Yarn, spandex, and dogs teach new tricks

As college students head back to campus, they may be surprised by how much thought some professors have put into making classwork just as fun and fascinating as a summer vacation. Far from doing everything by the book, the creative scholars featured here use everyday objects to put even the most abstract subjects within reach.

Even modern-art lovers sometimes run when they hear the dissonance of 20th-century experimental music. But Anthony Brandt, an associate professor of composition at Rice University in Houston, wants students to see how those scores reflect the era's radical changes in thought about the nature of such fundamentals as physics and consciousness. Modern music provides "one of our safest and most enjoyable ways of experiencing ... ambiguity," he says.

In his class on analysis of 20th-century music, Professor Brandt challenges students to think about music from the ground up. They have one month to invent an instrument out of an ordinary object and create a composition for it. He's seen everything from toothbrushes to popcorn poppers transformed. "It's one of my absolute favorite days when they come in ready to perform their pieces," he says.

One standout: A student who "performed" the blackboard. (No, not the fingernail screech.) He alternated using an eraser to make soft puff sounds and chalk to make quick percussive dots. "You watched the composition at the same time you heard it," Brandt says. "The class was mesmerized. For about a minute, you thought the blackboard was an instrument."

Another student composed with feedback from baby monitors placed at varying distances apart. One woman used a metronome and kept returning from the rhythmic clicks to the fixed tuning note - not realizing that her concept was similar to that of a famous solo for oboe.

And then there was the toilet piece. To change the length of the flushes for his composition, the student manipulated the toilet so the water refilled to different levels. He couldn't perform in class, of course, but he turned in a recording that gave his professor plenty to chuckle about.

A visiting scholar at Cornell University has taken both the math world and, more recently, the art world by storm with a touchable form of advanced geometry. Daina Taimina,a mathematician and crocheter, discovered a way to create durable and easy-to-use models of hyperbolic space. People have been attempting this ever since the concept emerged in the early 1800s and overturned Euclidean geometry's assumptions about parallel lines.

Students are excited by the crochet creations, some of which resemble curvy leaves of lettuce or kale. Straight lines of one color of yarn are stitched into models made of another color. By bending the forms, students see a new three-dimensional relationship between parallel lines.

"Maybe there's something psychological about using the yarns and the handicraft.... It just takes out the anxiety. It's not the same as looking at some abstract formula," Dr. Taimina says. "People say, 'It's very important that we can touch it - we can learn through feeling.' "

Taimina's models are on display at the Smithsonian in Washington, and a virtual exhibit with mathematical explanations is available at the Institute For Figuring, a Web-based educational organization directed by Margaret Wertheim in Los Angeles (www.theiff.org). "We tend to think of math as masculine, but this feminine activity has had a powerful role in illuminating it," says Ms. Wertheim, who has helped organized workshops for artists and other nonmathematicians.

Who would have thought that spandex would come in handy to help people grasp Einstein's theory of relativity?

Jim Borgardt, an associate professor of physics at Juniata College in Huntingdon, Pa., brings out a stretchy membrane to represent space and time, and places a marble or heavy ball on it to show how it's affected by various objects. Dr. Borgardt explains that the moon orbits the earth not because the two masses are attracted to each other, as Newton theorized, but because the earth forms a dimple in the space/time membrane, and the moon is trapped in that dimple. A heavier ball gets the class talking about black holes, because the ball sinks so deeply that it's surrounded by the membrane. "If you sit there and try to explain that just with words, some people's heads are spinning, whereas if you get something they can see - even though it's not a perfect metaphor - at least it gives them a road map," Borgardt says.

Elvis has left the building. Actually he's jumped into a lake to fetch a ball. Elvis is the Welsh Corgi that tours with math professor Tim Pennings to illustrate a calculus problem at schools and colleges.

When Dr. Pennings noticed Elvis running along the beach part of the way before jumping in to swim to a ball, he checked a hunch with some calculations. Based on Elvis's running and swimming paces, Pennings discovered his dog was instinctively choosing just about the quickest route to the ball, known in calculus as the "optimal solution." In the real world, factories calculate the optimal number of a product to put out each day to maximize profits, and NASA finds the optimal time to send out a spacecraft, says Pennings, who teaches at Hope College in Holland, Mich.

After talking about Elvis's approach, Pennings puts younger kids to the test by tossing a candy bar onto the lawn and telling them to walk on the sidewalk and crawl once they hit the grass.

Turns out the children's instincts for finding the optimal path aren't quite as good as Elvis's, but surely they have fun trying.