Getting Physical to Convey Mathematical Ideas

July 24, 2009 — Algebra is a hands-on subject for students in Michael D. Smith's graduate-level classes.

"We have used Jenga blocks, colored pencils and bulls-eyes, Frisbees, encrypted messages, cinder blocks, two-by-fours and contra dancing," said Smith, who is teaching his popular summer course, Proofs in Algebra, at the University of Virginia to graduate students and some advanced undergraduates.

Algebra is the study of the familiar logic rules governing the addition of real numbers and the search for other objects governed by the same rules, he said.

Smith, 29, a member of the summer faculty, uses contra dancing to show all the reflections and rotations of a square.

"Contra dances involve doing moves in a group of four, with one person on each corner of a square," he said. "Each move is an element in the group of symmetries of a square, and doing moves one after another is like 'adding' the symmetries together to get new symmetries. To make a dance progress, you need a list of eight symmetries whose 'sum' is the reflection over the x-axis."

Smith developed these techniques while a graduate student at U.Va. He first used his hands-on method to teach calculus, and when he was successful, he created similar methods for other mathematics classes.

"I hope students realize that math can be fun," he said, noting his class consists of graduate students and one master's in teaching student. "All of them have either taught math or will be teaching math in the near future. I hope my students leave with inspiration to be more creative in their own classes than they otherwise would have been."

In teaching calculus, Smith initiated a sack race and a wheelbarrow race to illustrate how competitors can determine a strategy to run the races in the shortest combined time.

Using calculus, students can "see that although wheelbarrow racing is much harder, the optimal strategy is never the strategy that involves the minimal amount of wheelbarrow racing," Smith said. "This is a life lesson because the path of least resistance is never the best path."

"Our class held the annual potato-sack and wheelbarrow relay race on the U.Va. Lawn on what we hoped was the muddiest Friday of the semester," Smith said.

Once he earned his Ph.D., Smith taught abstract algebra, statistics and foundations of math, inventing activities for each of these classes.

Students are usually receptive to the unusual teaching techniques. They find most of the activities useful, Smith said, and each activity has several students who find it helpful.

"There is usually a small period of shock when they realize 'Oh my gosh, he's really having us do this.' Afterward, students become comfortable with the activities," he said. "It has been easier at U.Va., because I have taught enough courses here that many students know of my unorthodox style coming in and are less surprised."

A native of Maine, Smith received his Ph.D. in mathematics from U.Va. in 2006 under the guidance of Kevin McCrimmon, a professor in the math department. Smith's work focused on non-associative algebra.

He has taught at Earlham College, Morningside College and the University of Massachusetts. He is beginning an appointment at Hollins University this fall.

"I enjoy seeing the connections between several different branches of math, inventing a sequence of games and activities that illustrate the math while building team spirit and rapport in the class," he said. "I also enjoy seeing students leave with less anxiety and a better understanding of math."

While he has seen some teachers use techniques similar to his, he has not seen anyone adopt his style completely.

"I modeled this approach off of my 11th-grade physics teacher, Mr. DeAngelis, who brought in several games and activities for his classes," Smith said. "Originally, I wanted to teach physics in Mr. D's footsteps, but he knew that math was my favorite subject and challenged me to invent my own stuff to teach math in the spirit of what he did."

— By Matt Kelly