Supporting Teachers’ Equity-Oriented Learning and Identities: A Resource-Centered Perspective

by Nicole L. Louie - 2017

Background/Context: Despite calls for equity in education, the dominant mode of schooling reproduces hierarchies, positioning some students as bright, gifted, or fast learners and others as lazy, in need of remediation, or slow. A number of studies have shown that teachers’ professional communities and networks can address this problem and enhance outcomes for all students. However, more research is needed not only to show the structure of supportive networks but also to explain the mechanisms through which they foster teacher learning.

Research Questions: This paper addresses three questions: (1) Where do teachers encounter resources that support their engagement with nondominant, equity-oriented teaching practice? (2) What kinds of resources support teachers’ engagement with nondominant teaching practice? and (3) How do different kinds of resources come together to support teachers’ patterns of engagement with communities of nondominant teaching practice?

Research Design: A multisite case study was conducted over the course of an academic year. The study involved extensive observations in routine teacher meetings and professional development settings, as well as classroom observations and teacher interviews.

Participants: Participants included 18 mathematics teachers from two diverse urban high schools. The mathematics departments at both schools expressed commitments to professional learning and collaboration in order to better serve their diverse student bodies, in particular, to support students who had previously been unsuccessful. Six teachers were selected as focal teachers for more in-depth observation and interviewing. This paper describes the contrasting cases of four teachers.

Findings:Two of the focal teachers maintained close engagement with nondominant, equity-oriented practice throughout the period of the study, while two did not. A comparison of teachers’ professional support networks showed that their patterns of engagement were related to their connections to sources outside their school-based communities and the access that these connections provided to four distinct types of resources. The two teachers who maintained their engagement with reforms were found to have abundant and personally significant identity resources, which were critical for their ongoing learning. Although the other two teachers had technical resources for engaging with reforms, they did not have such identity resources.

Conclusions/Recommendations: Research and practice have tended to focus on technical aspects of teacher learning in service of reform. This paper suggests that teachers need not only more resources, but also more kinds of resources in order to sustain the learning that student-centered, equity-oriented reforms require.

INTRODUCTION

Teachers, like most of us, often talk about students as high and low, bright and slow. These labels reflect an ideology of intelligence that describes individuals intellectual ability as innate, fixed, and quantifiable on a linear scale (Oakes, Wells, Jones, & Datnow, 1997). Many scholars have documented the particular dominance of this ideology in popular conceptions of mathematics ability (Ernest, 1991; Parks, 2010; Ruthven, 1987), noting, for example, the common belief that only geniuses and nerds can be good at math (Boaler, 2008b; Schoenfeld, 1988)and the negative impact this belief may have on students mathematics learning and identity development (Boaler & Greeno, 2000).

Redefining competence in more inclusive ways is thus an important aspect of advancing equity in mathematics classrooms and in education more broadly. This is not least because hierarchies of intelligence in general and of mathematical ability specifically are racialized, classed, gendered, and in other ways linked to classification systems that structure social inequality (Gutiérrez, 2002; Martin, 2009; Oakes et al., 1997). But the importance of redefining competence does not derive solely from these links. A society in which a demographically representative but still exclusive group of people was regarded as mathematically capable and intelligent would fall short of achieving equity; as philosopher Elizabeth Anderson argues (1999), the point of democratic equality is to create a community in which people stand in relations of equality to others (p. 289; see also Boaler, 2008a). It is not possible to achieve equality only for some.

But despite their expressed commitments to all students, the teachers in the study on which this paper reports frequently (though often subtly and unintentionally) reified ability hierarchies, positioning some students as smart and others as slow. In this respect, their cases mirror others; the tenacity of the traditional grammar of schooling (D. K. Cohen, 1990; Tyack & Cuban, 1995) and of hierarchical and exclusive ideologies of intelligence (Oakes et al., 1997) in the face of reform is well-documented. Elsewhere, I have analyzed the persistence of dominant discourses and ideologies in the daily work of the teachers in this study, examining how they come to life in teachers classroom instruction and collegial conversations (Louie, 2015).

Two of the teachers in this studyAmanda Pepper and Ryan Sower (all names are pseudonyms)were unusual in that they consistently framed mathematical competence in culturally nondominant ways, as something that all students possessed in a variety of forms. For example, launching a group task early in the year, Ryan used a multiple-ability orientation (as it is called in Complex Instruction parlance; see Tsu, Lotan, & Cossey, 2014):

Were gonna do a task today where youre gonna need all the different smartnesses of your group. You need people who are good at estimating, measuring, making conjectures, and seeing patterns (writing each smartness on the whiteboard as he says it). I know every single one of you is good at at least one of these. So everyone has something to offer your group.

Through statements such as this and a number of other instructional strategies, Ryan and Amanda disrupted dominant understandings of mathematical smartness in ways that created opportunities for each of their studentsincluding those would not have been viewed as good at math in most classroomsto engage with rigorous, challenging mathematics and to develop a sense of agency, authority, and competence (Louie, 2015).

Images of teachers in popular culture suggest that Ryan and Amanda are heroes, islands of hope (Gutiérrez, 1999) in a sea of dysfunction (think of the films Stand and Deliver, Dangerous Minds, and Freedom Writers, to name a few). Walking into either of their classrooms on any given day and observing their unusual instruction, it would be easy to conclude that they were reinventing their practice more or less alone. Following them out of their classrooms, however, it is obvious that though they deserve a great deal of credit, they have organized support networks that provided them with a rich mixture of resources to foster their ongoing engagement with nondominant practice. To better understand how these resources supported the teachersand how others might be supported to learn against the grain, in mathematics education or otherwise (Cochran-Smith, 1991)this paper addresses three questions:

1.

Where do teachers encounter resources that support their engagement with nondominant teaching practice?

2.

What kinds of resources support teachers engagement with nondominant teaching practice?

3.

How do different kinds of resources come together to support teachers patterns of engagement with communities of nondominant teaching practice?

That is, the paper examines the sources of support that the teachers in this study experienced, the kinds of resources that these sources provided, and the ways that these resources interacted to support or undermine teachers engagement with forms of teaching practice that deviate from the norm.¹

Ryan and Amandas cases show that it is possible for teachers to accomplish the kind of learning that the task of reframing mathematical competence requires. This paper analyzes how such learning can be supported, finding that it requires a diverse array of resourcesincluding resources that directly support teachers learning as well as resources that indirectly support learning by fostering teachers identity development. The ways in which learning and identity processes shape and inform each other, as different kinds of resources are made available to, taken up by, and sought out by teachers, are illustrated using the contrasting cases of four teachers (Ryan, Amanda, and two others). This analysis also highlights the interplay of environmental factors and teacher agency in seeking out and advocating for resources. Thus, it addresses a limitation of existing research on teacher learning communities, showing that important resources for learning may come from sources outside teachers school-based communities of practice. This may be especially relevant when the learning at issue involves not only the transformation of the individual practitioner but also the transformation of the practice itself. The paper therefore highlights the importance of building not only more resources but also more kinds of resources into teachers working lives, to better support ongoing engagement with equity- and reform-oriented learning.

THEORETICAL PERSPECTIVE

The idea that identity processes play a fundamental role in both mediating and constituting learning is a central premise of this paper. From this perspective, teacher learning cannot be understood in isolation from identity, because the two are dynamically intertwined. That is, both learning and identity are constantly taking shape, as opposed to being achieved and then fixed, and they take shape together in mutually informing and mutually constituting ways. Identity is both a cause and an effect of learning; learning is both a cause and an effect of identity. Yet the two are distinct; a person may appropriate cultural tools (an aspect of learning) without taking them on internally (an aspect of identity) (Nasir & Cooks, 2009, p. 44). For example, one teacher may learn to use instructional strategies associated with a particular reform and develop an identity as a reformer, while another may learn the same strategies but maintain an identity as someone whose work is peripheral to the reform.

This papers investigation of the relationships between teacher learning and identity processes is heavily informed by theories of learning in communities of practice (Wenger, 1998). A significant limitation of work on communities of practice, however, is its focus on the relatively strong relationships that arise through mutual engagement in a joint enterprise. I draw on theories of social networks, in particular Granovetters (1973) perspective on the strength of weak ties, to address this limitation and explore the resources that weak ties may provide for teacher learning and identity development.

LEARNING AND IDENTITY IN COMMUNITIES OF PRACTICE

As Wenger (1998) defines them, communities of practice are organized around mutual engagement in a joint enterprise, with a shared repertoire. Through everyday participation in communities of practice, people negotiate the nature of the enterprise and the shared repertoire that defines it, drawing onand constrained bythe history of the practice but also inevitably making it their own. This negotiation, Wenger argues, is simultaneously a process of learning and a process of becoming: our participation in social practices, he writes, shapes not just what we do, but also who we are and how we interpret what we do (p. 4).

Nasir and her colleagues (Nasir, 2012; Nasir & Cooks, 2009) build on Wengers analysis to reveal how learning and identity processes are linked to each other and to social practice through interactions around three types of identity resources: material, relational, and ideational. Nasir (2012) defines these resources as follows.

