Visual sustained attention and numerosity sensitivity correlate with math achievement in children

Visual form perception is fundamental for both reading comprehension and arithmetic computation

Visual perception has been found to be a critical factor for reading comprehension and arithmetic computation in separate lines of research with different measures of visual form perception. The current study of 1099 Chinese elementary school students investigated whether the same visual form perception (assessed by a geometric figure matching task) underlies both reading comprehension and arithmetic computation. The results showed that visual form perception had close relations with both reading comprehension and arithmetic computation, even after controlling for age, gender, and cognitive factors such as processing speed, attention, working memory, visuo-spatial processing, and general intelligence. Results also showed that numerosity comparison's relations with reading comprehension and arithmetic computation were fully accounted for by visual form perception. These results suggest that reading comprehension and arithmetic computation might share a similar visual form processing mechanism.

Dyscalculia and Typical Math Achievement Are Associated With Individual Differences in Number-Specific Executive Function

Deficits in numerical magnitude perception characterize the mathematics learning disability developmental dyscalculia (DD), but recent studies suggest the relation stems from inhibitory control demands from incongruent visual cues in the nonsymbolic number comparison task. This study investigated the relation among magnitude perception during differing congruency conditions, executive function, and mathematics achievement measured longitudinally in children (n = 448) from ages 4 to 13. This relation was investigated across achievement groups and as it related to mathematics across the full range of achievement. Only performance on incongruent trials related to achievement. Findings indicate that executive function in a numerical context, beyond magnitude perception or executive function in a non-numerical context, relates to DD and mathematics across a wide range of achievement.

Asymmetrical interference between number and item size perception provides evidence for a domain specific impairment in dyscalculia

Dyscalculia, a specific learning disability that impacts arithmetical skills, has previously been associated to a deficit in the precision of the system that estimates the approximate number of objects in visual scenes (the so called 'number sense' system). However, because in tasks involving numerosity comparisons dyscalculics' judgements appears disproportionally affected by continuous quantitative dimensions (such as the size of the items), an alternative view linked dyscalculia to a domain-general difficulty in inhibiting task-irrelevant responses. To arbitrate between these views, we evaluated the degree of reciprocal interference between numerical and non-numerical quantitative dimensions in adult dyscalculics and matched controls. We used a novel stimulus set orthogonally varying in mean item size and numerosity, putting particular attention into matching both features' perceptual discriminability. Participants compared those stimuli based on each of the two dimensions. While control subjects showed no significant size interference when judging numerosity, dyscalculics' numerosity judgments were strongly biased by the unattended size dimension. Importantly however, both groups showed the same degree of interference from the unattended dimension when judging mean size. Moreover, only the ability to discard the irrelevant size information when comparing numerosity (but not the reverse) significantly predicted calculation ability across subjects. Overall, our results show that numerosity discrimination is less prone to interference than discrimination of another quantitative feature (mean item size) when the perceptual discriminability of these features is matched, as here in control subjects. By quantifying, for the first time, dyscalculic subjects' degree of interference on another orthogonal dimension of the same stimuli, we are able to exclude a domain-general inhibition deficit as explanation for their poor / biased numerical judgement. We suggest that enhanced reliance on non-numerical cues during numerosity discrimination can represent a strategy to cope with a less precise number sense.

Dynamics and development in number-to-space mapping

Young children's estimates of numerical magnitude increase approximately logarithmically with actual magnitude. The conventional interpretation of this finding is that children's estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggests that logarithmic number-line estimates emerge via a dynamic encoding mechanism that is sensitive to previously encountered stimuli. Here we examine trial-to-trial changes in logarithmicity of numerosity estimates to test an alternative dynamic model (D-MLLM) with both a strong logarithmic component and a weak response to previous stimuli. In support of D-MLLM, first-trial numerosity estimates in both adults (Study 1, 2, 3, and 4) and children (Study 4) were strongly logarithmic, despite zero previous stimuli. Additionally, although numerosity of a previous trial affected adults' estimates, the influence of previous numbers always accompanied the logarithmic-to-linear shift predicted by D-MLLM. We conclude that a dynamic encoding mechanism is not necessary for compressive mapping, but sequential effects on response scaling are a possible source of linearity in adults' numerosity estimation.

