
Moving Beyond Sinking or Swimming: Reconceptualizing the Needs of Beginning Mathematics Teachersby Marcy B. Wood, Lisa M. Jilk & Lynn Webster Paine  2012 Background/Context: New teacher induction programs frequently focus on the struggles of teachers who are, metaphorically speaking, sinking rather than swimming in the challenging waters of actual classroom teaching. Also, research on induction and induction programs doesn’t typically address the subjectspecific needs of new teachers. Yet new teachers are not in generic environments in which content plays a minor role: They are negotiating the enormously complex context of teaching particular content to particular students in a particular moment. Purpose/Objective/Research Question/Focus of Study: Our goal is to add to and challenge the conversations about what learning to teach mathematics requires and how its complexity makes contentspecific induction and rich opportunities to learn not only desirable but also essential. We use a framing of teaching as a practice that is fundamentally about connection of students and content. Population/Participants/Subjects: We report on the cases of two wellstarted novice mathematics teachers. These cases come from a larger study of novice mathematics and science teachers in subjectspecific induction programs across the United States. Research Design: Exploratory case study based on interviews and observations. Findings/Results: One new teacher made considerable progress in thinking about and using knowledge of students to enhance mathematical learning. However, there were still several areas in which the new teacher missed opportunities to probe student understanding and further connect students to mathematics. And yet, because of the teacher’s competence relative to other new teachers, her support from her mentor was diminishing. This left her with few opportunities for support in reflecting on her practice. The second new teacher struggled to make sense of the role of the teacher. She was a strong beginning teacher in a context that provided support and encouragement for studentcentered teaching. However, she struggled to envision and enact a teaching role that either allowed students to make their own connections to mathematics or enabled her to learn how to work toward that role.
Conclusions/Recommendations: Our two new teachers made considerable progress in their teaching. Although there was much to celebrate about their progress, there was still much about the complexity of teaching, specifically teaching math, that the new teachers had to learn. This study suggests that we need to go beyond evaluation of visible performance or attention to instructional strategies to help new teachers think about how to simultaneously manage the complex relationships with students, with mathematical content, and with the connection between students and mathematics in ways that help them to continually teach and learn from teaching. The story of induction often starts with a challenge to the sink or swim metaphor: Teachers in their early years enter teaching by being tossed in the sometimes calm, sometimes stormy waters of classroom and school life. Induction, so the story argues, is a deliberate rejection of the view that teaching is learned through trial and error. But too often the argument of induction is told from the perspective of the program, those who conceive and lead it, or the districts that need it (for examples, see Gold, 1996; Kelley, 2004; Moir, 2005; Villani, 2002). At the same time, much of the literature on the needs of beginning teachers ignores subject matter (Gold; Veenman, 1984). As we listen to voices of new teachers, however, the prominence of subject matter understanding and needs comes to the fore. This article attempts to insert the experiences and voices of novice teachers into the discussions of induction. Our goal is to add to and challenge the conversations about what learning to teach mathematics requires and how its complexity makes contentspecific induction and rich opportunities to learn not only desirable but essential. Our analysis focuses primarily on the cases of two strong, new mathematics teachers, Helen and Ona. They are each working in a different context, each with the support of a wellregarded induction program. The experiences of the novices help us identify a range of needs—content as well as pedagogical content knowledge and knowledge about children and classrooms—for which they needed support to learn. We examine their teaching and their perceptions (as described in interviews, observations, correspondence, and logs) over a period of 15 months to document their experiences of opportunities to learn and the problem of induction programs having, too often, to attend to the sinking swimmer (to continue the metaphor), thereby leaving behind those who appear to have strong content knowledge but still have much to learn about teaching. We aim to make two points in our argument. First, the tendency to think of induction as helping those who might otherwise sink limits our ability to understand the situation of Ona, Helen, and the many other beginning teachers who, although not flailing in the water, still have very real needs. Learning to teach, as FeimanNemser (2001) suggested, requires pulling together strands of understandings and skills. This not only is hard but also can only be learned once in the swirling waters of actual classroom teaching. Looking at swimming (and not sinking) from the perspective of those doing the swimming has implications for induction providers and raises questions about the range of capacities and program supports that needs to be in place for novice teachers. It encourages us to rethink induction in ways that complicate the too simple notion that it is designed chiefly to avoid sinking. Second, in examining two novice mathematics teachers, we recognize that we are exploring learning to teach not just in any water, but also in the context of mathematics teaching. This article allows us to consider how contentrelated needs are an important and too often undiscussed part of the bundle of needs that beginning teachers bring to their early years of practice. FRAMING THE PROBLEM: LEARNING TO TEACH MATHEMATICS AS CONSTRUCTING A PRACTICE Learning to teach is hard. This is a truism for researchers, even if, in the tradition of Lortie (1975/2002), we recognize that the difficulty of learning to teach comes in part from teachers thinking that they don’t really have to learn much, having apprenticed throughout their prior schooling and having learned about teaching, partially and problematically, from experience (FeimanNemser & Buchmann, 1985). The recognition that learning to teach is hard is part of the justification for induction that narrates the literature on induction and mentoring. FeimanNemser, Carver, Schwille, and Yusko (1999) argued that much early discussion of induction has taken the period of induction, and programs of induction, as ones focused only on support rather than learning. Introducing the notion of learning to teach as a central problem of induction has been a powerful shift in the scholarship of induction. That literature, although shifting, nevertheless casts induction stories in terms of struggles and challenges (Flores, 2006). That rhetorical framing, along with the very real practical concerns of districts and other induction providers who work with limited resources in the context of big demand, leads to issues of induction often being considered against the background of potential failure. One small indication of this is the ways in which induction as policy problem is frequently characterized as motivated by a need to stem teacher attrition (Smith & Ingersoll, 2004). In other words, the gaze of induction is on the possibility of losing the struggling teacher. Popular discussions of the tasks for induction focus on how to help the novice get safely to the shore of his or her classroom work or how to gain calm in the turbulence of the early year(s) of teaching (Wong, 2002). Scholarly literature on new teachers has helped flesh out some mapping of a curriculum of novice teacher learning (Achinstein & Barrett, 2004; FeimanNemser, 2001; Wang & Odell, 2002; Wang, Strong, & Odell, 2004). Yet often induction literature has tended to treat content knowledge as a simple good—something desirable yet unexamined (Gold, 1996; Villani, 2002). FeimanNemser (2001) outlined the wide range of learning the beginner must undertake: “gaining local knowledge,” “designing responsive curriculum and instruction,” “enacting a beginning repertoire in purposeful ways,” “creating a classroom learning community,” “developing a professional identity,” and “learning in and from practice.” In investigating specific elements in this imposing list of the central tasks of new teacher learning, researchers have identified persistent challenges that novices face and argued for how they present possibilities for educative induction. Achinstein and Barrett (2004), for example, suggested that the shocks beginners experience in the classroom relate to both their understanding of students and their images of their roles as teachers; mentors can help, they suggested, in “(re)framing classroom contexts” (p. 716). Kauffman, Johnson, Kardos, Liu, and Peske (2002) and Wang and Odell (2002) reminded us that the curricular context of U.S. schools, especially in a period of emphasis on standards and accountability, adds to the difficulties for novice teachers in making wise decisions about what and how to teach; in addition to any role of policy for amelioration of this problem, Kauffman and his colleagues’ study advocated schoolbased supports for new teachers in this regard. Although these studies capture important foci of teacher learning for the beginning teacher, they tend to leave open questions about teaching within a particular subject area. There is, however, not much guidance on this issue within the subjectspecific literature. Literature on teaching mathematics has not identified particular issues related to the teacher early in her career and learning. Nevertheless, Ball and Bass (2000) do offer a complex view of knowledge for mathematics teaching that includes the need for “interweaving content and pedagogy.” Yet in their discussion, as is common within the subjectspecific scholarship of teacher learning, attention is focused on teacher education and professional development. There is no specific engagement with the crucial and complex period of induction, nor consideration of the implications for induction as program. In this article, we consider how a close focus on the learning of our two novice teachers, Helen and Ona, forces us to think about their (new teacher) learning not simply as a generic practice but as learning specific to teaching mathematics. What is challenging about learning this practice from the perspective of these new teachers, and what are the implications of these challenges for induction? Our question is framed by the work of FeimanNemser, Ball and Bass, and others about complexity and interconnection in teaching and teacher learning. Whether the originating question is about teacher learning (as in FeimanNemser’s 2001 piece) or about mathematics teacher knowledge (as in Ball & Bass, 2000), we are persuaded that the challenges are deeply related to the integrated aspect of teacher knowledge and the complexly interconnected nature of teaching as a practice. The framing of induction as understanding teaching and teacher learning as interconnected and as requiring all teachers—not just those struggling—to make connections requires a conceptualization that is often absent in the literature. We draw on Lampert’s (2001) notion of a “threepronged problem space” (p. 33) as a way to frame what beginners are swimming in, and what induction, well done, might help them recognize and work on. Lampert (2001) argued that the complex activity of teaching involves simultaneously facilitating three kinds of relationships: between the teacher and students, the teacher and content, and students and content (see Figure 1). Figure 1. A “threepronged problem space” 1) The first practice arrow represents the relationships that teachers develop and manage with their students. Connecting to students involves creating working relationships with them, understanding their personal histories, getting to know who they are and what motivates them, managing classroom behavior, and structuring relationships between students. 