Thursday, March 25, 1999
Bringing Calcullus Down to Earth
here's something about calculus.
Mathematicians turn misty-eyed at its very mention. Students cringe.
It's been praised as the highest intellectual edifice created by humankind, and dismissed as an absurdity that computes only "ghosts of departed quantities."
Physicists will tell you it's the mathematical scaffolding behind our entire technological society; former students will tell you it's used for calculating the volumes of bananas. A philosopher might say it's the mathematics behind counting the number of angels that can fit on the head of a pin.
In ever-more frantic efforts to bridge this gap, mathematicians are putting out dozens of new books and CDs designed to sell their favorite subject. These new approaches couldn't come at a better time.
These days, calculus is required for almost every even vaguely quantitative profession, from architecture to economics. Yet as many as a third of first-year undergraduates fail the course, according to mathematician Colin Adams of Williams College, one of the authors of the new book "How to Ace Calculus." The book is dedicated to "all the students whose lifelong ambitions were dashed on the cliffs of calculus."
Among those dropping over the edge, Adams says, are "a ton of students who have their sights set on, say, being a doctor. They take calculus, and then suddenly, boom, that's the end."
The calculus crisis has set off what UCLA math chairman Tony Chan calls a "war" among mathematicians over how to teach this essential subject. Yet more than a decade of reform has not solved the problem.
A Constantly Shifting World
What is it about calculus that causes such fear and loathing in some, and ardor in others?
What is calculus, anyway?
In a nutshell, calculus is the mathematics of moving targets. Most mathematics deals with things that hold still: numbers and triangles and points and quantities.
But the real world shifts constantly. Blood and electricity flow, temperatures rise, wind rushes, galaxies collide, empty space expands, continents drift. "Still life exists only in an art gallery," says mathematician Keith Devlin, dean of sciences at St. Mary's College of Moraga and creator of the CD and book "Electronic Companion to Calculus," as well as other popular interpretations.
Yet, until calculus was invented, mathematics couldn't get a grip on change.
Pinning down motion might not seem to be that much of a problem. But the very idea presented countless paradoxes to the ancient Greeks. "Some things were Greek, even to Greeks," David Berlinski opens his charming and instructive book, "The Tour of the Calculus."
The Greeks couldn't make mathematical sense of how a chicken could cross the road, for example. Obviously, chickens do cross roads, but ancient Greek mathematicians reasoned that a chicken has to cross half the road, then half of the half that's left, then half of that half, and so on and so forth, ad infinitum. It will always have half of some interval left to traverse. To be sure, the steps the chicken takes in this scenario will get infinitely small. But there are also infinitely many steps to take. Ipso facto, you can't get there from here. You can't get anywhere.
The flaw in the Greeks' logic was the assumption that you can't add up an infinite number of things and get a finite answer. Calculus showed that you can.
And once you can add infinite sums, you can do all sorts of amazing things--for example, calculate how an infinite number of angels can sit on the head of a pin.
"That alone is one reason mathematicians find it incredible," Devlin says. "It's the human, finite mind finding ways to handle the infinite. To me that's just staggering."
But the idea of adding up an infinite number of infinitely small pieces still boggles the mind. What does it mean, after all, to be infinitely small? What is the difference between infinitely small and nonexistent? "That was a really hard concept for people to deal with," Adams says. "Are [these infinitesimally small subdivisions] here or are they not here? Are they real or are they not?" These "infinitesimals" seemed so spooky to 18th-century British philosopher Bishop Berkeley that he dismissed them sarcastically as "ghosts of departed quantities."
Actually, calculus tames the infinite in two different, complementary, ways. First, it can pin down instantaneous rates of change--for example, how fast is your car moving down the freeway? Further, it can pin down rates of change of rates of change--for example, how fast is your car accelerating or slowing down?
By adding the total of an infinite number of infinitesimal changes, calculus also can compute the total amount of change. How much weight have you gained in your lifetime--despite all those ups and downs? Or, how much did you earn on that investment? Most important, calculus gives you a handle on how one thing changes as a function of something else. As master math expositor Martin Gardner puts it, calculus keeps track of how the number of toes in a family changes as a function of the number of persons the family contains.
Gardner explains all this in the recently reissued best-selling 1910 book, "Calculus Made Easy," by Silvanus Phillips Thompson, which Gardner still considers the best introductory calculus text on the market. The two faces of calculus grasp both the instantaneous flicker and the holistic sum.
The Symmetry of the Whole
Taken together, "there's a beautiful symmetry to it," UCLA's Chan says. "You can go back and forth, from one view to the other, and always get back where you started."
Why, then, don't students see the light?
"Calculus means different things to different people," Devlin says.
Students facing their first calculus courses tend to see it as a mismatched jumble of techniques for solving problems they'll probably never need in real life: slopes of graphs, areas under curves, velocity, acceleration, volumes.
Students get so lost struggling through the underbrush that they can't see the broader view, experts say. "It's like telling a joke without the punch line," says mathematician Deborah Hughes Hallett of the University of Arizona, who worked on the Harvard Calculus Reform program. "Generations of students have studied calculus without ever seeing its power."
Indeed, one of the reasons U.S. students do so poorly on standardized math tests in general, according to Hughes Hallett, is that they cover more topics more superficially than their foreign counterparts.
"One thing that makes students not enjoy a subject is the sense that they don't know what it means," she says.
These days, introductory calculus books try to cram in so much, Gardner says, that they typically weigh 5 pounds, are 3 inches thick and cost close to $100.
Are students being asked to cover too much too quickly? Isaac Newton was one of the inventors of calculus, and even he knew a lot less about the subject than students are required to learn today, Devlin says. Newton, Devlin says, "would probably be lucky to get a C" in today's typical calculus class. "If I were grading Newton's explanation of calculus now, I would put red ink all over it."
Because it took 200 years for the best minds in math to work out the details, he says, "we shouldn't be surprised if it takes students more than a semester."
Educators trying to inspire as well as instruct get caught between two sometimes contradictory goals. Teaching equations without context leaves calculus stuck in students' throats like dry crackers. On the other hand, preaching the magic of calculus without the often difficult mathematics that makes it work provides no substance.
In the end, Devlin says, any effective approach for bridging the calculus gap will have to appeal to aesthetics.
Fewer bananas, more angels.
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