On the origins of computationally complex behavior.

There is considerable evidence for computationally complex behavior, that is, behavior that appears to require the equivalent of mathematical calculation by the organism. Spatial navigation by path integration is perhaps the best example. The most influential account of such behavior has been Gallistel's (1990) computational-representational theory, which assumes that organisms represent key environmental variables such as direction and distance traveled as real numbers stored in engrams and are able to perform arithmetic computations on those representations. But how are these computations accomplished? A novel perspective is gained from the historical development of algebra. We propose that computationally complex behavior suggests that the perceptual system represents an algebraic field, which is a mathematical concept that expresses the structure underlying arithmetic. Our field representation hypothesis predicts that the perceptual system computes 2 operations on represented magnitudes, not 1. We review recent research in which human observers were trained to estimate differences and ratios of stimulus pairs in a nonsymbolic task without explicit instruction (Grace, Morton, Ward, Wilson, & Kemp, 2018). Results show that the perceptual system automatically computes two operations when comparing stimulus magnitudes. A field representation offers a resolution to longstanding controversies in psychophysics about which of 2 algebraic operations is fundamental (e.g., the Fechner-Stevens debate), overlooking the possibility that both might be. In terms of neural processes that might support computationally complex behavior, our hypothesis suggests that we should look for evidence of 2 operations and for symmetries corresponding to the additive and multiplicative groups. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Volumetric and surface characteristics of gray matter in adult dyslexia and dyscalculia.

Dyslexia, dyscalculia and their comorbid manifestation are prevalent disorders associated with well-documented behavioral manifestations. However, attempts to relate these manifestations to abnormalities in brain structure have yielded mixed results, with no clear consistency across a range of measures. In this study, we used a unique design including adults with dyslexia, dyscalculia, both disorders and controls, to explore differences in gray matter characteristics across groups. Specifically, we examined whether dyslexia, dyscalculia, or their comorbid manifestation could be related to volumetric and surface characteristics of gray matter, using voxel-based and surface-based morphometry. We demonstrate with Bayesian analyses that the present data favor the null model of no differences between groups across the brain, a result that is in line with recent findings in this field of research. Importantly, we provide detailed statistical maps to enable robust assessment of our findings, and to promote cumulative evaluation of the evidence. Together, these findings suggest that gray matter differences associated with dyslexia and dyscalculia might not be as reliable as suggested by previous literature, with important implications for our understanding of these disorders.

No evidence for systematic white matter correlates of dyslexia and dyscalculia.

Learning disabilities such as dyslexia, dyscalculia and their comorbid manifestation are prevalent, affecting as much as 15% of the population. Structural neuroimaging studies have indicated that these disorders can be related to differences in white matter integrity, although findings remain disparate. In this study, we used a unique design composed of individuals with dyslexia, dyscalculia, both disorders and controls, to systematically explore differences in fractional anisotropy across groups using diffusion tensor imaging. Specifically, we focused on the corona radiata and the arcuate fasciculus, two tracts associated with reading and mathematics in a number of previous studies. Using Bayesian hypothesis testing, we show that the present data favor the null model of no differences between groups for these particular tracts-a finding that seems to go against the current view but might be representative of the disparities within this field of research. Together, these findings suggest that structural differences associated with dyslexia and dyscalculia might not be as reliable as previously thought, with potential ramifications in terms of remediation.

Reading network in dyslexia: Similar, yet different.

Dyslexia is a developmental disorder characterized by reading and phonological difficulties, yet important questions remain regarding its underlying neural correlates. In this study, we used partial least squares (PLS), a multivariate analytic technique, to investigate the neural networks used by dyslexics while performing a word-rhyming task. Although the overall reading network was largely similar in dyslexics and typical readers, it did not correlate with behavior in the same way in the two groups. In particular, there was a positive association between reading performance and both right superior temporal gyrus and bilateral insula activation in dyslexic readers but a negative correlation in typical readers. Together with differences in lateralization unique to dyslexics, this suggests that the combination of poor reading performance with high insula activity and atypical laterality is a consistent marker of dyslexia. These findings emphasize the importance of understanding right-hemisphere activation in dyslexia and provide promising directions for the remediation of reading disorders.

Semantic processing of mathematical gestures.

