Review: Math from Three to Seven, by Alexander Zvonkin

It’s a somewhat flawed book review that takes a while to get started, but I did a search and didn’t see another good one for this book. From the blog post:

I’ve previously complained about how much I hate Russian mathematician Edward Frenkel’s book, but one thing it gets across well is just how important passion is to being a great mathematician, and passion was the thing the émigrés seemed to have a surfeit of. In college, the joke was that seminars by American professors would last an hour, whereas seminars by Russian professors would turn into boisterous debates lasting all night. […]

[I]n the interviews I’ve read with Soviet mathematicians and scientists, the things that comes up over and over again are “mathematical circles,” a practice that originated in the pre-revolutionary Russian Empire and then spread far and wide through the Soviet Union. A mathematical circle is an informal group of teenagers and adults who really enjoy math and want to spend a lot of time thinking and talking about it. They’re a little bit like sports teams, in that they develop their own high-intensity internal culture and camaraderie, and often have a “coach” who is especially talented or famous. But they’re also very unlike sports teams, because they don’t compete with each other or play in leagues or anything like that, and usually any given circle will contain members of widely varying skill levels. Maybe a better analogy is a neighborhood musical ensemble that gets together and jams on a regular basis, but for math.

The most important thing to understand about mathematical circles is that the math they jam on is completely unlike the math you study in school, and also completely unlike the “competition” math that bright kids in the United States sometimes do. […]

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The bread and butter of the mathematical circle is solving problems together, as a team. There is no time here for exercises; you can do that lame stuff at school. Sometimes the coach picks a problem for you, something just beyond your ability, just the thing you need to hone your edge. But sometimes the whole circle works together on a problem that nobody has the answer to and that challenges the very best members. These problems are the most important, because with them you see great minds, men older and more talented than you, stretched to the breaking point and occasionally beaten. You see them grind and grind and try every possible attack on a problem and sometimes lose anyway.

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This book is the story of one such mathematical circle. But it’s an unusual one because…it’s for preschoolers.

The “coach” of this circle is Alexander Zvonkin, a professional mathematician frustrated that his kids are having all the wonder and life and joy crushed out of them by the grey functionaries at their school. So he starts a circle for his son Dmitry and a few of the neighbors’ kids, most of whom are around three or four years old. That’s young enough that according to Piaget’s experiments there are cognitive modules related to number and volume that simply haven’t come online yet. Fortunately, Zvonkin is familiar with the latest research on developmental psychology, and turns lemons into lemonade by using the kids’ lack of numerical intuition to introduce them to some pretty deep ideas about when two sets have equal cardinality. […]

At this point I expect you are rolling your eyes, especially if you have experience with three-year-olds. It can be difficult enough to get them to sit still, never mind ponder deep questions about the cardinalities of sets. And what exactly does it look like to pit somebody against a problem who is barely potty-trained? This is where the genius of Zvonkin’s format kicks in — it’s not really a book, it’s a journal, and one that is barely edited. So it’s full of failure after failure, entries like, “today I had a cool idea for a puzzle but everybody just screamed instead and then one of the kids vomited.” And yet, slowly, wondrously, over the four years of the circle’s existence, his patience pays off and the kids start doing really incredible things.