The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by Jennifer Ouellette

  Different people learn in different ways. Some students respond well to how calculus is traditionally taught, while others, like Bisi (and me), don’t; but that doesn’t mean we lack the aptitude to learn. That was the viewpoint of an eighteenth-century educational pioneer named Johann Pestalozzi, whose ideas laid the groundwork for modern elementary education. Born in Zurich, Switzerland, Pestalozzi was the son of a physician who died when Johann was quite young. He was raised by his mother and grandfather in a rural village, and that experience gave him a lifelong empathy for the plight of the Swiss peasantry. While at university, he embraced the “natural” philosophy of Jean-Jacques Rousseau, even naming his son in the great thinker’s honor, and went on to become a schoolteacher.

  Many of his ideas were quite radical. Pestalozzi rejected the “tyranny of method and correctness” that pervaded Swiss schools of that era, declaring that he wished “to wrest education from the outworn order of doddering old teaching hacks as well as from the new-fangled order of cheap artificial teaching tricks, and entrust it to the eternal powers of nature herself.”54 He became the first applied educational psychologist, insisting that children begin with the concrete object before moving on to the underlying abstract concepts.

  Pestalozzi emphasized the individual, encouraging spontaneity and self-activity. His students were not given preset problems with ready-made answers but were encouraged to pursue their own curiosity. He also believed in creating a nurturing environment for students, abolishing in his school the then-common practice of flogging. And he worked hard to remove the “verbosity of meaningless words” from his system, preferring to emphasize concrete observation—a doctrine he called Anschauung (loosely translated as “sense perception” or “object lessons”). Yet the Anschauung must be bolstered with concrete action, Pestalozzi cautioned: “Life shapes us and the life that shapes us is not a matter of words but action.” The best way to achieve that action, he believed, was through repetition—not rote memorization, but mastering the action through practice within the context of the concrete object.

  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 way further into my heart.

  How did I become convinced that calculus was beyond my ken? No doubt part of it stems from gender bias. There is a well-documented prejudice against women in math and science dating back thousands of years, although history gives us the rare exception, such as the plucky Sophie Germain. Such women often have been dismissed as mere statistical anomalies, but evidence is mounting that there is no innate difference in the mathematical ability of girls and boys. Any gap in performance is due primarily to sociological factors. This is a controversial statement. We would prefer to believe that the overt sexism in math and science is a thing of the past, but the reality is that these attitudes persist, even in this enlightened age.

  A geometry teacher tells the entire class that the girls will probably do worse in his course because they lack spatial reasoning ability. A guidance counselor shunts female students into “practical math” classes where they learn how many ham slices each guest would need at a wedding. A physics professor insists on checking his female students’ work before they can leave the lab, yet doesn’t feel the need to check the work of his male students. A computer science professor dismisses any questions from female students as “lazy little-girl whining.” And a calculus teacher thinks it’s perfectly appropriate to measure his female students’ bodies and use those measurements as part of his volume calculations in class. One woman told of her high school math teacher who made the three female students sit in the front row, “because girls have a harder time with math than boys do.” It was really a flimsy excuse to ogle their cleavage and brush his crotch up against them suggestively during exams. “Guess which three people in that class were not about to be stuck in a basement computer lab with that dude?” she asked (rhetorically).

  I never experienced anything so horrific; my math teachers were kind and, if not openly encouraging, they certainly were not discouraging or hostile, nor was I ever sexually harassed. My parents were supportive of my intellectual pursuits, if a bit bemused by my headier inclinations. Nobody ever told me explicitly that girls weren’t as good as boys at math, yet somehow I absorbed that message anyway. Carol Tavris, a cognitive psychologist and author of several popular books (The Mismeasure of Woman should be mandatory reading for young women), explained to me that there are subtle, situational social cues that seep into our consciousness as if by osmosis, even if we never encounter overt negative messaging about gender.

  The phenomenon is known in psychological circles as stereotype threat, and it has been confirmed in more than a hundred scientific articles. For example, a 2007 study in Psychological Science found that female math majors who viewed a video of a conference with more men than women reported feeling less desire to participate in the conference and less of
a sense of belonging than female math majors who viewed a gender-balanced version of the video. The male math majors were immune to those subtle situational cues. That’s stereotype threat in a nutshell.

