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Unpacking personal identities for teaching mathematics within the context of prospective teacher education

Abstract

This article explores the personal identities of two prospective elementary teachers as they progressed from mathematics methods coursework into their capstone student teaching semester. Results indicate that both prospective teachers perceived contrasting obligations of effective mathematics teaching in the teacher education and student teaching contexts, yet came to reconcile these differences in different ways. Whereas one prospective teacher was able to oppose the obligations set forth in the student teaching context, the other complied with obligations in both the teacher education and student teaching contexts. Implications for teacher education include differentiated experiences and placements, methods design, and explicit foci on students’ mathematical thinking.

Introduction

To prepare highly skillful and knowledgeable teachers, teacher education programs commonly organize learning opportunities that help to develop specific competencies (e.g., instructional planning, classroom management, assessment, learner development). For most programs, the capstone student teaching experience includes significant fieldwork that seeks to deepen a wide variety of these competencies within the context of schools and classrooms. In her analysis of seven highly successful teacher education programs, Darling-Hammond (2006) notes the critical role field experiences play—not only in a capstone student teaching placement, but also as prospective teachers (PTs) develop knowledge and skills within traditional coursework. Yet, perpetual challenges exist in accounting for the “apprenticeship of observation” (Lortie 1975)—the conceptions of teaching and learning formed through more than a decade of taking part in classrooms as a student—alongside aligning the visions of teaching and learning present in teacher education programs with the visions, priorities, and realities of the K-12 classroom context (c.f. Hodges and Cady 2012). That is not to say that the only concerns in teacher education are the “book ends.” Teacher education program design varies widely and a design which ensures eventual expert teachers has not been empirically supported (Cochran-Smith and Zeichner 2010).

Despite the growing uses of student achievement data to assess the effectiveness of teacher education programs (e.g., Henry et al. 2013), programs commonly continue to rely on teacher candidates’ work samples to measure both the development of individual candidates’ skills and knowledge, as well as overall program effectiveness. However, there are significant forms of compliance that take place within teacher education programs. That is, it can be difficult to distinguish what types of knowledge and skills have been taken up by teacher candidates from artifacts, instructional designs, and reflections that are merely included for compliance. For example, teacher candidates may use certain instructional strategies and curriculum materials that reflect what they feel the teacher education program would value as good teaching; or if not aligned, candidates may follow the practices used in their host schools and classrooms. As Gresalfi and Cobb (2011) highlight in their analysis of practising teachers’ work within a professional development program, are teachers (a) compliant and merely meeting obligations to others; (b) valuing what counts as effective mathematics teaching and learning; or (c) opposing what counts as effective mathematics teaching and learning? In this article, we explore these questions by analyzing the experiences of two PTs who provide contrasting cases from which to understand the reasons behind teachers’ motivations to improve their instructional practice. We draw upon Cobb, Gresalfi, and Hodge's (2009) notion of personal identity to make sense of the work of these two teacher candidates during their capstone field experience. In doing so, we address the broader issue of how prospective teachers (PTs) in their preparation reconcile often conflicting conceptions of effective mathematics teaching and construct their own vision of what high-quality math instruction should entail. In particular, we ask: To what extent are PTs identifying with the vision of mathematics teaching and learning conveyed in teacher education as they participate in university and school-based contexts? As will become apparent, the two PTs on which we focus came to reconcile conflicting visions of effective mathematics teaching in differing ways.

The topic of teacher identity is significant given reasons related to both theory and practice. As a theoretical construct, identity serves as a tool to understand how, when, and why teachers are motivated to change their instructional practice. In the context of teacher preparation, the notion of identity offers a way to explore learning from the PTs perspective, focusing on aspects of the context and his/her valuations of those aspects. In terms of practice, identity can offer information that allows math educators to design learning experiences that build toward intended goals, which include coordination and alignment of the multiple pieces of university and school partnership contexts.

Development of teacher identity within the context of teacher preparation

The development of an identity may be characterized as a “process of increasing participation in the practice of teaching, and through this participation, a process of becoming knowledgeable in and about teaching” (Adler 2000, p. 37). Wenger (1998) refers to the forum in which this participation occurs as a community of practice. He describes identity as “a constant becoming” that characterizes who we are by:

the ways we participate and reify ourselves; our community membership; our learning trajectories (where we have been and where we are going); reconciling our membership in a number of communities into one identity; and negotiating local ways of belonging with broader, more global discourse communities. (p. 149).

The value of Wenger’s characterization of identity is the recognition of a dual focus on individual and collective practices. In Wenger’s view, the process of learning is linked to identity through new ways of belonging and being within a community of practice. Changes in the ways people participate in cultural activities, such as teaching, involve the evolution of roles and the development of new identities that are linked to new competencies (Kazemi and Franke 2004). Furthermore,

[a]s individuals come to participate in cultural practice, they negotiate an identity that is part of what they have come to view as consistent about themselves in their lives, part of what they perceive to be available to them in their practice, and part how they are perceived by others (Nasir and Hand 2006, p. 467).

