Yesterday I had a very difficult, but ultimately wonderful, teaching experience. Currently, in Math, Art, and Design, we have been learning about fractals. Yesterday, I planned to hit them with some big guns — Julia and Mandelbrot sets. To understand how these sets are created, it is necessary to understand how to “square” points on the plane, and for that, you need to think of each point on the plane as a complex number. So I showed the students a little intro video on complex numbers, gave them the formula for squaring, and we were off and running, doing some squaring practice as a prelude to Julia sets.
Except we weren’t really off and running. My students stopped the class because they didn’t understand what was going on. Questions were asked. I slowed down and backed up a bit. More questions were asked. The students were frustrated, and I couldn’t move the class forward. I want to tell you what was going on for me at the time, what I realized afterward, and what I think was happening for the class as a whole.
What was happening for the class as a whole? The students became mathematically liberated. They were involved and active, and they weren’t going to let me take the class to a place that they were uncomfortable with. They wouldn’t let me do things that they couldn’t understand, and they didn’t back down. I feel so proud of all of them for taking that power. I know that some students felt fine with how the lesson was going, either because they felt they understood things well or because they didn’t want to understand, just to follow my lead. But there were several that found their lack of understanding unacceptable, felt that they could understand the material, and felt powerful enough in the class to insist that I help them understand.
What was going on for me? I was frustrated, scared, and upset. The feeling that I had during part of that class is that the students were taking over, and that once they had control they were going to humiliate me. Isn’t that interesting? I was facing a classroom full of students, many of whom had felt humiliated and ashamed in math classes in the past. I know that because they wrote about their experiences, and we have talked about them on more than one occasion. In a typical classroom (and my classroom is often typical, even though I aspire to be different), the teacher holds all of the power. We determine what is talked about, for how long. We determine when something is done correctly. We even determine when we’ve provided sufficient explanation and its time to move one. I have been trying to give an increasing amount of control of those things to my students, and what I learned yesterday is that you have to be careful what you wish for, because I did not feel safe when the students had control. I’m not talking about my physical safety here, but rather that I had made myself vulnerable to the students and that they could really hurt me.
It wasn’t until the class was over, and I look at some of the student responses to the class* that I realized that they were trying to hurt me, and that they still liked me. The only thing that kept me going during the class was remembering that I had established an environment of trust in the classroom and that I just couldn’t violate it, no matter what. I couldn’t shut the students down, so I kept going. It wasn’t until I felt safe and was able to unpack what was happening for me emotionally that I realized how awesome the events of class really were.
As a final note, I realized last night where I went wrong in my attempt to move the class forward on the path to Julia sets. I gave them a completely arbitrary and inauthentic notion of squaring complex numbers. I may just as well have called it “some weird formula we do” instead of squaring. Now, that’s not necessarily a bad thing. As a matter of fact the way we were able to move forward is to stop talking about complex numbers, and just look at the process as a weird way to start with a point in the plane and get a different point. The real problem happened because I pretended that I was showing them something authentic and comprehensible. I gave them 2 minutes on complex numbers and acted as if that was enough to understand the squaring process. Of course it wasn’t, but I assumed the students would just trust me about the formula and move on to using it, rather than trying to understand it. In hindsight, I either should have gone carefully over complex numbers or not mentioned them at all, or perhaps mentioned them as a side comment with the suggestion that if they were interested in further explanation, I’d be happy to talk to them outside of class. I should not have pretended I was teaching them about complex numbers when was really doing nothing of the sort.
*I have the student write a few sentences about class after most classes, including what went well, what needs changing, and anything else they need to tell me.