The DC Math Circle

The DC Math Circle is an enrichment activity for highly motivated ninth-grade students who attend a public or public charter school in the District of Columbia. The Circle meets on Wednesdays from 3:30 to 5:00 in the School Without Walls, 2138 G Street NW, a couple of blocks from the Foggy Bottom metro station.
What are the defining characteristics of a math circle? How does a math circle differ from other sorts of ordinary activities such as math clubs or MathCounts or accelerated class work? There may not be any single universal answer to these questions, but here is the vision of the DC Math Circle: Engaging problems motivate the work. The students sit around a large table (in a circle!) and work together. It's fun. Solutions, ideas, guesses, conjectures, and counterexamples are shouted out. Debates ensue. Students feel comfortable to make mistakes or to exhibit their cleverness. The facilitator is constantly challenging the students with questions but rarely providing answers. Students are expected in the end to explain their reasoning and to try to persuade others. Every problem solved begets a further problem still to be solved.
Here is an example of a topic: How many squares are on a chessboard? Is it 64? What if we count k-by-k squares also for every k? What if the chessboard itself is n-by-n instead of the usual 8-by-8? How many rectangles are there? Can one get a “closed-form” expression for the answers? What if the chessboard itself is a rectangle? How does one sum the first n positive integers? The first n positive squares? Cubes? Can one generalize to 3 dimensions? What about four dimensions? What about counting triangles in a grid of triangles? The students will grow accustom to heuristic assistance from the facilitator: Can you do a smaller case? Have you seen something like this before? Can you make a guess or try an experiment? Can you convince your neighbor? Can you generalize the problem? What similar questions can you ask?