By material resources, I mean the ways that the physical environment, its organization, and the artifacts in it support ones sense of connection to a practice. Relational resources refer to the way in which positive relationships with others in the context can increase ones connection to a practice. Ideational resources refer to the ideas about oneself and ones relationship to and place in a practice and the world, as well as ideas about what is valued and what is good. (p. 110, emphasis in original).

These resources work together to foster trajectories of participation in particular practices and communities of practice. Thus, an instructional strategy (say, the multiple-ability orientation described above) might be a material resource that supports a teacher (such as Ryan) to participate in nondominant teaching practice. The strategy might also serve as an ideational resource, shaping his ideas about what constitutes good teaching to include responsibilities that he had not previously considered (e.g., expanding students ideas about smartness). Other ideational resources (e.g., a general commitment to supporting struggling students), relational resources (e.g., connections to more experienced practitioners who provide advice and encouragement), and material resources (e.g., a curriculum that presents mathematics as multidimensional) might simultaneously bolster his efforts to learn and develop his practice in equity-oriented, nondominant directions. In other words, these resources might support an inbound trajectory (Nasir & Cooks, 2009; Wenger, 1998) of deepening engagement with communities of nondominant teaching practice. On the other hand, insufficient resources might contribute to an outbound trajectory, in which the teacher leaves the profession because of a lack of support for learning and identity development.

I follow Nasir and Cooks (2009) in conceptualizing learning as shifts in use of artifacts (both cultural and cognitive) for problem solving, sense making, or performance (p. 44). When I talk about teacher learning in this paper, I am typically describing shifts in teachers use of artifacts that are directed at making sense of and solving the problems of equity-oriented, nondominant teaching practice. But it is worth noting that learning (e.g., to use traditional teaching methods such as lecturing) may lead teachers away from rather than toward more equitable teaching practice, though such learning is not the focus of this paper.

I conceptualize identity development in terms of two kinds of shifts surrounding participation in a particular practice: shifts in how important a persons participation in the practice is to who she is and wants to become, and shifts in how the person perceives her own competence or value as a participant. As she participates in nondominant teaching practice, for example, this practice might gradually become more central to a teachers sense of herself as an educator, even though she continues to view herself as an unskilled novice with respect to it. I draw here on Nasir and Hands (2008) concept of practice-linked identities, emphasizing aspects of identity that are linked to participation in particular social and cultural practices & and are fundamentally related to engagement (p. 147), extending this idea from their work with youth to consider the professional identities of practicing teachers. Because social participation in practice is at the heart of this approach to identity, I construe teachers sense of their own membership in communities of practice (both as they describe it and as they, with others, enact it) as key. This approach diverges from research that foregrounds aspects of teacher identity that, while related to practice, are more narrative in charactere.g., the stories White teachers tell themselves about their Whiteness (e.g., McIntyre, 1997) or the stories mathematics teachers tell themselves about their relationships with their discipline (e.g., Hodgen & Askew, 2007).

SOCIAL NETWORK THEORY

This paper draws on Wengers (1998) and Nasirs (2012; Nasir & Cooks, 2009) work to investigate the ways that teachers learning and identity processes relate to and potentially support one another, through teachers everyday interactions with the resources their communities of practice make available. I also make use of an insight from social network theory: that weak ties (e.g., acquaintances) provide people with access to resources that are both critical and not provided by strong ties (e.g., close friends or members of ones community of practice). Thus, in his seminal article on the strength of weak ties, Granovetter (1973) reported that recent job changers were more likely to have received information about their new jobs from contacts with whom they interacted occasionally or rarely than from people they saw often. Building on this perspective to study teacher development, L. Anderson (2010) found that one teachers network included not only teachers and staff members at her school but also teachers at other schools, researchers, and local business owners. To think of local business owners as engaged in the same enterprise as classroom teachers risks stretching Wengers (1998) concept of mutual engagement beyond all usefulness. But as Anderson describes, the business owners provided crucial support for the focal teachers professional development and for her sense of herself as an effective educator.

In recent years, interest in and use of social network analysis has grown tremendously (Daly, 2010). Two strengths of social network analysis are its ability to illuminate the structure of peoples networks in ways that are not confined to strong ties, and its ability to represent this structure in increasingly sophisticated ways. (A number of software programs have been designed to aid in the generation of representations; see, e.g., Daly, Moolenaar, Bolivar, & Burke, 2010). However, the mechanisms through which networks support changei.e., what kinds of resources flow across ties, and howhave received somewhat superficial attention.

This paper therefore takes up Wengers (1998) notion of learning and identity as intertwined through the negotiation of meaning and shows how it applies to networks that extend beyond the boundaries of communities of practice as they have traditionally been conceived. I continue to use the term community of practice, following Wengers definition, to refer to groups (such as subject-matter departments) that overlap but are not synonymous with the teachers professional support networks.

PRIOR RESEARCH

Much of the literature on teacher learning focuses on the privileged moments created by pre-service teacher education programs or professional development workshops. While these moments may be formative, research has also begun to attend more closely to the day-to-day learning that practicing teachers accomplish in their communities of practice. A major contribution of this research has been the identification of community and network characteristics that support teacher learning. With respect to reform-oriented learning in particular, scholars have used social network analysis to show that networks with high-depth interaction, strong ties, and high expertise support teachers in both adopting and sustaining ambitious instructional reforms (Coburn, Russell, Kaufman, & Stein, 2012; Daly et al., 2010; Penuel, Riel, Krause, & Frank, 2009). Research on professional learning communities (PLCs) has linked other kinds of characteristicse.g., shared values, collaboration around student learning, accountability structures, openness, and trustto support for teacher learning (Bolam, McMahon, Stoll, Thomas, & Wallace, 2005; DuFour, DuFour, & Eaker, 2008; Louis, Kruse, & Marks, 1996; McLaughlin & Talbert, 2001).

Fine-grained analyses of interactions between teachers illuminate learning processes in professional learning communities. For example, Horn and her colleagues (Horn, 2005; Horn & Kane, 2015; Horn & Little, 2010) have conducted detailed examinations of episodes of teachers pedagogical reasoning, finding that more opportunities for teacher learning than are typical are present in collegial conversations in which teachers focus their attention on problems of practice, coordinate general ideas about teaching with specific instances of practice, and use multivocal replays of classroom activity to render practice transparent. In a similar vein, Dobie and Andersons (2015) research indicates that open discussion, in which teachers express contrasting ideas with no dispreference for disagreement, likely & afford[s] greater benefits for teacher communities than discussions in which disagreement is avoided (p. 238). These studies draw on detailed observation to clarify what productive collaboration, openness, and trust can look like.

This paper extends prior research by exploring the mechanisms through which teachers participation in communities and networks supports their learning, taking resources for learning and identity as its unit of analysis. This approach bridges work at the level of moment-to-moment interaction (e.g., Horn, 2005; Dobie & Anderson, 2015) and work at the level of policy and organizations (e.g., Bolam et al., 2005; McLaughlin & Talbert, 2001). A resource-centered approach is more general than studies of interaction, allowing deeper theorizing about why some strategies for professional collaboration support learning better than others. At the same time, a focus on resources lends specificity to the structural features identified by studies taking an organizational approach, moving forward from these studies to show how structures that support learning function.

This paper also contributes to understandings of teacher learning by looking beyond the traditional boundaries of school-based networks and professional learning communities to locate critical but frequently overlooked sources of support. This strategy makes visible teachers agency in organizing supportive networks, illustrating how teachers and their social contextsboth within and beyond their school wallscoconstruct equity-oriented learning and growth.

Finally, the paper attends more closely to issues of professional identity than is typical in the literature on teachers professional communities. This illuminates the kinds of resources that are often missed in studies of teacher learning, which often focus on conceptual resources such as material artifacts and access to technical expertise (e.g., Coburn, Choi, & Mata, 2010; Lampert, Boerst, & Graziani, 2011). The paper also makes use of a practice-focused lens on identity and learning (Nasir & Hand, 2008). Through this lens, identity is not just a cause or an effect of learning (as it is commonly treated in research on mathematics teacher identity; see, e.g., Foote, Smith, & Gillert, 2011; Hodgen & Askew, 2007) but a process that both shapes and is shaped by learning in dynamic ways through everyday interaction.

METHODS

RESEARCH SITES AND PARTICIPANTS

This study was initially organized around the question: How do teachers professional communities support them to redefine mathematical competence? The literature on professional learning communities (e.g., Gutiérrez, 1999; Horn, 2005; Horn & Little, 2010; Little, 2002; McLaughlin & Talbert, 2001) led me to expect that teachers school-based communities, especially their departments and course teams (e.g., the Algebra team and the Geometry team), would be critical sources of support. I therefore sought to recruit mathematics departments that were collectively committed to redefining mathematical competence. I asked university-based researchers, teacher educators, and school district personnel to nominate schools with mathematics departments that fit this description.

