The author discusses major issues related to the teaching of elementary school mathematics and draws upon research results for his analysis. The elementary school mathematics curriculum is discussed in terms of selection of content, educational objectives and content arrangement. Within the realm of mathematics learning, discovery learning, the use of manipulative materials, student attitudes, the use of practical applications, and the need for real understanding in mathematics are examined. The author discusses the organization of classes and pupils for instruction. In this area he examines grouping practices for gifted and remedial students both within classes and school-wide as well as the usage of mathematics specialists in the classroom. Finally, specific questions regarding teaching practices are considered including computer-assisted instruction (CAI), programmed learning, and the effects of the "new mathematics" programs. (Author/CT)