Quizatzhaderac wrote:This is probably my biggest gap in my calculus instruction: when to actually use these techniques. In class you lean heavily on "we were just taught this technique, so we should probably use it".That seems a perilous thing to say for somebody actively teaching integration.pogrmman wrote:As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.
One student failed? "You can teach a monkey to differentiate, the editorial 'you"; obviously, you (specifically) can't teach that well."
Well, he could actually pull it off because he was (is) a fantastic teacher that did an awesome job addressing that particular gap that you mention — how to recognize when to do different things to integrate. He specifically challenged all of us to think about the “why” as much as the “how”. Obviously, being a high school class, we couldn’t learn the all the reasons behind quite a few of the things we did, but when it was possible, we did.
His main goal was to give us a thorough understanding of the material: often by challenging us a lot. I remember going into the AP worried about how hard it was gonna be, but being stunned by how easy it was. It turns out, whenever we’d been practicing, he’d specifically chosen some of hardest questions from both the AB and BC exams going back like 30 years, even though it was an AB class. When I took multivariable calc in college, I probably had a better understanding of the base material than most other people. The time we’d spent in high school on things like “OK, what actually is an integral? Why do they work the way they do?” and other things along those lines really helped me in college.