After conducting preliminary interviews and observations at three diverse urban schools, I settled on two: Union High and Boxer High. All of the mathematics teachers at both schools expressed a commitment to supporting all students, especially students who had not been academically successful in the past (in their conversations with each other and in research interviews). To that end, all of the teachers participated in an equity-oriented professional development (PD) program offered by their district. (The two schools were part of the same district.) The PD centered on Complex Instruction (CI; see E. G. Cohen & Lotan, 1997; Nasir, Cabana, Shreve, Woodbury, & Louie, 2014). CI posits that all students have important intellectual contributions to make to their classroom learning communities, and that teachers are responsible for drawing out every students strengths while also helping every student develop new strengths. The PD in which the teachers were participants targeted these beliefs as well as instructional strategies for equalizing statusi.e., for leveling hierarchies of perceived ability and worthsuch as the use of complex, open-ended tasks that require students to pool their various skills and work together. The PD also emphasized the development of robust, department-based professional communities, and teachers at Union and Boxer had dedicated time each week to collaborate around mathematics instruction. In addition, they participated in periodic CI trainings and coaching sessions. This array of resources made Union and Boxer rich contexts in which to examine supports for teacher learning around nondominant practice.

From the 13 mathematics teachers at Union and five at Boxer, I recruited six focal teachers: four from Union (Ryan Sower, William Barrett, Cyril Nazemi, and Luke Ziebler) and two from Boxer (Amanda Pepper and Rob Daly). Teachers were selected based on factors that appeared to be connected to the depth of their engagement with CI in my early observations. Specifically, I selected for range along two dimensions: years of experience teaching, and leadership roles in the district CI community.

DATA COLLECTION

I observed routine meetings of teachers department and course teams (30 hours of meetings at Union and 11 hours at Boxer) and Complex Instruction professional development sessions (70 hours) throughout the 20122013 academic year. I also conducted 48 classroom observations for each focal teacher. Audio recordings, field notes, and photographs of whiteboard inscriptions, worksheets, and other artifacts were produced for each observation. Informal interviews often accompanied observations. In addition, focal teachers were formally interviewed at the end of the school year following a semistructured protocol that focused on teachers goals and the challenges and supports that they experienced in relation to their goals. Formal interviews were audio recorded. I transcribed all audio recordings.

ANALYSIS TECHNIQUES

Locating Sources of Support

Recall the first research question guiding this study: Where do teachers encounter resources that support their engagement with nondominant teaching practice? To address this question, I analyzed interview transcripts with a focus on the ways that teachers themselves reported feeling supported by their colleagues and others. Responses to three interview questions were especially relevant:

"

Are there any experiences or people who you would point to as being especially important in your development as a math teacher? (How?)

"

Are there ways that your colleagueshere at [school name] or elsewheresupport your development as a teacher?

"

Are there ways that your colleagueshere at [school name] or elsewheresupport you to manage the challenges and dilemmas youve described?

Interview coding was not limited to these three questions, however; all talk about people and experiences that shaped or supported teachers thinking about their work was considered.

Teacher reports were used to generate egocentric network diagrams (L. Anderson, 2010; Borgatti & Ofem, 2010). These diagrams were a first step in visualizing each teachers network and the supports it provided. The focal teacher was placed at the center of the diagram and connected to each of his or her supporters with a line segment. The length of each line segment was used to show the frequency of teaching-related interaction between parties (based on teachers reports and on my observations). The shortest segments were used to represent daily interaction; longer segments were used to represent 1 or 2 times per week, 1 or 2 times per month, less than once a month, and never. In addition, supporters were clustered together based on setting (e.g., colleagues at the focal teachers school were placed near one another). I did not use social network analysis tools such as standardized survey instruments and network analysis software; rather, the network analysis that I conducted developed organically from interviews and observations.

Identifying Forms of Support

To address the second question in the studyWhat kinds of resources support teachers engagement with nondominant teaching practice?I used open coding (Emerson, Fretz, & Shaw, 1995) to analyze interview transcripts, again with a focus on teachers descriptions of how they had been supported by their colleagues and others. Emergent themes were coordinated with the literature on learning, identity, and meaning-making, in particular, with Wengers (1998) and Nasir and Cookss (2009) frameworks. This coordination of research questions, data, and the literature led to the creation of four categories of support for teachers equity-oriented learning and identity development. These four categories abstracted the phenomena that open coding had initially revealed, moving toward theorizing the relationships between each category and processes of learning and identity development (e.g., emotional support became relational resources, and support for nuts and bolts became technical resources). The four categories were then used to systematically code interview transcripts. I also examined field notes and transcripts from my observations to more fully understand the functions that various supports played, seeking both different perspectives on the supports that teachers themselves described and instances of the four categories that teachers did not mention. I paid particular attention to positional resources, reasoning that whereas all of the teachers alluded to the other types of resources in their interviews, the novelty of positioning as a concept may have made it difficult for them to notice or name this as a resource.

Network diagrams were then coded to show which resources flowed through each relationship, with a different color for each type of resource (represented in Figure 1 through the use of dotted and dashed lines).

Characterizing Teachers Patterns of Engagement

As a prelude to linking different constellations of resources to teachers patterns of engagement with nondominant teaching practice, I characterized the latter across two settings: their classrooms and their professional communities.

To characterize classroom engagement with nondominant teaching practice, I used both qualitative and quantitative analyses. In the hours I had spent in teachers classrooms, differences in instruction were almost tangible. I selected episodes from each teachers classroom for close analysis as a means of both clarifying what those differences were and searching for disconfirming evidence (i.e., of similarities). Episodes were selected to capture teachers practices around 1) adapting and assigning tasks and 2) responding to students struggles. The analysis examined how instruction framed mathematics as a discipline and students as learners of mathematics (as exemplified in Louie, 2015). I also sought to quantify differences in classroom practice. Taking the extent to which teachers instruction was student-centered (versus teacher-centered) as a measure of their engagement with Complex Instruction and related nondominant ideas, I coded time-stamped segments of classroom activity as teacher-led, including teacher presentations or class discussions in which student contributions were limited to brief responses to close-ended questions; student-led, including student presentations or work time in which students were presented with opportunities for problem solving; other work time, including work time in which students were presented with exercises identical to ones the teacher had demonstrated how to solve earlier in the lesson; and nonmathematical, including time spent on announcements and transitions between activities. I then calculated the percentage of class time each teacher spent in teacher-led versus student-led mode. A higher percentage of the latter and a lower percentage of the former were taken to indicate deeper engagement on the teachers part with Complex Instruction. For each teacher, I also calculated the median number of words in each student talk turn during whole-class discussions, associating a higher median with deeper engagement (arithmetic averages were also calculated, but median was selected as a more accurate measure of central tendency; averages were prone to distortion based on one or two unusually long talk turns).

To characterize teachers engagement in communities of nondominant teaching practice, I examined the frequency with which teachers attended meetings of their districts Complex Instruction network (which I attended) as well as teachers own reports of interaction with and connection to other educators who were linked to the regional Complex Instruction network (e.g., colleagues in other school districts who had attended CI-focused teacher preparation programs in the area).

Linking Patterns of Engagement to Network Resources

To address the third research questionHow do different kinds of resources come together to support teachers patterns of engagement with communities of nondominant teaching practice?I sought to understand the interplay between different kinds of resources in support of teachers learning and identity development. The network diagrams made a number of patterns visiblein particular, connections between the density of teachers networks and the depth of their engagement with nondominant practice, and differences in the distribution of each type of resource in different teachers networks. These patterns were then explored in narratives that I wrote to describe each teachers engagement with nondominant teaching practice and the resources that supported that engagement.

Member Checking

This paper focuses on four of the six focal teachers in the study: Ryan Sower, Luke Ziebler, and William Barrett at Union, and Amanda Pepper at Boxer. Data collected from all six teachers were analyzed to generate findings. The four cases presented here have been selected to showcase differences in teachers networks and trajectories of engagement with nondominant practice and radical change.

Completed narratives were shared with three of the four teachers featured in this paper (Luke Ziebler had since left the profession, and I was unable to reach him). Comments provided by the teachers have been incorporated.

LEARNING AND IDENTITY RESOURCES IN TEACHERS PROFESSIONAL NETWORKS

For the six focal teachers in this study, finding ways to reach students who had previously been unsuccessful in school was an explicit goal. For them, this entailed giving students access to mathematics content. But it also meant shaping students ideas about smartness and success so that all students could take ownership of their own learning and see themselves as competent. Three of the six focal teachers (Ryan, Luke, and Cyril) said that one of their goals was to make their students feel smart, using that exact phrase. Others said that they were trying to redefine what success is (William), to rethink & whats it mean to be smart at math (Rob), and to change [students] status of what they [think] about themselves as mathematics learners (Amanda). Yet their engagement with nondominant teaching practiceas evidenced by both their classroom instruction and their work with their colleaguesvaried widely.

