Blaw0013 wrote: “I cringe when the term "discovery" is used. It suggests to me that the teacher has a certain way of knowing or thinking in mind for the student to use. This seems to me to be a coercive way to interact with people. C. Kamii uses "re-invent" which seems better to me (don't know why). I most prefer invent--maybe because it is my way to remind myself to allow students to make the meaning that they do, it does not need to be mine.”
I have struggled with this question of how free the thinking should be in an ideal math class, and I think the first thing I realized is that I don’t teach in an ideal math class. There are time constraints, and there are people constraints. And I have a curriculum, ideas about what concepts I want the students to work with, a broad sequence that I think helps to build understanding in a productive way, and some goals for insights I want them to have.
I think of that as “mining student thinking.” I agree with Elizabeth: “I don't believe in playing ‘guess what I'm thinking’ because I find it psychologically and emotionally insulting.” I would add exhausting and usually discouraging, both for me and the students. I pose questions that highlight the ideas I am interested in, and then I accept whatever work they do in response, again highlighting the work I see as most pertinent. So it feels like a good balance of my leadership and their initiative. If something comes up that looks like an interesting detour, and we have time (that’s just the reality) then we will pursue it.
I also give the students opportunities to explore and investigate. We don’t spend nearly as much time in this open environment, partly because of the time, and partly because some students (I would say most) really struggle with the lack of structure. Clearly that is because they don’t get much practice with it, but there are also students who just aren’t that interested. Maybe that is a problem with schools, but as I said, I don’t teach in an ideal school. But students investigating is just one format for getting students to think about and discuss mathematics. Engaging the students in mathematical reasoning pretty much constantly is my broader goal, and that can occur in all sorts of ways.
I would caution beginning teachers from getting too caught up in trying to create an environment where the students follow a “pure” path of discovery. Students don’t have to discover all the math that they learn. But the process of discovering math should be a regular part of both what and how they learn.