The aforementioned perspective on identity is useful within the teacher education context as it places the development of identity against the norms and practices valued in the teacher education program and in related field work (e.g., student teaching). In each context, PTs reciprocally define what practices are affiliated with being viewed as a competent student and teacher. In reference to students in mathematics classrooms, Cobb, Gresalfi, and Hodge (2009) refer to this as a normative identity which “comprises both the general and the specifically mathematical obligations that delineate the role of an effective student in a particular classroom” (p. 43). We can then develop a parallel argument that PTs help to develop and are privy to normative identities associated with the teacher education programs in which they participate, as well as normative identities constructed as they participate in school-based communities. As Gresalfi and Cobb (2011) note, a normative identity is not a personal one, but a collective one that helps shape who is a competent teacher of mathematics and who is not. A personal identity is shaped as one comes to affiliate with particular practices valued within a community. This affiliation may align with or oppose what is valued within that context. Drawing a parallel from practising teachers to PTs, they may come to value the norms and practices evident within the teacher education program, they may oppose the norms and practices, or they may simply comply with others’ expectations for mathematics teaching and learning (cf. Cobb et al. 2009). As PTs enter field experiences, they again encounter normative identities which may or may not align with the normative identities constructed in the teacher education program. When these identities are not aligned, as is the case for the two prospective teachers selected for analysis here, PTs are left to reconcile conflicting normative identities and take up instructional practices which may or may not be valued in one or more of these communities (cf. Gresalfi and Cobb 2011). This perspective on identity is consistent with postmodern views, which describe identity as involving multiple sub-identities, which are ongoing in their construction and related to social contexts (Akkerman and Meijer 2011). The idea of personal identity provides structure to a narrative approach by focusing our attention on the (1) normative ways of acting and (2) valuations of these norms from a participant’s perspective. In this way, the notion of personal identity informs the design of teacher education coursework and field-based experiences by detailing whether or not and to what extent PTs identify with the vision of mathematics teaching conveyed through their teacher education program and with what obligations they are actually identifying or resisting.

The study

The cases presented here emerged from a research project that attempted to examine how teacher candidates learned to teach over the course of their elementary education program at a southern United States university. The larger project sought to understand the extent to which artifacts collected at particular points along the program, as well as during the capstone student teaching experience, reflected the priorities of the elementary level faculty associated with the teacher education program. The artifacts collected for the project included major assignments from methods courses and the student teaching portfolio which included four components: unit planning, unit implementation, unit reflection, and a leadership essay. There were also two interviews conducted by a research assistant during the student teaching experience, one conducted approximately midway through the student teaching experience and the other at the conclusion.

Participants

Approximately 20 PTs agreed to participate in the study. For the purposes of our work, we narrowed the sample to the seven PTs who selected mathematics as the content focus of their unit during their capstone field experience, which provided additional mathematics-focused artifact data. Furthermore, the lead author developed an interview protocol focused on participants’ mathematics experiences in teacher education in lieu of the more general interview protocol completed by other PTs in the study. Of these seven PTs focused on mathematics, three had between 18 and 24 credit hours in mathematics content. These additional mathematics experiences are significant given the critical role depth of content plays in effective mathematics instruction, as indicated by professional organizations (Conference Board for the Mathematical Sciences (CBMS) 2012) and the lingering questions that remain about the interplay of content and pedagogy (Hill et al. 2008). Further, additional mathematics coursework might well enable PTs to challenge “status quo” practices and develop a sense of agency for making instructional decisions. An initial review of the data indicated one PT that perceived alignment between the normative identities in the teacher education program and the local school. Elsewhere we have discussed the power of alignment between multiple communities of practice in supporting teachers’ implementation of reform practices in mathematics education (Hodges and Cady 2012). Here, our interest is in how PTs negotiated conflicting normative identities, which was the case for the other two PTs, Sarah and Nicole (pseudonyms). Consequently, this article discusses Sarah and Nicole, two PTs who were students in the same teacher education program, who had significant preparation in mathematics content, yet came to identify with mathematics teaching in contrasting ways.

Methods

We draw upon interview and artifact data to understand the personal identities formed by two PTs. Each PT completed mathematics methods coursework during the spring semester of her junior year, followed by part-time student teaching in the fall senior semester and full-time student teaching in the spring senior semester. Artifacts were collected from methods coursework and into the full-time student teaching semester, whereas interviews were conducted during the full-time student teaching semester. The following paragraphs delineate the specific data collected for this study.

Interviews

Semi-structured interviews served as a primary resource for understanding each PT’s characterization of the normative identities within their student teaching placement. While an individual teacher’s characterization of the student teaching setting may or may not be reflective of the true normative identity within the school, our focus was on each teacher’s perception of what counted as effective mathematics teaching and learning within a given context. Furthermore, we relied on interview data to help characterize the university normative identity for mathematics teaching. The interview data also supplemented and triangulated the artifacts collected. Each teacher participated in two interviews (see sample interview questions in Appendix). The first interview occurred roughly half way through the full-time student teaching experience and focused on perceptions of the normative identity within the student teaching placement and integration and usefulness of other teacher education experiences (e.g., coursework, practicum), and perceptions of what the teacher education program expected of them in their teaching. The final interview was a more open reflection on their teacher education experiences, allowing participants to expand upon statements made in the initial interview. This final interview provided a window into their perceptions of the normative teacher education identity and their personal identities.

