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I think that the math circle in Boston area is different, but the ones that I am familiar with are geared towards high school and advanced middle school kids. Post intermediate Algebra and Geometry.

Yes, the one in Boston takes children as young as 4. From what I hear it is devoted to teaching the kids to explore a problem together in the spirit of playing with ideas. I suspect it takes a very good teacher/moderator to make this work well, but when successful I'm pretty sure it is my ideal for the best kind of learning in any subject.

BB