| The goal of this paper is to provide an account of the development of multiplication skill and knowledge. I examined age-related changes and individual differences for Grade 4 and 6 children in (a) the computational skills used to solve simple multiplication problems including the procedures used and indices thought to reflect the representation of facts in memory, (b) concepts important for understanding multiplication, and (c) working memory important for mathematics. The use of multiple measures allowed for the opportunity to provide an integrated account of the development of multiplication including evaluating the developing relations between different areas important for multiplication. The performance of 60 children from Grade 4 and 60 children from Grade 6 was evaluated using a number of tasks. Specifically, children solved 28 multiplication problems. Accuracy and latency were recorded, as were immediately retrospective self-reports on how the problem was solved. As well, children solved a number of problems based on the following concepts: (a) commutativity, (b) relations between repeated addition and multiplication, (c) part-whole relations, (d) relative effects of operations on numbers, (e) relative magnitudes of numbers, (f) concepts important for solving word problems, and (g) relations between concrete manipulatives and symbolic representations of specific problems. Working memory was assessed using a backward digit span and an operation span task. Finally, children's performance on a test of mathematical achievement was evaluated. In terms of simple multiplication I found that: children use multiple procedures to solve multiplication problems, problem characteristics are important in predicting solution latencies, children use specific procedures on specific types of problems, and variables important for predicting solution latencies change with age. I found that conceptual understanding in multiplication is influenced by the context in which it is assessed, and that the development of concepts is uneven. Specifically, it appears that concepts which can be directly applied to solve problems are acquired first, followed by concepts used to enable solving novel problems. In terms of working memory, it appears that mathematical achievement is related to domain-specific working memory in Grade 4 but global working memory in Grade 6. Implication of the results are discussed in terms of the appropriate assessment of children's performance, models of performance in multiplication, principles of development, and the development course in the acquisition of multiplication skill and knowledge. |