A comparison of teachers professional support networks shows that their patterns of engagement are related to the distribution of four types of resources in their networks: orienting resources, which support teachers in envisioning the kind of practice they want to achieve; technical resources, which support teachers with the nuts and bolts of enactment; relational resources, which support teachers sense of belonging and identification with their practice; and positional resources, which support teachers sense of worth and competence as professionals (cf. Nasir & Cooks, 2009). This paper proceeds with an elaboration of each of these definitions, followed by a discussion of the contrasting cases of four teachers. These cases are used to illustrate how orienting, technical, relational, and positional resources came together to produce four distinct patterns of engagement with the work of redefining mathematical competence: two patterns of deep engagement, supported by different constellations of resources; a pattern of stable but peripheral engagement; and a pattern of disengagement with teaching practice. The cases also show the importance of resources that lie outside teachers local, school-based communities of practice.

ORIENTING RESOURCES

Orienting resources provide teachers with a vision of good teaching, presenting images of practice that inform teachers ideas of what their classrooms should look and sound like, what their roles should be, and whether or not they are successful. That is, they support teachers in orienting to their work in particular ways. Some orienting resources function by explicitly focusing teachers attention on fundamental principles and big ideas about teaching. For example, CI training attempted to orient teachers to the ways that status problems (perceived rather than actual differences in competence and worth) affect students participation and learning. Other orienting resources structure teachers thinking more tacitly, providing models to emulate and lenses through which to interpret their work. The apprenticeship of observation (Lortie, 1975), i.e., teachers own experiences as elementary and secondary school students, was thus an important source of orienting resources for many of the teachers in this study.

Orienting resources shape teachers learning in important ways, defining what it is that they should know and be able to do and providing lenses through which to understand the materials and strategies that are available for enacting their practice. Orienting resources can also be resources for learning insofar as they provide teachers with concrete examples to draw from and work toward (though they are not always specific enough to fill this function). And orienting resources shape teachers professional identities, setting standards against which teachers can measure themselves (discussed in more detail below) and supporting a sense of belonging to both real and imagined communities of educators with a shared, nondominant vision.

TECHNICAL RESOURCES

Teachers environments are full of technical resourcesthe materials and strategies that support them in enacting their practice on a daily basis. Participating in the exchange of materials (e.g., worksheets) and strategies (e.g., methods for managing small group interactions) is a way of garnering immediately usable information and learning tricks of the trade. It is also a way for teachers to identify themselves as part of a community of practice, as a contributor or as an accepting recipient. In other words, teachers develop new ways to participate in their practicei.e., learnas they interact with these resources. They also develop identities as professionals who are interested in or competent at (or both, or neither) using particular materials and strategies.

RELATIONAL RESOURCES

Relational resources are those connections with others that support a sense of belonging to and identification with a practice (Nasir & Cooks, 2009). Connections to colleagues, students, and others make teachers work livable and enjoyable, as teachers not only give but also receive the caring that some have characterized as central to their profession (Noddings, 2003). For teachers working to teach against the grain (Cochran-Smith, 1991), relationships with others with a shared mission can also bring a sense of solidarity that supports their experience of connection to their work. In addition, by connecting teachers to others, relational resources provide channels through which orienting and technical resources can flow. There is no guarantee that these resources will support teachers in engaging in change-oriented learning and identity development, however; some relational resources may feel supportive to teachers while supporting them in reproducing traditional practice.

POSITIONAL RESOURCES

Related to the caring embedded in teachers relational resources are positional resources that support teachers sense of their own worth and competence as professionals. Being included in a professional community and being recognized as a competent member of it do not always go together. In addition, teachers are part of multiple professional communities, with norms and standards of competence that are sometimes at odds. Consequently, they are sometimes positioned as highly skilled in one community of practice but not in another. For example, the veteran teachers in this study were positioned as experts at their school sites and as learners with respect to the districts Complex Instruction community.

Positional resources can both support and undermine teachers learning and identity development. Being positioned as an expert, for example, may suggest to a teacher that her own learning is unnecessary. However, this positioning can support her in taking up an identity as a competent professional, thereby supporting her continued engagement with her practice. On the other hand, being positioned as a novice can nudge teachers to pursue opportunities to learn while fostering feelings of incompetence that promote their disengagement with their practice. Or novice identities may be framed in terms of promise and potential in ways that support ongoing engagement. The cases that follow illustrate these various ways that positional resources are offered to and taken up by teachers.

RESOURCES FOR LEARNING, RESOURCES FOR IDENTITY

Orienting, technical, relational, and positional resources all support both learning and identity, but in more and less direct ways. Orienting and technical resources work in mutually reinforcing and interdependent ways to directly support teachers in learning and developing their practice by providing ideas (orienting resources) and tools for enacting those ideas (technical resources). Technical resources support orienting resources by making the technologies that operate within any expansive vision transparent and concrete; without them, orienting resources may prove too abstract to support shifts in teachers practice. At the same time, technical resources are rendered sensible by their connections to the big picture that orienting resources can provide.

Relational and positional resources, in comparison, support teachers learning by fostering identities that motivate and encourage learning. Learningespecially nondominant learningis hard work. Most of the teachers in this study experienced setbacks and doubts about whether they should continue to pursue nondominant learning over the course of the year. Teachers capacity to understand themselves as part of something bigger than themselves and as capable of continued learning, growth, and success supported them in persisting through challenges, advocating for additional resources, and finding support in their professional networks.

Of course, to delineate resources for learning and resources for identity so cleanly is an oversimplification. In practice, learning and identity development imply and depend on each other. Thus, learning to use a new instructional strategy (say, a multiple-ability orientation) is mediated by a teachers ideas about who she is as a professional and who she wants to become. At the same time, the teachers ideas about her identity may shift as she engages with the strategy and the field of possibilities that it brings into view. What matters is not drawing boundaries between learning and identity but understanding the interplay between them as teachers engage with their practice.

PATTERNS OF ENGAGEMENT WITH NONDOMINANT PRACTICE

Constellations of orienting, technical, relational, and positional resources work together to support or undermine teachers engagement with nondominant teaching practice inside and outside of their classroomsin other words, to produce patterns of engagement. I use the term pattern of engagement instead of trajectory because my data capture not so much change over time (which I did not see over the year of the study) as teachers closeness to equity-oriented practice and their apparent direction (whether inbound, outbound, or stable) at a particular moment or set of moments. This perspective is especially important in light of questions about sustainability; a teacher who appears to be on an inbound trajectory of increasing engagement with her practice one year might burn out the next. Thus, the concept of a pattern of engagement is not meant to predict where someone is headed; rather, its utility lies in its ability to help us understand the kinds of supports people have and might need in order to move in particular ways.

Two of the teachers in this study, Ryan Sower and Amanda Pepper, maintained active engagement with nondominant teaching practice throughout the period of the study. Their classroom instruction consistently framed mathematical competence in ways that expanded students opportunities to develop identities as powerful learners and doers of mathematics (see Louie, 2015). They were also core members of communities of practice that were organized around nondominant teaching practice, including recognizable ones like the districts Complex Instruction community as well as more figurative ones (cf. Holland, Lachicotte, Skinner, & Cain, 1998) that were joined (at least in their imaginations) by shared values around teaching and learning. The networks that Ryan and Amanda organized to support their work were both densely populated and rich in a variety of types of resourcessourced from a variety of places (see Figure 1[a] and [b]). But the particular arrays of orienting, technical, relational, and positional resources that each of them took up were different in important ways, showing that there are multiple ways to support teachers ongoing engagement with reform.

In comparison to Ryan and Amanda, William Barrett and Luke Ziebler had few resourcesespecially for engaging with nondominant teaching practiceat the time of the study (see Figure 1[c] and [d]). It was rare for either of them to attend district-sponsored CI gatherings or to take up other opportunities for learning and collaboration outside the school day. Williams resources were nonetheless sufficient to sustain his student-centered teaching style and his sense of connection to his workbut they were inadequate to support him in actively engaging with reframing mathematical competence. This work remained peripheral to his classroom instruction, and he himself remained at the periphery of the communities of nondominant teaching practice in which Ryan and Amanda were central. For his part, Luke did not have the resources to support continued engagement with either nondominant teaching practices or mathematics teaching more generally, and the year after the study concluded, he left the profession.

The character of teachers classroom instruction was part and parcel of their participation in communities of nondominant teaching practice. The resources that were available to and taken up by the four teachers in the study are thus reflected by their work with students. Table 1 gives a flavor of the differences between Ryans, Amandas, Williams, and Lukes instruction. (For detailed analysis of classroom episodes, see Louie, 2015.) The cases of these four teachers demonstrate the ways that learning and identity processes intertwine as orienting, technical, relational, and positional resources are offered to and taken up by teachers. In particular, they show how these four types of resources may support teachers in engaging in different patterns of participation in the work of reframing mathematical competence.

Table 1. Indicators of Teachers Classroom Engagement with Nondominant Teaching Practice

	More engaged		Less engaged
	Ryan	Amanda	William	Luke
% of class time spent on teacher-led presentations	10%	12%	22%	27%
% of class time spent on student-led worka	51%	40%	12%	14%
Median # of words per student talk turnb	5	6	1	2

Note. ^a Student-led work includes student presentations as well as group problem solving. Time spent having students work routine exercises (either in groups or individually) is not represented in this table. ^b Only data from whole-class discussions were used for this analysis.

Figure 1. Egocentric social network diagrams depicting the professional support networks of (a) Ryan Sower, (b) Amanda Pepper, (c) Luke Ziebler, and (d) William Barrett.

