The Calculus Diaries - Jennifer Ouellette [98]
I inadvertently adopted several elements of Pestalozzi’s method in my own adventures with calculus. For one thing, there was no flogging. For another, I avoided sources that relied too heavily on technical jargon—the “verbosity of meaningless words”—because I spent far too much time translating the terminology and not enough grappling with the essential concepts.
But the real key lay in the connections I was able to draw between the abstract equations and real-world examples. Don’t get me wrong: Mastering the abstraction is absolutely critical to fully grasping calculus; it’s just easier to see how the principles are applied if they are presented in many different familiar contexts. It’s the connection between the abstract and concrete that eludes most students. Until I had that mimetic moment—a realization that this abstract equation is connected to that real-world example—my understanding remained incomplete, even if I managed to crank out the “correct” answer to a textbook problem.
How did I make that critical connection? By observing the world around me and then by reinforcing that observation through practice (action). I abandoned the assigned problems in standard calculus textbooks and followed my curiosity. Wherever I happened to be—a Vegas casino, Disneyland, surfing in Hawaii, or sweating on the elliptical in Boesel’s Green Microgym—I asked myself, “Where is the calculus in this experience?”
The process of devising my own problems, rather than relying on existing ones, gave me insights into the discipline I would not have gained otherwise. It’s akin to taking apart a mechanical toy and figuring out how to put it back together again: That process teaches you more about how that toy works than simply reading a description about its operation. I still had to do the repetitive work to hone those nascent skills and make the lesson “stick,” but the repetitive process made more sense to me because it had a recognizable context.
It also helped me to see the hidden connections between seemingly unrelated phenomena. For instance, I never realized that an exponential decay curve can describe the rate at which a cup of coffee cools, and the rate at which wet clothing dries, as well as certain processes in astronomy, economics, and population dynamics. Those very different things nonetheless are related mathematically; they are described by the same kinds of equations. If you don’t “speak math,” it is much more difficult to see those connections.
Two years after beginning my journey, I can’t honestly say I love calculus, certainly not the way I love physics. It’s more of a grudging appreciation for the role calculus plays in describing our world. I am far from mathematically fluent: As with any foreign language, that fluency comes with years of practice and regular immersion in this brave new world. I only went from the equivalent of baby talk to sounding out “See Jane run.” But I have learned the history, the concepts, and the basic terminology and processes of calculus, which in turn have greatly enhanced my grasp of certain conceptual nuances in physics. More important, I am no longer reluctant to confront a simple equation, because I know it will yield a useful insight. The knee-jerk negative reaction and crippling fear are gone. And who knows? Learning is a lifelong process, so it’s possible that as I continue to dabble over time, mathematics will nudge its