Part of my goal for my students this year is to help them become comfortable being mathematically curious. In an effort to help students develop a growth mindset and to facilitate learning opportunities that foster this, I try to pose questions that allow time for exploration. This has been more difficult than I thought it would be, since I am somewhat attempting to undo 10-11 years of an unwritten didactic contract: teachers instruct, students passively absorb and regurgitate information. Repeat as many times as necessary.
Most of the trimester I have gotten questions like, “Is this what you are looking for?” or “Are we going to be tested on this?” I thought that as the trimester progressed, this would happen less often. I was wrong. I had to admit to myself that as long as I was in charge of giving these kids a grade and as long as their grades remained a driving force in their outlook on education, I didn’t see this changing much.
I wanted to make note, however, of the progress these kids have made toward being more mathematically curious. I’ve exposed them to some interesting graphs, and some students have shared how they expanded those ideas to make even more intricate versions of those graphs. (Ever wondered what y=xsin(1/x^13) looks like?)
I had another student explore the graph of y = 1/(x-2) + 1/(x-2) and wonder if there was an equation we could write that would isolate just the middle portion, between the two vertical asymptotes. He thought that it looked cubic, so he played around with a number of cubic functions, but couldn’t get the graph to fit quite right. I commended him for his efforts, because this was the type of student known for wanting his math straight to the point. I asked him the range of a cubic function compared to the range of the graph he was trying to match, and he quickly saw that his initial thoughts on a cubic function were incorrect. I challenged him to keep searching for an equation (or two) that would match the portion of the graph he wished to isolate. This was an important moment for both me and this student. I had gotten him to explore and wonder with something that had no external purpose. He did this for the meer wonderment of whether it would work or not.
This was fantastic and worth reflecting over for me as an algebra teacher. Much of high school algebra is taught in a dry, procedural manner. Unfortunately, the kids expect it this way, and the high achieving kids even want it this way. They’ve been successful with it so far, why change it? I hope as I continue to pose mathematical questions to these kids that they continue to push their understanding forward by exploring.