1 Introduction

Over the last 20 years our research has focused on the nature and meaning of mathematical understanding, and more specifically how the phenomenon can be characterized and described. In this work we have drawn extensively on the Pirie–Kieren (P–K) Theory for the Dynamical Growth of Mathematical Understanding (Pirie and Kieren 1989, 1992, 1994) both employing and also developing and refining the theory as a methodological tool (e.g. Martin 2008; Martin and LaCroix 2008; Martin and Towers 2014; Towers and Martin 2014). The Pirie–Kieren theory views mathematical understanding not as a static state to be reached or achieved, but as a dynamical, growing, and ever-changing process. It describes eight potential layers (ways of acting mathematically) for mathematical understanding and these can be represented pictorially in the model shown in Fig. 1. Using the model, the growth of understanding of a learner, for a particular mathematical concept or topic, can be mapped out diagrammatically through using a line to trace the changing actions of the learner. (A hypothetical pathway of growth is offered on Fig. 1.) The choice of focus is one that lies with the observer using the theoretical tool, but the intent of the theory is that one specifies what mathematical concept one is trying to describe the growth of, and for whom. Although the layers of the model grow outwards from the local to the general, this does not imply that understanding grows in any kind of hierarchical, linear way. Instead, and as suggested by the drawn pathway, growing mathematical understanding occurs through a complex movement backwards and forwards through the layers of understanding. Such recursive action, termed folding back, is a critical aspect both of the theory and of the growth of mathematical understanding and suggests that someone working at any layer of understanding will, when faced with a problem that is not immediately solvable, return to work at an inner layer of understanding (Pirie and Kieren, 1994; Martin, 2008).

Fig. 1
figure 1

Model for the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding

Full size image

Whilst it is not necessary here to fully elaborate the theory and its model, the notion of “images” is significant for our later discussion. The theory posits that as a learner’s mathematical understanding for a particular concept grows, he or she will make, hold and extend particular images as he or she works on mathematical tasks. In particular, it talks of the two nested phases of Image Making and Image Having as being key to developing an initial understanding of a concept. When Image Making, learners are engaging in specific activities aimed at helping them to develop particular initial conceptions and ideas for the meaning of a mathematical concept. By the Image Having stage the learners are no longer tied to actual activities, they are now able to carry with them a general mental plan for these specific activities and use it accordingly. This frees the mathematical activity of the learner from the need for particular actions or examples. At this layer, the learner has some understanding that is more general in its nature, and which does not require a specific action.

Our research has been located in sites of learning ranging from school classrooms, to clinical interview settings, to workplace training classrooms, and has predominantly employed video recordings (supplemented by other artefacts such as student work, observer notes, etc.) as the means and method of data collection. In this work we used the Pirie–Kieren Theory as our framework for conceptualizing the growth of mathematical understanding and as an analytic tool. While this theoretical perspective served us well, and allowed us to answer questions relating to the growth of personal understanding, we also increasingly found ourselves, during analysis of data, to be experiencing difficulty attributing particular mathematical understandings to individual students. Specifically, we often found it impossible to develop clear mappings of the growth of understanding for individual learners. However, it was also clear to us that growth was occurring, and that mathematical learning was taking place. Whilst this understanding was emerging from the actions of particular individuals, it was the ways in which these ideas were picked up, worked with, and developed by the group that was leading to the observed growth, and it was on these processes that we found we needed to focus. The difficulty in focusing theoretically on this collective activity led us to engage new theoretical frameworks to describe collective mathematical understanding and to elaborate and extend our existing perspectives on individual mathematical understanding (including the Pirie–Kieren Theory), which we still saw as useful and valid. As we sought to expand our framings we turned to contemporary theoretical discourses on collective cognition—including enactivism but also other frameworks, such as improvisational theory from jazz and theatre studies, that describe how participants engage with each other in group settings—and brought these to bear on the P–K Theory in an effort to map the structure of collective action in the field of mathematical understanding (see, e.g. Martin et al. 2006; Martin et al. 2012; Martin and Towers 2014; Towers and Martin 2014).

