The "Preparing Elementary Teachers to Teach Mathematics (PETTM) Project" (1988-1991) was a cooperative university/school effort to improve the teaching of mathematics by elementary teachers with its primary focus on improving the university training of preservice teachers (PSTs) in mathematics. This document, volume III of the project's 5-volume final report, describes the "Methods Component" of the project. The report is divided into four sections: an introductory section providing the background and rationale for the project and a description of its four components, and three sections describing respectively the early field experience associated with the mathematics methods course, the Math Tutorial Program designed to give volunteer PSTs the opportunity to gain additional experience with children while refining their teaching skills, and the design, construction, and utilization of the problem data bank developed for the course. Publications and presentations related to the methods component and a description of related instructional products are provided. (MDH)