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The Remaking of Math
Many scholars suggest a more real-world focus, but some worry about a loss of rigor
By Robin Wilson
Mathematics isn't exactly known as academe's most progressive discipline when it comes to curricular reform. Students are still taught to plug in numbers and chug through a formula, and some undergraduates never learn how calculus relates to other disciplines, much less the real world.
All of that, mathematicians say, is about to change.
Under the guidance of the Mathematical Association of America, mathematicians around the county are conducting a major review of the undergraduate curriculum. Their recommendations, which are scheduled to emerge in 2001, could form the basis for the biggest change in mathematics since calculus reform. It proposed new ways of teaching the subject and hit campuses a decade ago.
Curricular reform is usually a local phenomenon, taking place on individual campuses and within departments. So it is unusual for a discipline to pursue such change on a national scope. But what makes this review unique is that mathematicians aren't just talking among themselves -- they have asked professors from an array of other fields for advice on what mathematical knowledge students need.
Most students who take undergraduate math courses don't major in the subject; they major in physics, engineering, computer science, chemistry, biology, economics, or business. The revolution in information technology has prompted students to take more and more mathematics to help them use computerized models -- some chalk up enough credits for a math minor or even a double major. But the math curriculum at most universities was designed to appeal to traditional math majors, with a heavy focus on theory and formula manipulation.
"Undergraduate courses still have a lot of graduate-school influence from the math Ph.D., and may not be designed to attract mathematically oriented students in computer science or people who like math and theory but want to go into business," says Alan Tucker, a professor of applied mathematics at the State University of New York at Stony Brook.
The last time mathematicians decided their discipline was due for a face lift, calculus reform was the result. It swept college campuses during the early 1990's in response to educators' concerns that students knew how to use techniques to solve formulas but didn't understand what the formulas were for. Although universities did change course content -- story problems were introduced, and less importance was given to memorizing techniques and applying them repetitively -- the biggest changes were pedagogical. Universities traded large lecture courses for smaller sections of calculus, and students were asked to give oral presentations and write papers rather than manipulate formulas over and over.
Calculus reform evolved from a handful of visionary mathematicians, many of whom have spread their ideas through textbooks on new ways to teach the subject.
The curriculum review under way in mathematics now involves a much different process. It is an exercise in democracy being conducted by the mathematical association's Committee on the Undergraduate Program in Mathematics. Before the review is completed, it will have involved dozens upon dozens of discussions among mathematicians, as well as between mathematicians and professors in other disciplines. The mathematical association conducted a similar review in 1981, but it was much smaller in scale.
This month, when the discipline's three big societies -- the mathematical association, the American Mathematical Society, and the Society of Industrial and Applied Mathematics -- hold a joint annual meeting in Washington, 10 groups of a dozen mathematicians each will trade ideas about reforming the undergraduate curriculum.
Mathematicians have already begun holding workshops on campuses around the country to talk to professors outside the discipline. In October, they met at Bowdoin College with professors in physics and computer science, and in November, they met at the United States Military Academy at West Point to talk to physicists and engineers. Later this year, they'll speak to economists at the University of Arizona and to biological scientists at Virginia Commonwealth University.
The final report from the mathematical association will recommend curricular changes and may include models from institutions that the committee believes are doing a good job.
Universities need to examine their curricula because technology has changed so much in the last 20 years, says Thomas R. Berger, a professor of mathematics and computer science at Colby College and chairman of the committee that is leading the national review. Computers have made mathematical modeling and data analysis available to people in a broad array of professions. Technology has also eliminated the need for people to perform many basic computations that have formed the bedrock of college calculus courses.
"Mathematics is becoming universal," says Mr. Berger. "There is a need in math to serve everybody now. Students are picking up courses because the study of their main subject is becoming more math intensive."
Mr. Berger calls this phenomenon the "mathematicization of society." Engineers once built prototypes of cars and electronic gadgets, he says. Now, they can test their ideas on the computer using mathematical models, if they know enough mathematics to set up and run the models. Says Mr. Berger: "Almost everything we do today is modeled mathematically: our trips to the moon, the design of automobiles and Walkmen."
It's not happening only in the sciences and engineering. Even students in disciplines like art history and English are taking more math courses, partly because of new requirements in "quantitative reasoning" that many institutions have adopted in their core curricula lately.
Carol Geary Schneider, president of the Association of American Colleges and Universities, says 90 per cent of campuses have reviewed their general-education requirements within the last decade. Although she doesn't have exact figures, she says many colleges have either added mathematics requirements for the first time or eliminated basic algebra courses from the core in favor of new classes focusing on mathematical modeling and problem-solving.
Harvard University is one of the latest institutions to tinker with the role of mathematics in its core curriculum. Beginning with this year's freshmen, students must take a semester of quantitative reasoning, replacing a system that allowed undergraduates to test out of mathematics altogether.
So far, Harvard has developed six courses to fulfill the quantitative-reasoning requirement. A few are abstract mathematics courses, including "Introduction to Elementary Number Theory." But some are applied, such as "Health Economics," which looks at the economics of health-care policy, or "Counting People," which focuses on demographics and data analyses involving populations.
"Our hope is that we can take students from scratch and show them that they can think analytically," says Peter T. Ellison, a professor of anthropology at Harvard who will teach "Counting People" next academic year.
Many Harvard professors were wary of including math in the core, he says, because "math requirements scare people." But Mr. Ellison, who helped bring about the change, says: "We felt it was quite strange to purport to teach modes of reasoning without teaching quantitative reasoning. It has been such a fundamental part of human logic, thought, and attempts to understand the world and ourselves."