2) The second practice arrow represents teachers’ work in relationship to content. This includes school curriculum, content knowledge, mathematical topics and concepts, organizing lessons, and planning units of study over time, as well as the content about how to talk about and work with mathematical ideas in such a way that fosters more learning. 3) Although teachers “act on students” and “act on content” when they teach, student learning depends on the relationship between the students and the content. Therefore, the job of the teacher is to help students to engage with mathematical content in ways that result in learning. Lampert (2001) named this relationship between students and content the practice of “studying.” Studying includes engaging with “the ideas, processes and language they [the students] need to learn” (p. 31). Discussing a mathematical idea, justifying a solution, and reasoning through a process are just a few examples of the kinds of studying activities in which students could participate to further their learning. For teachers, the third practice arrow involves managing this relationship between students and content. Teachers do not work separately along each of the practice arrows. The work of teaching takes place simultaneously across the three as teachers use their relationships with students and with content to facilitate students’ engagement with mathematics (the middle arrow). In addition, teachers must also consider and manage relationships among students, supporting students in engaging productively with each other as they also engage with mathematical content. This is certainly a complex problem space. Time, personal histories, current social circumstances, local contexts, and students’ free will are just some of the many factors that contribute to the ongoing work of teaching (Lampert, 2001; Sizer, 1984). Using this vision of the complex work of teaching, we explore two themes that stood out as we examined Helen’s and Ona’s experiences: understanding students and understanding the role of the teacher. Both themes suggest parts of the threepronged problem space described by Lampert (2001). Certainly, each of these areas is typically a major part of one or more courses in a preservice teacher education student’s curriculum. Yet for the practicing (including the beginning) teacher, as Lampert argued, the challenge of teaching is to “proceed simultaneously in relations with students, with content and with the connection between students and content” (p. 33). For new teachers, this is especially difficult. Even if a beginner has skill and knowledge about one or more of these dimensions, it is entirely different to have developed a practice that entails the interaction of all three. Yet this, of course, is what teaching is all about. Induction is the moment when the novice teacher, for his or her own survival and success in the classroom, must pull these threads together. Herein we see the challenge for, as well as the role for, induction. We suggest that literature on induction has not considered these issues in terms of subjectarea teaching and that research on mathematics education has not devoted much attention to the particular period of learning in the early years of teaching. Hence, we argue that literature has not explored these issues in much depth for the beginning mathematics teacher, nor thought about the implications of this for teacher induction—policies, programs, and providers. Our goal is not to provide recommendations that address implications for policies, programs, and providers; rather, we aim to raise questions about misplaced emphasis in both the discussion of new teacher needs and the silences we note. We hope that providing this small window into the experiences of two teachers will encourage a more complex analysis of beginning math teachers’ learning. Before we consider each of the themes in turn, a brief note about the organization of the discussion that follows: Although our analysis explored examples of each theme from both Helen and Ona (as well as other teachers included in a larger study), our desire to provide an extended vignette and more detailed commentary led us to discuss only one teacher for each theme. We opted for crafting this discussion around elaborated vignettes because we aimed to consider the teacher from a more holistic perspective than has usually been reflected in research on beginning teachers’ needs. We recognize that one of the challenges of making the transition from preservice teacher education to induction is that the teacher is now not simply learning about discrete components of teaching (for example, planning, assessing, and working with parents) or particular topics or bodies of knowledge (for example, adolescent development, algebra, or the role of student error in learning mathematics). Instead, new teachers are constructing a practice at the same time that they are enacting it, and in both cases, that practice is characterized by its integrated character (Ball & Bass, 2000; FeimanNemser, 2001; Lampert, 2001). In addition, we chose to write with narratives of events, and events unfolding over time, to allow us to examine the ways in which these new teachers are indeed working with and trying to define a practice, one that for them is evolving. Finally, we hope that vignettes allow us to unpack abstract (and important) constructs—like understanding students, or the teacher’s role—and think about their meanings and complexities when teachers try to enact them. THE STUDY For us to be able to think about how visions of induction can be influenced by the experiences of beginners, we need to find ways to access those experiences and perspectives. In this study, we relied on data from the novice teachers—interviews, observations, and logs of activities—as well as the tools and resources they used (curriculum materials, textbooks, district policies) and those with whom they interacted as participants in school and in induction (mentors, math team members, principals, and so on). The cases of Helen and Ona came from a larger study that explored six different contentspecific induction programs from across the United States.^{1} For each site, we collected data about program goals and structure, activities, and networks involving beginning teachers. We interviewed program directors and program providers (such as workshop leaders and mentors), observed program activities, and studied program documents. Based on information we collected about the full population of novice teachers in an induction program, we identified 5–10 focal novice teachers at each site, choosing teachers who represented the range of different school settings, grade levels, and routes into teaching (traditional teacher preparation and alternate route). For each of these novice teachers (including Helen and Ona), we conducted three waves of data collection during which we interviewed and observed each of the new teachers. We scheduled our visits to teachers’ classrooms to capture typical teaching moments, visiting them in November of their first year, after the teachers had settled into a routine; in May, before the end of the school year; and again in the fall of their second year. Each of the three waves of data collection included one observation of the novice teaching one class period (during which we recorded field notes), and pre and postobservation interviews about the lesson, including their goals, their planning, decisions along the way and during instruction, and their thinking about the content and the learners in their class. The preinterview was short, typically lasting 15 minutes. The postinterview was longer, from 30 to 45 minutes. All interviews were audiotaped and then transcribed in their entirety. A summary of the classroom observation protocol and interview protocols is included in the appendix. In addition to classroom observations, each wave of data collection included a 60 to 90minute interview with the novice that was not lesson dependent. This interview asked the teachers to reflect on their experience of learning to teach, including their work with their mentor and participation in other induction activities. Each wave of interviews included a task that provided us with insights into the teachers’ thinking: about their subject matter and connections between the subject matter and students (Wave 1) and what practices the new teachers were working on and whom they turned to for support with those practices (Waves 2 and 3). In addition to classroom observations and interviews with the novice teachers, we also interviewed the mentor teacher, observed mentor–novice interactions, and observed any program activities that were occurring. Additional visits to programs were timed around opportunities for us to gain access to and observe program activities such as mentor training, novice teacher workshops, and summer institutes. Between data collection visits, additional data were collected via telephone interviews and email correspondence. See Table 1 for a list of all data collected for Helen and Ona.
DATA ANALYSIS We began our data analysis at the level of induction programs, first writing several memos (Miles & Huberman, 1994) that took us beyond information about the induction programs and the new teachers and mentors studied in each program, and helped us begin to analyze how induction programs characterize new teachers and their needs. Much prior literature has addressed, as we summarized in the preceding section, the psychological and managerial challenges that beginners face. We certainly saw evidence of this across the set of novices we were following. But as we continued to explore our data through initial coding and memos, we noticed other patterns in the struggles of new teachers that seemed to go unnoticed by induction programs and by the literature on new teachers’ needs. Helen and Ona caught our attention by quite directly and persistently asserting that they needed help, even though the needs they described did not fit the standard arguments of most induction literature. We probed further into these patterns by deciding to focus on these two new teachers, each from different induction programs. In selecting them for more detailed analysis, we consciously chose teachers who seemed to be doing quite well—they were highly regarded by their programs and clearly had fewer dramatic outcomes (such as leaving teaching) than some of the other teachers we studied, and yet they expressed frustrations and questions with their teaching. We began our more focused analysis of Helen and Ona by investigating each wave of data collection separately for each teacher. We first opencoded (Easterberg, 2002) each Wave 1 data source (i.e., all interviews, classroom observations, mentoring interactions, and other meetings or events that occurred before and during Wave 1). Consistent with Glaser and Strauss’s (1967) description of grounded theory, we did not begin coding with a list of categories, but rather labeled data with themes and categories that occurred to us as we coded. This opencoding resulted in the identification of multiple categories related to Helen’s and Ona’s areas of strength and areas of growth, both those areas specifically stated by the new teacher or the new teacher’s mentor and those areas perceived by the coder (see Table 2 for a partial list of codes). One of us took the lead on coding each wave of data for each new teacher and then presented our coding to the other two authors for peer debriefing (Marshall & Rossman, 2011). Once we were satisfied with our initial opencoding for the data in the wave, we wrote analytic memos (Easterberg), one for each teacher, which drew evidence from across all data sources in Wave 1 to describe possible patterns of new teacher strength and growth. Our use of multiple data sources provided one means of triangulating our findings (Denzin, 1970): Our confidence in the validity of our findings increased as we found similar themes across interviews, classroom observations, and other data sources.
