school’s regular math curriculum is insufficient or inadequate; it simply recognizes that some students want more.

Dr. Stankova was surprised that a similar system did not exist in the United States. (The first math circle in the United States was founded at Harvard by Robert and Ellen Kaplan in 1994; Stankova’s was the second.) Originally the Berkeley Math Circle was intended as a demonstration for a program that would move into secondary schools.

But the United States turned out to be different from Eastern Europe in important ways. Here, very few secondary school teachers had the knowledge, the confidence, or the incentive to start a math circle and keep it going. This was different from the situation in Bulgaria, where schoolteachers were compensated for their work with math circles. Although some U.S. math circles have flourished without a university nearby (for example, the math circle in Payton, Illinois), most have depended on leadership from one or more university mathematicians. For example, the Los Angeles Math Circle has very close ties to the math department at UCLA.

Other differences showed up over time. With circles based at universities, logistics—getting kids to the meeting, and finding rooms for them to meet in— became more difficult. At present the Berkeley Math Circle, with more than 200 students, literally uses every seminar room available within the UC Berkeley math department on Tuesday nights. Most universities offer little or no support to the faculty who participate. Administrators do not always realize that the high-school students who attend the math circles are potential future star students at their universities. In fact, some of them are already taking courses at the university. Stankova has often had to alert UC Berkeley faculty members to expect a tenth-grader in their classes who will outshine the much older college students.

One part of the math circles philosophy has, fortunately, survived its transplantation from Eastern Europe to America. Math circles encourage open-ended exploration, a style of learning that is seldom possible in high-school curricula that are packed to the brim with mandatory topics. Problems in a math circle are defined as interesting questions that one does not know at the outset how to answer—the exact opposite of “exercises.” They introduce students to topics that are almost never taught in high school: for example, circle inversion, complex numbers, continued fractions (the subject of O’Dorney’s Intel project), cryptology, topology, and mathematical games like Nim and Chomp.

Many participants in math circles have gone on to success in scholastic math competitions, such as the USA Mathematical Olympiad (USAMO) and the IMO. For example, Gabriel Carroll, from the Berkeley Math Circle, earned a silver medal and two golds in the IMO, including a perfect score in 2001. He participated in the Intel Science Talent Search and finished third. As a graduate student in economics at MIT, Carroll proposed problems that were selected for both the 2009 and 2010 IMO events. Ironically, the latter problem was the only one that stumped O’Dorney.

But not all students are interested in competitions. Victoria Wood participated in the local Bay Area Math Olympiad but did not like having to solve problems in a limited time. She liked problems that required longer reflection (as real research

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