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As the person at the front of the room, it is important to keep in mind the fundamental goal of a math circle session: to engage the kids. Capture their interest with an unexpected result. Challenge them with a series of interesting problems. Brainstorm as a group, or have students work individually while circulating to monitor progress. List good ideas, engender debate, and allow participants to present and defend their solutions. All of this activity should l take place within a structured environment, of course; providing a framework for exploration is one of the responsibilities of the leader. Ideally a math circle gathering will resemble a kindergarten classroom more than a college lecture.

Speaking of lectures, the astute reader will have observed by now that the words 'talk' and 'lecture' have been avoided entirely in this handbook when referring to a leader's activities at a math circle meeting. This is not to say that lectures have no purpose; their format provides one means of efficiently transmitting information from an expert to an assembled group of recipients of the knowledge. However, as Mark Saul (a long-time advocate and promoter of math circles) put it, "The objective of a math circle should never be to cover material. One develops material because it is interesting. It's an adventure without a definitive goal, necessarily." Tom Davis, one of the coordinators of the San Jose Math Circle, has this advice to give at his web site1, "Try not to lecture. Even though introducing new theory and techniques is an integral part of math circles, your sessions should be as interactive as possible. Score yourself: 1 point per minute you talk; 5 points per minute a student talks; 10 points per minute you argue with a student; 50 points per minute the students argue among themselves."

Effective math circles can follow many different formats, and speakers can employ a wide variety of presentation styles. Here are a few common elements of the most successful sessions. To begin, participants must become intellectually invested in the proceedings before they are able to realize a return on the mathematics being presented. So rather than beginning with definitions or background material, a presenter would be better served by finding a creative means of leading students into the topic for the day. This might take the form of a game or similar activity, such as the one used in the upcoming chapter on binary. Or it might be an elementary yet enticing problem which quickly leads to deeper waters. A group exploration can do the trick: which numbers from 1 to 100 can be written as the sum of two squares, and what can be said about the resulting list? Regardless, an engaging pathway into the material makes for a great way to begin the day.

Once the pump has been primed a presenter has a good deal of latitude in unpacking the topic for the day. There may be definitions to develop, examples to explain, or notation to agree upon. There is a simple maxim to adhere to that will keep this process lively: make students do as much of the work as possible. This is such sound advice that it is worth emphasizing: make students do as much of the work as possible. Rather than define a graph (from graph theory), have a volunteer offer a definition; then play devil's advocate by drawing in multiple edges, loops, graphs with no edges, and so on to force the group to think carefully about the object they are describing. Similarly, instead of simply presenting summation notation, ask if anyone knows of a compact way to write the sum 4+9+16+• • •+121. Find out what they already know! Then challenge them to come up with alternate answers, so that they appreciate that

\[ \sum_{n=2}^{11} n^2, \qquad \sum_{t=0}^{9} (t+2)^2, \qquad\mathrm{and}\qquad \sum_{a=1}^{10} a^2+2a+1 \]

are all legitimate answers. Getting students involved in this manner has several benefits: the participants continue to be invested in the unfolding mathematics, while the presenter can assess the level of background of the audience at the same time. Perhaps most importantly, this policy helps to prevent a speaker from inadvertently slipping back into a lecture style and thereby reducing any momentum that had built.

At the heart of any great session lies a selection of great math problems. The way in which a leader chooses to organize and present them is dictated by many factors, including the topic and personal taste, among other things. Thus one person holding forth on induction might choose to illustrate the technique with the entire group on a delightful sample question, then distribute a sheet of problems covering a range of subjects and difficulty levels with which to occupy the remainder of the time. Students might vote on which problems sound interesting or which ones are likely to be difficult before they dive in. The leader (and assistants!) could then work with students individually to listen to ideas and offer hints (never solutions) as appropriate. Periodically the group could reconvene to brainstorm, present solutions, or take a break. Another person might opt to lead the group through a series of carefully chosen problems which lead up to a single main result, such as the theorem on which integers can be written as the sum of two squares. This person might decide to hand out the list of problems only at the conclusion of the exploration in order to keep the group together. A third format for occasional use involves incorporating the questions as part of an exciting team game. This option requires a fair amount of organization and planning ahead, but can be great fun and stimulate focused problem-solving.

A survey of students attending the Stanford Math Circle revealed that they were most satisfied with the meetings in which the presenter reached some sort of concrete conclusion by the end of the session. This finding suggests that simply assembling a sheet of nice problems is insufficient. Paul Zeitz of the San Francisco Math Circle asserts that, "Cool math is not enough. One needs cool math packaged properly." So rather than simply setting kids to work on an assortment of problems, it would be far more effective to select the problems around a common theme, introduce the theme in a creative manner, have the audience identify several problems that especially appeal to them, then work towards understanding those problems as a group. This sentiment also informs the choice of topic to some extent: the most suitable topics are those which can lead to some satisfying result within the allotted time.

There is one final aspect to presenting mathematics at a math circle meeting that is hard to over-emphasize. (But I will try.)

Teach students how to ask good mathematical questions.

Students almost never have the opportunity to ask, "What if . . . ?" types of questions at any point during their secondary school careers. They have no idea that one of the most crucial skills acquired by a professional mathematician is the ability to ask productive questions; the sorts of questions that lead to new areas of research. All of their training suggests that mathematics is synonymous with solving problems; very few of them stop to wonder where the problems come from. Moreover, working on problems can become tiresome or frustrating. But contemplating new directions to explore, free from the burden of needing to answer all the questions that might arise (at least for the time being), is a marvelous, creative endeavor. Every student should be given the chance to practice this process.

So take a few minutes after wrapping up a nice problem to point out that the book is not yet closed on this particular idea, and ask students where it might lead. At first students may need a lot of coaxing. What happens if one uses different numbers or shapes? Is there an analogous result in higher dimensions? What if we allowed to three people to play this game, how would that look? Encourage any ideas or attempts; quite often once the first question or two is tentatively offered the floodgates are opened. Help students refine vague ideas into well-formulated questions. This activity can be as rewarding for the leader as for the students - it is exciting to see what they come up with, and invariably everyone leaves with new ideas to pursue.

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