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.)
After that, I learned to study for calculus by doing problems “blind”–solving a problem without looking at which chapter it came from. By thinking (hard!) before doing. By figuring out what the problem was before deploying a technique.
An accounting mid-term then offered me a less-costly refresher of this lesson: an annuity problem that didn’t self-identify as an annuity problem. I forget the specifics, but it was the reverse presentation of what we were used to. Tackling it as an annuity was simple and quick. Tackling it any other way was complex and way too long for an exam.
About half of us recognized it as an annuity-based problem and got full marks; about half did not and got zero, wasting considerable time to boot. Oops.
Life isn’t an exam, but it does present lots of (deliberately?) unlabelled problems.
Some kids don’t do well at school. Is it a social-adjustment problem or a learning disability problem or a problems-at-home problem?
Some adults don’t do well with mind-altering substances. Is it a self-discipline problem, a mental-health problem, a physical-predisposition problem, or an ease-of-access problem?
Some societies don’t do well with crime. Is it a problem caused by this particular set of values and policies, or that set?
The world doesn’t do well at fostering peace. Is it because we fail to prepare for war, or because we prepare all too well, or because this is just who we are?
I don’t need any more stuff: My house is full. As Christmas approaches this year, I want just one thing: a better understanding of even one problem.
If calculus will help me understand the current issues in the world, sign me up.
Tom – Sadly, I don’t think calculus offers that sort of help. But if you need to know the area under a mathematical curve, I used to know how to do that.
I took a slightly different approach to calculus (which was, as you note, a LOOOOONNNNNGGGG time ago). I started from the examiner, rather than the exam. I asked myself, “What is he expecting me to do here?” Secondly, “If I do that, what’s the answer likely to be?” After that it was simple — manipulate the data to achieve the desired result.
Unfortunately, that system doesn’t seem to work as well in life.
Jim T – LOL – that’s smart. I wish I’d thought of that at the time.
Isabel – it has been a long time and I still can’t do calculus.
John – For some reason, this triggered a “possible spam” filter.
I’m glad to know that engineers also struggle with calculus.
Life is a set of trade-offs.
Nuanced thinking leads to paralysis.
Any question leads to another.
Eat a peach.
May the New Year bring you lots of curious plot twists. 😀
Barbara – Not a fan of peaches but I’ll take the plot twists. Thanks!
U.S. peaches give all peaches a bad name — cardboard. But bite into a perfect (skinned) Canadian peach in August… it’s enough to bring tears to your (well, my) eyes.
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