In my Math, Art, and Design class today, an awesome thing happened — students were arguing with me about math! In this class right now, we’re looking at the concept of infinity. We’ve talked about hotel infinity, and understanding the size of a set through one-to-one correspondence, and defining an infinite set as one that is the same size as some part of itself. Today we talked through Zeno’s dichotomy paradox, and then we looped back to answer a question we asked last week. Suppose you have an infinite number of piles, each with one peanut. And suppose a friend also has an infinite number of piles, but she has one peanut in the first pile, two in the second, three in the third, and so on. Who has more peanuts all together?
I left this hanging as an open question last time, and we talked today about the answer. I showed them a method for making a one-to-one correspondence between the two sets (thus showing that they are the same size), and that’s when they started arguing with me and with each other. The thing that felt so wonderful to me is that so many of them felt they had the right to claim space in the discourse as their own. They saw themselves as arbiters of truth, rather than allowing me to dictate mathematical reality. In the end I think that some of them were convinced that the two sets were the same size and others weren’t, but I loved the quality of everyone’s ideas and the fact that we had a space in which multiple views were being heard and debated.
That feels hard to do in math class, with math’s emphasis on “right answers.” I struggle with allowing this kind of discourse to remain open, without forcing it to close with a final decree on my part that I force them to accept. As a mathematician, I’m uncomfortable with ambiguity and uncertainty. I know that I get frustrated at my students when they insist on narrowing everything down to one “right answer,” and not allowing for the complexity of real problems, but I realize that I also crave that certainty because it helps me know where I stand as a teacher.
One other great thing happened in class today. After we discussed a worksheet the students had done on Zeno’s dichotomy paradox (finding the sum of the infinite series 1/2+1/4+1/8+…), students spontaneously opened up a discussion on problem solving process and how they felt as they were solving the problems on the worksheet. It was a brief discussion, but it made me really happy.