Through the Mathematical Looking Glass
While it is not uncommon for students to find themselves in a class sitting around a large table and discussing a book they have been reading, it is an unusual occurrence in an upper-level mathematics course. Yet, that is exactly what happens in the “History and Philosophy of Mathematics” course taught by Maureen Carroll, Ph.D. Mathematics students are not accustomed to preparing for class by reading a new section of the book in advance. Even mathematics majors expect their professors to decipher the material and cut through the complexity for them. In Dr. Carroll’s History and Philosophy course this paradigm is thrown out. With a discussion-based seminar style format, the class chronicles 12 great theorems and their originators within their historical context. The format has proved successful, sparking many lively discussions and debates.
For a mathematician accustomed to lecturing, abandoning this traditional format was no easy task. To help prepare for the course, Dr. Carroll participated in the National Science Foundation-sponsored Institute in the History of Mathematics and Its Use in Teaching where she worked with the leading mathematics historians in the country. She designed this unconventional mathematics course as a way to share this expertise with her students. She has found the course to be a great way to convey the art and the humanity of mathematics while delving into some very challenging proofs and problems. Dr. Carroll feels, “the vastness of the history of the field and the open dialogue with my students can lead us down so many different paths that I learn something new each time I take the journey with a class.”
In addition to inspiring her teaching, Dr. Carroll’s interest in the history of mathematics has also influenced her scholarly pursuits. Her latest paper, coauthored with Steven Dougherty, Ph.D., and David Perkins, Ph.D., explores infinitesimal analysis, an area of mathematics that has existed for about 400 years, but was accepted by the mathematics community only 40 years ago. Dr. Carroll and her coauthors have revisited some well-known problems originally investigated by 17th century mathematicians Gulden, Torricelli and Roberval through the lens of this relatively new nonstandard analysis. One of the problems uses these new techniques to prove that the infinite object known as Gabriel’s Horn has finite volume.
When describing her research, Dr. Carroll noted that she has found new problems worthy of study in the usual ways, by reading a mathematics journal or having a conversation with a colleague, or more uncommonly when watching television or reading the newspaper. Two fields of interest to Dr. Carroll are game theory and geometry. A conversation with a colleague about a two-player game where the player with least amount of money in their wallet wins the other player’s wallet led to a game-theory paper that answered an unsolved problem in the field. She has also combined her interest in geometry with game theory in an article she coauthored with Dr. Dougherty based on the game of tic-tac-toe. They created a new version of the children’s game that can be played on infinitely many geometric planes and analyzed play from a game-theoretic viewpoint to determine when the first player has a winning strategy. In 2005, their article was awarded the Mathematical Association of America’s Merten M. Hasse Prize for a noteworthy paper appearing in an association publication. The high visibility of the paper and the national award translated into multiple opportunities for collaboration and outreach, including invited talks at conferences and colleges, a book chapter, tic-tac-toe contests at Scranton and other schools, and collaboration with a mathematician at Oxford University. The University’s Mathematics Club has taught interested students how to play tic-tac-toe on the affine plane of order four, and held single-elimination tournaments where students play this geometric game for prizes.How can watching television or reading a newspaper lead to mathematics research?
Watching the Olympic figure skating events led to an article that investigated the problem of determining overall rankings from a panel of voters with varying preferences. The resulting article was featured as the cover article in Math Horizons and was picked up by The Chronicle of Higher Education and Science magazine. Additionally, reading a newspaper article on Pennsylvania regulations regarding the installation of a dedicated left-turn signal at an accident-prone intersection inspired another of her articles.
Whether she is discussing her own mathematical scholarship or that of historical figures, Dr. Carroll hopes that by learning about the mathematicians who solved the problems and proved the theorems, her students will be inspired to indulge and develop their own intellectual curiosity and passion for learning.