Thanks for the suggestions! I will poke around at the things you mentioned.
We've done Julia Robinson problems, and he really likes them. I'm sure he'll continue to get joy out of those things. I should do some work to find problems that are matched with whatever he's learning (when I figure out what to teach him next).
He did a couple of math circles, but he talked too much and had to be shushed, and it didn't really satisfy him.
It would be nice if he could be in one with older kids, I think. (Or more-similar kids, but there's no way we could afford anything like epsilon camp! Although that looks super cool.)
He is able to solve some of the more challenging AoPS problems, but he's not especially good at them or anything. It would probably be worthwhile to at least convince him to learn how to approach these kinds of problems though. I don't really know why he doesn't especially like them and finds them difficult when he doesn't find certain other things difficult (he will happily explain Monty Hall to you). Maybe because they look too much like story problems? Probably if I just called them "puzzles", he would be more into it.
We do have a couple of the AoPS textbooks (prealgebra, and algebra we just started), but we've just done problems from them (for review and looking for gaps) because he already knew the lessons. He prefers to learn from books like Life of Fred (when he was little) and Cartoon Guide to Whatever. And "popular math" books (like Things to Make and Do in the 4th Dimension). Basically he is motivated by things that teach while showing the beauty and awesomeness of math, things that are entertaining (pictures, jokes), or by feelings of achievement and progress (like khan academy--he didn't do much of that because he uses up his "educational" screen time allotment on writing music and programming). He also read things like Complete Idiot's Guides. Maybe I should give an AoPS book about something he doesn't know yet a shot. (They're so expensive!)
Anyway, thanks, lots to investigate!
One other thing I would like to teach him is how to do formal proofs of math things? Not geometry proofs, but more general like number theory proofs, I guess? I'm not sure exactly what I'm looking for, but this kind of thing must exist (you can tell I don't have much math education myself!). He has good instincts about proving things (he had an interesting idea for goldbach, for instance, which of course didn't begin to work, and he has proved various cool things by drawing diagrams) but I think he would enjoy something formal and I haven't found anything that lays it out for you.
So those with older kids, what did your kids do when they were high school aged? Did your kid find a mentor at some point?
