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41
Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability
 Child Development
, 2007
"... Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n 5 15) and low achieving (LA, n 5 44) were identified. These groups and a group of typically achieving (TA, n 5 46) children were administered a ..."
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Cited by 16 (5 self)
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Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n 5 15) and low achieving (LA, n 5 44) were identified. These groups and a group of typically achieving (TA, n 5 46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M 5 6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed. Diagnostic criteria and thus the percentage of children with a learning disability in mathematics (MLD)
Promoting broad and stable improvements in lowincome children’s numerical knowledge through playing number board games
 Child Development
, 2008
"... Theoretical analyses of the development of numerical representations suggest that playing linear number board games should enhance young children’s numerical knowledge. Consistent with this prediction, playing such a game for roughly 1 hr increased lowincome preschoolers ’ (mean age 5 5.4 years) pr ..."
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Cited by 13 (4 self)
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Theoretical analyses of the development of numerical representations suggest that playing linear number board games should enhance young children’s numerical knowledge. Consistent with this prediction, playing such a game for roughly 1 hr increased lowincome preschoolers ’ (mean age 5 5.4 years) proficiency on 4 diverse numerical tasks: numerical magnitude comparison, number line estimation, counting, and numeral identification. The gains remained 9 weeks later. Classmates who played an identical game, except for the squares varying in color rather than number, did not improve on any measure. Also as predicted, home experience playing number board games correlated positively with numerical knowledge. Thus, playing number board games with children from lowincome backgrounds may increase their numerical knowledge at the outset of school. Children vary greatly in the mathematical knowledge they possess when they enter school. These differences in initial mathematical knowledge appear to have large, longterm consequences. Proficiency in mathematics at the beginning of kindergarten is strongly predictive of mathematics achievement test scores years later: in elementary school, in middle school, and even in high school (Duncan et al., 2007; Stevenson & Newman, 1986). This pattern is consistent with the general finding that initial knowledge is positively related to learning (Bransford, Brown, & Cocking, 1999), but the relations in math are unusually strong and persistent. For example, they were considerably stronger than the relations between initial and subsequent reading proficiency in the same six longitudinal studies reviewed by Duncan et al. (2007; average standardized beta coefficients of.34 vs..16). Given the strong and persistent relation between early and later mathematical proficiency, it is especially unfortunate that preschoolers and kindergartners from lowincome families enter school with far less numerical knowledge than peers from more affluent families. Being clear on the locus of this gap is crucial for understanding it. On nonverbal numerical
Sex differences in intrinsic aptitude for mathematics and science? A critical review
 American Psychologist
, 2005
"... for assistance, and Nora Newcombe and Elliott Blass for advice and comments on the manuscript. Above all, I am grateful to Ariel Grace and Kristin Shutts for their unending support and afterhours labor on this project. Draft, 4/20/05. This paper has not yet been peer reviewed. Please do not copy or ..."
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Cited by 11 (2 self)
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for assistance, and Nora Newcombe and Elliott Blass for advice and comments on the manuscript. Above all, I am grateful to Ariel Grace and Kristin Shutts for their unending support and afterhours labor on this project. Draft, 4/20/05. This paper has not yet been peer reviewed. Please do not copy or cite without author's permission. This report considers three prominent claims that boys and men have greater natural aptitude for highlevel careers in mathematics and science. According to the first claim, males are more focused on objects and mechanical systems from the beginning of life. According to the second claim, males have a profile of spatial and numerical abilities that predisposes them to greater aptitude in mathematics. According to the third claim, males show greater variability in mathematical aptitude, yielding a preponderance of males at the upper end of the distribution of mathematical talent. Research on cognitive development in human infants and preschool children, and research on cognitive performance by students at all levels, provides evidence against these claims. Mathematical and scientific reasoning develop from a set of biologically based capacities that males and females share. From these capacities, men and women appear to develop equal talent for mathematics and science.
Bootstrapping the mind: Analogical processes and symbol systems
 COGNITIVE SCIENCE
, 2010
"... Human cognition is striking in its brilliance and its adaptability. How do we get that way? How do we move from the nearly helpless state of infants to the cognitive proficiency that characterizes adults? In this paper I argue, first, that analogical ability is the key factor in our prodigious capac ..."
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Cited by 9 (1 self)
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Human cognition is striking in its brilliance and its adaptability. How do we get that way? How do we move from the nearly helpless state of infants to the cognitive proficiency that characterizes adults? In this paper I argue, first, that analogical ability is the key factor in our prodigious capacity, and, second, that possession of a symbol system is crucial to the full expression of analogical ability.
An evolutionary perspective on learning disability in mathematics
 Developmental Neuropsychology
"... A distinction between potentially evolved, or biologicallyprimary forms of cognition, and the culturallyspecific, or biologicallysecondary forms of cognition that are built from primary systems is used to explore mathematical learning disability (MLD). Using this model, MLD could result from defi ..."
