Doing math and talking school: Professional talk as producing hybridity in teacher identity and community

Linguistics and Education 55 (2020) 100766

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Doing math and talking school: Professional talk as producing hybridity in teacher identity and community Ian Parker Renga a,∗ , Frederick A. Peck b , Ricela Feliciano-Semidei c , David Erickson b , Ke Wu b a

Western Colorado University, United States University of Montana, United States c Northern Illinois University, United States b

a r t i c l e

i n f o

Article history: Received 20 December 2018 Received in revised form 3 October 2019 Accepted 10 October 2019 Keywords: Teacher community Teacher professional identity Essential tension Teacher talk Hybridity Math teachers

a b s t r a c t Teachers construct professional identities within community as they converse about their work and negotiate what it means to be a teacher. Grossman, Wineburg, and Woolworth (2001) suggest that such negotiation must account for an essential tension between focusing on pedagogical versus disciplinary concerns. How teachers navigate this tension and what this means for their joint production of identity and community is unclear. This gap in the literature became evident in our work with Math Teachers’ Circles (MTC), where we observed K-12 math teachers indexing instructional experiences and concerns despite the program’s explicit invitation to set teaching aside and do math problems together for pleasure. Drawing upon a community of practice framework and positioning theory, we consider the work this professional talk accomplished within MTC gatherings. We show how the teachers positioned themselves and established their community, thereby producing hybrid identities and MTCs as a kind of hybrid community. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Much attention has been given to teachers’ professional communities as powerful sites for sustained teacher learning that counteract the historically private and isolated nature of teaching (Little, 1990; Lortie, 1975). Depending their structure, purpose, and composition (Vangrieken, Meredith, Packer, & Kyndt, 2017), teacher communities can impact teachers’ construction of content knowledge (Borko, 2004; Brodie, 2013; Wilson & Berne, 1999), understanding of students’ needs (Bannister, 2015), beliefs (Brodie, 2014) and classroom practices (Jeanpierre, Oberhauser, & Freeman, 2005; Stein, Silver, & Smith, 1998). Teacher communities can also provide a supportive environment for teachers to develop as leaders (Caine & Caine, 2000) and be a space for professional renewal and wellbeing (Cochran-Smith, 2004; Owen, 2016). Such aims and purported benefits are interwoven with the identity work undertaken in community as teachers co-construct and renew their professional identities (Connolly, Hadfield, Barnes, & Snook, 2018; Pillen, Den Brok, & Beijaard, 2013). Teachers’ identities are person-

∗ Corresponding author at: Education Department, Western Colorado University, 1 Western Way, Gunnison, CO 81231, United States. E-mail address: [email protected] (I.P. Renga). https://doi.org/10.1016/j.linged.2019.100766 0898-5898/© 2019 Elsevier Inc. All rights reserved.

ally meaningful and complex, with a number of scholars arguing that being a teacher cannot be separated from who one is or seeks to become (Alsup, 2006; Arvaja, 2016; Day, Kington, Stobart, & Sammons, 2006). Identities are also consequential for teachers’ capacity to grow and adapt (Beijaard, Verloop, & Vermunt, 2000). Research on teachers’ identity construction has exposed the ontological work occurring within teacher communities as members discursively negotiate meaning and make sense of themselves as educators. Studies of teacher discourse reveal how professional identities are constructed dialogically through talk as teachers discuss facets of their work. In these interactions, teachers try out identities and explore the possibilities of living into or resisting particular educational narratives (Taylor, Vlach, & Wetzel, 2018; Ives & Juzwik, 2015). Talk positions interlocutors as a kind of teacher (e.g., ESOL) defined by certain values and beliefs (Yazan, 2017), with teachers’ discursive contributions signaling bids on particular professional identities that are either taken up or rejected by colleagues (Cohen, 2010). Identity construction within this dialogic space also entails negotiation of teachers’ participation across contexts (Hodges & Cady, 2012), transitioning roles and shifting expectations (Jones, Brown, Hanley, & McNamara, 2000; Losano, Fiorentini, & Villarreal, 2018), and larger professional discourses on teaching and teachers (Ma & Singer-Gabella, 2011; Neumayer-

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Depiper, 2013). As they interact, teachers use talk to introduce experiences, understandings, and broader discourses to the dialogic space, thereby making them consequential for their identities and communities. While this literature justifies the forging of teacher communities attentive to the identity work happening within them, it could benefit from more explicit consideration of how teachers navigate competing agendas for coming together. In their widely cited piece on the challenges and promises of teacher community, Grossman, Wineburg, and Woolworth (2001) identify an essential tension faced by members between (a) focusing on pedagogical practice on the one hand and (b) engaging in subject matter discussions and disciplinary practices on the other. They contend that both foci are essential elements in the joint enterprise of a teacher community and are best interwoven “in any successful attempt to create and sustain teacher intellectual community” so that the community is “equally concerned with student learning and with teacher learning” (p. 952). While not the only observable tension shaping teacher communities—tensions such as theory/practice and stability/change are also present (Barab, Barnett, & Squire, 2002)—this tension between pedagogy and disciplinary practices is under constant negotiation as teachers develop useful knowledge(s) for teaching (Ball, Thames, & Phelps, 2008; Berry, Friedrichsen, & Loughran, 2015; Shulman, 1987). Despite its central importance, the essential tension is taken for granted in the literature, with little empirical inquiry into how teachers negotiate it together in real time and what this means for their identities and the communities they form. Echoing calls to examine the interplay between microand macro-level perspectives in discursive constructive activity (Akkerman & Meijer, 2011; Anderson, 2009; Kayi-Aydar & Miller, 2018), we contend that there is much we do not yet understand about how teachers, in their moment-to-moment interactions with one another, position themselves and are positioned within this broader tension between the role’s instructional and disciplinary demands. This lack of knowledge became clear in our work with primary and secondary math teachers participating in Math Teachers’ Circles (MTC). The MTC program is unusual in how it addresses the tension between instructional practice and disciplinary practice by inviting teachers to mostly ignore the former and simply take pleasure in doing math problems. As defined within one chapter’s online advertisement, the MTC is an “engaging professional community that involves doing mathematics collaboratively” and “a space for math teachers to work as mathematicians do!” (Philadelphia Area Math Teachers, 2018). The improvement of pedagogical practices is therefore not a “core” activity (White, Donaldson, Hodge, & Ruff, 2013). In fact, an assumption of MTCs is that teachers are so consumed by the demands of their work—e.g., day-to-day operations, addressing standards, raising test scores, improving instructional techniques, etc.—that they lose sight of an important reason for teaching: a love of disciplinary engagement and learning. MTCs promise teachers an opportunity to renew their interest in teaching math through doing math, with the ambitious aim of inspiring teachers so they will inspire students. Despite an explicit invitation to set aside classroom concerns, MTC participants often initiate and spend a non-trivial portion of time engaging in professional talk that indexes teaching and schooling as they work together on math problems (Peck et al., 2017). This raised questions for us about the work such talk accomplishes in the micro-level negotiation of the essential tension and its resulting implications for the teachers’ joint production of identity and community. To address these questions, we used a case study approach and a combination of ethnographic and discursive analytic methods to investigate talk within and across multiple MTC gatherings. Though we examine math teachers in this paper, our study goes beyond the field of math teacher development to engage with

