You may not be able to teach an old dog new tricks, but you can get him to solve a calculus problem.
That's what Dr. Tim Pennings, associate professor of mathematics at Hope College, realized when he took his Welsh Corgi, Elvis, for an outing to Lake Michigan to play fetch with his favorite tennis ball.
"Most calculus students are familiar with the problem of finding the optimal path from point A to point B," he said. "Standing on the water's edge at Point A, I throw the ball into the water at Point B. Elvis runs along the shore a portion of the way, then plunges into the lake at a point of his choosing and swims diagonally to the ball. By the look in Elvis's eyes, it seems clear that his objective is to retrieve it as quickly as possible."
Pennings assumed, then, that Elvis unconsciously attempts to find a path that minimizes retrieval time--a problem typically solved using techniques from calculus. But he wanted to test his theory.
So, he clocked Elvis's running and swimming speeds and spent three hours collecting data measuring the distance between the ball and the shore and where Elvis jumped into the water to retrieve it. Plotting the results revealed that Elvis almost always chose a path which is in close agreement with the optimal path calculated mathematically.
In fact, given complicating factors--waves, the movement of the ball in the water and others--Pennings suggests that dogs may choose a path that is actually better than the calculated ideal.
"Although he made good choices, Elvis does not know calculus," Pennings said. "Though he does not know the calculations, Elvis's behavior is an example of the uncanny way in which nature often finds optimal solutions."
The topic was a natural for Pennings, who specializes in mathematical modeling--the way in which mathematics describes natural phenomena. As a teacher, he appreciates having a new opportunity to relate his discipline to everyday experience.
An article he has written about his work with Elvis has been published as the cover story of the May, 2003, issue of "The College Mathematics Journal," complete with a photo of the dog relaxing on the beach with tape measure and ball nearby.
In addition, Pennings, accompanied by Elvis, has also given talks on the subject at Hope and a variety of area schools.