There has long been dispute in mathematics between the drill and practice orientation that focuses primarily on memorizing mathematics as meaningless rote algorithms and the approach based on understanding and making creative use of mathematics. This 25-chapter book, based on a 7-year study at the University of Illinois, seeks to explain and diffuse this controversy by taking a broad view of the cognitive science approach to the teaching and learning of mathematics. This explanation of the processes of mathematical calculation owes much to the new approach to the study of knowledge, which has been developed from Jean Piaget's observation of child behavior and the broad range of new research into artificial intelligence. It is essentially concerned with providing a deeper understanding of the thought processes that are involved in mathematical thinking; what goes on inside children's heads as they learn mathematics and do mathematical problem-solving. Among the areas considered are: the cognitive science approach to mathematics education; deficiency in typical school curricula; the nature of representations; the "paradigm" teaching strategy; recognition problems; retrieval, construction, and mapping; and basic concepts used to facilitate the discussion of human information processing as it related to solving mathematical problems. (JN)