Knot Theory for Preservice and Practicing Secondary Mathematics Teachers

Project Summary

This proof-of-concept project addresses two sets of needs in mathematics teacher education. Following recommendations of the Conference Board on Mathematical Sciences, the project aims to develop curriculum in know theory which accomplishes two goals. First, that teachers "learn mathematics in a coherent fashion that emphasizes the interconnections among theory, procedures and applications," and second, that teachers "develop the habits of mind of a mathematical thinker" (CBMS, 2000). In addition, the project aims to test these materials in a distance education environment and in a face-to-face classroom setting. A comparison of discourse in these two settings will contribute to the knowledge base on distance education.

Specific goals of the project include

- the development and evaluation of curriculum modules in knot theory to be used in two contexts, Internet-based instruction and face-to-face classroom interaction,

- a comparison of learning and discourse between Internet-based instruction and face-to-face classroom interaction,
- research that addresses the mathematical knowledge and the beliefs about mathematics of preservice and practicing secondary mathematics teachers, and

- the development of a web site to make the curriculum materials available to other mathematics teacher educators.

Since knot theory draws on connections with diverse types of mathematics, this module will integrate well into courses on geometry, topology, combinatorics, discrete mathematics, or graph theory. The curriculum module we are proposing is based on the state-sum, or "lake and island" picture of knots proposed by Kauffman (1988). Java applications will be developed to illustrate the Jones polynomial in a dynamic and visual context.