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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.)

Children's failure on equivalence problems (e.g., 5 + 4 = 7 + __) is believed to be the result of misunderstanding the equal sign and has been tested using symbolic problems (including "="). For Study 1 (N = 48), we designed a nonsymbolic method for presenting equivalence problems to determine whether Grade 2 children's difficulty is due to the presence of symbols or to a more fundamental misunderstanding of equivalence. Children's superior performance on nonsymbolic versus symbolic problems suggests that children fail to map their understanding of equivalence onto problems presented with the symbols of arithmetic. For Study 2 (N = 32), we implemented a within-subject design to assess whether experience with nonsymbolic problems would facilitate performance on symbolic problems. This hypothesis was confirmed. Exposure to nonsymbolic problems may have enabled children to map their successful concepts and strategies to symbolic equivalence problems. (Contains 6 figures and 1 table.)

Knowledge and skill in multiplication were investigated for late elementary-grade students with mathematics learning disabilities (MLD), typically achieving age-matched peers, low-achieving age-matched peers, and ability-matched peers by examining multiple measures of computational skill, working memory, and conceptual knowledge. Poor multiplication fact mastery and calculation fluency and general working memory discriminated children with MLD from typically achieving age-matched peers. Furthermore, children with MLD were slower in executing backup procedures than typically achieving age-matched peers. The performance of children with MLD on multiple measures of multiplication skill and knowledge was most similar to that of ability-matched younger children. MLD may be due to difficulties in computational skills and working memory. Implications for the diagnosis and remediation of MLD are discussed. (Contains 6 tables and 3 notes.)

Investigates performance of 8, 10, 12 year olds and adults on cognitive tasks in terms of several processing-speed measures, each of which may change independently with age. Results underscore the complexity of developmental change in processing efficiency. (Author/RH)

A recognition memory experiment with 8-, 11- and 20-year-olds investigated the hypothesis that, with age, semantic encoding becomes increasingly important relative to acoustic encoding. (CM)

The goal of this project was to examine the profile of executive function (EF) deficits and age-related differences among children with FASD. Twenty-nine children with FASD (8 to 16 years of age) completed 8 tests from the Delis-Kaplan Executive Function System (D-KEFS). They had difficulty on many components of EF including cognitive flexibility, inhibition, some measures of verbal fluency, abstract thinking, deductive reasoning, hypothesis testing, problem solving, and concept formation. A distinctive profile emerged with performance being poorest on the card sorting test and relatively high on category fluency, design fluency, and the tower test, indicating relative strengths on some visual-spatial EF tasks. Older children with FASD showed more difficulty (relative to the norm) on some verbal tests of EF than younger children with FASD, suggesting that difficulty on some verbal EF tasks appears to become more pronounced with increasing age. [ABSTRACT FROM AUTHOR]

The principle of inversion, that a + b - b must equal a, is a fundamental property of conventional arithmetic. Exploring how children use and understand the principle of inversion can provide important insights about the development of mathematical thinking and about ways of optimizing instruction. Research on children's use and understanding of inversion has been focused primarily on whether they use inversion, with much less attention placed on what this understanding comprises and how it develops. To remedy this situation, we propose a framework in which understanding inversion is represented in terms of a matrix of possibilities. This framework is useful for highlighting the diverse ways in which children can show their understanding, for describing individual differences, for tracking changes in understanding, and for prompting investigations on the mechanisms that contribute to conceptual development. [ABSTRACT FROM AUTHOR]

Age-related change and patterns of individual differences in children's knowledge and skill in multiplication were investigated for students in Grades 4 and 6 (approximately ages 9 and 11, respectively) by examining multiple measures of computational skill, conceptual knowledge, and working memory. Regression analyses revealed that indexes reflecting probability of retrieval and special problem characteristics overshadow other, more general indexes (problem size and frequency of presentation) in predicting solution latencies. Some improvement in the use of conceptual knowledge was evident between Grades 4 and 6, but this change was neither strong nor uniform across tasks. Finally, patterns of individual differences across tasks differed as a function of grade level. The findings have implications for understanding developmental change and individual differences in mathematical cognition. [ABSTRACT FROM AUTHOR]

Abstract: We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in children’s acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e., speed of solutions to simple equations such as 3+4 and 8−5). Children in Grades 2 and 3 (N =157) completed executive attention and mathematical tasks. They repeated the mathematical tasks 1year later. We used structural equation modeling to examine the relations between executive attention and (a) concurrent measures of mathematical knowledge and arithmetic fluency and (b) growth in performance on these measures 1year later. Executive attention was concurrently predictive of both knowledge and fluency but predicted growth in performance only for fluency. A composite language measure predicted growth in knowledge from Grade 2 to Grade 3. The results support an important role for executive attention in children’s acquisition of novel procedures and the development of automatic access to arithmetic facts. [Copyright &y& Elsevier]