Sometimes it hits me: I’m kind of a bad parent.
Not whipped-them-all-soundly-and-sent-them-to-bed bad, just … a bit lazy. Neglectful. Inattentive to the shifting emotional barometric pressure of youth. When the girls are quietly dealing with some issue, it’s almost always their mom who picks up on it and helps them past that pinch point, while I offer perfunctory encouragement from my club chair.
We were facing one of those moments now. Our youngest was struggling in her math class, Pre-Calc 11.
She’d been doing okay till she hit the make-or-break part of the course: trigonometry. Trig is where you start really laying the foundation for calculus. The teacher moved quickly, and half the class was soon adrift. We tried hiring a tutor, a lovely young man named Johannes, but it was like he was attaching post-it notes to a dolphin. Not much stuck.
Our girl just could not find her Why. When would she ever need this, she a creature of words not numbers? The crude memorization struck her as unsoulful. Plugging in formulas, crunch crunch, grinding away like a machine. Life’s too short, man.
I’d watched her valiantly try to study. She’d lay a fresh sheet of paper on the table, write the word “Trig” at the top in pencil, and start, almost imperceptibly, to shake. I made the mistake of suggesting that if she passed the test she could go on to actual calculus, which might be more fun.
“I would rather cut off an ear,” she said evenly.
The exam was Monday. And she couldn’t even give the course her full attention because she had other assignments due.
It was time to Daddy up. If I could brush up on the basics by tomorrow we could blitz through a review right before the test, and maybe that little extra nudge would help her squeak through.
For a change, I seemed to be the right parent for the job. I crushed trig! Surely I could just dip back into the curriculum and all that teenage expertise would come flooding back.
The shock, as I began to review the material, was hard and swift. None of it was really ringing a bell. If this stuff was still in my noggin, the path to it had grown over. I would basically have to start from scratch.
A scientific calculator has been left out on the table. It occurred to me I don’t actually know how these work. Pretty sure in my day we didn’t use ’em. Wait, what did we use?
Ah, right. A slide rule.
Trigonometry is all about relationships – specifically between the angles and sides of triangles. The foundation of all of it is the circle, from which the triangles are sectioned out. (Leonardo’s Vitruvian Man looks like he’s giving a trig lesson in the nude – tracing reference angles with his arms and legs.) It seemed worth making time to try to understand at an intuitive level how it all fit together, why the ratios work the way they do. But I was already burning moonlight.
The theoretical physicist Richard Feynman claimed he had a foolproof recipe for learning any subject quickly:
Step one: Write down everything you already know about the topic.
Step two: Hit the books (these days the Net). Write down everything new you’re learning. When you feel like you’ve more or less “got it”…
Step three: Try explaining it to a five-year-old. Use simple language and terms. When you get stuck (and you will) go back to the source material and fill in the gaps.
Step four: Craft a new, smooth, organized story (for adults), based on your new, deeper understanding of the topic.
Step one didn’t take long. I drew a few triangles and quickly petered out. So: bring on the new learning.
It became clear, as I soldiered through some online tutorials, that I wasn’t exactly sponging this stuff up. Learning really does get harder as you age. Not only do you lose cognitive horsepower (so-called “fluid intelligence,” which is basically just processing speed and memory consolidation), you discover you’ve developed bad habits.
Fortunately, there are memory tricks to help you recall the formulas. I know the Pythagorean Theorem only because it was linked to a terribly offensive joke. (But then, that’s how the best mnemonics work – the more outlandish/inappropriate the image, the more likely it is to get filed securely away.) For the trig functions, there are several hacks to summon up which ratios correspond to the Sine, Cosine and Tangent functions. The most popular one is to just burn in the ear worm SOHCAHTOA (Sew-ka-TOE-ah): sine of the angle = opposite side over hypotenuse; cosine of the angle = adjacent side over hypotenuse; tangent of the angle = opposite side over adjacent side.
If your brain works better with stories, you can try:
Some Old Hippie Caught Another Hippie Tripping On Acid
Or, more poignantly:
Some Old Horses Can’t Always Hear Their Owners Approaching
But not everyone believes in these kinds of shortcuts. The fear is they’ll stop you from thinking, from truly grokking the beautiful gestalt of trig. “In complex systems,” as the leadership consultant Zafar Achi put it, “there is no recipe, only art.”
By the wee hours some of the old jam was returning. I’d Feynmaned out a basic understanding, and an m.o.: Draw the right triangle, fill in what you know and apply the appropriate formula to divine what you don’t. And if it’s not a right triangle? Then you have to use hammer to break glass and access one of the emergency tools: the fussier sine law or cosine law. (Warning: brute memorization required.) Or you can “drop a perpendicular,” make two right triangles and SOHCAHTOA till the cows come home.
Presently it came to me, my middle-of-the-night insight: The trick to breaking the back of trig is … you kind of have to fall in love with it.
Our girl has zero motivation in this area because she cannot see any relevance for trig in her life. This is where we have failed her. A parent’s job is to help connect the dots between things kids already like and useful things they might like if they really gave them a chance. Trigonometry goes back at least as far as the ancient Greeks, a kid-friendly era of myths of monsters. Those Greeks used trig to play God, in a sense, to apprehend the dimensions of the earth and heavens. Numbers and words are not mutually exclusive. Trig is literary. There are references in Moby Dick to “cycloids” – curves so distractingly beautiful they could (Blaise Pascal claimed) relieve the pain of a toothache. Those curves are mapped by calculus, and the way to get there is through trig.
For most of recorded history, the esteemed geometrician Sarah Hart reminds us, trigonometry was part of the mental furniture of every educated human. “Great literature and great mathematics satisfy the same deep yearning in us: for beauty, for truth, for understanding.” In her recent book, Once Upon a Prime, she quotes the pioneering Russian mathematician Sofia Kovalevskaya: “It is impossible to be a mathematician without being a poet in soul.”
Could not our girl be coaxed to see, in trig, a beauty on par with her beloved literature?
Well, probably not in the next twelve hours.
Sunup. We did a little bit of a review. I roughed out a problem with a skipper who motored away from port for ten miles, changed course twenty degrees and continued on until he found himself, alone at sea, x miles from home. (It’s like that old joke: “Don’t ask me to find my X. She left. I don’t know Y.”)
Formulas fought for real estate in my daughter’s skull. Math may be beautiful, but it can also be cruel. The frustration is, you can basically know what you’re doing but in one brief lapse of concentration you pooch a single calculation and boom: no soup for you. Score of zero. Adult life isn’t usually that unforgiving; one small error rarely torpedoes your whole effort. Math mirrors the winner-take-all game of teenage life, where a single social gaffe can lose you a friend, or your place in your friend group. That’s a lot of pressure. Life is surprisingly high-stakes for young people, and we should cut them slack when we can. If for no other reminder than that, this particular Big Day was worth it.
After she’d shuffled out the door to school, I noticed, in a corner of the room, a separate project she’d been working on. It was a presentation on communication strategies in humans and in animals. Animals don’t have language, she had noted, but they definitely have sophisticated ways to transmit ideas.
Take the honeybee. Turns out these little dudes can describe the location of a flower patch up to 15 km from the hive. How? By performing a “waggle dance.” One second of butt waggling equals one kilometre of linear distance. They calculate this, deep in their little bee brains, by triangulating on the angle of the sun.
In other words, they use trig.