I observed an honors-level geometry class several times this year in my middle school. I kept returning to this particular classroom on Thursdays because that is usually the day when the teacher steps aside. One geometry inspired problem is presented to the class and students are encouraged to grapple with it anyway they'd like, with anyone they'd like.
By choice, students stand, walk, move, congregate, and sit alone for the bulk of an entire period. And they talk about math while they grapple with math.
Some ping pong whys and what ifs and buts to one another out loud and some sit quietly and stare, and some sit quietly and scratch a pencil against a sheet of paper. Students work it out on laptops and students work it out on white boards. Some seems to work it out in their head. Some don't know how they worked it out, but they did. Almost all of the students used some combination of collaborative energy, social talk, walking (scurrying), debating, laughing, interrupting, pointing, blurting, listening, begging the teacher for a lifeline, et al. It was noisy at times. It was visually a whirlwind at other times. Yet, in the midst of this math party, pools of stillness and silence could be seen--some just sat and contemplated with an apparent deep focus as if nothing else was happening around them.
One thing I remember that my sketching captured in this group that my written words could not was the energy and joy for math. My inner child shudders because I can't remember having many positive math experiences when I was an adolescent or teenager. In my journal I scribbled, "So much physicality and a stream of social engagement" and that might be my happiest and most compelling revelation for me from this observation. And it makes me wish this is how I was taught math...immersed in the social energy of peers.