Numerical cognition is resilient to dramatic changes in early sensory experience

Humans and non-human animals can approximate large visual quantities without counting. The approximate number representations underlying this ability are noisy, with the amount of noise proportional to the quantity being represented. Numerate humans also have access to a separate system for representing exact quantities using number symbols and words; it is this second, exact system that supports most of formal mathematics. Although numerical approximation abilities and symbolic number abilities are distinct in representational format and in their phylogenetic and ontogenetic histories, they appear to be linked throughout development--individuals who can more precisely discriminate quantities without counting are better at math. The origins of this relationship are debated. On the one hand, symbolic number abilities may be directly linked to, perhaps even rooted in, numerical approximation abilities. On the other hand, the relationship between the two systems may simply reflect their independent relationships with visual abilities. To test this possibility, we asked whether approximate number and symbolic math abilities are linked in congenitally blind individuals who have never experienced visual sets or used visual strategies to learn math. Congenitally blind and blind-folded sighted participants completed an auditory numerical approximation task, as well as a symbolic arithmetic task and non-math control tasks. We found that the precision of approximate number representations was identical across congenitally blind and sighted groups, suggesting that the development of the Approximate Number System (ANS) does not depend on visual experience. Crucially, the relationship between numerical approximation and symbolic math abilities is preserved in congenitally blind individuals. These data support the idea that the Approximate Number System and symbolic number abilities are intrinsically linked, rather than indirectly linked through visual abilities.

Psychophysical evidence for the number sense

It is now clear that most animals, including humans, possess an ability to rapidly estimate number. Some have questioned whether this ability arises from dedicated numerosity mechanisms, or is derived indirectly from judgements of density or other attributes. We describe a series of psychophysical experiments, largely using adaptation techniques, which demonstrate clearly the existence of a number sense in humans. The number sense is truly general, extending over space, time and sensory modality, and is closely linked with action. We further show that when multiple cues are present, numerosity emerges as the natural dimension for discrimination. However, when element density increases past a certain level, the elements become too crowded to parse, and the scene is perceived as a texture rather than array of elements. The two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws, and show different dependencies on eccentricity, luminance levels and effects of perceptual grouping. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

This article is part of the discussion meeting issue ‘The origins of numerical abilities’.

Preschoolers' selective sustained attention and numeracy skills and knowledge

This brief report addresses preschoolers' selective sustained attention (SSA) and early numeracy skills and knowledge. Past research indicates that children's attention and early numeracy are positively associated, yet some concerns have emerged about the age appropriateness of tools used to measure preschoolers' SSA. This study used a new measure-the Track-It Task-that demonstrates strong psychometric properties. In total, 31 at-risk preschoolers (M_age = 46.6 months) participated and were assessed on SSA, nonsymbolic quantity discrimination, and symbolic quantitative skills and knowledge. The ability to sustain attention in the face of distractions was positively correlated with preschoolers' verbal counting and one-to-one correspondence, Arabic numeral recognition, and cardinal principle knowledge. SSA was not significantly associated with child age or performance on a memory task. This study provides preliminary evidence that SSA may facilitate the process whereby young children become reliable counters and learn that the symbol system of numbers represents specific quantities.

