[javajosh]

I don't recall where I read this, but a young Feynman motivated himself to learn trig by imagining that he had been challenged by a mysterious stranger to answer riddles. For example (I'm making this up) "You only have a protractor and the ability to measure your paces - tell me the height of that flagpole!" The operation, then, is to pace out a distance from the base of flagpole, sight down the protractor to the top of the flagpole to get an angle, and compute. (Since tan(y/x) = a, arctan(a) = y/x, y=x * arctan(a)). So the motivation was imaginary and concrete. And it's dramatic, because there's an obstacle, a chance of failure, and a chance for glory.

I can't help but see a parallel with magicians, who can dazzle us because they are willing to go further than most of us, in terms of practice. In the same way, math gives you the ability to dazzle with surprising answers, to do a lot with a little.