Once we felt that the memos captured our understanding of the data, we shared them with members of the larger research team. Our research colleagues raised questions, challenged our claims, and pushed us back into the data. We incorporated this feedback into our emerging understandings of the patterns in the data. We followed this same sequence of opencoding, memoing, presentation, and revision for the remaining two waves of data. Once we had systematically probed all the data, we generated a series of memos in which we both drew on our analytic work and returned to the data to explore, refute, and revise potential themes. We sought to find a thesis that withstood the rigor of the search for negative instances and that also added to our knowledge about new teachers. At the conclusion of this iterative analytic process, we established two broad themes: • Knowledge about students and context that impacts pace and structure of the mathematics of the course, including interpreting students’ alternative mathematical understandings • Emerging visions of the new teachers as teachers of mathematics and making sense of the role of the mathematics teacher For each theme and for each teacher, we reexamined the data set and wrote two summary memos (one for each new teacher) that organized examples of each theme (see Table 2 for information on how we connected the codes with the themes). These memos were shared across the authors and then revised to form the basis of this article. We then sought to add depth to our analysis by examining the fit between our findings and preexisting theoretical constructs (Miles & Huberman, 1994). As Miles and Huberman noted, the synergy between an extant construct and emerging themes adds explanatory power to the data and pushes the theoretical construct further. Our exploration of potential frameworks began with Berliner’s (1986) construct of expert pedagogues, but we quickly turned to Lampert’s (2001) description of prongs of practice once we realized the resonance between Lampert’s framework and our themes and the ways in which Lampert’s framework provided additional insights into the new teachers’ struggles and strengths. Finally, after this manuscript was in final form, we contacted both novice teachers and asked them to read and comment on the manuscript, and in particular on the accuracy of our assertions about their teaching. Both novices felt that our interpretations reflected their activities and intentions, commenting, “You were right on the money” (email from Ona) and “I was very impressed with the result and totally agree with the interpretation” (email from Helen). We do not interpret these endorsements to mean that there are no problems with our interpretations of the data. However, they do suggest that our assertions and conclusions resonate with these teachers’ perceptions of their experiences. Throughout this process, we have been mindful of the limitations of our data. We were limited in the number of observations we could have with each teacher. In the larger study design, we did not see individual teachers as case study subjects; rather, our focus was on programs, and the perspectives of new teachers were important to our understanding programs. Yet Helen and Ona both sought us out for extensive conversations and communicated with us beyond the formal waves of data collection. We also benefit from having the perspectives of their mentor teachers and other teachers working in their schools and districts, as well as written documentation that helps us understand the contexts in which they are teaching and learning. We do not presume to make these full case studies of individual teachers. We do see evidence of themes and challenges that their experience highlighted in the larger data set. We see this as an exploratory study that raises questions about how we might reframe our discussions of new teacher learning needs and the role of induction to support those. MEETING HELEN AND ONA: BEGINNERS WITH STRENGTHS, BEGINNERS WITH QUESTIONS Helen told us, “And, they keep saying, you’re doing a good job. Maybe they just think I don’t need them with the help. ‘Cause it seems like when I ask for it they, oh yeah, you don’t need it. But, I DO need it” (Postobservation interview, December, Year 2). Ona said, You could see me just pulling my hair saying, “What am I doing wrong? How can I explain this differently?” And what I would do, I would explain same thing three different ways. Or two, three, four different ways. And one that was getting one way, would get completely frustrated and lost if I introduced another way. And yet there were people that didn’t, couldn’t get either way. So, I wanted to go three way so you know? With this class I did have a lot of hard time as a teacher. What am I doing wrong? Why isn’t it working? Should I give them just one way and stick to it? Or should I introduce several ways and confuse them? (Postobservation interview, May, Year 1) Together these quotations remind us of how demanding it is to learn mathematics teaching as a practice and ways in which beginning teachers with strong mathematical content need support for learning how to teach mathematics, especially in their first few years of teaching. Helen and Ona each were well positioned to enter mathematics teaching and were assumed to be strong by mentors assigned to support them, but as these quotations suggest, they also encountered real struggles. HELEN’S PERSPECTIVE: UNDERSTANDING STUDENTS, STUDENTS’ FAMILIES, AND STUDENTS’ THINKING Helen’s context. Sheffield High School was a large urban public high school in a city located adjacent to a major metropolitan area in the eastern part of the United States. Sheffield’s students were predominantly African American (95% of the total student population), and 18% qualified for free or reducedprice lunch. The school was part of a very large school district with more than 135,000 students and over 9,000 teachers. In the fall of 2002, Helen Rogers was a new teacher at Sheffield High School. She was hired in November as one substitute in a long line of many who had attempted to fill the place of the “real” teacher who quit earlier that fall. Prior to teaching at Sheffield, Helen had earned a bachelor’s degree in mathematics and spent several years working at both the Mathematics Association of America (MAA) and the American Association of University Women (AAUW). Helen’s school district supported its new teachers with a threepart induction program that included teaching certification courses, general and contentspecific workshops, and mentoring. The certification courses were essential for the district; 40% of new teacher hires (including Helen) were uncertified. In addition to providing information on certification courses available through local colleges and universities, the district also ran its own alternative certification program and provided extensive support and coaching for the certification tests. New teachers also had access to general and contentspecific workshops during the summer and the school year (including a 4day new teacher orientation over the summer). Finally, the district ran an extensive mentoring program, employing 80 fulltime mentors and 70 peer mentors (fulltime classroom teachers who also mentored one or two new teachers). The fulltime mentors were placed in schools with poor standardized test scores. These mentors chose up to 15 teachers to mentor, making this decision based on their perceptions of teacher need. The mentors were to meet with each mentee at least twice a week to observe in his or her classroom and provide support and feedback. Helen’s mentor, Sandra, was a fulltime mentor working with 11 other new teachers at the school. Most often, Sandra’s interactions with Helen focused on moral support, ideas for setting up her classroom, and providing access to resources, in particular, mathematics teaching materials from Sandra’s personal collection. Sandra also observed Helen’s teaching on an almost weekly basis and provided her with feedback. Sandra felt that Helen was doing “fantastic” and, over the course of our visits, was spending less and less time supporting this quite strong new teacher. Helen’s strengths and struggles. Helen came to mathematics teaching with many strengths. During our first observation of Helen in December 2002, it was very clear that she had already created a stable and supportive classroom environment. She had a positive rapport with students. She was friendly and jovial and simultaneously maintained a clear sense of authority. Helen was assigned to teach algebra for six periods each day. She was “very confident” with the algebra content and she used the district’s adopted textbook and curriculum guide to plan her daily lessons. In our classroom observations of Helen, we learned that she was clearly able to make sense of the content she was required to teach, and she paced her lessons very well given the 45 minutes allotted for each class: She moved efficiently from the beginning of a lesson to its end without any major disruptions or classroom management issues. Helen’s lessons could be described as teacher centered.^{2} She had established a daily routine that included presenting warmup problems, delivering new information, assigning practice problems, and doing homework. Helen, however, was not satisfied with only “covering” the content laid out in the algebra textbook and the district’s curriculum guide, both of which were aligned with the state mathematics assessments. During our interviews with Helen, she continually articulated a desire to “bring the subject matter alive” for her students, to provide students with an understanding of “how important mathematics really is in daytoday life.” After only one month in the classroom, Helen told us that one of the things she felt least prepared to do was to work with the parents of her students. She identified a need to become more familiar with the larger social context in which her students lived. In our first interview with Helen, she explained, “So one of the goals that I want to do is to better understand the community, you know, better understand my students and the parents. I want to get a better hold, a better feel for the students in their environment, you know, for the social concerns they’re having” (postobservation interview, November, Year 1). This desire to learn more about her students, their parents, and the community is one that many might advocate as important for new teachers, given that research suggests that successful teaching requires an understanding of students’ families and their participation in outofschool communities (Banks, 2002; Gay, 2000; Moll, Amanti, Neff, & Gonzalez, 1992; Nieto, 1996). However, early in her career, Helen’s reasons for getting to know her students and their families were purely for relational purposes and separate from her teaching and learning goals. As a new teacher who entered this school partway into the beginning of a school year, Helen told us that she felt “prepared to come in here and teach this subject” but less prepared to “juggle the parents” and help students with the “social concerns that they are having.” She assumed that knowing her students better would support her in accomplishing these nonacademic goals. Helen’s choice of words in this first interview seems telling: She was prepared to teach mathematics (“this subject”), but she wasn’t thinking about teaching students or knowing students so she could teach math to her students. This theme of getting to know students was evident again during our second interview with Helen. However, this time Helen made a connection between this desire to know students and her ability to plan lessons that met students’ mathematical learning needs: The major thing for me is that all students, no matter what their skill level, can be in one classroom. And the major thing that I have learned from that is when you’re lesson planning and assessing, that you have to make sure you know your environment, you know your students and what level they’re at. So, you plan so that everyone can be on the same page, maybe at different speeds, but everyone has to be on the same page. (Postobservation interview, May, Year 1) Helen’s goals