Math professors David Pinchbeck and Scott Balcomb share a laugh while trying to explain the many-sided stellated icosahedron to the author.
"Heart" is not a term often paired with "math." But three years ago, two Saint Joseph's professors adopted a radically different approach to the math course for liberal arts majors, one where equations don't fill up the board and students don't try to memorize them. What they do is learn to think in a new way - one embraced in a book called The Heart of Mathematics.
Dr. David Pinchbeck calls the book a gem. In his restructured Contemporary Math course, students learn about big ideas in math, like infinity and the fourth dimension, like symmetry and chaos theory. Or how any type of vote counting system becomes imperfect once there are more than two candidates.
"They learn how to think mathematically, which turns out to be a practical skill," says Pinchbeck.
Sarah Knights '10, a history major from New Gloucester, Maine, says she hated math in middle and high school, but loved going to Pinchbeck's class.
"It was hands-on, and easier to understand, even though it was complex," she says. She also liked the small class size, felt Pinchbeck got to know her and paced the class at a comfortable speed.
Jennifer Jimenez ’09 of Freeport, N.Y., took the revised Contemporary Math course her first semester on campus. “The approach was very different,” she says.
Jennifer Jimenez '09 of Freeport, N.Y., also enjoyed the class. "The approach was very different," she says. "I didn't use the calculator once. It was like using your brain and applying it to life."
According to Jimenez, who did a presentation on string theory at the end of the semester, the math problems were more like problems from a board game or involved physically doing things to problem-solve.
Why was the curriculum changed to the new approach? "This is really exciting material. They'll remember it years from now," says Pinchbeck. He says more of his students are excited in class - something that never happened when he taught from the old book based on more practical, concrete applications.
In a sense, the new curriculum is more mathematical than the old. "We're learning the math that underlies everything. In calculus, they learn how to do the math, but not why," Pinchbeck says.
Another plus is that the book itself has a sense of humor. For example, it comes with a pair of 3-D glasses to add depth to the diagrams and shows a puzzle with knotted jello cubes to make a point about equivalency and distortion.
The chapters and subchapters have titles like "Finding aesthetics in life, art and math through the Golden Rectangle" or "Knots and Links: Untangling Ropes and Rings." Scott Balcomb, who has taught math for 17 years on campus, initially had doubts about the new approach. He now thoroughly enjoys teaching the revamped course. "It's truly a liberal arts course, because it's not mechanical," he says. "It's ideas and the ideas that drive other ideas." "These are the best ideas in the world," adds Pinchbeck. "There's nothing more interesting." But to consider these ideas, you have to let go of concrete visualizations normally prescribed by math. This math is more abstract, more conceptual, more philosophical. It teaches students to break down a difficult problem to its essential features and patterns. From there, they can generalize or interpolate to a higher level.
Dr. David Pinchbeck’s theorem proof, “Isomonodromic Flows for Fuchsian Connections on Riemann Surfaces,” was published in International Mathematics Research Notices. Although his work – which has to do with looped surfaces – has no practical applications as yet, it contains possible implications for physicists studying string theory and biologists studying DNA. Pinchbeck often collaborates with former SJC math professor John Fey.
Pinchbeck and Balcomb even teach this course differently. Since they want the students to think for themselves, they get the ball rolling, but can't control the way a class proceeds or plan the outcome as they would for a traditional class. When they give the students more freedom to think, the unexpected can happen - as when a student comes up with a new proof different than the professor's.
Even the tests are different: They require essay answers about why a problem is solved a certain way. "It can be uncomfortable for students used to moving numbers around in high school math," says Balcomb.
Now that they have each taught Contemporary Math twice with the new method, they wouldn't consider going back. Balcomb says a former student told him, "I miss that class. I'm sorry it was only one semester."
As Pinchbeck says, "If this is the last math course you'll ever take, it should be a great one."
- by Charmaine Daniels
A gem of a book: The Heart of Mathematics: An Invitation to Effective Thinking by Edward Burger and Michael Starbird