Note. Shorter lines indicate more frequent interaction. Darker shading represents overlapping membership in multiple communities (from the focal teachers perspective). A color version is available online at http://bit.ly/1LVYpT7

RYAN SOWER: A PATTERN OF ONGOING ENGAGEMENT

Throughout the year of the study, Ryan Sower was an active participant in a variety of communities that supported his engagement with nondominant teaching practice. These communities included the mathematics department at his school, his districts CI network, and a group of friends from the teacher education program he attended. His classroom practice reflected this engagement; he often used strategies associated with Complex Instruction, and the idea that mathematics classrooms should be intellectual communities in which all students are supported in contributing and learning permeated his instruction. Ryan himself was supported by a wide variety of resources, including strong orienting resources for envisioning CI in action, technical resources for enacting his vision, relational resources that sustained his sense of connection with relevant communities of practice, and positional resources that affirmed his identity as a valued professional. His case illustrates how relatively abundant resources can come together to foster mutually supportive learning and identity processes, producing a pattern of ongoing engagement with nondominant teaching practice and with reframing competence specifically.

Orienting Resources

Ryan drew on his memories of Tom and Ina, two of his first teachers in elementary school, to orient toward a vision of a teacher as someone who made every student feel loved, accepted, and treated as though he or she had something to contribute. But like the other teachers in this study, Ryan characterized the mathematics instruction he received as a student as traditional, emphasizing discrete skills and mathematical rules with little place for making sense of ideas and connections, especially in high school. Before he began his training as a teacher, this instruction was his primary resource for imagining his own practice; as he said, I was envisioning myself teaching much the same way that Id been taught. But through his teacher education program and the networks that the program opened up for him, Ryan gained access to orienting resources that supported him in learning new ways of conceptualizing the work of teaching secondary mathematics.

The teacher education program that Ryan attended was grounded in Complex Instruction, and it drew Ryans attention to issues of power and status in mathematics classrooms. His math methods instructor, Ruth, introduced him to a pedagogy that was designed to address these issues, linking somewhat abstract ideas he had had about accepting and valuing students to a concrete vision of mathematics teaching and learning. Ruth took her students seriously, really honoring all the different things that [they brought] to the table. She assigned his cohort complex mathematics problems and centered her instruction on students ideas instead of her own lectures, framing mathematics not as a fixed body of knowledge but as a rich terrain of big, interconnected ideas to be explored. Reflecting on his own mathematics learning through the lens of Ruths example, Ryan found that he had been poorly served by instruction that focused on following procedures without helping students to understand mathematics (even though he had earned good grades and been seen as a strong mathematics student in school). Thus, Ruths instruction oriented him firmly away from traditional teaching practice and toward an alternative in which redefining mathematics and mathematical competence were central.

Ryans learning around reframing mathematics and honoring all students continued throughout his year-long placement as a student teacher at Railside High School (the pseudonym given by Boaler & Staples, 2008; see also Nasir et al., 2014). At Railside, he and a fellow intern, Jasmine, were apprenticed to Guillermo Reyes, a nationally known leader in Complex Instruction (see, e.g., Horn, 2007; Horn & Little, 2010). Guillermos class provided a compelling image for Ryan: I just knew right away that I wanted to create that kind of classroom. I wanted to create that kind of space for kids to explore math, and feel smart, and be excited about working with their peers and their teacher.

At the time of the study, Ryan was in his second year at Union, which was also his second year as a full-time teacher. He appreciated many things about his colleagues, and they supported him in ways that will be described below, but he encountered limitations in their capacity to sharelet alone enrichhis perspective on what ideal mathematics instruction should look and sound like. In his view, this was not because they didnt share his fundamental values; on the contrary, it was his feeling that everybody [at Union] wants to get better at [CI] and were all really committed to it. But, he said, I just dont think weve had enough of a vision for what a CI department is. In this context, the images of good teaching that he drew from Ruth, Guillermo, Tom, and Ina continued to be important orienting resources for Ryan, supporting him in maintaining both his efforts to reframe mathematical competence and his identity as an educator for whom such reframing was central.

Technical Resources

Ryans work with Ruth and especially Guillermo also supported his practice by providing him with technical resources for its enactment. Through his year of student teaching, he learned to use curricular materials and instructional strategies in ways that became integral to his practice. His colleagues at Union also had access to these materials and strategies, but they were less able to make sense of them. For example, they had all experienced participation quizzes in the districts summer CI course, working together on math problems as the instructors circulated, listening and taking public notes on the helpful, productive, and smart things that they did (see Staples, 2008 for a detailed description). For many teachers, this was a powerful example of a strategy for developing and enforcing particular norms for groupwork. But it was only one example. In contrast, as an intern in Guillermos classroom, Ryan witnessed dozens of participation quizzes and led many himself, with feedback and support from Guillermo and Jasmine. In my year of observations at Union, Ryan was the only teacher I saw lead a participation quiz.

The broader meaning of teaching for Ryandeveloped through his array of orienting resourcesclearly informed his use of his technical resources. For example, empowering students to feel smart by honoring their contributions to their own and their peers intellectual growth played a central role in the participation quiz that I saw him lead. As he enacted the debrief portion of the quiz, for instance, he reported to the class that in Group 1,

there were some really good questions, a lot of people asking like, Is this line the base or the height, [and] how do you know? The part thats not on the bottom is the height, or is the base. They had a little bit of an argument but then they figured it out. And they did a really great job of making sure the new student got caught up right away, which was excellent. So like I would definitely give them an A+ right now if I gave them a grade.

As he continued to debrief, Ryan similarly highlighted specific actions and utterances that positioned each group as mathematically (and socially) competent, orienting toward the goal of reframing competence and disrupting hierarchies. At the same time, his use of technical resources like participation quizzes and other CI strategies provided Ryan with additional orienting resources. Employing these strategies gave him opportunities to look for and see the ways that his students were capable, smart contributors to each others learning, reinforcing his vision of good teaching as teaching that reframes mathematical competence. Strategies that were common in other teachers practice did not always make such opportunities available (for example, some teachers had students work in groups but jumped from one group to another offering assistance).

But technical resources were few and far between for Ryan by his second year at Union. In his first year, colleagues in the department shared their lesson plans, responded quickly to his questions and concerns about his students, and welcomed him into their classrooms to observe. William, who was his officially his coach, was a huge support, collaborating with him to plan lessons and even coteaching with him occasionally. At the end of that year, however, several experienced teachers left the school (including Guillermo, who was there for Ryans first year), and Ryan was slotted in to leadership roles. He spearheaded the math departments effort to rewrite the Advanced Algebra curriculum (in an effort to bring it into closer alignment with the state standardized test, under intense pressure to raise test scores) and took on more responsibilities on his grade-level team as well, knowing that if he didnt, his colleagues would be in untenable positions. For another teacher, or for Ryan at another time, these roles might have fostered learning and growth. But in this instance, he felt that they stymied it:

I was really hoping that my second year, Id be able to just focus on one or two aspects of my teaching, and really kind of dig into those. And I dont feel like Ive had the time or space to do that.

Thus, with more technical resources, Ryans engagement with equity-oriented practice might have deepened instead of staying constant. He might have learned to use Complex Instruction more effectively, for example, and found more space to nurture his identity as a competent member of a community of learners (versus as an expert practitioner, an identity that made him uncomfortable, as discussed below). The fact that he was able to maintain his engagement with the work of reframing mathematical competence despite limited technical resources suggests the power of other types of resources for processes of teacher learning and identification.

Relational Resources

Abundant relational resources supported Ryan in developing and maintaining an identity as a member of a community of nondominant teaching practice. He hadand took advantage ofmembership in many different professional communities. The math department at his school was one of the most important, not least because the department constituted his immediate working environment day in and day out. His colleagues in the department valued CI, as he did, and their solidarity on this point was an important relational resource that fostered Ryans sense of belonging in the department and enriched his connection to nondominant aspects of his practice. The district CI community, which met once or twice a month (with Ryan almost always in attendance), provided similar relational resources. Getting to know other teachers throughout the district who are also thinking really hard about this work supported Ryans sense that he was part of something worthwhile, something bigger than himself.

His teacher preparation program also gave Ryan an enduring professional community that affirmed his identity as an equity-oriented teacher. During the period of the study (his second year teaching), he continued to consult with his mentors and meet with his peers from the program a few times a year. Though it was hard for him to articulate exactly how this community helped him, its value can be understood in terms of the relational resources and sense of connection to a particular brand of practice that it provided. As Ryan said, I dont know that I get anything concrete out of it, you know. I dont get new ideas about teaching or anything. Its just like, a support network, where we talk about our work, and we just laugh and drink and do whatever. So thats been a big resource.

In addition to giving him a sense of connection to his teaching practice that fostered his ongoing engagement, Ryans relational resources linked him to other types of resources. Some of his relational resources doubled as orienting resources by reifying certain ideals; for example, seeing friends from his teacher preparation program was a reminder of shared experiences and ideas learned from Ruth and Guillermo, whether those two were present or not. And some relational resources doubled as positional resources, as described in the next section.

