Research on the psychological processes involved in early school arithmetic has now accumulated sufficiently to make it possible to construct a coherent account of the changing nature of the child's understanding of number during the early school years. This monograph presents an account of how number concepts are extended and elaborated as a result of formal instruction. A theory of number representation is outlined for three broad periods of development: (1) the preschool period, during which counting and quantity comparison competencies of young children provide the main basis for inferring number representation; (2) the early primary period, during which children's invention of sophisticated mental computational procedures and the mastery of certain forms of story problems point to two important expansions of the number concept; and (3) the later primary period, during which the representation of number is modified to reflect knowledge of the decimal structure of the counting and notational systems. (MNS)