OBJECTIVE: To examine whether or not university mathematics students semantically process gestures depicting mathematical functions (mathematical gestures) similarly to the way they process action gestures and sentences. Semantic processing was indexed by the N400 effect. RESULTS: The N400 effect elicited by words primed with mathematical gestures (e.g. "converging" and "decreasing") was the same in amplitude, latency and topography as that elicited by words primed with action gestures (e.g. drive and lift), and that for terminal words of sentences. SIGNIFICANCE AND CONCLUSION: Findings provide a within-subject demonstration that the topographies of the gesture N400 effect for both action and mathematical words are indistinguishable from that of the standard language N400 effect. This suggests that mathematical function words are processed by the general language semantic system and do not appear to involve areas involved in other mathematical concepts (e.g. numerosity).

Principles underlying the design of "The Number Race", an adaptive computer game for remediation of dyscalculia.

BACKGROUND: Adaptive game software has been successful in remediation of dyslexia. Here we describe the cognitive and algorithmic principles underlying the development of similar software for dyscalculia. Our software is based on current understanding of the cerebral representation of number and the hypotheses that dyscalculia is due to a "core deficit" in number sense or in the link between number sense and symbolic number representations. METHODS: "The Number Race" software trains children on an entertaining numerical comparison task, by presenting problems adapted to the performance level of the individual child. We report full mathematical specifications of the algorithm used, which relies on an internal model of the child's knowledge in a multidimensional "learning space" consisting of three difficulty dimensions: numerical distance, response deadline, and conceptual complexity (from non-symbolic numerosity processing to increasingly complex symbolic operations). RESULTS: The performance of the software was evaluated both by mathematical simulations and by five weeks of use by nine children with mathematical learning difficulties. The results indicate that the software adapts well to varying levels of initial knowledge and learning speeds. Feedback from children, parents and teachers was positive. A companion article describes the evolution of number sense and arithmetic scores before and after training. CONCLUSION: The software, open-source and freely available online, is designed for learning disabled children aged 5-8, and may also be useful for general instruction of normal preschool children. The learning algorithm reported is highly general, and may be applied in other domains.

An open trial assessment of "The Number Race", an adaptive computer game for remediation of dyscalculia.

BACKGROUND: In a companion article, we described the development and evaluation of software designed to remediate dyscalculia. This software is based on the hypothesis that dyscalculia is due to a "core deficit" in number sense or in its access via symbolic information. Here we review the evidence for this hypothesis, and present results from an initial open-trial test of the software in a sample of nine 7-9 year old children with mathematical difficulties. METHODS: Children completed adaptive training on numerical comparison for half an hour a day, four days a week over a period of five-weeks. They were tested before and after intervention on their performance in core numerical tasks: counting, transcoding, base-10 comprehension, enumeration, addition, subtraction, and symbolic and non-symbolic numerical comparison. RESULTS: Children showed specific increases in performance on core number sense tasks. Speed of subitizing and numerical comparison increased by several hundred msec. Subtraction accuracy increased by an average of 23%. Performance on addition and base-10 comprehension tasks did not improve over the period of the study. CONCLUSION: Initial open-trial testing showed promising results, and suggested that the software was successful in increasing number sense over the short period of the study. However these results need to be followed up with larger, controlled studies. The issues of transfer to higher-level tasks, and of the best developmental time window for intervention also need to be addressed.

Individual cognitive training of reading disability improves word identification and sentence comprehension in adults with mild mental retardation.

Reading therapy has been shown to be effective in treating reading disabilities (RD) in dyslexic children, but little is known of its use in subjects with mild mental retardation (MR). Twenty adult volunteers, with both RD and mild MR, underwent 60 consecutive weeks in a cognitive remediation program, and were compared with 32 untreated control subjects. The experimental group showed a significant improvement in word identification, as measured by oral production (p=0.0004) or silent reading (p=0.023), and sentence comprehension (p=0.0002). Adults with MR appear to benefit from new approaches in the field of RD.

Arithmetic and the brain.

Recent studies in human neuroimaging, primate neurophysiology, and developmental neuropsychology indicate that the human ability for arithmetic has a tangible cerebral substrate. The human intraparietal sulcus is systematically activated in all number tasks and could host a central amodal representation of quantity. Areas of the precentral and inferior prefrontal cortex also activate when subjects engage in mental calculation. A monkey analogue of these parieto-frontal regions has recently been identified, and a neuronal population code for number has been characterized. Finally, pathologies of this system, leading to acalculia in adults or to developmental dyscalculia in children, are beginning to be understood, thus paving the way for brain-oriented intervention studies.