I finally had a first math circle meeting today. I invited any and all Math League students to come. Around 25 showed up at the beginning...8 or 10 had to leave in the middle and 2 more stopped in toward the end. I didn't know most of them but I think they were mostly juniors and seniors (in precalc and calculus). 6 or so students completely dominated the conversation.
I started with "what's the biggest number?" which quickly morphed into "are there different sizes of infinity?" and stalled at "is infinity a number?". They never agreed on whether or not infinity was a number, and by the end decided that they didn't even know what a number was. That was kind of neat. They also kept looking at me imploringly and saying "IS THERE AN ANSWER?!" and you all would have been proud of my aikido-like deflections.
For my part, I tried to steer them to focusing on the countable sets first, and broached the idea of one to one correspondence, but no one picked up that particular ball and ran with it.
So I guess I need help with -
Managing such a big group? More than 20 seems too big to guarantee everyone gets involved. I have 2 other teachers who are interested in "helping" but one of them could only come for about half the time and the other just stopped in said hi and left.
Encouraging dominant students to be more inclusive and collaborative?
Suggestions for steering them toward Cantorian set theory and the proofs I have in mind? I want to get them at least through the diagonal proof of the nondenumerability of the continuum, so that they can see that there is more than one distinct cardinality. At the end I postulated that there were as many positive even integers as there were positive integers but I didn't want to do too much. How should I start next week? I tried to pick a topic where I already knew "how it went", maybe that was a mistake, maybe I should have picked something unfamiliar.