broader discussions of teacher identity, community, and education (Lutovac & Kaasila, 2018). 2. Conceptual framework We treat communities and identities as productively intertwined. By this we mean that they mutually produce and sustain one another such that each is only sensible in relation to the other. In the same way that a community exists and is defined by its people, we assume that communities and their norms, practices, and histories serve as the primary sites in which identity work takes place and identities gain meaning. Our assumptions are informed by Wenger’s (1998) notion of a community of practice, an argument for socially constructed identities, and an understanding of talk as both a form of social action and a cultural resource; we take each in turn. 2.1. Communities of practice We view MTCs through a community of practice lens, which posits that communities develop via three fundamental components: mutual engagement, joint enterprise, and shared repertoire (Wenger, 1998). Mutual engagement refers to the requirement that members jointly participate in the practice(s) that bind and define the community; joint enterprise refers to the purpose of the community; and shared repertoire refers to the objects that are naturalized in the community—those objects that are so natural to members so as to be taken-for-granted, but which may seem foreign or strange to outsiders (Bowker & Star, 1999). A community is established as participants interact with each other and with artifacts, as well as where norms of engagement, joint practices, and a shared repertoire emerge (Bannister, 2015; Dean, 2005; Lave & Wenger, 1991). In this paper, we focus on the joint enterprise of the communities in the observed MTC gatherings. While the joint enterprise can be designed for, as an emergent property of the community, it cannot be created by fiat (Wenger, 1998). As noted earlier, for many groups of teachers, the essential tension between pedagogy and discipline (Grossman et al., 2001) is a key consideration in this enterprise and must be negotiated by participants as they engage in joint activity. Of course, the same activity may attend to both foci of the tension and be understood by different people as attending to the foci differently. For example, one person might view engaging in a problem-solving activity as primarily about personal engagement in mathematical practice, while another person may understand the same activity as preparation for teaching. Thus, a focus on the joint enterprise of a community is not simply a focus on what participants do, but also on the meanings that participants give to their actions (Wenger, 1998). To give meaning to an action, people draw on historically constituted constellations of shared experiences, artifacts, and practices, collectively referred to as cultural models (Gee & Green, 1998). As such, meaningful engagement is an emergent and negotiated process that draws on both historical significances and activity-in-the-moment. The negotiation of meaning, in turn, shapes the joint enterprise as it “constantly changes the situations to which it gives meaning, and affects all participants” (Wenger, 1998, p. 54). 2.2. Identity as socially constructed through talk Following this understanding of community, we conceptualize identity in social and relational terms as an evolving construction that is neither fixed nor innate. Rather, we assume identities emerge as individuals position themselves and are positioned by others while they engage in joint practice (Dreier, 1999,2009). Positioning as we use the term describes “the discursive process

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whereby selves are located in conversations as observably and subjectively coherent participants in jointly produced storylines” (Davies & Harré, 1990, p. 48). Through talk, interlocutors jointly create one or more “storylines,” a mutual understanding of the world they understand themselves to be occupying. Furthermore, they position themselves (reflexive positioning) and are positioned by others (interactive positioning) into roles that are available in these storylines (Harré & Van Langenhove, 1999). Both individual positions and storylines are “mutually determined, pro tem unless challenged, by the speech acts people are heard to produce, and that in turn is mutually determined by the positions that they are taken to be occupying in the episode” (Harré, Moghaddam, Cairnie, Rothbart, & Sabat, 2009, p. 8). Identity can therefore be viewed as social, in that it is jointly accomplished as something the individual experiences but others help to construct (O’Connor et al., 2007). Identity is also relational, not just because it exists in the relationships between persons, but also because communities give it meaning (Hand & Gresalfi, 2015; Lave & Wenger, 1991; Packer & Goicoechea, 2000). As such, a “mathematician” identity takes on meaning within the specific context—the norms, practices, and histories—of a given community, and may look different in different contexts (Lave, 1988, 1997). Being a “mathematician” in a third grade classroom, for example, might be somewhat similar to being a university math scholar in that the practices may nominally look the same, but the elementary school context positions students differently, resulting in a different kind of mathematician identity available for construction. Over time, all of the local interactions accumulate into a relatively stable “sense of self” (Bamberg & Georgakopoulou, 2008; Davies & Harré, 1990; Holland & Lave, 2001; Wortham, 2005); this is how we, as people, are able to experience ourselves as singular and constant. However, that sense of self is always contested and under negotiation as people work to produce and reproduce their identity in moment-to-moment interactions and are held accountable by communities for what they produce (Hand & Gresalfi, 2015; Packer & Goicoechea, 2000; Wenger, 1998). Identity construction is therefore dialogic because it requires navigating the contradiction of being (or at least desiring to be) singular and continuous while also being multifaceted and dynamic (Akkerman & Meijer, 2011). Among teachers, this social negotiation of identity can be seen in teachers’ discursive attempts to position themselves and others with respect to personally meaningful understandings, beliefs, and values (Yazan, 2017) and to gain community validation for different interpretations of what it means to be a teacher (Cohen, 2010).

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discuss. Moreover, she is reflexively producing herself as a kind of person who “has” students, namely a teacher. Therefore, talk is a form of social action in the sense that it shapes the social context (Goodwin & Heritage, 1990; see also Searle, 1969). As Heritage and Clayman (2010) maintain, “persons are continuously creating, maintaining, or altering the social circumstances in which they are placed—regardless of how massively, even oppressively, ‘pre-defined’ those situations appear to be—and they do so in and through the [discursive] actions they perform” (p. 21). This is why the joint enterprise of a community of practice is arguably an emergent phenomenon; it is produced, moment-bymoment, through discursive actions including talk, no matter how pre-defined or designed it may seem to be. Identity can likewise be understood as forming and reforming within the discursive space of a given community as speakers work through talk to negotiate what it means to be a participant within the community (like a MTC). Often this discursive work draws on shared cultural models that have been developed in situations outside the local context. For example, consider the meanings that one might attach to the phrase, “my kids,” in the following two statements: “my kids woke me up early this morning” and “my kids were so good today in sixth period.” In the first utterance, one is likely to hear “my kids” as a reference to the speaker’s children, while in the second utterance, one is likely to hear the same phrase as a reference to the speaker’s students. Thus, the meaning of the phrase “my kids” is situated in that is it “assemble[d] ‘on the spot,’ as we communicate in a given context” (Gee & Green, 1998, p. 122). These situated meanings, in turn, rely on shared cultural models—of family life and school life, respectively—that are indexed in the utterances. In the first utterance, a model of family life is indexed through the phrase “early this morning”, whereas in the second utterance a model of school life is indexed through the phrase “sixth period.” Thus, through indexicality, talk serves as a resource through which cultural models are mobilized to produce situated meanings. Of course, in neither utterance is the meaning of “my kids” unambiguous, and interlocutors may engage in further discourse to negotiate the situated meaning of the phrase (Heritage, 1984). In light of this conceptualization of talk as social action and resource for situated meaning, we posed the following research questions: What work is being done by talk that indexes teachers’ professional experiences and concerns during the observed MTC gatherings? More specifically, how does such talk draw upon the cultural model of school and the roles therein to position participants as particular kinds of participants in the collective activity and establish MTCs as a particular kind of professional community?

