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Descriptors:
Intelligence; Mathematics Achievement; Numbers; Grade 6; Grade 7; Short Term Memory; Mathematics Skills; Predictor Variables; Achievement Gains; Mathematical Concepts; Computation; Mathematics Instruction

Abstract:
Competence with fractions predicts later mathematics achievement, but the codevelopmental pattern between fractions knowledge and mathematics achievement is not well understood. We assessed this codevelopment through examination of the cross-lagged relation between a measure of conceptual knowledge of fractions and mathematics achievement in sixth and seventh grades (N = 212). The cross-lagged effects indicated that performance on the sixth grade fractions concepts measure predicted 1-year gains in mathematics achievement (beta = 0.14, p less than 0.01), controlling for the central executive component of working memory and intelligence, but sixth grade mathematics achievement did not predict gains on the fractions concepts measure (beta = 0.03, p greater than 0.50). In a follow-up assessment, we demonstrated that measures of fluency with computational fractions significantly predicted seventh grade mathematics achievement above and beyond the influence of fluency in computational whole number arithmetic, performance on number fluency and number line tasks, central executive span, and intelligence. Results provide empirical support for the hypothesis that competence with fractions underlies, in part, subsequent gains in mathematics achievement. (Contains 1 table and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Intelligence; Mathematics Achievement; Short Term Memory; Grade 4; Grade 1; Problem Solving; Longitudinal Studies; Children; Cognitive Processes; Executive Function; Elementary School Mathematics; Addition

Abstract:
Children's (N = 275) use of retrieval, decomposition (e.g., 7 = 4+3 and thus 6+7 = 6+4+3), and counting to solve additional problems was longitudinally assessed from first grade to fourth grade, and intelligence, working memory, and in-class attentive behavior was assessed in one or several grades. The goal was to assess the relation between capacity of the central executive component of working memory, controlling for intelligence and in-class attentive behavior, and grade-related changes in children's use of these strategies. The predictor on intercept effects from multilevel models revealed that children with higher central executive capacity correctly retrieved more facts and used the most sophisticated counting procedure more frequently and accurately than their lower capacity peers at the beginning of first grade, but the predictor on slope effects indicated that this advantage disappeared (retrieval) or declined in importance (counting) from first grade to fourth grade. The predictor on slope effects also revealed that from first grade to fourth grade, children with higher capacity adopted the decomposition strategy more quickly than other children. The results remained robust with controls for children's sex, race, school site, speed of encoding Arabic numerals and articulating number words, and mathematics achievement in kindergarten. The results also revealed that intelligence and in-class attentive behavior independently contributed to children's strategy development. (Contains 3 tables and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Reaction Time; Grades (Scholastic); Mathematics Achievement; Short Term Memory; Long Term Memory; Grade 2; Grade 1; Arithmetic; Gender Differences; Problem Solving; Elementary School Mathematics

Abstract:
The ability to retrieve basic arithmetic facts from long-term memory contributes to individual and perhaps sex differences in mathematics achievement. The current study tracked the codevelopment of preference for using retrieval over other strategies to solve single-digit addition problems, independent of accuracy, and skilled use of retrieval (i.e., accuracy and reaction time [RT]) from first to sixth grades inclusive (N = 311). Accurate retrieval in first grade was related to working memory capacity and intelligence, and it predicted a preference for retrieval in second grade. In later grades, the relation between skill and preference changed such that preference in one grade predicted accuracy and RT in the next grade as RT and accuracy continued to predict future gains in preference. In comparison with girls, boys had a consistent preference for retrieval over other strategies and had faster retrieval speeds, but the sex difference in retrieval accuracy varied across grades. Results indicate that ability influences early skilled retrieval, but both practice and skill influence each other in a feedback loop later in development and provide insights into the source of the sex difference in problem-solving approaches. (Contains 3 tables and 4 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Learning Disabilities; Mathematics Achievement; Achievement Tests; Short Term Memory; Reading Ability; Nonverbal Ability; Arithmetic; Low Achievement; Scores; Identification; Elementary School Students; Cognitive Processes; Verbal Ability

