Later Conceptual Relationships Between Counting and Cardinality: Addition and Subtraction of Cardinal Numbers

Abstract

The focus of this chapter is the developmental progression of relationships between count meanings and cardinality meanings of number words that extends from the earliest and simplest situations of addition and subtraction through the solution of the most complex addition and subtraction situations with large multidigit whole numbers.¹ Articulating this progression requires us to consider changes in children’s representations of three differentiable aspects of addition and subtraction operations: (1) the addition or subtraction situation, (2) the solution procedure used to carry out the addition or subtraction operation, and (3) the relationships between count and cardinal meanings that are required by the solution procedure used. We hypothesize that each of these moves through a developmental sequence determined by the representational ability of the child. This representational ability shifts from a dependence on perceptually available entities to progressive internalization and abstraction of these entities. The key representational change at each developmental level involves either a change in the kind of unit item the child is able to conceptualize or in the complexity of the structure that the unit item can represent (an addend, a sum, or both simultaneously).