Book: OUT OF THE LABYRINTH: Setting Mathematics Free

Ellen & Robert Kaplan and The Math Circle


If you only read one book about mathematics teaching and learning this coming season, let me suggest that it should be OUT OF THE LABYRINTH: Setting Mathematics Free by Robert and Ellen Kaplan, founders of The Math Circle in Cambridge, MA.

I had the distinct privilege several years ago of attending a session Bob and Ellen led at Northwestern University's Math Club. We were taken through a short series of problems that led to the main question of the day: is it possible to cover a particular rectangle with non-congruent squares? The way things were led was truly masterful, perhaps the best teaching I've ever seen or experienced. The preliminary questions helped scaffold the main one, but once we entered into trying to solve the main problem, for which most in attendance seemed to believe the answer was "No," very few comments or questions were offered to help us. And yet those "hints" that were forthcoming seemed to be just the right ones needed to steer us out of ruts or stimulate our best thinking to move forward, and within about an hour, the students (who included undergraduate math majors, graduate students in mathematics, and some faculty members from Northwestern's mathematics department, none of whom seemed to be familiar with this problem or area of math) had successfully found a correct solution. The entire process was beautiful to see.

Afterwards, I introduced myself to the Kaplans and commented on how impressed I was with the way in which the lesson had been taught with such minimal but incisive questions and comments from them. Bob stated that they did the same problem recently with a group of 5th graders in Cambridge. Before I could express any skepticism or incredulity, he added, "Of course, it took about twelve weeks!" (The problem was presented as part of an on-going Math Circles course, and a great deal of time was spent building up the necessary tools so that the students could tackle the bigger question. I had no doubt, however, that the basic approach to helping these students was the same as what I experienced in Chicago.

When I asked Bob about who else taught in the Math Circle classes, he said that it was no problem finding people who knew the requisite mathematics. The difficulty was in finding such people who could also keep their mouths shut. And indeed, that is not only a problem in Cambridge, but nationally, including in my own teaching. I struggle mightily to resist the temptation to bail students out prematurely, to succumb to the pressure, both internal and external, to show what I know and to relieve the students of responsibility for thinking and learning mathematics the only way that really makes a difference, in my view: by doing it themselves. The cliche "Mathematics is not a spectator sport" is clever, but it is also very true. It's difficult to really "own" a piece of mathematics without struggling to wrap one's brain around it. This is not to say that we can gain nothing at all from a lecture, of course. But when the money is on the line and there's no one there to help us solve a challenging problem (where "challenging" is, of course, a very personal and relative term), will we do best by recalling a step we saw someone else use, seemingly pulled effortlessly from thin air, or by calling on methods we've used ourselves and built up sufficient experience with to sense that one might be of particular value in the current case?

The story the Kaplans have to tell in their new book is far more powerful than any argument I could offer here as to why it is so crucial for students to be led to believe correctly that they CAN do it themselves and, perhaps more importantly, that they SHOULD do it themselves, collaboratively, as a community of learners, without the usual competition to be the first or the best, but out of a collective passion to know.

I will say in all honesty that what I saw in Chicago had me nearly weeping with pleasure and frustration. Pleasure at the wonderful way in which the Kaplans taught, but frustration at the rarity of the experience and my knowledge that far too few Americans ever get to see and learn mathematics (or much of anything) in this powerful way. Reading their new book, I continue to experience these strong emotions. If you care about mathematics and really empowering people to DO it, go to the Math Circles web site: check into their books, look at some of the notes you can download from recent courses, look at the photos, and generally get a taste of what they do. You'll be very glad you did.