to make mathematics meaningful for her students and to help students recognize how math was connected to their lives had not changed. Helen still “envisioned seeing realworld situations” represented and posted on her classroom walls. However, after almost a full year of teaching, knowing her students had taken on a new meaning for Helen. Now Helen wanted to know students so she could support them intellectually. In Year 2 of her teaching, “knowing your environment” and “knowing your students” became necessary, from Helen’s perspective, not only because she wanted students to relate to the mathematics they were learning but also because she needed this information to support her teaching practices and her students’ learning. During this first year of teaching, we understood Helen to be working within Lampert’s (2001) threepronged problem space alongside the practice arrow that connects the teacher’s work with her students (see Figure 2). Inside this problem space, Helen wanted to use her students as a “resource to solve the problems of [her] practice” (Lampert, 2001, p. 31). She assumed that knowledge about her students, their families, and the communities in which they lived would positively contribute to her efforts to support her students’ mathematical achievement. Figure 2. Helen’s construction of teaching in Year 1 In addition to Helen’s desire to better understand her students so that she might better meet their academic needs, we also observed Helen making some real changes in her teaching practices over time. In the beginning, Helen’s lessons mostly consisted of delivering new information and giving students time to practice related problems. By the end of her first year of teaching, Helen spent less time talking and provided more time for students to directly engage with mathematics during class. In particular, Helen used warmups to start class and to provide students with an opportunity to volunteer and solve problems at the chalkboard. Helen told us that she had learned about this strategy from her mentor teacher and found that her students really enjoyed this activity. Furthermore, Helen had discovered that having students do the warmup presentations supported the progress of the lessons. “It helps with instruction because it just has a flow for the rest of the class period, having them engage in more of doing the warmups.” Additionally, Helen told us that she was “in better tune with her students.” When asked what this meant to her, Helen explained, “I want to say, the lay of the land, like the environment itself. Because some of my students this year, I had last year . . . and you know they didn’t know me, I didn’t know them . . . I know now when they’re upset, when they’re OK, when they’re having a good day, a bad day, that type of thing” (Postobservation interview, February, Year 2) . Although Helen did not articulate any explicit connections between her increased understanding of her students and the shift in her teaching practices, we do not believe that this change was coincidental. As Helen planned her lessons with more knowledge of her students in mind, she was better positioned to create learning opportunities that met their learning needs. This shift from delivering mathematical information to including students in the construction of mathematical knowledge was significant even if it only took place during the warmup activities. We made a final visit to Helen’s classroom at the end of her second year of teaching. At that time, Helen structured her lessons quite differently from what we had seen in the year prior. During this observation, Helen not only provided students with the opportunity to engage with mathematical content but also attempted to engage with students around their mathematical thinking and understanding. Rather than using students’ ideas as a springboard to launch her own lectures and presentations, Helen was now trying to build from students’ knowledge and use their ideas to support their mathematical learning. The following excerpt illustrates the change in Helen’s engagement with students. It comes from an observation of one of Helen’s algebra classes. One objective for this particular lesson was for students to understand whether a given graph was a function. A mathematical function is a particular kind of relation in which each value of the x variable (which is graphed along the horizontal axis) has only one corresponding value for the y variable (which is graphed along the vertical axis). This constrained relationship leads to the vertical line test: A relation is a function if you can draw a vertical line at any place on the graph and only cut through the graph in one place (see Figure 3). Figure 3. Graph of the third warmup problem. This graph is not a function because when x = 3, y = 0 and 3. The vertical line test shows the two corresponding yvalues for the value of x = 3. During Helen’s lesson, she asked students to solve three warmup problems and then made time for volunteers to present and discuss their solutions at the front of the room. The discussion that follows between Helen and Tyrone considers the third warmup problem: Use the vertical line test to determine whether the relation is a function: {(2, 1), (3, 3), (4, 1), (6, 3), (1, 1), (2, 2), (3, 0)}. (A solution for this problem is depicted in Figure 3.) Helen has asked Tyrone to explain his work on the problem. Tyrone: First I graphed everything on the graph and then I put an arrow through it, and it went through it twice. Ms. Rogers: What did you do? Tyrone: The thing. Ms. Rogers: What did you do? The vertical line test? This is not a function, because when you do a vertical line test it passes through two points. Tyrone: Those two points? [Ms. Rogers draws a coordinate graph on the white board, similar to Figure 3, and plots the points given to students in the warmup problem. She then draws many vertical lines across the graph.] In this conversation, we understand Helen to be making connections between her knowledge of students and her knowledge of content to facilitate a relationship between students and their mathematical understanding. Tyrone has been given an opportunity to explain how he solved the problem about whether this graph was a function. His explanation was vague but still correct. The “thing” Tyrone referred to was the vertical line test, which he knew to be a strategy for determining functionality. However, Tyrone was struggling with where to draw the vertical lines. He came up with different answers depending on where he drew the lines. Given our previous observations of Helen, we might have expected her to accept Tyrone’s answer, or restate Tyrone’s answer in her own words, and move on with the lesson. That Helen did not stop here and instead engaged Tyrone’s question represented a big shift in her stance toward teaching and learning. This new position required specific knowledge about Tyrone and the mathematical content at hand. For example, maybe Helen knew something about Tyrone’s understanding about functions that she wanted to build from. Maybe she wanted to use Tyrone’s explanation to help other students understand the problem. Maybe Helen considered this moment to be an opportunity to legitimize Tyrone’s reasoning so that she could build his confidence in mathematics. Whatever Helen’s purpose, she needed to know Tyrone and this particular mathematical content, and be able to simultaneously work with both in this moment to turn this conversation into a situation from which Tyrone could learn. When Tyrone saw that Helen was repeating the warmup problem, he asked a different question about using the vertical line test. Instead of focusing on the intersection of the vertical lines with the given graph, Tyrone asked about the placement of the vertical lines: “But last time I asked you if I could put the line on the other side [of the yaxis] and you said I couldn’t do that.” Tyrone did not seem to understand that the vertical lines used for the vertical line test must be contained within the domain (given xvalues) of the given graph. When Tyrone asked his question about placement of the vertical line, Helen realized that she had to do something different to support his needs in this moment. She then changed her approach from working with the warmup problem and created a new example better suited to Tyrone’s question: [Ms. Rogers draws a parabola that is symmetrical around the xaxis as an example of Tyrone’s question. See Figure 4.] Ms. Rogers: But that’s where the graph stops. Understand? [She points to the vertex of the parabola.] Tyrone: No. [Ms. Rogers draws several vertical lines through the parabola and tries to explain why the parabola stops. She does not compare the graph of the parabola with the graph of the line that was given in the warmup problem.] Andrea: How do you know if it’s a function? Ms. Rogers: How do we know if it’s a function, class? Tyrone: If it passes through once. Ms. Rogers: If it passes through the graph only once, but if you look at that, it passes through both of these points, [She points to two points that exist on the parabola, which are directly opposite each other] so it’s not a function. Figure 4. Representation of parabola that is symmetrical around the xaxis It is easy to read this excerpt from Helen’s classroom and notice that she did not really engage with Tyrone, his ideas, or mathematical understandings. An experienced mathematics teacher might use this opportunity to learn more about what Tyrone understood about functions and graphs. However, given the multiple challenges Helen faced as a new teacher, the messages she received about what it meant to be a “good” teacher, and the context of her school environment, she did more to try to listen to and respond to his question than she had in previous situations. In this scene, she was noticing the student’s understanding (or misunderstanding) of content. This represented a kind of shift in the prongs in Lampert’s model. Helen was moving from a focus on her relationship with students and her relationship to content to a focus on the relationship between students and content (see Figure 5). Figure 5. A shift in Helen’s practice from focusing on her relationship with students to focusing on the relationship between students and content By the time of this observation of and interview with Helen, she was not receiving any more support from her mentor. From Sandra’s point of view, Helen was a “good” teacher, and in the context of this large urban school, there were more than enough “squeaky wheels” who needed Sandra’s support more than Helen did. As long as Helen was able to successfully manage her students and keep them away from the principal’s office, she was left alone to teach, even though from our perspective, there was much Helen still needed to learn about how to teach. We can look more closely at this exchange with Tyrone to better understand Helen’s growth and the difficult challenges new mathematics teachers face in their efforts to support students’ understanding. First, it is important to note how unusual it is for Helen to use the warmup activity as an opportunity to discuss students’ mathematical ideas and processes. The idea for a warmup came from Helen’s mentor, Sandra, and it was meant to promote student engagement at the start of class. Interestingly, the warmups soon became a direct substitute for Helen’s lecture, with no time for questions or conversation. It was assumed that by doing the warmup problems, the students would also learn the mathematical content of the problems. This approach directly parallels Helen’s initial construction of teaching as the delivery of content. During our first interview with Helen, she told us, “Doing the lesson planning and coming in and doing the instruction, you know, actively planning the lesson, coming in and giving the instruction to the students, I feel like I can do that with my eyes closed. That’s not a problem.” Attention to the verbs Helen used to describe her teaching indicates that she was working with a construction of teaching as telling (Smith, 1996). She does instruction. She gives instruction. Knowing this, we understand Helen’s choice to entertain Tyrone’s question as a noticeable step toward a different