I think it's important to distinguish "learning the material" - learning new concepts and how to do routine things with them - from real mathematical ability, the ability to solve hard mathematical problems in which it isn't obvious what the steps to be taken are. Trouble is, such a vast majority of school time is spent on the former that it's quite hard to focus a comparison on the latter which is really more important and more interesting.
We don't have IQ test data so can't help with the MG/HG/PG stuff, but there is certainly a kind of child for whom there's just no friction with learning the material in this early maths. It isn't that it takes less time to learn - it takes no time at all to learn. All you have to do is explain what the words mean, and then they can do it. DS7 is still like this whether learning concepts intended for 16yos or for 20yo undergraduates.
So, when people only look at material knowledge, it looks as though DS is 9 or 13 or whatever years ahead of his age. Generally, I think how far ahead a mathematical child is with learning the material depends a lot more on exposure than talent. You can't demonstrate mastery of things you've never seen.
However, when you look at the stuff that matters, DS isn't "really" that far ahead - he can't solve the same mathematical puzzles as a good 20yo mathematics undergraduate (choosing words carefully here!) So how far is he ahead in what he can do with puzzle-type mathematical problems? Of course for this you have to compare him with older children who are mathematically interested and being stretched, since most childen, sadly, never get any significant exposure to this kind of challenge. He now finds things (e.g. national competition questions) aimed at mathematically-strong 11yos a bit too easy but I don't think we'd have to go that much further to find things that would be too hard (although this reminds me it is time to look for that boundary again, since it has moved). Say he's 5-7 years ahead for problem-solving, when compared with mathematically-inclined children.
*But* even this over-simplifies, because it's not that ability in problem-solving is independent of exposure, either. I don't know how to estimate or talk about phenomena like children who are intrinsically very mathematical but simply don't get encouragement to tackle hard problems. How fast do they catch up when they do get the exposure, how far is "intrinsic" mathematical ability really intrinsic and how far is it altered by exposure to hard problems? My guess is that it's unstable enough that if you took a bunch of children with little exposure to problem solving and tested them for 2 hours on problem solving, they'd probably be a lot better at problem solving at the end of the 2 hours than they were at the beginning! But then again, how far is my guess affected by having an unusual DS? I'm wittering, I'll stop.
Last edited by ColinsMum; 11/24/10 01:30 AM. Reason: clarity, expansion
