Introduction

Theoretical upper division mathematics courses may be interesting in and of themselves, but making them lively can be challenging. This article discusses the use of readings, student projects, and other creative endeavors to enhance understanding and make these courses come alive. A large part of the success of these extensions is providing the opportunity for students to discuss and explain what they have learned or accomplished during the course. Students react enthusiastically to the readings and projects. Upper division courses can become a natural place to introduce modern applications of mathematics. Specific examples of student research presented here are primarily drawn from number theory, discrete structures, group theory, and probability and statistics. Samples of suggested reading lists are provided. This article provides two examples of how the suggestions come together to form a cohesive course in the sections entitled Putting It All Together. Finally, the culmination of a project deserves special attention. At the end of the article is a section on Showcasing Student Accomplishments that presents rewarding opportunities for students to distinguish themselves.

As mathematics departments become more involved with capstone courses for their students, the information in this paper may be helpful in designing a capstone experience for your students. In this seminar each student, under the guidance of a faculty mentor, undertakes an independent project culminating in a presentation in the department. The topic may be suggested by the student or chosen by the mentor.