What is a mathematical circle?
On May 17, 2006, math department chairs from colleges and universities across the greater New York City area attended an organizational meeting held at the Courant Institute of Mathematical Sciences to take part in a mock math circle meeting and consider the possibility of starting one at their own institution. The event was organized by Mark Saul, a former administrator at the National Science Foundation and a long-time advocate of math circles. In a letter sent to departments announcing the meeting, Mark described math circles in this way:
: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.
: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 math 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.
The discussion, advice, and vignettes contained in these pages will convey a fairly complete picture of math circles. By the end most readers should be convinced that a math circle is a good idea for their institution, students, or child, depending upon their perspective. But that is not the true purpose of this document. More than just being informative, it is meant to function as a resource for individuals who are willing to take the leap of faith and actually initiate and maintain a math circle within their own community. These directors and coordinators, some of whom are featured in the Circle Snapshots sprinkled throughout the text, form another common thread among all math circles. Each person has a unique organizational style, but they all share the belief that presenting beautiful mathematics to motivated students in an engaging manner is a worthwhile, important enterprise.
There are as many compelling stories behind the launching of math circles as there are organizers. Perhaps a faculty member at a college or university owes their interest in mathematics to a circle they attended as a secondary student, and now wishes to provide a similar experience for students in their area. Since the math circle phenomenon is a relative newcomer to the United States, these professors tend to have arrived from a country with a rich tradition in math circles, such as Russia or Bulgaria. This was the case with Zvezdelina Stankova and the Berkeley Math Circle, for instance. Or perhaps a parent summons the courage to approach a local math department to forward the idea of partnering to establish a math circle, as occurred when Jennifer Jeffrey contacted Steven Krantz at the University of Washington, St. Louis. Bob and Ellen Kaplan, who both taught in the Boston area, became so tired of the negative attitude towards math ingrained in their students that they invited a group to meet in their living room to discover the true beauty of the subject. This informal gathering eventually grew to become The Math Circle.
A vibrant math circle can be a source of great inspiration to students and a rewarding enterprise for the mathematicians coordinating and leading them. However, just as with any sort of community, there is more to establishing a thriving math circle than meets the eye. Introducing students to exciting, accessible mathematics in a creative, engaging manner is certainly a key element. For this reason the entire second part of this handbook is devoted to laying out a wide variety of sample presentations, with a range of topics and difficulty levels, along with detailed presentation notes for the instructor and carefully developed problems and solutions. However, while solid mathematical content and delivery is certainly necessary for a successful math circle experience, it is not sufficient. Careful thought must also be given to the issues of how to attract and retain students, when and where to meet, how to structure the meeting time, whether to set up a web page for the circle, and more. Following a brief historical interlude, the upcoming sections will explore the various options available and consider how a number of successful math circles have addressed these issues.