Teaching and learning mathematics in the collective

Journal of Mathematical Behavior 32 (2013) 424–433

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Teaching and learning mathematics in the collective Jo Towers a,∗ , Lyndon C. Martin b , Brenda Heater c a b c

University of Calgary, Canada York University, Canada Calgary Board of Education, Canada

a r t i c l e

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Article history: Available online 28 June 2013 Keywords: Teaching Group learning Group cognition

a b s t r a c t In this paper we analyse and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiryoriented. We frame our discussion within enactivism—a theory of cognition that has helped us to understand classroom processes, particularly at the level of the group. We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective. Our analysis reveals teaching practices that value, capitalize upon, and promote group cognition, our discussion positions such work as teaching a way of being with mathematics, and we close by offering implications for teaching, educational policy, and further research. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Why do we gather students together into groups in schools to learn mathematics? In many (perhaps most?) schooling systems, the answer to this question likely has much to do with ideas of ‘efficiency’—a consequence of importing of turn-ofthe-century management practices into schools, based on the ideas of Frederick Winslow Taylor (Friesen & Jardine, 2009). With this efficiency mindset, though, comes a set of problematic assumptions about groups versus individuals—for example, and to list just a few, the assumption that the diversity (of learners) in a group is a problem to be managed not an asset, the assumption that learning can be efficiently sequenced and therefore a best teaching sequence isolated, the assumption that learners in groups need constant management and surveillance, and the assumption that, given the choice, individual one-on-one teaching would be better but, as it is not ‘efficient,’ grouping students into as small a group as can be afforded is an acceptable compromise. However, suppose we set aside such assumptions? Could it be that there is educational gain in group learning, that diversity is not automatically a problem, and that the search for the best sequence for the teaching of mathematical concepts is distracting teachers from other important aspects of preparing to teach? While in this paper we cannot immediately answer all of these questions, we begin by considering what difference it makes to gather high school students together to do mathematics in ways that we characterize as genuinely collective. What kind of mathematizing is possible under these conditions? And how might a teacher privilege, make use of, and promote the collective, rather than simply try to work within its perceived constraints?

∗ Corresponding author. Tel.: +1 4032207366; fax: +1 4032828479. E-mail address: [email protected] (J. Towers). 0732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmathb.2013.04.005

