The following dialogue is an excerpt from a roundtable discussion sponsored by the National Science Foundation and Discover magazine, held June 11, 2002, at the Hart Senate Office Building in Washington, D.C.
ERIC HASELTINE: When we went to school, learning mathematics was, especially in the lower grades, a rather rote, dull affair. Reform has swept through the classroom in the last 10 years as critics argued that students needed to learn how to solve problems rather than just practice the same old skill sets. In fact, now that calculators are ubiquitous, many school systems are thinking about only briefly teaching kids the hardest math they face in early education—long division. This sounds as if it might be liberating, but I wonder if the panel thinks it will work.
1 John Horton Conway is the John von Neumann Professor of Mathematics at Princeton University. He received his Ph.D. from the University of Cambridge in 1962 and remained there as a lecturer, reader, and, ultimately, professor until 1985. He was made a fellow of the Royal Society of London in 1981.
2 Keith Devlin (center) is executive director of the Center for the Study of Language and Information at Stanford University. Devlin has written more than 20 books, including The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. He earned his Ph.D. at the University of Bristol in 1971.
3 Michael Hawley (center) is director of special projects and founder of the Massachusetts Institute of Technology's Go Expeditions program. At MIT's Media Lab, Hawley held the Alex W. Dreyfoos professorship. He earned undergraduate degrees in music and computer science from Yale and his doctorate from MIT.
4 Brenda Dietrich is department manager, mathematical sciences, of the IBM Thomas J. Watson Research Center. She is the author of numerous publications and coauthor of Mathematics of the Internet: E-Auction and Markets. Dietrich joined IBM in 1984 after earning her Ph.D. at Cornell University.
5 Alan Chodos (left) is the associate executive officer of the American Physical Society, a position he has held since February 2000. He received his Ph.D. from Cornell University in 1970 and was a senior research physicist at Yale University for nearly 25 years before assuming the post at the Physical Society.
5 George Andrews (right) is an Evan Pugh Professor of Mathematics at Pennsylvania State University. He received his bachelor's and master's degrees from Oregon State University and his Ph.D. from the University of Pennsylvania. He is a critic of current trends in mathematics education.
6 Eric Haseltine (right), the moderator of this roundtable discussion, is associate director of research for the National Security Agency. From 2000 to 2002, he was head of research and development for the Walt Disney Company. He also writes the NeuroQuest column for Discover every month.
Photograph by Brad Hines
BRENDA DIETRICH: I have a son in high school, an eighth grader, and a fourth grader. I have supplemented their math education because I got very discouraged when my older son, who is very, very bright, came home and said, "You know, I really like science, and I really like history, and I like literature, but math is really boring. Why do I have to learn it?" To someone who makes her living doing mathematics, this was devastating. So I took it upon myself to find material beyond what's taught in school to make sure that my kids see the excitement of mathematics.
7 Sheila Tobias is an author and a consultant. She received a master's degree in history from Columbia University and holds eight honorary doctorates. She has written extensively on women and mathematics and is the author of Overcoming Math Anxiety, Succeed With Math, and Breaking the Science Barrier.
MICHAEL HAWLEY: There's a musical analogue to this. One of my least favorite composers is Igor Stravinsky, but he said something I'll never forget about teaching music. He saw musical conservatories as barely distinguishable from sausage factories. The problem with these places, he said, is that kids go through them playing scales and learning mechanics. What these places should be doing is teaching students to love music, to embrace the field with passion. That's missing from most of the math education I've seen, at all levels.
GEORGE ANDREWS: I am a severe critic of most of the reform efforts recently undertaken by the National Council of Teachers of Mathematics. Their idea that students can develop higher-order thinking skills in mathematics without learning arithmetic as seriously as it was learned before the age of calculators is ridiculous. Speaking as someone who has had countless college freshmen sit in my office, hopelessly out of their depth because their algebra and arithmetic skills were minimal, I believe the efforts to improve math education have failed. There is a strong trend, supported by companies that sell calculators, to make these tools standard in schools. I know it is difficult to learn mathematics. And teaching arithmetic is a difficult task for many who teach it now. There is, consequently, a strong temptation to toss out significant portions of elementary arithmetic—for example, long division. This would be a catastrophic mistake.
KEITH DEVLIN: I agree with practically everything George said—but let's be careful. There are two purposes of education. One is teaching people how to do things. The other, which is more important, is preparing people to live a life. Everything we do—our commerce, our entertainment industry, and so forth—is, root and branch, dependent upon mathematics. Do people want to be ignorant of the life they're leading? I think not. Mathematics is one of the greatest inventions of human creativity—an immensely rich piece of culture. If I went to an architecture class and I was taught just the nuts and bolts of bricklaying, I'd be very disappointed. When we teach math, we shouldn't concentrate solely on the nuts and bolts—but you can't just ignore them either. You cannot develop the ability to do mathematics unless you know an awful lot of things. The only way I know to make the brain understand numbers is to do boring, repetitive practice with numbers, just as the only way I know to become a good tennis player is to get out on that court and practice, practice, practice.
