Becoming a Teacher Educator: A Self-Study of the Use of Inquiry in a Mathematics Methods Course

Abstract

This article details the self-study of a beginning teacher educator in her first experience in teaching a mathematics methods course. The transition from teacher to teacher educator is explored through the experience of a course focused on inquiry. Inquiry is embedded within the course from two perspectives: mathematical inquiry and teaching as inquiry. The framework of teaching as inquiry is an important part of both the self-study and the mathematics methods course. Implications of the transition from teacher to teacher educator as well as the role of inquiry in mathematics teacher education are discussed.

Keywords::

• teacher educator
• self-study
• mathematics
• inquiry

This self-study describes my transition from teacher to teacher educator. It details my experience and struggles in teaching a mathematics methods course for the first time. As a novice in the field of mathematics teacher education, I relied heavily on my own experiences as a teacher candidate and classroom teacher, as well as the breadth of literature I had read in the course of my graduate studies to design my instruction. I used the methodology of self-study to help me uncover my assumptions, challenge my beliefs, frame my practice, and understand who I am as a teacher educator. Self-study became my vehicle for examining myself in my new role and understanding the teacher candidates' experiences in my methods course.

Based on my experiences as a classroom teacher and the research literature I read as a graduate student, I wanted to begin my career as a teacher educator “teaching and researching practice in order to better understand: [my]self; teaching; learning; and the development of knowledge about these” (Loughran, 2004, p. 9). I also wanted to clearly articulate and model that goal for teacher candidates. Like Munby and Russell (1992) and Nicol (2011), I wondered if it were possible to explicitly encourage teacher candidates to learn from experience and to inspire lifelong experiential learning with a specific focus on teacher inquiry as a way of building knowledge for teaching. I also wanted to engage in that kind of learning for myself. Self-study provided a systematic means to explore my beliefs about teaching mathematics and learning to teach while I transitioned from teacher to teacher educator.

During this transition, I explored how my beliefs about mathematics teacher education influenced the work of planning and teaching a course for the first time. In designing my mathematics methods course, I was guided by the belief that inquiry, defined by a variety of perspectives, could be a lens through which teacher candidates learn about and develop the skills required to deliver high-quality mathematics instruction. With this focus, I set out to answer the following questions:

• What can I learn about mathematics teacher education and about myself as a teacher educator through self-study of my transition from teacher to teacher educator?

• How can inquiry frame a mathematics methods course for preservice elementary teachers?

I begin this account of my self-study with a discussion of two types of inquiry – teaching as inquiry and mathematical inquiry – and I discuss the ways in which the two forms of inquiry are related for K-12 mathematics and teacher education. Next, I describe the self-study methods and data analysis techniques. In discussing my findings, I present the three major themes that I identified: transition, positionality, and tensions. To conclude, I discuss the implications of this work.

Conceptual Framework

Experiences in graduate-level coursework, as a classroom teacher, and in educational research inform my beliefs about mathematics teacher education and the process of learning to teach. Being both a student and a teacher in the USA, my experiences, education, and frames of reference are influenced by my geography. I agree with the philosophy of the National Council of Teachers of Mathematics (NCTM, 2000) that good teachers are lifelong learners who possess mathematical content knowledge and pedagogical content knowledge, as well as the knowledge and skills to approach their teaching in ways that are reflective, responsive, and flexible (Ball, Thames, & Phelps, 2008; NCTM, 2000; Nicol, 1997). The integration of different frameworks of inquiry into the methods course follows from my beliefs about mathematics teacher education.

The term inquiry is commonly used and widely interpreted in education. At their roots, the many interpretations of the term are guided by the principal definition that inquiry is an act of seeking or building knowledge. Inquiry-based learning is rooted in constructivist theories, which suggest that people learn best when they are actively engaged with subject matter, as opposed to learning being a transmission of knowledge from teacher to passive student (von Glaserfeld, 1991). This work is framed by two perspectives on inquiry: teaching as inquiry (Nicol, 1997) and mathematical inquiry (Richards, 1991). The following section details the definitions of each of these perspectives, the relationship between them, and their relevance to this research.

Teaching as Inquiry

Preservice teachers often view teaching as a technical endeavor because they have not yet experienced the “conscious decision making, deliberating, managing of dilemmas, and reflecting” (Nicol, 1997, p. 97) that teachers do on a daily basis. These novices lack an understanding of the complexity of good teaching and often believe that, “if they plan well, if they love children and speak clearly, if they go over what they want to do in their heads before they go into the classroom, everything will turn out fine” (Wineburg, 1991, p. 277). Wineburg (1991) argues that when this ideal scenario fails, beginning teachers are unequipped to learn from experience to build their knowledge for teaching. Preservice teachers need opportunities that facilitate “learning from teaching” in order to develop the skills for lifelong reflection and knowledge building (Cochran-Smith & Lytle, 2009).