Prioritization to visual objects: Roles of sensory uncertainty

Two hypotheses, attentional prioritization and attentional spreading, have been proposed to account for object-based attention. The attentional-prioritization hypothesis posits that the positional uncertainty of targets is sufficient to resolve the controversy raised by the competing attentional-spreading hypothesis. Here we challenge the sufficiency of this explanation by showing that object-based attention is a function of sensory uncertainty in a task with consistent high positional uncertainty of the targets. In Experiment 1, object-based attention was modulated by sensory uncertainty induced by the noise from backward masking, showing an object-based effect under high as compared to low sensory uncertainty. This finding was replicated in Experiment 2 with increased task difficulty, to exclude that as a confounding factor, and in Experiment 3 with a psychophysical method, to obtain converging evidence using perceptual threshold measurement. Additionally, such a finding was not observed when sensory uncertainty was eliminated by replacing the backward-masking stimuli with perceptually dissimilar ones in Experiment 4. These results reveal that object-based attention is influenced by sensory uncertainty, even under high positional uncertainty of the targets. Our findings contradict the proposition of attentional spreading, proposing instead an automatic form of object-based attention due to enhancement of the perceptual representation. More importantly, the attentional-prioritization hypothesis based solely on positional uncertainty cannot sufficiently account for object-based attention, but needs to be developed by expanding the concept of uncertainty to include at least sensory uncertainty.

Attention Contributes to Arithmetic Deficits in New-Onset Childhood Absence Epilepsy

Neuropsychological studies indicate that new-onset childhood absence epilepsy (CAE) is associated with deficits in attention and executive functioning. However, the contribution of these deficits to impaired academic performance remains unclear. We aimed to examine whether attention and executive functioning deficits account for the academic difficulties prevalent in patients with new-onset CAE. We analyzed cognitive performance in several domains, including language, mathematics, psychomotor speed, spatial ability, memory, general intelligence, attention, and executive functioning, in 35 children with new-onset CAE and 33 control participants. Patients with new-onset CAE exhibited deficits in mathematics, general intelligence, attention, and executive functioning. Furthermore, attention deficits, as measured by a visual tracing task, accounted for impaired arithmetic performance in the new-onset CAE group. Therefore, attention deficits, rather than impaired general intelligence or executive functioning, may be responsible for arithmetic performance deficits in patients with new-onset CAE.

Number Line Estimation Predicts Mathematical Skills: Difference in Grades 2 and 4

Studies have shown that number line estimation is important for learning. However, it is yet unclear if number line estimation predicts different mathematical skills in different grades after controlling for age, non-verbal cognitive ability, attention, and working memory. The purpose of this study was to examine the role of number line estimation on two mathematical skills (calculation fluency and math problem-solving) in grade 2 and grade 4. One hundred and forty-eight children from Shanghai, China were assessed on measures of number line estimation, non-verbal cognitive ability (non-verbal matrices), working memory (N-back), attention (expressive attention), and mathematical skills (calculation fluency and math problem-solving). The results showed that in grade 2, number line estimation correlated significantly with calculation fluency (r = -0.27, p < 0.05) and math problem-solving (r = -0.52, p < 0.01). In grade 4, number line estimation correlated significantly with math problem-solving (r = -0.38, p < 0.01), but not with calculation fluency. Regression analyses indicated that in grade 2, number line estimation accounted for unique variance in math problem-solving (12.0%) and calculation fluency (4.0%) after controlling for the effects of age, non-verbal cognitive ability, attention, and working memory. In grade 4, number line estimation accounted for unique variance in math problem-solving (9.0%) but not in calculation fluency. These findings suggested that number line estimation had an important role in math problem-solving for both grades 2 and 4 children and in calculation fluency for grade 2 children. We concluded that number line estimation could be a useful indicator for teachers to identify and improve children’s mathematical skills.

Visual Form Perception Can Be a Cognitive Correlate of Lower Level Math Categories for Teenagers

Numerous studies have assessed the cognitive correlates of performance in mathematics, but little research has been conducted to systematically examine the relations between visual perception as the starting point of visuospatial processing and typical mathematical performance. In the current study, we recruited 223 seventh graders to perform a visual form perception task (figure matching), numerosity comparison, digit comparison, exact computation, approximate computation, and curriculum-based mathematical achievement tests. Results showed that, after controlling for gender, age, and five general cognitive processes (choice reaction time, visual tracing, mental rotation, spatial working memory, and non-verbal matrices reasoning), visual form perception had unique contributions to numerosity comparison, digit comparison, and exact computation, but had no significant relation with approximate computation or curriculum-based mathematical achievement. These results suggest that visual form perception is an important independent cognitive correlate of lower level math categories, including the approximate number system, digit comparison, and exact computation.