Positional Resources

An important way that his multiple communities of practice supported Ryans engagement with nondominant teaching practice was by supporting him in taking up an identity as a skilled and valued member. At Union, he felt as soon as he was hired that there were a lot of people kind of looking out for me & [who] believed in my potential, perhaps partly because of his connections to Guillermo and Railside. And throughout his first two years, several of his colleagues were a big support in terms of just, kind of assigning me competence. In one meeting, for example, a department cochair reported (with light-hearted self-deprecation) that she had finished an observation in Ryans class depressed cuz hes so good. Its a great class, isnt it? Hes great. Though there were drawbacks to this kind of positioning (discussed below), it supported Ryans sense of connection to his practice.

Members of the district CI network also positioned Ryan as a high-status insider. For example, the CI coordinator chose to feature footage from Ryans classroom in the district-wide Video Club, which was structured to support strengths-based collective inquiry. The Video Club afforded technical resources for all of the teachers involved, as they worked to name strengths of Ryans and his students that they all could build upon in their own classrooms. It also afforded additional positional resources for Ryan, reinforcing his reputation as a great CI teacher. In this and other ways, Ryans positional resources created a snowball effect, amplifying his opportunities to learn and develop an identity as a worthy member of the CI community.

Yet Ryan himself was keenly aware that he still needed more support to grow as a teacher, and while he appreciated his colleagues recognition, he objected to being positioned as better than anyone else. He saw himself as a learner, first and foremost. In an interview, he said:

I sometimes feel like people are, some people are looking at me as someone who knows how to do CI, and Im like, yeah I dont know shit about how to make groupworthy tasks! Like I, all the groupworthy tasks I do come from Guillermo. I dont know how to do this stuff either.

Being positioned as an expert may not have been the greatest barrier to Ryans learning; the paucity of expertise in nondominant teaching (in terms of both orienting and technical resources) amongst his colleagues at Unionthe members of his support network that he had the most frequent contact withwas certainly quite significant as well. But it is worth keeping in mind that the positional resource of high status was potentially a double-edged sword for Ryan. It could clear a path to learning opportunities, but only in communities of practice in which high status was associated with continued learning.

Summary

Ryans case illustrates that when many types of resources are available, these resources can work in concert, not only reinforcing but also amplifying each other in ways that support teachers in developing practices and identities in which reframing mathematical competence plays a central role. It also demonstrates that important resources for learning may come from outside teachers school-based communities of practice, for teachers in strong school-based communities as much as for those who are not. At the same time, Ryans case shows that even a teacher who is embedded in multiple reform-oriented communities, who has strong connections to nationally recognized teachers, and who devotes an extraordinary amount of time to collaboration and professional development may still need additional support in order to do more than simply maintain his engagement with his practiceto continue to learn and grow. (And even simple maintenance might require more resources over time.)

AMANDA PEPPER: DEEP ENGAGEMENT SUPPORTED BY STRONG IDENTITY RESOURCES

Like Ryan Sower, Amanda Pepper had a rich array of resources sourced from a large and diverse support network. Asked to name people who had influenced her development as a teacher, she said, So its been a village. Theres been a lot of people thatve helped me. Because her math department was both smaller and less unified than Ryans, ties to people outside her school were especially important for Amanda. In addition, the forms that her resources took were somewhat different from Ryans, illustrating that there are multiple constellations of resources that can support the kind of teaching that both she and Ryan enacted. In particular, Amanda had fewer orienting resources, but she coordinated these resources with strong relational and positional resources to support her practice and her identity as a CI teacher.

Orienting Resources

Amanda got into teaching through Teach for America (TfA). As a TfA corps member, she did not receive much training. Her first experiences of deep engagement with equity-oriented approaches to mathematics instruction came from two workshops that she attended the summer after her first year. One was a training in College Preparatory Mathematics (CPM; Sallee, Kysh, Kasimatis, & Hoey, 2000), a standards-based reform curriculum that her school had adopted. The other was the Complex Instruction summer course. The workshops opened [her] mind to a different way of teaching, and she learned to reorient her practice to focus on student thinking, strengths, and successes. Following the workshops, Amanda actively sought out additional orienting resources. For example, she arranged to visit her CPM instructors classroom to see her teach; she was a regular attendee at district CI events; and she read books like Carol Dwecks (2006) Mindset. Although the orienting resources that these experiences provided were shallow compared to Ryans year of student teaching, Amanda stretched them by consciously working to imagine what her mentors and role models would say and do. She described asking herself in a moment of frustration, What would Lee [a CI coach] say in this situation. What would Jessica [another CI coach] say in this situation. How can we find some smart things that are happening? The images of teaching that she developed through CI and CPM training were thus constant resources for her practicein particular, for her efforts reframe mathematical competence by seeing and naming smart things.

Her day-to-day work environment seemed oriented against Amandas efforts to reframe mathematical competence, however. At Boxer, she found that many of her colleagues believe [in] one way of smartness, and its an easy thing to believe because its very easy to measure. She described her own doubts regarding the approach to teaching that she was pursuing, especially when it came to reframing competence:

Im constantly up against this traditional view of what smart looks like. And I think I still have it in my head. & I question what Im doing every day. Which is good, but also can drive me crazy, and make me have little confidence.

In this context, Amandas relational and positional resources were especially important.

Relational Resources

Challenges to her approach from members of her own department and others highlighted the nondominant nature of her practice for Amanda. Describing two of her colleagues, she said:

They were very questioning of CI. They still are. And so, sometimes Im like, is it because Im not critically thinking about it that I like CI, and Im just going and running with it? Or, like do I need to be thinking more critically about it? And I think that also gets back to my smartness. Like am I really, are [my students] really smart? Or am I just trying to compensate &

An important role that members of her support network played was therefore to provide relational resources in the form of reassurance. She relied especially heavily on Lee and Jessica to validate her vision of all students as smart:

Id like to see [my view of smartness] validated beyond my opinion, and I dont see it validated. Beyond my opinion of it. Except when Lee and Jessica come. Then Im like theyre gurus, theyve got it. Okay, now I can go back to believing it again. Like I need them to be confident for me.

Thus, Amandas relational resources affirmed for her that she was not alone, wrong, and crazy but a member of a larger community organized around a practice that was important precisely because it was not the norm.

Positional Resources

In addition to reinforcing her confidence in her perspective that all students are smart, members of her support network helped Amanda feel personally validated and to develop an identity as a good CI teacher. Amandas classroom (like Ryans) was featured in the districts CI Video Club, where her colleagues took her students and her teaching as a model of CI in action. The monthly meetings for CI teacher-leaders across the district were held after school in Amandas classroom, too, allowing her to be present even though she was not officially a teacher-leader and subtly underscoring her identity as a central member of the CI community. Amanda recognized that she had been given a lot of status and that this supported her practice as well as her sense of herself as a professional, saying, [I] have been told Im good at [CI], in different ways, which makes me want to continue to do it.

Technical Resources

Amanda put a great deal of effort into finding technical resources. After school, on weekends, and during the summer, she spent her time attending workshops, meeting with other teachers, reading books about intelligence, and so on. She leaned heavily on her textbook, which she trusted to align with her vision of good teaching, and on Dana, a teacher at another school who had attended the same CPM and CI workshops Amanda had. The two met regularly on Saturdays to plan lessons, and Amanda learned to use new materials and instructional strategies through their collaboration. Amanda was also bold about learning from her own practice, adopting materials and strategies for her own use with very little guidance. For example, she adopted the language of growth versus fixed mindset after reading Mindset (Dweck, 2006) largely on her own, and after observing Lee shift a typically disengaged students participation by publicly recognizing her as a table master (in response to the students effective use of a table of values), Amanda started to publicly identify all of her students as masters of something mathematical.

Summary

Despite her substantial efforts to garner resources and her identity as a good CI teacher, Amanda described feeling like she was riding on the seat of [her] pants most days, pointing to an ongoing need for further support. Like Ryan, she maintained her engagement with nondominant teaching practice, both inside her classroom and out. But, also like Ryan, she wished she had more resources not just to maintain her practice but also to push her and challenge her to grow. Such resources might have made a difference for Amandas engagement with equity-oriented mathematics teaching in the long run; two years after this study concluded, Amanda left the district, citing family reasons and inconsistent, unsupportive school leadership. Her case demonstrates the limits of strong identity resources without corresponding resources for learning, as well as the limits of external sources of support under difficult school or departmental conditions.

LUKE ZIEBLER: A PATTERN OF DISENGAGEMENT

Like Ryan, Luke encountered Complex Instruction in teacher training. But whereas Ryans program provided rich orienting, technical, and relational resources that paved the way to strong positional resources, Lukes appeared to leave him with fewer enduring ties and much thinner resources of all kinds. Luke took up an identity as a struggling and lazy teacher (to use his words), and instead of pursuing resources that would have supported his learning and shifts in his identity, he disengaged, taking as many vacation days as he could and questioning whether teaching was the right profession for him. In the middle of his third year teaching (the year following the study), he resigned from his position at Union and soon after took a job in another field. His case illustrates how teachers matrix of resources may support a nonteacher identity even as individual colleagues attempt to be supportive.