2.3. Talk as social action and a resource for meaning-making Following Gee and Green (1998), we understand talk as both a form of social action and a resource for the production of situated meaning. As a form of social action, talk is one of the means through which community and identity are produced, with particular utterances and conversations doing work that can be inferred through systematic analysis (Packer, 2018). Two aspects of talk are particularly relevant to our study: indexicality and reflexivity (Hanks, 1996). Through talk, we often index or make reference to particular cultural models using particular linguistic forms associated with a given model. Indexing a particular model makes it relevant or mobilizes it within the present situation (Silverstein, 1992). Additionally, talk is reflexive in the sense that it not only describes situations, but also produces them (Heritage, 1984; Potter, 1996). For example, if an interlocutor talks about “her students,” she is indexing a broader cultural model—schooling, in this case—where “student” is a relevant category. And in doing so, she is not only describing a past situation involving her students; she is also reflexively producing her present setting as one where formal schooling is appropriate to

3. Materials and methods of analysis We framed our inquiry using a comparative, constructivist case study approach (Bartlett & Vavrus, 2017; Yin, 2015). In line with our conceptual framework, rather than viewing them as bounded or static, we treated participant identities and local MTC gatherings as situated in broader sociohistorical narratives and under constant construction and renewal by participants. We thus aimed to build theory on teacher identity and community through a combination of ethnographic and discursive analytic methods that would account for the larger discursive framing of the MTC program when considering participants’ micro-level moments of talk and its positioning work (Anderson, 2009). 3.1. Study context Founded in 2006 by the American Institute of Mathematics (AIM), Math Teachers’ Circles bring K-20 math educators

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together to engage in collaborative mathematical activity. MTCs have expanded rapidly, and currently, there are over 110 MTCs in 40 states in the U.S.1 At the time of the data collection, the statewide MTC examined for this study held local gatherings in five locations across a state in the western United States (pseudonyms are used for all school and location names). These gatherings were facilitated by lead teams comprising 3–5 teachers and 2–3 faculty from local postsecondary institutions. Lead team members had previously attended a group training session facilitated by AIM. The lead teams designed and conducted their gatherings independently of each other, with financial support provided by the coordinators of the statewide MTC. Across all gatherings, there were 163 participants: approximately 80% were practicing teachers (20% elementary, 25% middle school, 35% high school), 10% were postsecondary faculty, and 10% were pre-service teachers. At a typical gathering, there were approximately 15 participants in attendance, including members of the lead team. Most gatherings were held in classrooms in postsecondary institutions; four were held in public community rooms. Despite their autonomy, lead teams developed their gatherings using a similar structure composed of a series of phases (Peck, Erickson, FelicianoSemidei, Renga, & Wu, 2018). Gatherings generally began with facilitators calling the participants together for introductions. Facilitators then reminded participants of the purpose of the gatherings and plans for future gatherings, a phase we called organizational work due to its focus on coordinating the local gathering and explicit attempts to connect those gatherings to the larger MTC initiative. After setting up the math problem for that gathering, facilitators prompted participants to work on the problem in small groups. Finally, facilitators convened the group toward the end of the gathering and wrapped things up with additional organizational work such as awarding door prizes. During the wrap-up, facilitators occasionally engaged participants in discussions of the relevance of the gathering’s math problem to classroom practice. 3.2. Data collection and analysis Our data for this analysis are from a larger data corpus collected as part of multiyear study of a statewide MTC initiative. Following our assumption that community and identity develop through interpersonal interactions, we had been using video and audio recorders to capture the naturally occurring interactions of participants as they engaged in activity during the gatherings (Silverman, 2016). The data used for this paper included video recordings of twelve, two-hour local MTC gatherings from five locations; this constituted all of the locations and gathering under investigation during 2016, the first year of the larger study. To analyze data we used ethnographic analytic and microanalytic techniques to build theory and analyze discourse. We began with a cyclical data analysis method (Corbin & Strauss, 2015; Glaser & Strauss, 1967) that relied on constant comparisons between our developing theory and our data. For each gathering, we engaged in the following process. First, we identified all of the excerpts in which participants indexed school in their talk. For this step, at least two members of the research team initially analyzed video and data logs for each gathering. The researchers independently identified excerpts that included indexes to school, which they coded as connection to school (see Appendix A for codebook). The sub-teams then met to discuss their independent codings, resolve disagreements,

1 Based in part on a growing body of research demonstrating the benefits of MTC participation on math teacher development (cf., Donaldson et al., 2014), the Conference Board of Mathematical Sciences have recommended MTCs as an exemplar professional development for teachers, and the American Institute of Mathematics has developed infrastructure to organize and promote MTCs.