Deficits in Approximate Number System Acuity and Mathematical Abilities in 6.5-Year-Old Children Born Extremely Preterm

Preterm children are at increased risk for poor academic achievement, especially in math. In the present study, we examined whether preterm children differ from term-born children in their intuitive sense of number that relies on an unlearned, approximate number system (ANS) and whether there is a link between preterm children’s ANS acuity and their math abilities. To this end, 6.5-year-old extremely preterm (i.e., <27 weeks gestation, n = 82) and term-born children (n = 89) completed a non-symbolic number comparison (ANS acuity) task and a standardized math test. We found that extremely preterm children had significantly lower ANS acuity than term-born children and that these differences could not be fully explained by differences in verbal IQ, perceptual reasoning skills, working memory, or attention. Differences in ANS acuity persisted even when demands on visuo-spatial skills and attention were reduced in the ANS task. Finally, we found that ANS acuity and math ability are linked in extremely preterm children, similar to previous results from term-born children. These results suggest that deficits in the ANS may be at least partly responsible for the deficits in math abilities often observed in extremely preterm children.

Impact of Attention Training on Academic Achievement, Executive Functioning, and Behavior: A Randomized Controlled Trial

Children with intellectual and developmental disabilities (IDD) experience significant difficulties in attention, learning, executive functions, and behavioral regulation. Emerging evidence suggests that computerized cognitive training may remediate these impairments. In a double blind controlled trial, 76 children with IDD (4-11 years) were randomized to either an attention training (n = 38) or control program (n = 38). Both programs were completed at home over a 5-week period. Outcome measures assessed literacy, numeracy, executive functioning, and behavioral/emotional problems, and were conducted at baseline, post-training, and 3-month follow-up. No training effects were observed at post-training; however, children in the training group showed greater improvements in numeracy skills at the 3-month follow-up. These results suggest that attention training may be beneficial for children with IDD; however, the modest nature of the intervention effects indicate that caution should be taken when interpreting clinical significance.

Effects of adaptation on numerosity decoding in the human brain

Psychophysical studies have shown that numerosity is a sensory attribute susceptible to adaptation. Neuroimaging studies have reported that, at least for relatively low numbers, numerosity can be accurately discriminated in the intra-parietal sulcus. Here we developed a novel rapid adaptation paradigm where adapting and test stimuli are separated by pauses sufficient to dissociate their BOLD activity. We used multivariate pattern recognition to classify brain activity evoked by non-symbolic numbers over a wide range (20-80), both before and after psychophysical adaptation to the highest numerosity. Adaptation caused underestimation of all lower numerosities, and decreased slightly the average BOLD responses in V1 and IPS. Using support vector machine, we showed that the BOLD response of IPS, but not in V1, classified numerosity well, both when tested before and after adaptation. However, there was no transfer from training pre-adaptation responses to testing post-adaptation, and vice versa, indicating that adaptation changes the neuronal representation of the numerosity. Interestingly, decoding was more accurate after adaptation, and the amount of improvement correlated with the amount of perceptual underestimation of numerosity across subjects. These results suggest that numerosity adaptation acts directly on IPS, rather than indirectly via other low-level stimulus parameters analysis, and that adaptation improves the capacity to discriminate numerosity.

Do Elite and Amateur Soccer Players Outperform Non-Athletes on Neurocognitive Functioning? A Study Among 8-12 Year Old Children

AIM

Research suggested a positive association between physical fitness and neurocognitive functioning in children. Aim of the present study is to investigate possible dose-response relationships between diverse daily physical activities and a broad range of neurocognitive functions in preadolescent children. Furthermore, the relationship between several sedentary behaviours, including TV-watching, gaming and computer time, and neurocognitive functioning will be investigated in this group of children.

METHODS

A total of 168 preadolescent boys, aged 8 to 12 years, were recruited from various locations, including primary schools, an amateur soccer club, and a professional soccer club, to increase variability in the amount of participation in sports. All children performed neurocognitive tasks measuring inhibition, short term memory, working memory, attention and information processing speed. Regression analyses examined the predictive power of a broad range of physical activities, including sports, active transport to school, physical education (PE), outdoor play, and sedentary behaviour such as TV-watching and gaming, for neurocognitive functioning.