way of thinking about teaching and learning. In each of our prior observations, Helen had either ignored students’ questions (most often because she felt pressured by time) or quickly provided a generic procedure that would lead to a correct answer if students followed it. In her exchange with Tyrone in this lesson, we saw Helen’s attempt as evidence of her effort to work with Tyrone’s mathematical ideas rather than talk at him or move past him altogether. Second, as Helen worked with Tyrone’s question, we watched as she tried to interpret his understanding about how to use the vertical line test to determine whether a graph is a function while she simultaneously attempted to generate a response that was appropriate for his question. This move takes considerable skill: It is a significant challenge for any mathematics teacher to understand a student’s thinking and generate a relevant example with which to work in the moment. We would have expected Helen to either reiterate the definition of a function and demonstrate how the vertical line test applied to the given graph, or just ignore Tyrone’s question altogether. Instead, Helen demonstrated that she understood Tyrone’s confusion and then built directly from his misunderstanding rather than imposing her own process and ways of thinking about the problem to move the lesson forward more quickly. Lampert’s (2001) model helps us clarify why Helen’s decisions and actions in this moment were more complicated than one might assume (see Figure 5). Helen understood the mathematical content of graphs and functions. She could clearly present this information and show students how to determine whether a graph was a function with the vertical line test. Helen also knew her students. She understood their social concerns and had met many of their families. However, mathematics teaching is more than managing one or the other. Teaching mathematics requires an ability to use knowledge about students and content in practice (Ball & Bass, 2000). Connecting students and content in practice means going beyond students’ social lives and family relationships and the teacher’s mathematical knowledge. Helen must also be able to recognize and specifically engage students’ emerging school knowledge of mathematics (which she begins to do in the excerpt with Tyrone). Finally, connecting students and content also involves recognizing and drawing on students’ mathematical funds of knowledge (Moll et al., 1992), explicitly connecting students’ outofschool mathematical expertise with the inschool curriculum. While Helen was becoming more aware of her students as individual learners, she was not yet noticing, pursuing, and valuing the mathematical experiences and skills they brought with them from outside the classroom. For Helen, her students still seemed to be mathematically empty vessels who needed to interact with her mathematical knowledge, rather than mathematical beings who could draw on their own resources to continue their mathematical learning. Noticing, eliciting, and engaging students’ prior subjectmatter knowledge can be particularly challenging for teachers of mathematics. Interacting with student knowledge requires the beliefs that students’ (potentially naïve) understandings are fruitful (even necessary) starting points for making sense of mathematics and that students can learn by exploring multiple solutions to a problem. However, these beliefs contradict widely held “common sense” notions of mathematics and mathematical learning. As Stodolsky and Grossman (1995) and Smith (1996) noted, many people (including mathematics teachers) believe that mathematics is a fixed body of knowledge that is best learned by first mastering basics and then building, in a specific sequence, to more complex knowledge. If mathematics is fixed and must be learned sequentially, then it makes little sense to indulge in studentinvented solutions, which are likely to distract from the mathematical canon; instead, the emphasis should be on conveying essential mathematical content in a clear and sequential manner. Given these pervasive messages about learning mathematics, we find much to celebrate about the changes in Helen’s construction of what it meant to teach and learn mathematics. What began as a model of teaching as telling grew to include knowledge of her students, not for the only purpose of knowing their names and recognizing them in the grocery store, but for the purpose of planning appropriate instruction and supporting their mathematical understanding. Unfortunately, there is also space for regret about Helen’s progress. As our observation in our third visit demonstrated, her teaching was marked by missed opportunities. She did not notice these, nor see them as problematic. With the advice from her induction program, which focused on students in ways that suggested they were somehow separate from content, Helen had few opportunities to reflect in new ways on what she is learning (and succeeding at) and where she still needs to learn. Ironically, as Helen grows as an educator, her opportunities for learning shrink. ONA’S PERSPECTIVE: UNDERSTANDING AND CONSTRUCTING THE MATH TEACHER ROLE Ona’s context. Ona Atanassov began her first year of teaching in the fall of 2003 at Shiffer Village School, a small public K–8 school located in a rural setting. Because of the school’s small size, Ona was the only middle school math teacher, and she was thus responsible for teaching fifth through eighth grade. Ona’s path to this teaching position was nontraditional. She grew up in Eastern Europe and immigrated to the United States after earning a bachelor’s degree in computer science. She worked for several years as a school aide and then decided to earn a teaching certificate. She enrolled in a oneyear alternative certification program and was placed as a student teacher in the classroom of her current mentor, Greta Aslan. When her mentor assumed a position as math teacher leader with the district, Ona was hired to replace her. As a consequence, Ona was teaching in the same classroom and with the same materials as her mentor teacher. The program supporting Ona’s teacher induction was quite different from Helen’s districtrun program. Ona’s program was region wide, including teachers from several states, and focused primarily on developing leadership capacity for mentor teachers. The program was content specific, targeting only mathematics and science teachers, and emphasized standardsbased teaching practices and how to mentor new teachers into these practices. Most of the work of the program occurred through workshops and meetings, although the program also maintained an extensive resource library and encouraged mentor teachers (and new teachers through their mentor teachers) to access these materials. The main event of the induction program was a weeklong summer institute during which the mentor teachers worked directly with standardsbased content and mentoring tools and the experts who designed those tools. We found it interesting that the program, because it primarily focused on the mentor teachers, was almost unknown to the mentees. The program required mentor teachers to select mentees but did not directly connect to the mentees. Thus, Ona, although she had many opportunities to work with her mentor (Greta), was largely unaware of the induction program, participating in just one statelevel meeting. However, Greta made use of many program resources and strategies, attending program wide and statelevel meetings. She used ideas from these meetings as well as books and other program resources in her work with Ona and her other mentees. Ona’s strengths and struggles. In front of students, Ona was confident and responsive. She was comfortable with mathematics content and easily led students through mathematical procedures. She had established classroom routines that students followed with little prompting. In addition, Ona was sensitive to student needs and requests and would revisit material, provide students with additional time, or allow them to move about the room or work with other students. When she talked about students, she celebrated their strengths and was analytical about their needs and weaknesses without using deficit language. Overall, as a firstyear teacher, Ona appeared to have many strengths in content, pedagogy, and student relationships. The case of Ona is particularly compelling because of the contrast between her confidence and poise in class and how she described her struggle to understand and enact her role as teacher. Ona appeared to be a competent beginning teacher who, at first glance, did not seem to struggle with her role in her classroom. However, in spite of the apparent ease with which she conducted herself with students, Ona had many concerns and questions about her role as a teacher. The urgent nature of Ona’s concerns was reflected by how quickly they emerged in our conversations. Within just a few minutes of our initial meeting, Ona expressed her frustration in making sense of her role: I’m not clear how to teach them, how much to teach them, what they have to give me and what I have to give to them. So, I’m not quite clear about that, but I have to do it anyways. That’s where I am and I’m going to do thinkaloud and hope that I don’t have to give them too much information, yet I will give them enough so that they can go on and do the problem on their own. (Preobservation interview, November, Year 1) Ona’s concerns about how much to support her students were repeated throughout our interactions with her, reflecting the ways in which defining and enacting the role of teacher of mathematics seemed hard to her. In this section, we explore Ona’s struggles to understand and define her teaching role. We first point out how Ona’s issues about her role are not the controloriented concerns of most firstyear teachers. We will then elaborate on the nature of her concerns and the implications for induction programs. As discussed earlier, Lampert (2001) suggested that teachers must simultaneously manage three different relationship “prongs”: teacher–students, teacher–content, and students–content. For many new teachers (like Helen) and for many induction programs, the primary focus is on the teacher–student prong. Both teachers and induction programs concentrate on the new teacher’s role in managing students. New teachers want to act as caregivers, friends, or gentle leaders (Britzman, 1991; Cooney, 1985; Veenman, 1984). They want to create a democratic and nurturing classroom. However, when they find that their role of comrade or gentle leader does not provide the desired control over students, they change to a more authoritarian style (Veenman). This negotiation between the desire to act as a gentle leader and the perceived need to act as an authoritarian is a significant and welldocumented role struggle for new teachers. In our observations and conversations with Ona, we found that she struggled to balance these two roles. However, Ona’s struggle between these two roles did not primarily focus on her role relative to students. Instead, Ona framed this struggle in terms of students’ relationship to content. Classroom management at times it can be hard because often times I let them discuss things and . . . the noise level goes up and then it gets into discussing, getting from what they’re discussing into some other conversation I have to get in. . . . I don’t think that’s hard to fix except I have to change my behavior and I have to be more intentional in how I act in a classroom so they would get the message clearly. But I also don’t find it priority for me to, to have an iron hand or have, you know, if they are discussing, if the noise level is up yet their discussion is about what they’re doing and it’s productive I don’t mind that. So it’s a balance. I don’t know where to find it yet between that freedom that they can discuss and enjoy math while they’re doing it and the behavior of where they have to stop whenever I speak and not waste five seconds before I can bring them together. . . I am still working on it. (Postobservation interview, November, Year 1) Ona did not want to “have an iron hand.” Instead, she permitted and encouraged her students to have