JOHN CONWAY: One big difficulty is that most teachers never really study mathematics. I would abolish departments of mathematical education altogether. They don't teach mathematics—they teach how to teach mathematics, which I think is probably a travesty. You can't learn mathematics unless you have a teacher who loves mathematics. I go out to a lot of schools to talk to kids. I want them to appreciate the joys of mathematics—and I get them dancing around, enjoying themselves, and seeing the pleasure of mathematics. But I also think it's important to be able to do elementary arithmetic reasonably well. If that goes away, it will be really terrible because it's the introduction to mathematics. If you are just taught how to press the buttons on your calculator, you'll never get an idea in your head.
HASELTINE: So do kids need to do long division over and over for a school year, as was once standard?
SHEILA TOBIAS: From my point of view, which is not that of a user of mathematics—I am an admirer of mathematics and an appreciator of mathematics—I think that studying long division for 38 weeks, as I did when I was in fourth and fifth grade, became a waste of time as soon as the calculator came in, just as learning to do square roots the long way is a waste of time.
ANDREWS: May I offer a few words on behalf of long division—and arithmetic in general? Two years ago I served on another panel where a question came up about whether we need to be able to do operations like long division. Someone in the audience suggested that 10 is a better answer than 8 to the question "What is 5 plus 3?" His reasoning was that it's important to know how to estimate, and the calculator can take care of getting an accurate answer. My response to that is this: Suppose you want to check out what 2,017 times 300,136 is. That's 2 times 10^3 times 3 times 10^5. If you add the exponents of 10^3 and 10^5, and you get 10^10 instead of 10^8, you're off by a factor of 100. In other words, arithmetic is a fundamental element that is not "just" designed to get answers. Arithmetic lays the foundation for what you will do in algebra, which then gets you ready for calculus. For example, if you're going to divide polynomials, you will see that if you've had experience with long division, it's more or less the same game.
HASELTINE: Sheila, you have world-class mathematicians here saying you cannot separate the process of math from the content of math.
TOBIAS: When I say long division doesn't matter, I mean that learning the techniques for doing long division is a waste of time. All it really does is force students to practice multiplication and subtraction and learn something about place value. Nobody with a $6.95 calculator does division the way we used to. The 38 weeks I spent learning long division could have been spent introducing simple probability and frequency distribution, not to mention interesting word problems.
ALAN CHODOS: I see value in what Sheila is saying. There's data to show that learning science is very helpful to people in learning math. I think one of the reasons is they begin to use math in ways that are not just, you know, what is 375 divided by 42. Suppose, for example, you looked at this panel of seven and noticed that three of them had beards and asked yourself, "What is the probability that the three people with beards happened to end up sitting next to each other?" Well, that's a lot more interesting, isn't it?
HASELTINE: What is that probability?
CHODOS: I didn't bring my calculator. I'm sorry. [Laughter]
HASELTINE: As a businessman, probability and statistics have played a much bigger role in my career than, say, trigonometry. What should we be teaching that we aren't right now?
DIETRICH: One of the things that needs a lot more focus is modeling. Learning arithmetic is great, learning algebra is great, but learning what sort of mathematics can be used to represent different physical situations, different decision processes, is rarely taught in the high school curriculum.
DEVLIN: I think there's a danger in Eric's question because it pushes you into deciding this bit of mathematics against that bit of mathematics. What distinguishes us as a species is that we live a life of the brain. We survive by thinking about things. We owe it to our children to prepare them to think about things in as many different ways as possible. One of those ways that is arguably more different than any other—because it's the most difficult—is mathematical thinking. Do you ever need to solve a quadratic equation in your everyday life? I doubt it. I never have. But I don't doubt that having learned how to do it has helped me become a better thinker.
CHODOS: There could also be a danger to teaching too much math. In many states, the proceeds from the lottery go to education. If you teach people about probability, they'll be much less likely to play the lottery. So paradoxically, the better you teach math, the less funding you may have to teach it.
HAWLEY: You know, we're the first generation of human beings to bioengineer wholly new species, and that's a Pandora's box. We're also the first generation of human beings to notice that we've significantly changed our world ecosystem, and that, too, is happening in ways we don't understand. The disciplines the next generation is going to need in order to grapple with these problems will include the mathematics and computer symbologies you need to tangle with those issues. That's a little different from just fretting over long division.
HASELTINE: Since we know we're going to need first-rate teachers of math, how do we go about getting them?
DEVLIN: We have to pay them more. Anyone who is good at mathematics can earn twice as much doing lots of other things besides teaching.
ANDREWS: Teaching is an art and not a science. That probably sounds like a cliché to most everyone, but it is definitely not how colleges of education look at the question. In my view, potential teachers would be much better off if they actually learned their subject matter first at the university and then spent a significant amount of time in apprenticeship instead of getting a master's degree in education. This is already acknowledged in a small sense by educators: They send students out for a portion of a year as student teachers. But there ought to be a great deal more practice, and indeed there was, early in the 20th century, before colleges of education realized that they would get more respect in the academic community if they became scientists. Becoming scientists meant they had to do research. And they got sidetracked into doing research on finding new ways of educating people. The fundamental role of apprenticeship got lost in all of this.
