A mathematics colloquium for sophomores

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Sep 2001 by Brabenec, Robert L

A MATHEMATICS COLLOQUIUM

FOR SOPHOMORES*

ABSTRACT: At Wheaton College, we introduced a department colloquium in 1997 as a required one hour course for the mathematics major. The purpose is to introduce our sophomore majors to the experiences of mathematical exploration, research, problem-solving, group work, and oral presentation in an informal setting of about 15 students and department faculty. This paper contains details for five different colloquium courses, along with a description of two openended problems we have used, and an assessment of the value of this experience.

KEYWORDS: Pythagorean triplets, muptiply perfect numbers, department colloquium

INTRODUCTION

We are always looking for ways to enrich the experience of our students as mathematics majors. Competitions, such as the Putnam Exam or the Mathematics Contest in Modeling (MCM), offer one valuable approach. Another involves doing mathematics in a different setting, such as the Budapest Semester or one of the NSF Summer Research Experiences for Undergraduates (REU). A third approach is to encourage the presentation of research results in a venue such as the national meeting of a mathematical organization that provides space for student papers. At Wheaton College, we have also used a supplementary reading program for many years [1]. We have since incorporated this program into our mathematics courses instead of treating it as a stand-alone requirement, and have been more satisfied with the results.

This paper deals with a different idea, namely the use of the department colloquium. Ours has assumed a variety of forms over a period of many years. In the late 1980s and early 1990s, we sponsored regular biweekly lectures and experienced good student attendance on a voluntary basis. But then attendance dropped to nearly zero, as the colloquium was forced to compete with an increasing number of extra-curricular activities. So in 1997, we introduced the department colloquium as a 1 hour required course for the major. It emphasizes such reform initiatives as the use of history, problem-solving, cooperative learning, applications, and technology. It meets for one hour a week on Thursday afternoons. We encourage department faculty to attend and interact with students in this informal setting (we meet in a seminar room around an oval arrangement of tables with snack food available). We typically have 10-15 students enrolled each semester.

While there is a place for the senior seminar, we want our students to take this requirement during their sophomore year, so they can have an early exposure to some of its unique positive experiences, such as the openended exploration of a problem, research in the literature or on the web, and the excitement of sharing new results with others. We have chosen to deemphasize grades, and want our students to remember this as a class in which they had fun and yet learned much about mathematics. One method I have tried is to guarantee a minimum grade of B to any student who attends every class and does the assigned work. There are no exams. A grade deduction is made for every absence. We have offered the course five times, with a different theme each time, partly for the benefit of faculty, so there will always be something new. Students may also take the course more than once for credit under this format. I will describe the five course themes we have used, present details of two successful open-ended problems, and conclude with some general comments.

FIVE POSSIBLE COLLOQUIUM TOPICS

Our first offering was based on William Dunham's book Journey Through Genius [4]. The format was largely discussion, based on a sheet of assigned questions that students were given each week. As a sample, the following questions were used for the chapter on set theory and Georg Cantor [4, pp. 245-266].

1. Describe the general nature of mathematics in the 19th century. How did it differ from the previous century? What changes occurred in the 20th century?

2. Discuss the comment about art and mathematics running parallel courses. Ask your art professor for his or her opinion.

3. What role did non-Euclidean geometry play in this development of mathematics in the nineteenth century?

4. What was meant by Dunham's comment on page 245 about "a deeper analysis of the logical foundations of mathematics that underlay the wonderful theories of Newton, Leibniz and Euler"?

5. Describe the family of Georg Cantor.

6. Provide support for the statement that there are more rational numbers than natural numbers. Next provide support for the statement that there are not more rational numbers than natural numbers.

7. Describe two different ways to decide whether one set is larger than another.

8. How did Cantor's view of infinity differ from that of his predecessors?

9. Describe the diagonalization process used by Cantor, and also tell what its purpose was.

10. Discuss the concept, including examples, of infinite cardinal numbers.

11. Discuss the nature of algebraic numbers and transcendental numbers.