Introduction
This chapter describes the actions taken to learn by a group of high school students during mathematics lessons in a Canadian school. The examples we offer are representative of the collective learning environment collaboratively created by the mathematics teacher and her students and we use them to show how a classroom collective can work to foster meaningful learning and to analyse what it might mean to learn, and to learn to act, within such a context.
Understanding Classroom Collectives
We will begin by explaining what we mean by a ‘classroom collective.’ Despite the fact that we gather bodies together to learn in schools, many of our current schooling structures draw on the Cartesian ideal of the self as solitary, coherent, and independent of context. The ideal knower in this frame, and hence the focus of most research and scholarship concerning learning, is the autonomous individual. In our work over the last 20 years, however, we have deliberately looked beyond a focus on the individual in an attempt to theorize and understand the collective as a learning unit.
It is in the interaction between the learner and environment that learning happens, not ‘because of’ the learning environment (including the teacher) or [the actions of] the learner him/herself. Simplistic interpretations of cause and effect in teaching and learning are therefore problematized in an enactivist interpretation. (Towers, Martin, & Heater, 2013, p. 425)
The starting point, then, for understanding classrooms as collectives is the severing of an attachment to the individual and a commitment to understanding what constitutes learning in groups.
Learning in Groups
students’ meaningful activities with material objects and mathematical objects need to take precedence over their dependence on the intellectual authority of teachers and texts. Instead of memorizing and applying information from authorities, students need to conjecture and experiment. In addition, they need to address each other’s argumentative positions…to advance argumentative claims. (p. 257)
Improvisational coacting…is a process through which mathematical ideas and actions, initially stemming from an individual learner, become taken up, built upon, developed, reworked and elaborated by others, and thus emerge as shared understandings for and across the group, rather than remaining located within any one individual. (Martin & Towers, 2009, p. 4)
In later examples we show how, through this kind of improvisational coaction, students can take actions as a collective in a mathematics classroom.
Methods
Data Collection
The study on which these ideas are based was designed to explore the nature of collective mathematical understanding—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, 2015; Martin, Towers, & Pirie, 2006). In particular we were interested in (1) exploring the classroom conditions within which collective mathematical understanding might emerge—including group characteristics, task structures, and teacher role and (2) determining how collective mathematical understanding relates to and enhances personal mathematical understanding. Data were collected in two classrooms in one high school in a large Canadian city. Mathematics lessons were videorecorded (with two cameras in each classroom) daily for several weeks 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 participant teacher were also videorecorded. For the purposes of this chapter we focus on data collected in one Grade 10 classroom.
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 recordings, 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. Stage Four involved examining these critical events to identify and construct a series of emerging narratives about the data. This perspective on the
Findings
We have detailed the structures and norms of the particular classroom on which we focus here quite extensively in another publication (Towers, Martin, & Heater, 2013). Below, we summarize relevant parts of that description with particular foci on the role of students in the collective and the conditions necessary for the students to learn, and then we present three examples. The first features a small group of students and is taken from a lesson towards the end of the semester in which the class has been asked to prove the veracity of a particular geometrical relationship. We use this example to explore the actions taken by the students to learn mathematics in this collective context as well as to explore the ways in which the teacher intervenes in the learning process. The second example is taken from an episode in which the teacher discovers that the students are missing a crucial mathematical technique that she expected they would know. We use this example to show how the teacher ensures that students are still required to take actions of their own to learn, even at moments when, in other classrooms, they might expect to be shown the steps of a procedure. The third example is taken from an episode in which the teacher speaks directly with the whole class about reviewing their learning. We use this example to show how the teacher shapes how students can learn to act and learn to learn in this context.
Classroom Structures and Processes
[While] work sometimes begins with the teacher writing a problem on the board…sometimes students are expected to continue working on a problem from the previous day despite now [working] with new group members. Most days, the students are…encouraged to get out of their desks to work on the many whiteboards… 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’ [and all amendments are
taken as friendly]… Work on the whiteboards [which is visible at all times for others to see] is treated as public property and belongs to the group… 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’ve just heard about, and sometimes (but not always) return to the group where they started… Disagreements about mathematical processes and solutions erupt and are resolved, usually without recourse to asking the teacher to intervene.… 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 participate in a whole-class discussion as a new topic is introduced or as a particularly thorny problem is explored. Sharon…shows great curiosity about every detail of the mathematizing… 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… 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. (Towers, Martin, & Heater, 2013, pp. 427–428)
Taking Actions to Learn
In the following excerpt, we attempt to give a sense of the flow of events and contributions that ultimately lead students to a solution of a particular problem. We do this through a narrative description of events rather than through the presentation of extensive excerpts of transcript interspersed with analysis of specific speech acts. We have purposely chosen, in order to be able to describe events without the aid of the video, an episode in which the action can be described by reference to just a small number of central characters, in which there are not too many comings and goings, and in which the teacher makes a brief appearance. In much of our data there is more movement of group members, more obvious “importing” of ideas from students from further away than just the two neighbouring groups, and often a disappearance of the original small-group members altogether as the groups fluidly dissipate and reform elsewhere. While such episodes could be said to also ‘represent’ the data—in fact, they may be more representative of the data we gathered—such events are extremely difficult to describe on paper and would likely make for frustrating reading. We therefore elected to choose a somewhat less complex example for the purposes of clarity in this medium.
