[I’m writing reflections on my student teaching, still.]
…But I should put more faith in my judgment of the kinds of lessons that can be created by following these sorts of provocative leads. If I had judged poorly, then I would have learned something about teaching mathematics. If I had judged well, then my students would come to expect that unusual but productive forays into thinking were the usual practice in our community. In either case, I would have demonstrated that mathematics is alive and that we, as students of mathematics, are willing to take risks, to challenge our assumptions, and to live a little in our math classroom. That means: we can and do make mistakes. This is not a new idea. It should be our practice to make mistakes as often as necessary, but no more, and not the same ones as before.