Education

Abstract

The Education section of this issue features two very different articles. The first is a statistical analysis of a venerable problem in academia: Can one determine the aptitude of a student and the difficulty of a course from the grades that are assigned? The topic and mathematics will be familiar to most readers. The second article is an exposition on Zeon algebra and its connection to the calculus of real functions. For many readers, this will be a first introduction to Zeon algebra and its connections to combinatorial identities. Readers may recall Robert Vanderbei's Education article on global warming two years ago. In “A Regression Approach to Fairer Grading,” author Robert Vanderbei returns to the Education section with co-authors Gordon Scharf and Daniel Marlow to tackle issues surrounding grade inflation. Their article explores two different models of grading that combine student aptitude and course difficulty, least squares and least absolute deviation. The authors deeply explore each issue on a small, illustrative data set, and then apply both methodologies to two semesters' worth of real encoded data from a small university. Both techniques separate student aptitude and course-level variations in grading. This self-contained article is appropriate for advanced undergraduates, graduate students, and instructors interested in statistics. However, the application itself is relevant to most institutions tackling issues of grade inflation and grade compression. The second article, “Zeon Algebra and Combinatorial Identities,” is an introduction to Zeon algebra, an associative algebra with the property that the squares of its generators are zero. The key idea is that since there are a finite number of generators, infinite combinations of products of elements become finite. Then one can define a real vector space over the Zeon algebra and move on to define a Grassman--Berezin integral or the Z-integral. The authors survey recent results recovering Stirling numbers, Euler, Fibonacci and Genocchi numbers, and a host of other identities all using Z-integral representations. Similar to the extension of the calculus of a single real variable to functions of a complex variable, the authors present this material as an extension of calculus to a Zeon algebra to obtain new generalizations and new representations of combinatorial identities. This module is accessible to graduate students and instructors interested in algebra, analysis, or combinatorics.