  These pressures are very real. Yet I can’t blame my ambivalence entirely on gender. After all, plenty of boys struggle with math, too. How we self-identify in our mathematical ability sets in at an early age and colors our perception from then on. “If ever I had an Achilles heel, mathematics would surely be it,” says Brian, who is studying to be an evolutionary biologist. Yet he keeps running afoul of the dreaded math classes and worries that his failures therein will dash his hopes of a career in science. “Nothing makes my blood run cold like an indecipherable word problem, and the very term ‘calculus’ is enough to give me nightmares,” he confesses, sounding just like many of the female students I encountered.

  Tavris bemoans our fascination in the United States with the notion of innate ability as the source of this kind of negative self-identification. We are born with certain built-in talents, this reasoning goes; you either have a gift for math or you don’t, and no amount of hard work can make up for that lack of innate ability. I certainly bought into this notion, assuming that because it didn’t come as easily to me as verbal skills, I lacked the “gift” of manipulating numbers. Yet it merely required a bit more effort on my part to learn the foreign language of mathematical symbols (vocabulary) and processes (the rules of grammar) until I became sufficiently conversant to solve basic problems. At heart, it is a foreign-language problem: Many students also struggle to learn French or German or Egyptian hieroglyphics.

  Consider Deborah, whose fourth-grade teacher held multiplication table competitions in class. Deborah was highly competitive, so she worked very hard on memorizing her multiplication tables and practicing at home. As a result, she excelled in these competitions and became known as being “good at math.” This had a significant impact on her later on: Whenever she struggled with an especially tough problem, she pushed through, thinking, “I should be able to do this because I’m good at math.” Yet her belief in her innate ability, and her success at math, were actually the product of a lot of hard work and repeated positive reinforcement in the classroom.

  Tavris also believes that American culture has an unhealthy attitude toward failure. It is considered a shameful thing rather than a natural stage of the learning process. Calla initially failed high school algebra. It shattered her confidence and instilled the telltale dislike of math that such failure so often brings. “I hated math for making me feel stupid, and because there was nothing enjoyable about it,” Calla said. “It was just there, like a big black wall I would run into every once in a while, not letting me know why it was there or why I should I care.” In reality, failure is how we learn. Take away the freedom to fail, and it is no wonder our students aren’t learning. Science, too, relies on failed experiments and null results just as much as its justly touted successes in order to advance human knowledge.

  The good news is that, regardless of the combination of factors that conspire to discourage any given individual from pursuing math and science, one good teacher can make up for all of it. I had Alan and Sean. Calla had a dedicated high school math teacher who literally changed her life. Everything changed when she took a class taught by a young woman who emphasized hands-on demonstration and applications for the math. It took some time for Calla to work through her mental blocks, but that teacher patiently guided her every step of the way with all kinds of creative approaches. They hammered away at the big black wall together until Calla finally broke through and realized she was “good” at math. She went on to major in physics in college.

  There are many excellent high school math teachers, laboring in the trenches for very little pay and even less appreciation. But they are fighting an uphill battle. The way calculus is so often taught is clearly not reaching a substantial fraction of students; more often than not, like my high school self, they end up solving problems by rote, with little comprehension of why they must perform these tasks—or get so frustrated at their inability to solve problems that they reject mathematics for the rest of their lives.

  Every teacher I know is heartened whenever they see that light bulb of genuine comprehension turn on in a student’s brain: “Oh! This is that!” In the same way that our favorite works of art, literature, music, or theater tend to be those with elements we recognize and can respond to emotionally, we tend to respond more to books, lectures, or classroom curricula that enable us to make similar connections between the abstract concepts of math and physics and our real-world experiences. If our emotions are engaged, even better: That excitement and enthusiasm serve to fuel students’ desire to persevere past the inevitable frustrating roadblocks in the quest for knowledge.

