Last night I had my first tooling session. In complete honesty, it was entirely my fault, I'd taken the fact that I was pretty good with numbers for granted, and assumed having taken Calc BC (and should've got credit for it) that it would be easy. Indeed, I knew exactly what lecture was about, and glancing over the pset I saw that there weren't too many problems, no big deal right? a bit of review right?
Oh so very wrong.
After working all night on that pset (literally, I got home at 7.a.m. today 0_0) I've realized what exactly makes this place so hard. The problems don't test your knowledge of the subject, they test your understanding. Now if I'd heard that last year I would've said "same thing", but now I understand the difference. Knowing how to do a problem means you can quickly identify similar problems and apply the same generic formula to solve them, the difference is that none of the problems on the homework are like examples in class, but they all use ideas presented in class.
Take the derivative for example, in class you might learn chain rule, product rule, power rule etc etc and find the derivatives for a bunch of functions. Easy huh? of course it is, it's just like what you know. But here, you might have a problem which asks you questions about the behaviour of the derivative, like this one particularly nasty problem last night. Essentially you had to find the "(p+q)th" derivative of a product of two functions raised to the P and Q respectively. The breakthrough that solved it was to realize that the nth derivative of a function with exponent n is n!, and thus the n+q derivative must be 0, because at n it's just a constant. That one little idea that was never explained meant that several terms in a series ended up as zero, and only the last term mattered.
That's the difference, application, a deep understanding. I don't just know how to solve problems, I know how to figure out how to solve them. And though it's ridiculously hard, and sometimes impossibly daunting at times, the best feeling is finally figuring out the one piece you're missing that makes it all fall in place. To make one more analogy, instead of asking you to connect-the-dots to draw a horse, MIT gives you some paper and says "draw the horse", it's up to you to figure out how you want to do that.
In related news, no longer will I be waiting until thursday to look at the second half of the pset. =)