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2. Enactivism In interpreting the basic idea of a group or collective we draw on enactivism, a theory of cognition that has its roots in biological and evolutionary understandings. Enactivism views human knowledge and meaning-making as processes that are understood and theorized from a biological and evolutionary standpoint (Proulx, Simmt, & Towers, 2009). Maturana and Varela (1992) note that species and environment co-adapt to each other, meaning that each influences the other in the course of evolution, a process they refer to as structural coupling (Maturana & Varela, 1992). Maturana and Varela (1992) explain that events and changes are occasioned by the environment but determined by the species’ structure: Therefore, we have used the expression “to trigger” an effect. In this way we refer to the fact that the changes that result from the interaction between the living being and its environment are brought about by the disturbing agent but determined by the structure of the disturbed system. The same holds true for the environment: the living being is a source of perturbations and not of instructions. (p. 96, emphasis in the original) Hence, we understand that changes do not reside inside either the organism or the trigger, they come from (and are dependent upon) the organism’s interaction with the trigger. This suggests that in teaching we cannot make the assumption that instructional properties are present in and of themselves in the tasks offered or the teaching moves made and that these will determine learners’ reactions. Nor, though, can it be assumed that the responsibility to learn rests solely within the learner and simply needs to be ‘facilitated.’ While enactivist thought helps to emphasize the critical role of the teacher as a trigger, it is in the interaction between the learner and environment that learning happens, not ‘because of’ the learning environment (including the teacher) or the learner him/herself. Simplistic interpretations of cause and effect in teaching and learning are therefore problematized in an enactivist interpretation and learning is seen as reciprocal activity—the teacher brings forth a world of significance with the learners (Kieren, 1995; Maturana & Varela, 1992). The teacher is a fundamental part of the learners’ processes (Proulx, 2010), and a “full participant in the emerging cognitive structure of the learning unit” (Towers & Martin, 2009, p. 47) that includes all players in the environment. Teaching is therefore “a complex act of participation in unfolding understandings” (Martin & Towers, 2011). Despite the fact that we gather bodies together to learn, many of our current schooling structures (e.g., assessment practices) rely more on Cartesian thinking, which has successfully emphasized the ideal of a modern self as solitary, coherent, and independent of context. Such thinking has then positioned this radical subject as the reference point for what is known (and worth knowing). The ideal knower, in this frame, has been the autonomous individual. Conversely, enactivist thought prompts a reorientation to the collective body, both in terms of what is known and of who is doing the knowing (Glanfield, Martin, Murphy, & Towers, 2009; Towers, 2011). Enactivism prompts attention to the structural dynamics of knowing and the co-emergence of knower/known. The starting point, then, for understanding classrooms as collectives is a severing of an attachment to the individual. As Froese (2009) notes, “cognition is a situated activity which spans a systemic totality consisting of an agent’s brain, body, and world” (p. 105) and this world includes other bodies and minds (as well as the typical tools of the classroom such as drawing instruments, whiteboards, and computers). Enactivism, then, has prompted us to pay attention to the relationship between things in a mathematical environment (ideas, fragments of dialogue, gestures, silences, diagrams, etc.), rather than to what each of those things might mean or represent in their own right and for the individual generating them. It has re-oriented our attention to collective activity. 3. Review of the literature on teaching groups of learners Recently, North American school mathematics curricula and policies have moved to valorize collaborative processes in learning (see, e.g., Alberta Education, 2007; WNCP, 2011), therefore pressing teachers to consider and adopt new teaching practices that respond to this expanded vision of how learning should be orchestrated in classrooms. There is a wellestablished and growing literature on group learning in the mathematics education field. Researchers recommend that teachers employ group processes in the classroom for various reasons, including to enhance students’ communication and reasoning skills in mathematics and to foster equity in the classroom (e.g., Boaler & Staples, 2008; Elbers, 2003; Esmonde, 2009; Zack & Graves, 2001). For example, Boaler and Staples (2008) report on a study of a secondary school in California where teachers taught heterogeneous groups in mathematics using a reform-based approach. The authors report that, compared with the other two schools in the study, students at this school learned more, enjoyed mathematics more, and progressed to higher mathematics levels. Furthermore, achievement gaps among various ethnic groups that were present on incoming assessments disappeared in nearly all cases by the end of the second year of the study. Many studies focus on the processes involved in interacting in groups (Armstrong, 2008; Cobb, Boufi, McClain, & Whitenack, 1997; Martin & Towers, 2009, 2011; Martin, Towers, & Pirie, 2006; Towers & Martin, 2009). Over the last decade, Cobb and colleagues have identified and explicated the sociomathematical norms that enable groups to learn mathematics through discourse in high-activity classrooms (e.g., Cobb, Stephan, McClain, & Gravemeijer, 2001). Davis and Simmt (2003) and Armstrong (2008) have explored the conditions within which complex systems, such as the interacting participants of a classroom community, operate, and much of our own recent work illuminates the ways in which groups are able to coordinate their speech and actions as they coact to generate collective mathematical understanding within small groups (Glanfield, Martin, Murphy, & Towers, 2009, Martin & Towers, 2009, 2011; Martin, Towers, & Pirie, 2006). Others have analysed individual contributions to, and outcomes of, such group learning (e.g., Barron, 2003).

