Another student in the class responded to this student's drawing with a friendly counterargument. In essence, he provided the student with a logical argument about why these five divisions were not squares. The student who had drawn the proposal on the board seemed to recognize the fault in his argument and we kept discussing other cases. Then, on his way out when class was over, I overheard him say, "I wonder what size starting square would make my five case possible?" It seemed to me that this student was still searching for ways to preserve his mental equilibrium; hesitant to make a (perhaps?) necessary accommodation.
All of this got me thinking about the ways our minds work and the role of the teacher in working with students. It's fascinating to me how people work with their ideas. I'm convinced that any lasting change must come from a student. I use the example to make a larger point about our goals in working with students. In the context of this puzzle-like problem, it's pretty easy to suggest that we just keep letting this student play with other size starting squares; to keep letting him play with his own ideas. I know that in my own teaching, it isn't always as easy to preserve this play-like quality with things that we are "supposed" to teach. But to hold students accountable to those things (or what we perceive to be their mastery of that Mathematics) is an oversimplification of the workings of the human mind and a coercive, unjust position to hold in education. Because if we were to force this student to abandon his ideas about that square, to insist that his thinking is wrong, he would inevitably start to reject his own thinking as mathematical. He would inevitably start to surrender his own agency.
I think it brings up some interesting questions about the goals of education. I've been reading The Having of Wonderful Ideas recently (which I highly recommend) and I think Eleanor Duckworth puts it quite nicely:
"Many people subscribe to the goals of the it's-fun (interest) and I-can (confidence) types, but when it comes to detail, almost never does one see a concern with anything other than the-way-things-are beliefs. Lesson-by-lesson objectives are almost without exception of this type, despite the fact that general goals very often mention things like interest, confidence, and resourcefulness. This is, of course, because it is difficult to produce a noticeable change in any of these in the course of one 50-minute lesson. But notice that as a result all the effort is put into attaining the objectives stated for the lesson."