Chapter 1: On Returning

There is a kind of self-annihilation in mathematical work that makes it profoundly difficult to return to the world as oneself; and at the same time to savor the mathematics with which one has been engaged. We had been exploring star-like shapes that could be drawn without lifting a pencil off the paper. We were working systematically, placing a number of points in a circle, starting at one point, and then drawing lines to points a fixed number away around the circle. With seven points connected one by one, we could produce a “routine” septagon; connecting two away led to a lovely star; connecting three away, an even pointier star; four away felt like the three-away star, but emerged “backwards”; five-away was “backwards” from the two-away pattern; six-away was a backwards one-away. In that moment, when fifth grader Glen twisted around and exclaimed, “You have to be at least two away to get a star, and not just a shape, and it’s gotta be possible that the two-away dot isn’t exactly half-way, so you don’t get an asterisk, so, so … so,… you gotta have a shape bigger than a square to get a star! “ And the entire class was looking at him, and listening, and understanding what he meant. We also had already returned to the world, were out of that moment, and back in the classroom where we could, of course, think about the implications of Glen’s conjecture, or capture it in writing, or continue to collect examples of stars with different numbers of vertices. And we were already losing that same moment, losing the non-self experience of mathematics; we were returning both to each other, out of the mathematics, and back into the world of tasks and objectives.