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While much of the research on teaching groups of learners focuses on small-group activity within the context of the larger classroom group, a significant section considers the classroom group as the unit of study (see, e.g., Jones & Tanner, 2002; Nathan & Knuth, 2003). Analysis of teaching and discourse in such classrooms consistently points to the significance of shared authority for learning in the classroom (e.g., Forman & Ansell, 2002; Ju & Kwon, 2007; White, 2003) and where this is authentically practiced, such as in the classroom we describe in this article, learning is recognized to be emergent and “collectively determined” (Sawyer, 2004, p. 13). In such classrooms, the teacher needs to be willing to share control, to allow events to unfold, and yet has a responsibility to act at critical moments. Knowing when and how to act in such a classroom requires a sophisticated improvisational competence that is unnecessary in a classroom dominated by more traditional modes of instruction—for example, the well-known IRE sequence, where a teacher deliberately funnels the talk and action in the classroom in such a way that there are a very limited number of possible responses to any question that is asked. [To authentically share authority for learning] the teacher must continually listen to, and re-connect with, the improvisational actions of students [and possess] a sophisticated capacity to step back until the collective action calls him or her forth. (Martin & Towers, 2011, p. 275) 4. Methodology and methods 4.1. Data collection The broad study on which these ideas are based was designed to explore the nature of collective mathematical understanding—a phenomenon that we have defined as acts of mathematical understanding that can not simply be located in the minds or actions of any one individual but instead emerge from the interplay of ideas of individuals as these become woven together in shared action (Martin & Towers, 2009, 2013; Martin, Towers, & Pirie, 2006)—and to develop and elaborate a theoretical framework for its growth. Data were collected in two high school classrooms (taught by the same teacher) in a single high-school in a Canadian city. The school has approximately 1500 students, drawn from a mixed blue-collar/middleclass neighbourhood and representing a wide ethnic diversity with the majority of students being of Asian descent. Many students in the school have a language other than English (the medium of instruction) as their first language, with over 20% having a special educational designation of ‘English Language Learner.’ Approximately 60 students participated in the study. Mathematics lessons were videotaped (with two cameras in each classroom) daily for an initial period of approximately 15 days at the beginning of the semester with a follow-up period of 5 daily lessons at the end of the semester, copies of student work and the teachers’ planning notes were collected, field notes were recorded daily during data collection periods, and initial planning meetings with the teacher before the semester began were also videotaped. In-classroom data collection did not begin until all student/parent consent forms had been returned a couple of weeks after the course started. Hence, some of the initial practices designed to initiate students into the culture and norms of this classroom were not captured on video and in analysing such practices we relied on the teacher’s self-reports of her preparatory practices, analyses of our videorecorded planning sessions in which the topic of initiating inquiries in these classrooms was discussed, and on our field notes from ongoing conversations with the teacher throughout the semester as we delved back into the origins of some of the practices that we observed. The level of interaction between the classroom teacher, researchers, and research assistants was high, with the researchers and research assistants sometimes engaging in co-teaching with the teacher in the two classrooms and actively interacting with students during lessons throughout the study. For the purposes of this article we focus on data collected in one of the two classrooms we studied—a group of approximately thirty-five Grade 10 International Baccalaureate students who were studying the Grade 11 Alberta Program of Studies for Mathematics. 4.2. Data analysis Data analysis proceeded through an adaptation of the approach proposed by Powell, Francisco, and Maher (2003). The first stage of analysis involved becoming familiar with the sessions in full, viewing the lessons in their entirety to get a sense of their content without imposing a specific analytical lens. In the second stage, the video data were described through writing brief, time-coded descriptions of each video’s content. In stage three the data (video tapes, time-coded notes, supplementary materials) were reviewed to identify “critical events” with regard to our objectives. Thus, we identified instances where collective growing understanding could be observed and instances where the teacher made explicit interventions in the learning process (such as problem-setting, direct teaching, and interacting with small groups). One such critical event, for example, is a 32-min episode in which a small group (initially of three students) works together at a whiteboard to prove an angle theorem involving circle geometry. The problem is ultimately solved through a collective process that includes multiple interventions by members of other neighbouring groups, one of the original group members moving in and out of the “action” as he withdraws to work with another group then returns to insert his ideas, and a brief interaction with the classroom teacher. Stage four involved examining closely these critical events to identify and construct a series of emerging narratives about the data, of which this perspective on the role of the teacher is one. Due to the high level of movement and interaction in this classroom (as described in the next section) transcripts of events in this classroom have proven less useful than those documenting events in a conventional classroom and we have