  Actor David Krumholtz plays a brilliant young mathematician on the hit TV series Numb3rs, and he bravely participated in a panel discussion at the 2006 meeting of the American Association for the Advancement of Science on the challenge of changing negative public perceptions of math and science. With disarming frankness, he readily admitted—before a roomful of scientists—that he had flunked algebra twice in high school.

  Numb3rs demonstrates the relevance of mathematics better than any pedagogical method I’ve yet encountered. Week after week, Charlie Epps (Krumholtz) helps his FBI agent brother crack a federal case using the tools of his trade. Math is a tough sell; couching it within the familiar crime-solving framework renders its abstract concepts not only palatable to nonscientists, but downright appealing. The show’s tagline sums it up perfectly: “We all use math every day.” Even Krumholtz confessed to developing a fascination for Pythagoras and the Fibonacci sequence because of their prevalence in nature and art—and were it not for his role as Charlie Epps, he might never have encountered those concepts outside of the classroom. This suggests that his struggles with math weren’t due to a lack of aptitude, but to how the subject matter was presented. Like many of us, he never understood why math was important or how it could possibly be of any use in our daily lives.

  There is much weeping and gnashing of teeth in academic circles about the sorry state of U.S. math-and-science education.

  I don’t pretend to have an easy answer to a sweeping, complex problem that confounds our best educational experts. Learning is profoundly individual, and what resonates with one student might not resonate with another. How can you systematize all those individual styles? But the power of mimesis to inspire young minds should not be ignored.

  Surely it is no accident that a similar interpretation of mimesis can be applied to key breakthroughs in physics: It’s that same creative impulse, finding inspiration in surprising connections. Albert Einstein credited his development of the theory of special relativity to a critical insight gleaned years before, as he sat on a train moving away from the station platform—namely that he would measure time differently from within the moving train than would someone standing on the platform (“this is that”).

  Watching an apple fall from a tree gave Isaac Newton his critical insight into gravity and his laws of motion: He realized the apple’s position, when plotted as a function of time, formed a parabolic curve, and connected motion with geometry and algebra (“this is that”). Archimedes found the solution to the problem of Hiero’s golden crown while soaking in the bathtub. My own modest breakthrough came on that fateful day in Santa Barbara, when I saw the connection between an abstract calculus equation and the motion of Saturn’s rings, and realized, à la Archimedes, “Eureka! This is that !”

  APPENDIX 1

  Doing the Math

  The only way to learn mathematics is to do mathematics.

  —PAUL HALMOS

  Tell me and I’ll forget. Show me and I may not remember. Involve me, and I’ll understand.

  —NATIVE AMERICAN PROVERB

  So, you’ve made it through The Calculus Diaries and feel as though you’re starting to get a handle on this whole calculus thing. Maybe you’re
even toying with the idea of delving a bit further into the topic. This appendix is here to help you take that next step. I deliberately avoided scary equations in the main text, but sooner or later one must bite the bullet and face the actual math head-on. Nothing here is intended to “teach” calculus—this is not a substitute for the experience of an actual class, textbook, and/or a private tutor—but it will give you a taste of how the concepts discussed in the text translate into the language of math. For those who really get bitten by the calculus bug and desire even more details, I recommend The Complete Idiot’s Guide to Calculus by W. Michael Kelley.

  Here are the most common terms and symbols you’ll encounter; this will help you “read” basic calculus equations:

  Function. The notation for a function is f(x). Whenever you see this at the start of an equation, you know you’re dealing with a function of some kind: For example, f(x) = x 2 tells you that x 2 is a function. However, just as it’s possible to convey the same meaning using different words, there can be more than one way to write an equation for a function. The function above can be written more generally as f(x) = ax 2, with a denoting “some constant.” It is also common to write f(x) simply as y. In that notation, x is the “independent variable” (it can be anything) and y is the “dependent variable” (it depends on x). This is important to remember when plotting points on a Cartesian grid (see page 268).

  The above function describes a parabola. The most general notation is f(x) = ax 2 + bx + c, where x is the independent variable, y is the dependent variable, and a, b, and c are constants. There is also the so-called vertex form: f(x) = a(x − h)2 + k. The vertex of the parabola is the point where it turns, and in this format, (h, k) delineates that point.