Using Lattice Theory to Promote Effective Communication of Mathematics

Bin Lu and Valerie Watts

University of Arizona

During the 2002-03 academic year at the University of Arizona, we ran a weekly undergraduate seminar on topics from lattice theory, meeting with students for one hour each week. Our goals for the seminar were threefold:

• To create an alternative learning environment as a complement to traditional courses offered by the Department of Mathematics;
• To develop students’ problem solving-skills and abilities to study math independently;
• To effectively communicate math verbally and in writing.

We achieved these objectives through close interaction between students and faculty, regular presentations by students and individual student projects.

We divided the seminar into two phases: studying the basics of lattice theory in the fall semester and working on special topics/projects in the spring semester. During the fall semester, the students gave lectures and worked problems individually and in groups to learn the basics of order and lattice theory. The lectures students gave included introducing concepts, giving examples and presenting proofs of theorems. During the spring semester, the students were assigned a special topic, such as formal concept analysis and representation theory, from lattice theory and then gave presentations and prepared a written report on their topic.

Our student participants were seniors and juniors with majors ranging from mathematics to physics and minors in biology and computer science. They had limited experience with mathematical proofs, in courses such as discrete mathematics and elementary linear algebra, but not in courses such as Introduction to Proof and Real Analysis.

Because of the background of the students, choosing the right topic was essential to the success of the seminar. Lattice theory does not require a lot of background material (linear algebra and discrete math are sufficient prerequisites) and was accessible to the students, even those who had not taken Introduction to Proof. Furthermore, the theory of order and lattices has applications to areas including logic, discrete mathematics and computer science. Since many math students are interested in applications of mathematics, we felt that lattice theory was an appropriate topic for this group of students.

One of our primary goals was for students to gain experience in effectively communicating mathematics. The effects of the seminar on achieving this goal are reflected in comments from the students:

· “I learned a lot, it is not only about learning how to be organized but get used to speaking in front of an audience;”

· “This is a good experience for me! The most valuable skill I have gleaned from it so far is the ability to present mathematical ideas to a group of people.”

However, despite the success of the seminar, there are areas that we feel can be improved and adjusted. These include the following:

· Clearer directions for presentations and discussion of the presentation with each student prior to their lecture;

· Provide some background on each topic and summarization after each topic was presented to motivate students’ learning and aid in their understanding;

· Indicate connections and interplays among different areas of mathematics and lattice theory.