[javajosh]

>these 2 skillsets have almost nothing in common

Are you sure about that? I know at least some number theory can help with quick mental calculations. For example, I remember that if the digits of a number add to 9 the number itself is divisible by 9. I'm sure there are lots of other relationships like that.

[timkam]

Of course there is some correlation between the two skillsets (with respect to ability): people who can abstract well can also understand the underlying ideas that are useful for fast calculation, and for both one needs to have a reasonably good memory (to rely on in an exam and when developing a mathematical intuition, respectively). But to be the best in high school 'math' (calculation) requires a degree of extrinsically-driven, dull discipline. One can probably be best in class without taking this degree to the extreme, but among the best in class, many excel at exactly this (and not at abstract thinking), which I find a bit troubling.

[jacobr1]

The bigger tragedy is how widespread the belief that calculation _is_ math. Even for the university educated, the capstone mathematical course of often some version of calculus ... it is even in the name!

Personally, my first introduction into the more abstract mathematical concepts was a trigonometry class in high-school and later was fortunate enough to take more theoretical classes at university. It would be wonderful to introduce some of the more accessible, elegant concepts to kids at an earlier age.

[mmmpop]

Limits are taught in a way that gives the learner a taste of higher math, but when I took university calculus at 18 I wasn't really enthused about the "beauty". 15 years later I finally get it and am working my way into some coursework that will require upward of real analysis, but I wonder what changed in my brain to care all of a sudden? How can I have been inspired earlier in my life/career?

[javajosh]

This has happened to me several times over the years, where some topic only half-understood in college suddenly becomes clear. It feels as if some part of my mind was slowly chewing on it all this time, and finally finished its work.