and come to a consensus. Talk that indexed teaching and schooling amounted to approximately 15 percent of the cumulative talk documented at all locations and MTC gatherings combined. The full group then analyzed the coded excerpts and used inductive coding to apply sub-codes to each excerpt according to concepts embedded within them. This resulted in preliminary interpretations, which were tested against data from the next gathering, and so on, for all twelve gatherings. Codes 2-7 (see Appendix A) emerged in this phase, including adopt school roles and relate to school math. We then engaged in axial coding to relate the codes to each other (Strauss, 1987). In this phase, we identified two super-ordinate categories, related to (a) talk that positioned participants and (b) talk that framed community (Davies & Harré, 1990; Harré et al., 2009). Though we report on these themes separately in the findings, we did not view them as mutually exclusive given our understanding of community and identity as intertwined. In fact, we treated many excerpts of talk as doing both kinds of work. Take the following utterance by a teacher, which occurred as participants shared “takeaways” from the evening’s mathematical activity: We thought it was nice how you would be able to differentiate this—even just these puzzles to different, classes. Right? So depending on the level of the students you could change the time (brief pause) frame, like, ‘what’s the shortest amount of time?’ and see the different scenarios that each group [of students] brought. By relating the activity to schooling, the interlocutor both positions herself and the other participants as teachers and frames the community as one concerned with finding better ways to engage students. Finally, we produced a data display (Miles, Huberman, & ˜ 2014) in the form of a matrix that crossed the phases Saldana, of the gathering with the two super-ordinate categories identified through axial coding (positioning participants and framing community). Several representative excerpts of the positioning and community-building work were identified for each cell in the matrix. We transcribed these excepts and engaged in ethnographic microanalysis of interaction (Erickson, 1992; Erickson, 1995) to get a deeper understanding of the work done by the talk (Packer, 2018), checking this analysis of representative talk against the other excerpts in a given cell. When necessary, we transcribed and analyzed additional talk to ensure full representation. In our analysis of interaction, we analyzed interactive sequences to account for how actions are produced and evaluated in turns of talk (Heritage & Clayman, 2010). We analyzed these turns by examining the way that they were designed, taken-up, and sequenced in interaction by the participants. For example, in examining how participants introduced themselves at the start of gatherings (see Transcript 2 below), we analyzed the design of Turn 1, noting the use of the imperative mood, the action it implies, and the ways that it positions the speaker and hearers. We then analyzed the subsequent turns to determine how this action was taken up by the other interlocutors, including how they endorsed or resisted the positioning. 3.3. Researcher positionality Before proceeding, we need to acknowledge that three of authors of this paper are the coordinators of the statewide MTC initiative (Peck, Erickson, and Wu). The coordinators thus felt affection for and attachment to the MTC gatherings, which they see as a promising way to address the persistent challenge of retaining quality math teachers in rural schools (Lowe, 2006; Monk, 2007). Even so, this group worked to take a more neutral position, remain open to critical insights, and seek critical lessons from the observed activity. As such, our team followed the data to conjec-

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Fig. 1. Talk within the context of particular phases of the gathering appeared to be positioning participants between teaching and doing math, resulting in a temporary hybrid identity of teachers doing math.

tures and conclusions, even those that were uncomfortable for the MTC coordinators. Also worth noting is that two members of the research team (Renga and Feliciano-Semidei) are not involved in implementing MTCs, which enabled an outsider perspective during analysis. 4. Findings

seen as working to position participants on a micro-trajectory of identification (cf., Wortham, 2005) from (1a) teachers of mathematics to (2) teachers doing mathematics and back again to (1b) teachers of mathematics. This trajectory followed the structure of gatherings as facilitators moved participants through the phases of the gathering and engaged them in particular kinds of activities and discussions (Fig. 1).

Our analysis illuminated that talk indexing the professional context of teaching served to establish a hybrid identity for participants at the observed MTC local gatherings. More specifically, it could be

4.1.1. Initial positioning as teachers of math Participants presumably entered MTC gatherings with complex identities constructed within and across a range of social contexts. Even so, participants selectively indexed their respective connections to K-12 teaching as current or prospective math educators or as teacher educators. During the introductory phase of the MTC gatherings, such talk thus appeared to position the participants largely as teachers of math. This was evident as participants socialized prior to the start of meetings, where they would often share with each other aspects of their work in schools. At the start of one gathering, for example, three participants sat down at a table with plates of food from a buffet. After a brief silence, Participant A initiated the conversation shown in the following excerpt.3

2 Grossman et al. (2001) persuasively argue that the concept of community is often used inappropriately to refer to any group of teachers when, in fact, community formation is a process. We agree, though we see communities as in a perpetual state of renewal, with no clearly defined moment at which a group becomes a community and ceases to develop. For parsimony, we refer to the MTC groups as communities, though it would be more accurate to describe them as emergent or communitiesin-development.

3 Talk here is transcribed using standard punctuation: commas denote short pauses, periods denote longer pauses after a falling intonation, and question marks denote pauses after a rising intonation. Underlining denotes vocal emphasis, and a hyphen denotes a restart. Vertically-aligned square brackets denote overlapping talk, while double parentheses denote non-verbal action. Bracketed ellipses denote spoken words that have been excluded from the transcription to preserve analytic focus.

Our analysis of talk indexing schooling resulted in two broad findings: (1) talk as producing a hybrid identity; and (2) talk as establishing the community as a hybrid space in which disciplinary engagement becomes intertwined with the teachers’ professional concerns.2 These findings are aggregate portraits of the positioning work inferred from the talk, which we now discuss in turn. 4.1. Talk as producing a hybrid identity

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Assuming a shared professional identity of teacher, Participant A presumed the others’ professional status as teachers, which was affirmed and followed by participants jointly refining their teaching identities with respect to grade band and subject taught. This professional talk was intertwined with topics such as personal histories (“I’m from Milford. . .”) and desired futures (“I’d love teaching

in a smaller school.”). This seamless intertwinement of school talk with other topics of conversation was a common occurrence across gatherings. Facilitators further positioned participants with respect to their shared professional identity as they started meetings by soliciting introductions, as illustrated in the following excerpt.

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As with the prior excerpt, the initiating speaker introduced professional categories that are meaningful for teachers and positioned the participants as the sorts of people who identify with such markers. Participant D obliged the facilitator’s request, thus accomplishing two pieces of work. First, because the introduction was unmarked, it served to validate the facilitator’s request as reasonable and unproblematic, which reified the categories of identification. Second, by slotting herself into the categories, Participant D reflexively positioned herself as a teacher and refined the positioning by noting that she taught geometry. Even when the normative construction broke down in turn 14, the broader categorical referent for identification with the group did not. While Participant E introduced herself as a student, she provided an account for her presence by clarifying that she was studying to be a teacher; she then speculatively slotted herself into the professional category of grade level (middle school). Thus, when one could use the phrase “I teach,” no further accounting was necessary. But when this phrase was not available, one needed to provide an account for why (“I’m a student . . . I hope to teach. . .”) in a way that positioned the speaker relative to the community’s shared professional identity. Across gatherings, this expectation to identify with teaching was so strong that, even when facilitators did not explicitly ask participants to position themselves as teachers of math during introductions, they did so anyway. Facilitators sometimes reinforced this identity by holding drawings for prizes such as books on math instruction.

