Developmental changes in numerosity and area perception in school-age children

Humans can quickly estimate the quantity of objects in their environment through the Approximate Number System (ANS). However, the developmental trajectories of numerical and spatial perception in school-aged children are not well understood. This study aimed to address this issue by examining the performance of 7-year-olds, 9-year-olds, and 11-year-olds in the dot-number and dot-area tasks. In each trial, participants were presented with pairs of dots array simultaneously and were asked to indicate which array had a greater quantity of dots in the dot-number task, and which array had a larger overall area of dots in the dot-area task. We manipulated number cues and area cues of the dots array in these two tasks, creating three different relationships between dot-number and dot-area: congruent, neutral, and incongruent. Our results showed school-aged children's ability to estimate numerosity improved significantly after the age of 7, with no apparent improvement observed between the ages of 9 and 11. This indicates a marked growth in acuity in the perception of numerical quantities. Conversely, the capacity to estimate area showed consistent stability across the various age groups examined. Additionally, our results demonstrated a pronounced difficulty among participants in ignoring numerical cues when assessing dot quantity or dot area, as opposed to non-numerical cues. This highlights a preferential sensitivity to numerical information in cognitive processing. These findings provide valuable insights into the cognitive development underlying quantity perception and may offer practical guidance for educational practices.

Factors influencing the role of inhibitory control in non-symbolic numerical processing

Previous studies have found that inhibitory control plays an important role in non-symbolic numerical processing. However, this role may be influenced by the visual cue control method or the stimulus' presentation time. We investigated these questions by conducting three experiments using a priming paradigm to compare the level of inhibitory control in a sequential dot comparison task with single-dimensional and multi-dimensional control of visual cues under two presentation time conditions (300 ms and 1500 ms). We found that neither the method of visual cue control nor the presentation time of dot arrays affected the level of inhibitory control in the dot comparison task. These results reveal a stable role of inhibitory control in non-symbolic numerical processing, providing further evidence for integrating numerical and visual information during non-symbolic numerical processing.

Number sense deficits in children with developmental dyscalculia, dyslexia, co-occurring disorder and their typically developing peers

The aim of this study was to explore a number sense deficits in children with developmental dyscalculia, dyslexia, co-occurring disorder and their typically developing peers. A non-symbolic quantity comparison task was used in this study to examine whether children with dyscalculia have number sense deficits. Children aged 10-11 years old from nine primary schools in Taif city, Saudi Arabia, were selected to participate in this study. The children were divided into the dyscalculia group (n = 62), the dyslexia group (n = 60), and co-occurring disorder group (n = 65), and the typically developing peers group (n = 100).4 groups (dyscalculia, dyslexia, co-occurring disorder and typically developing peers group) × 2 stimulus ratio (6:7; 8:12). There were significant differences in non-symbolic quantity comparison tasks between children with dyslexia, co-occurring disorder, and typically developing peers. These results indicate that children with dyscalculia do have number sense deficiencies, but number sense deficiencies are not specific to children with dyscalculia.

Electrophysiological correlates of temporal numerosity adaptation

Introduction

Much research has revealed the human visual system is capable to estimate numerical quantities, rapidly and reliably, in both the spatial and the temporal domain. This ability is highly susceptible to short-term plastic phenomena related to previous exposure to visual numerical information (i.e., adaptation). However, while determinants of spatial numerosity adaptation have been widely investigated, little is known about the neural underpinnings of short-term plastic phenomena related to the encoding of temporal numerical information. In the present study we investigated the electrophysiological correlates of temporal numerosity adaptation.

Methods

Participants were asked to estimate the numerosity of a test sequence of flashes after being exposed to either a high or low numerous adapting sequence. Behavioral results confirmed the expected underestimation of test stimulus when this was preceded by a high numerous sequence as compared to when preceded by a low numerous sequence.

Results

Electrophysiological data revealed that this behavior was tightly linked to the amplitude of the steady-state visual evoked (ssVEP) response elicited by the test stimulus. When preceded by a high numerous sequence, the test stimulus elicited larger ssVEP responses as compared to when preceded by a low numerous sequence with this pattern being robustly correlated with behavior. Finally, topographical maps showed that this difference was mostly evident across two antero-posterior distributed clusters of electrodes and correlated with changes in functional connectivity.

Discussion

Taken together, our results suggest that visual plastic phenomena related to the encoding of temporal numerosity information reflect changes in rhythmic evoked activity that are likely related to long range communications between distinct brain regions.

Inter-individual, hemispheric and sex variability of brain activations during numerosity processing

Numerosity perception is a fundamental and innate cognitive function shared by both humans and many animal species. Previous research has primarily focused on exploring the spatial and functional consistency of neural activations that were associated with the processing of numerosity information. However, the inter-individual variability of brain activations of numerosity perception remains unclear. In the present study, with a large-sample functional magnetic resonance imaging (fMRI) dataset (n = 460), we aimed to localize the functional regions related to numerosity perceptions and explore the inter-individual, hemispheric, and sex differences within these brain regions. Fifteen subject-specific activated regions, including the anterior intraparietal sulcus (aIPS), posterior intraparietal sulcus (pIPS), insula, inferior frontal gyrus (IFG), inferior temporal gyrus (ITG), premotor area (PM), middle occipital gyrus (MOG) and anterior cingulate cortex (ACC), were delineated in each individual and then used to create a functional probabilistic atlas to quantify individual variability in brain activations of numerosity processing. Though the activation percentages of most regions were higher than 60%, the intersections of most regions across individuals were considerably lower, falling below 50%, indicating substantial variations in brain activations related to numerosity processing among individuals. Furthermore, significant hemispheric and sex differences in activation location, extent, and magnitude were also found in these regions. Most activated regions in the right hemisphere had larger activation volumes and activation magnitudes, and were located more lateral and anterior than their counterparts in the left hemisphere. In addition, in most of these regions, males displayed stronger activations than females. Our findings demonstrate large inter-individual, hemispheric, and sex differences in brain activations related to numerosity processing, and our probabilistic atlas can serve as a robust functional and spatial reference for mapping the numerosity-related neural networks.

Neural underpinnings of processing combinatorial unstated meaning and the influence of individual cognitive style

We investigated the neurocognitive mechanisms underlying the processing of combinatorial unstated meaning. Sentences like "Charles jumped for 5 minutes." engender an iterative meaning that is not explicitly stated but enriched by comprehenders beyond simple composition. Comprehending unstated meaning involves meaning contextualization-integrative meaning search in sentential-discourse context. Meanwhile, people differ in how they process information with varying context sensitivity. We hypothesized that unstated meaning processing would vary with individual socio-cognitive propensity indexed by the Autism-Spectrum Quotient (AQ), accompanied by differential cortical engagements. Using functional magnetic resonance imaging, we examined the processing of sentences with unstated iterative meaning in typically-developed individuals and found an engagement of the fronto-parietal network, including the left pars triangularis (L.PT), right intraparietal (R.IPS), and parieto-occipital sulcus (R.POS). We suggest that the L.PT subserves a contextual meaning search, while the R.IPS/POS supports enriching unstated iteration in consideration of event durations and interval lengths. Moreover, the activation level of these regions negatively correlated with AQ. Higher AQ ties to lower L.PT activation, likely reflecting weaker context sensitivity, along with lower IPS activation, likely reflecting weaker computation of events' numerical-temporal specifications. These suggest that the L.PT and R.IPS/POS support the processing of combinatorial unstated meaning, with the activation level modulated by individual cognitive styles.

