Decades ago, when I was in school and also had children in school, I slogged through two courses that were new to me: calculus and accounting. Lots of people conflate mathematics and accounting but their newness to me was about all they had in common.
About all, but not quite all. They both taught me that memorizing techniques wasn’t enough: I had to actually understand what I was doing. Calculus was an especially effective teacher in this regard. After acing two mid-terms–one devoted to differentiation, one to integration–in which I happily applied the technique-du-test to implicitly labelled problems, I failed the final when I had to independently select a technique to solve deliberately unlabelled problems. I was, at least, consistent: I chose the wrong technique every time, or almost. Oops. A spectacular oops. (As a side note, why the professor passed me, I’ll never know. It looks like it was just a numbers game: I had enough points from the first two tests to pass and so I did, even though clearly I had no idea what I was doing.)
