The Calculus Diaries - Jennifer Ouellette [4]
Calculus boils down to two fundamental ideas: (1) the derivative (differential calculus), which is a way of measuring instantaneous change, such as finding the speed of a car when you only know its position; and (2) the integral (integral calculus), which describes the accumulation of an infinite number of tiny pieces that add up to a whole and can be used, for instance, to determine the distance a car has traveled when only its speed is known. Everything else is just a variation on these two themes. The derivative and the integral are like the two ends of a hammerhead: One is for pulling out the nails, and the other is for pounding them in. The first is a process of subtraction and division; the second, a process of multiplication and addition. Each “undoes” the work of the other. And not every math problem requires a hammer; sometimes a screwdriver works best. So calculus is just one tool in a broad arsenal of mathematical instruments, applicable to specific kinds of problems.
I explained this to Allyson, who responded with an incredulous, “That’s it? Why can’t math teachers just say that?” In fairness, they probably do; they just say it in a foreign language. Galileo famously observed, “Nature’s great book is written in mathematical symbols.” Unfortunately, to the untrained eye and ear, that language resembles ancient Sanskrit, and math teachers may as well be speaking gibberish. Most of us never get past the strange symbols and jargon, and thus meander through life without any quantitative tools beyond basic arithmetic. We can balance a checkbook, but have no grasp of statistics, compound interest, or probability, for example—and this puts us at the mercy of those who do understand them, and thus can manipulate us at will. Knowledge is power, and we forfeit that power when we choose to remain willfully ignorant.
The inability to grasp basic algebra and calculus also can be a stumbling block to many students who otherwise would wish to become scientists. Take my friend Lee, whose struggles with algebra in high school—despite top grades in all her other classes—reduced her to tears, and kept her from becoming a marine biologist. She still loves science, but has a visceral hatred of mathematics to this day. “It wrecked my self-confidence in a way nothing else ever did, and still knots my stomach,” she told me. “I’m not totally innumerate, but anything that looks like an equation makes me break out into a cold sweat and run screaming in the other direction.”
Ironically, given my distaste for the subject, I succeeded at math, at least by the usual evaluation criteria: grades. Yet while I might have earned top marks in geometry and algebra, I was merely following memorized rules, plugging in numbers and dutifully crunching out answers by rote, with no real grasp of the significance of what I was doing or its usefulness in solving real-world problems. Worse, I knew the depth of my own ignorance, and I lived in fear that my lack of comprehension would be discovered and I would be exposed as an academic fraud—psychologists call this “impostor syndrome.”
I might have gone through the rest of my life cringing compulsively at the mere sight of an equation. But I became a science writer and fell in love with physics—not the math part, mind you. I loved the rich history, the people, the funky experiments, and the big ideas. One fateful day, I asked a physicist named Alan why it was true that all objects fall at the same rate, regardless of mass, when casual observation would seem to indicate the opposite. It seemed counterintuitive to me.
This is the basis of a famous experiment proposed by Galileo. If you drop a coin and a feather under normal (atmospheric) conditions, the coin will hit the ground first. But Galileo reasoned that another