From vision to cognition: potential contributions of cerebral visual impairment to neurodevelopmental disorders

Vision has a crucial role to play in human development and functioning. It is, therefore, not surprising that vision plays a fundamental role in the development of the child. As a consequence, an alteration in visual function is, therefore, likely to hinder the child's development. Although ocular disorders are well known, diagnosed and taken into account, cerebral visual impairments (CVI) resulting from post-chiasmatic damage are largely underdiagnosed. However, among the disorders resulting from an episode of perinatal asphyxia and/or associated with prematurity, or neonatal hypoglycaemia, CVIs are prominent. In this article, we focus on the role of the possible effects of CVI on a child's learning abilities, leading to major difficulty in disentangling the consequences of CVI from other neurodevelopmental disorders (NDD) such as dyslexia, dyscalculia, dysgraphia, attention-deficit/hyperactivity disorder (ADHD), developmental coordination disorder (DCD) and autism spectrum disorders (ASD). Although we focus here on the possible overlap between children with CVI and children with other NDD, De Witt et al. (Wit et al. Ear Hear 39:1-19, 2018) have raised exactly the same question regarding children with auditory processing disorders (the equivalent of CVI in the auditory modality). We underline how motor, social and cognitive development as well as academic success can be impaired by CVI and raise the question of the need for systematic evaluation for disorders of vision, visual perception and cognition in all children presenting with a NDD and/or previously born under adverse neurological conditions.

To what extent is dot comparison an appropriate measure of approximate number system?

Introduction

Number sense has been systematically measured using dot comparison tasks. However, recent studies have reported that performance on dot comparison might be influenced inhibitory control and visual properties of dot arrays. In the present study, we analysed the influence of continuous magnitude, inhibitory control, and numerical ratio on the dot comparison performance of preschool children.

Methods

Participants were 517 preschool children from 13 different schools in Chile. Children completed a dot comparison and two inhibitory control tasks. Gebuis and Reynvoet method was used to create well-controlled dot arrays for use in the dot comparison task. A logistic mixed effects model was conducted to predict participants' dot comparison accuracy. Continuous magnitude and ratio were entered as level-1 predictors and inhibitory control as level-2 predictors.

Results

The results showed that all predictors made a significant contribution to dot comparison accuracy. Furthermore, a significant double interaction (inhibitory control x continuous magnitude) and a triple interaction (inhibitory control x continuous magnitude x ratio) showed that the contribution of inhibitory control skills in dot comparison accuracy depends on the continuous properties of dot arrays and ratio.

Discussion

These findings suggest that preschool children rely more on continuous magnitudes than numerosity in dot comparison tasks. They also indicate that the greater children's inhibitory control, the more able they are to respond based on numerosity in fully incongruent trials, particularly when ratio is low (easiest items). Taken together, the above findings support the competing processes account provided that both ANS and inhibitory control skills influence performance on dot comparison tasks.

EEG-representational geometries and psychometric distortions in approximate numerical judgment

When judging the average value of sample stimuli (e.g., numbers) people tend to either over- or underweight extreme sample values, depending on task context. In a context of overweighting, recent work has shown that extreme sample values were overly represented also in neural signals, in terms of an anti-compressed geometry of number samples in multivariate electroencephalography (EEG) patterns. Here, we asked whether neural representational geometries may also reflect a relative underweighting of extreme values (i.e., compression) which has been observed behaviorally in a great variety of tasks. We used a simple experimental manipulation (instructions to average a single-stream or to compare dual-streams of samples) to induce compression or anti-compression in behavior when participants judged rapid number sequences. Model-based representational similarity analysis (RSA) replicated the previous finding of neural anti-compression in the dual-stream task, but failed to provide evidence for neural compression in the single-stream task, despite the evidence for compression in behavior. Instead, the results indicated enhanced neural processing of extreme values in either task, regardless of whether extremes were over- or underweighted in subsequent behavioral choice. We further observed more general differences in the neural representation of the sample information between the two tasks. Together, our results indicate a mismatch between sample-level EEG geometries and behavior, which raises new questions about the origin of common psychometric distortions, such as diminishing sensitivity for larger values.

Visual P2p component responds to perceived numerosity

Numerosity perception is a key ability for human and non-human species, probably mediated by dedicated brain mechanisms. Electrophysiological studies revealed the existence of both early and mid-latency components of the Electrophysiological (EEG) signal sensitive to numerosity changes. However, it is still unknown whether these components respond to physical or perceived variation in numerical attributes. We here tackled this point by recording electrophysiological signal while participants performed a numerosity adaptation task, a robust psychophysical method yielding changes in perceived numerosity judgments despite physical numerosity invariance. Behavioral measures confirmed that the test stimulus was consistently underestimated when presented after a high numerous adaptor while perceived as veridical when presented after a neutral adaptor. Congruently, EEG results revealed a potential at around 200 ms (P2p) which was reduced when the test stimulus was presented after the high numerous adaptor. This result was much prominent over the left posterior cluster of electrodes and correlated significantly with the amount of adaptation. No earlier modulations were retrievable when changes in numerosity were illusory while both early and mid-latency modulations occurred for physical changes. Taken together, our results reveal that mid-latency P2p mainly reflects perceived changes in numerical attributes, while earlier components are likely to be bounded to the physical characteristics of the stimuli. These results suggest that short-term plastic mechanisms induced by numerosity adaptation may involve a relatively late processing stage of the visual hierarchy likely engaging cortical areas beyond the primary visual cortex. Furthermore, these results also indicate mid-latency electrophysiological correlates as a signature of the internal representation of numerical information.

Temporal dynamics of face adaptation

The appearance of a face can be strongly affected by adaptation to faces seen previously. A number of studies have examined the time course of these aftereffects, but the integration time over which adaptation pools signals to control the adaptation state remains uncertain. Here we examined the effects of temporal frequency on face gender aftereffects induced by a pair of faces alternating between the two genders to assess when the aftereffects were pooled over successive faces versus driven by the last face seen. In the first experiment, we found that temporal frequencies between 0.25 and 2.00 Hz all failed to produce an aftereffect, suggesting a fairly long integration time. In the second experiment, we therefore probed slower alternation rates of 0.03 to 0.25 Hz. A rate of 0.0625 Hz (i.e., 8 seconds per face) was required to generate significant aftereffects from the last presented face and was consistent with an average time constant of 15 to 20 seconds for an exponentially decaying integration window. This integration time is substantially longer than found previously for analogous effects for alternating colors, and thus points to a potentially slower mechanism of adaptation for faces compared with chromatic adaptation.

