In a Math Circle, a leader presents a vague but interesting math problem. The participants ask clarifying questions, pose conjectures, invent methods to test conjectures, justify or reject conjectures, again and again, to the eventual ends of generalizing, abstraction,understanding truth, solving problems, and proof. The problems are low-floor, high-ceiling so that students can enter at and progress to varying levels of concepts rooted in advanced mathematics. The leader acts more like a secretary, in a supporting role, as the participants work collaboratively to solve the problem.