Devlin's Angle

June 2005

Staying the course

"I tried, professor, I really did, but I just couldn't see how to do it." Surely, every college or university mathematics instructor has faced this lament as a student hands in an incomplete homework assignment. Sometimes the student is probably being honest. He or she did try. But what does that "try" amount to? Five minutes? Ten? Half and hour? In my experience, it is rare these days to encounter a student who will spend more than a few minutes on a math problem, let alone the several hours - or more - it might require. Most students don't know what it means to niggle at problem - to worry it - on and off for days or weeks on end. In their eyes (if they think about it at all), those of us who do mathematics for a living are some kind of alien species, born with a weird brain that finds math easy. We're not, of course. Our brains are not that different from theirs. Any mathematician who says she or he finds math easy isn't tackling sufficiently challenging problems. The fact is, what most of our students don't realize is that mathematicians are not people who find math easy. We don't. We find it hard. The key factor is that we recognize that, given enough effort, and enough time, it is nevertheless possible.

Not that mathematics is particularly unusual in requiring effort to succeed. Most things do. Take writing, for example. I had a student once in a math course for nonscience majors, which I had structured in large part around writing about mathematics. She came to me to complain about the grade I had given her for an essay. I had deducted marks for the poor structure of her paper, which in truth amounted to little more than a list of points on which an essay might be based. Some of her points were good, and I gave her credit for those. But what she handed in didn't come close to being a completed essay. She couldn't see it, and she was indignant that I, a mathematician, would dare to evaluate her writing. I pointed out to her that just because mathematics was my primary professional interest did not mean I was not able to evaluate an essay. And as it happens, I said, writing about mathematics was something I had built a second career around.

I asked her what her major was. English, she proclaimed proudly, adding that she was a writer and wanted to have a career in professional writing. I asked her how many drafts she had gone through in order to produce the work she had handed in. She seemed not to understand my question. To her, writing an essay was a one-shot deal. No period of prior thought and reflection. No first, second, and maybe further, drafts. No revisions. I pointed to a small pile of books in the corner of my office, the complementary copies of my latest book, Life by the Numbers, that had just been sent to me by the publisher. (This was back in 1996.) "You see that book?" I said. "If you took every draft I wrote for that book, and piled them one on top of another, the stack would probably rise to shoulder height. I went through dozens of drafts, trying to get it right." It soon became clear that I was wasting my time. To her, telling her that it took me so many revisions to produce a version that I - and my editor - felt happy sending to the printers, simply showed what she already assumed: that I was a poor writer.

Faced with students who think that if you don't get something right immediately, you might as well give up, what hope is there to teach them any mathematics? How could such a student even begin to appreciate the many years - not days or weeks or months, but years - it took San Jose State University professor Dan Goldston and his colleagues to make their recent breakthrough discovery about the pattern of the prime numbers, which I described in a recent news article on MAA Online. Would any of them be able to understand how Goldston could recover from the collapse of what seemed like a correct proof that he announced in 2003, go back to square one, and try again. And keep trying, until the problem finally yielded? Or not; for in mathematics it is always on the cards that a problem will never yield.

I find it a paradoxical feature of American youth that large numbers of them bring a feverish intensity to sporting endeavors, putting in endless hours of dedicated training to become the best in their school, their district, their country, or even the world, yet only a few will put in the same kind of effort to mastering mathematics.

As recently as twenty-five years ago, the situation was very different. Early in my academic career, when my home base was in the UK, I used to come over to the USA frequently for a semester at a time to collaborate with colleagues at various universities, funding my trips by teaching courses as a visiting faculty member. I used to look forward to those trips not only because of the research activities they afforded, but because of the students I would teach. In contrast to most of the students I dealt with at home, many of my American students were highly motivated, hard working, fiercely competitive, and determined to show they were the best in the world. They would go to heroic lengths to avoid being defeated by a problem. Two decades later, living in the US now, I still encounter such students from time to time. But they no longer seem to be in the majority. For most of the young people I meet, the spark I used to see in their predecessors seems to be absent. What has led to this change? Why do so many of them seem to give up so easily? And is there anything we can do about it?

Devlin's Angle is updated at the beginning of each month.

Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published recently by Thunder's Mouth Press.