April 8, 2008
Thinking like a mathematician
If you add infinity plus one, does infinity get bigger? Can there be multiple infinities, of different sizes?
Ask the students in Great Ideas in Mathematics. They’re history majors, student filmmakers, would-be diplomats and others who, truth be told, may not have signed up for a math class if it hadn’t been required. But they’re mulling the big questions as they’re introduced to high-level topics from number theory to topology to probability and chaos theory.
“Most people don’t see any of this stuff until grad school. But it’s more like what mathematicians do than quadratic equations,” says Professor Michael Keynes, director of undergraduate studies, Department of Mathematics and Statistics.
Suppose you are in a polygonal mirrored room. Will there always be a point in the room from which, if a light were placed, that light would illuminate the entire room? Are there two (non-bald) people in the world that have the same number of hairs on their body? How could you figure this out? In the 1939 movie The Wizard of Oz, when the brainless scarecrow is given the confidence to think by the Wizard (by merely handing him a diploma, by the way), the first words the scarecrow utters are, “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” An isosceles right triangle is just a right triangle having both legs the same length. Suppose the right triangle has legs each of length three. What is the length of the hypotenuse? Is the scarecrow’s assertion valid? This question illustrates the true value of a diploma without studying. (From Burger and Starbird, Heart of Mathematics: An Invitation to Effective Thinking, second edition) |
Some students fulfill AU’s general education requirement in math by taking a statistics course required for their majors. Others satisfy it with an algebra course that builds on things they’re familiar with from high school.
But for those who want to delve into different territory, there’s also Great Ideas in Mathematics, pioneered two years ago by Keynes, whose research interests include mathematics education. It’s a challenging 4-credit course, but the emphasis isn’t on computation. It’s on learning to think like a mathematician while looking at the kind of abstract, mind-bending, and sometimes wryly comical problems that fascinate mathematicians.
The idea for the course grew out of conversations that Keynes had when he arrived at AU. He learned that general education math courses were not, to put it bluntly, always a lot of fun to teach. Many of the students hadn’t liked math in high school, and only took math at AU because they needed it to graduate. It would be the last math course most ever took.
So Keynes started to rethink the purpose of college-level math education. Does every student really need to spend a semester learning or relearning how to solve, say, quadratic equations?
Granted, science majors, premed students, business majors, and many others won’t get far without it. And granted, even those who don’t envision a future as doctors should have learned algebra in high school. In fact, if they’re at AU, they did learn it, because they must have done well on their SATs.
But honesty compels Keynes to admit that some students are quite good at memorizing and forgetting the things they don’t really like. They’ll dutifully memorize for one semester, and then erase it from their brains as they head off to courses that intrigue them more, and a future in which quadratic equations really won’t reappear.
For those students, what is the benefit of a college-level math course? Is it truly needed? If so, what need does it serve?
Keynes believes that every college student should study math. The reason math is valuable for everyone on a college level is that it teaches a clear and logical way of thinking that, ideally, can prepare their minds for any challenge. Whether they’re a lawyer or a policymaker or journalist, they’ll be faced many times with the need to define a problem, examine the information they have to solve it, and decide on an approach.
And there’s another reason, as well.
Imagine the following scenario. It’s happened fairly often, or something like it. A math professor is at a party and meets someone new.
Stranger: So, what do you do for a living?
Professor: I’m a math professor.
Stranger: Oh, I hate that!
Such is the popular attitude towards mathematics in U.S. culture. Keynes can’t imagine a doctor hearing, “Oh, I hate medicine!” But somehow, “Math is one subject it’s socially acceptable not to like.”
For some reason, students are often left with a bad taste for math. So when it comes to the course that, for many, will be the last math course they ever take, he wants them to leave with a different attitude. “I want someone coming out saying, ‘You know, math is kind of cool.’”
They won’t become mathematicians. But they’ll understand math on a deeper level. “The math I feel I do is like poetry,” Keynes says. “But too often, what we teach is grammar. I understand you need the grammar. But you need to see the poetry, too.”
(Photos by Jeff Watts)
