This document is a revision of the 1961 report of the same title, taking into account the many changes which have occurred in school mathematics in the past decade. Four objectives of mathematics training are identified and discussed: (1) understanding of concepts and structure; (2) facility with applications; (3) ability to solve problems; and (4) development of computational skills. Specific course recommendations are made at four levels: (I) elementary school teachers (grades K-6); (II-E) specialist mathematics teachers, coordinators and middle school mathematics teachers (grades 5-8); (II-J) junior high school mathematics teachers (grades 7-9); and (III) high school mathematics teachers (grades 7-12). Appendices give details of the content of the proposed courses, including alternative sequences for Level I. Special features of these courses are an emphasis on mathematics as a unified subject (no separate courses in algebra or geometry are recommended for Level I); a constant reference to the importance of the applications of mathematics; the use of flow charts and computers at various levels; and a second course in geometry for high school teachers, using vectors and transformations. (MM)