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worked, in developing our analyses, exclusively from the video data. Hence, in the following section, rather than present slices of transcription which might be incoherent to the reader, we have elected to describe life in this particular classroom in narrative form so as to paint a picture of the typical structures, activities, and norms at play in the learning environment. 5. Data 5.1. Life in the classroom The physical arrangement of the classroom is conventional, with single-student desks—sometimes two or three clustered together—all in rows facing the front. At the front is a teacher’s desk and a trolley holding a laptop and projector. Unusually, perhaps, there are whiteboards covering the entire length of three of the walls. It is a small room and a very tight fit for the 35+ students who populate this class. As in many Canadian high schools, in this school many teachers, particularly those new to the school and therefore with less ‘seniority,’ move from classroom to classroom throughout the teaching day and so the classrooms are often set up (and have to be left at the end of each lesson) in a standard organization and boards must be wiped clean at the end of every class. Though an experienced teacher, Sharon1 is in only her second year teaching at this particular school and so her timetable requires that she move classrooms for almost every period. A typical lesson in Sharon’s classroom begins with random assignment of seating. She uses a variety of methods to enact her desire to mix up the seating arrangements of students from day to day, perhaps the most unusual documented during the study being the request that every student remove a shoe as they entered the room, shoes then being randomly distributed one per desk to designate the seat the owner should occupy at the start of the lesson that day. None of the students seem uncomfortable with this daily process and they participate good-humouredly in the ever more creative strategies to vary the seating plan. Neither does any student show any reluctance to work with any other member of the class, a feature we find interesting given the social stratifications and gender norms that typically operate in high schools. While the physical organization of desks and chairs remained static throughout the period of our study, norms established by the teacher meant that the students spent relatively little time actually sitting in seats. Even when students are seated they frequently move chairs to facilitate working with a neighbouring group across the aisle, or with students seated in the row behind, or they simply get up and wander around the room seeking inspiration, direction, cues, solution paths, mathematical devices, or conversation. Work sometimes begins with the teacher writing a problem on the board, but sometimes students are expected to continue working on a problem from the previous day despite now being seated with new group members. Again, none of the students appear flustered by this expectation. As we describe more fully in a moment, the students operate as a collective and are not ‘tied’ to a particular set of partners, even for a single problem or task. Most days, the students are offered a problem and encouraged to get out of their desks to work on the many whiteboards. Considerable numbers of whiteboard pens are made available to them and they jostle for position, some students writing on the board, others offering suggestions about what to write or draw. Students often add to one another’s drawings, or erase all or parts of a drawing someone else has created. No-one objects to such ‘interference.’ Students in this classroom move constantly. They participate in a discussion here, move off to add something to a drawing over there, listen in on another group as they pass by (sometimes contributing but just as often not), pass on an idea they have just heard about, and sometimes (but not always) return to the group where they started. One student we come to nickname “the pollinator” for his peripatetic spreading of ideas around the room. Another student is a copious note-taker and her notes are valued and shared by the group. If you were to walk into this classroom on a typical day, you would find every student standing, clustered in two’s, three’s or more around a slice of whiteboard that they have etched out for their own use. The room is crowded and students are shoulder to shoulder and two or three deep from the boards. There is a hub-bub of noise and lots of movement. Everyone is working on the same problem or set of inter-related problems. All work is visible for others to see. At any one time, perhaps fifteen students are writing or drawing on the boards. Others are leaning in to point to, or erase, something. Some are moving from group to group, watching, listening, or intervening with suggestions or questions. Many, but not all, students are carrying a pad and pen and sometimes they record something, but not always. Students jostle for position and the whiteboard pens are passed from person to person as a member of the small group decides they have something to add, or when a student from a different group arrives and has something to contribute. Sometimes a student watches in silence for an extended period before suddenly exploding with an excited lunge to grab a marker pen from a peer and start writing, but sometimes after a long period of silence the student simply walks away to join another group. Both of these actions are unremarkable to the learners and in either case activity continues uninterrupted. Disagreements about mathematical processes and solutions erupt and are resolved, usually without recourse to asking the teacher to intervene. If the teacher ‘interrupts’ (perhaps thinking that the student have struggled long enough with a tricky problem) and offers a solution, she is shouted down with calls of “I want to prove it,” “I want to figure this out,” and “I want to solve it, though.” The teacher invariably acknowledges the will of the group and gives them more time. If, as an observer, you focus in on one small group’s activity, you will see that students from the neighbouring groups to the right and left frequently intervene in ‘your’ group’s solution-building. Hands appear from one side or the other to point at your focus group’s drawing or mathematical symbolism. Sometimes

1

All names are pseudonyms.

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the intervening hand is wielding a marker pen and something is added—with or without an explanation offered. Sometimes the hand wields an eraser and something is erased. Your focus group perhaps questions the addition/deletion, but perhaps not—just assuming it is intended to assist and carrying on from the amended position. All amendments, it appears, are taken as friendly. Work on the whiteboards is treated as public property and belongs to the group. The teacher might pass by and stop at your group’s section of the board. She might watch and listen and pass on without interrupting, or she might ask a question, make a suggestion, offer a different strategy, explicitly teach a required concept, or lead the group forward towards a solution through a Socratic dialogue. From time to time, students are asked to return to their seats. They might be assigned some practice questions to reinforce a concept that has emerged in the problem-solving, or be asked to make their own notes on their findings, or they might review, as a class, a test taken the day before, or participate in a whole-class discussion as a new topic is introduced or as a particularly thorny problem is explored. 5.2. Teaching in this environment Planning for teaching and learning in this kind of space requires a preparation that goes well beyond the textbook and teachers’ guide. Sharon plans for teaching in a flexible manner, starting with the curriculum guide to ‘locate’ herself for the unit, drawing on a range of resources or creating her own problems and activities, and always reflecting on the previous lesson to determine where to begin the next. The following excerpt from Sharon’s journaling shows some of her thinking about preparing to teach. Sharon:

