Animated Computer Graphics Give a New Angle to Math Education : Learning: The goal is to teach the TV-generation in an engaging, visual way. A Caltech professor helped put the video in motion.


One of the best-selling items at the Caltech Bookstore in Pasadena is not even a book.

It's a videotape, according to bookstore manager Judy Capron, or rather a series of computer-animated videotapes on mathematics for high school classrooms.

They are so popular, in fact, that the shop hired a person to help handle the orders.

Tom Apostol, retired professor of mathematics at Caltech, is co-director of Project Mathematics!, the nationwide producer of the tapes. He creates them with James Blinn, a lecturer in computer science at Caltech and a computer animator.

About 50,000 tapes have been sold in three years (at cost, for $11.25 each), Apostol said.

Profit is not the motive. It's the desire to bring mathematics to students in ways that engage and excite them. The pair used the money from a corporate grant to send more than 6,000 free tapes to high school teachers nationwide.

They also have an agreement with the departments of education in 36 states to supply a master tape so that state officials can make and distribute copies. Two teachers associations list them in their catalogues. A version narrated in Korean is nearing completion.

But the tapes can be freely copied for educational purposes in the United States, Apostol said.

Altogether, he thinks, more than 2 million students have seen the tapes.

Not bad, when you consider that the subjects include such matters as, "In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs."

Extensive computer animation is what makes the subjects lively. The animation was designed and executed by Blinn, 42, the project's tall, bearded and bespectacled co-director and one of the world's leading computer animators.

Blinn, who is known for his Voyager planetary fly-by simulations for the COSMOS series on PBS, including Voyager II's journey through Saturn's rings, was awarded a MacArthur "genius" fellowship of $265,000 in 1991.

"He put it in the bank," Apostol said, "He didn't even buy a new car."

Under the aegis of Caltech and the Southern California Consortium, a company that makes educational videos, Blinn created eight hours of animation for "The Mechanical Universe", a series that was awarded the Japan Prize in 1987 for outstanding achievement in educational programming.

"The Mechanical Universe" is a 52-part physics course for college freshmen. It was produced with a $5.8-million grant from the Annenberg/CPB Project. The National Science Foundation (NSF) provided an additional $2.5 million to adapt the course for high school students.

Apostol, an adviser on that series, thought it would be a good idea to do a similar one on mathematics.

After two years, he and Blinn were able to get a grant of $1.1 million from the NSF to support Project Mathematics! NSF granted an additional $1.3 million in September, 1991, and that two-year grant is expected to be supplemented by another $694,884 this year.

The videotapes are distributed primarily to high school teachers.

Self-contained, they are not part of a curriculum, but cover various important topics of mathematics ranging from the story of pi to the theorem of Pythagoras, similarity and polynomials.

Teachers use them in different ways, Blinn said. They show the whole tape or bits and pieces that are relevant to their classroom topics. Some send the videotapes home with their students.

Gwen Roberts has used the tapes to teach math at Pasadena High School for four years.

"What I love," she said, "besides the graphics and the rich mathematics is that you can pause. You can show a little bit, then stop and say 'Wow!' together.

"The tapes are a visualization of an idea. We learn by hooking ideas onto ideas, and the more hooks the better."

Roberts has been invited to speak on her use of the tapes at the upcoming International Conference on Technology in Education in London later this year.

An accompanying workbook is designed to stimulate discussion and encourage interaction between students and teacher.

"Kids are brought up on commercials and professional TV," Apostol said. "They won't settle for less in the classroom. TV gives the teaching credibility."

The computer animation also helps students who have difficulty with English or with hearing, he said.

It takes Apostol and Blinn four to five months to produce each 20-minute tape. (A 30-second computer-animated TV commercial can take three months and cost $100,000 or more). They are writing and teaching at the same time.

"Tom writes a script," Blinn said, "and I tinker with ways of putting visuals on a screen. Most of the time is spent thinking and coming up with visual metaphors. A lot of it has to do with geometric ideas, calculating areas, chopping up and rearranging things."

In a tape demonstrating the relationship between algebra and geometry, for example, Blinn uses a little crank. Turning the crank to adjust numbers in an algebraic equation changes a geometric figure. The tape on similar shapes uses an "Expando Ray" and a "Shrinko Ray" to illustrate concepts.

The narration is breezy, to match the visual metaphors, and the background music is often Mozart or Chopin. Blinn lifts a page from Lake Wobegon to end one segment: "That's the news from the Mechanical Universe, where the forces are strong, the equations are good-looking and all the concepts are above average."

Blinn said his first encounter with computers was in a class at the University of Michigan in 1966. By his junior year, he was teaching graduate-level computer courses.

In 1975, he went to the University of Utah and worked with Ivan Sutherland, a pioneer in computer graphics. Both later came to Caltech.

Blinn created his fly-bys and astronomical images for Jet Propulsion Laboratory.

"A lot of my work in the early days," he said, "was in figuring out how to make more realistic pictures. A lot had to do with applying textures to surfaces and with simulating the shapes of highlights and shining reflections."

"The biggest problem now," he said, "is finding something that hasn't been done yet. All the easy problems have been solved."

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