2 Enactivist principles

In interpreting the basic idea of a group or collective we draw primarily 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 (Maheux and Proulx, 2015; Proulx et al. 2009). Di Paolo et al. (2011) identify five core ideas that define the enactive paradigm: autonomy, sense-making, embodiment, emergence, and experience. While each of these constructs underlies our understanding of the data we present later, in this writing we will be concerned primarily with ideas of sense-making, embodiment, and emergence as they relate to the growth of students’ collective mathematical understanding. To begin our explanation of these ideas we note that “a key principle of the enactive approach is that the organism is a centre of activity in the world” (De Jaegher and Di Paolo 2007, p. 487), but the organism is not seen as isolated from its environment. Species and environment co-adapt to each other, meaning that each influences the other in the course of evolution, a process referred to as structural coupling (Maturana and Varela 1992). Maturana and Varela (1992) explain that as organisms become structurally coupled, events and changes are occasioned or “triggered” (p. 96) by the environment but determined by the species’ structure. “The same holds true for the environment: the living being is a source of perturbations and not of instructions” (p. 96). Reid and Mgombelo (2015, this issue) provide a compelling example of how this reciprocal engagement occurs, showing how species and environment become coupled over time. Hence, from an enactivist point of view, we understand that changes do not occur inside either the organism or the trigger, they are brought forth by (and are dependent upon) the organism’s interaction with the trigger—a phenomenon that Reid and Mgombelo (2015, this issue) explain is central to the enactivist framing of the concept of learning. Simplistic interpretations of linear cause and effect in teaching and learning are therefore problematized in an enactivist interpretation and learning is seen as reciprocal activity—the teacher, or, more broadly understood, the environment, brings forth a world of significance with the learners (Kieren 1995; Maturana and Varela 1992).

A key idea within this formulation of learning is the notion of emergence. Emergence describes “the formation of a novel property or process out of the interaction of different existing processes or events” (Di Paolo et al. 2011, p. 40). As Froese (2009) notes, cognition or learning “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, smartboards, whiteboards, and computers. Enactivist thought emphasizes this dynamic interdependence of individual and environment and has the tendency to coalesce the observer’s attention on the coordination of action in such environments. In fact, as Coles (2015) points out, structural coupling entails co-ordination of action. For the researcher, noticing how elements of an environment become coordinated means paying attention to the twists and turns of how joint action and meaning are seen to emerge. For us, then, a focus on the emergence of mathematical understanding has prompted a reorientation to the collective body, both in terms of what is known and of who is doing the knowing (Glanfield et al. 2009; Towers 2011), rather than the individual cognizing agent. Observing and theorizing about the emergence of such collectivity calls forth an interpretation of classroom life that runs counter to prevailing discourses of schooling that rely more on Cartesian thinking, which has successfully emphasized the ideal of a modern self as solitary, coherent, and independent of context. The ideal knower, in this traditional frame, has been the autonomous individual. Such thinking has then positioned this radical subject as the reference point for what is known (and worth knowing, and, indeed, worth studying). Enactivism therefore challenges interpretations of learning and teaching that privilege individual cognition.

While the human individual is no longer necessarily the focus of enactivist attention, the body remains of significance in any analysis of human cognition. Many authors (e.g. Lakoff and Johnson 1999; Lakoff and Núñez 2000) have provided evidence that “higher-level cognitive skills, such as reasoning and problem solving, mental image manipulation, and language use depend crucially on bodily structures” (Di Paolo et al. 2011, p. 43). An enactive approach to cognition therefore posits that bodily actions (gestures, facial expressions, utterances, etc.) are intimately part of the cognitive process, not simply expressions of an already-generated, brain-based cognition. These actions, within the kinds of collectivity that interest us, are also, typically, coordinated between the participants. Hence, we see that joint sense-making emerges from this coordination through a “correlation between the behaviours of two or more systems that are in sustained coupling….[expressed as] a coherence in the behaviour of [the] systems over and above what is expected, given what those systems are capable of doing” (De Jaegher and Di Paolo 2007, p. 490). Such coordination of action in mathematics learning environments has become our focus and in allowing our research to be reoriented in this way we have been prompted 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. The enactivist imperative, then, for understanding groups of learners as collectives is a severing of an attachment to the individual, a reorientation that is also gathering momentum in the wider mathematics education literature as researchers turn their attention to social processes and collaborative negotiation of meaning in the classroom (see, e.g. Davis and Simmt 2003; Gellert and Steinbring 2014; Kieren and Simmt 2002; Steinbring 2005).

