Orpheus, on May 8 2003, 10:58 AM, said:
I agree that sometimes one simply ends up working by rote, expecially stuff like iintegration by parts ( the bit about § U dV = UV- § V du), but back when I was learning, I really did have a great deal of luck with the "area under a curve" and "filling a bathtub" analogies. I knwo you have the concepptual abilities - did you ever sit down and visualize the analogy for each of the basic transformations?
Well, I don't quite remember what the analogies are -- it's been so long since I had to think about it. The only time in the past decade that I've tried to do anything with integrals was a few months ago when I was trying to figure out an equation for the average time dilation of an accelerating ship. I got what I thought was the appropriate formula from my old calc text, but eventually figured out I'd done something wrong when I was getting inconsistent or illogical results. But I couldn't figure out what I'd done wrong, and I had to ask for help online.
I dunno, I did okay in 11th-grade calculus class, but when I took it in college I was lost. Maybe part of it was the teachers; my high-school calc teacher was a no-nonsense, straightforward guy (or so I thought -- my opinion of him cooled when female friends pointed out to me how sexist he was), while my college calc teacher was a classic milquetoast, a really sweet avuncular guy but not exactly attention-grabbing as a lecturer. But I think it was more to do with the fact that when I took college calc I was suffering from a debilitating case of unrequited love and kind of lost the ability to concentrate on classes.
Actually most students at my high school didn't get calc until 12th grade, but I was in the accelerated program. Which turned out badly for me -- normally seniors took a college-level course in Prob & Stat, but the professor was on sabbatical or something that year, and they brought in a substitute who knew less about the subject than we students did. It was a total failure as a Prob & Stat class, and was soon retooled into a generic and rather pointless "Topics in Math" class. Worse, the only way this special course could be fit into the schedules of all the accelerated-math students was to hold it in "zeroth bell," before first period -- after which I had two or three study hall periods in a row. It became the only high school class that I ever persistently skipped.
Quote
I should add that for the AP physics exam, you only need to understand the first chapter or two of each group well (you'll see the groupings of 3-5 chapters readily, even when they're not explicitly stated) and a decent gloss of the remaining chapters in each group will more than suffice. The test is really very easy: it's mostly checking on breadth and foundation, not depth and intricacy.
Of course ideally the goal of education is to actually
learn stuff, not just to pass tests. The tests should be a means to the end of gaining knowledge, not vice-versa.