Part of what is driving mathematicians to reach out to students from other disciplines is fear. Mathematicians at the University of Rochester learned the hard way that maintaining an ivory-tower discipline primarily for mathematics majors can be dangerous. Four years ago, administrators at Rochester proposed shutting down the mathematics Ph.D. on the campus and cutting the number of math professors in half. The administration thought the department had grown lazy: Although most freshmen took math courses, the department focused on math majors and did little, officials said, to insure that math appealed to anyone else.
Mathematicians at Rochester did some scrambling and managed to maintain their Ph.D. program and limit the cuts in their department's faculty, primarily by forging ties with other departments to broaden the undergraduate math curriculum.
The crucial question scholars in mathematics must answer now is how far universities should stray from the theoretical mathematics traditionally taught on American campuses.
Don Small is a professor of mathematics at West Point, which many say is ahead of the curve in reforming its math curriculum. He offers an answer that lies at one extreme of the debate. Calculus, he argues, is on its way out as the "umbrella" under which college-level mathematics is organized. Mathematical modeling and inquiry, which focus on the application of mathematics to solve real-life problems, are gaining predominance over the ability to differentiate formulas, he says.
Someday, argues Mr. Small, calculus may not be taught as a separate subject at all. Instead, he predicts, calculus topics -- such as change and accumulation -- will be taught in computer-modeling courses, as needed. "The focus must be on problem-solving, not on developing tools for problem-solving," Mr. Small wrote in a paper he delivered at the math association's workshop on his campus in November.
Most mathematicians aren't willing to go that far. They worry that a shift away from calculus will leave students unprepared to do the kind of modeling that Mr. Small describes. "There is more emphasis on applications, and that's refreshing," says Dusa McDuff, a professor of mathematics at SUNY's Stony Brook campus. "But you can't forget the calculating side. Students make the most horrendous mistakes. They will scribble and think it doesn't matter because the machines will take care of it."
Arthur Mattuck, who has taught college calculus for 40 years and is a professor of mathematics at the Massachusetts Institute of Technology, says that teaching mathematics through modeling is more difficult than it sounds. Typically, he says, models must be quite sophisticated to capture students' interest. But the more complicated the models are, the more time students must spend understanding the models themselves, and the less time on mathematics, he says. "In the end, very little mathematics gets done," he adds.
Many professors believe that traditional mathematics encourages logical reasoning and mental rigor. "Mathematics courses are sometimes said to build mental discipline, introduce the mathematical method, and teach students how to think," Paul Zorn, a professor of mathematics at Saint Olaf College, said in a paper he delivered at West Point. "This sort of talk sounds a little passe nowadays. But the old mental-training agenda is still perfectly valid."
The conversations about the future of mathematics have only just begun. But those participating in the review are willing to point to some institutions that they believe are on the cutting edge.
Professors say West Point is one of them. It leads a consortium of 12 universities that have a $2-million grant from the National Science Foundation to make stronger connections between math and other disciplines. The N.S.F. made the money available through a $12-million program it began in 1995 called Mathematical Sciences and their Applications Throughout the Curriculum.
The West Point consortium has developed a book called Interdisciplinary Lively Application Projects (The Mathematical Association of America, 1997). The text features multiproblem projects that are several pages long and ask students to use mathematical modeling and problem-solving to calculate the health benefits of exercise, for example, or to determine how long it would take before fish in mercury-poisoned water would be safe to eat.
West Point has also established ways to help students link calculus with physics and engineering. The academy arranges the syllabus in its mathematics courses so that students get the calculus they need just before they will have to use the concepts in physics and engineering. That kind of coordination is possible because some of West Point's math professors serve as liaisons with science and engineering departments, a practice that is rare.
Dartmouth College received a $4-million grant from the N.S.F. program and is trying similar experiments. Dartmouth couples its calculus and physics courses, teaching them together in the same semester. A mathematics professor teaches calculus and then sits in on the physics course, and vice-versa for the physics professor. The point is to help students weave the two subjects together and to make the calculus more directly applicable.
Not all students at Dartmouth learn math and physics this way. The college has established the program as an alternative to the standard track, in which the two disciplines are taught separately. But Dorothy I. Wallace, a professor of mathematics at the college, says the coordinated courses have had an impact. First-year engineering students who take the new course sequence are more likely to remain engineering majors at the end of their freshman year than are their counterparts on the standard track, she says.
Dartmouth has also created courses like "Mathematical Science Fiction," which looks at math in literature, and "Pattern," which links mathematics and studio art. Ms. Wallace, who has taught the latter course, says students produce artwork with repeating patterns and study the mathematics of them. "Students who wouldn't have, are now taking courses with math in them," she says.
John Mackey, a postdoctoral student in mathematics at Dartmouth, has developed a course called "Applications of Calculus in Medicine and Biology," primarily for premed students. His wife, who is finishing medical school, took many calculus courses as an undergraduate only to discover that she never used it after that. So Mr. Mackey decided to offer a calculus course that would explore the mathematics that his wife and her colleagues in medical school thought were useful.
Undergraduates in the class, which Mr. Mackey is teaching this semester, study the mathematics of how a decrease in the diameter of a blood vessel leads to an exponentially greater decrease in blood flow. And they examine the mathematical probability of a sexually transmitted disease being spread by someone who wears a condom.
"Students are hungry for this and want to know, How can math help me?" says Mr. Mackey.
Jerry P. Becker Dept. of Curriculum & Instruction Southern Illinois University Carbondale, IL 62901-4610 USA Fax: (618) 453-4244 Phone: (618) 453-4241 (office) (618) 457-8903 (home) E-mail: firstname.lastname@example.org