Undergraduate research: It's a SMART thing

by Joyce Ritchie

Amanda Schermer, an Indiana Univer sity South Bend sophomore, and her faculty mentor, Michael Kinyon, assistant professor of mathematics, are sharing a "Moment in Mathematical Research"--the computer-assisted three-dimensional exploration of geometric figures called fractals. Shermer undertook the research project last summer and has yielded very interesting results--but not what the two expected.

"It turned out that we were trying to push a diamond-shaped peg into a round hole," says Kinyon.

Translation: Fractals are complex designs which are created by repeating a simple mathematical rule. They expected to see a cone shape. Instead, some results showed images with wavy boundaries, stringy filaments and mysterious dots. The "why" is a matter for future investigation. The "why" of such research investigation at all, at the undergraduate level in particular, and at universities in general, is currently a question of hot popular debate.

Schermer's research project yielded almost as many questions as answers. Yet it also serves as a key to understanding the intrinsic interconnection of research, teaching and learning, and the application of knowledge and skills developed in the university to the real problems and concerns of people and communities.

For example, to the non-mathematical mind's-eye, fractals are delightfully colorful designs. To Schermer, the project was an opportunity to"enjoy a pure exploration of a mathematic model in itself."

For Kinyon: "The original motivation for this project came not from the realm of just pure research, but from an interest in teaching a particular subject."

For laboratory scientists, fractals are used in the study of predictability of long-term behavior of biological systems: Their presence indicates chaos. For the rest of us, it's enough to know that one practical application of fractals is in helping to better understand and find cures for epilepsy, because the chaotic nature of changes in the brain is similar to changing patterns that emerge in fractal research.

"Mathematics seems to me to be the search for patterns because they are interesting in themselves," says Schermer. "Yet, when a mathematician comes up with a seemingly useless theory, it seems that within a year or two a new scientific application is discovered," she says. "My main goal was to learn about Mandelbrot and Julia Sets, old information about fractals. It's a beautiful side of mathematics you don't often get to explore."

However, the results of her exploration were quite different. "As it turned out, the project gave me a chance to do research that hadn't been done before," she says. "I can say with certainty that I understand the material I worked with, that I helped expand knowledge and that I have shared new information with colleagues here at IUSB and at other places."

One of those places was the IU Undergraduate Research Conference held annually in Indianapolis at the IUPUI Conference Center.

"I loved going to the IU research conference," says Schermer. "I met all kinds of people doing research in the field."

Kinyon attended also, her mentor there as at every other step of the project. "Coming from the mentoring side," says Kinyon, "I don't see these aspects of my research as separate from my teaching. I'd been playing with the idea when Amanda got interested. I'll use the results in my classes next year.

"The idea that research is separate from teaching strikes me as ridiculous. Undergraduate research has brought new ideas and information into the field, and I've really warmed to it," he says. "I believe now that it is a misperception widely held that in heavily theoretical math or physics, it's too difficult to come up with projects for undergraduate research.

"At the IU conference, Amanda's was the only mathematics project. All the physics undergraduate projects were experimental," he continues.

"Especially in research of this type, the increased use of computers brings new opportunities and makes the subject more accessible for undergraduate students."

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