The neural correlates of retrieval and procedural strategies in mental arithmetic: A functional neuroimaging meta-analysis

Mental arithmetic is a complex skill of great importance for later academic and life success. Many neuroimaging studies and several meta-analyses have aimed to identify the neural correlates of mental arithmetic. Previous meta-analyses of arithmetic grouped all problem types into a single meta-analytic map, despite evidence suggesting that different types of arithmetic problems are solved using different strategies. We used activation likelihood estimation (ALE) to conduct quantitative meta-analyses of mental arithmetic neuroimaging (n = 31) studies, and subsequently grouped contrasts from the 31 studies into problems that are typically solved using retrieval strategies (retrieval problems) (n = 18) and problems that are typically solved using procedural strategies (procedural problems) (n = 19). Foci were compiled to generate probabilistic maps of activation for mental arithmetic (i.e., all problem types), retrieval problems, and procedural problems. Conjunction and contrast analyses were conducted to examine overlapping and distinct activation for retrieval and procedural problems. The conjunction analysis revealed overlapping activation for retrieval and procedural problems in the bilateral inferior parietal lobules, regions typically associated with magnitude processing. Contrast analyses revealed specific activation in the left angular gyrus for retrieval problems and specific activation in the inferior frontal gyrus and cingulate gyrus for procedural problems. These findings indicate that the neural bases of arithmetic systematically differs according to problem type, providing new insights into the dynamic and task-dependent neural underpinnings of the calculating brain.

Deprivation of Auditory Experience Influences Numerosity Discrimination, but Not Numerosity Estimation

Number sense is the ability to estimate the number of items, and it is common to many species. Despite the numerous studies dedicated to unveiling how numerosity is processed in the human brain, to date, it is not clear whether the representation of numerosity is supported by a single general mechanism or by multiple mechanisms. Since it is known that deafness entails a selective impairment in the processing of temporal information, we assessed the approximate numerical abilities of deaf individuals to disentangle these two hypotheses. We used a numerosity discrimination task (2AFC) and an estimation task, in both cases using sequential (temporal) or simultaneous (spatial) stimuli. The results showed a selective impairment of the deaf participants compared with the controls (hearing) in the temporal numerosity discrimination task, while no difference was found to discriminate spatial numerosity. Interestingly, the deaf and hearing participants did not differ in spatial or temporal numerosity estimation. Overall, our results suggest that the deficit in temporal processing induced by deafness also impacts perception in other domains such as numerosity, where sensory information is conveyed in a temporal format, which further suggests the existence of separate mechanisms subserving the processing of temporal and spatial numerosity.

Numerosities are not ersatz numbers

In describing numerosity as "a kind of ersatz number," Clarke and Beck fail to consider a familiar and compelling definition of numerosity, which conceptualizes numerosity as the cognitive counterpart of the mathematical concept of cardinality; numerosity is the magnitude, whereas number is a scale through which numerosity/cardinality is measured. We argue that these distinctions should be considered.

The Sense of Number in Fish, with Particular Reference to Its Neurobiological Bases

Simple Summary

The ability to deal with quantity, both discrete (numerosities) and continuous (spatial or temporal extent) developed from an evolutionarily conserved system for approximating numerical magnitude. Non-symbolic number cognition based on an approximate sense of magnitude has been documented in a variety of vertebrate species, including fish. Fish, in particular zebrafish, are widely used as models for the investigation of the genetics and molecular mechanisms of behavior, and thus may be instrumental to development of a neurobiology of number cognition. We review here the behavioural studies that have permitted to identify numerical abilities in fish, and the current status of the research related to the neurobiological bases of these abilities with special reference to zebrafish. Combining behavioural tasks with molecular genetics, molecular biology and confocal microscopy, a role of the retina and optic tectum in the encoding of continuous magnitude in larval zebrafish has been reported, while the thalamus and the dorso-central subdivision of pallium in the encoding of discrete magnitude (number) has been documented in adult zebrafish. Research in fish, in particular zebrafish, may reveal instrumental for identifying and characterizing the molecular signature of neurons involved in quantity discrimination processes of all vertebrates, including humans.

Abstract

It is widely acknowledged that vertebrates can discriminate non-symbolic numerosity using an evolutionarily conserved system dubbed Approximate Number System (ANS). Two main approaches have been used to assess behaviourally numerosity in fish: spontaneous choice tests and operant training procedures. In the first, animals spontaneously choose between sets of biologically-relevant stimuli (e.g., conspecifics, food) differing in quantities (smaller or larger). In the second, animals are trained to associate a numerosity with a reward. Although the ability of fish to discriminate numerosity has been widely documented with these methods, the molecular bases of quantities estimation and ANS are largely unknown. Recently, we combined behavioral tasks with molecular biology assays (e.g c-fos and egr1 and other early genes expression) showing that the thalamus and the caudal region of dorso-central part of the telencephalon seem to be activated upon change in numerousness in visual stimuli. In contrast, the retina and the optic tectum mainly responded to changes in continuous magnitude such as stimulus size. We here provide a review and synthesis of these findings.

Bidirectional Mapping Between the Symbolic Number System and the Approximate Number System

Abstract. Previous studies have discussed the symmetry of bidirectional mapping between approximate number system (ANS) and symbolic number system (SNS). However, these studies neglected the essential significance of bidirectional mapping in the development of numerical cognition. That is, with age, the connection strength between the ANS and SNS in ANS-SNS mapping could be higher than that in SNS-ANS mapping. Therefore, this study attempted to explore the symmetry of bidirectional mapping by examining whether the connection between the ANS and SNS is the same. Using two types of dot array materials (extensive and intensive) and sequence priming paradigms, this study found a stable negative priming effect in the ANS-SNS priming task, but no priming effect in the SNS-ANS priming task. In addition, although sensory cues (extensive and intensive) could affect performance in the ANS-SNS mapping task, these cues did not affect performance in the ANS-SNS priming task. In general, this study provides valuable insight into the symmetry of bidirectional mapping.

Symbols Are Special: An fMRI Adaptation Study of Symbolic, Nonsymbolic, and Non-Numerical Magnitude Processing in the Human Brain

How are different formats of magnitudes represented in the human brain? We used functional magnetic resonance imaging adaptation to isolate representations of symbols, quantities, and physical size in 45 adults. Results indicate that the neural correlates supporting the passive processing of number symbols are largely dissociable from those supporting quantities and physical size, anatomically and representationally. Anatomically, passive processing of quantities and size correlate with activation in the right intraparietal sulcus, whereas symbolic number processing, compared with quantity processing, correlates with activation in the left inferior parietal lobule. Representationally, neural patterns of activation supporting symbols are dissimilar from neural activation patterns supporting quantity and size in the bilateral parietal lobes. These findings challenge the longstanding notion that the culturally acquired ability to conceptualize symbolic numbers is represented using entirely the same brain systems that support the evolutionarily ancient system used to process quantities. Moreover, these data reveal that regions that support numerical magnitude processing are also important for the processing of non-numerical magnitudes. This discovery compels future investigations of the neural consequences of acquiring knowledge of symbolic numbers.