I usually begin preparing a topic by starting with the curriculum—what is it we are to work around? Then I try to consider the big ideas. The curriculum document is written in bits so I look for relationships, connections and possible problems and questions to pose. I have a general idea of the where we need to go but I honestly wait to see what happens in the [previous] class, then I consider where we are at the end of the day—what has come up, what are the possibilities in their wonderings or struggles related to the curriculum and where can we go? Then I begin to think or search for a problem or activity that will require input from all group members, not just a situation where the ‘smart kid’ can just answer it and the rest write it down. No two classes are ever the same from one year to the next so there is no way to predict what will happen. Often, they outrun the curriculum, like we saw them dance around calculus [during one of our data collection sessions in the class], so then I have to consider how far do we go, especially since they are so interested. I have some books, websites, and past experiences that I draw on for ideas. The textbook can be a good source if you see that you can rewrite a question to open up the possibilities.

During lessons, Sharon’s flexible and wide-ranging planning for teaching shows itself in the kinds of activities she chooses and in the ways in which she engages with both the students and the mathematics. Sharon is active in offering prompts for learning (e.g., choosing and writing a focus problem on the board), engaging in conversations with individuals and groups about the emerging mathematical concepts, directing students to relevant resources, teaching particular concepts to a small group or the whole class, or looking up information on the web and directing students’ attention to relevant sites. She is relaxed and on good terms with the students. They seem familiar with Sharon’s family life and eagerly anticipate stories about her young son and his escapades in school. Students casually share jokes and tidbits about their own lives and experiences. Things are calm—until the mathematics begins and from then on there is feverish activity. As she moves around the room while students are working, Sharon is adept at reading the environment. She has a depth of working knowledge of mathematics for teaching that allows her to quickly recognize what strategy the students are using and whether it is fit for the purpose to which it is being applied. She shows great curiosity about every detail of the mathematizing. She listens patiently to circuitous arguments, never hurrying a student whom she believes needs to articulate his or her thinking, but by the same token will not tolerate shoddy mathematical argumentation, symbolism, or documentation if she knows that the group knows more than they are showing. When she sees an incorrect solution emerging in one area of the classroom, she rarely acts immediately to redirect the group, trusting the collective and giving students time to self-correct. Her authority is not in doubt—the students have great respect for her—but it is won through co-participation. She actively engages in doing mathematics at the whiteboards, not simply checking the students’ mathematics, because the nature of the problems she sets means that there are always new avenues to explore and student approaches that are novel to her. 6. Analysis Our analysis reveals significant insights into the ways in which a collective such as the one described above might be orchestrated in the high-school setting. We have clustered these findings into two groups: (1) those that address the (teaching) structures necessary for initiating and sustaining such a collective, and (2) those that reveal the kind of relationship with mathematics that is fostered in this environment. We draw together these two sets of findings in the discussion section by describing the collective practices as a way of being with mathematics. 6.1. Structures necessary for initiating and sustaining a collective Planning for teaching in this setting is much more a matter of good preparation than step-by-step planning. The teacher pays careful attention to unfolding events each day and makes a decision to move forward with a new problem or into a new conceptual area, or to dwell for longer, or even step back to re-teach a previous idea that seems to need reinforcement,

