When we were planning our trip I was very excited about going to Bend, OR. It has the reputation as being an outdoor mecca, the kind of place where you can climb a mountain in the morning and kayak a river in the afternoon, so I wanted to make sure we got to enjoy as much of it as we could. Unfortunately, a heat wave had settled in over the Pacific Northwest a few days before we got there. It was 100 degrees on Wednesday, which was far warmer than anything we were used to and not at all conducive to doing everything I had hoped. Still, we did get ouside for a while, giving the kids their first-ever opportunity to wade in a river, which in turn gave a me a little more time to sit and think (and to do some teaching).

My thoughts turned again to the “What if?” questions. As I said in my last post, I’d been thinking about how to pose the right kind of “What if?” questions, the kind that challenge students to deepen their understanding as they try to answer them, when another question occurred to me: what if I’ve been thinking about this the wrong way?

The way I think about teaching math is based on the idea that unlike the other disciplines, which have a manifestation in the physical world, mathematics can exist purely in the mind. One of the marvelous things about math is that we see it manifested in the world around us, or at times motivated by the world around us, but mathematics would exist even if we never observed it. Because of this, we can know things in mathematics with a certainty that we can never have in other disciplines. Scientists, for example, never prove something true; the best they can do is repeatedly fail to prove something not true. In mathematics, however, logic allows us to prove something is true. A proper liberal arts education, then, must include an understanding of pure mathematics simply because pure mathematics exists.

The implication of this argument is that while it’s fine to teach applications of math, we must teach pure math. The corollary to this is that we should focus on pure math, including applications only after we’ve taught the theory. This belief is evident in most major textbooks and most high school classrooms today. A second corollary is that applications should only be the focus of classes for students who aren’t “strong enough” to be able to do pure mathematics, which is another belief that is evident by looking at most high school curricula. So that leads me back to my question, and to related questions:

- What if we didn’t focus on pure mathematics first?
- What if “strong” high school math students were those who could apply the math well instead of those who could prove things?
- What if students used ideas first and proved them later?
- What if our primary goal was to graduate students who are mathematically fluent?
- What if we replaced one term of our existing curriculum with a course in critical reasoning or (non-symbolic) logic?
- What if our goal was for students to take AP Statistics instead of, or in addition to, AP Calculus?

Let me be clear: I don’t have answers to any of these questions, they’re just the questions that occurred to me when I started considering the theory-application dichotomy. What I like about these questions is that they’re along the same lines as the question, “What if water didn’t evaporate?” that got me started on this train of thought. I still have a lot of thinking to do, and a lot more of *The Falconer* to read. The next few weeks should be very interesting.