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Having expended considerable time and energy to create an opportunity for students to broaden their mathematical horizons, it is enormously affirming to see a solid group of kids taking advantage of this opportunity week after week. Assuming that there is a sufficiently large population of students in the area who are genuinely interested in what a math circle has to offer, it is not hard to establish a core group of faithful participants. There are a few common sense principles to observe which will help to ensure that attrition never becomes an issue. (It can be one of the more discouraging issues—to be avoided if possible.) A few of these principles have already been touched upon, such as choosing an appropriate meeting time based on the intended audience or asking families to invest in the circle financially by encourage donations or charging a nominal tuition. The remaining suggestions are all related to activities that occur at the math circle itself as opposed to decisions made during the planning phase.

In the absence of extra incentives, students will attend an event such as a math circle if they (or their parents) feel that they are part of a worthwhile activity and that their presence is significant. The quality of the mathematical content and its presentation is the subject of the second part of this handbook; for now let us consider ways in which students can be made to feel that they are meaningfully connected to the math circle. The task is to turn a collection of students of various ages from a number of schools with differing mathematical backgrounds into a cohesive community. This process is more of an art than a science; as such, it is not necessarily the forte of every mathematician. However, the following ideas can be implemented by any organizer who wishes to build a thriving math circle.

One of the most powerful means for accomplishing this goal is to establish a thread of continuity from one meeting to the next. Adhering to a uniform format, location, and meeting time is essential. If possible, the coordinator should attend every meeting in order to welcome students, make announcements, and introduce speakers. The regular presence of the person in charge helps to make the circle seem more familiar and accessible to students. Better yet, arrange to have speakers come for two or more weeks in a row, thereby halving the number of new faces that students must contend with and allowing students and instructors to become acquainted. (Recall that one should be prepared to offer non-trivial honoraria in this case.) Another effective strategy is to involve students in endeavors that span the course of several weeks or an entire semester. For example, at the first meeting students could be given a “Math Circle Challenge” consisting of a sheet of problems of varying levels of difficulty. The first ten minutes of each session could be devoted to presenting solutions, with the promise that a prize for the entire group would be awarded on the final meeting of the semester commensurate with their collective progress.

It would be hard to overstate the impact that is made by an organizer who takes the time to greet students by name as they arrive, asks them if they have come across any interesting mathematics lately, or lends them an interesting play with. Name tags for everyone at the first several meetings might help, although this approach is more likely to be effective at the middle school level. Use the initial moments of the meeting to run activities that help students connect with one another. For instance, one could instruct everyone to write down a secret positive integer from 1 to 20, with the caveat that they should select a number which they think that nobody else has chosen. (Adjust the range as necessary based on the size of the group.) Those who perform the task successfully win chocolates, those who don’t will have at least created a small bond with those other members of the circle who read their minds and spoiled their chance for a treat. By the way, this simple game quickly leads to some interesting mathematical questions in probability and expected value.

Along the same lines, incorporate plenty of time for mathematical and social interaction. Encourage speakers to set aside time for kids to work together on problems. Many topics can be motivated by simple games or numerical investigations upon which students can embark in pairs. Have students present ideas and solutions on the board, thus giving them a stake in the proceedings. Schedule time during or after the session for snacks and chatting. Plan a holiday party (with a mathematical theme, of course) or an ice-cream social for the second half of the final meeting of each term. The group could also arrange to attend an event of mathematical significance together. While the social aspect of a math circle should certainly not become its focal point, a healthy amount of social interaction is important to its well-being.

Incorporating special events into the calendar also serves as an effective antidote to attrition. Math circles in the San Francisco Bay Area all turn their attention to promoting the Bay Area Math Olympiad (BAMO) in late February and helping students to prepare for this intense proof-oriented competition. Everyone subsequently attends a lavish awards ceremony hosted by MSRI during the second weekend of March to hear a fabulous guest lecture, honor the high scoring students, and enjoy a nice lunch. Most circles have various special events spaced throughout the year, ranging from parties to special guest speakers. These events break up the routine of weekly presentations, provide a source of focus and anticipation about which to rally, and add momentum to the year, all of which contribute to healthy attendance.

Doubtless an organizer will want to drum up enthusiasm for an upcoming special event, which leads naturally to yet another means of retaining students: staying in regular touch. Several methods of communicating with participants have already been advanced—choose the method that best fits the group, then put it to good use. (But not overuse.) It is helpful for a student who has missed a few sessions to receive an email announcement regarding the need to bring a ruler and compass to the next gathering or reminding them of the special guest lecture for AMC-12 contest preparation. Students inevitably have legitimate reasons for missing several sessions in a row, at which point a potential barrier to renewing regular attendance materializes. The goal is to counteract this hurdle by offering students good reasons to come to at least one more math circle session. High school students in particular are bombarded with many demands on their time, and often need an extra nudge to attend an event that they would sincerely enjoy and benefit from once they arrived.

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