Abstract:
Using 4 years of mathematics achievement scores, groups of typically achieving children (n = 101) and low achieving children with mild (LA-mild fact retrieval; n = 97) and severe (LA-severe fact retrieval; n = 18) fact retrieval deficits and mathematically learning disabled children (MLD; n = 15) were identified. Multilevel models contrasted developing retrieval competence from second to fourth grade with developing competence in executing arithmetic procedures, in fluency of processing quantities represented by Arabic numerals and sets of objects, and in representing quantity on a number line. The retrieval deficits of LA-severe fact retrieval children were at least as debilitating as those of the children with MLD and showed less across-grade improvement. The deficits were characterized by the retrieval of counting string associates while attempting to remember addition facts, suggesting poor inhibition of irrelevant information during the retrieval process. This suggests a very specific form of working memory deficit, one that is not captured by many typically used working memory tasks. Moreover, these deficits were not related to procedural competence or performance on the other mathematical tasks, nor were they related to verbal or nonverbal intelligence, reading ability, or speed of processing, nor would they be identifiable with standard untimed mathematics achievement tests. (Contains 6 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Intelligence; Mathematics Education; Reading Achievement; Mathematics Achievement; Short Term Memory; Grade 5; Grade 1; Longitudinal Studies; Anxiety; Problem Solving; Prediction; Elementary School Mathematics

Abstract:
The study's goal was to identify the beginning of 1st grade quantitative competencies that predict mathematics achievement start point and growth through 5th grade. Measures of number, counting, and arithmetic competencies were administered in early 1st grade and used to predict mathematics achievement through 5th (n = 177), while controlling for intelligence, working memory, and processing speed. Multilevel models revealed intelligence and processing speed, and the central executive component of working memory predicted achievement or achievement growth in mathematics and, as a contrast domain, word reading. The phonological loop was uniquely predictive of word reading and the visuospatial sketch pad of mathematics. Early fluency in processing and manipulating numerical set size and Arabic numerals, accurate use of sophisticated counting procedures for solving addition problems, and accuracy in making placements on a mathematical number line were uniquely predictive of mathematics achievement. Use of memory-based processes to solve addition problems predicted mathematics and reading achievement but in different ways. The results identify the early quantitative competencies that uniquely contribute to mathematics learning. (Contains 6 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Brain; Problem Solving; Children; Memory; Arithmetic; Computation; Cognitive Development; Multivariate Analysis

Abstract:
Cognitive development and learning are characterized by diminished reliance on effortful procedures and increased use of memory-based problem solving. Here we identify the neural correlates of this strategy shift in 7-9-year-old children at an important developmental period for arithmetic skill acquisition. Univariate and multivariate approaches were used to contrast brain responses between two groups of children who relied primarily on either retrieval or procedural counting strategies. Children who used retrieval strategies showed greater responses in the left ventrolateral prefrontal cortex; notably, this was the only brain region which showed univariate differences in signal intensity between the two groups. In contrast, multivariate analysis revealed distinct multivoxel activity patterns in bilateral hippocampus, posterior parietal cortex and left ventrolateral prefrontal cortex regions between the two groups. These results demonstrate that retrieval and counting strategies during early learning are characterized by distinct patterns of activity in a distributed network of brain regions involved in arithmetic problem solving and controlled retrieval of arithmetic facts. Our findings suggest that the reorganization and refinement of neural activity patterns in multiple brain regions plays a dominant role in the transition to memory-based arithmetic problem solving. Our findings further demonstrate how multivariate approaches can provide novel insights into fine-scale developmental changes in the brain. More generally, our study illustrates how brain imaging and developmental research can be integrated to investigate fundamental aspects of neurocognitive development. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Mathematics Education; Nonverbal Ability; Cognitive Ability; Short Term Memory; Grade 1; Elementary School Students; Computation; Word Problems (Mathematics); Language Skills; Attention; Problem Solving; Listening

