`I try to set up a framework where students work hard because their teacher is working hard'
Standing in front of Math 35, Tom Banchoff looks like a puppeteer, arms extended, twisting a set of four hinged sticks into squares, rectangles and other quadrilaterals while peppering his 27 students with questions about the functions associated with calculating each shape's area.
Use of the sticks allows Banchoff to offer real-world illustrations of how calculus, the study of functions, works. Banchoff knows that students often draw the line at studying mathematics. Given the chance, many students will subtract math classes from their curriculum in high school, even if the move cuts them off from different career options. Banchoff considers it his job to provide a mathematical experience as a valuable part of the liberal education offered to Brown students.
For his three decades of making math come alive to Brown students, Banchoff is the 1997 Rhode Island Professor of the Year, an award bestowed by the Carnegie Foundation for the Advancement for Teaching. He will be honored by acting President James Pomerantz Friday, Oct. 24, at 6:30 p.m. during ceremonies to dedicate the Harriet W. Sheridan Center for Teaching and Learning.
"I try to set up a framework where students work hard because their teacher is working hard," said Banchoff, the fifth Brown professor to win the award since its establishment in 1981. "I try to get students to be conscious of their development. I do so in a timely way by giving them a lot of work but also providing extensive feedback."
Banchoff exploits the fact that students explore math at many different levels of learning. He employs various methods to engage students. As a researcher at the forefront of computer graphics and four-dimensional geometry, he embraces computer use to stimulate student-professor interaction.
Try logging-on to the Math 35 website. There you will find this challenge problem offered the first day of class: "Was there ever a time in your life when your height measured in inches was equal to your weight in pounds?"
Within three hours of that posting, Banchoff's students began to respond, sparking discussion on use of graphs to represent height and weight changes. With each input, the mathematical language sharpened. Sparking collaborative thinking and providing opportunities for students to generate ideas based on those of their cohorts are techniques at the heart of Banchoff's teaching style. Even more striking, glance at the height-weight challenge problem, and you find that Banchoff personally answered each student.
According to Dan Margalit '98, web TA for Math 35 (Honors Calculus), Banchoff reads and responds to every student's homework assignment. "Starting with the first day of class, he knows how each person is doing," Margalit said. "This way, he reinforces students who are doing well and makes helpful suggestions to students who aren't.
"Every day, Professor Banchoff is trying to do something better, whether in the way he uses computers or in his teaching style. He is constantly perfecting his art."
John Hughes agrees. In 1980, Hughes was a graduate student at the University of California, Berkeley, where Banchoff had received both his master's degree and doctorate.
Hughes met Banchoff when the professor returned to attend a conference in honor of his doctoral advisor's 60th birthday.
"Tom Banchoff is one of those people who is devoted to his own teachers, and those teachers become devoted to him," Hughes said. "Tom believes in close relationships between students and teachers and in maintaining those relationships."
Hughes, now an assistant professor of computer science at Brown, says that in their brief 1980 chat, Banchoff helped him, an uncertain graduate student, with a mathematical proof, while displaying confidence in his work. Hughes never forgot.
Banchoff has a gift for making math appealing, Hughes said. His ability to use calculus and linear algebra to create and display curves, surfaces and other visual phenomena turns math into an experimental experience.
"In my discussions with Tom, I am amazed at how often he asks me for ways to best present information to introductory calculus students," Hughes said. "Tom finds teaching math to be the best part of his job."
Banchoff says that teaching is communication. It is a group enterprise, where the classroom experience is co-created by students and teachers. The interactive nature of computers allows Banchoff to make the classroom experience even more effective.
"You can put the learning experience right out in front by using computers," Banchoff said. "That's so much better than putting a bunch of papers in the library. Students can see the course take shape as they participate. An effective class sparks individual learning plus a constructive shared experience."
With every class of incoming freshman, Banchoff meets a new set of young people, whom he challenges with questions such as what are the largest box shapes that meet U.S. Postal Service regulations of package length plus girth not exceeding 108 inches? Solving the problem entails understanding multivariable calculus, which examines functions that depend on two or more variables.
This semester, one of those students is Adam Leventhal '01. After studying how his classmates were responding to the U.S. Postal Service question, Leventhal thought the ultimate proof to the challenge was to construct a cylindrical package and mail it from the Post Office in Faunce House to Banchoff's office in Kassar House.
Leventhal persuaded a skeptical postal worker that the well-taped box, 3 feet tall and 6 feet in circumference, did indeed meet postal regulations. The employee measured the box twice before declaring it fit for mailing. Leventhal handed over the $2.75 postage.
Why go through the hassle? "I thought it was something fun to add to the challenge problem," Leventhal said. "A challenge problem lends itself to more student creativity. We're not just solving textbook examples, but working on real-world situations."
Leventhal's box has taken its place in Banchoff's office amidst an array of objects built by his students over the years to better visual mathematical concepts. Banchoff works at a desk surrounded by spheres, parcels, cells and other models constructed out of cardboard, wood, wire and tape. There sits the first model he received, in 1968. It's a small corkboard laced with parabolic wires and strings. Nearby is a two-bladed knife designed for cutting spherical cakes, designed by another student in his first honors class at Brown.
Banchoff likes to talk about the objects because they allow him to admire student creativity. Take Leventhal's package.
"Creating the box showed a certain amount of self-coincidence and free-spiritedness," Banchoff said. "It's one thing to solve the problem. It's another thing to make a dramatic display."
In front of Math 35, eyes bright behind dark-rimmed glasses, Banchoff wields the hinged sticks into a bow, striking an archer's pose for a brief second. His questions to the class about volume and variables come rapid fire. More and more students volunteer answers.
Responding to an inquiry about one computation, Banchoff places the sticks down in front of him.
"As a professor it's my job to work with you to test those algorithms," he said. "How do we do it? By testing hard cases. How do we know what a hard case is? By experience."