This book is meant to describe innovative approaches that have been used successfully by a variety of instructors in the undergraduate mathematics courses that follow calculus. These approaches are designed to make upper division mathematics courses more interesting, more attractive, and more beneficial to our students. The authors of the articles in this volume show how this can be done while still teaching mathematics courses. These approaches range from various classroom techniques to novel presentations of material to discussing topics not normally encountered in the typical mathematics curriculum.

One overriding goal of all of these articles is to encourage students to stretch their mathematical boundaries. This stretching can be done in a variety of ways but there is one common theme; students expand their horizons not merely by sitting and listening to lectures but by doing mathematics.

This book can be used by:

- Instructors seeking new ways to approach the courses they teach
- Individuals looking for ideas to incorporate into a specific course
- Teachers who want to expose their students to current mathematical activity

This book is meant for the instructor. It will be very useful to anyone teaching a course beyond first year calculus. These would include: abstract algebra, applied mathematics, biostatistics, differential equations, linear algebra, mathematical biology, module theory, multivariable calculus, number theory, probability, real analysis, statistics and topology. Also, three capstone courses are mentioned and there are interdisciplinary applications cited that involve biology, computer science, economics, engineering, physics and the social sciences.

### Table of Contents

**1. Papers Covering Several Courses**

Introduction

1.1 Using Writing and Speaking to Enhance Mathematics Courses, *Nadine Myers*

1.2 Enhancing the Curriculum Using Reading, Writing, and Creative Projects, *Agnes Rash*

1.3 How to Develop an ILAP, *Michael Huber and Joseph Myers*

1.4 The Role of the History of Mathematics in Courses Beyond Calculus,* Herbert Kasube*

1.5 A Proofs Course That Addresses Student Transition to Advanced Applied Mathematics Courses, *Michael Jones and Arup Mukherjee*

**2. Course-Specific Papers**

Introduction

2.1 Wrestling with Finite Groups; Abstract Algebra need not be a Passive Sport,* Jason Douma*

2.2 Making the Epsilons Matter, *Stephen Abbott*

2.3 Innovative Possibilities for Undergraduate Topology, *Samuel Smith*

2.4 A Project Based Geometry Course*, Jeff Connor and Barbara Grove*

2.5 Discovering Abstract Algebra: A Constructivist Approach to Module Theory,* Jill Dietz*

**3. Papers on Special Topics**

Introduction

3.1 The Importance of Projects in Applied Statistics Courses,* Timothy O’Brien*

3.2 Mathematical Biology Taught to a Mixed Audience at the Sophomore Level,* Janet Andersen*

3.3 A Geometric Approach to Voting Theory for Mathematics Majors, *Tommy Ratliff*

3.4 Integrating Combinatorics, Geometry, and Probability Through the Shapley-Shubik Power Index,* Matthew Haines and Michael Jones*

3.5 An Innovative Approach to Post-Calculus Classical Applied Math, *Robert Lopez*

About the Editor

### About the Editor

**Richard Maher** received his PhD from Loyola University, Chicago. He served as department chair at Loyola from 1977 through 1985. His initial mathematical research involved the theory of lifting, and he has published a number of papers on this subject appearing in *Advances in Mathematics, Les Annales de L’Institut Fourier, the Journal of Mathematical Analysis and Applications, Rendiconti del Circolo Matematico Di Palermo*, and *Studies in Probability and Ergodic Theory*. During his tour as chair, Professor Maher became interested in problems affecting undergraduate mathematical education, particularly those involving effective classroom methods. He has written a number of papers on this issue appearing in the *Notices (AMS), CBMS Issues in Mathematics Education, UME Trends*, and the MAA Notes series. He is also the author of the Calculus text, *Beginning Calculus with Applications*.

Teaching is the critical component of his professional activity, and for the past fifteen years, it has fueled his scholarly activity and interest.