Abstract:
The purpose of this study was to examine the interplay between basic numerical cognition and domain-general abilities (such as working memory) in explaining school mathematics learning. First graders (N = 280; mean age = 5.77 years) were assessed on 2 types of basic numerical cognition, 8 domain-general abilities, procedural calculations, and word problems in fall and then reassessed on procedural calculations and word problems in spring. Development was indexed by latent change scores, and the interplay between numerical and domain-general abilities was analyzed by multiple regression. Results suggest that the development of different types of formal school mathematics depends on different constellations of numerical versus general cognitive abilities. When controlling for 8 domain-general abilities, both aspects of basic numerical cognition were uniquely predictive of procedural calculations and word problems development. Yet, for procedural calculations development, the additional amount of variance explained by the set of domain-general abilities was not significant, and only counting span was uniquely predictive. By contrast, for word problems development, the set of domain-general abilities did provide additional explanatory value, accounting for about the same amount of variance as the basic numerical cognition variables. Language, attentive behavior, nonverbal problem solving, and listening span were uniquely predictive. (Contains 4 footnotes, 5 tables, and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Evolution; Psychologists; Psychology; Misconceptions; Individual Differences; Comparative Analysis; Hypothesis Testing; Models; Homosexuality; Birth; Correlation

Abstract:
Comments on Evolutionary psychology: Controversies, questions, prospects, and limitations by Confer et al. We applaud Confer et al.'s (February-March 2010) clarifications of the many misconceptions surrounding the use of evolutionary analyses in psychology. As they noted, such misunderstandings are common and result in a curious tendency of some of our colleagues to criticize evolutionary psychology without a firm understanding of evolution itself. Confer et al. also did an admirable job acknowledging current unresolved issues among evolutionary psychologists (e.g., the relative importance of group selection on humans). The above said, we disagree with their view that a current limitation of evolutionary psychology is its inability to explain phenomena "that appear to reduce an individual's reproductive success, and cannot be explained by mismatches with, or hijacking of, our psychological mechanisms by modern-day novel inputs" (Confer et al., 2010, p. 122). Mismatches between modern environments and environments of evolutionary adaptedness are only one set of explanations for seemingly maladaptive traits (Nesse, 2005). Another set involves evolutionary trade-offs. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Mathematics; Problem Solving; Young Children; Elementary Education; Instructional Effectiveness; Educational Practices; Guides; Mathematical Concepts; Comprehension; Cognitive Processes

Abstract:
This practice guide presents five recommendations intended to help educators improve students' understanding of, and problem-solving success with, fractions. Recommendations progress from proposals for how to build rudimentary understanding of fractions in young children; to ideas for helping older children understand the meaning of fractions and computations that involve fractions; to proposals intended to help students apply their understanding of fractions to solve problems involving ratios, rates, and proportions. Improving students' learning about fractions will require teachers' mastery of the subject and their ability to help students master it; therefore, a recommendation regarding teacher education also is included. Appendices include: (1) Postscript from the Institute of Education Sciences; (2) About the Authors; (3) Disclosure of Potential Conflicts of Interest; (4) Rationale for Evidence Ratings; and (5) Evidence Heuristic. A glossary and index are also provided. (Contains 5 tables, 10 figures, and 250 endnotes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Numeracy; School Readiness; Short Term Memory; Concept Formation; Grade 1; Cognitive Ability; Word Problems (Mathematics); Prediction; Mathematical Concepts; Mathematics Skills; Elementary School Students

Abstract:
Contributions of domain-general and domain-specific numerical competencies were assessed on first graders' number combination skill (NC) and word-problem skill (WP). Students (n = 205) between 5 and 7 years of age were assessed on 2 aspects of numerosity, 8 domain-general abilities, NC, and WP. Both aspects of numerosity predicted NC when controlling for domain-general abilities, but domain-general abilities did not account for significant additional variance. By contrast, when controlling for domain-general abilities in predicting WP, only precise representation of small quantities was uniquely predictive, and domain-general measures accounted for significant additional variance; central executive component of working memory and concept formation were uniquely predictive. Results suggest that development of NC and WP depends on different constellations of numerical versus more general cognitive abilities. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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