Mathematical Logic in the Undergraduate Curriculum

Joint Mathematics Meetings
Atlanta, GA, January 5-8, 2005

Wednesday January 5, 2005, 2:15 p.m.-5:05 p.m
AMS-MAA-MER Special Session on Mathematics and Education Reform, II

Mathematical Logic in the Undergraduate Curriculum

Martin D. Davis

Experts in mathematical logic can be found in three di erent academic departments: mathematics, philosophy, and computer science. Some knowledge of technical logic is important for majors in each of these subjects. This creates problems and opportunities. Which topics are important for majors in these di erent subjects? Can courses be devised to serve the needs of all three? These questions will be addressed in this talk.

Logic for Undergraduates at Notre Dame

Julia F. Knight

Notre Dame has a universal requirement of two semesters of mathematics. Students intending to major in science, engineering, or business need at least two semesters of calculus. Arts and Letters students have some other options. One is a logic course. The goal is to give students some formal tools for analyzing arguments. There is an emphasis on formal proofs, and we cover both propositional logic and predicate logic. Such a course is standard in philosophy departments. However, in our course, homework is done in groups, and students also write a group paper, on some topic in logic that is not covered in the lectures.

For mathematics majors at Notre Dame, there are several junior/senior electives. One of these is on various kinds of automata, and what they can and cannot do. Such a course is standard in computer science programs. However, there
are challenging problems that encourage mathematical thinking, and the mathematics majors who choose the course seem to respond well to the material.

Logic as Compass. Preliminary Report

Iraj Kalantari

For most early students of mathematics, deep understanding of intricate mathematical concepts is a challenging charge. Absorbing and reproducing proofs, as well as authoring one's rst set of proofs, because it is a combination of subject and logic, is a major discouragement for many capable but inexperienced students.

At Western Illinois University, where we have about 100 mathematics majors half of whom are pursuing a career in teaching, we meet the students' needs by requiring every mathematics major to take, post Calculus II and before proof embedded courses, a course in logic.

I will describe the history, the nature, the success, and the structure of our course arguing for existence of a similar course in all institutions of higher education including those whose students generally absorb most of the material of such a course implicitly while taking advance mathematics topics.

Discovery Learning in Logic Education

Walker M White

Mathematical logic is often seen as a technical discipline with little relevance to how "mathematics is done". But this is a complaint about logic education -- which often focuses on formal syntax -- and not the end itself. With discovery learning in logic education, students can learn how logic and mathematics relate to their everyday use of language.

In this talk, we show how to introduce discovery learning by focusing on natural language, and deemphasizing formal syntax. Students learn to use natural language to prove theorems and construct models for axiom systems. They also learn how to use nonstandard models as counterexamples, insights into a proof, or the motivation of new axioms.

By itself, however, this is not enough for discovery learning. The structure of the course must motivate and reward this type of learning. We will also dicuss how institute portfolio grading, and how this can encourage students to attempt
challenging and unfamiliar problems.

Throughout the this talk, we will examine the opportunities and challenges that have been encountered in these types
of courses over the past 30 years. We show how it has been successfully adopted at all levels of the curriculum, from
bridge courses for majors to liberal-arts mathematics classes.

Symbolic Logic in a Proofs Course: Finding the Right Balance. Preliminary report

Connie M Campbell

Symbolic logic plays a critical role in a rst course in proof writing (a bridge course) by providing students with some of the fundamental skills necessary for developing logical arguments. However, when teaching a bridge course, the goal is not only to teach students how to read and write mathematical proofs, but also to move them from algorithmic thinking to critical, and even creative, thinking. It has been the experience of the author that this transition can be greatly hindered by either an overemphasis, or an underemphasis, on symbolic logic. She will discuss problems that she has experienced from bridge courses which erred on either of these two extremes, as well as discuss what she has found to be the right balance for her classroom. A course outline and classroom activities will be discussed as well.

Teaching Logic to Prospective Elementary and Middle School Mathematics Teachers. Preliminary Report

Gregory D. Foley

According to renowned mathematics educator Bob Davis, mathematics is what you have left over after you have invented ways to solve problems and re flected on those inventions. With this as his motto, the presenter taught an undergraduate course in the Introduction to the Logic and Structure of Mathematics during Spring Semester, 2004 to a group of prospective teachers of Grades K-8, using a modified R. L. Moore method. Much time was devoted to students presenting their work to the class. The course focused on problem solving, exploring, conjecturing, reasoning, and communicating using number systems and algebraic systems as the content. The talk will focus on how this course helped the pre-service teachers enrolled in the class to learn about logic and mathematical reasoning.