Neural state space alignment for magnitude generalization in humans and recurrent networks

A prerequisite for intelligent behavior is to understand how stimuli are related and to generalize this knowledge across contexts. Generalization can be challenging when relational patterns are shared across contexts but exist on different physical scales. Here, we studied neural representations in humans and recurrent neural networks performing a magnitude comparison task, for which it was advantageous to generalize concepts of "more" or "less" between contexts. Using multivariate analysis of human brain signals and of neural network hidden unit activity, we observed that both systems developed parallel neural "number lines" for each context. In both model systems, these number state spaces were aligned in a way that explicitly facilitated generalization of relational concepts (more and less). These findings suggest a previously overlooked role for neural normalization in supporting transfer of a simple form of abstract relational knowledge (magnitude) in humans and machine learning systems.

Synergy between research on ensemble perception, data visualization, and statistics education: A tutorial review

In the age of big data, we are constantly inventing new data visualizations to consolidate massive amounts of numerical information into smaller and more digestible visual formats. These data visualizations use various visual features to convey quantitative information, such as spatial position in scatter plots, color saturation in heat maps, and area in dot maps. These data visualizations are typically composed of ensembles, or groups of related objects, that together convey information about a data set. Ensemble perception, or one's ability to perceive summary statistics from an ensemble, such as the mean, has been used as a foundation for understanding and explaining the effectiveness of certain data visualizations. However, research in data visualization has revealed some perceptual biases and conceptual difficulties people face when trying to utilize the information in these graphs. In this tutorial review, we will provide a broad overview of research conducted in ensemble perception, discuss how principles of ensemble encoding have been applied to the research in data visualization, and showcase the barriers graphs can pose to learning statistical concepts, using histograms as a specific example. The goal of this tutorial review is to highlight possible connections between three areas of research-ensemble perception, data visualization, and statistics education-and to encourage research in the practical applications of ensemble perception in solving real-world problems in statistics education.

Transparent number-naming system gives only limited advantage for preschooler's numerical development: Comparisons of Vietnamese and French-speaking children

Several cross-sectional studies have suggested that the transparency of the number-naming system of East Asian languages (Chinese, Japanese) facilitates children's numerical development. The Vietnamese number-naming system also makes the base-10 system very explicit (eleven is "mười một," literally "ten-one," and thirty is "ba mươi," literally "three-ten"). In contrast, Western languages (English, French) include teen words (eleven to sixteen) and ten words (twenty to ninety) that make their counting systems less transparent. The main question addressed in this paper is: To what extent does a language's number-naming system impact preschoolers' numerical development? Our study participants comprised 104 Vietnamese and 104 French-speaking Belgian children between 3½ and 5½ years of age, as well as their parents. We tested the children on eight numerical tasks (counting, advanced counting, enumeration, Give-N, number-word comparison, collection comparison, addition, and approximate addition) and some general cognitive abilities (IQ and phonological loop by letter span). The parents completed a questionnaire on the frequency with which they stimulated their child's numeracy and literacy at home. The results indicated that Vietnamese children outperformed Belgian children only in counting. However, neither group differed in other symbolic or non-symbolic abilities, although Vietnamese parents tended to stimulate their child at home slightly more than Belgian parents. We concluded that the Vietnamese number-naming system's transparency led to faster acquisition of basic counting for preschoolers but did not support other more advanced numerical skills or non-symbolic numerical abilities. In addition, we extended the evidence that both transparent number-naming system and home numeracy influence young children's counting development.

Hemispheric asymmetries in processing numerical meaning in arithmetic

Hemispheric asymmetries in arithmetic have been hypothesized based on neuropsychological, developmental, and neuroimaging work. However, it has been challenging to separate asymmetries related to arithmetic specifically, from those associated general cognitive or linguistic processes. Here we attempt to experimentally isolate the processing of numerical meaning in arithmetic problems from language and memory retrieval by employing novel non-symbolic addition problems, where participants estimated the sum of two dot arrays and judged whether a probe dot array was the correct sum of the first two arrays. Furthermore, we experimentally manipulated which hemisphere receive the probe array first using a visual half-field paradigm while recording event-related potentials (ERP). We find that neural sensitivity to numerical meaning in arithmetic arises under left but not right visual field presentation during early and middle portions of the late positive complex (LPC, 400-800 ms). Furthermore, we find that subsequent accuracy for judgements of whether the probe is the correct sum is better under right visual field presentation than left, suggesting a left hemisphere advantage for integrating information for categorization or decision making related to arithmetic. Finally, neural signatures of operational momentum, or differential sensitivity to whether the probe was greater or less than the sum, occurred at a later portion of the LPC (800-1000 ms) and regardless of visual field of presentation, suggesting a temporal and functional dissociation between magnitude and ordinal processing in arithmetic. Together these results provide novel evidence for differences in timing and hemispheric lateralization for several cognitive processes involved in arithmetic thinking.

Predicting individual differences in reading, spelling and maths in a sample of typically developing children: A study in the perspective of comorbidity

We examined reading, spelling, and mathematical skills in an unselected group of 129 Italian fifth graders by testing various cognitive predictors for each behaviour. As dependent variables, we measured performance in behaviours with a clear functional value in everyday life, such as reading a text, spelling under dictation and doing mental and written computations. As predictors, we selected cognitive dimensions having an explicit relation with the target behaviour (called proximal predictors), and prepared various tests in order to select which task had the best predictive power on each behaviour. The aim was to develop a model of proximal predictors of reading (speed and accuracy), spelling (accuracy) and maths (speed and accuracy) characterized by efficacy also in comparison to the prediction based on general cognitive factors (i.e., short-term memory, phonemic verbal fluency, visual perceptual speed, and non-verbal intelligence) and parsimony, pinpointing the role of both common and unique predictors as envisaged in the general perspective of co-morbidity. With one exception (reading accuracy), the proximal predictors models (based on communality analyses) explained a sizeable amount of variance, ranging from 27.5% in the case of calculation (accuracy) to 48.7% of reading (fluency). Models based on general cognitive factors also accounted for some variance (ranging from 6.5% in the case of spelling to 19.5% in the case of reading fluency) but this was appreciably less than that explained by models based on the hypothesized proximal predictors. In general, results confirmed the efficacy of proximal models in predicting reading, spelling and maths although they offered only limited support for common predictors across different learning skills; namely, performance in the Orthographic Decision test entered as a predictor of both reading and spelling indicating that a single orthographic lexicon may account for performance in reading and spelling. Possible lines of research to expand on this approach are illustrated.

