Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. We hypothesize that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. Here we show that ninth grade students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low-, typically-, and high-achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relationship persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low- from typically-achieving students, suggesting an ANS deficit that is specific to MLD.

Researchers of mathematics learning disability (MLD) commonly use cutoff scores to determine which participants have MLD. Some researchers apply more restrictive cutoffs than others (e.g., performance below the 10th vs. below the 35th percentile). Different cutoffs may lead to groups of children that differ in their profile of math and related skills, including reading, visual-spatial, and working memory skills. The present study assesses the characteristics of children with MLD based on varying MLD definitions of math performance either below the 10th percentile (n = 22) or between the 11th and 25th percentile (n = 42) on the Test of Early Math Ability, second edition (TEMA-2). Initial starting levels and growth rates for math and related skills were examined in these two MLD groups relative to a comparison group (n = 146) whose TEMA-2 performance exceeded the 25th percentile. Between kindergarten and third grade, differences emerged in the starting level and growth rate, suggesting qualitative differences among the three groups. Despite some similarities, qualitative group differences were also observed in the profiles of math-related skills across groups. These results highlight differences in student characteristics based on the definition of MLD and illustrate the value of examining skill areas associated with math performance in addition to math performance itself.

The Approximate Number System (ANS) is a primitive mental system of nonverbal representations that supports an intuitive sense of number in human adults, children, infants, and other animal species. The numerical approximations produced by the ANS are characteristically imprecise and, in humans, this precision gradually improves from infancy to adulthood. Throughout development, wide ranging individual differences in ANS precision are evident within age groups. These individual differences have been linked to formal mathematics outcomes, based on concurrent, retrospective, or short-term longitudinal correlations observed during the school age years. However, it remains unknown whether this approximate number sense actually serves as a foundation for these school mathematics abilities. Here we show that ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicts performance on school mathematics at 6 years of age. In contrast, ANS precision does not predict non-numerical cognitive abilities. To our knowledge, these results provide the first evidence for early ANS precision, measured before the onset of formal education, predicting later mathematical abilities.

This paper is a descriptive report of findings from a prospective longitudinal study of math disability (MD). The study was designed to address the incidence of MD during primary school, the utility of different MD definitions, and evidence of MD subtypes. The results illustrate the dynamic properties of psychometrically derived definitions of MD. Different groups of children meet criteria for MD depending on which measure(s) are used for identification. Over time, a given individual may not continue to meet MD criteria, even when using the same assessments. Thus, the findings lead to cautions regarding the single-tool/ one-time assessment for a clinical diagnosis of MD. Twenty-two of 209 participants demonstrated "persistent MD" (MD-p), or MD for more than one school grade. Reading disability was relatively more frequent in this MD-p subgroup than in the remaining participants (25 percent vs. 7 percent). Reading-related skills were correlated with math achievement, as were select visual spatial skills. There was minimal overlap between groups who met either a "poor achievement" criteria or an "IQ-achievement discrepancy," and the latter was far less stable a measure over time than the former. The results highlight the complexities of defining MD and illustrate the need for more research in this area.

Many middle-school students struggle with decimals and fractions, even if they do not have a mathematical learning disability (MLD). In the present longitudinal study, we examined whether children with MLD have weaker rational number knowledge than children whose difficulty with rational numbers occurs in the absence of MLD. We found that children with MLD failed to accurately name decimals, to correctly rank order decimals and/or fractions, and to identify equivalent ratios (e.g. 0.5 = 1/2); they also 'identified' incorrect equivalents (e.g. 0.05 = 0.50). Children with low math achievement but no MLD accurately named decimals and identified equivalent pairs, but failed to correctly rank order decimals and fractions. Thus failure to accurately name decimals was an indicator of MLD; but accurate naming was no guarantee of rational number knowledge - most children who failed to correctly rank order fractions and decimals tests passed the naming task. Most children who failed the ranking tests at 6th grade also failed at 8th grade. Our findings suggest that a simple task involving naming and rank ordering fractions and decimals may be a useful addition to in-class assessments used to determine children's learning of rational numbers.

The aim of the present study was to address how to effectively predict mathematics learning disability (MLD). Specifically, we addressed whether cognitive data obtained during kindergarten can effectively predict which children will have MLD in third grade, whether an abbreviated test battery could be as effective as a standard psychoeducational assessment at predicting MLD, and whether the abbreviated battery corresponded to the literature on MLD characteristics. Participants were 226 children who enrolled in a 4-year prospective longitudinal study during kindergarten. We administered measures of mathematics achievement, formal and informal mathematics ability, visual-spatial reasoning, and rapid automatized naming and examined which test scores and test items from kindergarten best predicted MLD at grades 2 and 3. Statistical models using standardized scores from the entire test battery correctly classified ~80-83 percent of the participants as having, or not having, MLD. Regression models using scores from only individual test items were less predictive than models containing the standard scores, except for models using a specific subset of test items that dealt with reading numerals, number constancy, magnitude judgments of one-digit numbers, or mental addition of one-digit numbers. These models were as accurate in predicting MLD as was the model including the entire set of standard scores from the battery of tests examined. Our findings indicate that it is possible to effectively predict which kindergartners are at risk for MLD, and thus the findings have implications for early screening of MLD.The primary aim of the present study was to examine whether cognitive data obtained from psychoeducational or neuropsychological assessments of kindergarteners can effectively predict math achievement outcome in third grade. Although this study of predictors was not a screening study per se, the findings from the present study may have direct implications for early screening of risk for math learning disability (MLD). The findings may also contribute toward efforts to identify core deficits that underlie MLD. Relative to knowledge about reading disability (RD), there is less known about the core deficit(s) of mathematics disability, and there are few evidence-based screening tools available for detecting risk for poor achievement in mathematics. The proposed project was designed to help address these gaps in researchbased practice. The Need for Early Identification of MLDIt is important to identify risk for MLD, because-like poor reading achievement-poor math achievement is a risk factor for negative outcomes in both childhood and adulthood. Education level achieved in adulthood is associated with math performance level (Delazer, Girelli, Grana, Requests for reprints should be sent to Michèle Mazzocco, KKI West Campus, 3825 Greenspring Avenue, Painter Building, Top Floor, Baltimore, MD 21211. Electronic inquiries may be send to mazzocco@kennedykrieger.org. NIH Public Access Author ManuscriptLearn Disabil Res ...

Fragile X (fra(X)) males with a standardized IQ score of 70 or higher represent a high functioning (HF) or nonretarded fra(X) male group. This group, which does not include nonpenetrant males, has received little research attention to date. Of 221 fra(X) males who had been evaluated through The Children's Hospital in Denver since 1981 and had completed cognitive or developmental testing, 29 (13%) were high functioning by the above definition. We found that HF males on the whole had a lower cytogenetic score and were younger than retarded fra(X) males, but there was no difference between these two groups in the number of typical fra(X) physical manifestations present. FMR-1 DNA testing was performed on 134 fra(X) males and methylation status was determined for 51 of these. A greater percentage of HF males had a mosaic pattern or an incompletely methylated full mutation than did retarded males. A unique DNA pattern, an unmethylated fully expanded mutation, was discovered in 3 of the highest functioning fra(X) males. Protein studies performed on 2 of these males demonstrated the presence of FMR-1 protein, albeit at lower levels than normal. FMR-1 protein was not present in retarded fra(X) males. Significant FMR-1 protein expression may be responsible for higher cognitive functioning in the 2 males with unmethylated fully expanded mutations compared to retarded fra(X) males.