Dave Rusin <firstname.lastname@example.org> wrote: > > Sure. (I've never "memorized" a piece at the piano either. As you > suggest, "memorizing" comes from constant practice.) But some students > believe that "understanding" is equivalent to "exposure", or something.
Something of which I have to remind myself constantly is that "understanding" isn't always the same as "learning." Sometimes I read through a proof and understand each step, and how each follows from the previous step. But the real test is whether I can come back to it later, after I've forgotten the details, and reproduce it myself. If I can't, I haven't really *learned* it. I need not only to be able to follow the reasoning, but to grasp the overall direction of the proof and understand why *that* particular approach was taken. If I can't understand both the details and the "big picture" then I'm just wasting my time. All too often I read through a chapter and "understand" everything in it, but have only a hazy idea of the concepts a few days later. That's why I often put a book aside for a while and then come back and re-read the last couple of chapters I've read, to be certain I'm really ready to go forward.
-- Wayne Brown (HPCC #1104) | "When your tail's in a crack, you improvise email@example.com | if you're good enough. Otherwise you give | your pelt to the trapper." "e^(i*pi) = -1" -- Euler | -- John Myers Myers, "Silverlock"