In this paper, I analyze the impact of culture and metaphor on cryptology education. I will compare and contrast the historically grounded metaphors of cryptology-is-warfare and encryption-is-security to a set of counter-metaphors: cryptology-is-privacy and encryption-is-communication. Using this explicit understanding of conceptual metaphor, I…

In the precalculus curriculum, logistic growth generally appears in either a discrete or continuous setting. These actually feature distinct versions of logistic growth, and textbooks rarely provide exposure to both. In this paper, we show how each approach can be improved by incorporating an aspect of the other, based on a little known synthesis…

Creativity plays an important role in mathematics, but it can be overlooked in undergraduate courses. This article describes three ways an existing first-year math seminar was modified to increase its focus on creativity: introducing readings and discussions related to creativity, drawing connections to creativity in other disciplines, and adding…

The purpose of this article is to share a new approach for introducing students to the definition and standard examples of Abelian groups. The definition of an Abelian group is revised to include six axioms. A bullseye provides a way to visualize elementary examples and non-examples of Abelian groups. An activity based on the game of Jenga is used…

In this paper we present a short tutorial on creating mini-videos using two platforms--PCs and tablets such as iPads--and software packages that work with these devices. Specifically, we describe the step-by-step process of creating and editing videos using a Wacom Intuos pen-tablet plus Camtasia software on a PC platform and using the software…

Linear algebra students are typically introduced to the problem of how to convert from one coordinate system to another in a very abstract way. Often, two bases for a given vector space are provided, and students are taught how to determine a transition matrix to be used for changing coordinates. If students are successful in memorizing this…

In this article, we share a model of flipped instruction that allowed us to gain a window into our students' mathematical thinking. We depict how that increased awareness of student thinking shaped our mathematics instruction in productive ways. Drawing on our experiences with students in our own classrooms, we show how flipped instruction can be…

We provide in-depth information and analysis on three activities for use in History of Mathematics courses taught either in a traditional semester format for undergraduates or in a summer professional development course for middle school teachers. These activities require students to be active participants in their own learning. They also…

The method of variation of parameters can be found in most undergraduate textbooks on differential equations. The method leads to solutions of the non-homogeneous equation of the form y = u[subscript 1]y[subscript 1] + u[subscript 2]y[subscript 2], a sum of function products using solutions to the homogeneous equation y[subscript 1] and…

In this article, I outline how integral equations and Green's functions are used in the courses that I teach, and how I believe my students benefit from exposure to these topics.

Based on the Lobatto quadrature, we develop several new closed form approximations to the mathematical constant "e." For validating effectiveness of our approximations, a comparison of our results to the existing approximations is also presented. Another objective of our work is to inspire students to formulate other better approximations by using…

For over a decade it has been a common observation that a "fog" passes over the course in linear algebra once abstract vector spaces are presented. See [2, 3]. We show how this fog may be cleared by having the students translate "abstract" vector-space problems to isomorphic "concrete" settings, solve the "concrete" problem either by hand or with…

Preliminary report of the results of a project to introduce peer instruction into a multi-section first semester calculus course taught largely by novice instructors. This paper summarizes the instructional approaches instructors chose to use, and the subsequent results of student performance on common exams throughout the course of the term.…

This article addresses the enhancement of the learning environment by using Java applets in the mathematics classroom. Currently, the first year mathematics program at the United States Military Academy involves one semester of modeling with discrete dynamical systems (DDS). Several faculty members from the Academy have integrated Java applets…