Originally posted by: DrPizza
Ahhh, after reading what you wrote, Literati...
There are two schools of thought: one is that some people are naturally gifted in their ability to do math, the other is that everyone can learn to the same level of math, given good enough teachers. I'm somewhere in the middle, but leaning toward "naturally gifted." There's something called "mathematical insight" that some people seem to possess and others don't possess. This is the ability to look at a novel problem (one you haven't seen before) and figure out the best way to approach it, starting from step one.
Only about half of the students (if that) who make it into my pre-calculus or calculus classes seem to possess this "gift". The majority of those students can arrive at a solution, out of sheer stick-to-it-iveness. An example problem that I gave as a challenge problem this week: There is a 30x35x10 inch box on the floor, with its largest face against the floor. A bug wants to get from one corner on the floor to the opposite corner on the floor. He may crawl in any direction on any surface or edge, but cannot crawl through or under the box. What is the shortest route? Immediately, virtually every student I have was able to quickly calculate a route for going along the edges at the bottom (30 + 35 = 65), then crawling up the edge 10 inches, diagonally (pythagorean theorem), and down the opposite side (10 more inches). This route was slightly longer. Then, some realized the bug could crawl diagonally across the side, diagonally across the top, and diagonally across the other side; yet they couldn't figure out where the bug would have to reach the top edge when crawling upward diagonally (one student guessed it was the center of the edge). The insight that would have made this problem very easy for them was to unfold the box (making a big cardboard plus)
It's not a matter of having seen someone else solve a similar problem but with different numbers before; it's just an ability to "see" things.
The reason I said that I'm somewhere in the middle of those two groups of thought is that I believe that most people can develop this ability; I'm just uncertain if everyone has the same potential for its development. Take the time to attempt math puzzles/ math problems of the week/ math problems of the month, etc. There are numerous sites all over the internet dedicated to such problems. Many of the people who are good in math enjoy these puzzles as fun challenges and see them as a hobby or way to relax while exercising their gray matter. I think that in addition to providing a recreational activity for these people that the puzzles also help them develop more mathematical insight.
Through working out a great deal of problems, these are the people who see "cos 2x = 2cos^2x - 1 and can "see" cos .5x = +/- sqrt((1+cosx)/2). Thus, they may "memorize" the double angle formula for cos 2x, but having that in their mind, they can write down the half angle formula. Then again, they don't "memorize" the double angle formula. They "know" it, because many of these people have derived it repeatedly from the cos(x+y) formula which they "memorized"...
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