HASELTINE: Let me ask another big question: How much do we need to teach? Several of you have talked about how arithmetic lays the foundation for algebra, which lays the foundation for calculus, which lays the foundation for higher concepts, which raises these questions: How far do you think the average student should go with mathematics? Do we really believe that everybody needs to know calculus?
ANDREWS: When you're dealing with children who are learning arithmetic, is it fair to tell them, "Well, you're certainly not going into science, so you don't really need to know any of this"? Everyone should have the chance, especially in this scientific age, to master enough skills that they will not be shut out later because they don't have the least clue how to do mathematics.
CONWAY: You really don't want to cut a young child off from becoming a great physicist or a competent astronomer.
HASELTINE: All right, let me make the concept more outrageous. First there was the abacus, then the calculator, now computers. One can imagine the day when you have an assistant that's an artificial thing, a character of some kind on your computer screen, and you say, "Balance my checkbook" or "Go out and find me the best investments." You won't need to know the meat and potatoes of anything anymore, because you'll have this assistant that will do it all for you. Technology could create something like that. So, will math skills become, for the average person, even less relevant?
CONWAY: So maybe we'll also have history machines. Imagine a personal history machine with essentially all of human history in it—we've got it on the Web now. Should we stop teaching history?
ANDREWS: We have spell checkers and grammar checkers, but everyone knows that they are just assistants. No one suggests that we don't need to know any English.
HAWLEY: Let me give you another musical example. We live in a world where music is available all the time. A few goons in Hollywood produce the stuff and squirt it out through broadcast channels. But in the 19th century, every upper-middle-class home had a piano, and almost everybody knew how to play it. One of the charming things about making music yourself—a charm it shares with other forms of serious play—is that you have to invest time and energy in order to get good at that instrument. The more you practice, the better you get; the better you get, the better you feel about yourself. The process lifts your self-esteem. And that's totally infectious: When a family member makes live music, everybody else in the room gets lifted up by the experience. When you walk into the living room and your son or daughter says, "Hey, Mom, listen to this," and pushes a button on the CD player, it does not have the same effect.
HASELTINE: Are you suggesting that the family can sit around while Junior does differential equations?
HAWLEY: Sure. I am suggesting that you can have lots of fun with these sorts of things because what we're talking about is making things with your mind. It might be manipulating musical symbols or mathematical symbols—but this kind of stuff is part of what lifts us up.
HASELTINE: Well, before we end this discussion, allow me to restate the questions yet again: How far do we want to lift up our students when it comes to math? Do we need to aim our sights higher because we don't know what the future holds but we do know math will have a lot to do with it? For example, someone once said that if you want to know the future, just look at what's hot in mathematics.
CONWAY: That's right. Who would have thought that the business of factorizing would become so important that it's turned into a billion-dollar industry? Factorizing numbers, when I was a student at Cambridge, was a subject only suitable for mathematicians. Suddenly, it's become practical.
HAWLEY: There's a rich body of lore, mathematically speaking, that is still waiting to be applied to interesting domains. For example, tensor calculus is a relatively obscure prong of the calculus tree. Einstein found it very handy when he was working on his general relativity theory. There are very few people who actually understand the mathematics and the tensor work that Einstein applied, so it hasn't been used all that much in other domains. But it resurfaced not too long ago. Jim Blinn, now at Microsoft Research, has found tensor calculus very handy for expressing the dynamics of fabrics blowing in the breeze and hair blowing around and stuff like that. The lesson there is that it really pays to be kind of a polyglot and embrace what is known in the literature of mathematics because that gives you the armamentarium you need to go out and understand new things.
CONWAY: There are countless stories like this. Invariant theory is a branch of mathematics that went baroque in the 1920s. I mean, it was richly developed, and then it sort of went out of favor, but it's come back with all sorts of applications, like information transmission and many other things. But there's a danger in this line of reasoning. You don't have to know everything about mathematics. You do have to know some mathematics. People in general need not be frightened of mathematics. They need to enjoy and love it. And they will if they get taught properly. And they won't get taught properly if they're just getting taught because "they'll need to know it."
TOBIAS: I'd like to remind everybody that we have different populations to serve with mathematics. There are young people who will grow up to advance the field. Many of you were those young people; you are sitting around the table. There are others who will use mathematics, higher mathematics and physics, or medium-level mathematics, if you like, in business. Then there's a large number of people who need to appreciate the power of mathematics without necessarily being able to do it, because they sit on the funding boards or design curricula or are superintendents of schools. Those people can't be left out of the equation. But it doesn't mean that they learn the same kind of mathematics as the other two populations.
HASELTINE: Thank you all very much.
National Council of Teachers of Mathematics Web site: www.nctm.org.