Inferring uncertainty from interval estimates: Effects of alpha level and numeracy

Interval estimates are commonly used to descriptively communicate the degree of uncertainty in numerical values. Conventionally, low alpha levels (e.g., .05) ensure a high probability of capturing the target value between interval endpoints. Here, we test whether alpha levels and individual differences in numeracy influence distributional inferences. In the reported experiment, participants received prediction intervals for fictitious towns’ annual rainfall totals (assuming approximately normal distributions). Then, participants estimated probabilities that future totals would be captured within varying margins about the mean, indicating the approximate shapes of their inferred probability distributions. Results showed that low alpha levels (vs. moderate levels; e.g., .25) more frequently led to inferences of over-dispersed approximately normal distributions or approximately uniform distributions, reducing estimate accuracy. Highly numerate participants made more accurate estimates overall, but were more prone to inferring approximately uniform distributions. These findings have important implications for presenting interval estimates to various audiences

Arousal Modulation in Females with Fragile X or Turner Syndrome

The present study was carried out to examine physiological arousal modulation (heart activity and skin conductance, across baseline and cognitive tasks, in females with fragile X or Turner syndrome and a comparison group of females with neither syndrome. Relative to the comparison group, for whom a greater increase in skin conductance was associated with poor arithmetic performance and less risk taking behavior, females with fragile X displayed a minimal increase in heart activity that was nevertheless associated with poor performance on mental arithmetic. In contrast, no arousal–cognitive performance relationship emerged for the group with Turner syndrome. Taken together, our findings suggest that distinct profiles of arousal modulation might be associated with cognitive deficits in these syndrome populations

Variation in early number skills and mathematics achievement: Implications from cognitive profiles of children with or without Turner syndrome

Individuals with Mathematics Learning Disabilities have persistent mathematics underperformance but vary with respect to their cognitive profiles. The present study examined mathematics ability and achievement, and associated mathematics-specific numerical skills and domain-general cognitive abilities, in young children with Turner syndrome compared to their matched peers. We utilized two independent peer groups so that group comparisons would account for verbal skills, a hypothesized strength of girls with Turner syndrome, and nonsymbolic magnitude comparison skills, a hypothesized difference of girls with Turner syndrome. This individual matching approach afforded characterization of mathematics profiles of girls with Turner syndrome and girls without Turner syndrome that share potential key features of the Turner syndrome phenotype. Results indicated differences in mathematics ability and nonsymbolic magnitude comparison tasks between girls with Turner syndrome and peers with similar levels of verbal skill. Mathematics ability and mathematics achievement scores of girls with Turner syndrome did not differ significantly from their peers with similar levels of accuracy on a nonsymbolic magnitude comparison task. Cognitive correlates of mathematics outcomes showed disparate patterns across groups. These quantitative and qualitative differences across profiles enhance our understanding of variation in mathematics ability in early childhood and inform how mathematics skills develop in young children with or without Turner syndrome

Challenges in mathematical cognition: a collaboratively-derived research agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition

Preschoolers' Precision of the Approximate Number System Predicts Later School Mathematics Performance

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

Challenges in Mathematical Cognition: A Collaboratively-Derived Research Agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education,psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into sixbroad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition.</p

The nature of the association between number line and mathematical performance: An international twin study

Background: The number line task assesses the ability to estimatenumerical magnitudes. People vary greatly in this abilityand this variability has been previously associated with mathematical skills. However, the sources of individual differences in number line estimation and its association with mathematics are not fully understood. Aims: This large scale genetically sensitive studyuses a twin design to estimate the magnitude of the effects of genes and environments on: (1) individualvariation in number line estimation and (2) the co-variation of number line estimation with mathematics. Samples: We used over3,0008-16 years-old twins from US, Canada,UK, and Russia, and a sample of 1,456 8-18 years-old singleton Russian students. Methods: Twins were assessed on: (1)estimation of numerical magnitudes using a numberline task and (2) two mathematics components: fluency and problemsolving. Results: Results suggest that environments largelydrive individual differences in numberline estimation.Both genes and environments contribute to different extents to the number line estimationandmathematics correlation, depending on the sample and mathematics component. Conclusions: Taken together, the results suggest that in more heterogeneous school settings, environments may be more important in driving variation in number line estimation and its associationwith mathematics, whereas in more homogeneous school settings, genetic effects drive the covariation between number line estimationand mathematics. These results are discussed in light of development and educational settings

Persistent consequences of atypical early number concepts

How does symbolic number knowledge performance help identify young children at risk for poor mathematics achievement outcomes? In research and practice, classification of mathematics learning disability (MLD, or dyscalculia) is typically based on composite scores from broad measures of mathematics achievement. These scores do predict later math achievement levels, but do not specify the nature of math difficulties likely to emerge among students at greatest risk for long-term mathematics failure. Here we report that gaps in 2nd and 3rd graders’ number knowledge predict specific types of errors made on math assessments at Grade 8. Specifically, we show that early whole number misconceptions predict slower and less accurate performance, and atypical computational errors, on Grade 8 arithmetic tests. We demonstrate that basic number misconceptions can be detected by idiosyncratic responses to number knowledge items, and that when such misconceptions are evident during primary school they persist throughout the school age years, with variable manifestation throughout development. We conclude that including specific qualitative assessments of symbolic number knowledge in primary school may provide greater specificity of the types of difficulties likely to emerge among students at risk for poor mathematics outcomes