4.1.2. Transitioning participants to become doers of math The next phase observed across the gatherings transitioned participants into math problem solving, which precipitated a shift in identity. Facilitators within this phase invited participants to swap the educator’s hat for that of a math doer who takes pleasure in solving problems. This was initiated in part by facilitators stating the purpose of MTCs and providing a rationale for positioning oneself differently with respect to math activity. One facilitator explained how she and the other facilitators of the gathering were “not trying to teach you something here to use in your classroom.” Instead, she

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explained, their aim was to “enjoy math” and “enjoy each other.” As another facilitator explained to participants at his local gathering, We don’t have any expectations except you walking away with a good feeling. We’re not trying to prove anything. This is just for us. We’re not trying to say, “And now, fourth grade math achievement will go up because—” It has nothing to do with it. What we’re trying to do is just, be a group that likes mathematics. In this turn, the facilitator indexed a cultural model of schooling through the phrase “fourth-grade math achievement.” The turn worked to put schooling, and especially the attendant professional concerns about student achievement, into opposition with engagement in mathematical activity. This can be seen in the use of the adverbs “not” and “just” to modify the verb “trying” in the second half of the turn. In particular, the use of the word “not” in “we’re not trying to say” negates professional concerns, which is reinforced with the exclusionary “just” in reference to mathematical activity. We documented similar constructions across gatherings in facilitators’ efforts to frame the group’s doing of math. In the common construction of these turns, teaching math is indexed to surface and then to temporarily diminish its prominence. Notably, these turns did more than describe the intent of the gatherings; they also did identity work by repositioning participants from the previously established identity of teachers of math to a new identity: doers of math. This repositioning was further accomplished as facilitators set up the problems to be addressed in their gatherings as mathematical problems with little if any indexing of potential classroom applications (across gatherings, approximately 80% included no mention of teaching during problem set up). During the problem solving phase of gatherings, participants engaged in sustained mathematical discourse. The following excerpt began after the facilitator established the mathematical activity—the “game of Stomp”—as the primary “event” of the gathering. As he explained, the object of the activity was not so much to win the game as to solve a mathematical optimization problem by finding a minimum number of moves. Transcript 3 begins as one group is working on the activity.

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Such talk accomplished identity work as speakers positioned themselves and their groupmates as the kinds of people who do math, which served to reinforce the participants’ becoming doers of math. 4.1.3. Positioning participants as teachers doing math Even as participants engaged in sustained mathematical discourse, they collectively indexed teaching and schooling on average at least three times per gathering during the problem solving phase, or approximately once every 25 min. In some cases, this talk explicitly (re)positioned participants as teachers. For example, as a group worked together to solve a “taxman” problem, one participant appealed for help from her group by stating, “Oh geez, I don’t want the taxman to get my money so . . . I’m trying a new strategy, so—math teachers help me: should we take low?” Of interest here

is the way the participant addressed her group not as “mathematicians” or “problem solvers” or another signifier of a doer of math, but specifically as “math teachers.” This brief index to schooling worked to retain the participants’ initial positioned identity with respect to their shared professional role as teachers even as they were doing math. Often such references were not as explicit in their positioning of participants. More commonly, talk that indexed the cultural model of schooling during problem solving arose as participants made their identities as teachers relevant to their ongoing mathematical work in a way that was seamlessly interwoven into the mathematical discourse. The following interaction, pulled from the ongoing conversation of three participants working together on a problem that involves a sequence of numbers, is representative of this sort of talk.

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In turns 2 and 7, Participant J positioned all three group members (including herself) as math teachers when she asks about teaching sequences. These turns accomplished positioning work by associating the recipients with the actions and specific knowledge of math instruction. Neither participant overtly resisted this positioning in

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their uptake; rather, both simply answered the question with a reference to their classroom practice. Likewise, in turn 11, Participant J reflexively positioned herself as a teacher when she talked about textbooks, which are artifacts of teaching practice (and were taken up as such—e.g. Participant I’s uptake in turns 12 and 14).

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Even as this positioning took place, the three participants continued to discuss the math problem. In this way, their math talk positioned them as doers of math even as their teacher talk positioned them as teachers of math. The two kinds of talk, and related identities, were temporally interwoven. The intertwinement or coupling of identities was further underscored in turns 11 and 15 as Participant J explicitly brought her work as a doer of math into coordination with her work as a teacher of math by indicating that, even though she taught from a book that “really hammers” sequences, she was not prepared to see the sequence “as a quadratic.” This coupling was commonplace during the problem-solving phase of MTC gatherings, as illustrated by the following excerpt that occurred right after a group of participants reached a solution to their math problem.

Here the math doer and teacher identities are intertwined to different effect. Whereas Participant J’s identity as a teacher of math informed her identity as a doer of math, in this excerpt we see the opposite, with Participant M’s math work having informed her identity as a teacher. Such examples illustrate how, despite facilitators’ attempts to position participants as “pure” doers of math—an identity that was explicitly contrasted with one’s identity as a teacher—participants worked to make their professional identities relevant while engaged in mathematical activity. The identity being negotiated during this phase can thus be understood as a hybrid of doer of math and teacher of math, or a teacher doing math.

4.1.4. Repositioning participants as teachers of math As the end of the gathering approached, facilitators would typically reconvene the whole group. Where talk indexing teaching and school during the problem-solving phase was infrequent and punctuated long periods of conversation centered on doing math, talk in the wrap-up phase would often directly address teaching math and make explicit connections to schooling. Mirroring the introduction phase, this sometimes entailed subtle cues such as a drawing math teaching books for prizes. Other times it involved giving participants materials for doing a math activity with students. At one gathering, for example, facilitators handed out thumb drives with the documents required for implementing the activity from the gathering, accompanied by a suggestion that they laminate the printable pieces for classroom use. The connections to math teaching occurred most commonly, though, as facilitators explained how they used the math problem from the gathering in their own classrooms or when they engaged participants in thinking about how they might frame and use the problem with students. All but one of the 12 gatherings concluded in one of these two ways. As an example of the former, a facilitator wondered aloud: “The teacher brain within me was asking, ‘What mathematical practices does this problem use?’ Because I did use this with my kids and they were super into it.” As an example of the latter, a facilitator proclaimed: [Math] is power. And just to be able to take an activity like this and hook your students- you can see you’re not doing computations, you can see you’re not having them come up on the board and try to be in front of everybody, to try to write out a solution, or an algorithm here. But the ability for Mary [a facilitator] to be able to show you her powers as a math- mathemagician, I

guess- that- that’s the word that’s out there. You can get kids just eating out of your hands, with some of these activities. In these excerpts, facilitators described various hypothetical and real classroom scenes using pronouns to position facilitators (“I did this with my kids”) and participants (“You can get kids”) as teachers. Such references to math teaching arguably transitioned participants from doers of math back to teachers of math. The facilitators and participants thus concluded their gatherings by repositioning themselves into the professional identity and initial orientation to math activity with which they began. Even so, comments by facilitators suggested the hope was for a transformed professional identity through doing math, or at least sufficient disruption of the status

quo to foment a search for novel ways of seeing oneself as a math teacher—perhaps as a “mathemagician” in the classroom. As one facilitator explained: So the idea is not necessarily to give you something to use in your classroom; the idea is more to look at why we find math interesting. You know? Is it really the fact that we can work through an algorithm? Or is it the fact that we can problem solve and get that “Aha!” moment? Hopefully that’s what you’re bringing back to your classroom. Perhaps, the speaker suggested, experiencing a hybrid identity during MTC gathering—as simultaneously being a teacher and a doer of math—might lead participants to see math differently and reimagine their classroom identity and aims.