Artifacts

Approximately one year prior to student teaching, each PT completed an elementary mathematics methods course. Artifacts from this course that were included in data analysis were: (a) mathematics autobiographies; (b) lesson sequence assignment; and (c) final reflection on the autobiographies. The mathematics autobiography was written at the beginning of mathematics methods coursework. Here PTs articulated a belief about what mathematics was, high and low points in mathematics, and their beliefs about what entailed high-quality mathematics instruction. The lesson sequence assignment required PTs to conduct interviews with students during a co-requisite field experience and then design a series of connected mathematics lessons to build upon the strengths and address weaknesses in students’ mathematical thinking. The final reflection called upon PTs to reflect on their autobiographies in light of their experiences in the mathematics methods course and co-requisite field experiences.

As a part of licensure requirements, each PT was required to complete a teacher work sample portfolio comprised of four parts: (a) lesson plans; (b) implementation and assessment; (c) reflection; and (d) leadership essay. Five sample lesson plans from a single unit of study are included in the teacher work sample. Sample student work, pictures taken during lessons, and results from assessments serve as evidence of implementation and assessment. A three to five page reflection on the unit is written at the conclusion of implementation. Finally, PTs construct an essay highlighting what teacher leadership means to them, as well as provide examples of their own school-based leadership.

Data analysis

Transcribed interviews and artifacts were combined into a single data set which was ordered chronologically. That is, artifacts from mathematics methods coursework preceded interview transcriptions and artifacts collected during student teaching. This ordering allowed researchers to document the PTs change in views concerning the obligations in different contexts and their thoughts concerning these obligations. The primary unit of analysis across all artifacts and interview data was an extended phrase. An extended phrase is a phrase relevant to the research question, including contextualized information surrounding the phrase such that meaning can be attributed. Three types of extended phrases were considered: frequent phrases, missing phrases, and declarative phrases. Frequent phrases demonstrated patterned regularity (Wolcott 1994) in PTs’ perceptions of the communal obligations and in the evolution of their personal identities. Missing phrases are items that were anticipated by the researchers, but never appeared in the data (LeCompte 2000). For instance, we might anticipate that PTs describe connections between teacher experiences during the student teaching semester and earlier coursework. Yet, when asked for such connections, the inability to describe connections is significant. We foreground in our work as teacher educators and researchers the centrality of theorizing from practice, making explicit teacher candidates’ abilities to explicate the dynamic relationship between theory and practice as they move in and out of teacher education and school contexts (Hodges and Mills 2014). As such, missing phrases provide evidence of gaps in theory–practice connections. Finally, declarative phrases are those which the researchers found as significant, but did not occur with any regularity. The attention to declarative phrases created opportunities to consider participants’ valuations of any aspect of their learning. Further, declarative phrases offered a way to get at situations that did not occur on a regular basis, but that informed the obligations required of participants in different contexts.

The coding consisted of two phases. First, the lead author and an additional coder independently identified in the data extended phrases related to dimensions of personal identity previously mentioned. In particular, the analysis focused on PTs characterizations of themselves as teachers of mathematics in relation to their perceptions of what counts as effective mathematics teaching and learning in a given context. The researchers independently applied the coding scheme to each identified extended phrase. Second, as researchers analyzed the data chronologically, each made conjectures about the PTs developing personal identities. In this way, the back and forth between conjectures and refutations addressed the trustworthiness of the analysis. Disagreement between the researchers was discussed until consensus was reached. Next, the authors used the coded data to document (a) PTs’ perceptions and description of the norms and obligations of their student teaching placements and (b) the valuations PTs placed on these norms and obligations as they simultaneously participated in the teacher education program and their student teaching placements.

Results

Teacher education coursework and classroom experiences provided Sarah and Nicole with opportunities to expand their understanding of what counts as effective mathematics teaching. How they came to define effectiveness and the extent to which they found themselves as competent in this definition were salient in their written and verbal reflections on practice. At times, each PT articulated perceived tensions between the normative identities established in teacher education and in their student teaching placements, which provided demarcations of each normative identity. In the following sections, we describe the PTs personal identities by delineating the norms and the PT’s valuations of these norms in the university and school contexts.

The teacher education context: high-quality mathematics teaching

Three themes emerged with patterned regularity in analyses of Sarah’s and Nicole’s mathematics methods coursework samples related to the teacher education context. These themes also appeared as declarative phrases in interviews conducted during the internship experience. The themes included: (a) a focus on problem solving, (b) making content relevant for students, and (c) encouraging students’ explanations and justifications of thinking. These themes generally align with reform recommendations in mathematics education (National Council of Teachers of Mathematics 1989, 2000, 2007; National Research Council 2001), which emphasize reasoning and sense making about mathematics concepts over simply memorizing facts and procedures. While the statements the PTs made were in relation to what they came to understand as a consequence of participating in the mathematics methods courses, such statements were taken as an indication of what the teacher education context would view as high-quality mathematics instruction as perceived and described by the PTs. That is to say, at the very least, they were complying with what is valued by the teacher education program when completing assignments embedded within the methods courses.

