| I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "form" of a problem and applying an algorithm appropriate for that form, drawn from the most recent chapter. In the TFA, the natural log in the integrand was a dead give-away because it only comes from one place in the standard order of topics in calculus class. Is this what we call intuition? The students called this the "trick." Many of them had come from high school math under the impression that math was subjective, and was a matter of guessing the teacher's preferred trick from among the many possible. For instance, all of the class problems involving maxima and minima involved a quadratic equation, since it was the only form with an extremum that the students had learned. Every min/max problem culminated with completing the square. I taught my students a formula that they could just memorize. The whole affair left me with a bad taste in my mouth. |
The thing I hated about integration was which approach would work and the best option for each approach were much more "do a lot and see what's right" and I was too lazy :).