The Daily Bulletin Archive

By Mary A. Durlak

Daniel W. Cunningham, associate professor of mathematics, spends his time exploring the mathematical universe. During his fall 2008 sabbatical, he discovered a truth within that universe, and prepared a paper about his research that is slated for publication in the Notre Dame Journal of Formal Logic. He also completed a draft of a mathematics textbook for undergraduates.

Cunningham came to Buffalo State as an assistant professor of mathematics in 1991 after earning his Ph.D. at the University of California, Los Angeles. “I’m an L.A. native who loves Buffalo,” he said.

A personal search for truth brought him, by way of logic and philosophy, to the study of pure mathematics. There he found a universe that is no less real for being a universe of the mind, a universe that offers new discoveries without end, and a universe in which Pilate’s famous question, “What is truth?” can be answered with certainty.

Cunningham’s sabbatical discovery is something called a “covering lemma” by mathematicians. With this new information, Cunningham, whose research interest is set theory, adds to the description of a particular smaller universe within the mathematical universe. This smaller universe, according to Cunningham, is the playful universe, in which games take place. The nature of this playful universe is revealed by who wins and who loses the games.

This discovery is based in part on his own research, which was published earlier in the Annals of Pure and Applied Logic and Archive for Mathematical Logic. Given that the name of his latest paper is “A Covering Lemma for HOD of K(R),” it is apparent that details of his discovery can only be shared with fellow explorers who speak the language.

Cunningham hopes that the textbook he has written will help undergraduates become fluent in that language. “I love teaching,” he said. “It’s fun to explain these things.” Over the last 20 years, however, he has observed that students seem to have more difficulty making the transition from lower-division to higher-division mathematics.

“To be successful,” Cunningham has written, “students must possess three essential skills: the ability to read, to understand, and to communicate in the language of mathematics.” The purpose of his text is to help students acquire those skills, and to construct mathematical proofs.

Mathematical proofs and theorems go hand in hand. Most students learn the Pythagorean Theorem in high school. Cunningham explained that a theorem is a mathematical statement that has been verified to be true by a mathematical proof. One of the key components in Cunningham’s textbook is the development of a methodology to lay bare the structure underpinning the construction of a proof, much as diagramming a sentence lays bare its grammatical structure. His classroom experience indicates that this strategy helps students develop the skills to write mathematical proofs, a prerequisite to advanced mathematics.

In fact, the earliest drafts of the textbook came into being as Cunningham taught his own classes. His dedication to teaching is demonstrated not only by the courses he has developed and revised, and the Ph.D. candidates he has inspired, but also in his willingness to mentor honor students. He has also presented a talk to junior high school students, “Every Function Tells a Story.”

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Read previous Focus on Sabbatical stories:

Felix Armfield
Betty Cappella
Ann Colley
Michael De Marco
Rob Delprino
Mark Fulk
Musa Abdul Hakim
Katherine Hartman
David Henry
Andrew Nicholls
Wendy Paterson
M. Stephen Pendleton
Stephen Phelps
John Song
Carol Townsend
Jonathan Thornton
Mark Warford
Michael Zborowski