RESULTS

Time spent in sports significantly accounted for the variance in inhibition, short term memory, working memory and lapses of attention, where more time spent in sports was associated with better performance. Outdoor play was also positively associated with working memory. In contrast, time spent on the computer was negatively associated with inhibition.

CONCLUSIONS

Results of the current study suggest a positive relationship between participation in sports and several important neurocognitive functions. Interventions are recommended to increase sports participation and to reduce sedentary behaviour in preadolescent children.

A Systematic Investigation of Accuracy and Response Time Based Measures Used to Index ANS Acuity

The approximate number system (ANS) was proposed to be a building block for later mathematical abilities. Several measures have been used interchangeably to assess ANS acuity. Some of these measures were based on accuracy data, whereas others relied on response time (RT) data or combined accuracy and RT data. Previous studies challenged the view that all these measures can be used interchangeably, because low correlations between some of the measures had been observed. These low correlations might be due to poor reliability of some of the measures, since the majority of these measures are mathematically related. Here we systematically investigated the relationship between common ANS measures while avoiding the potential confound of poor reliability. Our first experiment revealed high correlations between all accuracy based measures supporting the assumption that all of them can be used interchangeably. In contrast, not all RT based measures were highly correlated. Additionally, our results revealed a speed-accuracy trade-off. Thus, accuracy and RT based measures provided conflicting conclusions regarding ANS acuity. Therefore, we investigated in two further experiments which type of measure (accuracy or RT) is more informative about the underlying ANS acuity, depending on participants’ preferences for accuracy or speed. To this end, we manipulated participants’ preferences for accuracy or speed both explicitly using different task instructions and implicitly varying presentation duration. Accuracy based measures were more informative about the underlying ANS acuity than RT based measures. Moreover, the influence of the underlying representations on accuracy data was more pronounced when participants preferred accuracy over speed after the accuracy instruction as well as for long or unlimited presentation durations. Implications regarding the diffusion model as a theoretical framework of dot comparison as well as regarding the relationship between ANS acuity and math performance are discussed.

Spontaneous perception of numerosity in humans

Humans, including infants, and many other species have a capacity for rapid, nonverbal estimation of numerosity. However, the mechanisms for number perception are still not clear; some maintain that the system calculates numerosity via density estimates-similar to those involved in texture-while others maintain that more direct, dedicated mechanisms are involved. Here we show that provided that items are not packed too densely, human subjects are far more sensitive to numerosity than to either density or area. In a two-dimensional space spanning density, area and numerosity, subjects spontaneously react with far greater sensitivity to changes in numerosity, than either area or density. Even in tasks where they were explicitly instructed to make density or area judgments, they responded spontaneously to number. We conclude, that humans extract number information, directly and spontaneously, via dedicated mechanisms.

Numerosity but not texture-density discrimination correlates with math ability in children

Considerable recent work suggests that mathematical abilities in children correlate with the ability to estimate numerosity. Does math correlate only with numerosity estimation, or also with other similar tasks? We measured discrimination thresholds of school-age (6- to 12.5-years-old) children in 3 tasks: numerosity of patterns of relatively sparse, segregatable items (24 dots); numerosity of very dense textured patterns (250 dots); and discrimination of direction of motion. Thresholds in all tasks improved with age, but at different rates, implying the action of different mechanisms: In particular, in young children, thresholds were lower for sparse than textured patterns (the opposite of adults), suggesting earlier maturation of numerosity mechanisms. Importantly, numerosity thresholds for sparse stimuli correlated strongly with math skills, even after controlling for the influence of age, gender and nonverbal IQ. However, neither motion-direction discrimination nor numerosity discrimination of texture patterns showed a significant correlation with math abilities. These results provide further evidence that numerosity and texture-density are perceived by independent neural mechanisms, which develop at different rates; and importantly, only numerosity mechanisms are related to math. As developmental dyscalculia is characterized by a profound deficit in discriminating numerosity, it is fundamental to understand the mechanism behind the discrimination. (PsycINFO Database Record