Teaching as inquiry (Nicol, 1997) is a constructivist model of mathematics teacher education that addresses the importance of reflection and continuous learning in teaching. In this model, the teacher educator works from a perspective of teaching as inquiry and models this in ways parallel to mathematical inquiry, “the pedagogy of mathematics instruction envisioned in reform documents” (p. 97). Nicol's framework engages both teacher educators and preservice teachers in an exploration of what is desirable and possible in mathematics teaching and learning to teach.

When teaching as inquiry guides teacher education, the philosophy and methods of mathematical inquiry are woven together with the philosophy and methods of teacher research, highlighting the ways in which they complement and enhance one another. Teacher education coursework based on teaching as inquiry engages teacher candidates in, “learning about and participating in an inquiry into their own understandings of mathematics and students' understanding of the mathematics as well as the discipline of mathematics and the teaching of mathematics” (Nicol, 1997, pp. 96–97). As teacher candidates learn about how children come to know and understand mathematics, they develop their own understanding of mathematical content and pedagogy, developing a reflective stance toward teaching.

Through teacher education practice that “authentically represents the nature of teaching” (Nicol, 1997, p. 97), teacher educators can help preservice teachers develop what Cochran-Smith and Lytle (2009) refer to as an “inquiry stance” toward teaching. When teachers take an inquiry stance toward their work, the result is “enriched learning opportunities for preK-12 students” (Cochran-Smith, 2006, p. xii) and the evolution of a knowledge base for teaching (Cochran-Smith & Lytle, 1993). Teaching as inquiry (Nicol, 1997) reflects the importance of reflective teaching, the role of inquiry in learning to teach, and the importance of inquiry in the teaching and learning of mathematics.

Mathematical Inquiry

Inquiry-based learning is rooted in constructivist theories that suggest that people learn best when they are actively engaged with subject matter (von Glaserfeld, 1991). Mathematical inquiry is inquiry-based teaching and learning specifically related to the development of mathematical knowledge and understanding (Richards, 1991; von Glaserfeld, 1991). It is an approach to mathematics education in which students learn by engaging in mathematical discussions, listening to mathematical arguments, proposing conjectures, asking mathematical questions, and solving unfamiliar problems (Richards, 1991). It is the vision of mathematics instruction outlined by the reform documents (Nicol, 1997).

In the USA, mathematical inquiry differs from traditional mathematics instruction that often seems to be focused on procedures and memorization. Mathematical inquiry focuses on building knowledge and deep conceptual understanding through carefully selected tasks chosen by the teacher (NCTM, 2000). The goal is to “develop a repertoire of general heuristics and approaches that can be applied in many different situations” by engaging in habits such as looking for patterns, experimenting, describing one's work, visualizing, conjecturing, guessing, thinking about the big picture, thinking about the specific details of a particular case, seeing multiple points of view, using mathematical language, and utilizing inductive and deductive reasoning (Cuoco, Goldenberg, & Mark, 1996, p. 378). Teaching and learning through mathematical inquiry requires instruction that incorporates:

the use of mathematics to solve problems; the application of logical reasoning to justify procedures and solutions; involvement in the design and analysis of multiple representations to learn, make connections among and communicate about the ideas within and outside of mathematics. (NCTM, 2008, p. 1)

These processes facilitate the development of conceptual understanding and, “provide students with a connected, coherent, ever expanding body of mathematical knowledge and ways of thinking” (NCTM, 2008, p. 1). “Without facility with these critical processes, a student's mathematical knowledge is likely to be fragile and limited in its usefulness” (NCTM, 2008, p. 2). The framework of mathematical inquiry aligns with the NCTM (2000) Process Standards and the Standards for Mathematical Practice in the Common Core State Standards Initiative (2010).

Mathematical Knowledge for Teaching

Teachers who engage students in mathematical inquiry and engage themselves in inquiry about their teaching are working at the intersection of teaching as inquiry and mathematical inquiry. These teachers must understand the mathematics that students know and need to know, draw flexibly on their mathematical knowledge in their teaching, create and select experiences and tasks that help students learn, be thoughtful and skillful in their instructional decisions and choices of assessment, be committed to their students, and possess the skills to challenge and support students in their learning (NCTM, 1991, 2000). In addition, effective teaching involves reflection and continued self-improvement (NCTM, 2000). Effective mathematics teachers are always working to improve their instruction and to understand the importance of mathematical inquiry (Richards, 1991) and teaching as inquiry (Nicol, 1997) in their daily work.

Inquiry in the mathematics classroom provides children with tools to develop a deep understanding of mathematical concepts and provides teachers with tools to understand and manage the complexity of teaching the subject well (Cochran-Smith, 2006). In order to create a classroom environment in which students engage in mathematical inquiry, teachers need a special type of knowledge: Mathematical Knowledge for Teaching (MKT; Ball et al., 2008). Parallels can be drawn between the knowledge development encouraged through teaching as inquiry (Nicol, 1997) and the specific types of MKT (Ball et al., 2008). MKT is a multifaceted knowledge framework that organizes mathematical content knowledge and pedagogical content knowledge into six knowledge types: common content knowledge, specialized content knowledge, horizon content knowledge, knowledge of content and students, knowledge of content and teaching, and knowledge of content and curriculum (Ball et al., 2008). Teachers use this knowledge as they design instructional activities fostering mathematical inquiry and opportunities for the development of conceptual understanding.