4.2. Talk as establishing a hybrid community Along with positioning their identities within the gatherings, participants’ professional talk appeared to shape the joint enterprise of the group, thereby establishing MTCs as a kind of professional community. We explicate this finding in two parts. First, we show that, through talk, participants mobilized schooling as a cultural model to develop situated meanings related to their participation in MTCs. We then show that, in doing so, talk that indexes teaching and schooling worked to produce the MTCs as hybrid communities that are concerned, at least in part, with the profession of teaching (Fig. 2).

4.2.1. Participants use schooling as a resource to develop situated meanings Participants indexed teaching and schooling to make sense of three aspects of their participation in MTCs: (a) the mathematics, (b) the aims of the activity, and (c) the social organization of gatherings.

4.2.1.1. Indexing teaching and schooling to make sense of the mathematics. A common way that participants made sense of the mathematics they were doing was to consider it in relation to their school-based experience. Take, for example, the following con-

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Fig. 2. Participants used schooling to develop situated meanings related to their participation in MTC gatherings, which works to establish the community as a kind of hybrid professional community.

versation between three participants working on a problem that involves summing an arithmetic sequence of integers.4

In turn 2, Participant O used the indexical “this” to reference the local math work of the group and then brought the local math work into coordination with schooling over the course of turns 2, 5, and 7. The key move comes in turn 7 as he explained his strategy by referencing something that “he just taught.” The word “taught” indexed teaching and schooling as a cultural model, thereby serving to situate the local mathematical activity within a familiar knowledge domain. In doing so, the model becomes a resource through which the participant made sense of the group’s activity and attempted to convey his understanding to his colleagues. This use of teaching and schooling as a sense-making resource is echoed in turn 8 as Participant P connected the group’s strategy to a particular

4 We infer that the participant is referring to Carl Friedrich Gauss, whose method for finding the value of an arithmetic series by identifying pairs of integers with equal sums matches the method he is describing.

method that she used in Mathcounts, a school-based math club and competition. Another common way that participants used schooling to make sense of the mathematics was to consider it in relation to grade band (e.g., elementary, middle, or high school). For example, in another gathering, as the facilitator passed out a sheet of problems, he stated that “[problem] three might be better if you’re in elementary school.” While working on this problem, participants sometimes indexed the facilitator’s instruction, using grade band to make sense of the mathematics. The following conversation (Transcript 7) between a group of participants discussing their problem-solving strategies serves as illustration.

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Participant R asked a question of Participant Q’s strategy that was designed to favor a negative response, serving as an implicit critique of the strategy reinforced by the participants’ responses in turns 3 and 4. Participant Q’s negative uptake in turn 3 was unmarked, indicating it was the preferred response. Participant S’s follow-up suggested the problem has not been solved, a tacit rejection of the proposed strategy for solving the problem. Notably, Participant R’s critique was not directed at the correctness of Participant G’s mathematics but at a perceived misreading of the grade bands—elementary, middle, and high—prescribed by the cultural model of schooling. Participant R did not see the relevance of the solution to an “elementary kid” and those who taught this grade band. Encoded within these grade bands are normative expectations for their respective student populations—what they can be expected to know and do in math. Participant R’s question about whether an “elementary kid” would “use symmetry” arguably drew on these normative expectations of “elementary” students. Her construction of a negative-preferred question served as a shorthand for the following logical chain: elementary kids would not use symmetry, this is an elementary problem, therefore symmetry is not an appropriate strategy for this problem. In both of these examples, participants employed professional talk that indexed teaching and schooling to assemble situated meanings of the mathematics they were engaged in. In Transcript 6, the group made sense of their own mathematical activity by locating it with the school-based mathematics they taught. In Transcript 7, the group used school grade-band expectations as a criterion for judging the appropriateness of their mathematical work. 4.2.1.2. Indexing teaching and schooling to make sense of the activity aims. As noted earlier, MTC participants sometimes employed professional talk while reflecting on the purposes and takeaways of the math activity. For example, at the end of a session that involved a set of logic puzzles, a facilitator invited participants to share “what they took away” from doing the puzzles together. Participants discussed the prompt in small groups and then shared a summary of their conversation with the whole group. For most groups, the publicly shared “takeaway” involved implementing the activity with students, as the next excerpt illustrates. We thought it was nice how you would be able to differentiate this—even just these puzzles to different, classes. Right? So depending on the level of the students you could change the time (brief pause) frame, like, ‘what’s the shortest amount of time?’ and see the different scenarios that each group [of students] brought. Here, the participant mobilized a cultural model of schooling through the references to differentiation and students to develop a situated meaning for the aims of the activity. She situated the meaning of the math done at the gathering within the model and

her role as teacher, employing the impersonal you and the subjunctive mood (“you could”) to describe her group’s imagining of how the activity might be implemented in a classroom with students.

4.2.1.3. Indexing teaching and schooling to make sense of the gathering’s social organization. In many respects, the MTC gathering looked like the typical classroom. Most gatherings shared a similar social and physical organization with participants seated in groups at tables while facilitators stood and walked around. Participants talked in these small groups as facilitators often interrupted, issuing prompts and addressing the whole group using loud voices. Furthermore, participants only shared with the larger group when invited to by facilitators. Participants made sense of these roles and the social organization by making connections to schooling. One facilitator, for example, began her gathering by stating: So we’re going to do the first problem together, and then I’ll kind of set you loose and you guys can work through the rest of them together. (brief pause) I’m programmed to teach middle schoolers, so I’m just going to treat you guys kind of like middle schoolers. During the pause and through the hedge “kind of”, the facilitator appeared to recognize the awkwardness of assuming a position of authority with respect to her peers and seemed compelled to acknowledge it. To make sense of this social organization, she mobilized a cultural model of schooling, noting the parallels between the roles enacted in the gathering (facilitator and participant) and the roles typical of school classrooms (teacher and student). This instance was one of the rare moments where the MTC participants’ enactment of the cultural model of schooling was explicitly identified through talk.

4.2.2. Talk as establishing the group as a kind of professional community Viewed collectively, the evidence of participants indexing teaching and schooling to develop situated meanings of the MTC experience arguably reflected their efforts to make sense of the joint enterprise of the community (What do we do here? What kind of community is this?) and their participation (Why am I here?). Crucially, however, this talk seemed to accomplish more than sense making; its reflexivity constitutes the group as a hybrid community, one in which “doing what mathematicians do” became intertwined with teaching math. Indeed, efforts to frame the MTC gatherings against or as a break from teachers’ normative professional expectations were countered by professional talk and the gatherings’ school-like social organization, which kept those expectations part of the MTC community and its joint enterprise.

