The methods of
calculus are a lot like linear regression: they’re purely mechanical, your
calculator can carry them out, and it is very dangerous to use them
inattentively. On a calculus exam you might be asked to compute the weight of
water left in a jug after you punch some kind of hole and let some kind of flow
take place for some amount of time, blah blah blah. It’s easy to make
arithmetic mistakes when doing a problem like this under time pressure. And
sometimes that leads to a student arriving at a ridiculous result, like a jug
of water whose weight is −4 grams.
If a student
arrives at −4 grams and writes, in a desperate, hurried hand, “I screwed up
somewhere, but I can’t find my mistake,” I give them half credit.
If they just
write “−4g” at the bottom of the page and circle it, they get zero—even if the
entire derivation was correct apart from a single misplaced digit somewhere
halfway down the page.
Working an
integral or performing a linear regression is something a computer can do quite
effectively. Understanding whether the result makes sense—or
deciding whether the method is the right one to use in the first place—requires
a guiding human hand. When we teach mathematics we are supposed to be
explaining how to be that guide. A math
course that fails to do so is essentially training the student to be a very
slow, buggy version of Microsoft Excel.