Doing math is about problem solving

The final paper in the second mathematics methods course required students to reflect upon their mathematics autobiographies written at the beginning of the course. Both PTs indicated that high-quality mathematics instruction includes a focus on problem solving. A problem-solving focus included consideration for the cognitive demand of mathematics problems, as well as emphasizing students’ mathematical thinking over solutions to procedurally oriented problems. The following statements are indicative.

Sarah: I now realize that not all word problems are quality mathematics. The Mathematics Task Analysis Guide highlighted that although many word problems involve scenarios, often “there is little ambiguity about what needs to be done and how to do it.” I’ve learned that rather than assuming a problem is cognitively demanding simply because it is a word problem, I should be critical about the tasks I choose for my students to do.

Nicole: I really had no clue what all math included. I had a stereotype of getting a problem, plugging it into an equation and getting an answer … There is so much more to problem solving than solutions. You have to see how they are thinking about the problems.

Both Sarah and Nicole emphasized the role problem solving plays in quality mathematics instruction. However, neither teacher provided any indication of a balance between conceptual and procedural understanding or where in the sequence of instruction problem-solving tasks might occur, despite a stated focus on these elements of problem solving by mathematics teacher educators in the program. These omissions are critical as they limit the extent to which Sarah and Nicole can take up identities aligned with the teacher education program if they in fact do not know these are valued by the teacher education context. Furthermore, the focus on problem solving reflected in the PT’s comments is critical for two reasons. First, their comments indicate that there is an emphasis on problem solving regardless of the specific topic and classroom context. This somewhat generic perspective on problem solving can lead to difficulties for PTs in applying this recommendation during student teaching, leaving PTs with an “all or nothing” view of enacting problem solving. Second, the emphasis on problem solving also detracts from the local decision-making that is part of most, if not all, methods courses. This kind of decision-making in design, instruction, and assessment is based on careful observations of students’ thinking that can be leveraged for future instruction.

High-quality mathematics instruction makes content relevant for students

Sarah and Nicole regularly made statements that indicated high-quality mathematics instruction includes contexts that are relevant to students’ lives. During an interview conducted at the conclusion of the student teaching semester, Sarah stated the following:

[The mathematics education instructors] all stress that you should relate what you are teaching to real world experiences. Because if the kids don’t see it as relevant why should they be learning it. I haven’t used any of the problems they’ve given us. Just relating to real world experiences and think through problems rather than it being straight forward.

Furthermore, Sarah provided students opportunities to make connections to real world use of measurement during the introduction of a lesson plan developed during methods coursework. Sarah called on students to explain their use of measurement outside of school and then supported student to categorize those uses according to attributes (e.g., length, weight, capacity, temperature).

While she was less specific, Nicole noted the importance of hands-on instruction that students’ could then connect to their out-of-school lives:

… the way I was taught is not necessarily the best for these students today. More hands-on, more let them think and then see how they can apply it to their lives.

Congruent with Sarah, analyses of Nicole’s lesson plans developed in conjunction with the mathematics methods courses indicate further emphasis on making mathematics content relevant to students. For example, in a lesson plan introducing integers to 6th grade students, Nicole began the lesson with several contextualized problems involving football, golf, money, and temperature.

Student explanations and discussions are important

Each PT indicated the value of having students discuss mathematical ideas with one another, with varying degrees of specificity. Sarah made specific references to mathematics methods course activities and readings, while Nicole indicated the importance of discussions without any specific references. The following quotes are indicative.

Sarah: According to social cognitive learning theories, students learn best when they hear students their own age model or explain their behavior or process, rather than from a teacher. Having students explain their thinking like the author of “Never Say Anything a Kid Can Say” suggests, can provide more enhanced learning and attention from the students. Also, a student will typically explain the material in a way that may be more relevant and meaningful for other students in the class.

Nicole: Students have different ways of learning. If you give them a task before presenting the information, then they have time to develop their opinions and feelings of how to carry out the task. Then, by sitting them in groups, these activities will foster discussion about math.

In Sarah’s lesson plans written during the mathematics methods courses, she often made use of think-pair-share instructional sequencing when posing questions to students. For example, in a lesson on standard units for linear measurement, Sarah posed the question, “Describe some situations when measuring by yards would make sense. Do the same for miles.” She first required students to write out their own ideas and then allowed time for students to share at their tables. The discussion concluded with sharing and challenging the ideas generated at each table.

Nicole was less specific about how these discussions would play out with students within the classroom. Lesson plans developed during the methods courses provided some indication that discussions between students were planned, yet details on how those discussions would unfold or probing questions she might ask were not included in the plans. For example, two lesson plans contained the phrase, “students will discuss,” but no further elaboration of how that discussion would be constructed.

The student teaching context: lecture-oriented instruction

Each PT articulated an explicit focus on lecture-oriented instruction provided in their host classrooms during student teaching. This focus was evident during interviews conducted with the PTs, but did not appear in any written artifacts collected during student teaching.

