Understanding arithmetic concepts: Does operation matter?

- Most research on arithmetic concepts focuses on one concept and one operation at a time.
- Grade 5, 6, and 7 children's understanding of additive and multiplicative versions of six concepts was examined.
- Identity and negation were strong, followed by commutativity, equivalence, inversion, with associativity the weakest.
- Understanding of additive concepts was better than of multiplicative concepts.
- There were no grade differences in conceptual knowledge but there were marked individual differences.

Most research on children's arithmetic concepts is based on (a) additive concepts and (b) a single concept leading to possible limitations in current understanding about how children's knowledge of arithmetic concepts develops. In this study, both additive and multiplicative versions of six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and associativity) were investigated in Grades 5, 6, and 7. The multiplicative versions of the concepts were more weakly understood. No grade-related differences were found in conceptual knowledge, but older children were more accurate problem solvers. Individual differences were examined through cluster analyses. All children had a solid understanding of identity and negation. Some children had a strong understanding of all the concepts, both additive and multiplicative; some had a good understanding of equivalence or commutativity; and others had a weak understanding of commutativity, equivalence, inversion, and associativity. Associativity was the most difficult concept for all clusters. Grade did not predict cluster membership. Overall, these results demonstrate the breadth of individual variability in conceptual knowledge of arithmetic as well as the complexity in how children's understanding of arithmetic concepts develops.