Global Visual Motion Sensitivity: Associations with Parietal Area and Children's Mathematical Cognition

Sensitivity to global visual motion has been proposed as a signature of brain development, related to the dorsal rather than ventral cortical stream. Thresholds for global motion have been found to be elevated more than for global static form in many developmental disorders, leading to the idea of "dorsal stream vulnerability." Here we explore the association of global motion thresholds with individual differences in children's brain development, in a group of typically developing 5- to 12-year-olds. Good performance was associated with a relative increase in parietal lobe surface area, most strongly around the intraparietal sulcus and decrease in occipital area. In line with the involvement of intraparietal sulcus, areas in visuospatial and numerical cognition, we also found that global motion performance was correlated with tests of visuomotor integration and numerical skills. Individual differences in global form detection showed none of these anatomical or cognitive correlations. This suggests that the correlations with motion sensitivity are unlikely to reflect general perceptual or attentional abilities required for both form and motion. We conclude that individual developmental variations in global motion processing are not linked to greater area in the extrastriate visual areas, which initially process such motion, but in the parietal systems that make decisions based on this information. The overlap with visuospatial and numerical abilities may indicate the anatomical substrate of the "dorsal stream vulnerability" proposed as characterizing neurodevelopmental disorders.

Predicting Children's Reading and Mathematics Achievement from Early Quantitative Knowledge and Domain-General Cognitive Abilities

One hundred children (44 boys) participated in a 3-year longitudinal study of the development of basic quantitative competencies and the relation between these competencies and later mathematics and reading achievement. The children's preliteracy knowledge, intelligence, executive functions, and parental educational background were also assessed. The quantitative tasks assessed a broad range of symbolic and nonsymbolic knowledge and were administered four times across 2 years of preschool. Mathematics achievement was assessed at the end of each of 2 years of preschool, and mathematics and word reading achievement were assessed at the end of kindergarten. Our goals were to determine how domain-general abilities contribute to growth in children's quantitative knowledge and to determine how domain-general and domain-specific abilities contribute to children's preschool mathematics achievement and kindergarten mathematics and reading achievement. We first identified four core quantitative competencies (e.g., knowledge of the cardinal value of number words) that predict later mathematics achievement. The domain-general abilities were then used to predict growth in these competencies across 2 years of preschool, and the combination of domain-general abilities, preliteracy skills, and core quantitative competencies were used to predict mathematics achievement across preschool and mathematics and word reading achievement at the end of kindergarten. Both intelligence and executive functions predicted growth in the four quantitative competencies, especially across the first year of preschool. A combination of domain-general and domain-specific competencies predicted preschoolers' mathematics achievement, with a trend for domain-specific skills to be more strongly related to achievement at the beginning of preschool than at the end of preschool. Preschool preliteracy skills, sensitivity to the relative quantities of collections of objects, and cardinal knowledge predicted reading and mathematics achievement at the end of kindergarten. Preliteracy skills were more strongly related to word reading, whereas sensitivity to relative quantity was more strongly related to mathematics achievement. The overall results indicate that a combination of domain-general and domain-specific abilities contribute to development of children's early mathematics and reading achievement.

The effect of oppositional parietal transcranial direct current stimulation on lateralized brain functions