Orienting and Technical Resources

Lukes career as a teacher grew out of his tutoring experiences in college. He had always really enjoyed math and been a successful math student, and he liked helping others, so tutoring and then teaching math seemed natural. As a student teacher, what stood out to Luke was his cooperating teachers strict discipline and focus on math, math, math all the time. Luke oriented toward his cooperating teachers example while struggling to find his own style, without being something that Im not. Luke had a gentle, relaxed way of interacting with his students, which supported them in taking ownership of their work. But he also had trouble controlling his classroom, and student learning was often compromised by a lack of focus. For Luke, Complex Instruction hinted at how he might support students engagement with mathematics, but he found that it had the potential to be so bad, too, because students would take advantage of the opportunity to talk to each other and get off task easier.

In the absence of a clear alternative, Luke turned to the images and techniques of the teachers he had looked up to as a high school student. On some level, he recognized that the orienting and technical resources that these role models provided were at odds with the student-centered instruction that he wanted to achieve. But at times, he wasnt sure what else to do. He described reading through the student-centered curricular resources that his colleagues at Union had developed and thinking,

Im not sure how to run it, or Im not sure what questions to ask to get them to be successful. And I felt like well, I could either do [the group activities] and be totally unsure that theyll get what they need? Or I can do [a lecture], and at least I told them what they need [to know]. & As the year went on, I found myself more and more like doing that, just like lecture, and then Id [catch myself] talking for thirty minutes and be like, holy shit.

Thus, in the face of a blurred vision of how to run groupwork, Luke turned to lecturing as an instructional strategy more often than he would have liked. He also did more and more practice problems, drawing on his textbook because he felt that practice was missing from the resources that his colleagues shared. Lukes example demonstrates a common phenomenon: teachers own schooling often supplies them with tools for enacting their daily instruction that function as technical resources for reproducing the status quo. Importantly, in turning away from student-centered instruction, Luke did not give up on the CI vision of all students as competent. He just didnt know how to break down students perceptions of themselves while supporting their learning. As he said,

I sort of refuse to believe that any, like theres this one student whos just like not capable, of doing [math]. Its just other things that I think, and so, my, like I feel challenged in breaking that stuff down.

Luke occasionally saw CI-based instructional strategies modeled, and that modeling revealed glimpses of equity-oriented teaching practice. Instead of supporting his practice, however, this potential technical resource interacted with his positional resources to discourage him from engaging with nondominant teaching practice.

Positional Resources

Lukes colleagues seemed to treat him as a smart and promising new teacher. On the Geometry team, he had a reputation as a math whiz; during one meeting, when the group was working on a geometry puzzle together, Margaret looked at his solution and said, Youre just like the brain box in the group, I can tell there Luke, with that. His ideas about teaching were also met with glowing (if vague) praise in the rare times that he shared them. For example, when it was his turn to take the lead on writing curriculum, he presented his outline for the unit and his colleagues called it fantastic and awesome. Margaret added, Im going to need you on my team next year, no matter what Im teaching, thatsso you can do more of the same.

But Luke himself felt that he was too new. In interviews, he said, I dont feel like my opinion has any value yet & [my colleagues] couldve done it a million times, they know it doesnt workfor example if I try to suggest a solution to something. He also interpreted his struggles in the classroom as indications that teaching was the wrong profession for him. For instance, he described his reaction to seeing one of his students presenting to the class after a brief interaction with Jessica, his CI coach:

[This] one student had always told me hes not very comfortable, like he didnt want to come up. Id ask him, I was like yeah you want to do it? Like yeah, you got the right answer, whatever. Um. And so I sort of took him to be a very shy student & but Jessica said something to him, and he was up at the board, and it was like, night and day. Like the way that he acted with me was like totally totally different. Which was cool, but then it puts me in a spot where Im just thinking like what, like. You know, Im just not, somethings not going right.

Similarly, he described the frustration he encountered when he tried to use participation quizzes (an instructional strategy from Complex Instruction, described above):

I think I did maybe in the first two grading periods, I did a couple participation quizzes, and then that was it. You know? And I saw the benefits of it, but I just, I couldnt run it like Ive seen other people run it, and that frustrated me and so instead of, you know, trying, I sort of gave up on it.

Thus, some of the opportunities that he had to watch other educators at workwhich could have served as orienting and technical resources in a different contextfor Luke fit into a narrative about himself as unsuccessful. Sometimes, he did find them helpful, as when William (who was officially assigned to provide support to new math teachers at Union) visited his classroom and asked questions or gave a minilecture in a different way than Luke would have. But often, instead of taking up learning opportunities or figuring out how to build a more supportive network, Luke understood his struggles as reflections of his personality. Im not necessarily the type of person to ask for a lot of help, like maybe when I need it, he said, and he repeatedly called himself lazy.

Lukes perception of himself as lazy was reinforced by structures and norms at Union. For example, he knew that it was beneficial to have time for faculty collaboration, and he appreciated that it was a priority at Union (where teachers spent several hours each week meeting with department and grade-level teams). But he almost kind of env[ied] a colleague who taught at a school where meetings were sporadic and brief, because my personality just doesnt want it. Just doesnt want to be here, doing those types of things for that long. In much the same way, he felt that achieving the kind of focused student engagement that his cooperating teacher had would require him to be someone that he wasnt. There were several ways, then, that Luke had difficulty bringing his views of himself into alignment with the expectations he perceived for teachers at Union.

Relational Resources

There were ways that Lukes colleagues succeeded in communicating that he was one of them, despite his overall assessment that he did not belong. In particular, he appreciated their responses to his struggles, saying,

Theyve totally been supportive. And theyre willing to sort of back up what they say, and come in and help out, or give advice, and just spend the time. & It was like, yeah, we all sort of have been through this, and still go through it sometimes, and let me help you out. You know, not like, you cant do your job.

But this kind of solidarity and camaraderie were insufficient to support Luke either in learning or in developing a strong connection to his practice.

Summary

Lukes case highlights the importance of positional resources in supporting teachers learning and identity development. Whereas Amandas strong positional resources encouraged her to seek out resources that were not immediately accessible and to persist in the face of challenges, Luke interpreted his struggles not as natural components of his professional learning or as indicators that he needed more support, but as indicators of flaws intrinsic to his personresulting in a nonteacher identity. He was not supported in developing a vision of teaching that built on his strengths, needs, and ideas about himself, and this limited his capacity to learn and identify as a teacher at all, let alone a nondominant one.

WILLIAM BARRETT: LIMITED RESOURCES, LIMITED ENGAGEMENT

William Barrett was a caring and reflective educator, well respected by his colleagues at Union and at other schools. He played a central role in the Union math department, especially on the Geometry team. He constantly worked to improve his and his colleagues practice, experimenting with new activities in his classroom and pressing his colleagues to think deeply about how to meet their students needs. But William also set deliberate boundaries around his work, which kept him at the periphery of the district CI communityand kept the work of reframing mathematical competence peripheral to his practice. His case raises questions about how to provide resources for teacher learning in ways that respect teachers accomplishments and are sustainable for them as professionals.

Orienting and Technical Resources

William came to teaching after a brief career in human resources. Part of the meaning that he found in his work as a teacher was connected to making a difference in an unjust world. He got his first teaching job at Union, and 10 years in, he was still proud to teach at a school where every kid in the city is welcome & [and] there is this explicit expectation that if a kid is on your roster, you teach them. And you find a way to reach them. Yet this goal was moderated by his orientation toward realistic expectations for success, inspired by a mentor (Hiro) who he remembered saying, If you devote yourself fully for a whole year, and if you make a difference for just one or two kids, its been a good year. By carefully circumscribing his ambitions and getting that save-the-world sort of idealism out of my system, William hoped to sustain his participation in the profession for many years to come:

I would like to do this until I retire. And, I think without hopefully settling or diminishing my hopes and dreams for my impact, too much. [So] I also am careful about not overextending myself or even, just putting the bar so high up there that I can never reach it. And then just, being disappointed all the time, bummed out, and [feeling that its] time to move on.

Thus, William oriented toward a vision of good teaching in which working steadily for incremental changeand being careful not to overextend and burn outwere central. This vision may have been effective in supporting his longevity in the teaching profession, but it also restricted his access to learning and identity resources that would have engaged him more centrally with nondominant teaching practice.

For example, Complex Instruction provided William with orienting resources, presenting a more transformative view of teaching than he had previously encountered. But he took up the CI vision of equitable instruction in quite limited ways. The chills he occasionally felt when he saw CI working were accompanied by challenges that he found disheartening:

[M]ore times than not what happens [when I assign a group task] is, the kids who have, you know, better access to it love it and go for it, and the kids that dont tend to do a lot of copying, and I dont know how engaged they are. & so Im sort of jaded, or a little disheartened.

William explained the challenges he faced in using CI in terms of his technical resources and skills:

A lot of itI think a big part of it is my skill level. & I do believe every kid is smart. I totally believe in multiple intelligences and all sortlike I really really believe that is true. I think its, again its a skill thing.