Parametric Representation of Tactile Numerosity in Working Memory

Estimated numerosity perception is processed in an approximate number system (ANS) that resembles the perception of a continuous magnitude. The ANS consists of a right lateralized frontoparietal network comprising the lateral prefrontal cortex (LPFC) and the intraparietal sulcus. Although the ANS has been extensively investigated, only a few studies have focused on the mental representation of retained numerosity estimates. Specifically, the underlying mechanisms of estimated numerosity working memory (WM) is unclear. Besides numerosities, as another form of abstract quantity, vibrotactile WM studies provide initial evidence that the right LPFC takes a central role in maintaining magnitudes. In the present fMRI multivariate pattern analysis study, we designed a delayed match-to-numerosity paradigm to test what brain regions retain approximate numerosity memoranda. In line with parametric WM results, our study found numerosity-specific WM representations in the right LPFC as well as in the supplementary motor area and the left premotor cortex extending into the superior frontal gyrus, thus bridging the gap in abstract quantity WM literature.

Prevalence of Mathematical and Visuospatial Learning Disabilities in Patients With Posterior Cortical Atrophy

Importance

Increased prevalence of language-based learning disabilities (LDs) has been previously reported in patients with primary progressive aphasia (PPA). This study hypothesized that patients with focal neurodegenerative syndromes outside the language network, such as posterior cortical atrophy (PCA), would have a higher rate of nonlanguage LDs, congruent with their mainly visuospatial presentation.

Objective

To investigate the prevalence and type of LD (language and/or mathematical and visuospatial) in a large cohort of patients with PCA compared with patients with logopenic variant PPA (lvPPA) and amnestic Alzheimer disease (AD).

Design, Setting, and Participants

This case-control study reviewed 279 medical records from a university-based clinic and research center for patients with neurodegenerative diseases for LD history, including patients with PCA (n = 95), patients with lvPPA (n = 84), and a matched cohort with amnestic AD (n = 100). No records were excluded. The study compared cognitive and neuroimaging features of patients with PCA with and without LDs. A review of the records of patients presenting from March 1, 1999, to August 31, 2014, revealed 95 PCA cases and 84 lvPPA cases. Then 100 patients with amnestic AD from this same period were chosen for comparison, matching against the groups for age, sex, and disease severity. Data analysis was performed from September 8, 2013, to November 6, 2017.

Main Outcomes and Measures

Prevalence of total LD history and prevalence of language and mathematical or visuospatial LD history across all cohorts.

Results

A total of 179 atypical AD cases (95 with PCA and 84 with lvPPA) and 100 disease control cases (amnestic AD) were included in the study. The groups were not statistically different for mean (SD) age at first visit (PCA, 61.9 [7.0] years; lvPPA, 65.1 [8.7] years; amnestic AD, 64.0 [12.6] years; P = .08), mean (SD) age at first symptom (PCA, 57.5 [7.0] years; lvPPA, 61.1 [9.0] years; amnestic AD, 59.6 [13.7] years; P = .06), or sex (PCA, 66.3% female; lvPPA, 56.0% female; amnestic AD, 57.0% female; P = .30) but differed on non-right-hand preference (PCA, 18.3%; lvPPA, 20.2%; amnestic AD, 7.7%; P = .04), race/ethnicity (PCA, 88.3% white; lvPPA, 99.0% white; amnestic AD, 80.0% white; P < .001), and mean (SD) educational level (PCA, 15.7 [3.2] years; lvPPA, 16.2 [3.3] years; amnestic AD, 14.8 [3.5] years; P = .02). A total of 18 of the 95 patients with PCA (18.9%) reported a history of LD, which is greater than the 3 of 100 patients (3.0%) in the amnestic AD cohort (P < .001) and the 10.0% expected rate in the general population (P = .007). In the PCA cohort, 13 of 95 patients (13.7%) had a nonlanguage mathematical and/or visuospatial LD; this rate was greater than that in the amnestic AD (1 of 100 [1.0%]; P < .001) and lvPPA (2 of 84 [2.4%]; P = .006) cohorts and greater than the 6.0% expected general population rate of mathematical LD (P = .003). Compared with the patients with PCA without LDs, the group with LDs had greater preservation of global cognition and a more right-lateralized pattern of atrophy.

Conclusions and Relevance

Nonlanguage mathematical and visuospatial LDs were associated with focal, visuospatial predominant neurodegenerative clinical syndromes. This finding supports the hypothesis that neurodevelopmental differences in specific brain networks are associated with phenotypic manifestation of later-life neurodegenerative disease.

Implications of Change/Stability Patterns in Children's Non-symbolic and Symbolic Magnitude Judgment Abilities Over One Year: A Latent Transition Analysis

Non-symbolic magnitude abilities are often claimed to support the acquisition of symbolic magnitude abilities, which, in turn, are claimed to support emerging math abilities. However, not all studies find links between non-symbolic and symbolic magnitude abilities, or between them and math ability. To investigate possible reasons for these different findings, recent research has analyzed differences in non-symbolic/symbolic magnitude abilities using latent class modeling and has identified four different magnitude ability profiles residing within the general magnitude ability distribution that were differentially related to cognitive and math abilities. These findings may help explain the different patterns of findings observed in previous research. To further investigate this possibility, we (1) attempted to replicate earlier findings, (2) determine whether magnitude ability profiles remained stable or changed over 1 year; and (3) assessed the degree to which stability/change in profiles were related to cognitive and math abilities. We used latent transition analysis to investigate stability/changes in non-symbolic and symbolic magnitude abilities of 109 5- to 6-year olds twice in 1 year. At Time 1 and 2, non-symbolic and symbolic magnitude abilities, number transcoding and single-digit addition abilities were assessed. Visuospatial working memory (VSWM), naming numbers, non-verbal IQ, basic RT was also assessed at Time 1. Analysis showed stability in one profile and changes in the three others over 1 year. VSWM and naming numbers predicted profile membership at Time 1 and 2, and profile membership predicted math abilities at both time points. The findings confirm the existence of four different non-symbolic-symbolic magnitude ability profiles; we suggest the changes over time in them potentially reflect deficit, delay, and normal math developmental pathways.

