Chapter 4: A Tender Coming-of-Age Story (complete with wolves and bunnies)
Every tour guide knows how to fill those three hours on the bus trip between landmarks: tell a good story. If I’ve learned anything in my years on both sides of the mathematics classroom, it’s that the more impersonal and crystalline the subject matter, the more we crave the “I-Thou” connection between teachers and learners that comes from the telling of our stories. I’ve had my share of textbook teachers over the years—the ones who stick to the subject at hand without giving away the slightest whiff of their own heartbeat or dream life. But I’ve also been blessed by those willing to tell me their memories, their own metaphors, their own glimpses out the window on their journey through math. I suppose the least I can do for you is share a bit of my own.
Do you remember learning to drive? Do you remember what motivated you to learn?
Teens have a wide range of feelings about getting behind the wheel for the first time. My big sister Vivian, always the bold adventurer of the partnership, couldn’t wait to get her learner’s permit and got her license the instant she turned sixteen. I, on the other hand, had the good sense (and enough neuroticism) to be properly terrified of what my day-dreamy, clumsy self could accomplish in a two-ton steel cage. I dragged my feet, and with Vivian around to drive me to church youth group and everywhere else, what was the rush? My parents urged me to learn, but it took the allure of something very, very special to get me mobile. It took a math class.
As a homeschooled kid who loved math, I had finished Calculus I at fifteen, but I didn’t want to wait until college to take more math. So, I talked my parents into letting me take Calculus II, and then Differential Equations, at Rice University, a beautiful campus of live oaks and stunning architecture in my native Houston. Only problem was, we lived half-an-hour from campus—a good hour in Houston’s notorious rush-hour traffic—and the route to campus involved some of the most hair-raising freeways imaginable. Vivian drove me at first but had the good sense to kick the baby bird out of the nest, so I was soon on my own. I memorized exactly the two spots where I had to change lanes, a fear-inducing maneuver every time, and otherwise kept my eyes fixed ahead, driving my grandmother’s aging white Buick Skylark and listening to the Newsboys on cassette. What can I say? People will do crazy things when in love.
Differential Equations was taught in a large lecture hall whose basement contained Valhalla, the graduate student bar. Every Monday morning the courtyard was littered with partially-full beer cups and cigarette butts, a phenomenon that made my sheltered little homeschool heart giddy with tantalizing promises of maturity and independence ahead. The professor, Leslie Ward, was a tall, lithe Australian who showed up each morning at 8 a.m. in jeans, with wet hair, which felt very modern and casual and sophisticated. Leslie, as she wanted us to call her, was an exquisite teacher. She brought everything into sharp focus and kept us entertained with gimmicks like naming an equation “Bob” for easy reference.
The subject matter itself was so wonderful, though, that really no gimmicks were needed. Differential Equations is about how things change and grow over time, a fitting course of study for a seventeen-year-old in the spring of her senior year, on the cusp of all that would soon unfold and change and blossom. To model and predict how things grow, DiffEq—“diff-ee-q,” as we call it—makes use of the Calculus concept of a derivative, which is rate of change expressed as a function. In plain English: suppose there are some predators and some prey in the forest—we’ll talk about wolves and rabbits. If you study these populations over several years, you’ll observe some interesting phenomena: the rabbit population might start to increase (rabbits are known to be quite competent in this area), and as the availability of prey increases, so will the wolf population that can be supported. But then eventually the wolf population might start to outpace the availability of food, and as the rabbits become scarcer, this will drive the wolf population down again. Of course in the real world, a million other variables (climate, other populations, etc.) will impact the numbers, but DiffEq gives us a method to build models of varying complexity that make sense of all this data and let us make surprisingly accurate predictions about what’s to come. When you pause to consider that this technique applies not just to wolves and rabbits, but to just about every fluctuating quality in the world, you can catch a glimpse of how powerful and amazing these ideas are.
Chapter 5: Into the Clouds
Sometimes when you love to read, you rush through the canon at a young age and grow up and realize how much you missed when you first read Shakespeare and Tolstoy.
Unfortunately, math is even less forgiving than literature. If you can imagine such a thing as formalized “Shakespeare II” and “Shakespeare III,” you can understand my predicament as a college freshman when confronted with a course in theoretical statistics that called on advanced calculus skills I hadn’t taken time to develop well.
By mid-semester, I was failing.
I could sense this was a matter of not being able to see the forest for the trees; while drowning in the abstraction of probability density functions, multiple integration techniques, and mysterious concepts like independence and unbiasedness, I couldn’t find a point of contact to the “real world” probability and stats that I knew at the beginning. I knew before starting the class that if you flip a coin a hundred times you can expect to get roughly fifty heads; I knew that you could get more accurate information by surveying a thousand people instead of a hundred. I knew that if you measured the heights of your twenty-five classmates and found the average, most people would be pretty close to the average, and only a few would be very tall or very short. These are all very foundational concepts in probability and statistics, but they were nowhere to be found, in any recognizable form, in the pages of my textbook.
I went to see my prof, Dr. Mann, a short, cranky, Harvard-educated engineer-turned-math professor who made me nervous. He wasn’t impressed by my prodigious acceleration through math in high school, knowing that faster doesn’t mean “better” and that my foundations had some weak spots. Dr. Mann was all business, and tried to answer my questions, but the problem was I was so lost that I didn’t know what questions to ask. His practical engineering-style outlook clashed fantastically with my pie-in-the-sky forays into speculative philosophy. I remember myself at the end of a session in his office, babbling incoherently and happily about the statistical distribution of leaves on trees, about the mystery and wonder of statistics in the air all around us, even though I still couldn’t make sense of the homework problems.
I’m here to tell you, miracles do happen and wishes do come true. Somehow, by the grace of God, by the end of the semester there was a convergence of my intuition and the symbolic formalization I needed, and I passed the class with a good final exam score.
But even after this success in starting to link the concepts and equations of statistics, I have to confess that I still approach the subject with a lot of trepidation and struggle. Maybe that’s a good thing though. Every mountain climber loves the idea of a difficult peak shrouded in mist. It’s sort of ironic—probability and stats may sound like the most pedestrian, sterile, downright boring area of math you can imagine. “Statistics” has all the allure of “data entry” or “accounting.” But I want you to know that it might, if your experience is anything like mine, be the area of math you find to be most steeped in mysticism. Maybe that’s just the personal prejudice of my own lack of understanding, but I find that in statistics, more than in any other area of math, even when I’ve come to understand a concept in a technical sense, I’m still baffled by it on some deeper level. We see through a glass darkly, but the darkness always pushes us to keep looking for light.
I hope you’ve enjoyed this little tour of my adopted country. It seems like the hardest part of showing people around is not deciding what to include, but deciding what has to be left out. There are more stories I could tell and many more areas of math full of beauty, surprise, and wonder; but I guess my job was never to show you all of it, just whet your appetite to learn more someday. I hope there’s been something in these reflections that has intrigued you or made you lower or defenses or question your assumptions about math. If so, the pleasure has been all mine.