Nicole included a description of her student teaching context early in the semester in which she portrays her classroom as a “comfortable and fun mix of 21 students,” a place where

[t]echnology is easily interwoven into daily classroom instruction. The classroom has 11 computers, a document camera, Smart Board, video cameras, and email for each student, which makes working with technology a common activity for students. The classroom also has hundreds of fiction and nonfiction books and magazines, as well as many manipulatives for math and science. The classroom is very large with additional room left after the space occupied by the students’ desks, teachers’ desks, tables, and multiple bookshelves.

Yet, when asked to describe how she made instructional decisions with her cooperating teacher, she noted a lack of a common vision alongside a lack of conversations about planning and teaching.

Sarah: I’ve really tried to avoid that rote, do the algorithm all the time that the teachers at my school seem to always use. I think what I am doing is new to a lot of teachers at my school. We don’t have a relationship where he talks [to] me. We don’t sit together and plan. That’s how it always is.

Sarah’s teacher work sample lesson plans included tasks where students made connections to art when exploring parallel and perpendicular lines and a grid city inspired by an activity Investigations in Number, Data, and Space—an NSF-funded curriculum organized around the National Council of Teachers of Mathematics standards documents. Sarah’s perception that such instruction was “new” to teachers and different from how teachers typically taught mathematics was an indication of the nature of instruction provided by her cooperating teacher.

When Nicole was asked a similar question about planning with her cooperating teacher, she stated that her cooperating teacher “would look at my lesson plans and say this is how I do it. My teacher always encourages me to look at the state website for problems that go along with the standards.” As she reflected on her teacher work sample unit, Nicole stated that she attempted to work with her cooperating teacher to plan each lesson, noting that her cooperating teacher has significantly more experience teaching this material. Nicole went on to describe her teaching as primarily lecture-oriented instruction within the unit. Analyses of lesson plans confirmed this focus, where Nicole would routinely model how to solve several problems, followed by individual seatwork competed by students. In Nicole’s written reflection on her unit, she noted a lack of student engagement within the lessons, which she believed could be remedied with more experience.

I would change some things so that the students were more engaged and more excited about what they were learning. I feel with experience I will gain more ideas and a wider range of things I can do with my students.

The lack of student engagement in instruction and narrow set of strategies for teaching was consistent with her reliance on lecture-oriented instruction throughout the unit.

PTs’ thoughts on obligations in the two contexts: opposing, complying, and valuing

As Cobb, Gresalfi, and Hodge (2009) note, a personal identity can be understood through the obligations an individual has in a particular context. How PTs value their obligations in a context provides an indication of whether the PTs are identifying with the normative identity or merely complying with such obligations. I looked for evidence that PTs found obligations empowering and attainable, or unreasonable and constraining (cf. Gresalfi and Cobb 2011). Resisting obligations can be observed when a PT deliberately opposes the normative identity in a particular context.

The PTs valuations of the two contexts of coursework and student teaching were apparent in written reflections constructed at the end of methods coursework and in interviews conducted during student teaching. Evidence of opposition, compliance, and value was apparent across the contexts for the two PTs. I begin by describing their valuations of the teacher education context, followed by the student teaching contexts.

Personal identities in relation to the teacher education context

For Sarah, much of what she came to understand about mathematics teaching and learning was deepened by and solidified during teacher education coursework. Sarah felt congruence between her existing conceptions of mathematics teaching and those espoused in mathematics methods coursework, evident in the declarative statement, “My current vision for a mathematics environment reflects much of same beliefs expressed prior to the semester.” Yet, she went on to acknowledge the role that teacher education coursework played in deepening her understanding of mathematics instruction.

I believe the role of mathematics is not to fill students will endless algorithms, but rather tools to help critically solve problems in the classroom as well as their everyday life. Yet, I now feel that I have a greater understanding of how to approach developing higher order thinking in my students than when I first [started].

In an interview conducted during student teaching, Sarah made explicit theory–practice connections in her vision of effective mathematics instruction. These statements served to further illustrate the alignment between her personal identity and the normative teacher education identity. The following interview quote is illustrative:

I think the best way to teach anything is to throw kids into that disequilibrium that Piaget talks about. Make them solve it and figure it out for themselves. I feel like its harder to do it in math than science or reading, but we’ve done it.

Furthermore, Sarah noted that the individual seatwork associated with conventional textbook lessons was not engaging for students in her internship setting, stating that students were not interested in “flipping through a textbook constantly … so you have to make it appealing to them.” Sarah went on to say,

They [students] can learn a lot from their peers. I assign them a partner who will challenge them to think a different way. I have some students who are extremely good at learning a concept first thing, but maybe their not good at creative thinking. Then I have some students that are good at thinking outside the box, although they may not catch a concept right away. So I try to pair student who learn differently together.

Consistent across written artifacts collected during methods coursework and in later interviews, Sarah constructed a personal identity that came to value the obligations of effective mathematics teaching established in the teacher education context. Her consistent reference to teacher education experiences as she described her views of effective mathematics teaching provided salient connections to the intended priorities of mathematics education faculty—namely, the broad vision of reform-oriented instruction in mathematics education.

Nicole, however, never provided salient connections between her vision of effective teaching and the normative identity established in teacher education. In fact, where Sarah indicated what she knew and believed about mathematics instruction, Nicole regularly provided statements of what she “should” do, or what she was expected to do. The following statements made during reflections on coursework and during an interview one year later, respectively, are indicative:

  • I knew that giving notes on a topic before a class and then doing an activity was golden, but I need to remember to present in a variety of ways, relate it to make sense situations and not to bore the students with worksheets everyday of 100 problems.