This self-study was guided by teaching as inquiry (Nicol, 1997) and mathematical inquiry (Richards, 1991) and informed by the framework of MKT (Ball et al., 2008). Teaching as inquiry provided a structure for me, as a teacher educator, to meld mathematics and teacher inquiry and to construct both a course and a self-study. Mathematical inquiry was used to engage teacher candidates with mathematical content. The frameworks of teaching as inquiry and mathematical inquiry framed the methods used to conduct this self-study and to analyze results.

Methodology

Many mathematics teacher educators have engaged in self-study as a means of exploring, understanding, and improving their own practice and the field of mathematics education (Schuck & Pereira, 2011). This research was designed to explore my own transition from teacher to teacher educator. Inquiry became a major theme of my work as I examined how mathematical inquiry and teaching as inquiry could be used to frame both my experiences of transition and my teaching of a mathematics methods course.

Why Self-Study?

Self-study is an opportunity to engage in a, “critical examination of the self's involvement both in aspects of the study and in the phenomenon under study” (Kosnik & Beck, 2008, p. 117). This methodology provided me with a means to engage in both teaching and research within the framework of teaching as inquiry (Nicol, 1997). I was “learning from teaching” by taking an inquiry stance toward my work as a teacher educator (Cochran-Smith & Lytle, 1993, 2009; Nicol, 1997). Like Kosnik and Beck (2008), I found that as I engaged in inquiry, my discoveries changed both my teaching and me. Self-study afforded me the space to discover myself as a teacher educator and introduced me to the tools that will help me continue to study, change, and grow throughout the course of my career in teacher education. In my methods course, I hoped to provide the same space and tools for the teacher candidates as they discovered themselves as teachers.

Course Design

A faculty member and I designed a course syllabus with inquiry as a stated course objective. Our goal was to design a course that would help teacher candidates develop an approach to teaching mathematics that engaged them in career-long learning. Through teaching as inquiry (Nicol, 1997), we wanted students to develop an inquiry stance (Cochran-Smith & Lytle, 2009) toward teaching wherein they learned to build knowledge for teaching throughout their career instead of striving to obtain it all in one semester of a methods course. We articulated this goal to students with the following course objective: “Take an inquiry stance toward the teaching of mathematics, which means studying one's own practice in order to build local and global knowledge through examination and reflection of the events happening in your own classroom” (course syllabus).

Informed by the frameworks of teaching as inquiry, mathematical inquiry, and MKT, we compiled a set of assignments, readings, and in-class activities that engaged teacher candidates in explorations of mathematical concepts, pedagogical issues, and the “thinking practice” (Lampert, 1998) of teaching. Texts were carefully selected to address certain mathematical and pedagogical ideas while honoring the complexity and emotions of teaching and, in many cases, highlighting examples of teachers taking an inquiry stance toward their work. Several course activities modeled some type of inquiry, including “teacher voice” readings (real accounts of teaching written by classroom teachers), course assignments, exit cards from each course session, and teacher candidates' personal reflections written after each assignment.

Assignments were constructed to engage teacher candidates in real activities of teaching, such as writing and delivering a lesson, looking at student work, evaluating commercial curriculum materials, and building a classroom website. In addition, they were required to analyze and ask questions about the work they did in their practicum placements. They considered questions such as the following. How would I teach that lesson differently in the future? What does this particular child understand and not understand about subtraction? How do the activities and lessons in this textbook meet the needs of my students? And how can I communicate my philosophy and teaching style to parents through a classroom website? With notes, audio recordings, and the time and space to reflect, teacher candidates were able to “stop time” and truly investigate what transpired in their work throughout the course (Ballenger, 2009). They had data to help them answer their questions about themselves as teachers and the work of teaching. They also had a framework for asking new questions in the future.

Data Collection and Analysis

I employed qualitative methods of data collection and analysis within the larger methodology of self-study. Data were drawn from students' course assignments, reflections, and other class artifacts. In addition, I kept a journal where I documented my own thinking during the course. The study was conducted over the course of two years, including a spring and summer of planning the course, a semester of teaching the course, and several semesters of reflection and data analysis.

A total of 14 teacher candidates (13 female, 1 male) in the methods course consented to have their work used for this research. In the months immediately following the completion of the course, two participants agreed to be interviewed about their experiences. All 14 participants were pursuing licensure in elementary education and/or special education in M.Ed. (13 participants) or B.A. (1 participant) programs and taking the methods course as a degree program requirement. A total of 11 participants were engaged in a practicum experience or working in a classroom setting in some capacity while taking the course.

