[pgustafs]

I think problem solving math is definitely fun and can be a huge source of confidence, but I don't see why "racing" through the standard curriculum is a negative. Why should a smart kid do a million multiplication/division problems for 5 years when they would have a ton more fun and get a lot more long-term utility from learning some stat/algebra/geometry? If a kid demonstrates mastery of a concept, it's a lot more bizarre and potentially damaging to force them to relearn the same material over and over.

[opportune]

I think you misunderstand OP’s point and maybe don’t know what problem solving math the OP is referring to. “Art of problem solving” and contest math is not rote repetition or basic arithmetic at all. It’s a way of challenging students with difficult math problems that are approachable without comprehensive study of high level math but require combining concepts, applying logic and deduction, solving world problems in a way that don’t fit a “type” that’s covered in class. Try searching for USAMO problem sets for example.

The reason racing through a math curriculum can be problematic is… what’s the goal? If it’s not “look as advanced as possible to a non-mathematician to get into a tier 2 college” and instead something like “expose kid to as much math as possible because they enjoy it/find it challenging” or “be a top mathematician for their age so they get into a tier1 college because mathematicians see promise in them” you don’t actually want to cram in subjects like typical community college or undergrad calculus, stats, and linear algebra at all. Those are not nearly as helpful for pursuing advanced mathematics as learning how to prove things, apply theorems to problems/reduce problems to theorems, or just generally becoming excellent at “lower” math like in contest math. In fact it might turn a kid off of math to get that far and still be doing mostly rote computational problems, and it won’t help that much in becoming a mathematician because those classes typically focus on the applied aspect (outside of particularly selective math courses at certain universities).

[pgustafs]

I have nothing against contest math (I was a USAMO qualifier in high school), but contest math isn’t enough if a kid has to sit through years of tedium during regular classroom hours. Also, there is a large difference between contest problems which focus on cleverness and real-world problems which focus on conceptual understanding. Many kids prefer one or the other, and I think it’s a mistake to assume that contest math works for all kids who might be mathematically inclined.

Re: goals —- the goal is to let the kid learn as fast as they want assuming they have solid foundations. If they like proofs let them do proofs, if they like applications let them do that. Just don’t force them to sit in a classroom doing busywork for the most formative years of their lives.

[whimsicalism]

I think it's more that there are massive large fields of math that are just left out in the race to calculus.

I had a friend who grew up in Japan and he said he learned a lot more number theory-style stuff.