Children's understanding of addition and subtraction concepts.

After the onset of formal schooling, little is known about the development of children's understanding of the arithmetic concepts of inversion and associativity. On problems of the form a+b-b (e.g., 3+26-26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a+b-c (e.g., 3+27-23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.