National Association of Math Circles Wiki
The Fountain of Youth

Legend has it that Ponce de Le´on discovered Florida while searching for the Fountain of Youth. While this tale is almost certainly not true, the belief in an elixir that will restore youthful vitality or even reverse the aging process continues to persist. And while most math circle coordinators would welcome a rejuvenation of mathematical energy among its participants, not everyone is interested in decreasing their actual age. Yet this is precisely what can and does happen in many instances, at least to the average age of the students attending.

The groups most vulnerable to this Fountain of Youth effect are those aimed at high school students. As noted previously, this age group is attractive to coordinators in a college math department for several reasons. For instance, students are closer in age and mathematical maturity to those whom the coordinator are accustomed to teaching. In addition, the list of potential mathematical topics which could be presented to such a group includes many more fascinating ideas than would be available to a middle school audience. Why then are math circles devoted solely to high school students so scarce?

The Stanford Math Circle provides a typical illustration of how the Fountain of Youth effect operates. Prior to its first meeting the circle was advertised only and teachers. Topics selected during the fall (including Gaussian integers, geometry of complex numbers, and topology) were tailored for an older audience. However, word quickly spread among parents looking for mathematical enrichment for their middle school student with a scientific inclination and light homework load. A tally taken of all students who attended the Stanford Math Circle at any point during the winter session revealed that approximately a quarter of them were in eighth grade or below. Several factors help to guarantee their presence: middle school students typically have fewer responsibilities and hence more free time to devote to such activities, they are often bright enough to understand enough of what is presented to be truly captivated by the material, and they are still substantially influenced by their parents’ decisions regarding their activities and reliant upon them for transportation. Indeed, some of the most consistent participants that winter were also the youngest.

Once a significant proportion of the audience falls into the under-fourteen age range, at least two things are prone to happen. First, high school students may start to feel self-conscious about hanging out with younger kids. Expressing a public interest in math is uncool enough at many schools; doing math with middle school students might be too much. It doesn’t help when younger siblings tag along, and the situation becomes even more uncomfortable when there is a little hotshot who is actually as quick at answering questions and coming up with bright ideas as the older students. At a more significant level, speakers may also start to unwittingly lower their level of exposition in response to the young faces and elementary questions. Since the younger students are often the most enthusiastic to raise their hands, this can be an issue even for speakers who are presenting more advanced material. Math circle coordinators are especially susceptible to watering down material since they are the most familiar with the group and most eager to boost attendance by ensuring that the mathematics is accessible to everyone.

There are several remedies for this situation which help middle school students feel welcome without discouraging the older students. One of the simplest is to encourage parents or other adults to attend, thus artificially raising the average age. Once the math circle setting feels less like a school classroom and more like an event of interest to the general mathematical public, the issue of age range will become less prominent. It is also important to maintain a steady diet of advanced topics (but not too advanced!), even at the risk of leaving behind some of the middle school students. They are not as easy to intimidate as most coordinators believe, and the ones who are willing to be stretched will benefit a great deal from the presentations regardless. It is good policy to set clear expectations during the first meeting with respect to how much of the material students are likely to absorb. At the Berkeley Math Circle newcomers are advised that to follow even a third of any given talk is a significant achievement.

Occasionally students who attend for several years may reach the point where they feel they have mastered the math circle curriculum, such as it may be. Thus a Stanford Math Circle parent raised the concern, in a very diplomatic and constructive manner, that material was being pitched too low for his Olympiad caliber daughter. The discussion was productive and led to the conclusion that, rather than raising the level of exposition and leaving the majority of the remaining students behind, it was time for this particular girl to move on to more advanced offerings. (These might include top flight summer programs, regional or national olympiads, or college courses, for instance.) In a sense this scenario is ultimately the goal of any math circle: to attract students interested in mathematics and develop them into such proficient and experienced problem-solvers that they are ready for greater challenges.

Yet another recourse for math circles with a sufficiently large audience involves splitting the group into two or more tracks based on some combination of age and mathematical experience. This has proven to be a successful approach in Boston, Berkeley, San Diego, and San Francisco, for example. Many aspects of coordinating such a circle, such as filling the speaking schedule, become twice as taxing as before, so be sure to muster additional staffing support. Ideally a secondary coordinator could take over the middle school group, as occurred at the Stanford Math Circle.

It is worth mentioning that the Fountain of Youth effect doesn’t automatically resolve itself over time as students grow older. In other words, middle school math circle participants don’t necessarily become high school math circle participants. A math circle might be the only outlet for younger students interested in science; as they grow older more opportunities become available and the same students may opt for the robotics team or become engrossed in developing their school’s web site. Similarly, as students become increasingly independent of their parents they may decide that they are actually more interested in the a cappella group or drama club than in math. Yet another factor at work to discourage high school attendance is the fact that participation in a math circle is not perceived to carry nearly as much weight on a college resume as does recognition of achievement in competition, even though these prizes are often much less meaningful. Some math circles have incorporated olympiads or other types of productive contests into their circles, not only to give kids an arena in which to concentrate on solving challenging problems, but also to be able to make awards to their top students. Another organizer is planning to act as a mentor to older students interested in developing math projects for science fairs or papers for talent searches. Students often need direction when developing fruitful problems to tackle. They would then work independently to discover and present solutions.

To be fair, there are many excellent reasons to arrange a math circle for younger students, say in grades five through eight. This window of time is arguably the most effective one for spreading the mathematical gospel, since many students make up their minds in terms of whether they “like math” or are “good at math” during these years. In fact, individuals on a mission to spark interest in the subject will probably want to focus their efforts on a middle school audience. The point is that coordinators interested in reaching older kids must be proactive in finding ways to keep a math circle targeted for high school students attractive to that age group.

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