This handbook is intended to contribute to the development of a theory of remedial instruction that can be easily understood and assimilated by teachers. The document evolved from research and workshop activity over a period of several years. It is designed for use by those with instructional or supervisory reponsibilities in elementary schools, grades K-5. It is divided into three distinct and quite different sections. The first section describes four competencies consistently demonstrated by effective mathematics teachers in these grade levels. Those competencies are: (1) knowledge of the content and structure of mathematics; (2) knowledge of cognitive/developmental theories; (3) knowledge of and ability to diagnose students' errors; and (4) knowledge of and ability to select instructional strategies appropriate for reteaching mathematics skills and concepts. The second section is devoted to the diagnosis of specific errors frequently encountered in grades K-5. In addition, the development of strategies to effectively remediate these errors is included. The handbook concludes with an extensive primer on the concept of the training of trainers. (PK)