Or: How I Learned To Stop Worrying and Try Again at Math
Let me tell you the story of two classrooms. One was about ten years ago in a nontraditional high school with startling purple walls. The class was Algebra II, although this particular curriculum integrated algebra and geometry. I was a sophomore in high school, and a little advanced beyond my peers in the material. It had been some time since math had been intuitive to me while English, history, and civics continued to be quite second-nature. I didn't exactly struggle in math. I could follow the procedures laid out for me, but I had less and less idea of what I was actually doing. And I can still remember the concept that shut the door and turned me into a math-hater.
Sine, cosine, and tangent.
I remember my math teacher Ms. Gordon* going over the concept. Neither I nor the other students really got it. Her explanation of the process was straightforward enough (SOHCAHTOA and all that). We kept asking questions of what sine, cosine, and tangent even were. What are they used for? What's this number even mean? Why do we need this? Ms. Gordon got impatient. "Don't make it more complicated than it needs to be. Divide opposite over hypotenuse and punch the 'sine' button," she said. Maybe she thought we were being difficult.
Math is just a thing you have to do. It's useful for everything. That was what I'd heard for years. I'd heard it that time in second grade that we visited the firehouse on a field trip; here it was again in this classroom in 2004. Math was a thing you just had to do. And here, as I really and truly tried to figure out what on earth these bizarre trigonometric terms were telling me, I heard it again. Just punch the numbers in. It's just a thing you have to do.
"What does this even mean? It's just nonsense.," I pressed. I got some laughs from some equally-confused classmates. This emboldened me to push in more. I loaded a silver bullet I'd been sitting on for a few months. "For that matter, why are there 360 degrees in a circle? Why not 400? Why not 12 like a clock? And why does every triangle have to be 180?"
"Because it's half a rectangle."
"But why? Why can't a triangle exist by itself? Isn't that like saying that no triangle is happy until it's married to another triangle?" The chuckles continued as the teacher got a little more flustered. Now I was a burgeoning comedian on a roll. "Triangles can find fulfillment by themselves! They don't need no man to make 'em happy!"
It was immature. It was simplistic. It was a distraction from finishing the lesson. Most teenagers probably aren't asking for the Euclidean proofs underlying the calculations. And yet behind the thin layers of teenage insecurity masked as comedy lay a very real struggle: What is this thing? What does it mean? And before I even started playing amateur hour at the Laugh Factory I had already checked out when I heard that deadly phrase: "Just mash the button." Sine, cosine, tangent; even the degrees in a figure were just steps to memorize to get an answer. Just like formulas in geometry. Just schlock you have to vomit out on the SAT. Just another thing you have to do so you can get into the college you want.
Years of pragmatism and lowest-common-denominator thinking wrought a dichotomy in my thinking. Literature, history, and civics were classes where you got to do the fun stuff, wrestling through a dilemma or uncovering the mysteries of what it means to be human. Math was a chore by contrast. The humanities became like taking flight for the first time, each rush of discovery like the rush of a loop or a dive. Math became the canvas bag I relieved my airsickness into when I came down. Mind you, this was the same year I was reading The Count of Monte Cristo in World Literature and Korematsu v. United States in Civics. In literature I kept wondering, How can Dantes see himself as a hero while he makes these people miserable? And even worse, why am I rooting for him? As I read Korematsu, I shuddered when I realized that the presidential power of imperium has only revolution to check it. As I learned the formula for the surface area of a sphere, I wondered when I'd ever have to tile the outside of a sphere or why anyone made such a reckless purchase to begin with.
So I did what everyone told me. I learned enough math to get into a liberal arts college, majored in a humanity, and graduated with Elementary Statistics and Personal Finance as my only math courses.
In another classroom ten years later, I find myself a somewhat reluctant science teacher. I was always pretty decent at chemistry, but I never touched physics since I'd heard it was math-heavy. I took the class primarily so I could work with the same students I did last year. It's a physical science class, which ends up being an introduction to chemistry and physics. The material is more challenging than anything I faced in the eighth grade. I also find that middle school curiosity goes far beyond what one can prepare (even if you know the material very, very well).
Yet I had an enlightening experience last month. We were going over chemical reactions for the third time, specifically combustion reactions. A hydrocarbon plus O2 gas yields CO2 and H2O (and heat). A student asked an off-handed question about light. "What elements is light made out of? How do we account for it in a chemical equation?" I was sorta surprised because this wasn't a question I expected. I answered, "Light isn't a compound. It's a form of energy."
"Huh?" James asked.
"It's a wave and/or a particle that is produced by exothermic reactions like combustion."
"Yeah, like setting things on fire or explosions."
The boy-heavy classroom erupted into excitement.
