The typical evening at math circle goes something like this. Students arrive, check in, and take a seat and begin quietly working on the problems that have been posted on the Internet for the week. About an hour later pizza arrives and everyone takes a break, both eating a slice of pizza and talking with friends. Some twenty minutes later, students begin discussing the problems that were worked on during the first hour of the meeting.
It is during this discussion of the problems that the bulk of the learning actually takes place and it progresses in an inquiry fashion. A student may share an approach to a problem that they thought was promising, and another student may share a better approach, allowing for questions and greater understanding in the investigation of what the best approach may have been. This back and forth discussion runs for as long as it seems that there is still interest in the problems selected for that night.
As the interesting nature of the problems begin to get exhausted through familiarity, we usually switch gears in the last half hour and work on something more competitive, such as an old ‘GUTS’ round from a Harvard MIT Math Tournament. GUTS rounds are set up for students to work collaboratively in small groups competing against other small groups. This collaboration and competition can be very transformative for the dynamics in a group.
An example of this transformation that I have observed, would be when a ‘quiet’ group, perhaps composed of somewhat shy students suddenly comes alive in hot pursuit of the ‘win’. One evening, I had become aware of just such a group when it occurred to me that things were rather quiet during the discussion phase of the evening. It struck me that perhaps they hadn’t introduced themselves to each other, but at the very least some ice breaking seemed to be needed. I decided to circulate with the candy supply and conduct introductions as I went from table to table where it seemed they were needed. That was helpful for this particular groups for a little while, but it seemed that it was not the ‘fix’ I was hoping for.
I took notice of this table again during the GUTS round and was astonished to see the group had taken on a completely new enthusiastic character. The competitive and collaborative nature of the GUTS round had encouraged this group of students to drop their inhibitions and seriously pitch in working on very challenging problems. I have thought about what I witnessed that night on a few occasions this semester, and in different contexts. Would creating a shared ‘crisis’ in the form of solving problems under competitive pressure be helpful for educators in creating a cooperative group dynamic in the classroom, and when can that be useful?
In addition to the manufactured ‘crisis’ resulting in group cohesion though, I also considered how those students had created a shared learning space using the challenge of GUTS as a means to facilitate their interaction similar to Nemirovsky’s ‘lived-in space’. In the Nemirovsky article “Body Motion and Graphing” the observations made by the researchers detailed how a student and a facilitator working with a motion sensor that was hooked to a computer and monitor, had created a ‘lived-in space’ by means of their interaction. According to the authors,
“we want to highlight three properties of lived-in spaces, namely, that they are relational, intentional, and creative. By relational, we mean that changes, even if they are physically circumscribed to a particular aspect, affect the lived-in space as a whole. Lived-in spaces are intentional in the sense that they are places to do things and to accomplish purposes….Finally, by saying that lived-in spaces are creative, we emphasize that they are not set and fixed but always subject to and constituted by the ongoing drift of the life experience (Nemirovsky 153).”
The math circle students learning space was similarly relational, intentional, and creative as stated by Nemirovsky. It was relational because of the group dynamic that they had managed to create, if there had been a change in the number of group members or in member composition, it would have had an impact on their group dynamic. The learning space was intentional because of the purpose the space served in problem solving. Although the space was not particularly designed exclusively for the purpose it was being used for, one could say in an abstract way, the immediate environment of the students served in the same capacity. Finally, the space was creative as problems were distributed and solved and as new problems became old problems and were replaced with new problems, the ongoing progress formed a life process experiences.
Nemirovsky, Ricardo, Cornelia Tierney, and Tracey Wright. “Body Motion and Graphing.” Ethics & Behavior 16.2 (1998): 119-72. Print.