Math computer labs give students big picture

By KEVIN O'KELLEY
Office of the Faculty Chair

CHAPEL HILL -- It's common knowledge by now that American students do worse in math than students in almost any other industrialized nation.

Results of the Third International Mathematics and Science study released in February, showed that U.S. 12th-graders ranked 19th among students from 21 countries.

The problem isn't limited to secondary education since many of those same poor math students go on to college. Some of them want to become physicists, economists or business people. Even those who don't have to satisfy some sort of core curriculum math requirement in college.

Two University of North Carolina at Chapel Hill math professors are trying to help.

They want their freshman students to get a solid background in math so they can go on and become doctors or computer programmers or whatever they want. In a project funded with Chancellor Michael Hooker's Instructional Technology Awards, professors Sue Goodman and Jane Hawkins are using computer technology to help students catch on to math faster. The project relies on a distinct strength of computers: they can do routine tasks quickly.

Goodman is planning how she'll use instructional technology in Math 30, a calculus prerequisite. Much of the course is spent graphing trigonometric functions, polar coordinates or conic sections. Ideally, all this problem-solving gradually accumulates into a grasp of general mathematical concepts. But in real classrooms, that doesn't happen nearly as often as math teachers hope.

"Typically, when a student plots a graph, it's a very slow, laborious process that gives them little conceptual insight into what is going on," said Goodman.

For much math modeling, the student needs to learn the basic shape of things, the broad trend, she said.

"In mathematical modeling for economics, for example, you want to know where you hit the low spot," Goodman said. "You may need to plot a hundred points to do the complete graph and find that low spot, but you don't really care about all those hundred points. And it's precisely those hundred points they get lost in."

Such a pedagogical problem has consequences beyond a grade in Math 30.

"One of the things you do in chemistry courses is to take data points when you're doing an experiment, go and plot all of the points and graph the points," said Suzanne Buchta, Goodman's graduate assistant.

"The graph would be a picture of the function that summarizes what happened in the chemical experiment," Buchta said. "The more comfortable you are with this thing called a function and the notion that graphs are pictures that reflect what happens in a function the better you understand economics, chemistry or physics."

The key to achieving that familiarity with the relationship between a problem and a graph is a software program called "Mathematica," devised by mathematician Stephen Wolfram and used by professional engineers, mathematicians and physicists worldwide.

Goodman and Buchta are creating math workbooks using "Mathematica" software and placing them on the World Wide Web.

"Mathematica's" usefulness lies in its speed and graphics.

Goodman's students will do functions and draw graphs by hand. Once they've mastered that, they'll go to the online workbooks Goodman and Buchta wrote for the broader view.

"With the computer you can plug in the numbers, and see the graph instantly," Goodman said. "When you activate the animation feature you can see the graphs changing shape, corresponding to changing values the students have put in. It simply illustrates the qualitative changes much better than I ever could on a blackboard in front of the class."

With the "Mathematica" materials available at any campus computer terminal, Goodman will require her students to explore the concepts on their own. The idea is to make math a subject in which you do experiments, just like biology or chemistry.

Goodman never stops stressing the math as the goal of computer lab work. This is still a math class, not a computer class, she emphasized.

"We're trying to make the "Mathematica" software as invisible as possible," she stressed.

Math 18, Hawkins' course, is far-more computer-oriented, but the basic emphasis -- learning math by experimentation -- is the same.

Hawkins spends less effort de-emphasizing the computer component because she wants her students to approach math with tools that they're already familiar with and will use later in college. Most undergraduates encounter some math and lots of computer work.

"But they may not know what mathematics have to do with computers and what mathematical problems they encounter might be solved by computer," Hawkins said. "So the goal of the course is to show you how the computer can be used as a tool to solve mathematical problems."

One exercise Hawkins poses to students originated in ancient Greece and involves asking them how often they will see a multiple of three among positive numbers.

"They'll say, 'one-third of the time,'" Hawkins said. "I will ask, 'how often do you see a prime number?' And it stumps them."

Hawkins uses the computer to count the prime numbers occurring in the numbers one through 100 and divides that number by 100 - which shows the proportion of primes in the first 100 counting numbers. Then she does the same exercise for 1,000 and 1 million.

"They start to see a pattern developing and they grasp the general principle: this is about how to find the frequency of something in a larger group," she said.

Such techniques can help solve problems outside the classroom, too. Suppose an airline employee is asked to examine the number of passengers ordering vegetarian meals.

"You remember you once had to do this problem about how often prime numbers occur, and you get the total number of people who flew on the airline in a given year and then the number of vegetarian meals in that year," she explained. "You repeat the exercise for a number of years and reproduce the results graphically and you can see if there is a trend."

Using the same approach, Hawkins leads her students into sophisticated mathematical topics such as derivation of fractals from polynomials. (A fractal is a geometric shape that's neither one-dimensional nor two-dimensional, but something in between.)

"These are kids who don't even think they're interested in math and it's incredible that they can do this," Hawkins said.

Hawkins stressed that despite the computer emphasis, her students are still "doing" mathematics, just as much as math majors who use calculators.

"We do some of this by hand first," she said. "I don't want my students running a television program. They have to know what they want to solve mathematically and then make the computer work for them."

Kate Goldstein, an English major from Winston-Salem, was surprised both by how much she liked the class and by how much she learned. "I came out with a greater confidence in approaching the general concepts of math," she said.

Goldstein quickly credited the "big picture" approach championed by Goodman and Hawkins for the class' success.

"We got beyond the typical minutiae," Goldstein said. "We approached programs holistically, which is something I had never done in a math course before."

The course turned out to contain some pleasant surprises for Hawkins herself.

As a mathematician used to working with students with substantial backgrounds in math, teaching a classroom full of math novices was a satisfying although different experience.

"Their mathematical lexicon was limited but not their creativity in approaching solutions," she said. "For example, many didn't know what the word algorithm meant, but they sure could produce one that worked."

Hawkins' class taught her to look at some areas of mathematics with new eyes.

"I was forced to think as a non-mathematician to explain the beauty and usefulness of various types of math problems. I learned a lot from them."

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Contact: Mike McFarland