Some memories have staying power, and feel vivid and fresh like they happened yesterday. In this post I want to reminisce about the first problem I remember solving and give some context to it, which hopefully would be more interesting than the problem itself.
I was truly fortunate to begin my mathematical education in the unique environment of St. Petersburg in the late 80s (the city was called Leningrad back then, but I’ll stick with the current name). The basic unit of extracurricular math instruction was a circle—an entity consisting of motivated and self-selected kids of the same grade level led by several adults, typically young mathematicians or PhD students. A circle was a living and breathing organism that grew, thrived, matured and then withered away over the course of several years.
To call attending these circles a rite of passage is an understatement—it has been defining experience for several generations of St. Petersburg mathematicians. Both recent Fields medalist from St. Petersburg—Grisha Perelman and Stas Smirnov—progressed through all stages of this system from fifth-graders to lecturers.
This “system of math circles” as it is commonly known is documented in the excellent book by Masha Gessen “Perfect Rigor” in the chapters on early education of Grisha Perelman. Without attempting to give a comprehensive picture of how these circles operated, I mention here three core principles that sustained the system for more than 30 years.
The first was casting a wide net, bringing in all interested middle schoolers (11–13-year-olds). The attendance in the first year could easily exceed a hundred kids, split into several groups of about 30. The number of students naturally diminished over time, with only a handful making it to the last year of high school. Upon graduation, alumni were often co-opted into the mentoring ranks, and this practice of training its own instructors proved surprisingly resilient even in the face of mass exodus of Russian mathematicians in the late 80s and early 90s.
The second principle was emphasis on hearing students out on every one of their solutions. A typical meeting of a circle began with students explaining their solutions to homework problems in one-on-ones with the instructors. It was followed by presentation of complete solutions on the blackboard by an instructor and a lecture that could be part of a longer course (such as introduction to graph theory or complex analysis).
The third principle was what now would be called gamification of mathematical instruction. The math circles were integrated with the system of mathematical Olympiads, and essential component of their formula was practicing on competitive math problems. The connection ran deeper: competitiveness was encouraged by keeping track of the students’ individual accomplishments, and math Olympiads were important milestones of the circle’s annual cycle. Even more significantly, much of the standard math curriculum, roughly corresponding to the requirements of a typical math major in the US, was taught through a series of small, tractable problems rather than lengthy lectures. There was a place for lectures too but they were mainly used to introduce definitions and discuss examples (the appropriate buzz word du jour is “flipped classroom”). Two outstanding books, translated from Russian, exemplify this approach to mathematical instruction: “Elementary Topology” by Viro et al. and “Abel’s Theorem in Problems and Solutions” by Alekseev. This really conveyed a sense of mathematics as something to be discovered (and discoverable!) rather than passively imparted or handed down from on high.
That was the backdrop (to which I was blissfully oblivious) to the first session of the math circle I attended in March of 1988 at twelve years of age. A set of problems was distributed, and we were asked to report solutions at the next meeting. One problem, which many readers will recognized as an old classic, read as follows:
A standard chessboard has two opposite corners removed. Each of the 62 remaining squares is occupied by an ant. On cue, all ants crawl to adjacent squares (sharing a side with the square they have come from). Prove that two ants will end up in the same square.
At the next meeting I began explaining my “solution” to a student volunteer, who had to endure some inarticulate nonsense along the lines of “I tried arranging the ants every each way, and it didn’t work”. He gently offered a few hints, and finally asked whether I could think of some characteristic specific to the chessboard and suggested that I drew it in my notebook. That was the moment when the light bulb went off: “The squares are black and white, and there are 30 blacks and 32 whites!” Indeed, each ant has to move to a square of the opposite color, and there are more white squares than black. Even writing this, I get choked up, recalling the sensation of suddenly seeing a crisp and elegant solution instead of a swarm of ants crawling all over the place. This is how the pigeonhole (Dirichlet) principle was introduced, and I got hooked and stayed with the circle.
Twice-weekly sessions of the math circle led to a summer camp, with three weeks of several hours of math per day. This became an annual ritual until I graduated from the high school, which recruited from the same math circle, and then came back as an instructor. Intellectually it was a fairly cloistered environment but it did lead to many wonderful things in my life, and it was more than balanced by seismic shifts happening in the world outside (the country ceased to exist, for one).
All math circles reflected personalities of their leaders. The hallmarks of mine was a high number of girls, due to Anna Bogomolnaia’s efforts, and its bias towards mathematical analysis, fueled by professional interests of other instructors—Evgueni Abakoumov, Lev Parnes (who tragically died in 1993), Evgeny Dubtsov, and Fedor Nazarov. They all shared passion for teaching, learning, and practicing mathematics, which proved to be infectious for many of my fellow members.
I fell under the spell of computers and CS while still attending the math circle, but that’s a different story.
Update: This post would be incomplete without referencing two books: one very recent, “Mathematical Circle Diaries, Year 1: Complete Curriculum for Grades 5 to 7” by Anna Burago, and one slightly older, aptly titled “Mathematical Circles: Russian Experience” by Dmitri Fomin, Sergey Genkin (my math-circle grandfather and a fellow Microsoftie) and Ilia Itenberg.