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Mathematical enrichment activities in the United States have been around for at least thirty years, in the form of residential summer programs, math contests, and local school-based programs. The concept of a math circle, on the other hand, with its emphasis on convening professional mathematicians and secondary school students on a regular basis to solve problems, has appeared only within the past twelve years. This form of mathematical outreach made its way to the U.S. most directly from Russia and Bulgaria, where it has been a fixture of their mathematical culture for decades. (The first ones appeared in Russia during the 1930's; they have existed in Bulgaria for a century.) The tradition arrived with *emigres* who had received their inspiration from math circles as teenagers. Many of them successfully climbed the academic ladder to secure positions within universities, and a few pioneers among them decided to initiate math circles within their communities to preserve the tradition which had been so pivotal in their own formation as mathematicians. The Mathematical Sciences Research Institute (MSRI) in Berkeley, California became involved at an early stage by supporting the Berkeley Math Circle. Not long after, Steve Olson highlighted this math circle in his book *Countdown*, since a couple of members of the 2001 U.S. International Mathematical Olympiad (IMO) team attributed their success in part to the problem-solving sessions offered at Berkeley. In this and other ways math circles began to attract national attention as a means for encouraging students to enjoy, explore, and excel in mathematics.

The author was part of a delegation from the United States that recently visited Russia to observe the secondary school math culture in Moscow and St. Petersburg first-hand, including visits to a number of active math circles. These circles constitute only one piece of a wide array of programs aimed at identifying and cultivating mathematical talent amongst the Russian student population, at least those privileged enough to live near an urban center with a university or institute supporting such activities.

Russian students grow up within an educational system that promotes problem solving (as opposed to computation or textbook exercises) to a much greater degree than in the United States. Therefore it is a natural step for these students to begin taking olympiads and attending math circles during their middle school years. (In contrast to a math contest, an olympiad typically asks students for an explanation rather than a computation. These olympiads are offered at all levels and geographic scales; the most prestigious is the All-Russian Olympiad which helps to determine their IMO team. It is the Russian analogue of the USA Math Olympiad.) The olympiads and circles feed into one another and serve as the entry point to the school-level mathematical culture in Russia. From there the stronger students might be admitted to one of several "specialized schools" with advanced math classes taught by teachers whose mathematical pedigree is comparable to those at the best American high schools. Students who wish to pursue mathematics may also attend a selection of summer programs and participate in various special events such as "math battles" or "math fests."

It is illuminating to contrast this system with that currently in place in the United States. The most notable difference concerns the general perception of mathematics as a domain for investigation versus a tool for science or business. This is due in large part to the significantly higher level of interaction between research mathematicians and secondary school students in both Russia and Bulgaria. It is not uncommon for graduates with degrees in mathematics to enter the teaching work force as well as to continue active research at some level. It is also standard for well-known mathematical figures to join the staff at summer programs. The Moscow Center for Continuous Mathematics Education, which coordinates a wide array of activities for school-level students, is housed in the same building as the Independent University of Moscow and has ties with Moscow State University as well. Most striking, perhaps, is the incredible level of involvement of a considerable number of undergraduate and graduate students who spend hours each week helping out at math circle meetings, grading olympiads, or otherwise supporting the effort. Because of their faithfulness in repaying their debt to the prior generation of university students, kids attending math circles receive an unparalleled amount of personal attention. It is standard operating procedure to devote the bulk of a meeting to individual discussions with mentors, in which participants present their solutions to problems which had been distributed during the previous session.

At the same time, Russian mathematicians working to encourage students with an interest in math face many of the same challenges that their American counterparts do. There are state-mandated multiple choice tests for all students intended to guarantee minimal competency in math which divert attention from high performers. Those who conduct mathematical research often find themselves at odds with those who study mathematics education. Many students who would benefit greatly from attending math circles are geographically out of reach. Time and funding for worthwhile projects is often in short supply. But it is heartening to note that because of these similarities each nation can learn from successful approaches discovered and employed by the other.

- From *Circle in a Box*