  • It would be nice to use [manipulatives] on a daily basis in my future classroom … but you’ve really got to be creative for that and I lack creativity.

While she did not oppose the normative teacher education identity, she did articulate that this normative identity was unattainable given her lack of “creativity.” Such statements were considered to be indicative of her compliance with the normative teacher education identity. That is, she indicates that these are important considerations when designing high-quality mathematics lessons, but on the other hand, fails to indicate how such instructional expectations are empowering and attainable for her.

Personal identities in relation to student teaching context

The contrasting descriptions of the obligations established in teacher education and student teaching provided the PTs with opportunities to reconcile these differences in a variety of ways. Again, I looked at the obligations each PT had and the extent to which she saw those obligations as valuable and attainable, or whether she opposed and resisted obligations. Sarah provided numerous statements that indicated her opposition to obligations in her student teaching contexts. Those statements focused on two aspects of instruction: reliance on the textbook and overemphasis on procedural fluency. The following interview quotes are indicative of the patterns observed:

  • At some points my host teacher has been real helpful and suggested that I use my textbook, but I’ve really tried not to touch that too much

  • [Teaching] starts too early with that rote memory and it’s torture for so many people and I know I didn’t want it to be like that.

  • I know I didn’t want to be a textbook-based teacher, that I wanted to think outside the box. Challenge them in ways they were not used to and bring other elements into the mathematics curriculum and connect it to real life.

The phrases across these quotes, “I’ve really tried not …” and “I know I didn’t want …” indicated her opposition to the student teaching normative identity. Further, these statements are consistent with statements Sarah’s made in relation to her personal identity in the teacher education context.

Nicole also suggested that her host teacher was supportive in offering suggestions for instruction, but characterized her response to those suggestions differently. This was apparent when she said, “I was given my [teacher work sample] topic by my CT. She helped me a lot. My CT would look at my lesson plans and say this is how I do it … so I did it like that.” During interviews, she was asked what aspects of her lesson development were self-selected. Nicole was unable to articulate lesson plans in which she played a central role in design. In the reflection on her teacher work sample, Nicole justified her reliance on her host teacher for making instruction decisions when she said,

… asking for help is one of the best things you can do as a beginning teacher. I shouldn’t be afraid to ask others for their advice or opinions if they have done this a ton of times before me. They usually will know ways of what works, what doesn’t.

However, Nicole is also critical of her use of the instructional practices encouraged by her host teacher, noting limitations in student engagement:

My kids seem to sometimes, well it’s my fault, like I don’t have them [their attention] all the time. I don’t think I have the best method all the time. It’s something I need to work on a lot.

Such statements indicate that Nicole has not come to value the instructional practices in the student teaching context—that is, she does not see the normative identity as empowering and attainable, but rather limiting the extent to which students can engage in her lessons.

Summary

Both PTs indicated the teacher education context valued instructional practices broadly consistent with reform recommendations in mathematics education, with a particular focus on problem solving, making content relevant for students and encouraging students to justify and explain their thinking. Further each PT perceived the student teaching context as predominantly focused on teacher-directed, lecture-oriented mathematics instruction. In particular, each PT noted the reliance on textbooks and state policy documents (e.g., state standards website) to inform the vast majority of instructional decisions.

Despite these similarities, the personal identities each PT developed were quite different. Sarah came to value the teacher education context and oppose the student teaching context, whereas Nicole complied with both the teacher education and student teaching contexts. It is important to note that both PTs demonstrated significant competencies in both methods coursework and student teaching. That is, as external demands were placed upon them, both were able to demonstrate an ability to plan for and reflect upon reform-oriented mathematics instruction.

Discussion and implications

While each student displayed similar competencies in teacher education coursework, the application of Gresalfi and Cobb’s (2011) framework provides a lens for teasing out the nature of the identities PTs develop within teacher education by looking at artifacts at particular benchmarks (e.g., methods coursework, field experiences) known to be significant (Jong and Hodges 2015). While each PT recognized conflicting mathematics teaching obligations in the two contexts of teacher education coursework and student teaching, the process for reconciling these conflicts differed significantly. Whereas Nicole sought to meet obligations within each context, Sarah challenged the normative identity established within her school context by taking on practices that differed from her characterization of what counted as good instruction within the school. One alternative interpretation of these findings is that Sarah had previous experiences that supported her in “seeing” and “understanding” the value of university obligations. That is, she had resources that helped her construct this identity.

The data we have shared in this analysis convey the usefulness of drawing on personal identity to inform learning for prospective teacher learning across university and school-based contexts. The two aspects of personal identity allow us to examine whether or not PTs identify with the vision of high-quality mathematics teaching reflected through university coursework and what this vision entails from their perspective. This is important for how teacher educators design courses and significant learning experiences along a trajectory. The data also show the potential of personal identity in understanding the different obligations in which PTs participate in order to be successful in a teacher education program. This focuses attention on PTs’ perspective, but also places emphasis on the obligations of “what mathematics teachers do” in order to be effective.