Both the M.Ed. and B.A. programs require students to take a variety of courses in pedagogical methods, including the mathematics methods course. In addition to methods courses, students in both programs are required to take courses on applied child development and social contexts of education, as well as inquiry seminars and practicum experiences. Most participants in a M.Ed. program were in their first semester of coursework and taking other methods courses during the same semester as the mathematics methods course. The undergraduate was in the third year of coursework.

Data for this study are drawn from course syllabi, pre-semester surveys of participants' attitudes and beliefs about mathematics and teaching, mid-semester course surveys, participants' assignments and reflections on assignments, exit cards from each class session, and a personal teaching journal that I kept throughout the planning and teaching of the course. Mid-course surveys were completed anonymously, but all other data, including pre-course surveys, assignments, reflections on assignments, and exit cards, were not. Prior to data analysis, all participants were given nonidentifying codes and all work was coded to protect students' identity.

I used provisional and open coding to analyze the data (Saldaña, 2009). Provisional coding establishes and utilizes a predetermined set of codes to analyze data. In this case, the codes of observing, questioning, and acting were chosen because they are representative of the conditions of action research (Carr & Kemmis, 1986), the genre within which teacher research is most often categorized. Cochran-Smith and Lytle (2009) cite a clear relationship between an inquiry stance and a continuous cycle of “observing, questioning, and acting” (Carr & Kemmis, 1986). As a result, the parts of the cycle were used in the coding scheme as a means of coding when teacher candidates were engaged in inquiry during their experiences in the course. For example, when a teacher candidate described differences in the way she learned mathematics and the way she was learning to teach mathematics, this was coded as observing. When participants asked questions about how to handle situations that confused them in their practicum experiences, this was coded as questioning, and when participants described the teaching moves they made in response to questions they had, these were coded as acting.

As I coded the syllabus and course assignments, I noted the opportunities that teacher candidates had to take part in the actions of “questioning, observing, and acting” and the evidence that they engaged in and acknowledged the importance of such actions in the work of teaching. Such evidence included instances where teacher candidates posed questions specifically related to their teaching experiences, made relevant observations related to those questions, and discussed their actions and the corresponding actions of students in response to those questions and observations. In course assignments and reflections, teacher candidates often discussed what they saw in practicum and other classroom experiences and posed thoughtful questions with follow-up action. I also examined my comments to teacher candidates on these assignments and the ways in which I highlighted their engagement in teaching as inquiry and the development of their inquiry stance toward teaching. In addition, I looked for instances in which I attempted to push them forward in their thinking and their actions.

In pre- and mid-course surveys, I looked for changes in participants' beliefs and attitudes toward teaching, specifically noting references to inquiry. In my own journal, I looked for evidence that I was truly engaged in teaching as inquiry as I posed my own questions, made my own observations, and recorded my own actions. In the mining of these data, I sought to better understand teacher candidates' experiences with and perspectives of the course and its goals and objectives related to inquiry. I explored how well my objectives for their long-term learning came through in the course, in their work, and in my work as I explored whether or not I “practiced what I preached” (Loughran, 2004, p. 10) in regard to both mathematical inquiry and teaching as inquiry.

Finally, I used my journal as a record of my work as a new teacher educator. I examined my own cycle of “questioning, observing, and acting” (Carr & Kemmis, 1986) through reflections on the course. I used the syllabus as a record of my values as I started teaching the course and analyzed the ways in which I did or did not express those values to my students through my instruction. In reviewing and coding my pre-course research and design notes, I was able to analyze my practice and how I lived the values expressed in the syllabus in my teaching. Reexamining my goals during the planning of the course and the writing of the syllabus provided an opportunity to improve my work as well as develop knowledge about mathematics teacher education. I also explored the tensions between my goals for the course and teacher candidates' goals as expressed in their coursework and reflections. I mined their reflections and my personal journal for instances of this chasm, coding each instance as a tension.

Findings

I set out to learn two things from this self-study: (1) what it means to be a part of the field of mathematics teacher education and who I am as a teacher educator; and (2) how inquiry can be used to frame a mathematics methods course. These findings are organized around three themes: transition, positionality, and tensions.

Transition

The methods course facilitated an important transition both for the teacher candidates and for me. Without it, we would have had neither the experiences nor the opportunities for reflection necessary to tackle our new roles. We built knowledge and learned about our new roles by engaging in inquiry. Personally, it was teaching as inquiry and the methodology of self-study that helped me make sense of my transition. The teacher candidates acknowledged mathematical inquiry and teacher inquiry as the means to learning throughout their transition. In many ways, our experiences ran parallel to one another.

The transition from teacher to teacher educator played a major role in my experience of designing and teaching the course and conducting this self-study. The research became a way for me to uncover and work through the tensions I was experiencing in this transition. Like Dinkelman, Margolis, and Sikkenga (2006), I found that much of the time I spent in graduate-level coursework and then planning and teaching the methods course forced me to confront the nagging question at the back of my mind: “Why did I leave teaching?” (p. 13). The methods course became an avenue to identify and address what was missing in my work as a classroom teacher and what I believed teacher candidates needed from teacher education. As I engaged in teaching as inquiry, I frequently circled back to the question, “If I could go back and be a pre-service teacher again, what would I want to know about teaching and learning?” I tried to incorporate the answers to those questions into the methods course.

