I am learning again to enjoy teaching integral calculus. Second semester calculus, at least the portion of it relating to integration techniques, is a wonderful garden of delights, filled with hemlock, nightshade, and oleander. Yet while it is glorious to see the expressions on the students' faces when you first show them volume and surface of revolution, and the formulas therein, I'm thinking of simpler pleasures.
If I had a nickel for every time a student in my class differentiated instead of integrated, or integrated by parts or substitution incorrectly, I could fund a month-long tropical vacation. And it's sad, really, that it gives them so much trouble. After all, they spent the better part of a semester practicing every single permutation and contortion of derivatives. The integral is posed as an antiderivative, integration by substitution is just the chain rule in reverse, and integration by parts is nothing more than the product rule backwards. So with all the work they've done taking it one direction, you'd think it wouldn't be so hard to just reverse the process.
And yet, I suppose I really shouldn't be surprised. Don't think I mean this from the point of view that the reversal of many mathematical processes is highly nontrivial (which it is most certainly not, as evidenced by the wealth of research currently being done in various flavors of inverse problems). No, instead I'm talking about the fact that these children have been stymied by simple reversals as long as they've been doing the most basic arithmetic.
Ask almost anyone who teaches math what sort of mistakes crop up most frequently, and you can almost guarantee you will hear variations of things involving negatives, fractions, and division. Negatives and subtraction, arise as the inverse of addition. Fractions and division similarly arise as the inverse of multiplication. The panic rises whenever students have to do anything with logarithms, the inverse functions for exponentials. They can add, multiply, and exponentiate all day long. But ask them to back up and go the other way, and they look at you like you've told them to grow wings on a dog.
Over and over these students stumble whenever asked to do anything in reverse, no matter how little or big, no matter how much preparation and coaxing they are given. It's sad, as it points to their lack of even having a single dimension of motion in their thinking. They can only go forwards, never backwards, and heaven help you if you ask them to look left or right.
But that's okay, I will continue to enjoy pulling them forward, shoving them backwards, pushing them left and right, and delighting in their confusion as all the symbols and ideas swirl around them. Other teachers can spend all their time and energy trying to "reach" the students and have them understand. I know they are wasting their time, and I will instead focus my energies in better, and more entertaining, directions.