From the enactivist perspective, a key idea in understanding sense-making in the collective is to recognize cognition—mind—as a process, not an object. This orientation occasions new ways to think about interacting bodies. Drawing on enactivist thought, interactions (between students, lets say) consist of reciprocal perturbations (Maturana and Varela 1992, p. 75). Such perturbations trigger mutual changes in the system and a history of recurrent interaction leads to structural congruence or coupling between the participants. This system is emergent—its properties emerge at a certain level of complexity that may not be present previously. Much of our recent work has been concerned with investigating this complexity—trying to understand the characteristics of, and conditions for, collective (rather than individual) mathematical understanding. By collective mathematical understanding we mean the kinds of shared mathematical actions, thinking, and learning that we may see when a group of learners, of any size, work together on a piece of mathematics. We thus see the growth of collective mathematical understanding as a dynamical ever-changing interactive process, where shared understandings exist and emerge in the discourse of a group working together. We use the word shared not to suggest that understandings “happen to overlap and their intersection is shared” nor that “some individuals communicate what they already knew to the others” but that understandings emerge and are “interactively achieved in discourse and may not be attributable as originating from any particular individual” (Stahl 2006, p. 349).

In seeking to more clearly characterize the process through which such collective understanding emerges, we developed the notion of coactions as a means to describe and account for the kinds of structural couplings we saw occurring among the learners in our data. 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 and Towers 2009, p. 4). Coacting is a contingent, emergent process that places as “much responsibility on those who are positioned to respond to an action as on the originator” (Martin and Towers 2009, p. 4). An implication of this, from a methodological perspective, is that “a coaction is a mathematical action that can only be interpreted in light of, and with careful reference to, the interdependent actions of the others in the group” (Martin et al. 2006, p. 156). Elsewhere, and in reference to other forms of learning such as early mother–infant interaction (Fuchs and De Jaegher 2009) and story-telling (Caracciolo 2012), this kind of collaborative activity has been labelled participatory sense-making—“the process of generating and transforming meaning in the interplay between interacting individuals and the interaction process itself” (Fuchs and De Jaegher 2009). Our initial difficulties in observing and accounting for acts of mathematical understanding emerging through this kind of process prompted us to seek out and develop new methods of data collection and analysis. In this paper we examine in particular the ways in which we have used a blend of familiar and new analysis techniques for handling data in pursuit of these research aims, and which have allowed us to more precisely observe, describe, and account for coaction and the growth of collective mathematical understanding.

3 Methods and examples

The severing of our attachment as researchers to tracing individual cognition, while initially being simply philosophically attractive, became theoretically compelling as we struggled to trace individual threads of mathematical thought and expression in particular data examples. At the time, we were using the P–K Theory (Pirie and Kieren 1994) to trace the mathematical thinking and actions of individual students working together on a piece of mathematics. However, one or two troublesome pieces of data began to gnaw away at us. In these extracts, try as we might we could not tease apart the actions and dialogue to create separate, coherent, individual traces of understanding. It was as though the mathematical understanding in these episodes was emerging not at (or not only at) the level of the individual but, significantly, at the level of the group. In taking seriously the enactivist prompt to loosen our hold on the assumption of individual streams of thought and action, we were drawn to consider the troublesome data episodes not as interacting individual mathematical monologues but as a single coherent voice, emerging through a coactional process.

To illustrate our point we share below several versions of a brief extract of transcript from a small-group, task-based interview with three Grade 6 students conducted at the end of the school year. The three students had spent their Grade 6 year together in one classroom. Their teacher considered them to be strong mathematics students and had distributed their ‘expertise’ into different groups within the classroom, so prior to this task-based interview these three students had not had the opportunity of working together as a single group. Any small-group collaborative structures revealed by the data, therefore, could be considered to be generated in the moment, though influenced by their shared history of participating in a classroom environment. In the interview, the students were presented with a shape that they had not previously studied (a parallelogram) and invited to try to figure out how to find its area. They had no prior knowledge of methods for finding the area of a parallelogram, but had worked during the year on finding areas of rectangles, squares, and triangles. They worked collaboratively for a considerable time on this problem. The section we present below shows their first foray into solving the problem and at the end of the episode we share here the students have reached an incorrect solution based on an erroneous assumption. Our interest here, though, is in discussing their cognitive processes as their understanding grows. As researchers, as we repeatedly watched the video of the students working together, trying to tease apart three separate threads of mathematical understanding, we began to feel that, rather than three threads, we were compelled to ‘read’ just one thread—the group’s developing growth of mathematical understanding. In the first iteration of our presentation of data below we document the group’s speech in a traditional transcript format with each speaker’s words presented on a new line for each speech act. In one small departure from convention, though, we use the ellipsis not, as is usual, to represent omitted speech but instead to represent a continued line of thought (even though several voices may contribute to its emergence).

