Published on *Hope College* (http://www.hope.edu)

An article co-written by mathematician Dr. Tim Pennings of the Hope College faculty concerning the way that his Welsh Corgi Elvis demonstrates mathematical principles is receiving national recognition from the Mathematical Association of America (MAA) for its effectiveness in teaching about calculus.

The MAA is presenting Pennings and his co-author Dr. Roland Minton of Roanoke College with its George Polya Award for their article "Do Dogs Know Bifurcations?" They will receive the recognition during the association's annual summer meeting, MathFest, being held in Madison, Wis., on Thursday-Saturday, July 31-Aug. 2.

Pennings, who is a professor of mathematics at Hope, and Elvis have also been invited to present the paper during the meeting, which is geared toward mathematicians, professors, undergraduate students, graduate students, high school teachers and others who enjoy mathematics.

The George Polya Award is given for articles of expository excellence published in the "College Mathematics Journal," and includes a prize of up to $500. Published in the journal's November 2007 edition, the article by Pennings and Minton, who is author of a popular calculus textbook, considers how Elvis responds in retrieving a ball when he is in the water and the ball is thrown into the water down shore. The citation that the MAA has prepared in conjunction with the award praises the article for its accessible presentation of calculus principles.

"This paper gives an engaging presentation of a nice optimization problem that calculus students can easily visualize and understand," the citation notes. "It presents some lovely intuitive arguments that confirm the mathematical conclusions and a limiting case of the problem that gives rise to a pictorial proof of the arithmetic mean-geometric mean inequality."

Pennings appreciates that the award is named in honor of Dr. George Polya (1887-1985), who he noted is renowned among mathematicians for thinking about the creative discovery process and trying to help students learn how to do mathematics.

"I am grateful and honored to receive an award bearing the name of George Polya. Professor Polya spent his career not only doing mathematics, but also thinking about how we do mathematics," he said. "Thus, there is a pleasant irony in the fact that Elvis shows that even dogs have instinctive creativity and problem-solving skills."

The article examines the options available to Elvis in retrieving the ball in the water and how their effectiveness can be analyzed mathematically.

One possibility would be to swim directly for the ball. Another possibility would be to swim to shore, run along the shoreline to get closer to the ball and then to swim out to it. Which route is optimal - defined as the fastest - depends on a variety of factors, including the distance from shore and the distance to the ball. The point at which one strategy or the other should be employed is called the bifurcation point, and can be calculated. The calculations, Minton and Pennings, reflect choices that Elvis makes instinctively.

Pennings specializes in mathematical modeling, the way in which mathematics describes natural phenomena, and has been sharing his experiences with Elvis as a living example for several years. His original explorations focused on how Elvis responded when standing at the shoreline when a ball was thrown diagonally into the water, determining the optimal combination of running some of the distance along the beach and then swimming on an angle to reach the ball.

They have shared their experience with a variety of audiences as an accessible way of highlighting serious mathematical concepts, including by speaking at area schools. They have been featured by multiple news organizations and in professional publications, including the BBC and as the cover story of the May 2003 issue of "The College Mathematics Journal." In April 2007 Pennings discussed his work with Elvis and the topic at a National Science Foundation-funded conference at DukeUniversity, and in November of this past year he and Elvis were featured keynote speakers during the annual "Research by Undergraduates in Mathematics Boston University Symposium."

Founded in 1915, the MAA is a forum for educators, students, professionals, and mathematics enthusiasts to share ideas, keep abreast of developments in the mathematical community, enhance their careers and make new friends. The organization's membership includes more than 25,000 individuals and institutions nationally.

**Links:**

[1] http://www.hope.edu/2008/06/05/article-elvis-dog-and-calculus-wins-award