I found that many of the teacher candidates were experiencing similar tensions and asking questions about their experiences as students of mathematics as they engaged in mathematical inquiry. They often asked, “Why didn't I learn about math this way when I was in elementary school?” (classroom discussion). This was a question I saw as similar to my own musings about why I did not learn to teach “this way,” with an inquiry stance toward my work in the classroom. As teacher candidates completed assignments and reflections, they often discussed their experiences of learning mathematics as students and how different they were from the ways they were engaging with mathematics as they transitioned to the role of teacher. In a parallel experience, I was asking questions about my experiences as a classroom teacher and how they were different from the ways I was engaging with teaching and learning to teach as a teacher educator.

Throughout the course, the teacher candidates and I explored mathematical content by engaging in mathematical inquiry to build knowledge. This resulted in many a-ha moments for teacher candidates in their understanding of the content of elementary school mathematics and effective ways to teach that content. They exhibited a change in their orientation toward mathematics teaching and learning, noting that their engagement with the content in the methods course showed them what “math is made up of” (mid-semester survey) and how it was preparing them to teach that mathematics to children. One teacher candidate noted, “I think ‘doing math’ in class helps us to recognize different situations that might happen” (mid-semester survey). This comment and others like it suggest a change in teacher candidates' approach to teaching mathematics, belief in the effectiveness of mathematical inquiry, and a growing understanding that knowledge can be built through experience. Similarly, I was finding that engagement in teaching as inquiry was changing my orientation toward the teaching methods that were most effective for teacher candidates in methods coursework.

The transition from student to teacher had a significant impact on teacher candidates' experiences in the course. Some students cited their inexperience with teaching in their reflections and the impact of that inexperience on their work in the course. Alison (all names are pseudonyms), a first-year graduate student with no prior teaching experience or education coursework, stated that “maybe it's just because I'm still a student, but I haven't been able to reflect as much on the math that they're doing” (student interview). She went on to discuss feelings of uncertainty and acknowledged that her coursework and practicum were important parts of her transition to teacher, but also said that at times she did not “even know what questions to ask” (student interview) to help herself along that journey. Alison felt that her inexperience in the field of teaching sometimes prevented her from taking full advantage of those opportunities to best prepare herself for the role of teacher.

Brenda, a first-semester graduate student, also spoke at length about her transition to teacher. Having worked as an instructional aide in a special education classroom, Brenda had some experience in the classroom but discussed feelings of change and transition into the role of teacher and the impact of coursework on that transition. She wrote:

I think I definitely started to look at how to teach math differently to kids than I was doing before. And it also helped me in other subjects too I think. Or just as one of my first classes for my grad school, like it's definitely helpful to get us going to thinking as a teacher … I think before I just was doing what I was supposed to and then I just started to work in my school like as if I was actually a teacher. (Student interview)

With Brenda's transition to teacher came feelings of greater responsibility and greater impact. Talking about her work as an instructional aide, she commented:

Before I started actually going to grad school … I was just going through the motions, I didn't think of myself as a teacher really … but now I think of myself as more important, as making more of a difference. (Student interview)

Despite still being months or years from the completion of the transition to teacher, Brenda noted that the course was influencing her change. She adds that, “everything played a small role in me being more comfortable with the idea of teaching and then with my responsibilities as a teacher” (student interview).

Like Brenda, I saw my own experiences with the course changing my approach to my new role. Before I taught the course, I, like Brenda, did not think of myself as a teacher educator. Despite coursework preparing me for the role, I did not see myself having the expertise of a teacher educator. Thus, my experiences in the course, both positive and negative, helped me to see myself differently. When teacher candidates treated me like an expert, I began to see myself making a difference in their transition. Teacher candidates came to me to ask questions about the mathematics being taught in their practicum, sought advice on how to help a particular student, and challenged information presented in class with their own experiences. Through my responses, I began to see myself transitioning to teacher educator. The role of expert felt uncomfortable and I struggled with who I was throughout the work of this self-study. Herr and Anderson's (2005) continuum of positionality helped me to examine my positionality and make sense of my changing place in teacher education.

Positionality

In my transition to teacher educator, I struggled to define my positionality. I experienced my transition from teacher to teacher educator as both an insider and an outsider. As I examined my own positionality in the analysis of my data, I saw a shifting perspective and place for myself within the research. I was in-between in so many realms. I was “in the process of becoming” (Dinkelman et al., 2006, p. 6), no longer a teacher, but not quite a teacher educator. I could not identify a neat position for myself. In thinking about this after the course, I found, as Herr and Anderson (2005) explain, that “one's positionality doesn't fall out in neat categories and might even shift during the study” (p. 32). Identifying and exploring my positionality helped me to make sense of my experiences upon completion of the course. It helped me to begin to understand the tension and transition that I experienced and informs who I am becoming as a teacher educator.