On the particular day we have chosen to describe, the topic of study was geometry. The class was asked to prove that angle AOB is twice angle ACB (see Figure 5.1).
Three students, Simon, Cerys, and Leah, who were seated together for the brief introductory remarks from the teacher, move to a section of the whiteboard to begin work. All around the room, other groups are working on the same problem on other sections of whiteboard. All work is visible for others to see. Cerys draws the figure onto the whiteboard, however her diagram shows point C as vertically above O (Figure 5.2), which the teacher’s did not (see Figure 5.1).
Cerys and Leah begin talking about which line segments on the diagram are equal and which angles are equal. For the moment, Simon remains a step behind the two girls and watches their progress. Cerys and Leah add a dotted line to the diagram (AB) and establish that AO=BO and that angles OAB and OBA are equal to one another. They propose that angles CAO and CBO are also equal, but quickly dismiss this claim, recognizing that “no, they’re not necessarily [equal].” Cerys adds various lines and several annotations indicating perpendicularity and equivalence (see Figure 5.3).
For several seconds the group seems stuck and Leah steps away from the board. Cerys draws Simon into the conversation and they have a brief discussion about angles that might be equivalent, then a student from a neighbouring group to the left leans over to point at the diagram—noting that point C is not necessarily vertically above the centre, O. Cerys insists that “it doesn’t really matter” but the intervener
At this point (approximately 20 minutes into the problem-solving process) Sharon asks the class to pause in their work and offers them the opportunity to see the way she solved the problem. She stresses that there are many ways to prove the relationship and hers would be only one way and that they could continue with their work afterwards, but the offer is met with vehement protests. Students call out “I want to prove it” and “I want to figure this out.” Cerys adds, “I want to solve it, though.” Sharon acknowledges the will of the group and gives them more time. Nevertheless, the break seems to signal a moment for refocusing and Simon asks Cerys and Leah to consider the approach his new group has taken. He erases the
Cerys and Leah engage, Cerys challenging some of Simon’s proposals for angle equivalencies and Leah challenging the fact that Simon is still drawing the figure as though C were vertically above the centre and therefore making some (erroneous) assumptions based on this. After a few moments, Cerys and Leah turn back to their diagram (Figure 5.6) and continue work, but their focus now seems to be on angle relationships rather than lengths of line segments, although they do not seem to have a systematic way of notating angles—some they mark with an arc, some they assign a letter, some they give a symbol such as a small dot. Meanwhile, Simon calls the teacher over to ask a question and while Sharon is nearby she glances over at Cerys and Leah’s diagram and suggests that they consistently use small letters to denote angles. They re-annotate their diagram. The teacher then asks the students to identify symbolically the angles at O and they add further annotations (Figure 5.8).
Cerys and Leah now begin a proof, starting by expressing the 360 degree angle at O. They continue by simplifying the expression. With one further intervention from the teacher (who points out that an expression they create during this simplifying process (90-a) is exactly half of an expression representing one of the angles of interest (angle AOB, which is labelled 180-2a), they are able to effect the final line of their proof.
We see in this extract many examples of students taking actions of their own to learn. To begin, we note the students were highly active rather than passive in their learning. Conditions for learning were carefully orchestrated such that the environment was structured by the teacher to make space for students’ creativity in engagement and for their diverse ways of knowing and learning. For example, the random groupings, that changed from day to day, afforded many opportunities for students to engage and re-engage with others and provided the mechanism for collective actions and understandings. Students were called upon to articulate their own understandings while coacting to accommodate those of others. The changing group structures meant that each student was responsible for carrying
A second feature of the environment that is critical to understanding how students took actions of their own to learn is the consistent use of whiteboards as a “thinking space.” All work in this classroom was public, and so students learned to take responsibility for what they wrote or drew on the whiteboards, as such material was constantly available for scrutiny, questioning, and challenge. This is not to say that such material became fixed because it was public. On the contrary, students in this classroom learned that knowledge is mutable and open to interrogation and can be seen from multiple points of view, and, as such, mathematical work was frequently erased or revised as new(er) understandings came into play.
