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

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).

Re: ‘what is the goal’—- avoiding wasting time, dying of boredom and killing interest in math with years of unnecessary repetition. Yes, proof-based math curricula are better, but even standard applied tracks are far more interesting than a million reps of long division.

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.

I don't think this has to be bad, but it's not the "gifted acceleration" and can be tough on these students if expectations are that they are "fast".

Their parents don’t care, on the contrary they want it to be tough on their kids. They aren’t interested in raising happy, well adjusted kids. What they want is rich kids (that marry the correct spouses.)

Pretty presumptive.

If school covers the advanced topics, there wouldn't need any cram center. Case in point, my teacher gave me such challenging problems when I grew up, that tutoring schools were never necessary.

Also, not every tutoring school is about cramming. Instead, they are simply accelerating. The Art of Problem Solving is one of such schools. Another example would be those math camps, such as Math Path. Maths are fun, powerful, beautiful, and leads kids towards a promising future. American schools simply don't care enough about it.

There will always be the nature vs nurture problem. And I agree with you: do not focus on one area without trying others first, since the very nature of a gifted child is, that there is positive manifold. Even excelling in one particular area does not mean you have a special interest there. Offer a lot of input and see what sticks.

Gifted children are a special topic. I highly advise to consult trained professionals (psychologists, usually necessary for IQ testing) as well as consulting with an organization like Mensa.

Potential problems from childhood do not magically disappear during adulthood. People usually do not react happily in competitive situations: "Wow, someone way more intelligent than I am! How cool!" There is a lot of envy and vice versa, there is also a so called "feeling alien" problem.

Many gifted people I know have different interests and also shunned Math later in life, because of missing learning strategies. Not everything comes easy. And then it is important to have the right support.

You sound like you aren't a parent, or you are a parent of very exceptional children who have enough willpower and ability to organize their schedule to actually do something like learning math.

My experience, especially with kids until high-school and even into the high-school is that unless there's an external framework which tells them how and when and from what book and so on they won't do a thing.

Of course it would've been easier if left to their own devices children could just somehow organize their curriculum, but that isn't going to happen. So, the adults need to create the curriculum. Not just that, mathematical problems are very hard for independent study. They absolutely require guidance. It took many generations to prove theorems we find in introductory math books. So, the question is less about the students and more about the teachers -- how do they structure and plan their guidance to be more effective. And for that matter it'd be fair to make a simplifying assumption that children are more or less the same, since the guidance and the curriculum are orders of magnitude more important.

Now, as to how actually do it -- I don't know. I tried my hand a little, and it was a miserable failure. But, to put this into perspective, Wittgenstein, the author of "The Tractatus", one of the founding figures of modern mathematics, tried to be a school teacher for a while, and also failed to the point of being hated by his school students.

When I reflect on the school curriculum from a college perspective, I cannot help but to shrug about how most subjects and problems seem like a worthless dead-end stuff... and yet when I tried to work with students through more important mathematical stuff -- it just didn't work. Zero retention and very little understanding to begin with. Things I thought elegant or simple don't generate anything similar to my excitement when introduced to the students. No number of "creative" ways to explain this stuff ever made it more appealing. If anything, it sometimes causes resentment because, compared to their peers, the students may think they aren't getting the "good stuff" because they cannot do some worthless thing their peers were taught the automation of.