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based on her careful assessment of the group’s learning each day. This requires a deep and connected sense of mathematics that allows the teacher to pick up upon the right concept to introduce at the right moment yet keep an eye on the broader mathematical territory. It is the kind of disciplinary knowledge highlighted by Friesen and Jardine’s concept of “field[ing] knowledge” (2009) and requires the conceptual agility and depth identified as critical by many of those currently studying teachers’ disciplinary knowledge (Adler & Davis, 2006; Ball & Bass, 2000; Davis, 2010). It is clear from Sharon’s comments in the journaling excerpt we included in Section 5.2 that she understands that she brings a connected sense of mathematics to her work. (“I try to consider the big ideas. The curriculum document is written in bits so I look for relationships, connections.”) A deliberate fluidity also ‘structures’ this classroom. Seating assignments are purposefully random. This structure creates the expectation that everyone engages with, and should expect to learn with and from, everyone else. The group as a whole is considered the learning unit. Further, students move freely around the room once the task is initiated and students actively push the boundaries of this structure—seeking out multiple working partners, multiple solution strategies, new ideas, and interesting mathematical detours. The structure of the physical environment is also of significance. The students make active use of the whiteboard space, and these boards are not only a place to record their working but a source of ideas. We regularly noticed students inching their way around the crowded room, not copying notes from other places as might be the response of a student in a more traditional class who suddenly finds him- or herself surrounded by others’ solutions, but simply seeking a way to think about the problem. When our data show students returning after such a mission, they reveal that students rarely brought back a clear solution and much more often returned with an idea about how to structure the work or a clue about what aspect of the information in the problem to focus upon. Students in this classroom use the opportunity offered by potential (and sometimes actual) interaction with every class member for problem-solving. They are able to ‘pick up’ their previous day’s processing and continue it with interchangeable working partners because the group as a whole is the cognitive unit here. Ideas promulgate around the room (sometimes rapidly, sometimes slowly over multiple days) and such sharing of intellectual property is considered not only to be legitimate but expected, by both teacher and students. In addition, the collective is valued as a means through which mathematical errors are caught and modified, ideas are questioned and exposed to challenge and verification, and mathematical conventions are agreed upon and then reinforced. 6.2. Relationships with mathematics fostered in the collective In this classroom, the teacher and the students, and, in fact, knowledgeable others such as classroom visitors and electronic sources, all constitute mathematical authorities. The teacher purposely engages such resources actively, as do the students. The teacher also encourages students to verify their own results with reference to the (classroom) collective. When students complete a solution, they typically set off around the room looking at other students’ work, thereby verifying their own strategies and also interpreting other approaches. The teacher often directs attention to multiple solution strategies. In this sense, Sharon views (and expects her students to view) the classroom (and every interaction and material object constituting it) as a text to be interpreted and she actively models what such an orientation to learning mathematics looks like. The kind of mathematics learned in this classroom is created, not transmitted, knowledge. Further, it is created in and by the collective. It has a focus on fundamental concepts, proof, and reasoning, but it is also a mathematics that is open to interpretation and challenge by the community. Mathematics in this space is not a static body of knowledge but a territory to be explored and expanded. When creating mathematics, no-one objects to ‘interference’ from other class members because in this classroom the objects of study are a means to enter the discourse not a property of individuals or small groups of individuals. Students use the whiteboards as a collective thinking space. They draw, erase, and re-draw constantly and they do so with their own and with others’ work. They recognize that to participate in mathematics in this space they must contribute to it as well as consume it. 7. Discussion—teaching a way of being (with mathematics) Maturana (in Gumbrecht, Maturana, & Poerksen, 2006) notes that, “teachers do not simply transmit some content; they acquaint their pupils with a way of living. In the process, the rules of arithmetic, the laws of physics, or the grammar of a language will be acquired” (Gumbrecht, Maturana, & Poerksen, 2006, p. 26). So, we might infer that what students learn when they interact with each other and a teacher in a mathematics classroom is primarily a way of living—a way of being with mathematics—and that specific mathematics knowledge (the very thing that most teachers are trying to teach) is a kind of by-product of this other, more significant, process. What can we interpret about a way of being with mathematics in the classroom we have described? Firstly, we note that in this classroom, “what is not prohibited is permitted” (Varela in Gumbrecht, Maturana, & Poerksen, 2006, p. 43). Students move freely around the room once the task is initiated. “There are natural limits but there is no densely woven, blocking, and stifling system of rules” (Varela in Gumbrecht, Maturana, & Poerksen, 2006, p. 43). In this classroom, doing mathematics is a public not a private process. Looking at other students’ work is not forbidden—in fact it is expected, as is questioning and challenging other students’ solutions and processes. Secondly, it is expected that (mathematical) things are interesting and anything interesting can be pursued. Students seem to believe that interesting things will happen (mathematically/socially/structurally) in this class. As Maturana (in Gumbrecht,