"So... what is fire?" asked Jack.
"It's... I guess technically what you're seeing and feeling are the compounds turning into super-hot gas and particles in the smoke. And lots of energy in the form of heat."
James was taken aback. "Whoa... that's what all those symbols on the board mean?"
In that moment, I saw something I had never witnessed in this context. It clicked. I was used to seeing clicking all the time in my history and government lectures. I hadn't seen (or experienced) it in a STEM subject (science, technology, engineering, mathematics) in years. For at least half the class, two weeks of puzzling and confusion had given way to a breakthrough. These weren't just runes tossed up on a board arbitrarily. It was a meaningful statement of real process we observe all the time. It was a really real explanation of a common phenomenon--all the way down to the molecular level--that proved that matter was neither created nor destroyed but changing form.
In a piece entitled "Understanding the Current Condition", Andrew Elizalde from the Veritas School in Richmond, Virginia, discusses the ineffective reforms pushed through curricula since the Sputnik launch. Attempts are consistently made to "teacher-proof" curricula, especially in math and science where America is perceived to be falling behind international competitors (be they China today or the Soviets in 1957). He notes that these attempts at reform have ironically made math education worse, not better. Because publishers are trying to market their books as widely as possible, they cover much more material than necessary. Really, it's more than any teacher really could cover in a year. This means that math teachers must emphasize certain topics and minimize or ignore others. This makes the transition between courses rough. He writes that this minimizes or eliminates "experiences of investigation, adaptation, discovery, contemplation, collaboration, perseverance and creative problem-solving." If a student doesn't get a concept, he will ask the teacher to do it on the board for him. Math teachers may be rushed to cover everything demanded them in a given year--or worse, to achieve a certain pass-rate on a standardized test--without devoting time to the struggle students need to learn it.
He continues, "This fast-paced learning experience leaves little time for exploration, experimentation, discovery, and argumentation--the processes that were necessary for the development of mathematics across history and are still at the very core of what real mathematicians do." So students need to be able to ask questions, make mistakes, try and try again. They need to fight to understand a problem, learn a plan of attack, and really wrestle with a concept in order to have any hope of understanding any material--be it STEM or humanities. A truncated pace that deprives students of the struggle is short-sighted; it gives them procedures but not real understanding.
Struggling is easier in the humanities because 1) it's harder to objectively manage and 2) it lends itself to discussion. Students are encouraged to throw out lots of answers both individually and as a group until they find the explanation or solution that sticks. They can wrestle with a text both individually and corporately, chew it over, try and fail and eventually succeed in synthesizing and solving a dilemma without easy resignation. Teachers are less likely to feel a pressure to fast-forward through material because few are anxious about how our poetry compares with Japan's or China's.
I think there's also a necessary component of pragmatism preferred over theory. Alexis de Tocuqeville said our American addiction is toward practice rather than theoretical science. Notice, for example, that theoretical physics is not in the STEM acronym. Giving students formulas might get them the right answer, but it doesn't inspire them. It can't capture their attention or give them insight. It tells them how to complete a process, but not why that process works or even necessarily what all the symbols on the page mean. That comes only through the arduous process of struggle.
In spite of disparaging, contrary remarks I have made in the past, STEM subjects are not boring, rote chores or the refuges of uninspired pragmatists. They only become that when they are procedures for profit or application rather than for the joy of the discovery. The best way to learn something is to let it capture you, perhaps even to briefly obsess you. Now not everything can or will catch you. It isn't necessary for learning something; I never recall shoe-tying holding me under its thrall. Nevertheless, leaders and innovators in any field should be (and almost always are) sorta quirky and weirdly obsessed about their subject. I want my neurosurgeon to have a bedside manner and have memorized the surgical procedure; I want the neuroscientist who taught him to have brains in jars all over his creepy science lair.
I also don't mean to say that practical application of theoretical concepts is bad. It is only a problem when the practical dominates the theoretical. Science and math are best studied for their own sake; if you make money using them, fine. Just don't try to inspire a generation of students with the promise of dollar signs. Show them instead that there's something intrinsically fascinating about quantifying the amount of energy they burn running a 5k or how to isolate variables in an equation.
So maybe after all these years, it's time for me to reexamine sine, cosine, and tangent. Maybe the problem-solving skills presented there really do correspond with reality. Maybe Euclid does have an explanation as to why all rectangles are really triangles who finally decided to settle down. It could even be that a 361st degree in a circle is tantamount to dividing by zero. Guess it's time for me to find out. It should be at least as interesting as Jack Burden's character progression in All the King's Men. And if not to me, then maybe I'll at least see why someone else gets stoked over it.
*Name changed to protect the guilty.