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5. Discussion Our findings offer further evidence for how teachers jointly construct identity and community, in part, through discourse. Within the context of the MTC gathering, teachers indexed broader cultural models of disciplinary activity and schooling in their momentto-moment talk as they sought to make sense of themselves and the activity. In doing so, participants could be seen negotiating Grossman et al.’s (2001) essential tension of teacher communities. Through this discursive negotiation, participants produced hybridity with regards to schooling and the MTC community. In more conventional professional development focused on improving teachers’ instructional practice (cf., Heredia, Furtak, Morrison, & Renga, 2016), talk indexing teaching by teachers would seem unremarkable. Indeed, it is unsurprising that a group of math teachers gathered together would make connections to their classrooms and instruction. But in the context of gatherings devoted to engagement in disciplinary activity, this talk stood out and drew attention to teachers’ efforts to constitute and negotiate the essential tension between instructional and disciplinary practices in real time. As such, it made strange an otherwise commonplace situation: teachers gathering for professional development. With such situations that are “massively, even oppressively, ‘predefined”’ (Heritage & Clayman, 2010, p. 21), it strikes us as crucial to study participants’ social negotiation to appreciate its intricacy and better understand the way it produces particular identities and communities. Along with the essential tension, the repeated emphasis on doing math for its own sake pointed to the presence of another tension under negotiation as participants made meaning of their shared experience—a tension between engaging in the joint enterprise for personal pleasure versus for practical usefulness. If the essential tension of teacher community arises largely from teachers determining What domain of practice am I in? and Who am I here? while making sense of their joint enterprise, then this other tension attends to their asking the questions Why am I here? and What do I hope to get out of it?. Participants appeared to be negotiating both tensions as they worked to position themselves within the math activity and the MTC community. This negotiation was tricky and might explain the facilitators’ attempts to transition participants from teachers to doers of math and contradictory assertions of “doing the math for us” but also “doing it for students and teaching.” Indeed, despite their efforts to establish gatherings as about doing math for personal pleasure, participants’ professional talk routinely established the gatherings as being useful for acquiring classroom resources. We see such tensions as productive rather than as design flaws or signs of failure within the MTC or any other teacher community. Grossman et al. (2001) labeled their tension as “essential” because, in their view, both of its foci (pedagogy and disciplinary practice) are likely to emerge and must be addressed by any professional community of teachers. We agree, although we would add that tensions in general are essential because they are inherent in the negotiation of meaning and, as such, in the co-development of identity and community. For individuals within a community, the joint enterprise is emergent—rather than preordained—precisely because it is forged through members’ negotiation of various tensions (Barab et al., 2002; Barab, Schatz, & Scheckler, 2004). MTCs may therefore advertise themselves as a particular kind of community, but it is through the participants’ navigation of the intersection of teaching and math and their competing desires for pedagogical improvement and relief from that concern that the community’s character emerges.

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Similarly, the negotiation of tensions is the mechanism that drives individuals’ identity development. As Packer and Goicoechea (2000) explain, “typically membership is the start of a struggle for identity [as] people actively strive to come to terms with the practices of their community, adopting an attitude, taking a stand on the way membership of a community has positioned them” (p. 234). In this stance taking, which is often accomplished discursively, identities are (re)constituted and take shape. Thus, even as facilitators sought to create distance for participants from their professional identities (e.g., by inviting them to transition to mathematicians or doers of math), participants seemed to resist or at least modify this positioning to retain a teacher identity. This negotiation, we suggest, is the means through which the hybrid identity of teachers doing math emerged. Embracing this hybridity and what can be learned from it strikes us as necessary for helping teachers to grow, transform, or renew passion while navigating the complexity of being in community. Similar to work on small stories (cf., Taylor et al., 2018; Ives & Juzwik, 2015), we see productive possibilities in leveraging discourse spaces to provide teachers with opportunities to make temporary shifts in identity—to explore kinds of hybrid identities—so they might see themselves differently with respects to the disciplines they teach. Following from Cohen (2010), we suggest that designers of teacher community structure members’ identity bids (and counterbids) with knowledge of how the essential tension and other tensions render certain identities as more or less appealing. We recommend making those tensions visible to participants and allowing time for sharing answers to questions such as Who am I here? and Why am I here?. As a final thought, we want to acknowledge how teachers’ communities and professional identities are constructed within macro-level discourses on teaching. Such discourses position the profession in ways that can prompt collective resistance as teachers seek to position themselves differently. This is true of MTCs, which were created as a response to a perceived imbalance in the professional discourses and expectations of teachers. The program’s primary aim of inverting the essential tension to prioritize disciplinary engagement establishes an intriguing space for teachers to examine and (re)negotiate their professional identities. We see two complications to this project that are likely to keep the instructional side of the tension ascendant in discussions of teacher communities. First, it is worth considering how teachers acquire certain disciplinary knowledge and practices and the challenge this poses to their disaggregation of doing the discipline from teaching it. This dilemma is especially pronounced for many math teachers given how their practice-linked identities (Day et al., 2006) are arguably informed by their extensive experiences learning and teaching math in schools (Lave, 1997). As Lave (1988) observes, math practices learned in school should be seen as interwoven with classroom practices and are thus bounded by schooling’s roles, limitations, and demands. She explains how math in the typical classroom is closed rather than open, distinct from other forms of problemsolving activity in its emphasis on “small-scale demands for an acquiescent problem solver to operate on the information given by a problem giver using algorithms or formal inferential reasoning to match a correct or ideal answer” (p. 35). The math activity at the observed MTC gatherings mostly fit this description, with participants, much like students, working to solve pre-selected problems administered by a “problem-giver”. The story of doing math and doing school may thus have been tightly linked for the teachers. Breaking this linkage is likely to require opportunities for teachers to do math in ways more similar to how it is done by mathemati-