"Approximate number system" training: A perceptual learning approach

Recent research suggests that humans perceive quantity using a non-symbolic "number sense." This sense is then thought to provide a foundation for understanding symbolic numbers in formal education. Given this link, there has been interest in the extent to which the approximate number system (ANS) can be improved via dedicated training, as this could provide a route to improving performance in symbolic mathematics. However, current evidence regarding the trainability of the ANS comes largely from studies that have used short training durations, leaving open the question of whether improvements occur over a longer time span. To address this limitation, we utilized a perceptual learning approach to investigate the extent to which long-term (8,000+ trials) training modifies the ANS. Consistent with the general methodological approach common in the domain of perceptual learning (where learning specificity is commonly observed), we also examined whether ANS training generalizes to: (a) untrained locations in the visual field; (b) an enumeration task; (c) a higher-level ratio comparison task; and (d) arithmetic ability. In contrast to previous short-term training studies showing that ANS learning quickly asymptotes, our long-term training approach revealed that performance continued to improve even after thousands of trials. We further found that the training generalized to untrained visual locations. At post-test there was non-significant evidence for generalization to a low-level enumeration task, but not to our high-level tasks, including ratio comparison, multi-object tracking, and arithmetic performance. These results demonstrate the potential utility of long-term psychophysical training, but also suggest that ANS training alone (even long-duration training) may be insufficient to modify higher-level math skills.

The measurement of approximate number system acuity across the lifespan is compromised by congruency effects

Recent studies have highlighted the influence of visual cues such as dot size and cumulative surface area on the measurement of the approximate number system (ANS). Previous studies assessing ANS acuity in ageing have all applied stimuli generated by the Panamath protocol, which does not control nor measure the influence of convex hull. Crucially, convex hull has recently been identified as an influential visual cue present in dot arrays, with its impact on older adults’ ANS acuity yet to be investigated. The current study therefore investigated the manipulation of convex hull by the Panamath protocol, and its effect on the measurement of ANS acuity in younger and older participants. First, analyses of the stimuli generated by Panamath revealed a confound between numerosity ratio and convex hull ratio. Second, although older adults were somewhat less accurate than younger adults on convex hull incongruent trials, ANS acuity was broadly similar between the groups. These findings have implications for the valid measurement of ANS acuity across all ages, and suggest that the Panamath protocol produces stimuli that do not adequately control for the influence of convex hull on numerosity discrimination.

A Cognitive Dimensional Approach to Understanding Shared and Unique Contributions to Reading, Math, and Attention Skills

Disorders of reading, math, and attention frequently co-occur in children. However, it is not yet clear which cognitive factors contribute to comorbidities among multiple disorders and which uniquely relate to one, especially because they have rarely been studied as a triad. Thus, the present study considers how reading, math, and attention relate to phonological awareness, numerosity, working memory, and processing speed, all implicated as either unique or shared correlates of these disorders. In response to findings that the attributes of all three disorders exist on a continuum rather than representing qualitatively different groups, this study employed a dimensional approach. Furthermore, we used both timed and untimed academic variables in addition to attention and activity level variables. The results supported the role of working memory and phonological awareness in the overlap among reading, math, and attention, with a limited role of processing speed. Numerosity was related to the comorbidity between math and attention. The results from timed variables and activity level were similar to those from untimed and attention variables, although activity level was less strongly related to cognitive and academic/attention variables. These findings have implications for understanding cognitive deficits that contribute to comorbid reading disability, math disability, and/or attention-deficit/hyperactivity disorder.

Color Afterimages in Autistic Adults

It has been suggested that attenuated adaptation to visual stimuli in autism is the result of atypical perceptual priors (e.g., Pellicano and Burr in Trends Cogn Sci 16(10):504-510, 2012. doi: 10.1016/j.tics.2012.08.009 ). This study investigated adaptation to color in autistic adults, measuring both strength of afterimage and the influence of top-down knowledge. We found no difference in color afterimage strength between autistic and typical adults. Effects of top-down knowledge on afterimage intensity shown by Lupyan (Acta Psychol 161:117-130, 2015. doi: 10.1016/j.actpsy.2015.08.006 ) were not replicated for either group. This study finds intact color adaptation in autistic adults. This is in contrast to findings of attenuated adaptation to faces and numerosity in autistic children. Future research should investigate the possibility of developmental differences in adaptation and further examine top-down effects on adaptation.

Resting-state functional connectivity and motor imagery brain activation

Motor imagery (MI) relies on the mental simulation of an action without any overt motor execution (ME), and can facilitate motor learning and enhance the effect of rehabilitation in patients with neurological conditions. While functional magnetic resonance imaging (fMRI) during MI and ME reveals shared cortical representations, the role and functional relevance of the resting-state functional connectivity (RSFC) of brain regions involved in MI is yet unknown. Here, we performed resting-state fMRI followed by fMRI during ME and MI with the dominant hand. We used a behavioral chronometry test to measure ME and MI movement duration and compute an index of performance (IP). Then, we analyzed the voxel-matched correlation between the individual MI parameter estimates and seed-based RSFC maps in the MI network to measure the correspondence between RSFC and MI fMRI activation. We found that inter-individual differences in intrinsic connectivity in the MI network predicted several clusters of activation. Taken together, present findings provide first evidence that RSFC within the MI network is predictive of the activation of MI brain regions, including those associated with behavioral performance, thus suggesting a role for RSFC in obtaining a deeper understanding of neural substrates of MI and of MI ability. Hum Brain Mapp 37:3847-3857, 2016. © 2016 Wiley Periodicals, Inc.

Numerosity estimation benefits from transsaccadic information integration

Humans achieve a stable and homogeneous representation of their visual environment, although visual processing varies across the visual field. Here we investigated the circumstances under which peripheral and foveal information is integrated for numerosity estimation across saccades. We asked our participants to judge the number of black and white dots on a screen. Information was presented either in the periphery before a saccade, in the fovea after a saccade, or in both areas consecutively to measure transsaccadic integration. In contrast to previous findings, we found an underestimation of numerosity for foveal presentation and an overestimation for peripheral presentation. We used a maximum-likelihood model to predict accuracy and reliability in the transsaccadic condition based on peripheral and foveal values. We found near-optimal integration of peripheral and foveal information, consistently with previous findings about orientation integration. In three consecutive experiments, we disrupted object continuity between the peripheral and foveal presentations to probe the limits of transsaccadic integration. Even for global changes on our numerosity stimuli, no influence of object discontinuity was observed. Overall, our results suggest that transsaccadic integration is a robust mechanism that also works for complex visual features such as numerosity and is operative despite internal or external mismatches between foveal and peripheral information. Transsaccadic integration facilitates an accurate and reliable perception of our environment.

Inhibitory control and decimal number comparison in school-aged children

School-aged children erroneously think that 1.45 is larger 1.5 because 45 is larger than 5. Using a negative priming paradigm, we investigated whether the ability to compare the magnitude of decimal numbers in the context in which the smallest number has the greatest number of digits after the decimal point (1.45 vs. 1.5) is rooted in part on the ability to inhibit the “greater the number of digits the greater its magnitude” misconception derived from a property of whole numbers. In Experiment 1, we found a typical negative priming effect with 7^th graders requiring more time to compare decimal numbers in which the largest number has the greatest number of digits after the decimal point (1.65 vs. 1.5) after comparing decimal numbers in which the smallest number has the greatest number of digits after the decimal point (1.45 vs. 1.5) than after comparing decimal numbers with the same number of digits after the decimal point (1.5 vs. 1.6). In Experiment 2, we found a negative priming effect when decimal numbers preceded items in which 7^th graders had to compare the length of two lines. Taken together our results suggest that the ability to compare decimal numbers in which the smallest number has the greatest number of digits is rooted in part on the ability to inhibit the “greater the number of digits the greater its magnitude” misconception and in part on the ability to inhibit the length of the decimal number per se.

