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Monday, February 26, 2007

Teaching the powerful ideas of mathematics

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“I was a first-generation college student in Australia,” said Adrian Gentle, assistant professor of mathematics. “I went to what is equivalent to a large American state university in Melbourne. There was no opportunity for interaction with professors until the upper-level classes.”

Gentle was searching for an institution of higher education with a teaching focus when he interviewed at USI. He published and did research, but he was looking for an intellectual environment that offered the one-on-one interaction with students.

“My agenda is to teach people about the powerful ideas of mathematics,” Gentle said. “I’m an applied mathematician. I have a passion for it.”

This semester Gentle teaches Calculus III, Linear Algebra, and Differential Equations at USI. Students who want to teach mathematics at the secondary level enroll in his classes as do students interested in careers that use mathematics.

Some individuals who complete degrees in math work in pure (or theoretical) mathematics while others work in applied mathematics. Students with a good background in fundamental mathematics have more options in terms of career possibilities.

“I encourage students to consider dual majors,” said Gentle. “Math is a good match with finance, computer science, engineering, or other fields. The quantitative skills like problem solving and logical thinking that are learned in math are appreciated by employers.”

USI junior doing student research
Over recent weeks Gentle has been working with USI junior Kyle Besing of Newburgh, Indiana.

Gentle said, “Kyle would come to the office to talk. During one visit he said, ‘Do you have student research like other majors? My girlfriend is collecting data under the guidance of psychology faculty. Do you have anything like that in mathematics?’”

Gentle is now working with Besing on a project in numerical relativity, the computational brand of Einstein’s famous theory. Besing is also learning about general relativity this semester in a Special Topics course taught by Shadow Robinson, assistant professor of physics.

Gentle said, “Undergraduates don’t generally have the background knowledge to jump straight into a mathematical research project. As Kyle undertakes the computational part of the project, we will develop the mathematical concepts which underpin the calculations. I think he’ll benefit intellectually from the research.”

Besing plans to attend graduate school, earn a doctoral degree, and teach upper-level math at a university. His minor is physics.

Besing said, “Math is so much more than an answer to a math problem. I want to use math to find ways to describe the world.”

Mathematical ability
Last semester Gentle taught a math modeling class, the first time it was offered recently. “The modeling class was about making assumptions and seeing where they lead, and less about finding an absolutely correct answer,” he said.

In math modeling, techniques from algebra, calculus, probability, and other areas of mathematics are employed to solve problems arising in the biological, physical, and social sciences. Students start with a hypothesis of how a system could work such as a prediction about population growth, a disease outbreak, or consumer behavior toward a new product. Their mathematical formulation of the system can then be used to make predictions. Comparing the predictions to observations can then help improve the model.

Gentle said, “I see students cringe about math classes. But mathematics is more than arithmetic and algebra. There are some truly profound mathematical ideas.”

He said, "To get from A to B you must first travel half the distance. But to get to the halfway point you need to first travel one quarter of the distance. To get a quarter of the way, you need to make one eighth of the total journey, and so on. In fact, to get from A to B you must make an infinite number of these smaller journeys, each of which takes time. Since there are an infinite number of them, we can argue that it's actually impossible to move from A to B. This is Zeno's Paradox, and it can be resolved mathematically."

Students of mathematics are sought for postgraduate study in almost every quantitative field. After he earned his Ph.D. at Australia’s Monash University, Gentle took a Post doctoral (Postdoc) position at Los Alamos National Laboratory in New Mexico. The Postdoc Program offers the opportunity for candidates to perform research in a scientifically rich research environment.

Gentle’s research is related to a revolutionary new branch of astronomy with the advent of a world-wide network of gravitational wave detectors. He explains that gravity is the weakest of the four fundamental physical interactions, and yet is almost solely responsible for the large-scale structure and evolutions of the universe. Einstein’s geometric theory of gravity predicts the existence of black holes and gravitational waves. Using gravitation waves, researchers would be able to investigate the nature of the hypothetical dark matter and dark energy, thought by some to account for the majority of the mass in the universe.

Math across disciplines
The theoretical basis of the modern computer stems from the work of John von Neumann, Alan Turing, and other brilliant mathematicians of the 20th century.

Gentle said, “In the coming century the hottest area of applied mathematics will be in the biosciences. This exciting new interdisciplinary field is still in its infancy, but provides great opportunities for talented students.”
The increasing pervasiveness of mathematics in every area of human activity, together with the enormous advances in the subject itself, indicates that mathematics in the next decades will be an exciting field.

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