  1. 1.

    Natalie: Can try it. Measure sides. (She begins to measure one of the shorter sides).

  2. 2.

    Stanley: Ten.

  3. 3.

    Natalie: Mm hm. And that’s ten right? (She points to the opposite side). They’re both ten. And then the top one is longer, and that is…

  4. 4.

    Thomas: …it’s eighteen.

  5. 5.

    Natalie: Yep. So that’s ten…and eighteen. (She writes the numbers on the sides). Ok, so we could…

  6. 6.

    Thomas: …multiply…

  7. 7.

    Natalie: …the area, yeah ten by eighteen…and then see what we get.

  8. 8.

    Thomas: It would be this…(writing 180)

  9. 9.

    Natalie: yep (pause) and then if we wanted to do…so that would be the area then?

  10. 10.

    Stanley: But look at the shape.

  11. 11.

    Natalie: I know that’s what I’m saying that can’t be right cause that—that’s a little bit…

  12. 12.

    Stanley: Okay. Wait, I know….draw a straight line, here and here, you get triangles, and squares. (He adds lines to the parallelogram, see Fig. 2)

    Fig. 2
    figure 2

    Annotated parallelogram

    Full size image
  13. 13.

    Thomas: Oh I know…

  14. 14.

    Stanley: Now these triangles…I think…

  15. 15.

    Thomas: is half?

  16. 16.

    Natalie These triangles will make up that square though. So then if we just measure that….

  17. 17.

    Stanley: and times by two.

  18. 18.

    Natalie: Yeah. Because these two, these triangles make that square, right? (They have concluded (wrongly) that the two outer right-angled triangles are equivalent in area to the inner rectangle).

  19. 19.

    Thomas: That’ll work.

  20. 20.

    Stanley: I think…

  21. 21.

    Natalie: So if we…what is…this it’s still, it’s not eighteen though, anymore. Because we’re cutting it… it will be twelve. (She measures the sides of the rectangle that has been created).

  22. 22.

    Thomas: and then ten

  23. 23.

    Natalie: and this side is…no it wouldn’t be ten. (She is referring to the width of the rectangle, i.e. the perpendicular height of the parallelogram).

  24. 24.

    Thomas: Like, eight?

  25. 25.

    Natalie: this side is eight, yep. So…

  26. 26.

    Stanley: and then 12 times 8…

  27. 27.

    Thomas: …is…ninety…

  28. 28.

    Stanley: Yeah, ninety-six. (This is the area of the inner rectangle).

  29. 29.

    Thomas: …six

  30. 30.

    Natalie: So ninety-six times…

  31. 31.

    Stanley: So ninety-six times two, so uh ninety-s….

  32. 32.

    Thomas: a hundred…I mean no, not a hundred…yeah a hundred eighty-uh… three?

  33. 33.

    Stanley: ninety-three…

  34. 34.

    Natalie: ninety-six. One ninety-six.

  35. 35.

    Stanley: Okay.

  36. 36.

    Natalie: How does that work? Two times six is twelve. One ninety…two.

  37. 37.

    Thomas: So it’s one ninety-two.

  38. 38.

    Stanley: Yeah, one ninety-two…

  39. 39.

    Natalie: …is the area of that. So it’s the area of the whole…

The above transcription, while technically accurate, does not, for us, fully convey the interweaving, coactional nature of the speech (and therefore the cognition) that is so compelling when viewing the video record. Lost in this transformation of data is the students’ sense of purpose and shared project that is much more evident in the video record. In addition, the ability to trace a single, coherent development of understanding for the group as a whole is muted. In the second iteration of the transcript below, in order to emphasize what, for us, became important, we have erased the names of speakers and the starting points of each of their utterings. This iteration helped us to see the single thread of mathematical understanding that builds around the idea of calculating the area of a parallelogram by splitting the figure into smaller figures.

