A Mathematician's Apology

First, the apology: I started a summer job last week and it's taking up a huge amount of time. So posts will be infrequent until my job ends in August.

But since it's on my mind, I'd like to share a bit about this job. I'm assisting in the PROMYS for Teachers program. The goal of the program is to give math teachers an experience similar to the way mathematicians do math.

We mathematicians approach math differently from the way anyone else does. Most people learn math by watching a teacher explain a concept and demonstrate some examples. Students then apply these concepts to some practice problems, and that's pretty much it.

Mathematicians are in the business of discovering new mathematics, not reproducing what is known. To do this, we do what all scientists do: we experiment. Except that for us, experimentation involves only a pencil and paper (computers are sometimes used, but not as much as you might think.) We take numbers or shapes or other mathematical objects and play around with them. This can be a frustrating and fruitless process, but eventually we hope to discover something interesting about the way these objects work.

The next step after discovery is to describe our discovery. This is harder than it sounds, because mathematical language is very precise. One's first few attempts to describe a discovery are often wrong in some way; perhaps an important qualifier has been left out. Sometimes we have to invent new language to describe a discovery.

Finally, we try to justify our discovery by proving it from first principles or from other established theorems. This process can range from easy (a few minutes of thinking) to moderately hard (a few weeks) to epic (a few centuries.) The most famous mathematical theorems come from discoveries that are simple to describe but surprisingly difficult to prove.

In the PROMYS for Teachers program, we try to give teachers a taste of this experience. We give them numerical problems that hint at deep mathematical patterns. We then ask them to describe these patterns precisely and prove them if possible. This is often frustrating for them, since they haven't been shown how to do the problems or proofs beforehand. But by making their own discoveries, they take ownership of the mathematics, and when the process works it is tremendously exciting.

This program is modelled on the Ross program at Ohio State University, which I attended as a high school student. That program basically made me into a mathematician, so it is very rewarding to me to be able to share this experience with teachers.