Volume 24, Issue 4, Pages 339-494 (2009)
Atypical Development of Numerical Cognition
Edited by Ann Dowker and Liane Kaufmann
Atypical development of numerical cognition: Characteristics of developmental dyscalculia
Ann Dowker, Liane Kaufmann
Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching?
Karen C. Fuson
This article provides an overview of some perspectives about special issues in classroom mathematical teaching and learning that have stemmed from the huge explosion of research in children's mathematical thinking stimulated by Piaget. It concentrates on issues that are particularly important for less-advanced learners and for those who might be having special difficulties in learning mathematics. A major goal of the article is to develop a framework for understanding what effective mathematics teaching and learning is, because doing so is so important for struggling students and for research about them. Piaget's research had a fundamental influence on the on-going tension between understanding and fluency in the classroom, supporting efforts toward increasing understanding. But in some countries, misinterpretations of Piaget led to practices that are counterproductive for children, especially struggling learners. Such misinterpretations are identified and a more balanced approach that also draws on Vygotsky is described—a learning-path developmentally-appropriate learning/teaching approach.
Co-occurrence of developmental disorders: The case of Developmental Dyscalculia
Five to seven percent of children experience severe difficulties in learning mathematics and/or reading. Current trials that are focused on identifying biological markers suggest that these learning disabilities, known as Developmental Dyscalculia (DD) and Dyslexia (for reading), are due to underlying brain dysfunctions. One ongoing controversy concerns the extent to which arithmetic impairments are specific to DD or shared with other developmental disorders such as Dyslexia. This review explores and develops a hypothesis for cases of DD + Dyslexia. Three factors warrant consideration: (a) the behavioral factor, including definitions of the disabilities and assessment tools; (b) the cognitive factor, including whether co-occurrence of DD and other developmental disorders such as Dyslexia derive from similar or different cognitive risk factors; (c) the biological factor, including consideration of static vs. developmental neuropsychology. Better understanding of the causes of co-occurrence of DD and Dyslexia, or other developmental disorder such as Attention Deficit Hyperactivity Disorder (ADHD), can have an important influence on research that examine the two disorders, including research on therapy and etiology.
Basic number processing deficits in developmental dyscalculia: Evidence from eye tracking
K. Moeller, S. Neuburger, L. Kaufmann, K. Landerl, H.-C. Nuerk
Recent research suggests that developmental dyscalculia is associated with a subitizing deficit (i.e., the inability to quickly enumerate small sets of up to 3 objects). However, the nature of this deficit has not previously been investigated. In the present study the eye-tracking methodology was employed to clarify whether (a) the subitizing deficit of two boys with dyscalculia resulted from a general slowing in the access to magnitude representation, or (b) children with dyscalculia resort to a back-up counting strategy even for small object sets. In a dot-counting task, a standard problem size effect for the number of fixations required to encode the presented numerosity within the subitizing range was observed. Together with the finding that problem size had no impact on the average fixation duration, this result suggested that children with dyscalculia may indeed have to count, while typically developing controls are able to enumerate the number of dots in parallel, i.e., subitize. Implications for the understanding of developmental dyscalculia are considered.
Numerical distance effect in developmental dyscalculia
Sarit Ashkenazi, Nitza Mark-Zigdon, Avishai Henik
Children in third and fourth grades suffering from developmental dyscalculia (DD) and typically developing children were asked to compare numbers to a standard. In two separate blocks, they were asked to compare a number between 1 and 9 to 5, or a two-digit number between 10 and 99 to 55. In the single-digit comparisons, DD children were comparable to the control group in reaction time but showed a difference in error rates. In the two-digit number comparisons, DD children presented a larger distance effect than the controls. In addition, they were more influenced by the problem size than controls were. Assuming an analog representation of quantities, this suggests that quantities are less differentiated in those with DD than in typically developing children.
Use of derived fact strategies by children with mathematical difficulties
339 children aged 6 and 7 at Oxford primary schools took part in a study of arithmetic. 204 of the children had been selected by their teachers as having mathematical difficulties and the other 135 children were unselected. They were assigned to an Addition Performance Level on the basis of a calculation pretest, and then given Dowker's (1998) test of derived fact strategies in addition, involving strategies based on the Identity, Commutativity, Addend + 1, Addend - 1, and addition/subtraction Inverse principles. The exact arithmetic problems given varied according to the child's previously assessed calculation level and were selected to be just a little too difficult for the child to solve unaided. The technique was used of giving children the answer to a problem and then asking them to solve another problem that could be solved quickly by using this answer, together with the principle under consideration. The children were also given the WISC Arithmetic subtest and the British Abilities Scales Basic Number Skills Subtest. Performance on the standardized arithmetic tests was independently affected by both Addition Performance Level and group membership (unselected children versus those with mathematical difficulties). Derived fact strategy use was affected by Addition Performance Level, but there was no independent effect of group membership.
