By far the most important matter that the organizer of a new math circle must settle concerns its intended participants, as this decision will inform choices to be made on all the other issues raised in the upcoming sections. The arrangement that naturally springs to mind, and hence the most common model, is to set up a math circle targeting motivated students interested in math within a certain geographical area. In this case the common denominator among members of the audience is each individual's (or their parent's) knowledge of and desire to attend the circle, independent of school affiliation or other background. Although this is likely to be the appropriate choice in most cases, there are other possible models for math circles that target teachers or underserved populations. While they are not addressed separately here, information pertaining to these models appears throughout the text. In particular, the San Francisco Math Circle snapshot and the sample grant proposal in the appendices contain helpful ideas regarding math circles for these intended audiences.
For most new coordinators, the first truly substantial decision concerns the level at which to pitch the math circle. Given the organization of grades in the United States, the natural options to consider are the middle school level, high school level, one size fits all, or two or more levels offered simultaneously. (Although The Math Circle in Boston reaches students as young as four years old.) The final option is more ambitious than is advisable for a new circle, but it is a healthy development among circles which outgrow their original charters. In terms of content, material intended for middle school students may assume basic algebra such as factoring and the quadratic equation, techniques for solving simple word problems, standard area and volume formulas from geometry, sequences or patterns, and similar topics. High school students will have also been exposed to Euclidean geometry, polynomials, exponential and logarithmic functions, trigonometry, elementary counting techniques, and other precalculus topics. Therefore middle school students would certainly presentations introducing graph theory, basic probability, iterated functions, elementary number theory, and so on. However, more than likely they would become quickly lost if this material were presupposed. Topics that are better suited for a high school group might include combinatorial proofs of Fibonacci identities, Gaussian integers and sums of two squares, inversion in Euclidean geometry, complementary sequences, and a host of other exciting mathematics. Further thoughts on selecting topics are assembled in the chapter on leading math circles.
The range of mathematics available to high school students, as opposed to middle school students, turns out to be disproportionately broader than a few extra grade levels of mastered curriculum might suggest. There is no denying that older kids have a greater base of knowledge, which permits a wider variety of potential topics. But there is another more significant sociological factor at work. If a high school student attends a math circle it is usually because that student has answered the question, "Am I interested in and good at math?" in the affirmative, has independently studied advanced mathematics, and is likely to be substantially beyond the average high school student in terms of mathematical experience. On the other hand, late elementary and middle school students (and their parents) are usually in the midst of exploring this question, and attendance at the local math circle can be part of the process of finding the answer. These students have not had nearly as many years in which they knew they wanted to focus on mathematical thinking. The point is, a high school group is usually further above grade level in mathematical experience, on average, than a middle school group. This fact should not preclude the presentation of challenging material that will stretch middle school student's minds; presumably they are attending the circle because they are interested in discovering new mathematical horizons, even if they do not completely understand the material on the first pass. But it is a good idea to keep this distinction in mind.
The rich variety of interesting topics available at the high school level would seem to make this age group far more desirable, especially to a college or university math department. Therefore it may come as a surprise to learn that there are very few math circles which primarily serve high school students. (The Mobile Math Circle is one of the exceptions.) There is a counterintuitive force at work that has the effect of lowering the average age of a math circle over the years. This phenomenon will be discussed in more detail when dealing with the issue of retaining students, but it is important to note at this stage that a math circle which intends to target older students should be prepared to actively resist this tendency. For the same reason, organizers who decide to make their circle accessible to all age levels should commence such an undertaking with realistic expectations; practically every circle that begins in this fashion metamorphoses into a middle school math circle soon after.
Next: Location, Location, Location