My emphasis on research in mathematics education has focused on two areas: the use of technology (graphing calculators, CAS, etc..) and the introduction of concept tests. I have written and administered several Louisiana Board of Regents Support Fund grants to provide Centenary classrooms with technology (including Maple, a dedicated computer lab, and graphing calculator equipment). You can find a paper showing some of the software I developed in Matlab for our multivariable calculus class (back before the days of built-in GUIs).
I have also created a set of concept tests (multiple choice questions that test mathematical concepts rather than calculation) following the example of Eric Mazur. You can read a paper on my use of conceptests, download my collection of 89 conceptests for multivariable calculus, or visit Eric Mazur's site at galileo.harvard.edu to see conceptests from other fields. I also have a paper in the upcoming MAA volume on concept tests.
My most recent research has been work with students in combinatorial game theory. Jeffery James (Centenary 09) and I wrote and published a paper on variants of nim where a player cannot repeat the last move. You can find the paper in the games section of volume 8 of the online journal Integers. The sequence of Grundy numbers we found is in the Online Encyclopedia of Integer Sequences.
I also worked on the game Snort with Stacey Stokes (Centenary 08) and obtained some preliminary results on games of Snort on cycles. I am currently working with Mark Goadrich (our computer science professor) on analysis of the game Babylon.
My field of specialization in graduate school was model theory, a branch of mathematical logic. Basically, model theory determines which objects follow a specific set of rules (or axioms). More specifically, I've have looked at results from classification theory and stability theory and explored how those results can be extended when looking at permutation groups of objects (that is, the group of automorphisms of an object). I have several results in this area, including:
Sona sand drawings are drawings created by the Chokwe people of Africa with some interesting mathematical properties. (A good introduction to them can be found in Paulus Gerde's book Geometry from Africa: Mathematical and Educational Explorations.) These types of drawings can be seen as a subset of figures known as mirror curves. (You can find a definition and examples at Slavik Jablan's page at http://members.tripod.com/~modularity/mir.htm.) I published one paper in Mathematics Magazine called "Permutations in the Sand" and another article "How to Create Monolinear Mirror Curves" in the online journal Visual Mathematics.
About a decade ago, I was looking for a mathematical way to rank teams in the NFL. I came across an article by James Keener (The Perron-Frobenius Theorem and the Ranking of Football Teams, SIAM Review, Vol. 35, No. 1, pp 80-93, March 1993) and used what he calls the direct method (basically finding the principal eigenvector of a nonnegative matrix by using the power method). You can find more details in this short essay I wrote.
The mathematics of this method bring up some interesting questions involving matrix algebra and graph theory such as:
For more wide-ranging information on sports ranking and mathematics, I recommend this article by Ivars Peterson and this site explaining the theory behind some other ranking systems.
With Ken Aizawa, I wrote a paper on the ealry mathematical work of Walter Pitts and how that fed into the groundbreaking paper "A Logical Calculus". You can find a link to the article at Synthese.
Based on my work at the Viewpoints conference, I wrote a paper for the College Mathematics Journal entitled "How to View a Flatland Painting". You can find an illustrative webpage with Geometer's Sketchpad applets that explains the results.
I've done some work explaining the debunking of the Bible Code results. You can look at the slide show I've presented to Centenary undergraduates.