What Math for Biology?

At the banquet I attended recently, the high school senior I sat next to asked me two very interesting questions about his future education, one I had thought a lot about and one that I hadn't.

I'll tackle the first one now: what math should he take in college?

My own B.S. program required only Calculus. To be honest, I've never actually had to use calculus in my studies; the one time I attempted to integrate a function it turned out to be one without a computable derivative! But I would hesitate to ditch Calculus altogether, because it is a powerful concept and the concept of integrating shows up so often.

The easy recommendation was statistics. Of course, one needs a good statistics course -- the one course I took was awful. The main problem it had been dumbed down to meet the perceived weaknesses of the business majors for which it was a requirement, spreading over three semesters what should have been no more than 3/4 of a semester. It was only later I realized that I should have taken the Psych department's stats class, as it emphasized experimental design. Ideally, a modern stats course for biology would give a heavy exposure to Bayesian statistics and touch on topics that may see limited relevance in other areas (such as the Extreme Value Distribution -- important to search programs).

IMHO biologists would also be served well by a survey of various mathematical disciplines, a survey that would emphasize understanding of the key concepts over being able to execute all the calculations. I realize that is probably heresy in many math circles, but it would be very useful. For example, basic topology and graph theory -- everyone should understand the difference between a Directed Acyclic Graph and an Undirected Cyclic Graph without being able to prove theorems about them. You're going to run into eigenvalues constantly in the bioinformatics literature; demystifying them early is important. At home we have a great textbook (my mother collected them for tutoring) called Mathematics, A Human Endeavour, which has just the right flavor.

It should come as no surprise that I believe every science student should have an exposure to computer programming. Even if you don't ever write code for fun or profit, the underlying thinking is very useful. Understanding the fundamental data structures (tree, linked list, etc) and some common algorithms should simply be part of everyones intellectual foundation.

Given the opportunity (make me education czar!), I would, of course, like to see these things pushed far lower. My own math education, which I think is representative of typical in the U.S., was far too unaggressive and bored the heck out of too many people too quickly. Subjects like plane geometry need to be distilled down to their essence, with the remaining time filled with a taste of programming, which involves similar thinking to mathematical proof execution. It appalls me now that at an early age I learned how to compute mean and median, but nowhere in grade school was it pointed out when one might be superior to the other (especially since this distinction is so routinely ignored by the media).

One last comment: if my math education was representative, then too many math programs fail to stimulate real excitement in math. I do feel it is important to learn to compute certain things, and you really need to have the basic multiplication tables memorized to have any luck at algebra, but too much drilling flattens the excitement. I was disappointed, but not surprised, that my tablemate had never heard of the Seven Bridges of Koenigsburg. I doubt he had been exposed to snowflake curves (a simple to sketch fractal) either, a concept that blew my mind back in 3rd grade or so -- but it wasn't in the curriculum. A little less drilling, a little more mind expansion & math education would be on the right track.