A Mathematical Prelude and Fugue

I spent the latter half of last week moving as little as possible. The fabled Conference Infection (also known as the nerdobacter) has hit me very hard, and I was barely able to complete my articles for Ars Technica, much less write anything else. I’m only just now beginning to recover, thanks to antibiotics and self-enforced rest.

In the spirit of David Dobbs and Douglas Hofstadter, I shall structure this post musically, drawing on my favorite composer, Johann Sebastian Bach. The post is in two parts: a book review of The Calculus Diaries by Jennifer Ouellette (prelude), followed by some thoughts on math, its public perception, and how gender stereotypes continue to totally screw everything up (fugue).

Prelude: The Calculus Diaries

The Calculus Diaries by Jennifer Ouellette

I was one of the lucky people to win a free copy of Jennifer Ouellette’s book at the Science Online 2012 conference last weekend, and since she was actually in attendance, I got her to sign it for me. (However, I totally forgot to take my copy of Written in Stone for Brian Switek to autograph. Maybe next year!)

First, a caveat: I am not the book’s intended audience. As a physicist, I have more in common with Jennifer’s husband, the cosmologist Sean Carroll, than I do with her and her primary intended readership. (If her stories are true, I am dramatically less likely to bring up Fourier transforms during romantic interludes. Though that makes me start thinking of dirty jokes, so I should probably stop now.) This means I’m reading the book from the perspective of many years of math classes and using high-level math in my research; more on that in the “Fugue” section below. I would really like the opinion of someone whose background is not as math-centric to go along with mine, but I found the book to be wonderful, an easy read, and provides a compelling answer to the question of why studying calculus is good.

Jennifer Ouellette is a science writer whose focus is primarily physics, but her background is in English literature. She never took calculus in high school or college, which isn’t an unusual story: unless it’s explicitly required for the major or (as with some colleges like Johns Hopkins) as a graduation prerequisite, I imagine most people don’t take calculus. Similarly to physics, the reputation of the subject is bad: it’s considered “hard”, and its practicality is given a backseat to the stigma against all math. Jennifer, however, wants to get over that hurdle, not least because she’s a physics writer with a deep love of her subject matter: calculus and physics are tied together through history and their shared conceptual framework.

The Calculus Diaries has a lot of history in it, which is great: mathematicians are human beings, after all, and some of them lived very colorful lives. The usual suspects (Archimedes, Newton, Liebniz, the Bernoulli family, and so forth) make their appearances, but also some  figures like Heaviside I have certainly heard of, but know little about. However, this isn’t just a history: Ouellette doesn’t shy away from the real terminology. She explains the basic concepts—derivatives and integrals—both in their abstract versions, but also in a wide variety of specific real-world applications. (Here I suspect she is not helped by her editor and/or illustrator: I would love to have seen lots more diagrams to show exactly what all her excellent examples are doing.)

Though anecdotes, conversations with scientists of various sorts, and history again, Ouellette shows specifically where derivatives and integrals are used. Besides physics, calculus infuses economics, probability theory, and a variety of topics in biology and ecology, all of which Jennifer describes briefly in turn. Each one could be (and sometimes is) treated in a separate book, but wisely Ouellette decides to keep the pace brisk. The length of The Calculus Diaries is very nice, and each chapter is also a manageable length, leaving the reader curious for more specifics while not feeling incomplete. Jennifer even brings up important “advanced” topics like differential equations (essential for physics and other fields), Fourier transforms, vector calculus, and (one of my favorites) the calculus of variations, which shows why some paths and shapes are preferred over others in real-world systems.

Again, I would like to have the opinion of someone who isn’t thoroughly familiar with calculus. I may be missing things at times, since I can fill in gaps with my own experiences, but I found the book to be very insightful, interesting, and full of the reasons scientists love what they do. And all of this brings me to the final chapter, entitled “Epilogue: the Memetics of Math”….

Fugue: #LearningToMath

Voice 1

Jennifer Ouellette:

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 memetic 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.

Voice 2

Math and science are not “easy” for me: I work very hard at them, even now. I always double-check, and still make silly minor errors that nevertheless require going over pages of equations to locate. I am envious of those for whom math comes as easily as speaking; I will always have an accent when I speak math, I suspect.

My own “memetic moment” came a little earlier in life than Jennifer’s: I took all the math required of me, and since I always intended to be a physics major in college, I enrolled in calculus concurrently with calculus-based introductory physics. I struggled through my calculus classes, though there were moments where I glimpsed the true meaning of Christmas calculus beneath the day-to-day problem solving we did. Differential equations really brought me fully around on math, however: that’s where my memetic moments began to arrive in earnest, as they fully connected what I was learning in my physics courses. From that point on, I felt I understood math far better, and that understanding traveled backward in time to previous classes I had.

Voice 3

Now it’s your turn. After my post Fear of a Math Planet, I threw out a question on Twitter asking when they discovered their love for math—when they had their own mimetic moment (borrowing that phrase again from The Calculus Diaries). I’m especially interested in hearing from women and teens of all genders, since of course I’m in the category (white male in the sciences) where it’s “normal” to like math. As research shows more and more that there is no gender gap in math abilities, it seems more and more important to emphasize our own personal discoveries.

If you want to participate in the conversation, even if it’s to say that the memetic moment has eluded you so far, feel free to email me, leave a comment, or tweet using the hashtag #LearningToMath . If you want your story shared, I will post it on this site later on. But let’s widen the conversation started between author (Jennifer Ouellette) and reader (me): please add your voice.

2 responses to “A Mathematical Prelude and Fugue”

  1. Calculus never worked for me – After 3 semester of it in college I still didn’t have any good grasp of it, and with DiffEQ on the horizon I fled, sure that failure was nigh. I think it is something I might be more prepared to tackle now. Now I know how to ask questions about things I don’t understand; specific questions to help me get to where I need to go.

    That being said I absolutely love doing algebra, and have tutored folks in it for years just because I think it is fun and logical.

  2. I’m not your target audience (sorry), but I remember quite clearly my own “memetic moment,” or at least when I began to understand “intuitively” how to attack and solve physics problems: midway through sophomore year in college. Prior to that I’d always used the “plug and chug” method: OK, we’re given these variables and we want that one, and these equations seem to have most of those variables in them, so let’s plug in and see how far we get. It wasn’t quite that I didn’t know they were connected, but I didn’t have a truly solid, gut-level grasp of what they meant and how to use them to get where I wanted to go.

    On the other hand, I never did get comfortable with abstract algebra (group theory, etc.). Non-commutative groups, rings, cosets…ugh. I need to be able to draw pictures.

    (Btw, I too am a white male; I took freshman calculus as a senior in high school–very handy since I could devote 90% of my attention to that one class–and majored in physics/astrophysics/math in college.)

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