In the first chapter Brownell critically examines the psychological bases of the three most common theories of arithmetic instruction: drill, incidental learning, and meaning. In chapter 2 the results of a nation-wide survey of actual teaching practices are reported. Chapter 3 presents a contrast between "informational arithmetic" and "computational arithmetic." In chapter 4 "social utility" is defined much more broadly than just "computationally useful," and implications for arithmetic instruction are discussed. Following this is a survey of opportunities for use of arithmetic in an activity program, with specific examples. Chapter 6 discusses practices in the teaching of fractions and decimals, followed by the report of a study on transfer by Overman. Reported in chapter 7 are the results of a survey on current practices in teacher-training courses in arithmetic. Two chapters are devoted to transfer of training in arithmetic and types of drill; using these as a basis, David Eugene Smith expounds on the past, present, and future of instruction in arithmetic. In chapter 11 arithmetic is considered from a mathematical viewpoint as contrasted to a pedagogical viewpoint. Next Gestalt psychology is discussed, with implications for mathematics teaching. The last chapter compares the efficiency of different methods for division. (LS)