Can try it. Measure sides. (Begins to measure one of the shorter sides). Ten. Mm hm. And that’s ten right? (Points to the opposite side). They’re both ten. And then the top one is longer, and that is it’s eighteen. Yep. So that’s ten and eighteen. (Writes the numbers on the sides). Ok, so we could multiply the area, yeah ten by eighteen and then see what we get. It would be this (writing 180). Yep (pause) and then if we wanted to do so that would be the area then? But look at the shape. I know that’s what I’m saying that can’t be right ‘cause that that’s a little bit. Okay. Wait, I know. Draw a straight line, here and here, you get triangles, and squares. (Adds lines to the parallelogram, see Fig. 2). Oh I know. Now these triangles, I think is half? These triangles will make up that square, though. So then if we just measure that and times by two. Yeah. Because these two, these triangles, make that square, right? That’ll work I think. So if we, what is this? (indicating the longer dimension of the newly created rectangle in the middle) It’s still, it’s not eighteen though, anymore. Because we’re cutting it. It will be twelve (measures the sides of the rectangle that has been created) and then ten and this side is, no it wouldn’t be ten (referring to the width of the rectangle, i.e., the perpendicular height of the parallelogram). Like, eight? This side is eight, yep. So and then twelve times eight is ninety. Yeah, ninety-six. (This is the area of the inner rectangle) Six. So ninety-six times. So ninety-six times two, so uh ninety-s….a hundred, I mean no, not a hundred, yeah a hundred—eighty-uh three? Ninety-three. Ninety-six. One ninety-six. Okay. How does that work? Two times six is twelve. One ninety-two. So it’s one ninety-two. Yeah, one ninety-two is the area of that. So it’s the area of the whole.

The above transcription affords several nuances for our understanding of the students’ process of growing mathematical understanding in the collective. First, this transcription technique affords a view of the interaction as truly one of actively taking up and building upon what is offered by others so that a shared conceptual structure can be observed through the coactional discourse. This can be seen, for instance, in the coherence of sentence structures. Despite the fact that we know three ‘voices’ are contributing, for the most part the text can be read as a monologue. There are no sharp breaks in flow, no overlapping speech, no disagreements, no evidence that one person is needing to explain or re-explain their thinking to others, etc. A reviewer of an earlier draft of this paper asked whether our chosen transcript is one where there is an unusual amount of agreement between the students and what a similar analysis would look like if the participants disagreed more or did not reach a resolution. We agree that there is perhaps an unusual amount of agreement within this group; however, these were precisely the kind of data that interested us. Episodes such as this one had resisted our attempts to trace individual pathways of growth of understanding. It was these kinds of episodes that prompted our search for new theoretical frameworks and new methodological approaches. It is clear to us now that transcriptions of interactions that do not have a shared character cannot be read smoothly and do not convey a shared image of the developing mathematics when organized as a monologue. Instead, such interactions are characterized by distinct trains of thought that intersect and/or overlap but do not cohere.

In the example we have shared, the ‘monologue’ technique also emphasizes that the growth of collective mathematical understanding intimately includes the shared artefact (the diagram), towards which all the speech relates. In episodes in which the growth of collective mathematical understanding is not exhibited, conversation often does not consistently incorporate a shared image or artefact and/or participants may refer to their own individual artefacts, drawings, textbooks, writings, etc., which may differ from those used and referred to by others in the group.

While the above transcription emphasizes the coherence of the shared image making in which the three students participated, it does have limitations. Through it we lose the capacity to understand details of how the discourse unfolds as a complex weft of multiple voices—and hence we no longer see how ideas and actions originating with one speaker are picked up and built on by others, and how subsequent actions are contingent upon those prior. In a third iteration of our method of analysing the data we developed the technique of using colour as a device to reveal the interweaving, coactional, character of the discourse. Here is the same transcript again, but now the three speakers’ contributions are colour-coded—Nathalie’s contributions in green, Stanley’s in blue, and Thomas’ in red:

This transcript shows much more readily the way in which the students’ thinking interweaves. For example, notice that throughout the episode the participants complete one another’s sentences—or, indeed, insert words in the middle of each other’s sentences—with no loss of flow or meaning. Early in the episode we see one such example: “Ok, so we could multiply the area, yeah ten by eighteen and then see what we get”, which is spoken predominantly by Natalie but with the single word “multiply” inserted by Thomas at precisely the right moment and with no interruption to the meaning of the sentence. Natalie does not pause and wait for this offering—she does not in any way seem “stuck” for the correct operation—instead, the word “multiply” naturally appears in the sentence and it seems of no import to any of the students that the contribution was inserted by Thomas rather than Natalie. Similar examples follow later—“Oh I know. Now these triangles, I think is half” spoken by Thomas and Stanley, “These triangles will make up that square, though. So then if we just measure that and times by two?” spoken by Natalie and Stanley, and “That’ll work I think” spoken by Thomas and Stanley. The colour-coded transcription device therefore affords the ability to pay close attention to how an idea, the seed of which is initially perhaps offered by one voice, is taken up and worked upon by others. It also shows how a single, coherent sentence that moves the mathematics forward can be formed by multiple voices (e.g. “Ok, so we could multiply the area, yeah ten by eighteen, and then see what we get” and “Now, these triangles, I think, is half”), a phenomenon that is much less obvious when the sentence in question is split onto three or more separate lines, as it would be in a traditional transcription (such as the first one offered in this paper).

There are also other aspects of the episode that are emphasized through the colour-coding but that remain more hidden in a traditional transcription. For instance, a key structure that strikes most readers when looking globally at this colour-coded transcript is the relative amount of green text in comparison with blue and red text. For some analyses, this might be an important feature and therefore our colour-coding device (particularly with speaker names stripped and the discourse allowed to run on in narrative form as we illustrate) provides researchers with a visual tool for such global analyses. For our particular conceptual focus, an initial analytical response to this visual impact is to assume that the speaker whose words are in the majority is somehow leading the mathematical problem-solving. However, a deeper analysis of the interweaving character of the discourse and the nature of the unfolding mathematical understanding shows that while one voice (green) may indeed seem dominant in terms of number of words spoken, this voice has more of a role of narrating and binding together the collective idea for determining the area of the parallelogram rather than one of leading the mathematical problem-solving. Even though Natalie initiates the problem-solving process by beginning the process of measuring the parallelogram, the process seems immediately to be understood as a joint project by Stanley and Thomas who both insert elements of the measurement (e.g. reading off the dimensions on the ruler even though the ruler is physically held and manipulated by Natalie). It is certainly not possible to say that Natalie is ‘responsible’ for shaping the emerging solution and this recognition that volume of speech is not necessarily correlated with leadership and/or understanding of mathematics within group problem-solving has become an important dimension of our analyses of collective mathematical understanding. The prevalence of green text in this episode actually prompted us to engage in deeper analyses of the structures of how collaborating groups narrate their evolving understanding and of how coherence is brought to group meaning-making processes, leading us, for instance, to develop the construct of “improvisational coaction” and its characteristics (Martin and Towers 2009)—a means of illuminating and describing the joint action of group members working together on mathematical tasks.

4 Discussion

From an enactivist point of view, the above manipulations of data successively reveal the complex ways in which students—their words, actions, and movements—and environment—the physical materials, generated objects, other bodies and speech—are structurally coupled and coordinated. A clear characteristic of the discourse, highlighted in particular by the coloured transcript, is the interweaving of partial fragments of the emerging image for area of the parallelogram, as we described in the previous section. This type of coaction is also characteristic of structural coupling in other fields of study. For example, in talking about musicians, Monson (1996) wrote:

when you get into a musical conversation, one person in the group will state an idea or the beginning of an idea and another person will complete the idea or their interpretation of the same idea, how they hear it. So the conversation happens in fragments and comes from different parts, different voices. (p. 78)