At the beginning of the course, I identified with the concept of being an “outsider within” (Herr & Anderson, 2005), an idea I had read about in my doctoral coursework. Herr and Anderson (2005) describe an “outsider within” as an observer in the margins with a unique vantage point on a situation. This description seemed to fit, as I saw myself very much on the margins of teacher education but with a unique vantage point because of my role as a teacher and graduate student.

As I dug deeper into my experiences, I began to see myself as more of an insider than I had originally perceived myself to be. Despite my status as a graduate student, to the teacher candidates, I was the professor. I was one of many doctoral students teaching their courses, and they appeared to see me much as they saw full-time faculty members. Despite their knowing that I was a student, like them, when I began class and gave the instructions I became the professor. I felt this tension of being both within and outside the graduate student community. I felt, as Herr and Anderson (2005) describe, how “each of us as researchers occupies multiple positions that intersect and may bring us into conflicting allegiances or alliances within our research sites” (p. 44). The management of these tensions influenced the instructional decisions I made, such as when to end a class early to accommodate the busy student schedule or be flexible with assignment requirements and due dates. This tension of being both inside and outside of the student community shaped my experiences in teaching the course.

Further examination of my positionality showed me that I was, in fact, also more of an insider in teacher education than I had initially realized. In addition to the knowledge I gained in my graduate-level coursework, I had observed and been a teaching assistant in multiple sections of the mathematics methods courses at the university. The coursework I had done gave me perspective on the field of teacher education and the practices used in teacher formation. I was not, as I had originally thought, a true outsider to the teacher education community, as I had considerable prior experience and education that informed my construction of the reality of mathematics teacher education. Upon reflection, I believe that many would have categorized me as an insider in teacher education, despite the fact that I may not have categorized myself in that way. This is an important realization about positionality, that often we occupy positions that could categorize us in one way while we identify with another (Herr & Anderson, 2005).

To make sense of my experience in teaching the course, I spent a great deal of time wrestling with the issue of my positionality. In the end, I believe that this tension influenced my development as a teacher educator. I choose to use the specialized knowledge I have because of my insider and outsider statuses to foster organizational learning, building knowledge for myself and my practice as a teacher educator.

Wrestling with the tension of shifting positionality taught me much about my journey from teacher to teacher educator. Upon exploring the experiences of teacher candidates in the course, I found that many of my students reported similar feelings of being in-between, and they also reported struggles with positionality in the K-12 classroom. They were not students in the same sense as their pupils and not teachers like their mentors. I realized that we were all uncomfortable in our new positions. In addition, analysis of my students' experiences in the course revealed another set of tensions centered on the tension between the goals of teacher education identified by teacher candidates and those identified by teacher educators.

Tensions

In my own experiences and in the research literature, there is evidence of a variety of tensions between the short-term goals of teacher candidates and the long-term goals of teacher educators. In my doctoral coursework, I read about how different teacher educators address these tensions and was strongly influenced by the work of Chin (1997) and Ball, Sleep, Boerst, and Bass (2009). Chin (1997) discussed the tension between immediate needs of teacher candidates and long-term objectives of teacher education. He has long-term goals for his students, goals that focus on professional growth through reflection that he balances with students' goals for the course. Ball et al. (2009) used the MKT framework to build and teach a methods course that seeks to begin to develop preservice teachers' MKT in a course while giving them the tools to continue to develop the knowledge over time. Both had a vision of learning beyond a one-semester methods course and acknowledged the challenges of balancing the tension between teacher candidates' short-term objectives and the long-term goals of mathematics teacher education.

Teacher educators in many fields document preservice teachers' short-term objectives for teacher education. Teacher candidates desire tips and knowledge they can apply immediately in practicum settings as opposed to skills that would help them develop a broader body of knowledge over time (Chin, 1997; Wineburg, 1991). I identified with this need for tips and tricks and recognized it as an attitude that I had toward coursework when I was a teacher candidate. In addition, in my transition to teacher educator, I desired some of those same shortcuts that I could use in my teaching at the university level. As I read the work of Chin (1997) and Ball et al. (2009), their shared goal of preparing teacher candidates to be reflective and engage in a lifetime of professional growth influenced my own goals and objectives for my course. I saw that reflective practice and the pursuit of continuous professional growth were what had been missing in my own teaching and what I wanted to be a part of my teacher candidates' work in classrooms from the very beginning.

