Children's understanding of additive concepts.

Most research on children's arithmetic concepts is based on one concept at a time, limiting the conclusions that can be made about how children's conceptual knowledge of arithmetic develops. This study examined six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and addition and subtraction associativity) in Grades 3, 4, and 5. Identity (a-0=a) and negation (a-a=0) were well understood, followed by moderate understanding of commutativity (a+b=b+a) and inversion (a+b-b=a), with weak understanding of equivalence (a+b+c=a+[b+c]) and associativity (a+b-c=[b-c]+a). Understanding increased across grade only for commutativity and equivalence. Four clusters were found: The Weak Concept cluster understood only identity and negation; the Two-Term Concept cluster also understood commutativity; the Inversion Concept cluster understood identity, negation, and inversion; and the Strong Concept cluster had the strongest understanding of all of the concepts. Grade 3 students tended to be in the Weak and Inversion Concept clusters, Grade 4 students were equally likely to be in any of the clusters, and Grade 5 students were most likely to be in the Two-Term and Strong Concept clusters. The findings of this study highlight that conclusions about the development of arithmetic concepts are highly dependent on which concepts are being assessed and underscore the need for multiple concepts to be investigated at the same time.

Conceptual knowledge of arithmetic for Chinese- and Canadian-educated adults.

This study investigated whether Canadian- and Chinese-educated adults differ in their understanding of simple arithmetic concepts. Participants (n = 21 per group) solved 3-term addition and subtraction (e.g., 5 + 22 - 22 and 3 + 24 - 26) and multiplication and division (e.g., 2 × 28 ÷ 28 and 4 × 39 ÷ 13) problems. All problems could be solved more easily if conceptual knowledge of the relationship between the 2 operations in each problem was understood and applied. Accuracy, solution time, and immediately retrospective self-reports of problem-solving strategy data were collected. Participants also completed a timed arithmetic fluency task. Chinese-educated participants demonstrated stronger conceptual understanding of arithmetic on all problems and outperformed Canadian-educated participants on the fluency task. A cluster analysis revealed 4 groups of individuals: weak concept users, who rarely used conceptual knowledge to aid their problem solving; strong concept users, who almost exclusively used their conceptual knowledge to facilitate problem solving; addition and subtraction concept users, who frequently used conceptual knowledge except on difficult multiplication and division problems; and multiplication and division concept users, who frequently used conceptual knowledge except on difficult addition and subtraction problems. Chinese-educated participants were more likely to be in the strong concept clusters, and none were in the weak concept cluster, providing further evidence of stronger conceptual knowledge of arithmetic. These results demonstrate for the 1st time that there are strong cross-cultural differences in conceptual knowledge of simple arithmetic, even in adulthood. (PsycINFO Database Record