Reference: Kelton, M. L., & Ma, J. Y. (2018). Reconfiguring mathematical setting and activity through multi-party, whole-body collaboration. Educational Studies in Mathematics, 98(2), 177–196. https://doi.org/10.1007/s10649-018-9805-8
Summary: Kelton and Ma's study helps explain how designs for mathematical activity enhance or lessen mathematical possibilities and relations by demonstrating the effect that the social learning environment has on the organization, opportunity and productivity of embodied mathematics in learners. The researchers focused on a few key distinctions outside of "traditional" (or "container") mathematics:
- Full-body movements in mathematics (as opposed to gesturing with hands-only).
- Mathematical environment = arena, but the arena is experienced differently by each individual so an individuals mathematical environment = setting.
- Mathematics as a non-hierarchical and non-dualist conceptualization of mind-body relations. So, math is not a transcendent ideal that is separate from the body, but instead it can be performed with the participation of bodies.
- Multiple bodies as a whole interacting in a space together, rather than emphasis on an individuals single body for mathematics.
They present two cases of mathematical multi-party, whole-body activities for analysis and comparison: The walking scale number line (WSNL) and Whole + Half (W+H). They conclude that repurposing space could create resonances or dissonances depending on the history built in that space and this consideration should be taken into account while designing learning experiences involving whole body multi-party mathematical activities.
Stop 1: "Students' moving bodies also became meaningful aspects of the setting themselves and each other, beyond simply performing individual quantities moving and operation along the number line. Quantitative relationships were understood and talked about as spatial relations between students' home positions and bodies" (Kelton & Ma, 2018, p. 184).
Upon hearing about this walking scale number line (WSNL) activity, I didn't think it was particularly interesting. However, reading about the interesting operations and movements students were able to do in relation to each is generating some intrigue. I imagine every child standing on their own "home-base" (ie. a point on a numberline without numbers) and then being asked to do operations. Addition and subtraction would be viewed as everyone marching together up or down the numberline. Multiplication would see an expansion of coordinated bodies from centre. I am now curious as to how this activity would look on a large scale.
Student buy-in is a concern for me, but seeing that grade 8 students participated in this number line aspect and seemed to enjoy this full-body activity, there could further be value added when taking this to include imaginary numbers too with a rotation about the x-axis. That would be an interesting extension onto this activity having two intersecting axis and using a complex number.
Another extension idea would be to replace the blue tape with a flexible knotted rope. Arrange the rope in a spiral, or create a connected ring and see what kinds of mathematical relationships arise when thought of as a cycle.
Stop 2: "Thad suggested that if he held onto Morgan (two to his right) and Kian (two to his left) with either hand, he could just turn around and rotate them to their opposites. He then revised this to include the whole group... Maggie and Thad solve the problem from their respective physical and mathematical perspectives in the material arrangements of the space" (Kelton & Ma, 2018, p. 186).
This was very interesting. Thad was centre on the numberline, and swinging his partners around him allowed for a "multiply by -1" scenario. He suggested everyone link arms and swing instead of each student individually count. This physical (re)arrangement without the need for counting placements could've only been accomplished with this space-oriented, full-bodied activity. Very interesting connection.
Stop 3: "...Jeff, was presenting a whole interval to Ms. Collins that involved crossing his hands. Jeff explained that , in this case, half would need to 'go to the other side of the world' in order to complete the task. This solution imaginatively expanded beyond the walls of the classroom, wrapping and bending the whole interval in a great circle around the world while entailing an impossible journey for half." (Kelton & Ma, 2018, p. 191)
While reading about this partnered activity of half (H) keeping their hand in between whole's (W) interval, I had this exact wonder... where would H place their hand to keep in between? I wondered about he limits of the boundaries, assumptions, etc. Claire's alternative solution of putting the hand behind W's back was an interesting and more practical solution incorporating realistic boundaries. How creative!
Stop 4: "Bringing this comparison into broader dialog with the field, we suggest that researchers and practitioners attend more closely to the ways in which different patterns of mobilities in mathematical activity might affect the negotiation and development of reconfigured mathematical practices in any instructional design" (Kelton & Ma, 2018, p. 193).
These whole bodied activities produced some interesting and unanticipated results that developed from the use and limitations of the students' bodies (not necessarily the mathematics part) leading to interesting conversation that were brought about organically. There is much value in this kind of discovery through experimentation. I would attribute this quality to higher level students if they could do this kind of hypothetical thinking and pose a problem with pen and paper... but with whole-body movements, each child is capable of this higher level thinking by associating their movement with the concept and asking "what does this mean?" and "what now?". My big take-away from this is whole-bodied mathematics is an accessible way to engage mathematics concepts at exploratory/experimental (higher) levels for a wider range of students when done right.
Wonders: In this article, they have used really very dense vocabulary to describe some of these ideas, but aren't explicit in the concerns that they have for space and how it may influence movement-based mathematicians. Are there spaces, based on past experiences/history, in and around the school where movement mathematics would be severely hindered by their space? (I'm imagining the examples from the paper where a gym was used which, based on experiences, encouraged the students to move, walk and explore, whereas the classroom experience for W+H activity a student had to be coaxed out of her desk to start the exploration in a crowded room with its limitations actually leading to more theoretical imaginings.) I couldn't imagine even an area, even a home economics classroom with kitchens to impede whole-body mathematics.
Perhaps they are mentioning something more on a personal level? Each student has had an experience in a space, and that will influence how they interact in it. Probably why the researchers used new terms like "arena" and then "setting" that required "editing" (or certain ways that students interacted in the environment).
I resonate with your concerns about buy-in with older students, but after your summary, I wonder if combining the idea of space with narrative could create conditionals that would be easier for students to get started with embracing embodiment in mathematics. For example, math classrooms have not traditionally been spaces for movement, so buy-in here may be harder. But, if we were to go to an art room, where creativity is “supposed to happen”, and combine that with the narrative that whole body movement allows us to see patterns in different ways and uncover new understandings that paper pencil might not show us (like the spin for multiplying -1) maybe students would be more willing to try it out? As math teachers, we know math is creative and that creativity can come out in any setting. But maybe students that have been steeped in traditional teaching ways need a bridge until a new norm can be established?
ReplyDeleteI had the same concerns with Nichola, and I am thinking about my school now. Our math classes are generally large groups, and small rooms. So this type of activity is definitely not happening in the math classroom. It is not happening in the gym either, as it is always FULL with physical education classes, unless they go outside (this would require some organizing with Phys. Ed. teachers - OK). I think I would prefer to just take my class outside when weather permits for this type of activity! We have a very large green campus and lots of asphalt, so if something was needed to be drawn on the ground, we could use chalk! if not, we could go on the grass. We are currently a growing population, for years our school had ~500 students, and we have exponentially grown to ~750 in 2 years, so space is difficult sometimes!
DeleteThanks, Oliver, for the very interesting consideration of the detailed accounts in this piece! It is really striking that so many deep insights arise for most or all students through these thoughtfully-designed embodied activities -- things that only advanced students would be able to imagine with only pencil and paper.
ReplyDeleteAnd yes, most math classrooms are not designed for movement and have too little space and too much furniture! I've booked multipurpose rooms, drama rooms, dance studios, gyms, and just taken classes outdoors to the grass or paved areas when the weather is amenable. I think you're right, Nichola, that there is more buy-in once you get away from the forest of math room desks and into an arts-related room where creativity is 'supposed to' happen!