But whereas Amanda put significant effort into garnering resources that were not readily available to her, William did not. He was a willing learner, but as a veteran with 10 years experience, he was no longer on the receiving end when it came to support for learning at Union. Indeed, William periodically raised questions and dilemmas with his colleagues, but as a group, their vision of CI seemed too abstract or idealized (to use Williams words) to drive their practice toward reframing mathematical competence. And Williams prioritization of sensible limits meant that he rarely accessed resources beyond his immediate circle at Union. Doing so would have required an investment of time and energy outside of the regular school day, and he had childcare issues and other concerns, including protecting himself from burning out. I cant complain so much about supports, he said, because

there are ample opportunities to collaborate, with all these people [across the district]. This is more just my decision to set up boundaries. & For the sake of making it at least another ten years, I say no a lot. And thats a good thing for me.

Thus, William deliberately decided to limit the energy he put into cultivating his professional network, and the decision made sense given the context of his life and his goals for himself as a teacher. But he had fewer resources for developing his practice and his identity as an equity-oriented teacher as a result.

Relational Resources

Williams professional network may have been small, but in terms of relational resources, he felt really well cared for by his colleagues at Union. The math departments solidarity around core values was especially important to him:

Its not just CI, the practice of CI. Its the values behind CI. Like how do you viewdo you view every student as a contributor to the learning environment. I mean. So I, I do think that everybody here believes that. And at, I mean thats kind of a baseline. But thats pretty huge. & We have some differing opinions, I think, around [how we carry it out]. But I really trust the intentions of the teachers in the department? And I think, overall, we have each others backs.

William also highlighted enduring connections with a few students from each class, saying half-jokingly that staying in touch with them and keep[ing] involved in their lives was his plan to save the world.

The relational resources provided by his rapport with his colleagues and his students supported Williams sense of connection to Union and his sense of identification with his practice more generally. In the presence of richer orienting and technical resources, Williams relational resources could have provided a foundation with the potential to support his ongoing learning.

Positional Resources

Despite his low engagement with out-of-school networks, William was highly involved in Unions math department. He hosted and often facilitated the Geometry teams weekly meetings; he wrote most of the Geometry homework assignments (classwork was divided between team members); he coached new teachers; and in part because his classes were generally a few days ahead of his colleagues classes, he often gave advice and fielded questions about upcoming lessons during team meetings. He saw himself as the veteran with a fair number of tools in my toolbox, and he accepted leadership as his duty. He saw his colleagues as responsive to his influence (a view that Ryans and Lukes descriptions of his support for their work confirmed). Thus, he took up a position at the core of his department but on the margins of broader CI networks. This position supported his identity as a competent professional but afforded limited opportunities for him to learn.

Summary

Williams case highlights the tension that teachers face, especially in under-resourced and high-need schools, between preserving their own lives outside of work and participating in their practice in ways that support them in learning. William was a thoughtful and reflective person, and over the course of 10 years, he had developed enough skill to be regarded as a leader in his department and to feel confident about many aspects of his practice while remaining open to continued improvement. But none of these factors were enough to support radical shifts in his practice in the absence of readily accessible resources for developing a clear vision of an alternative to traditional practice (orienting resources), for enacting such a vision (technical resources), and for seeing himself as a competent and valued member of communities of nondominant teaching practice (positional resources). This is not to say that William was not learning or developing his practice in any ways that were significant for him or for his students, just that the extent to which he engaged in reframing mathematical competence was limited. Nor is it to say that his engagement with nondominant practice cannot deepen. But he would require more resources (like the ones that Ryan and Amanda got from attending Video Club, from observing and coteaching with teachers who were centrally engaged with CI, and from participating in coaching that highlighted their strengths) in order to support such deepening.

IMPLICATIONS

This article began by asserting the force of narrow, exclusive ways of understanding what it means to be good at math, which continue to pervade American classrooms and American culture more broadly. The four teachers described in this paper all cared deeply about their students, and they shared a commitment to the idea that all students are mathematically capable. Yet traditional view[s] of what smart looks like were still present in [their] head[s] and in their practice (to repeat Amandas phrasing).

Nonetheless, the cases of Ryan Sower and Amanda Pepper show that it is possible for teachers to engage in meaningful ways with the work of redefining mathematical competence and, through such engagement, to learn to enact forms of mathematics instruction that expand students opportunities to see themselves as powerful mathematical thinkers. This finding suggests that such learning is not a magical process that only superheroes can perform but a challenging and countercultural task that requires many kinds of support. The importance of four particular kinds of resourcesorienting, technical, relational, and positionalin turn illustrates the ways that learning and identity development inform each other in teachers engagement with nonnormative shifts. For Ryan and Amanda, learning to enact new practices was enmeshed with becoming a new kind of teacher and coming to belong to communities of nondominant teaching practice. In contrast, for Luke and William, unsuccessful attempts at learning were both a cause and a result of identities as peripheral members of communities of nondominant practice, in a mutually reinforcing cycle of inadequate resources and disengagement with the work of redefining mathematical competence.

In focusing on the resources that are provided to, sought out by, and taken up by teachers, this analysis situates teachers workboth when it successfully reframes mathematical competence and when it reproduces hierarchiesin social contexts (e.g., networks) that support or fail to support their engagement with nondominant practice. At the same time, it illustrates some of the ways that teachers participate in shaping these contexts, pursuing some supports but not others and making their own sense of varied and conflicting resources.

A resource-focused analysis has implications at many levels. For individual teachers who are interested in sustaining or deepening their engagement with nondominant practice, a typology of resources may point to supports that are worth seeking out. For policy makers and teacher educators, the analysis suggests that whereas research and practice have often attended exclusively to resources that can be characterized as orienting or technical, more attention ought to be given to the ways that teachers access to and pursuit of these resources is mediated by relational and positional resources, as learning and ongoing identity work intertwine. In other words, those who are interested in supporting teacher learning ought to think creatively not only about how to provide teachers with more resources, but also about how to provide them with more kinds of resources.

Although the focus of this paper has been on learning to redefine mathematical competence, it seems probable that these implications are relevant for supporting teacher learning more broadly. If learning and identity development are generally intertwined, this study suggests the particular importance of the links between the two when the learning at issue is difficultas teacher learning often is. The work of teaching is marked by endemic uncertainty, requiring teachers to constantly adapt to ever-changing students and circumstances. The current context presents teachers with the additional challenges of learning to enact ambitious, student-centered forms of instruction that they themselves are not likely to have experienced beforeto find ways forward along paths that are not well worn while abandoning those that are familiar. Explicit attention to relational and positional resources, as well as orienting and technical ones, might support more teachers to meet these challenges.

Further investigation of the resources that support teacher learning and identity is warranted to address questions such as: Are orienting, technical, relational, and positional resources all equally important? In what contexts, and for whom? Are other kinds of resources also necessary?² Under what circumstances are the resources provided by ties outside ones local community of practice worth the time and energy that those ties consume? And perhaps most importantly, how can we design environments that make the resources teachers need more accessible in order to improve teacher learning and, ultimately, to make richer learning experiences available to all students?

Acknowledgement

The work reported here was supported in part by a dissertation fellowship from the Spencer Foundation and a training grant from the Institute for Education Sciences (IES) to the University of California, Berkeley (R305B090026). The views expressed are those of the author and do not represent these organizations. The author wishes to thank the Spencer Foundation and IES, as well as Alan Schoenfeld, Nailah Nasir, Evra Baldinger, Judith Warren Little, and the teachers involved in her research for their thoughtful engagement with drafts of this paper.

Notes

1. I use the term nondominant to describe teaching practice that aims to redefine mathematical competence, highlighting the direct opposition of such practice with culturally dominant views of mathematical ability as something that is distributed across populations in grossly unequal ways, leaving some people intrinsically good at math and others intrinsically bad at math. The nondominant aspect of the practice I describe is particularly significant in the context of this paper because of the challenges that navigating dominance creates for teachers as they negotiate shifts in their practice. I conjecture that my findings are relevant to the learning of many forms of nondominant practice, not only to the project of learning to redefine mathematical competence in math classrooms.

2. The data collected for this study might obscure some resources that are potentially quite important, not because of flaws or limitations in the study design but because of the range of phenomena that exist to be studied. For example, their content knowledge did not seem to support the teachers in this study in redefining mathematical competence. In fact, the teacher with the highest mathematics degree (Rob) seemed most rigid in his views of both the discipline and his students abilities, whereas Ryan and Amanda seemed to work more from a general orientation to students as smart than from content knowledge that was remarkable in any way. But it seems reasonable to expect that some form of content knowledge would support teachers in developing their practice in ways that would support the redefinition of mathematical competence. Thus, the hypothesis that other types of resources (or subtypes of orienting, technical, relational, and positional resources) matter for supporting teachers engagement with nondominant practice warrants further investigation.

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Cite This Article as: Teachers College Record Volume 119 Number 2, 2017, p. 1-42 https://www.tcrecord.org ID Number: 21665, Date Accessed: 10/22/2021 3:29:41 PM Purchase Reprint Rights for this article or review