Adaptation to numerosity requires only brief exposures, and is determined by number of events, not exposure duration

Exposure to a patch of dots produces a repulsive shift in the perceived numerosity of subsequently viewed dot patches. Although a remarkably strong effect, in which the perceived numerosity can be shifted by up to 50% of the actual numerosity, very little is known about the temporal dynamics. Here we demonstrate a novel adaptation paradigm that allows numerosity adaptation to be rapidly induced at several distinct locations simultaneously. We show that not only is this adaptation to numerosity spatially specific, with different locations of the visual field able to be adapted to high, low, or neutral stimuli, but it can occur with only very brief periods of adaptation. Further investigation revealed that the adaptation effect was primarily driven by the number of unique adapting events that had occurred and not by either the duration of each event or the total duration of exposure to adapting stimuli. This event-based numerosity adaptation appears to fit well with statistical models of adaptation in which the dynamic adjustment of perceptual experiences, based on both the previous experience of the stimuli and the current percept, acts to optimize the limited working range of perception. These results implicate a highly plastic mechanism for numerosity perception, which is dependent on the number of discrete adaptation events, and also demonstrate a quick and efficient paradigm suitable for examining the temporal properties of adaptation.

Improving Preschoolers' Arithmetic through Number Magnitude Training: The Impact of Non-Symbolic and Symbolic Training

The numerical cognition literature offers two views to explain numerical and arithmetical development. The unique-representation view considers the approximate number system (ANS) to represent the magnitude of both symbolic and non-symbolic numbers and to be the basis of numerical learning. In contrast, the dual-representation view suggests that symbolic and non-symbolic skills rely on different magnitude representations and that it is the ability to build an exact representation of symbolic numbers that underlies math learning. Support for these hypotheses has come mainly from correlative studies with inconsistent results. In this study, we developed two training programs aiming at enhancing the magnitude processing of either non-symbolic numbers or symbolic numbers and compared their effects on arithmetic skills. Fifty-six preschoolers were randomly assigned to one of three 10-session-training conditions: (1) non-symbolic training (2) symbolic training and (3) control training working on story understanding. Both numerical training conditions were significantly more efficient than the control condition in improving magnitude processing. Moreover, symbolic training led to a significantly larger improvement in arithmetic than did non-symbolic training and the control condition. These results support the dual-representation view.

Leader-follower consensus and synchronization in numerosity-constrained networks with dynamic leadership

In this work, we study leader-follower consensus and synchronization protocols over a stochastically switching network. The agents representing the followers can communicate with any other agent, whereas the agents serving as leaders are restricted to interact only with the other leaders. The model incorporates the phenomenon of numerosity, which limits the perceptual capacity of the agents while allowing for shuffling with whom each individual interacts at each time step. We derive closed form expressions for necessary and sufficient conditions for consensus, the rate of convergence to consensus, and conditions for stochastic synchronization in terms of the asymptotic convergence factor. We provide simulation results to validate the theoretical findings and to illustrate the dependence of this factor on system parameters. The closed form results enable us to study the factors affecting the feasibility of consensus. We show that agents' traits can be chosen for an engineered system to maximize the convergence speed and that protocol speed is enhanced as the proportion of the leaders increases in certain cases. These results may find application in the design and control of an engineered leader-follower system, where consensus or synchronization at the fastest possible rate is desired.

Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review

It is widely accepted that human and nonhuman species possess a specialized system to process large approximate numerosities. The theory of an evolutionarily ancient approximate number system (ANS) has received converging support from developmental studies, comparative experiments, neuroimaging, and computational modelling, and it is one of the most dominant and influential theories in numerical cognition. The existence of an ANS system is significant, as it is believed to be the building block of numerical development in general. The acuity of the ANS is related to future arithmetic achievements, and intervention strategies therefore aim to improve the ANS. Here we critically review current evidence supporting the existence of an ANS. We show that important shortcomings and confounds exist in the empirical studies on human and non-human animals as well as the logic used to build computational models that support the ANS theory. We conclude that rather than taking the ANS theory for granted, a more comprehensive explanation might be provided by a sensory-integration system that compares or estimates large approximate numerosities by integrating the different sensory cues comprising number stimuli.

Dyscalculia and the Calculating Brain

Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to "many" with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.

It all adds up …. Or does it? Numbers, mathematics and purpose

No chimpanzee knows what a square root is, let alone a complex number. Yet not only our closest ape cousins but even some invertebrates, possess a capacity for numerosity, that is the ability to assess relative numerical magnitudes and distances. That numerosity should confer adaptive advantages, such as social species that choose shoal size, is obvious. Moreover, it is widely assumed that numerosity and mathematics are seamlessly linked, as would be consistent with Darwinian notions of descent and modification. Animal numerosity, however, involves sensory processes (usually vision, but other modalities such as olfaction can be as effective) that follow psychophysical principles, notable the Weber-Fechner law. In contrast, mathematics may require sensory mediation but is an abstract process. The supposed connection between these processes is described as supramodality but the mechanisms that allow humans, but not animals, to engage in even simple mathematics are opaque. Here, I argue that any resolution will depend on proper explanations for not only mathematics, but language and by implication consciousness. In this light, concepts of purpose are not intellectual mirages but legitimate descriptions of the worlds in which we are embedded. These are both visible (and tangible) and invisible (and although intangible, equally real).

Dot Display Affects Approximate Number System Acuity and Relationships with Mathematical Achievement and Inhibitory Control

Much research has investigated the relationship between the Approximate Number System (ANS) and mathematical achievement, with continued debate surrounding the existence of such a link. The use of different stimulus displays may account for discrepancies in the findings. Indeed, closer scrutiny of the literature suggests that studies supporting a link between ANS acuity and mathematical achievement in adults have mostly measured the ANS using spatially intermixed displays (e.g. of blue and yellow dots), whereas those failing to replicate a link have primarily used spatially separated dot displays. The current study directly compared ANS acuity when using intermixed or separate dots, investigating how such methodological variation mediated the relationship between ANS acuity and mathematical achievement. ANS acuity was poorer and less reliable when measured with intermixed displays, with performance during both conditions related to inhibitory control. Crucially, mathematical achievement was significantly related to ANS accuracy difference (accuracy on congruent trials minus accuracy on incongruent trials) when measured with intermixed displays, but not with separate displays. The findings indicate that methodological variation affects ANS acuity outcomes, as well as the apparent relationship between the ANS and mathematical achievement. Moreover, the current study highlights the problem of low reliabilities of ANS measures. Further research is required to construct ANS measures with improved reliability, and to understand which processes may be responsible for the increased likelihood of finding a correlation between the ANS and mathematical achievement when using intermixed displays.

