Sociomaterial configurations for mathematics: Some notes on Multi-Party, Whole Body Interaction in Mathematical Activity

When my family and I moved to New York City 5 years ago, my child Sammy had just turned 2. This meant that she was beginning to demand levels of independence that we were not prepared to support, and her demands were often buttressed by remarkably effective tantrums that brought all activity to a standstill. At the same time, our family was developing new routines, which included learning to navigate the city without a car. Sammy’s daycare center was in Chinatown, about a mile and a quarter away from our apartment– a perfect walk on nice days, but when the weather was bad, we took the subway. These days were a continual source of anxiety and frustration for me. If you’re not familiar with New Yorkers, most of them do not have the patience to wait while a two-year-old stops all foot traffic to and from a subway station during rush hour because she NEEDS TO CLIMB THE STAIRS HERSELF. One rainy afternoon, as Sammy and I approached the crowded entrance to the East Broadway subway station, I saw a woman walking with a toddler, holding hands. As they approached the stairs, I watched as their hands separated, and their bodies turned to orient toward each other as they walked. As the toddler turned toward the woman, he looked up, leaned back a little, and raised his arms up, and the woman scooped him up to sit on her hip just as she stepped down onto the first step. The graceful coordination with which this was accomplished amazed me. It was like a beautiful, if brief dance between adult and child. Researcher-Jasmine wished I’d caught it on film. Harried-Mother-Jasmine was awed and more than a little jealous.

The bodies of that woman and toddler, moving so effortlessly and in synchrony, in contrast with my struggles with Sammy’s two most effective maneuvers (screaming-while-thrashing and screaming-while-limp), have become symbolic to me of the possibilities for how we, as educators, might think about the relationship between bodies and learning activity. For me and Sammy, the activity of getting into the subway station was primarily about management of a little body—trying to get it quickly and safely down the stairs without enraging either of us, or those around us. The demands and limits of that little body, juxtaposed with my goals, were such a source of frustration for both of us that the activity essentially became about managing the body, rather than simply getting into the subway station. Are you starting to see where I’m going with this?

The image of that woman and that toddler replays in my mind often as I interact with my children and with others in my everyday life, much of which is spent in meetings and classrooms where folks are sitting—down, still, and separately—for much of the time. So much of formal school learning (and teacher preparation) is often about managing children’s bodies and their behavior, as if it is separate from the rest of learning activity. If we can just get those kids’ bodies under control, the thinking goes, if we could get those bodies out of the way, then we could get down to real business: mathematics. Some of you might be thinking, “Well, this is the same as getting into the subway station, because one step in the process is navigating stairs, which toddlers can’t do properly (for rush hour norms, anyway). So, we need to teach them to walk faster or agree to be picked up.” I argue, however, that the idea is not that we need to discipline our children’s bodies so that they all compliantly enact the scripts that we (adults) impose. That’s certainly an assumption we could make about how the choreography I witnessed between the woman and child developed. I prefer to use the image as a reminder of how multiple bodies, acting together, can constitute activity smoothly; when the coordination works (the woman and toddler), it’s invisible to the participants of the activity, because things do not go wrong. In contrast, when it does not work (me and Sammy), it becomes the focus of attention, detracting from everyone’s ability to focus on the actual goal of activity.

I begin with this lengthy explanation as a way to frame my article in Cognition & Instruction, titled “Multi-Party, Whole-Body Interactions in Mathematical Activity”. One way to think about the overarching question guiding the piece is that it’s an exploration of one possibility for putting bodies into interaction in mathematics learning so that they are not a separate component of the learning environment to be managed, but rather part and parcel of the mathematics itself. In computer science terms, I wanted to know what might happen if we think of learners’ bodies as a feature, rather than a bug. I emphasize that I wanted to know what might happen if we deliberately put them into interaction with each other—in analogy to the way math educators have increasingly realized that students’ interactions through talk can be incredibly productive for learning, maybe students’ embodied interactions might also be designed to be productive, rather than disruptive, as we typically think of them.

Another way to ask the question, one I find to be incredibly valuable, is this: What are the new ways in which we might see what mathematics activity and learning, when we take into consideration what bodies, in interaction, are doing? If we take for granted that mathematics is about abstractions and mental activity that primarily takes place in one’s brain–and perhaps written notations, other representations, and some talk can support this activity–then we might not pay much attention to bodies, except to try to make them as undisruptive as possible. However, if we make a different assumption, that what bodies are doing together help shape mathematics activity and learning, we might see something pretty different. Here are some possibilities for what we might see:

In more “typical” classroom configurations that include desks or tables and chairs with some narrow spaces for walking around, students are expected to be sitting, and much of the action is designed to take place on tabletops or on a shared space at the front of the room (white board or projection).

In some classrooms individual desks might be pushed together for group work, or students may sit at tables that allow them to face each other for most of class. In this case the configuration orients students’ bodies toward each other, supporting interactions that might include talk, gesture, and the sharing of ongoing work on the surface of the desks or table.

In Walking Scale Geometry (WSG), students are outside, and groups of three or four are constructing geometric figures at large scale using their bodies, ropes, and other materials. They sit, stand, walk, and run. They hold and manipulate materials individually and together as they negotiate strategies through talk and physical (inter)action for their constructions.

In each of the scenarios above what counts as mathematics activity, and what learning looks like, turns out to be pretty different. The third is what I set out to explore in the article. I use microanalytic methods to carefully look at two different episodes of WSG activity, and how talk, physical (inter)action, and emergent mathematical goals come together to shape mathematics and the learning. You’ll notice that the analysis sometimes goes into minute details of bodily action—it quickly became clear to me in this work that even the tiniest of movements by one student could have significant impact on the others. In the second episode, in particular, I tried to describe how a cascade of these movements by members of the group built toward a new idea for representing vertices of figures in WSG. To me, following interactions, in talk and in action, at this microanalytic level is important for really understanding the contributions of students to their own learning. How their bodies and ideas interact has something important to tell us, both about how they understand what they are doing, and also about our own assumptions about what they should be doing. My hope is that these kinds of accounts of how bodies can come together in coordination to negotiate and execute problem solving strategies will serve to spark the imagination of more mathematics educators to leverage our students’ bodies as integral to mathematics activity, rather than distractions to be managed.

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