Turning the focus toward the different visions of mathematics teaching experienced by PTs broadens the development of PTs to include not only a recognition of and competence in reform practices in mathematics education, but also a willingness to oppose what counts as effective mathematics teaching and learning. At several points, it was clear that Sarah opposed the instructional practices in her internship setting in favor of practices that more closely aligned with her evolving personal identity for teaching mathematics. Lord’s (1994) notion of “critical colleagueship” suggests that improving instruction necessitates developing norms for teacher discourse that encourages and embraces professional critique. Consequently, one implication of this study is that developing critical colleagueship should not be reserved for practising teachers, but nurtured within teacher education programs such that PTs can develop a voice for critiquing existing practices. The preparation of mathematics teachers is achieved through the interaction of individuals in multiple contexts. Differences in the vision of high-quality mathematics instruction will obviously emerge, given all the relevant participants involved in the process. The key from our point of view is to dialogue in partnership about these differences and to be explicit about what we collectively are supporting PTs to know and to be able to do. This kind of dialogue can benefit PT learning and the university and school cultures by encouraging and embracing PT advocacy and decision-making. This kind of open dialogue is also potentially beneficial for PTs as they seek to be successful in educational contexts that do not concretely value or readily accept inquiry-based practices that are central to high-quality mathematics instruction. The conversation with PTs can then address specific strategies for being successful in a system that one is also seeking to change. This role is a demanding one for all teachers, but especially challenging for teachers who are beginning their professional careers. For beginning teachers like Nicole whose acceptance of inquiry-based practices is quite tenuous, the opportunity to confront established norms and be intentional and systematic in the design of mathematics instruction may well be a central element of her continued professional development.

Supporting mechanisms that create space for open and critical dialogue for prospective and practising teachers might well afford PTs opportunities to develop professional agency. Approach such as the Critical Friends methodology (Bambino 2002) is an example of a support mechanism that has the potential to provide structure in discussions through group norms of interaction. This approach has been used to facilitate meaningful, open dialogue. This study finds that developing a personal identity that extends beyond compliance toward the obligations of the teacher education is critical. That is, it is doubtful that Nicole is positioned to challenge the obligations wrapped up in her student teaching placement when she has yet to value obligations that oppose and resist the obligations of her school placement.

While teacher education research indicates the need for a shared vision of effective teaching between universities and school partners (Darling-Hammond 2006), shared visions are not universal. As such, if the instructional practices suggested in mathematics teacher education are to be “taken up” by PTs, there must be some comfort in enacting the kinds of oppositional views displayed by Sarah. This may require a shift in experiences within teacher education to enact goals for establishing practices that run counter to normative identities established in school contexts. The challenge for teacher education is to provide a set of tools for and develop a sense of agency in PTs that afford them the ability to enact oppositional views in school settings. One promising approach involves changing the design of methods coursework prior to students teaching experiences. Moyer and Husman (2006) found that when methods coursework was offered at an elementary school and integrated into experiences with elementary students, PTs were “future-focused” and better able to develop a mastery-oriented stance toward learning goals. Integrating site-based methods courses within real classrooms with real children grounds theories of learning and pedagogical skill development by placing practice at the center of teacher education. Here, PTs are offered opportunities to grow in their understanding of how to “think mathematically while simultaneously teaching kids how to see and act on the world as mathematicians” (Hodges and Mills 2014, p. 258). Personal identities that represent compliance are implicitly discouraged when methods coursework is grounded in experiences with children, limiting a PT’s ability to merely meet course requirements when they are confronted with children looking to them for scaffolded support.

Findings from this study suggest that Sarah and Nicole are likely on two different trajectories in their enactment of reform practices and consequently are in need of different types of support. A teacher education model of differentiated instruction might well be needed so that each teacher’s unique needs can be addressed. For example, Nicole’s statement that she does not “have them all the time” and that she struggles to find effective instructional designs is in contrast to Sarah, who stated that kids would not be interested in “flipping through a textbook constantly … so you have to make it appealing to them.” Furthermore, Sarah noted the value of heterogeneously grouping students based upon what she sees as assets students bring to a particular task. Sarah’s characterization of students is consistent with the use of complex instruction in mathematics classrooms (c.f. Wood 2013), where teachers assign high demand tasks to heterogeneously organized groups under the assumption that each member brings unique and valuable competencies to the group’s task. Organizing students in this way necessitates a deep understanding of learner development alongside children’s mathematical thinking. Consequently, Nicole’s differentiated support might include increased attention to student’s thinking within mathematics lessons, as well as specific strategies that promote student engagement in high-quality mathematics tasks. Attention to students’ mathematical thinking when designing instructional tasks is supported by a wealth of mathematics education research (e.g., Carpenter et al. 1996; Fennema et al. 1996; Vacc and Bright 1999). However, it may be that enacting practices that run counter to normative identities is predicated on, or at least enhanced by, a deep knowledge of how students reason about mathematics.

