FOM: Definition of mathematics
charles silver
silver_1 at mindspring.com
Tue Jan 4 11:23:07 EST 2000
Here's an article from "The Chronicle of Higher Education." Will this
make the subject matter of mathematics more empirical?
Does the book _Interdisciplinary Lively Application Projects_ sound
enticing?
Charlie Silver
*************************************************************************
>From the issue dated January 7, 2000
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.
It's up to mathematicians, professors are learning, to teach them how.
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