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Maturana, & Poerksen, 2006) notes, “everything is interesting once you are interested in it” (p. 26) and one powerful aspect of the teacher’s practice is to treat mathematics as an authentic and intrinsically interesting discipline. This orientation sets the tone for the classroom and encourages authentic interest in the discipline on the part of students. Our analysis of Sharon’s strategies for preparing to teach indicates that her choice of problems that connect to one another and that draw upon, or through their solution point to, fundamental mathematical ideas, is one important feature. Another is her insight in choosing problems that no one student can solve individually—a feature of tasks that we have noticed, in our previous research, is significant in prompting collectivity (Martin & Towers, 2009). One further feature of note is Sharon’s rewriting of textbook questions with the aim of “open[ing] up the possibilities.” Such efforts, we believe, return the discipline of mathematics to its original difficulty (Caputo, 1987), thereby treating the discipline as though it were intrinsically interesting and ensuring that all students in the classroom have a place to begin problem-solving as multiple possibilities for inquiry are offered with every problem. Thirdly, in this classroom, mathematical reasoning is emphasized and students act as though mathematics is meant to make sense. It is our impression, gleaned from the many hours of video data collected in this classroom, that the students seek understanding, sometimes going so far as to resist help and hints offered by the teacher (who occasionally offered to be more ‘teacherly’ than the students desired). Finally, we note that there is a tangible sense of collective purpose in this classroom, with students’ actions directed to helping all members of the group learn. The atmosphere is non-competitive and collegial, an environment deliberately fostered and structured by the teacher. As Varela (in Gumbrecht, Maturana, & Poerksen, 2006) notes, understood through the lens of enactivism, in such an environment “actions. . .are not primarily intended to strengthen the individual personality but always serve to build up relationships with other people” (p. 47). Actions serve to strengthen the collective, which provides the sustenance for the individual to flourish. There is mutual determination between organism (learner) and environment—each enhances (and adapts to) the other. In this classroom, the students’ actions indicated that they saw their presence in the flow of mathematical interactions as legitimate—one in which their contributions mattered both to the moment-by-moment unfolding of mathematical understanding and to the way in which subsequent lessons would be shaped. Sharon reflects this legitimizing of the student role in the mathematizing when she describes always judging at the end of a lesson “where we are. . .what has come up, what are the possibilities in their wonderings or struggles related to the curriculum and where can we go?” Student voice is, then, in a very real way, a component of planning for teaching in the collective. Sharon’s planning for teaching shows that she understands that teaching mathematics requires us to listen to what particular situations call forth from us, rather than “requiring of all situations, all speech, all children, and all activities that they live within the boundaries that we have already drawn” (Jardine, 1998, p. 61). In this classroom, students were granted “a space of legitimate presence. . .in the flow of interactions” (Maturana in Gumbrecht, Maturana, & Poerksen, 2006, p. 28). 8. Implications 8.1. Implications for teaching In viewing teaching as a way of being with mathematics, our analysis of this classroom suggests that teaching in the collective includes elements such as thoughtful preparation (see Section 5.2), genuine respect for, and capitalizing on, student thinking and presence in the classroom (see Sections 5.2 and 7), and deep and connected mathematics knowledge (see Sections 5.2, 6.1 and 6.2). It also suggests that such work calls forth dispositions for teachers that include tolerance for ambiguity, patience in meeting curricular objectives and teaching goals, trust (in students), and (mathematical) curiosity, and that these must be cultivated and exercised despite the conditions under which teachers must practice. Significantly, this study also shows that teaching in the collective is possible even within the constraints of a typical high-school setting, with its rigid schedules and classroom layouts and assessment regimes. This is an area with scope for much further research that documents the ways in which traditional schooling structures allow for, or interrupt, teachers’ efforts to teach authentically in the collective. We also acknowledge that there are challenges in working in this way in high-school. Sharon has expressed some of these to us and they include the difficulty of assessing joint problem-solving (in the context of high-school and provincial policies for individual standardized assessments), the challenge of communicating about her practices to other teachers, and the difficulty of knowing what to encourage students to record as ‘evidence’ of their activity. Each of these areas offers scope for other teachers to contribute to the literature by documenting their own practices as they teach collectives (rather than collections of individuals). 8.2. Implications for policy If, as Maturana (in Gumbrecht, Maturana, & Poerksen, 2006) has suggested, students primarily learn from their teachers a way of living (with mathematics) and that it is only through such a process that specific knowledge is acquired, this study prompts us to ask why educational policies in most jurisdictions direct attention to monitoring the knowledge generated not the ways of being that are being fostered in classrooms? As studies have found, while Canadian students typically fare well on cross-national comparisons of their mathematics knowledge (Knighton, Brochu, & Gluszynski, 2010), qualified students