A third way in which this example offers insights into the actions that students (can) take to learn is through an examination of the help-giving behavior that we observed. In the above example we see that help-giving by peers in this classroom is (1) often not invited, but gifted anyway, (2) taken as a friendly amendment to the mathematics, not as a “correction,” and certainly not as a critique of the person or group whose mathematics is amended or questioned, and (3) listened to with hermeneutic intent—always with an ear to what might possibly be true about what is being offered, even if it seems at odds with the current mathematical path. In this way, groups serve as a foil for each other and as an internal verification mechanism.
Learning to Take Actions to Learn
The self-reliance that is evident in students’ actions in the above example does not, of course, simply happen. Many high school students have been conditioned by their school experiences to be passive learners and to expect to be shown every detail of the content to be learned and every step of the procedures to be followed. In contrast, Sharon had high expectations for student agency throughout the semester. The two examples we offer in this section show how Sharon carefully scaffolded challenge in her classroom such that students needed to be active in order to learn.
The first of these two examples occurred approximately six weeks into the semester. The class had been working on determining the factors of polynomial expressions and Sharon noticed that many students were factoring using long division. She queried whether they knew how to do synthetic division. Some students made tentative noises in the affirmative but it seemed to be clear to Sharon that some students did not know this technique and others may have only partially remembered or understood it from previous years. The following interaction ensued:
Sharon then proceeded, in silence, to work through the synthetic division for (2x3 + 3x2 – x + 4) divided by (x + 1). See Figure 5.9.
Sharon now writes a new problem for the students to solve on the board using synthetic division [(2x4 – 3x3 + x2 – x + 7) divided by (x − 2)]. They move to the whiteboards and begin working. The groups we observed proceeded in much the same way—one student began laying out the solution, using the same synthetic division framework offered by the teacher, with other group members watching the unfolding mathematics and intervening in the process to question a step, change a value, or discuss what should happen next. In one group, a student who had used the procedure before confidently began working through the problem. She erroneously, though, chose −2 rather than +2 as the initial root. Her group members, both of whom were meeting this procedure for the first time, watched for a while as she explained the first few steps and then participated in generating the last few elements of the division. Cerys, who we met earlier, was one of the group members who hadn’t met the procedure before and as they put the final touches to their procedure she glanced around the room and seemed to notice something about other groups’ solutions. She says, “Wait, no, you guys, this is wrong, this is wrong, because this [pointing to the −2 they have used as one of the roots] is not the root.” The other group members quickly acknowledge the amendment and, without further explanation from Cerys
In the next classroom excerpt, we show how the students learned to be active in documenting their learning. We do this by carefully examining how the teacher treated “note-taking” as a mechanism for learning. In traditional high school mathematics classrooms, notes are often provided by the teacher and copied down by the students. In this classroom, however, as we noted earlier, the teacher rarely provides a worked example (and never, during our data collection period, provided notes for students) and so the question of how students made notes, and about what, drew our attention. The following extract is taken from a lesson in which Sharon asked the students to make their own notes for the topics that the class had been working on in previous lessons. She begins by emphasizing that how students put together their review notes is going to be entirely up to them, and then writes three key ideas on the board—function, solving equations, and inequalities.
In the above extract we are able to see that Sharon took care to ensure that students were pointed to the key pieces of mathematics that she knew they would have encountered in her classroom, but she did not make notes for the students, nor did she specify precisely how they should document the ideas. In addition, her advice about note-taking privileges the big ideas of the topics, not particular techniques or procedures. For example, she notes that the students should pay attention to how
Discussion
The above examples reveal several aspects of a classroom that operates as a collective, where each member is responsible to all other members, where work is public and shared, and where the teacher is a fundamental part of the learning environment and brings forth a world of (mathematical) significance with the learners (Kieren, 1995; Maturana & Varela, 1992). As we can see from the first example we offered, this positioning of the teacher shifts his/her role away from being sole arbiter of mathematical truth or “corrector” of mistakes, and leaves him/her free to engage in doing mathematics with the students, something that Sharon did daily and with relish. The active help-giving behaviour of the students that was evident in the first example also transformed the help-giving behaviour of the teacher so that, as we described above, help-giving by the teacher that is perceived as simply “telling” students how to solve the problem is soundly rejected, but the students accept the teacher’s help-giving when it engages the specific mathematizing at play (e.g., Sharon’s suggestion to be consistent in labeling angles, and her pointing to the significant relationship between two algebraic expressions—90-a and 180-2a).