Non-symbolic division in childhood

The approximate number system (ANS) underlies representations of large numbers of objects as well as the additive, subtractive, and multiplicative relationships between them. In this set of studies, 5- and 6-year-old children were shown a series of video-based events that conveyed a transformation of a large number of objects into one-half or one-quarter of the original number. Children were able to estimate correctly the outcomes to these halving and quartering problems, and they based their responses on scaling by number, not on continuous quantities or guessing strategies. Children's performance exhibited the ratio signature of the ANS. Moreover, children performed above chance on relatively early trials, suggesting that this scaling operation is easily conveyed and readily performed. The results support the existence of a flexible and substantially untrained capacity to scale numerical amounts.

Near-optimal Integration of Magnitude in the Human Parietal Cortex

Humans are often observed to make optimal sensorimotor decisions but to be poor judges of situations involving explicit estimation of magnitudes or numerical quantities. For example, when drawing conclusions from data, humans tend to neglect the size of the sample from which it was collected. Here, we asked whether this sample size neglect is a general property of human decisions and investigated its neural implementation. Participants viewed eight discrete visual arrays (samples) depicting variable numbers of blue and pink balls. They then judged whether the samples were being drawn from an urn in which blue or pink predominated. A participant who neglects the sample size will integrate the ratio of balls on each array, giving equal weight to each sample. However, we found that human behavior resembled that of an optimal observer, giving more credence to larger sample sizes. Recording scalp EEG signals while participants performed the task allowed us to assess the decision information that was computed during integration. We found that neural signals over the posterior cortex after each sample correlated first with the sample size and then with the difference in the number of balls in either category. Moreover, lateralized beta-band activity over motor cortex was predicted by the cumulative difference in number of balls in each category. Together, these findings suggest that humans achieve statistically near-optimal decisions by adding up the difference in evidence on each sample, and imply that sample size neglect may not be a general feature of human decision-making.

Stable measures of number sense accuracy in math learning disability: Is it time to proceed from basic science to clinical application?

Math learning disability (MLD) or developmental dyscalculia is a highly prevalent and persistent difficulty in learning arithmetic that may be explained by different cognitive mechanisms. The accuracy of the number sense has been implicated by some evidence as a core deficit in MLD. However, research on this topic has been mainly conducted in demographically selected samples, using arbitrary cut-off scores to characterize MLD. The clinical relevance of the association between number sense and MLD remains to be investigated. In this study, we aimed at assessing the stability of a number sense accuracy measure (w) across five experimental sessions, in two clinically defined cases of MLD. Stable measures of number sense accuracy estimate are required to clinically characterize subtypes of MLD and to make theoretical inferences regarding the underlying cognitive mechanisms. G. A. was a 10-year-old boy with MLD in the context of dyslexia and phonological processing impairment and his performance remained steadily in the typical scores range. The performance of H. V., a 9-year-old girl with MLD associated with number sense inaccuracy, remained consistently impaired across measurements, with a nonsignificant tendency to worsen. Qualitatively, H. V.'s performance was also characterized by greater variability across sessions. Concomitant clinical observations suggested that H. V.'s difficulties could be aggravated by developing symptoms of mathematics anxiety. Results in these two cases are in line with the hypotheses that at least two reliable patterns of cognitive impairment may underlie math learning difficulties in MLD, one related to number sense inaccuracy and the other to phonological processing impairment. Additionally, it indicates the need for more translational research in order to examine the usefulness and validity of theoretical advances in numerical cognition to the clinical neuropsychological practice with MLD.

Topology-defined units in numerosity perception

What is a number? The number sense hypothesis suggests that numerosity is "a primary visual property" like color, contrast, or orientation. However, exactly what attribute of a stimulus is the primary visual property and determines numbers in the number sense? To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for low-level features (e.g., orientation, color, and size). Subjects performed discrimination, estimation, and equality judgment tasks in a wide range of presentation durations and across small and large numbers. Results consistently show that connecting/enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. In contrast, grouping based on color similarity had no effect on numerosity judgment. We propose that numbers or the primitive units counted in numerosity perception are influenced by topological invariants, such as connectivity and the inside/outside relationship. Beyond the behavioral measures, neural tuning curves to numerosity in the intraparietal sulcus were obtained using functional MRI adaptation, and the tuning curves showed that numbers represented in the intraparietal sulcus were strongly influenced by topology.

Numerosity processing is context driven even in the subitizing range: An fMRI study

Numerical judgments are involved in almost every aspect of our daily life. They are carried out so efficiently that they are often considered to be automatic and innate. However, numerosity of non-symbolic stimuli is highly correlated with its continuous properties (e.g., density, area), and so it is hard to determine whether numerosity and continuous properties rely on the same mechanism. Here we examined the behavioral and neuronal mechanisms underlying such judgments. We scanned subjects' hemodynamic responses to a numerosity comparison task and to a surface area comparison task. In these tasks, numerical and continuous magnitudes could be either congruent or incongruent. Behaviorally, an interaction between the order of the tasks and the relevant dimension modulated the congruency effects. Continuous magnitudes always interfered with numerosity comparison. Numerosity, on the other hand, interfered with the surface area comparison only when participants began with the numerosity task. Hemodynamic activity showed that context (induced by task order) determined the neuronal pathways in which the dimensions were processed. Starting with the numerosity task led to enhanced activity in the right hemisphere, while starting with the continuous task led to enhanced left hemisphere activity. Continuous magnitudes processing relied on activation of the frontal eye field and the post-central gyrus. Processing of numerosities, on the other hand, relied on deactivation of these areas, suggesting active suppression of the continuous dimension. Accordingly, we suggest that numerosities, even in the subitizing range, are not always processed automatically; their processing depends on context and task demands.

Mathematical difficulties in developmental coordination disorder: Symbolic and nonsymbolic number processing

At school, children with Developmental Coordination Disorder (DCD) struggle with mathematics. However, little attention has been paid to their numerical cognition abilities. The goal of this study was to better understand the cognitive basis for mathematical difficulties in children with DCD. Twenty 7-to-10 years-old children with DCD were compared to twenty age-matched typically developing children using dot and digit comparison tasks to assess symbolic and nonsymbolic number processing and in a task of single digits additions. Results showed that children with DCD had lower performance in nonsymbolic and symbolic number comparison tasks than typically developing children. They were also slower to solve simple addition problems. Moreover, correlational analyses showed that children with DCD who experienced greater impairments in the nonsymbolic task also performed more poorly in the symbolic tasks. These findings suggest that DCD impairs both nonsymbolic and symbolic number processing. A systematic assessment of numerical cognition in children with DCD could provide a more comprehensive picture of their deficits and help in proposing specific remediation.