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Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality--that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting. (Contains 1 figure and 1 table.)

Presented preschoolers and first graders with 3-term inversion problems such as 3 + 2 - 2 and similar standard problems to examine whether children used the inversion principle and if use was based on qualitative identity, length, or quantity. Found that both age groups showed evidence of using inversion in a fully quantitative manner, indicating that this principle is available in some form prior to extensive formal mathematics instruction. (Author/KB)

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.jecp.2005.01.004 Byline: Carmen Rasmussen, Jeffrey Bisanz Keywords: Working memory; Arithmetic; Mental model; Representation Abstract: Working memory has been implicated in the early acquisition of arithmetic skill, but the relations among different components of working memory, performance on different types of arithmetic problems, and development have not been explored. Preschool and Grade 1 children completed measures of phonological, visual-spatial, and central executive working memory, as well as nonverbal and verbal arithmetic problems, some of which included irrelevant information. For preschool children, accuracy was higher on nonverbal problems than on verbal problems, and the best and only unique predictor of performance on the standard nonverbal problems was visual-spatial working memory. This finding is consistent with the view that most preschoolers use a mental model for arithmetic that requires visual-spatial working memory. For Grade 1 children, performance was equivalent on nonverbal and verbal problems, and phonological working memory was the best predictor of performance on standard verbal problems. For both age groups, problems with added irrelevant information were substantially more difficult than standard problems, and in some cases measures of the central executive predicted performance. Assessing performance on different components of working memory in conjunction with different types of arithmetic problems provided new insights into the developing relations between working memory and how children do arithmetic. Author Affiliation: Centre for Research in Child Development, Department of Psychology, University of Alberta, Edmonton, Alta., Canada T6G 2E9 Article History: Received 7 July 2004; Revised 10 January 2005

Children in grades 3 through 5 and adults performed a number-matching task in which they were required to verify the presence of a target digit (i.g., 7) in a previously presented pair (e.g., 2 + 7). For adults, targets that were the sum of the initial pair (e.g., 9) and targets that were close to the initial pair (e.g., 6) were rejected more slowly than unrelated digits. Activation of sums is consistent with models of arithmetic knowledge in which addition facts are stored in a network representation. Activation of close targets (the distance effect) may reflect the counting-based origins of number representations. Consistent with these hypotheses, 8- to 10-year-old children showed minimal effects of sum activation, but large effects of distance. Even at this age, however, distance effects were smaller for subjects with higher levels of arithmetic skill. Thus, changes in the strength of arithmetic connections occurred with development and accounted for individual differences among adults. For children, however, individual differences were related to changes in the strength of number-line connections.