discussions. For her, these student conversations were an important tool in learning and enjoying math. Yet she knew that she needed to set boundaries to help students focus on mathematics. Thus, although Ona struggled with democracy and autocracy, her main concern in this struggle was not control of students. Instead, she was more concerned with her role around students’ learning of mathematics. To return to Lampert’s terminology, Ona was not primarily focused on the teacher–student prong; she instead worried about her role in managing the third prong, the connection between students and content. For some new mathematics teachers, like Helen for example, the challenge of managing the connection between students and content arises from questions about what that connection should look like and how the connection facilitates mathematical learning. Ona’s experiences learning math in Eastern Europe may have included more traditional or teachercentered mathematics experiences, which could have resulted in these same questions about the connections between students and content for Ona. However, examination of Ona’s questions shows that she was not focused on whether students should be actively engaged in learning mathematics. Instead, she worried about what she should do to support students in their active struggle to learn mathematics. Consider the following quotes from Ona, from across our three interviews with her. During the first interview (as quoted earlier), she stated, “I’m not clear how to teach them, how much to teach them, what they have to give me and what I have to give to them.” During our second visit (also quoted), she described herself as pulling her hair and asking, “What am I doing wrong? How can I explain this differently?” Finally, during our last visit, she stated (as quoted later in this article), “I question myself and my methods and my teaching . . . and I’m thinking what can I do differently.” These statements reflect Ona’s concern over her activity and what she needs to do to support her students in learning mathematics, rather than questions about whether her students should be actively involved in mathematics. It is possible that Ona was more focused on her role than on the appropriate student–content relationship because of her considerable experience in standardsbased classrooms after her arrival in the United States: She had spent the previous 6 years as a teacher’s aide or student teacher at Shiffer Village School, working with teachers who were using standardsbased mathematics curricula. This exposure to standardsbased mathematical teaching may have supported Ona in developing an image of how students should connect to content, which then allowed her to focus on her role in managing that connection. We think it is important that Ona was able to recognize and frame concerns about her role relative to the connection of students and content. This focus demonstrates her ability to frame the other two Lampert prongs, teacher–content and teacher–students, in productive ways: Her background in mathematics was adequate for making sense of the concepts she taught. In addition, her understanding of students allowed her to interpret their behaviors as rational and provided her with productive structures for evaluating her concerns about students. Being able to manage these two prongs is no small feat for a new teacher and attests to Ona’s strengths as a novice. However, her struggles with the third prong, defining her role in connecting students and content, remind us that even though Ona was a strong novice, she still had areas of concern that would be important for an induction program to address. Ona had many questions about her role in establishing the connection between students and content. She repeated these questions throughout our first visit with her: As I mentioned earlier, I don’t know how much to give them, how much not to, what to model, what not to, how much they’re ready for it. . . . When I start from scratch, they complain, “Oh, you know, can you just let us go?” You saw them. They wouldn’t take an answer. “We don’t want somebody to tell us. Let us try.” But yet I see so much frustration and I don’t know how much time to give them, when to stop and bring someone in and give them modeling and that’s hard. I don’t know if I need experience, Greta [her mentor] says the more I do it, the more comfortable I will be with it. So, I’m hoping by the time it’s the end of the year, they will do some real good stuff for me and I will be happy. (Postobservation interview, November, Year 1) She wondered about what mathematics she should teach, and how and when she should teach it. She wondered whether to let students struggle or whether to intervene, whether to listen to student requests or whether she might be able to use some “trick” to reduce student frustration. These questions reflect Ona’s belief that it was part of her role to ensure that students and content were connected. They also reflect her uncertainty about how to enact that role. These concerns persisted during Ona’s first year and into her second: During our March phone interview and May visit, she continued to wonder how much input to give students and how many different ways to explain. By November of her second year, her concerns were no longer about students in general; she felt “quite confident and comfortable” meeting the needs of most of her students. Instead, her questions about her teaching role centered on supporting students with special needs. With seventh graders it’s a different story because [they are] such a small class and 40% of them are struggling. There are four of them that have special needs and out of that only 20% are with me, so I question myself and my methods and my teaching (inaudible: cost) and I’m thinking what can I do differently? What am I not doing right? . . . . So I don’t know. With them, I’m struggling just as well as they are. And the kids that are strong are getting bored and you know, they’re getting confused because the others are pulling them different directions. I don’t know. (Postobservation interview, November, Year 2) Although Ona still wondered what to do differently and what she wasn’t doing right, her thinking about connecting students and content had become more sophisticated in her second year. Instead of worrying generally about students and content, she recognized that her students had different needs, and she wondered about how to simultaneously help individual students connect with content at different levels. Ona’s concerns seem generic and disciplinefree: She didn’t embed any of her talk about concerns in a particular mathematical concept or problem. However, the challenge of connecting students to mathematical content has components that are unique to the discipline of mathematics. Not only did Ona need to decide how to support students in connecting to mathematical ideas, but she also needed to help them see how to do this in a mathematical way. For example, mathematics has particular ways of reasoning and conjecturing (Lampert, 1990), and Ona needed to consider how her moves would help students learn this mathematical discourse. Also, the logical and sequential nature of mathematics might present a second struggle. Drawing on Dewey’s (1902) distinction between the logical and psychological, it might seem intuitive for Ona to push students toward a mathematically logical and parsimonious next step even though that next step might not be psychologically appropriate for her students. For example, in some instances, it might seem logical to give a student a definition for a mathematical term (such as a function). However, if the student is struggling to make sense of functions, the definition, which is a generalization of multiple instances of functions, might not help the student grasp the notion of function as well as a series of different examples. Ironically, the powerful push to connect students to the mathematical canon (Smith, 1996; Stodolsky & Grossman, 1995) might undermine the teacher move that might best improve student understanding. In spite of her many questions about what she should do, Ona had a vision for teaching mathematics: I want to be a teacher who is not standing in front of the classroom and thinking of kids as an empty vessel and filling them up. “Here, I know all and here’s what you have to figure out.” I would like to be a facilitator more than a lecturer. I don’t see that right now and that’s my goal that kids can, that I can make a classroom culture so that I can be out of the school and kids will carry the class. I don’t know the recipe how to get there. I don’t know exactly what would it take but that’s my goal. (Postobservation interview, November, Year 1) Ona did not think that her job was to give students all the mathematics they were to learn. Instead, she imagined that she needed to facilitate their learning and that this facilitation meant she should set up students so they could then learn without her. In many respects, Ona’s vision of mathematics teaching reflected the standardsbased mathematics pedagogy encouraged by her district. She rejected the notion that mathematical knowledge could be simply received by students. She recognized that students needed to be actively involved in their learning and that students could learn from each other. For example, she frequently had students present their work to the class, and she allowed students to work together on some problems. Yet, in spite of her use of standardsbased techniques and her denunciation of students as empty vessels, Ona’s construction of her role still predominantly relied on traditional notions of teaching and learning mathematics. Ona’s many questions about her role may arise in part because of a contrast between Ona’s vision for what was possible and her teaching enactments. Examination of Ona’s teaching reveals her tendency to serve as the central focus of mathematical activity. For example, the teaching vignette that follows was from our second teaching observation of Ona (May of her first year). Ona and the students were discussing homework problems on experimental probability. To determine the experimental probability of an event (such as a coin landing heads up), the students needed to divide the number of times the event (landing heads up) occurred by the number of trials (number of times the coin was tossed). Ona began this teaching moment by inquiring about the homework. Ona talks to the class, “You said the homework is hard. What problems were hard?” As students call out numbers of homework problems, Ona makes a list on the board. She then puts information from the first problem on the board. This problem asks students to explain how the experimental probability of heads was calculated if 10 is the number of tosses, 4 is the number of heads, and 0.4 is the experimental probability of getting heads. Ona writes 4 10 and then 4/10 = 0.4. As she writes this, she answers the homework question by saying, “Heads over the number of tosses.” The class then moves on to the next homework problem, determining the experimental probability that a spinner will stop on “B.” The students have the following data table:
Ramona, a student, volunteers, “It’s 25 out of 60. I added it up.” Ona follows up on Ramona’s comment by stating, “Each time you spin the spinner is an experiment. You did 60 experiments. How many times landed on B? 25.” (The number of experiments comes from adding 17 + 25 +18.) Jess, a different student, asks, “I don’t know how you get percent.” Kelly, another student, responds, “60 times 5 equals 300. 25 times 5 equals 125. Then divide the top and bottom by 3.” (This solution appears to be mathematically correct. Kelly seems to be trying to calculate percent by converting 25/60 into a fraction with 100 in the denominator.) Kelly then realizes that she made a mistake as she tried to divide 125 by 3 so she doesn’t want to finish. Other students get out calculators and input 125 divided by 3. Kelly tries solving the problem another way, using long division. Ona asks her, “What percent [did you get]?” Kelly responds, “240%” to which Ona replies, “You can’t have that.” Another student has calculated 41.6 and announces this to the class. Ona writes on the board “42” and asks students why she wrote 42. One student responds, “Because it’s greater than 5” (meaning that the .6 from 41.6 was greater than .5 so 41.6 should be rounded to 42). The class then moves on to another problem. Throughout this vignette, Ona was the focus for most of the thinking and explaining that occurred. She presented her response to the first homework problem (the experimental probability of getting heads) and did not encourage student input or discussion. As the class worked on the second problem (about the spinner), Ramona volunteered an answer that was then restated by Ona. Ona allowed Kelly to respond to Jess’s request, but she did not support Kelly in explaining her ideas to Jess or to the class. Kelly had an interesting approach to finding percent: Through multiplication and then division, she could find an equivalent fraction that would easily lead to the percent. Ona could have scaffolded Kelly’s presentation and encouraged her to explain why her actions would lead to an equivalent percent. Instead, Ona focused on finding and then evaluating numerical answers so that when Kelly arrived at the answer 240%, Ona dismissed the answer without probing Kelly’s thinking. Through this series of moves, Ona lost opportunities to make student thinking explicit so that she and the other students could learn more about students’ mathematical ideas. She also missed the opportunity to encourage students to be responsible for their learning and the learning of their peers. This vignette was typical of the interactions we observed between Ona and her students: She frequently responded to student questions by providing explanations herself, evaluating student responses, or restating student answers. When students explained their answers, they usually explained their procedures but did not typically attempt, and were not usually required, to justify their thinking or mathematical choices. At first glance, these teachercentered practices were in stark contrast to Ona’s description of the ideal teacher. Ona hoped to be “like a facilitator more than a lecturer.” She wanted to craft a classroom culture such that she “can be out of the school and kids will carry the class.” She did not want to be the director of student learning. Instead, she saw that students can and should direct their own learning. Although Ona aspired to a more studentcentered classroom, her vision of her teaching role in the classroom did not include what she might do to make the classroom more student centered, nor did it include an image for her role after the classroom had become more student centered. Indeed, Ona’s vision of the ideal classroom is actually more consistent with the description of a teachercentered classroom than a studentcentered classroom. In a traditional mathematics classroom, the teacher is successful when she has adequately conveyed her knowledge to her students, such that they can practice mathematical procedures and work mathematics problems without the intervention of the teacher. In other words, when the teacher has done her job well (conveying mathematical knowledge), she is no longer needed, and, in Ona’s words, the “kids will carry the class.” In contrast, studentcentered or standardsbased mathematics pedagogies rely on the teacher to support, extend, and challenge student thinking (Fraivillig, Murphy, & Fuson, 1999). Student thinking is centered, but students are not left alone to think. Instead, the teacher plays a critical role in teaching students how and what to think by engaging their thinking. Ona did not yet see how her teaching role involved working with student thinking. As a consequence, she was unsure what she should do to engage students in mathematics. The tension between Ona’s enactment and her vision of teaching might also be tied to tensions between what is valued in mathematics and beliefs about how students learn mathematics. As mathematicians search for pattern and order, they place a premium on elegance and efficiency. As a result, a shorter solution is usually a more desirable solution. Other, longer solutions may be logically valid but not as valued as tidy manipulations involving as few steps as possible. This preference for efficiency is in contrast to the many mistakes and messy procedures students may develop as they work to make sense of mathematical ideas. In traditional teaching, this tension between elegant, efficient mathematics and messy, inefficient student learning is minimized as teachers demonstrate the mathematically desirable solution and then provide opportunities for students to master the procedure through practice that mimics the teacher’s elegant and efficient solution. As teachers move to more standardsbased teaching practices in which student ideas are foregrounded, they must first persuade students that they can engage in their own mathematical investigations. Then teachers must decide how to balance students’ messy and inefficient explorations with the need to share mathematical notation and conventions of efficiency. Ona did not face the challenge of convincing students to engage in mathematical investigations: Her students wanted to work on their own and were resistant to hearing other solutions before they had found their own answer. For example, during our first observation, Ona suggested to her students that some students who were further along on a particular problem might show the others what they had done so far. The students asked for more time to work on their own before anyone presented. One student explained, “I think what everyone is trying to say is that they don’t want the answer, they want to solve it themselves.” Rather than trying to persuade students that they could work on their own, Ona needed to persuade students that they could learn from each other and even from her. She noted (in a quote cited earlier), “I’m not clear how to teach them, how much to teach them, what they have to give me and what I have to give to them.” This statement points at one of Ona’s central struggles: She knew that she needed to pass along mathematical conventions to her students, which in some cases means modifying or even rejecting students’ messy solutions, and she struggled to determine when and how to do this. Ona was clearly dissatisfied with her teaching, but her enactment of her role as math teacher prevented her from interacting with students in ways that could help her resolve her questions and concerns about her teaching. As described earlier, her tendencies toward traditional, teachercentered mathematical teaching practices placed the emphasis for presenting thinking and ideas on herself rather than on the students. As a result, she had few opportunities to examine what students were thinking beyond their numerical answers and procedural explanations. In addition, Ona’s construction of the ideal teacher role as a facilitator further limited her exploration of student thinking. She constructed the ideal teacherfacilitator as someone who initiated the mathematics lesson and, if successful, was not needed as the students carried the class. As a result, most of her contact with students was with struggling students who requested additional support. For example, she explained how she was going to introduce a lesson: “We’ll do one example together and the discussion and then they’ll be on their own working. If they are stumbling on something, I’m circulating, they’ll raise their hands and I’ll go over try to explain, try to ask questions, see what they’re not understanding” (Preobservation interview, November, Year 1) Because Ona focused primarily on struggling students, she did not have many opportunities to examine the work or probe the thinking of students who were successfully connecting their ideas to the mathematical content. As a result, she could not use the work of successful students to help her understand how students might connect to content. This focus on her role as the support for struggling students also meant that she did not use students to support the mathematical learning of other students as often as she might. When we map Ona’s teaching enactment and teaching vision on Lampert’s prongs, we find that her teaching enactment places her between students and content. Instead of serving as a mediator of the relationship between students and content, Ona becomes the conduit through which content must pass in order for student learning to occur (see Figure 6). In contrast, as illustrated in Figure 7, her vision of the ideal teacher erases the teacher from the picture altogether so that students are directly linked to content, but without any mediating or supporting influence from the teacher. Lampert’s model suggests that Ona would benefit from support in understanding how she can (and must) support student learning without serving as the conduit for that learning. Figure 6. Ona’s construction of her mathematics teaching practice, as enacted. The teacher is the conduit between the students and the content. Figure 7. Ona’s vision of mathematics teaching practice. Students are directly linked to content without the intervention of the teacher. Ona struggled to make sense of her role as a teacher of mathematics. Although she had some concerns that paralleled typical new teacher issues around controlling students, her primary worry was what she should do to connect students to mathematical content. Unfortunately, Ona’s decisions about her teaching did not provide her with opportunities to learn more about her students’ mathematical sensemaking. As a result, she had limited resources to help her resolve her questions and concerns about her teaching. In many ways, Ona was well positioned to become a strong standardsbased, studentcentered mathematics teacher. Her school and district encouraged standardsbased pedagogies. Her mentor modeled standardsbased teaching and provided teaching suggestions that made students responsible for learning. Ona also had a standardsbased curriculum, and her students experienced standardsbased curriculum throughout the school. Her induction program had an explicit commitment to developing standardsbased teaching, and that orientation guided the work with the mentors and the activities in the program. Finally, and perhaps most important, Ona was dissatisfied with her own teaching and was asking questions about student thinking that could lead her to define her role in a way that moved students to the center of mathematical learning. Ona’s difficulty in enacting a more standardsbased teaching role in the midst of so much support points out the challenges of making sense of role and leads us to suggest that experience and supportive context are insufficient guides for the complexities of understanding the teaching decisions. For teachers, like Ona, to define, understand, and enact their ideal math teacher role, they need more than support for their practice; they also need purposeful and contentspecific guidance in examining their practice and construction of role. PULLING IT ALL TOGETHER Helen and Ona both began their teaching with real promise. Not ‘squeaky wheels” or swimmers going under, they brought strengths to their first months of practice and, as our vignettes suggest, made real changes in a short amount of time. Helen was able to shift in her understanding of what it means for teachers to understand their learners, as she moved from a simple desire to know the backgrounds and families of her students to thinking about how her particular students needed to be taken into account as she planned her mathematics lessons. By the end of her second year, Helen was making an effort to understand a young person’s mathematical thinking and confusion. But it was clear that her recognition of the importance of connecting students with content did not mean that she was able to, in effect, pull it off. Such work is hard, especially because it has to happen in the moment, and in a mathematics classroom, there are many minds, not just a single learner’s, that need to be connected with particular content. Of course, pushing against Helen’s growing enactment of a practice that realized that her learners had ideas, questions, and unique understandings were the pressures outside the classroom