The tensions between teacher candidates' goals for the methods course and the goals of teacher education were obvious in my work. I found myself in a constant, semester-long struggle with the tension between giving teacher candidates what I believed they needed and what they wanted from my course and from their teacher education program as a whole. Teacher candidates were often discouraged by assignments and class activities that did not fit their own definitions of useful experiences. They were looking for shortcuts – activities and lessons they could implement right away – as evidenced in their reflections and on their exit cards at the end of class each week. In my journal, I found myself asking, “Is it more fruitful for me to worry about what they get out of it, what they think of it, or the value I see in it?” (personal journal). As I reexamined my journal, I found my attitude toward teacher candidates and teacher education changing over time. After some particularly tough class sessions where teacher candidates did not see the value in course activities and complained loudly about it, I wrote that “I see students so much more as ‘customers’ than I ever did before” (personal journal). I felt pressure to give them what they wanted in order to keep them engaged in the course and happy with their experience in the teacher education program, but this pressure was at odds with my long-term objectives for the course and what I knew to be important for their future work as teachers.

A striking example of the tensions between their course objectives and my goals involves the tension that arose around the ways I grouped teacher candidates during class. From the beginning of the course, teacher candidates were grouped according to the grade levels in which they were student teaching, including a group focused on special education. In designing the course, I felt that if teacher candidates focused their work on a specific grade level, then they might be able to dig deeper into student work, course readings, and practicum experiences. I envisioned these groups as modeling an experience they would likely have in their future work, where they would be with colleagues on a grade-level team. The teacher candidates did not see the grade-level groups in this way. When I asked for feedback on the course, some teacher candidates talked about not being able to relate to other group members and having difficulty in building relationships. Others reported that they preferred not to work with the same group each week and felt stuck with certain classmates. In addition, they did not share their experiences or question one another in the ways I had envisioned when I assigned them to groups. Perhaps this is evidence that teacher candidates are, as Chin (1997) suggests, so worried about themselves and their own work that they are not ready to think about acting together as a group to do the best teaching. It is possible that anxiety and low levels of confidence prevented students from feeling comfortable with working together in a group. Tasks that teacher candidates did report as useful were the reading of case studies and the viewing of video of real classroom teaching. In these instances, teacher candidates engaged in discussion around the common experience of reading or viewing the same classroom vignette.

Teacher candidates' dissatisfaction with grade-level groups and their enjoyment of reading case studies of classroom scenarios and watching video of pupils and teachers in real classrooms could suggest that I need to find common experiences that could be discussed by the whole class rather than forced sharing of individual experiences in assigned groups (Wineburg, 1991). When they read classroom scenarios or watched videos, the teacher candidates had a common experience to share and discuss as opposed to reporting back to their group on events and scenarios that happened in their student teaching or work placements. These common experiences eliminated the anxiety that could accompany reporting back on one's own teaching and moved the focus from the teacher candidates as the unit of study onto an unknown teacher and group of students.

Inquiry in the Methods Course

Despite the tension between my goals for the course and teacher candidates' objectives, many of them did report seeing value in inquiry, which was a main component of the course. I attempted to model how taking an inquiry stance toward one's teaching can build knowledge about teaching. By the end of the course, many teacher candidates reported believing in the value of opportunities to engage in both mathematical inquiry and teacher inquiry.

At the beginning of the semester, most teacher candidates' goals for the course were to “learn everything about teaching math” (pre-course survey). Instead of attempting to teach everything in one course, which was an unrealistic goal, I sought to help teacher candidates develop the skills and attitudes necessary to take an inquiry stance toward their teaching and to continue to build knowledge over time, like Ball et al. (2009). Like Alderton (2008), my goal was “to prepare student teachers who are reflective practitioners and who engage in the construction of teacher knowledge” (p. 98). With this in mind, I assigned readings and tasks that promoted this long-term goal.

Through their reflections and discussions, teacher candidates demonstrated an awareness of the value of inquiry-based learning. One teacher candidate, Phillip, noted the importance of learning through experience, saying that teaching “happens in the classroom and with students every day” (student reflection). He felt that the course assignments, which mimicked the real work of teaching, were important because they highlighted the work of teaching that textbooks could not capture. Phillip demonstrated an understanding that the assignments allowed him to learn through real or simulated teaching experiences, and he saw those as more meaningful than reading about teaching in a textbook. Brenda noted the importance of the required reflections as she talked about how they changed the impact of an assignment. In her interview, she spoke about writing reflections: “I did enjoy it because, usually I finish an assignment and put it away and don't think about like how it was to actually do it.” This course requirement showed her how experience can build knowledge.

Over the course of the semester, the teacher candidates demonstrated a growing belief in the value of inquiry in teaching and made consistent mention of the acts of “observing, questioning, and acting” (Carr & Kemmis, 1986) as important aspects of teaching. One teacher candidate, Susan, provided the following insights after analyzing student work for an assignment:

The ability to step back, review, and analyze his work and the thinking behind it provided valuable insight into his strengths and weaknesses. I think that this is an important process for teachers to employ in their classrooms in order to get a more holistic perspective of not only their students but of themselves as teachers. (Student reflection)

Comments like Susan's show an understanding that there is great value in learning through the work of teaching and development of knowledge through experience.