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Cited by 6 (4 self)
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A distinction between potentially evolved, or biologicallyprimary forms of cognition, and the culturallyspecific, or biologicallysecondary forms of cognition that are built from primary systems is used to explore mathematical learning disability (MLD). Using this model, MLD could result from deficits in the brain and cognitive systems that support biologicallyprimary mathematical competencies, or from the brain and cognitive systems that support the modification of primary systems for the creation of secondary knowledge and secondary cognitive competencies. The former include visuospatial longterm and working memory and the intraparietal sulcus, whereas the latter include the central executive component of working memory and the anterior cingulate cortex and lateral prefrontal cortex. Different forms of MLD are discussed as related to each of the cognitive and brain systems. When viewed from the lens of evolution and human cultural history, it is not a coincidence that public schools are a recent phenomenon and emerge only in societies in which technological, scientific, commercial (e.g., banking, interest) and other evolutionarilynovel advances influence one’s ability to function in the society (Geary, 2002, 2007). From this perspective, one goal of academic learning is to acquire knowledge that is important for social or occupational functioning in the culture in which schools are situated, and learning disabilities (LD) represent impediments to the learning of one or several aspects of this culturallyimportant knowledge. It terms of understanding the brain and cognitive systems that support academic learning and contribute to learning disabilities, evolutionary and historical perspectives may not be necessary, but may nonetheless provide a means to approach these issues from different levels of analysis. I illustrate this approach for
Moving along the number line: Operational momentum in nonsymbolic arithmetic. manuscript submitted for publication
, 2006
"... Can human adults perform arithmetic operations with large approximate numbers, and what effect, if any, does an internal spatial–numerical representation of numerical magnitude have on their responses? We conducted a psychophysical study in which subjects viewed several hundred short videos of sets ..."
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Cited by 6 (4 self)
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Can human adults perform arithmetic operations with large approximate numbers, and what effect, if any, does an internal spatial–numerical representation of numerical magnitude have on their responses? We conducted a psychophysical study in which subjects viewed several hundred short videos of sets of objects being added or subtracted from one another and judged whether the final numerosity was correct or incorrect. Over a wide range of possible outcomes, the subjects ’ responses peaked at the approximate location of the true numerical outcome and gradually tapered off as a function of the ratio of the true and proposed outcomes (Weber’s law). Furthermore, an operational momentum effect was observed, whereby addition problems were overestimated and subtraction problems were underestimated. The results show that approximate arithmetic operates according to precise quantitative rules, perhaps analogous to those characterizing movement on an internal continuum. Human adults possess an ability to estimate and manipulate approximate numerical magnitudes, which has been termed number sense (Dehaene, 1997). This ability appears to be largely independent of language and other symbol systems, since it is present in both infants (Xu & Spelke, 2000) and other animal species (Brannon & Roitman, 2003;
Origins of Mathematical Intuitions  The Case of Arithmetic
 THE YEAR IN COGNITIVE NEUROSCIENCE
, 2009
"... Mathematicians frequently evoke their “intuition” when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced parad ..."
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Cited by 5 (0 self)
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Mathematicians frequently evoke their “intuition” when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of “core knowledge” associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.
Representations of the magnitudes of fractions
 Journal of Experimental Psychology: Human Perception and Performance
, 2010
"... We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and den ..."
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Cited by 4 (0 self)
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We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and denominators as separate whole numbers. However, atypical characteristics of the presented fractions might have provoked the use of atypical comparison strategies in that study. In our 3 experiments, university and community college students compared more balanced sets of singledigit and multidigit fractions and consistently exhibited a logarithmic distance effect. Thus, adults used integrated, analog representations, akin to a mental number line, to compare fraction magnitudes. We interpret differences between the past and present findings in terms of different stimuli eliciting different solution strategies.
Natural Number and Natural Geometry
"... How does the human brain support abstract concepts such as seven or square? Studies of nonhuman animals, of human infants, and of children and adults in diverse cultures suggest these concepts arise from a set of cognitive systems that are phylogenetically ancient, innate, and universal across human ..."
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Cited by 3 (1 self)
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How does the human brain support abstract concepts such as seven or square? Studies of nonhuman animals, of human infants, and of children and adults in diverse cultures suggest these concepts arise from a set of cognitive systems that are phylogenetically ancient, innate, and universal across humans: systems of core knowledge. Two of these systems—for tracking small numbers of objects and for assessing, comparing and combining the approximate cardinal values of sets—capture the primary information in the system of positive integers. Two other systems—for representing the shapes of smallscale forms and the distances and directions of surfaces in the largescale navigable layout—capture the primary information in the system of Euclidean plane geometry. As children learn language and other symbol systems, they begin to combine their core numerical and geometrical representations productively, in uniquely human ways. These combinations may give rise to the first truly abstract concepts at the foundations of mathematics. For millenia, philosophers and scientists have pondered the existence, nature and origins of abstract numerical and geometrical concepts, because these concepts have striking features. First, the integers, and the figures of the Euclidean plane, are so intuitive to human adults that the systems underlying them are called “natural number ” and, by some, “natural geometry”