Young children's representations and strategies for addition

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.

Patterns of knowledge in children's addition

Patterns of conceptual and procedural knowledge of addition were examined in 5- to 8-year-olds (N = 80). Conceptual knowledge was measured by assessing children's responses to problems in which addends were reordered or decomposed and recombined. Problems were presented using abstract symbols, numbers, and physical objects. Children were more successful in noticing that addends had been reordered rather than decomposed and in noticing the decomposition of addends presented with objects rather than with symbols. Distinct profiles of procedural competence were derived from an analysis of children's problem-solving accuracy and strategies. Profiles were associated with patterns of conceptual knowledge and with age, although age and conceptual knowledge were not related. Findings highlight the usefulness of identifying profiles of procedural and conceptual knowledge for understanding the development of children's knowledge of addition.