ABSTRACT
Two contrasting ways of thinking about relations among quantities are reflected in the operations of addition and subtraction; and multiplication and division and the contrast between the two hinges on differences in the way units are used. This chapter discusses a number of profound differences between additive relations and multiplicative relations. It considers the major findings concerning the development of additive reasoning and of multiplicative reasoning, respectively. The discussion of additive reasoning focuses on children's understanding of the part-whole relations that are the conceptual core of both addition and subtraction. The main goal of the discussion of multiplicative reasoning is to clarify the profound conceptual differences between it and additive reasoning and at the same time to illuminate the conceptual connections between the two. The chapter concludes with a brief discussion of the developmental and instructional ramifications of understanding the relation between additive and multiplicative reasoning in terms of different ways of thinking about relations between quantities.