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in Canada and elsewhere gravitate away from post-secondary study of mathematics, particularly at the graduate level and especially women (Brown, Brown, & Bibby, 2008; Fenwick-Sehl, Fioroni, & Lovric, 2009). Clearly, students’ school experience of mathematics, even though it may have provided them with a residue of mathematical skills, has left them uninterested in the discipline and unwilling to pursue it into university and beyond. This is, of course, a serious economic concern, which is not limited to the Canadian context and is paralleled in many other Western nations (see, e.g., Smith, 2004). A (policy-driven) re-orientation to understanding how young people are living mathematically in classrooms, rather than gathering data on what individuals might ‘know’ (often only briefly) as a result of this living, could be an important first step for schools in helping to expand teaching practices and therefore strengthen students’ relationships with mathematics as a discipline. 8.3. Implications for research We see several pathways for continued research emerging from this study. Firstly, other studies that parallel ours and that aim to document expressions of teaching in the collective are needed as a means to show what is possible within the structures of regular schools and programmes and to help broaden the literature on teachers’ repertoires and approaches to whole-class teaching. We have tentatively begun such work by offering an expanded vocabulary to describe teaching actions (Towers & Proulx, 2013). Secondly, although our study did not measure individual student knowledge resulting from learning in this classroom, the findings suggest that studies that aim to link interpretations of collective activity to individual student competencies are needed so that we better understand how such classrooms contribute to students’ capabilities when called upon to represent their (shared) knowledge individually. There are a few such published studies (see, e.g., Barron, 2003) but this is certainly an area where further work is warranted. Next, we note that Sharon showed through her planning and teaching activities that she has particular dispositions suited to collective action, such as a deep respect for student thinking and presence in the classroom, and an engaged mathematical curiosity. Our work suggests that teacher dispositions is an under-researched component of teacher selection and preservice preparation and that further studies that explicitly interrogate and explicate this concept in the domain of mathematics teaching would strengthen the field’s understanding of the conditions necessary for sustaining collective action in classrooms. Finally, our analysis has revealed a need for enhanced methodological tools for studying group learning in high-activity classrooms. From a research perspective, we discovered very rapidly that documenting learning in this particular classroom was going to be a challenge, even with two video cameras and often multiple researchers present every day. Students’ movements were so frequent that it was difficult to know where and on whom to focus the cameras and microphones. Analysis of the collected data was equally challenging. We see ample scope for the development of new data collection and analysis tools and approaches to facilitate the understanding of learning patterns in these kinds of high-activity classrooms. As we develop these, it is important to emphasize the nature of our roles as researchers in this environment. Students invited us (researchers) into their conversations and explorations and sometimes we took on ‘teaching’ roles in the classroom. It was assumed by all present that we were there to be part of the collective, not simply to observe it. The researchers therefore adopted the same mode of engaging as the teacher—moving around the room, interacting with students, watching, listening, responding to student questions, asking questions about the emerging mathematics, and conversing about what we were noticing as things evolved. Hence, we became part of the collective. A reviewer of an earlier draft of this manuscript raised a question about how we might “tease out” our involvement from that of the teacher, given that in this piece we are focusing on the role of the teacher in the collective. However, our point is precisely that no “teasing out” is necessary in our formulation. As we noted earlier (Section 2), enactivism calls upon us to sever our attachment to the individual as the object of study—and this includes the teacher as individual. Teaching, then, becomes an experience of shared cognition, not an act attributable purely to the designated ‘teacher’ in the environment. For the designated teacher in the classroom we studied, other knowledgeable adults in the classroom were considered an asset to the collective rather than a source of distractions and we were made to feel that our presence was valued and significant but not “special”—in other words, in principle no different from the usual collective assembled any other day in that room, a collective that included multiple ‘teachers’ (including students) and ‘learners’ (including the teacher and researchers). Just as on other days, the teaching (including ours) on the days we were present unfolded in response to the environment and the environment (including the researchers) responded to the teaching, which came in different forms, initiated and sustained by different “bodies” (student bodies, research team bodies, etc.). Research methodologies that privilege objective observation, or even those that conceive of the researcher as somehow separate from the context, are inappropriate for studying the kind of classroom we describe here. We see no inconsistency in positioning this piece as a reflection on the role of the teacher while simultaneously noting that the actions of the designated classroom teacher cannot be separated from the actions of others present whose words and deeds may have contributed to the “teaching” in the classroom. Any future methodological tools that are developed to study classroom collectives will need to honour this sensibility. Acknowledgements Funding for this study was generously provided by the Social Sciences and Humanities Council of Canada (SSHRC), Standard Research Grant # 410-2009-0383. SSHRC exercised no oversight in the design of the research, the collection,

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analysis, and interpretation of data, or the writing of this report. We wish to thank the participating teacher and the many students who allowed us to study their learning processes as part of this project.

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