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cians, which will require a different structure and content than school math (Bass, 2011; Lave, Smith, & Butler, 1988; Lockhart, 2009; Movshovitz-Hadar & Kleiner, 2009). But as it was, the math in the MTC gatherings bore sufficient resemblance to K-12 classroom problem solving that discussions of practical relevance may have seemed appropriate, even natural. Second, the current policy climate keeps instructional practice at the forefront of teachers’ professional concerns. Increasingly in the U.S. and around the world, teachers have become immersed in an accountability discourse that prioritizes high stakes evaluations and test-based accountability (Lingard, 2014; Lingard & Lewis, 2016; Moses & Nanna, 2007). Under this policy regime, contextual nuance has given way to the demands of international benchmarking (Lewis & Hogan, 2019) and curriculum has narrowed as teachers in high poverty schools especially feel pressure to cover what is on the test rather than engage students in exploring the richness of various disciplinary fields (Au, 2007; Berliner, 2011; Roth McDuffie et al., 2017). Likewise, with their instructional practice under intense scrutiny, teachers are pushed to focus more on improving technique (doing what works) than on understanding their core values or motivations (Cochran-Smith & Lytle, 2009; Zeichner & Liston, 2013) or on cultivating their educational passions (Burch, 2004; Garrison, 1997; Renga, 2017). This seems likely to affect how teachers negotiate the essential tension and the kinds of hybrid identities and communities they produce or aspire to produce. 6. Study limitations We want to stress that the local teacher communities and participant identities were more complicated than our analysis suggests, which warrants three important caveats. First, our case study approach admittedly glossed over the ways individuals positioned themselves with respect to the joint activity. Presumably, some participants bought into the negotiated meanings more than others did. Understanding individual trajectories and meanings, perhaps based on career stage (e.g., new or veteran), could be helpful for making sense of differences in participant contributions. Second, we assume that doing math with fellow math teachers fulfilled purposes not captured by our analysis. For example, there are mathematical aims such as the building of a new mathematical understanding, sense of purpose for math engagement, or appreciation for the beauty of mathematics (cf., Bleiler-Baxter, Wanner, & Strayer, 2019). Easier to take for granted, perhaps, are the social aims of friendship or enlarging one’s professional network, as well as emotional aims such as wanting the warmth of shared accomplishment. These and other purposes for community participation would benefit from further inquiry. Third, the data for this study was limited to a single year, limiting our capacity to ascertain trends in community and identity construction over a longer period of time. Our findings represent a snapshot of the continual process of constructing meaning. Although we would expect talk that indexes teaching and schooling to persist, and the hybrid identity of teachers doing math to remain fairly constant in MTC gatherings as the program grows and codifies its aims, structures, and desired outcomes, we cannot be sure without a longitudinal analysis.

7. Conclusion There is compelling reason to see teacher communities as important sites for teachers’ construction of professional identities. Those communities, however, are not fixed entities and are under construction by members as they negotiate tensions like the essential tension between engaging in disciplinary activity on the one hand, and focusing on professional concerns such as teaching and learning on the other. In this paper, we contribute to the literature on teacher community and identity by exploring how teachers navigate the essential tension and what it means for their joint production of community and identity. To do so, we examined the work done by professional talk in teacher communities that were ostensibly organized around engagement in disciplinary mathematics activity. Our main finding is that within MTC gatherings, teachers’ professional talk indexing schooling worked to produce hybrid identities and communities. We documented how professional talk propelled participants along a micro-trajectory of identification, producing a negotiated identity comprising a hybrid of teacher of mathematics and doer of mathematics. With respect to community, we found that participants used professional talk to develop situated meanings related to their participation in the community. Specifically, participants indexed teaching and schooling to make sense of the mathematics they were doing, the aims of the activities, and the social organization of the gatherings. Taken together, this sense-making worked reflexively to produce the communities as a kind of hybrid space in which disciplinary activity and professional concerns became intertwined and hybrid identities emerged. Future research should explore how this and other hybrid identities may thicken (Holland & Lave, 2001; Wortham, 2004) or stabilize over time and become consequential for teachers and their work in classrooms. It would also be worth examining how facilitators of teacher professional development navigate, contribute to, and shape the production of hybridity within teacher community. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We are grateful to the MTC participants and facilitators who graciously opened their gatherings to our study. We also want to thank George Kamberelis, Carrie Allen, Sara Heredia, Joanna WeidlerLewis, and the two peer reviewers for their invaluable feedback on the manuscript. This work was supported by grants from The American Institute of Mathematics, The Montana Office of the Commissioner of Higher Education [grant number S367B140023-14A], and The University of Montana. Appendix A. Codebook

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Code

Description

Examples

1. Connection to School

Participants or facilitators explicitly bring up anything to do with students, teaching, learning, school, or classrooms. Note that this is not used to identify places where a facilitator does something that reminds the researcher of a “teacher move” or some other “school like” behavior happens in a gathering. Rather, it is for where someone explicitly indexes teaching or something related to teaching or school. Participants or facilitators explicitly index their professional role (their own, or someone else’s) related to schooling (e.g., as a teacher, paraprofessional, etc.).

Small group working on problem on whiteboard. Participant says to another, “You don’t have middle school kids, do you?” She replies, “No, I teach high school.” First participant says, “Because the term I was thinking of was a middle-school term.”

2. Professional Roles

3. Adopt School Roles 4. Use with Students

Participants are positioned as students, and/or facilitators are positioned as teachers Participants or facilitators discuss implementing the math activity with students, including retrospectively describing how they have implemented the activity with students OR prospectively discussing how they might implement the activity with students

5. Relate to School Math

Participants relate the activity to school problems, instructional techniques, or school level

6. Purpose of MTCs Connected to School

Participants or facilitators index school in talk that discusses the purpose of MTCs,

7. Teaching Artifacts

Artifacts related to teaching (e.g., books, lesson plans, websites) are introduced, shared, or discussed

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Example of a participant indexing their own professional role: “I’m {name}, I teach at {school}, mostly geometry.” Example of a participant indexing someone else’s professional role: Facilitator refers to another facilitator as a “[f]irst class math teacher.” Facilitator says to her group, “I’m programmed to teach middle schoolers, so I’m just going to treat you guys kind of like middle schoolers.” Example of a retrospective description of implementing the activity with students: Facilitator says to her group, “The teacher brain within me was asking, ‘What mathematical practices does this problem use?’ Because I did use this with my kids and they were super into it.” Example of a prospective imagining of implementing the activity with students: Facilitator says, “We thought it was nice how you would be able to differentiate this—these puzzles. Right? So depending on the level of the students you could change the time frame, like, ‘what’s the shortest amount of time?’ and see what each group [of students] brought.” Example of relating math activity to school problems: Participant says, “This is like that problem where you discount by 10% and raise the price by 10%” Example of relating math activity to instructional techniques: Facilitator connects a strategy to “how we teach multiplication.” Example of relating math activity to school level: Facilitator explains, “Some of this math is elementary, some is high school level.” Facilitator explains, “So the idea is not necessarily to give you something to use in your classroom; the idea is more to look at why we find math interesting. You know? Is it really the fact that we can work through an algorithm? Or is it the fact that we can problem solve and get that “Aha!” moment? Hopefully that’s what you’re bringing back to your classroom. Participant mentions that she does a similar activity with students and shares a resource. Participant replies, “I’ve been here for 20 min and already got two resources.”

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