It's interesting to watch my eight year old's approach to accumulating math. He's getting above level math problems, particulalry complex word problems, at school which is helping with his attention to detail and general problem solving acumen.

At home, purely self-directed, he watches a wide range of math videos from algebra to statistics to calculus. He has a couple of random algebra books and a geometry book he's reading, and he likes to do timed multiplication and such in various (various here to include excessive) iPad apps. It's interesting that he gets derivitives and integration, but hasn't picked up quadratic equations.

I think his approach points out that there is a conceptual layer different than processes and such. There should be a place in school for gifted kids to get a survey/tour of the range of mathematics.