We see a similar evolution of mathematical ideas in the above data. There is a breadth of possibilities open to each participant at each moment and the pathways taken are dependent on what has gone before, what each word, gesture, drawn line, facial expression, etc., suggests to other hearers/viewers and what moment-by-moment response is made as a further action. Each of these subsequent offerings “triggers” (Maturana and Varela 1992, p. 96) the growth of the shared image making. The triggers for growth include (but are certainly not limited to) the impulse to: measure the figure (promoted initially by Natalie but taken up by both Stanley and Thomas), record measurements (physically carried out by Natalie), suggest a procedure (Thomas suggested multiplying), conduct calculations (Natalie suggests dimensions to multiply, Thomas carries out the calculation), propose a refinement to the diagram (Stanley), visually interpret the affordances of the altered diagram (all three students), estimate areas (all three students), determine what new measurements to make and carry these out (Natalie and Thomas), and conduct further calculations (all three students). Each of these more observable triggers is supplemented in the moment by nuances that our data collection tools may or may not allow us to perceive (minute facial expressions, sub-vocalizations not picked up by the microphone, and perhaps even the excretion of pheromones at moments of significance when excitement builds around a particular idea—invisible to the researcher but perceivable by participants’ bodily cognitive systems). Each of these elements of problem-solving is dependent on the contingent ‘actions’ (interpreting this broadly as the above list suggests) immediately preceding it and each requires that the students listen to the “group mind” (Sawyer 2003, p. 47), pay close attention to emerging mathematics, and be willing to alter what they are doing “in response to tiny cues that suggest a new [mathematical] direction that might be interesting to take” (Becker 2000, p. 172). Such contingent action is a reminder that understanding is emergent and “arises in the moment-to-moment interaction” (Fuchs and De Jaegher 2009, p. 466) of the participants and this understanding includes such components as “bodily resonance, affect attunement, coordination of gestures, facial and vocal expression and others” (p. 466). We have already shown in detail the coordination of vocalizations that occurred between the participants in our example, but we also note here the compelling effect of watching the video-recording of this episode. The three students’ bodily actions are also highly coordinated. For example, when the idea of splitting the parallelogram into smaller shapes begins to coalesce (lines 12–18 of the first transcription above), we see a physical shift of the three students as they lean out of their seats, clustering towards one another and gathering more closely around the drawn artefact. Additionally, they use their bodies to point to the drawn amendments on the diagram, emphasizing their shared image.

Cognition, as expressed in these moments, is understood as embodied action—the appropriate and ‘intelligent’ (Varela 1999) expression of knowing and doing in context. However, it is important to recognize that such coordination of action

does not have to be absolute or permanent. There are degrees of coordination and coupled systems may undergo changes in the level of coordination over time….Coordination can be like a swaying into and out of states that are close to stable but not quite. Eventually it may break down altogether. (De Jaegher and Di Paolo 2007, p. 491)

Consequently, while the participants in our example are closely coupled and their actions coordinated for the duration of this extract (and indeed for some time afterwards), at times the collectivity wanes as individual preoccupations come to the fore in the problem-solving process. At other times, the group coalesce again upon particular ideas and eventually a correct solution to the general problem of finding the area of a parallelogram is reached. Recognizing these periods of coaction in an otherwise ordinary interaction between problem-solvers working on a basic geometry problem would not have been possible for us without the theoretical and methodological orientation engendered by an immersion in a body of thinking and literature that emphasized the fundamental inextricability of all things. While we acknowledge that other theoretical perspectives (e.g. social constructionism, cultural-historical activity theory, etc.) may have offered other researchers the opportunity to re-orient to the collective in meaningful ways, it was the enactivist-inspired possibility of group mind as a mathematical process that occasioned for us new ways to think about interacting bodies, prompting us to recognize such “bodies” as reciprocal perturbations (Maturana and Varela 1992, p. 75) leading to structural coupling and coacting in a complex system and provoking us to develop new ways of looking at and transforming our data to bring us into closer contact with our phenomenon of interest.

5 Conclusion

As Sriraman and English (2005) and others (e.g. Schoenfeld 2008) note, mathematics education researchers have adopted a wide range of theories to guide their studies, and, in concert with these theoretical positionings, have developed a comprehensive suite of methodologies and methods of study. However, in the narrowed field of our area of interest—observing and analysing the phenomenon of mathematical understanding, be it individual or collective—qualitative analysis methods evident in the literature have remained somewhat limited, typically involving direct and painstaking transcription of audio-recorded or video-recorded data and its faithful re-presentation in accurate and elaborate detail. We have seen few experimental and/or innovative uses of transcription data in the literature and the variations we offer here are intended to show how a willingness to play with data, prompted by an enactivist perspective that drew us to pay attention to the interplay between things rather than their autonomous constitution, can shed light on an emergent phenomenon. The development of enactivist theory and methodology therefore go hand in hand in our work.