Another form of differentiated support could focus on targeted field experience placements based upon each student’s needs. Tripp and Eick (2008) demonstrated the power of individually targeted placements in their application of a Myers–Briggs type inventory to assign student teachers to classroom mentors based on personality constructs measured by the instrument. Much of what was uncovered about PTs’ personal identities in relation to the teacher education context came from artifacts written in conjunction with mathematics methods coursework. Such information may be used to provide targeted field placements, designed to support the ongoing process of identity construction. For example, whereas Sarah was able to enact an oppositional view in the student teaching context, Nicole might benefit from a placement that is more aligned with the teacher education context. In such a placement, Nicole can further solidify a personal identity that extends beyond compliance, without having to reconcile contrasting images of mathematics teaching and learning.

Finally, the artifacts collected to assess PTs’ competency alongside semi-structured interviews provided salient images of normative and personal identities. However, these same artifacts and mathematics methods coursework also indicated that each teacher held similar levels of competence for teaching mathematics. Again, it can be difficult to discern what is valued versus practices included out of compliance. Using Cobb, Gresalfi, and Hodge's (2009) framework for personal identities within the context of prospective teacher education afforded an analytic lens capable of teasing out issues of opposition, compliance, and value. Given shifts in teacher accountability associated with adoption of the Common Core State Standards for Mathematics (Common Core State Standards Initiative 2010), the advent of new accreditation standards for teacher education (e.g., Council for the Accreditation of Education Programs), and mechanisms for determining PT effectiveness (e.g., edTPA), an analytic lens that can uncover issues of compliance and value remains critical. While measuring what “counts” is challenging, it appears that teachers who can develop a critical stance toward teaching, willing and able to enact oppositional views, and with a particular eye toward students’ mathematical thinking, might be best positioned to enact reform-oriented mathematics instruction in elementary schools in substantive ways. Future studies that include an observational component that affords detailed analyses of practices that foster identification with the teacher education obligations may deepen and extend our findings here.

In addition, more studies that examine the detailed resources that contribute to the development of different kinds of personal identities in relation to mathematics teaching would be helpful in advancing our understanding. In considering the relationship between identity and learning, Nasir and Cooks (2009) presented a model on how learning settings provide resources for the development of identities in a practice-based setting. Their study focused on how members of a track and field team were offered and took up identities through resources made available to them in their practice setting. They highlighted three kinds of resources that contributed to the identity development of their participants. First, they defined material resources as the physical artifacts of a setting that support one’s sense of connection to the practice. Second, they defined relational resources as the positive relationships with others in the context that can increase connection to the practice. Third, ideational resources were defined as ideas about oneself and one’s place within the practice and the world, as well as ideas about what is valued or good. The researchers found that all types of resources contributed to their participants’ identities as track athletes as well as supported their learning in this setting. Effective teacher education must simultaneously consider what is taken up in teacher education as identity-making resources alongside what it means to do mathematics teaching in particular school contexts.

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Appendix

Appendix

Sample interview questions

  1. 1.

    (Pre-teaching only) Please tell me some of the most important things you thought about as you created your teacher work sample.

    1. (a)

      What made you think ____ was an important thing to consider?

    2. (b)

      (At intervals), please give me an example of how those thoughts about __ show up in your plans.

  2. 2.

    (Pre-teaching only) How did these ideas relate to your experiences in your courses?

    1. (a)

      Were there particular experiences in your math methods courses that are reflected in your teacher work sample?

  3. 3.

    (Pre-teaching only) Describe your assessment plan?

    1. (a)

      What most influenced your thoughts on your assessment plan?

  4. 4.

    Picture walking into a classroom and seeing the best teacher teaching mathematics.

    1. (a)

      What is s/he doing?

    2. (b)

      What are the students doing?

    3. (c)

      What kinds of materials and resources might this teacher/students be using?

    4. (d)

      What do you think this best teacher’s plans for instruction look like?

    5. (e)

      How do you think this best teacher assesses students?

  5. 5.

    What aspects of this “best” teacher do you think are reflected in your teacher work sample?

  6. 6.

    What aspects of this “best” teacher do you think are not reflected in your teacher work sample?

    1. (a)

      What keeps you from being able to do these things?

  7. 7.

    (Pre-teaching only) What role, if any, did the following resources play in planning instruction and assessment for your teacher work sample?

    1. (a)

      University supervisor

    2. (b)

      Other university faculty

    3. (c)

      Cooperating teacher

    4. (d)

      Other faculty in host school

    5. (e)

      Adopted textbook

    6. (f)

      Other print resources

    7. (g)

      State curriculum or other Standards

    8. (h)

      Research/literature in mathematics education

    9. (i)

      Internet searches

    10. (j)

      Other web-based resources

    11. (k)

      Materials from coursework (if yes, ask for specific courses and items)

    12. (l)

      Your own knowledge and experiences as a K-12 student

  8. 8.

    (Reserved for post-teaching interview protocol) Did you make any changes to the unit plan after you started teaching it?

    1. (a)

      If so, what prompted you to make those changes?

    2. (b)

      If so, what changes did you make?

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Hodges, T.E., Hodge, L.L. Unpacking personal identities for teaching mathematics within the context of prospective teacher education. J Math Teacher Educ 20, 101–118 (2017). https://doi.org/10.1007/s10857-015-9339-2

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Keywords

  • Prospective teacher education
  • Identity
  • Mathematics education