Moore is better

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Sep 2003 by Brewer, J Patrick

ABSTRACT: This article describes a sophomore transition-to-advanced-math course taught via a modified Moore method at Lebanon Valley College.

KEYWORDS: Moore method, active learning.

INTRODUCTION

R. L. Moore developed his teaching method in the teens and twenties of the last century to train future mathematicians, and it was wonderfully successful, producing many doctorates and influencing generations of mathematicians. In its original and perhaps most familiar form, an instructor gives her students a list of definitions and theorems on the first day of a class, and the students spend the rest of the classes presenting proofs of the theorems while the instructor observes and offers guidance only rarely. In that form the Moore method is probably seldom appropriate for undergraduate courses, but modified versions of it can be tailored to match the goals of many undergraduate classes. This article describes one such modification. For more information about the Moore method and its history, please see [1] and [2].

At Lebanon Valley College our transition-to-advanced-math course was originally pitched to sophomores and designed to help them make the transition from computation-based courses to proof-based courses. To that end the students studied proof techniques and then applied them to fundamental mathematical ideas such as functions, relations, and cardinality. The goals are unchanged, but in the last five years, we have shifted the emphasis from the content to the process. A modified-Moore approach allows us to concentrate more on reading and writing. Before continuing with a more detailed description of our course, I first discuss Carol Schumacher's textbook Chapter Zero [3].

CHAPTER ZERO

If you have considered teaching a (modified) Moore method class or some flavor of active learning course but decided against it because of the work required to produce the course materials, then Chapter Zero may be a solution. It includes the topics common to most transition texts but in a unique setting. Where most authors provide examples after a definition and proofs after theorems, Professor Schumacher leads students to construct their own examples and to create their own proofs. She includes enough motivation and discussion to make the book accessible to sophomores while providing a wonderful framework for active learning.

OUR COURSE

What follows is a brief, then more detailed, description of the latest version of Foundations, our sophomore transition course.

At the end of each class period, I give a homework assignment (usually from the book but sometimes supplemented); at the beginning of the next class period students turn in their homework and take a short quiz based on that material. After the quiz, students begin to discuss the assignment. Then I announce what problems are available for presentation credit and ask for volunteers. We vary from that procedure for exams but that is the way the course usually runs. By design, I am rarely at the front of the room; the most valuable communication occurs among the students.

Grading

The course grade is based on homework (5%), class participation (10%), quizzes (20%), 3 exams (45%), and a cumulative final (20%).

Homework

If the students engage the material between classes, the course will probably be successful; if they do not, they will have nothing to say during the following class period, and the course can easily degenerate into a hesitant, stumbling lecture format. To encourage students to wrestle with the material before class, I assign, collect, and grade homework daily. Everything in the book that is not already complete is an exercise, so the homework from the text may consist of examples, problems, proofs, etc. Compared to more traditional transition courses, some may conclude that we cover less material, but only if you measure coverage by counting pages. If you consider the amount of work that students are doing, the coverage is quite similar. In a modified Moore course the instructor does not read the text to the students; they first address the material themselves. Surely that is as important as the number of pages.

At this stage of student development, and especially given the novel approach, I believe frequent feedback is vital. Prom each assignment I choose two problems to grade carefully, and I try to focus on their writing. To maximize their benefit from the homework, students must be able to make mistakes without significant grade consequence, so the homework is worth only 5% of their grade, thus the impact of any one assignment is epsilon.

In our department we have learned that unless we collect and grade homework, most of our students will not engage the material between classes, but that is not the case everywhere. For example, at Kenyon College, Carol Schumacher relies less on homework than we do and her students excel, so you have to tailor your course to your students.

Participation

Participation includes asking and answering questions and presentations. The grade weighs them equally.