Our second example shows that students cannot simply be abandoned to the mathematics and expected to take actions of their own to learn in the absence of a support structure that helps them learn how to take actions of their own in learning mathematics. In our example, Sharon was careful to expose the students to a new
The third example offers a glimpse into an aspect of high school students’ learning that is under-researched—the practice of note-taking. Often, high school classrooms are structured around the very act of note-taking—the teacher reviews the previous day’s homework, demonstrates the new technique for the day, which students carefully copy down in their notes, and then students practice the technique, often using textbook exercises. In the classroom we describe here, though, classroom activities are more fluid and there is rarely a structured example worked through by the teacher that students might copy down. Hence, the question of what students ought to ‘note’—what counts as noteworthy in this space—is a matter of negotiation. As we can see from the third example, the teacher uses this indeterminacy as a moment for teaching, helping students to learn how they should learn from what transpires in the class. Several times we noted students using their agency—taking actions of their own to learn—by discussing with peers what should constitute the content of their notes for a particular topic. Hence, content was reviewed naturally and organically (and continually) rather than artificially as (only) a summing up exercise before moving to a new topic. The question of what was worthy of learning therefore became part of the learning event.
Conclusion
In examining the phenomenon of students taking actions of their own to learn as part of a collective, we have drawn attention to the critical role of the teacher in setting up an environment in which taking actions to learn is not only helpful but required in order to participate in unfolding understandings. We also note, though, that not just any kind of action is desirable in this context. The actions students take
Acknowledgments
We gratefully acknowledge the support of the Social Sciences and Humanities Research Council of Canada (SSHRC), Grant # 435-2009-383. SSHRC exercised no oversight in the design of the research, the collection, analysis, and interpretation of data, or the writing of this report.
References
Forman, E. A., & Ansell, E. (2002). Orchestrating the multiple voices and inscriptions of a mathematics classroom. Journal of the Learning Sciences, 11(2–3), 251–254.
Jones, S., & Tanner, H. (2002). Teachers’ interpretations of effective whole class interactive teaching in secondary mathematics classrooms. Educational Studies, 28(3), 265–274.
Ju, M.-K., & Kwon, O. N. (2007). Ways of talking and ways of positioning: Students’ beliefs in an inquiry-oriented differential equations class. Journal of Mathematical Behavior, 26(3), 267–280.
Kieren, T. E. (1995, June). Teaching mathematics (in-the-middle): Enactivist views on learning and teaching mathematics. Paper presented at the Queens/Gage Canadian National Mathematics Leadership Conference, Queens University, Kingston.
Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. (2000). The inter-related development of inscriptions and conceptual understanding. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms (pp. 325–360). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Martin, L. C., & Towers, J. (2009). Improvisational coactions and the growth of collective mathematical understanding. Research in Mathematics Education, 11(1), 1–20.
Martin, L. C., & Towers, J. (2015). Growing mathematical understanding through collective image making, collective image having, and collective property noticing. Educational Studies in Mathematics, 88(1), 3–18. doi:
Martin, L. C., Towers, J., & Pirie, S. E. B. (2006). Collective mathematical understanding as improvisation. Mathematical Thinking and Learning, 8(2), 149–183.
Maturana, H. R., & Varela, F. J. (1992). The tree of knowledge: The biological roots of human understanding (2nd Rev. ed.). Boston, MA: Shambhala Press.
Nathan, M. J., & Knuth, E. J. (2003). A study of whole classroom mathematical discourse and teacher change. Cognition and Instruction, 21(2), 175–207.
Powell, A., Francisco, J., & Maher, C. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data. Journal of Mathematical Behavior, 22(4), 405–435.
Proulx, J. (2010). Is “facilitator” the right word? And on what grounds? Some reflections and theorizations. Complicity: An International Journal of Complexity and Education, 7(2), 52–65.
Proulx, J., Simmt, E., & Towers, J. (2009). The enactivist theory of cognition and mathematics education research: Issues of the past, current questions and future directions. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd annual meeting of the international group for the psychology of mathematics education (Vol. 1, pp. 249–252). Thessaloniki: PME.
Sawyer, R. K. (2003). Group creativity: Music, theater, collaboration. Mahwah, NJ: Lawrence Erlbaum.
Sawyer, R. K. (2004). Creative teaching: Collaborative discussion as disciplined improvisation. Educational Researcher, 33(2), 12–20.
Towers, J. (2011, May). The class as a body that learns: Theoretical and methodological issues. Invited presentation at the Doyal-Nelson Symposium, The biology of cognition and its implications for curriculum and pedagogy, University of Alberta, Edmonton.
Towers, J., & Martin, L. C. (2009). The emergence of a ‘better’ idea: Preservice teachers’ growing understanding of mathematics for teaching. For the Learning of Mathematics, 29(3), 44–48.
Towers, J., Martin, L. C., & Heater, B. (2013). Teaching and learning mathematics in the collective. Journal of Mathematical Behavior, 32(3), 424–433.