While teacher candidates saw the value in approaching teaching with an inquiry stance and an ability to look at teaching as an experiment, they were less comfortable with the possibility that the experiment might not work. They frequently discussed their fears about what to do if a lesson or activity failed, without appearing to have a Plan B in many cases. They showed, as Wineburg (1991) noted, an inability to learn from failures. They frequently cited learning from experiences that went well (as they had planned) and not knowing what to make of experiences that had not gone well. This was the source of many questions from students about how to prepare a Plan B. I encouraged teacher candidates to engage in inquiry as a means of exploring those what-ifs and developing their knowledge in order to continuously improve their teaching over time.

While my findings did not provide me with all the answers, what I learned as a result of this self-study contributed to my own knowledge for teaching. Self-study enabled me to approach the transition to teacher educator as a learning opportunity. In doing so, I recognized the parallels between my transition and that of the teacher candidates in my course, and that recognition will inform my future work in teacher education. Through the exploration of my positionality (Herr & Anderson, 2005), I am better prepared for the next course that I teach. Seeing myself as both an insider and an outsider reveals the assumptions and beliefs associated with the multiple positions I occupy and informs my instructional decisions and how I address them in my work. The opportunity to recognize and work through the many tensions in teacher education will certainly benefit future teacher candidates in my courses.

Conclusions and Implications

For researchers, policy-makers, teacher educators, and teachers, the call to reexamine the mathematical education of our nation's teachers is hardly new. Prior to the passage of the No Child Left Behind Act, researchers and teacher educators such as Ball (2000) and Lampert (1998) acknowledged that one course on teaching mathematics was insufficient but often the norm in elementary teacher education programs; they also called upon teacher educators to help students explore more than just content and pedagogy in their courses. Ball (2000) urged researchers to look not only at the content teachers know and need to know but also at how they understand and know the content and what they can do with it in the classroom. The NCTM (2000) promotes the value of teachers engaged in inquiry in the Teaching Principle in their statement of Principles and Standards for School Mathematics; they assert that “using appropriate instructional tools and techniques, and engaging in reflective practice and continuous self-improvement are actions good teachers take every day.” In order to support these tenets of good teaching, mathematics methods courses need to provide opportunities for preservice teachers to practice these skills and develop the tools to do mathematics, to understand how children engage with the mathematics, to teach the mathematics effectively, and to engage in reflective practice and teacher research that builds knowledge and facilitates self-improvement and student achievement.

Nicol's (1997) framework of teaching as inquiry and the idea that learning to teach is a career-long journey are needed in teacher education. Long-term goals related to reflection, career-long learning, and professional growth were what I felt were missing in the courses I had taken as a teacher candidate, observed as a graduate student, and worked in as a teaching assistant. This tension between the short-term goals of teacher candidates and the long-term goals of the faculty was striking, but in my reading I sensed that a compromise was being sought in the field. Inquiry in its many forms could be this compromise. Inquiry as a framework for the mathematics methods course could provide teacher candidates with “a broad range of opportunities to think about the broad spectrum of their work” (Richert, 1992, p. 187), coursework focused on the complexities of the work of teaching, and a long-term view of building teacher knowledge.

A similar tension might also be noted in the formation of teacher educators. Reflection, career-long learning, and professional growth are as important for teacher educators as they are for K-12 teachers. Self-study was a powerful experience for me as a teacher educator and a means to explore my teaching practice at the university level. Today, the exploration of one's own practice is not commonplace in higher education. This self-study suggests to me that reflection, inquiry, and self-study of teaching practices should be the norm, not the exception, in schools of education.

The integration of different forms of inquiry into methods courses is an attempt to better prepare teachers for the work of teaching mathematics. Teachers need to know how to approach their teaching in a way that is reflective, responsive, and flexible. Rather than learning a set of specific methods and practices for teaching, teachers need to be able to assess children's needs and develop and try new ways to improve learning. They need to take an inquiry stance toward their teaching. Through mathematical inquiry, teachers and pupils can deepen their understanding of mathematical content. Through teacher research, teachers can expand their ability to meet the needs of all pupils. They will be able to assess and respond to pupils' needs based on their observations, questioning, and reflection. They will have tools that allow them to grow and develop as professionals over their careers instead of specific methods and activities that may work for only some pupils some of the time. Through teaching as inquiry, teacher educators must model this practice for teachers.

If a goal of teacher education is to help teachers learn to approach their teaching in ways that are reflective, responsive, and flexible, what does that mean for teacher education? What do teacher educators need to know in order to achieve that goal? How do we train teacher educators to prepare them for this work? These questions must be addressed in future research. This self-study makes public the knowledge built and created through experience, including my experiences as a new teacher educator and the experiences of my teacher candidates as they explored mathematical content through a lens of mathematical inquiry and began to engage in the craft of teaching with assignments and coursework that modeled and encouraged an inquiry stance. This work adds to the knowledge of the work of teacher educators engaged in self-study and teachers engaged in teacher research, promoting “the questions of knowledge creation and the extension of discourse beyond the local” (Dalmau & Gudjonsdottir, 2000, p. 49), which are important for both K-12 mathematics education and mathematics teacher education.