### Free Math

I started reading non-fiction recently, due in large part to my friend Mara’s choice of birthday gift. Through the course of several bus trips I have read through Flowers for Algernon, Out of Africa, and Le Petit Prince. It’s a big change from me since I hardly remember reading any non-required non-math books in the last 5 years. After these three, all really enjoyable, I set my sights on a book I bought a while ago but barely touched, “Out of the Labyrinth: Setting Mathematics Free.”

The authors of the book, Robert & Ellen Kaplan, started running “Math Circles” in 1994 at Harvard, wherein a leader meets with ~10 students for ~2 hours per week to do mathematical explorations. (They had classes with students as young as 5 and going up to the late teens, sometimes with adult participants.) The general philosophy is to give the courses a minimal amount of structure, thereby allowing hands-on experience with mathematical argument, intuition, proof, and communication.

The book itself was extremely interesting to me. It is the first I’ve read that combines thick mathematical philosophy with a highly personal tone of voice. In order to properly express the importance of their Math Circles model, the Kaplans spend a lot of time discussing what it is that mathematicians actually do, and what constitutes mathematical research. This was comforting to me since I’ve recently sunk a lot of time into a journal version of a conference paper; the comfort was roughly “yes, there is a such a thing as good research, and it’s worth the time making your writeup reflect it; make absolutely sure it’s correct; use enough detail so that people can fully understand it if they try to read it; but don’t worry about spelling out every detail you’ve considered.”

I had mixed feelings on a chapter that spoke somewhat negatively of mathematical competitions. I enjoyed these when younger, but think they don’t fully prepare you for research, and agree that they can lower self esteem for people who don’t do as well as they would hope. Further, from my emerging viewpoint on the world of math research, I would say that excessive competition can be counterproductive for adult mathematicians.

I have been involved with “Math Circles” at Waterloo in the last two years and they run in much larger groups (upwards of 40) but in a patchwork fashion, with different people volunteering only for a few weeks at a time. There’s a certain economy of scale here since I don’t think Waterloo could recruit the 4x as many people needed to reduce the circles to a reasonable size. Still, I hope to apply some of the philosophy from the book to my next jaunt in the circles.

Edit: there’s an really interesting article by W. Thurston if you want a taste of the book without devoting yourself to hundreds of pages.

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