First-grade predictors of mathematical learning disability: A latent class trajectory analysis
David C. Geary, Drew H. Bailey, Andrew Littlefield, Phillip Wood, Mary K. Hoard, Lara Nugent
Kindergarten to third grade mathematics achievement scores from a prospective study of mathematical development (n = 306) were subjected to latent growth trajectory analyses. The four corresponding classes included children with mathematical learning disability (MLD, 6% of sample), and low (LA, 50%), typically (TA, 39%) and high (HA, 5%) achieving children. The groups were administered a battery of intelligence (IQ), working memory, and mathematical-cognition measures in first grade. The children with MLD had general deficits in working memory and IQ and potentially more specific deficits on measures of number sense. The LA children did not have working memory or IQ deficits but showed moderate deficits on these number sense measures and for addition fact retrieval. The distinguishing features of the HA children were a strong visuospatial working memory, a strong number sense, and frequent use of memory-based processes to solve addition problems. Implications for the early identification of children at risk for poor mathematics achievement are considered.
The trajectory of mathematics skills and working memory thresholds in girls with fragile X syndrome
Melissa M. Murphy, Michèle M.M. Mazzocco
Fragile X syndrome is a common genetic disorder associated with executive function deficits and poor mathematics achievement. In the present study, we examined changes in math performance during the elementary and middle school years in girls with fragile X syndrome, changes in the working memory loads under which children could complete a cognitive switching task, and the association between these two areas of function, in girls with fragile X syndrome relative to their peers. Our findings indicate that the trajectory of math and executive function skills of girls with fragile X differs from that of their peers and that these skills contribute to predicting math achievement and growth in math performance over time. Also, changes in math performance were associated with incremental increases in working memory demands, suggesting that girls with fragile X have a lower threshold for being able to perform under increasing task demands. Still, we found improvement in executive function performance between 10 and 12 years in girls with fragile X rather than a performance plateau as has been reported in other studies. The findings implicate the importance of early intervention in mathematics for girls with fragile X that addresses poor calculation skills, the supporting numerical skills, and deficits in executive functions, including working memory.
Computer-assisted intervention for children with low numeracy skills
Pekka Räsänen, Jonna Salminen, Anna J. Wilson, Pirjo Aunio, Stanislas Dehaene
We present results of a computer-assisted intervention (CAI) study on number skills in kindergarten children. Children with low numeracy skill (n = 30) were randomly allocated to two treatment groups. The first group played a computer game (The Number Race) which emphasized numerical comparison and was designed to train number sense, while the other group played a game (Graphogame-Math) which emphasized small sets of exact numerosities by training matching of verbal labels to visual patterns and number symbols. Both groups participated in a daily intervention session for three weeks. Children's performance in verbal counting, number comparison, object counting, arithmetic, and a control task (rapid serial naming) were measured before and after the intervention. Both interventions improved children's skills in number comparison, compared to a group of typically performing children (n = 30), but not in other areas of number skills. These findings, together with a review of earlier computer-assisted intervention studies, provide guidance for future work on CAI aiming to boost numeracy development of low performing children.
An electro-physiological temporal principal component analysis of processing stages of number comparison in developmental dyscalculia
Fruzsina Soltész, Dénes Szucs
Developmental dyscalculia (DD) still lacks a generally accepted definition. A major problem is that the cognitive component processes contributing to arithmetic performance are still poorly defined. By a reanalysis of our previous event-related brain potential (ERP) data (Soltész et al., 2007) here our objective was to identify and compare cognitive processes in adolescents with DD and in matched control participants in one-digit number comparison. To this end we used temporal principal component analysis (PCA) on ERP data. First, PCA has identified four major components explaining the 85.8% of the variance in number comparison. Second, the ERP correlate of the most frequently used marker of the so-called magnitude representation, the numerical distance effect, was intact in DD during all processing stages identified by PCA. Third, hemispheric differences in the first temporal component and group differences in the second temporal component suggest executive control differences between DD and controls.
Numerical and non-numerical ordinality processing in children with and without developmental dyscalculia: Evidence from fMRI
L. Kaufmann, S.E. Vogel, M. Starke, C. Kremser, M. Schocke
Ordinality is – beyond numerical magnitude (i.e., quantity) – an important characteristic of the number system. There is converging empirical evidence that (intra)parietal brain regions mediate number magnitude processing. Furthermore, recent findings suggest that the human intraparietal sulcus (IPS) supports magnitude and ordinality in a domain-general way. However, the latter findings are derived from adult studies and with respect to children (i.e., developing brain systems) both the neural correlates of ordinality processing and the precise role of the IPS (domain-general vs. domain-specific) in ordinality processing are thus far unknown. The present study aims at filling this gap by employing functional magnetic resonance imaging (fMRI) to investigate numerical and non-numerical ordinality knowledge in children with and without developmental dyscalculia. In children (without DD) processing of numerical and non-numerical ordinality alike is supported by (intra)parietal cortex, thus extending the notion of a domain-general (intra)parietal cortex to developing brain systems. Moreover, activation extents in response to numerical ordinality processing differ significantly between children with and without dyscalculia in inferior parietal regions (supramarginal gyrus and IPS).
Technorati Tags: psychology, school psychology, educational psychology, neuropsychology, cognition, cognitive abilities, intelligence, mathematics, dyscalculia, number sense, numerical cognition, special education, LD, learning disabilities