discourse. Her work has recently been published in Mathematical Thinking and Learning: Horn, I. S. (2008). Turnaround students in high school mathematics: Constructing identities of competence through mathematical worlds. Mathematical Thinking and Learning, 10(3), 201239.