Many math classes go something like this: Teacher reviews a homework problem or two, introduces a new idea, gives a couple of examples, and then assigns a problem set to students. Students try those problems after class, at home, and what seemed to make sense in class is suddenly a complete mystery. As teachers, we often forget that the first time we try a problem, it isn’t always easy. In my case, of course Algebra I appears easy for me when I do it. I’ve been playing around with basic algebra skills for more than 25 years, so it would be pretty inexcusable for me to falter. And yet, when I first started teaching, I used to go out of my way to be sure I would appear infallible to my students. I selected problems that worked out nicely and quickly, perpetuating the idea that for some of us, math is just easy. As I progressed, I realized that students needed to see understand the true work of a mathematician, and I had several examples from my pre-teaching career to draw upon. But all the stories in the world were no substitute for seeing me make mistakes. I’d then fake mistakes, but no matter how good my acting, I knew it was fake, and students surely figured it out as well.
Now, I seek out truly challenging problems, the kind of low floor/high ceiling problems that are easy to explain, but difficult to solve, and most importantly, new to me. On a few occasions, I have introduced some of these problems to my classes, either as challenge problems or problems that arise in the context of a lesson. A recent example from my geometry class involved trying to find the area of a trapezoid in a different way than normal. I suspected that we would be able to derive the common formula, but I wasn’t positive the solution would be there, nor was I sure how we could arrive at the solution. Still, this gave my class the opportunity to see me in a vulnerable state, wondering out loud to them what we could do, soliciting ideas from students, and exploring every possible pathway. In the end, this turned into a really nice derivation. (That’ll be another blog post, don’t worry).
So, my challenge to myself is to continue to not just show math to students, not just to ask them to do math, but to actually collaborate with them on math problems for which I don’t already know the answer. Because I want them to see how we actually do math as mathematicians, and to see why the focus needs to be on the process, and not just the answer.