Mathematical circles are a form of outreach that bring mathematicians into direct contact with pre-college students. These students, and sometimes their teachers, meet with a mathematician or graduate student in an informal setting, after school or on weekends, to work on interesting problems or topics in mathematics. The goal is to get the students excited about the mathematics they are learning; to give them a setting that encourages them to become passionate about mathematics.[MSRI Circle in a Box 1]
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Math circles can have a variety of styles. Some are very informal, with the learning proceeding through games, stories, or hands-on activities. Others are more traditional enrichment classes, but without formal examinations. Some have a strong emphasis on preparing for olympiad competitions; some avoid competition as much as possible. Models can use any combination of these techniques, depending on the audience, the mathematician, and the environment of the circle. Athletes have sports teams through which to deepen their involvement with sports; math circles can play a similar role for kids who like to think. One feature all math circles have in common is that they are composed of students who enjoy learning mathematics, and the circle gives them a social context in which to do so.[MSRI Circle in a Box 1]
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.[MSRI Circle in a Box 2]
Decision about content are difficult for newly forming math circles and clubs, or for parents seeking groups for their kids.
Project-based clubs may spend a few meetings building origami, developing a math trail in their town, or programming a mathy computer game together. Math-rich projects may be artistic, exploratory, applied to sciences, executable (software-based), business-oriented, or directed at real contributions to local communities. Museums, cultural and business clubs, tech groups, online networks, artists/musicians/actors active in the community, and other individual professionals can make math projects especially real and meaningful. Increasingly, math clubs invite remote participation of active people (authors, community leaders, professionals) through webinar and teleconferencing software.
Problem-solving circles get together to pose and solve interesting, deep, meaningful math problems. Problems considered "good" are easy to pose, challenging to solve, require connections among several concepts and techniques, and lead to significant math ideas. Best problem solving practices include meta-cognition (managing memory and attention), grouping problems by type and conceptual connections (e.g. "river crossing problems"), moving between more general and abstract problems and particular, simpler examples, and collaboration with other club members, with current online communities, and with past mathematicians through the media they contributed to the culture.
Research mathematicians and connecting students with them can be a focus of math circles. Students in these circles appreciate and start to attain the very special way of thinking in research mathematics, such as generalizing problems, continue asking deeper questions, seeing similarities across different examples and so on. [1]
Topic-centered clubs follow math themes, such as clock arithmetic, fractals, or linearity. Club members write and read essays, pose and solve problems, create and study definitions, build interesting example spaces, and investigate applications of their current topic. There are lists of time-tested, classic math club topics, especially rich in connections and accessible to a wide range of abilities. The plus of using a classic topic is the variety of resources available from the past; however, bringing a relatively obscure or new topic to the attention of the club and the global community is very rewarding, as well.
Applied math clubs center on a field other than mathematics, such as math for thespians, computer programming math, or musical math. Such clubs need strong leadership both for the math parts and for the other field part. Such clubs can meet at an artists' studio, at a game design company, at a theater or another authentic professional setting. More examples of fruitful applied math pathways include history, storytelling, art, inventing and tinkering, toy and game design, robotics, origami, and natural sciences.
Most circles and clubs mix some features of the above types. For example, the Metroplex Math Circle [2] has a combination of problem-solving and research, and the New York Math Circle [3] is some combination of a problem-solving circle and a topic-centered club, with vestiges of a research circle.
One can expect problem-solving groups to attract kids already strong in math and confident in their math abilities. On the other hand, math anxious kids will be more likely to try project-based or applied clubs. Topic-centered clubs typically work with kids who can all work at about the same level. The decision about the type of the club strongly depends on your target audience.
Math competitions involve comparing speed, depth, or accuracy of math work among several people or groups. Traditionally, European competitions are more depth-oriented, and Asian and North American competitions are more speed-oriented, especially for younger children. The vast majority of math competitions involve solving closed-ended (known answers) problems, however, there are also essay, project and software competitions. As with all tests requiring limited time, the problems focus more on the empirical accuracy and foundations of mathematics work rather than an extension of basic knowledge. More often than not, competition differs entirely from curricular mathematics in requiring creativity in elementary applications—so that although there may be closed answers, it takes significant extension of mathematical creativity in order to successfully achieve the ends.
For some people, competition carries with it a negative connotation and corollary of greed for victory rather than an appreciation of mathematics. However, those who run math circles centering mostly on competition rather than seminars and lessons attest that this is a large assumption. Rather, participants grow in their appreciation of math via math competitions such as the AMC, AIME, USAMO, and ARML.
Some math circles are completely devoted to preparing teams or individuals for particular competitions. The biggest plus of the competition framework for a circle organizer is the ready-made set of well-defined goals. The competition provides a time and task management structure, and easily defined progress tracking. This is also the biggest minus of competition-based mathematics, because defining goals and dealing with complexity and chaos are important in all real-world endeavors. Competitive math circles attract students who are already strong and confident in mathematics, but also welcome those who wish to engage in the mathematics competitive world. Beyond the age of ten or so, they also attract significantly more males than females, and in some countries, their racial composition is disproportionate to the country's demographic.
Collaborative math clubs are more suitable for kids who are anxious about mathematics, need "math therapy" because of painful past experiences, or want to have more casual and artistic relationships with mathematics. A playgroup or a coop that does several activities together, including a math club, usually chooses collaborative or hybrid models that are more likely to accommodate all members already in the group.
Most math circles and clubs combine some competitive and some collaborative activities. For example, student-founded and student-run Orange County Math Circle [4], while largely centering on competitions, hosts seasonal tournaments as well as infuses their competition seminars with fun mathematical lessons.