that encouraged her to think that covering the material at a particular pace was the goal. Novices, even strong ones like Helen, need help in pulling together this complex bundle of commitments and understandings, thinking about what that means for mathematics teaching, carrying this out in the moment (and over moments), and doing this within a particular school context. Similarly, Ona showed great insight as a beginning teacher who was ready to think about her role beyond the narrow framing of student control. She was trying to figure out her role around student learning and not just classroom management. But as her repeated comments suggested and as her actions in the classroom demonstrated, she had conflicting understandings of how she could construct her role as a mathematics teacher. The construction she relied on in these first years, as facilitator, didn’t help her understand how to connect students and content. The role she actually enacted in her teaching was at odds with other aspects of her intentions. Ona, like other new teachers, needed the guidance and insight of others to help her question her position, consider alternatives, and, as a mathematics teacher, develop a practice. Sharon FeimanNemser (2001) argued that the needs of beginning teachers in the period of induction are unique and reflect the particular position of that moment: Teacher induction is often framed as a transition from preservice preparation to practice, from student of teaching to teacher of students. As these phrases imply, induction brings a shift in role orientation and an epistemological move from knowing about teaching through formal study to knowing how to teach by confronting the daytoday challenges. Becoming a teacher involves forming a professional identity and constructing a professional practice. (p. 1027) For Helen and Ona, as well as other new math teachers, these early years of learning to teach were, as FeimanNemser suggested, “inherently paradoxical. Like all beginning professionals, they must demonstrate skills and abilities that they do not yet have and can only gain by beginning to do what they do not yet understand” (p. 1027). They must, in multiple senses of the common phrase, pull it all together. That is, they must connect threads of knowledge—about mathematics, students, and teachers. And they have to do this in convincing ways, in the moment, in the very public gaze of a classroom full of young people. Learning to teach is complicated enough, as many research and personal narrative accounts of teaching document. Especially in the settings in which new mathematics teachers today work, this learning is particularly challenging. Ona and Helen worked in contexts in which mathematics was a highstakes enterprise, with external pressures for accountability that helped shape the goals, scope, and pace of teaching. Knowing how to navigate the teacher’s need to define her own purposes and her need to support students’ success as defined by the particular context is difficult. In addition, standardsbased mathematics teaching calls for understandings of mathematics, children, and the teacher, and managing those relationships that, as Heaton (2000) documented, require teachers to engage in a kind of learning that goes beyond the already complex demands on any beginning teacher. Helen’s and Ona’s perspectives on their experiences remind us that this learning of beginning mathematics teachers is fraught with difficulties. If we only look at the surface of their teaching, both of these teachers look like competent swimmers (to extend our metaphor). Their mentors, in both cases, see them as successful novices, ones who can move through the water with relative ease, and, at least in Helen’s case, the mentor decides, given the many demands on her own time, that Helen can do well enough on her own by her second year. But both of these teachers, as the quotations early in the article remind us, describe their own sense that they are struggling. The waters in which they are swimming are challenging. And although they seemed to be doing relatively well for beginners, from their vantage point, far from any comfortable shore, teaching appeared an uncertain and confusing practice. For the purposes of this article, we have chosen to focus on these two novices because we had relatively fuller case data available for each of them. However, our analysis of the larger set of data suggests that the patterns we discuss next for these two are in fact ones that characterize the experiences and challenges of many of the novice teachers we studied. For Helen and Ona, and their counterparts elsewhere, to thrive and grow as mathematics teachers, sustained conversation and activities that allow them to think more deeply about and develop ways of constructing what Lampert described as the three prongs of teaching are important. Such support means that induction programs and mentors need to be able to recognize the distance that the new teacher experiences between intentions and reality. FeimanNemser (1998, 2001) and Schwille (2008) made strong cases for a vision of “educative mentoring.” The perspective of “successful” beginners like Ona and Helen reinforces the need for a vision of induction that entails mentoring as “professional practice” (Schwille) and sees induction as helping novices construct an interconnected, subjectspecific practice. One needs to go beyond simply the visible performance or attention to instructional strategies and think, with the new teacher, about proceeding “simultaneously in relations with students, with content and with the connection between students and content” (Lampert, 2001, p. 33). Our purpose in writing was to begin the conversation about the difficult challenges faced by even wellstarted and strong new mathematics teachers. Although we have said much, our work is preliminary and offers only a starting point for future research. Although Helen and Ona had similar needs that were reflected across other new teachers in our large data set, there is still much that could be learned through a more comprehensive examination of needs of new mathematics teachers and then a comparative examination of how these needs are different from the needs of new teachers from other disciplines. Research might also explore what happens to all new teachers—swimmers and nonswimmers—when induction shifts to focus more on swimming and succeeding as a mathematics teacher. We are also unsure about the best ways to support new teachers in engaging in the challenging work of teaching mathematics. In this article, we’ve outlined two struggles, and we suggest the difficult connections that new teachers struggle to make, but there is still much to be explored about the strategies induction programs might implement so that strong swimmers, like Helen and Ona, become even more adept at the challenging work of teaching mathematics. Notes 1. The data set was collected as part of the Mathematics and Science Teacher Induction study (MSTI), a National Science Foundationfunded study (Grant No. DRL0207623) jointly conducted by researchers at Michigan State University and WestEd. The opinions expressed herein are those of the authors, not the National Science Foundation. 2. Throughout this article, we will use traditional teaching practices interchangeably with teachercentered practices to refer to instruction that primarily emphasizes skills and transfer of knowledge of the mathematical canon from the teacher to the student. We will use standardsbased teaching practices or studentcentered practices to refer to instruction that primarily focuses on engaging student ideas and providing scaffolding for supporting students in making connections to mathematical processes and content. References Achinstein, B., & Barrett, A. (2004). (Re)Framing classroom contexts: How new teachers and mentors view diverse learners and challenges of practice. Teachers College Record, 106, 716–746. Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83–104). Westport, CT: Ablex. Banks, J. A. (2002). 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Mentornovice conversations about teaching: A comparison of two U.S. and two Chinese cases. Teachers College Record, 106, 775–813. Wong, H. K. (2002). Induction: The best form of professional development. Educational Leadership, 59(6), 52–54. APPENDIX Summary of Classroom Observation and Interview Protocols New Teacher Classroom Observation Preobservation Interview This interview asked about who would be in the classroom, how the lesson fit within a unit, what the new teacher hoped students would learn, and how the lesson would progress. Postobservation Interview This interview asked how the lesson went: what was surprising, which students understood, which students struggled, how they made specific instructional decisions in the moment, and how typical the lesson was. Observation WriteUp Field notes for each observation consisted of information about the class, including a sketch of the room and any readily observable demographic information about students. The school and the school community were also described. The lesson was scripted and then, at the conclusion of the visit, the classroom observer responded to a series of analytic questions about subject matter, students, classroom discourse, norms, reasoning about practice, and learning to teach. Wave 1 Interview With New Teacher This interview occurred during the fall of our first year of observations. During this interview, we asked about the new teachers’ background, experiences and participation in the induction program, experiences with their mentor, other opportunities to learn to teach, how they are thinking about themselves as teachers, and their subject matter preparation. We then gave teachers examples of student work on a mathematics task and asked them to comment on the work and whether/how they might use the task with their own students. Wave 1 Interview With Mentor During this interview, we asked about the mentors’ teaching and mentoring background, their views of mentoring, their work as a mentor, and their experiences with the induction program. Wave 2 Interview With New Teacher This interview occurred during the spring of our first year of observations. We produced a map of our current understanding of the induction program and asked the new teachers to talk about their understanding and experiences of the program components on the map. We then gave them a list of mentoring activities and asked them to talk about any activities they had done so far with their mentor. We also asked about other opportunities to learn, how they were thinking about themselves as teachers, and any changes in their teaching situation. Wave 2 Interview With Mentor We used the same map we used with the new teachers and asked the mentor teachers to comment on which activities they had done with their new teacher. We also gave them the list of mentoring activities and asked them to talk about which activities they engaged in with their mentee. We also asked about what they felt was hard for their mentee and how their work with their mentee had changed over time. Wave 3 Interview With New Teacher This interview occurred during the fall of the second year of our observations. We started by asking the new teachers about their current teaching assignments. We then asked about their views of themselves as teachers. We gave them a list of teaching practices and asked them to tell us whom they know who might be an expert in each practice. We then gave the new teachers a stack of cards with teaching practices and asked them to divide them into piles based on how much they were working on each practice. For the teaching practices they were most focused on, we also asked them whom they went to for help with those practices. Wave 3 Interview With Mentor We started with the map of the induction program and asked them to comment on what they were doing with their mentee and how that differed from the previous year. We then showed them the list of mentoring activities and asked them to comment on activities they were doing with their mentee. We then asked them for their perspective on what a successful teacher looks like and what their goals were for their mentee.