Cognitive functions such as numerical processing and spatial attention show varying degrees of lateralization. Transcranial direct current stimulation (tDCS) can be used to investigate how modulating cortical excitability affects performance of these tasks. This study investigated the effect of bi-parietal tDCS on numerical processing, spatial and sustained attention. It was hypothesized that tDCS would have distinct effects on these tasks because of varying lateralization (numerical processing left, spatial attention right) and that these effects are partly mediated by modulation of sustained attention. A single-blinded, crossover, sham-controlled study was performed. Eighteen healthy right-handed participants performed cognitive tasks during three sessions of oppositional parietal tDCS stimulation: sham; right anodal with left cathodal (RA/LC); and right cathodal with left anodal (RC/LA). Participants performed a number comparison task, a modified Posner task, a choice reaction task (CRT) and the rapid visual processing task (RVP). RA/LC tDCS impaired number comparison performance compared with sham, with slower responses to numerically close numbers pairs. RA/LC and RC/LA tDCS had distinct effects on CRT performance, specifically affecting vigilance level during the final block of the task. No effect of stimulation on the Posner task or RVP was found. It was demonstrated that oppositional parietal tDCS affected both numerical performance and vigilance level in a polarity-dependent manner. The effect of tDCS on numerical processing may partly be due to attentional effects. The behavioural effects of tDCS were specifically observed under high task demands, demonstrating the consequences of an interaction between stimulation type and cognitive load.

Number As a Primary Perceptual Attribute: A Review

Although humans are the only species to possess language-driven abstract mathematical capacities, we share with many other animals a nonverbal capacity for estimating quantities or numerosity. For some time, researchers have clearly differentiated between small numbers of items--less than about four--referred to as the subitizing range, and larger numbers, where counting or estimation is required. In this review, we examine more recent evidence suggesting a further division, between sets of items greater than the subitizing range, but sparse enough to be individuated as single items; and densely packed stimuli, where they crowd each other into what is better considered as a texture. These two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws and show different dependencies on eccentricity and on luminance levels. But provided the elements are not too crowded (less than about two items per square degree in central vision, less in the periphery), there is little evidence that estimation of numerosity depends on mechanisms responsive to texture. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

Mechanisms for perception of numerosity or texture-density are governed by crowding-like effects

We have recently provided evidence that the perception of number and texture density is mediated by two independent mechanisms: numerosity mechanisms at relatively low numbers, obeying Weber's law, and texture-density mechanisms at higher numerosities, following a square root law. In this study we investigated whether the switch between the two mechanisms depends on the capacity to segregate individual dots, and therefore follows similar laws to those governing visual crowding. We measured numerosity discrimination for a wide range of numerosities at three eccentricities. We found that the point where the numerosity regime (Weber's law) gave way to the density regime (square root law) depended on eccentricity. In central vision, the regime changed at 2.3 dots/°2, while at 15° eccentricity, it changed at 0.5 dots/°2, three times less dense. As a consequence, thresholds for low numerosities increased with eccentricity, while at higher numerosities thresholds remained constant. We further showed that like crowding, the regime change was independent of dot size, depending on distance between dot centers, not distance between dot edges or ink coverage. Performance was not affected by stimulus contrast or blur, indicating that the transition does not depend on low-level stimulus properties. Our results reinforce the notion that numerosity and texture are mediated by two distinct processes, depending on whether the individual elements are perceptually segregable. Which mechanism is engaged follows laws that determine crowding.

Time, number and attention in very low birth weight children

Premature birth has been associated with damage in many regions of the cerebral cortex, although there is a particularly strong susceptibility for damage within the parieto-occipital lobes (Volpe, 2009). As these areas have been shown to be critical for both visual attention and magnitudes perception (time, space, and number), it is important to investigate the impact of prematurity on both the magnitude and attentional systems, particularly for children without overt white matter injuries, where the lack of obvious injury may cause their difficulties to remain unnoticed. In this study, we investigated the ability to judge time intervals (visual, audio and audio-visual temporal bisection), discriminate between numerical quantities (numerosity comparison), map numbers onto space (numberline task) and to maintain visuo-spatial attention (multiple-object-tracking) in school-age preterm children (N29). The results show that various parietal functions may be more or less robust to prematurity-related difficulties, with strong impairments found on time estimation and attentional task, while numerical discrimination or mapping tasks remained relatively unimpaired. Thus while our study generally supports the hypothesis of a dorsal stream vulnerability in children born preterm relative to other cortical locations, it further suggests that particular cognitive processes, as highlighted by performance on different tasks, are far more susceptible than others.

Numerical Estimation in Children With Autism

Number skills are often reported anecdotally and in the mass media as a relative strength for individuals with autism, yet there are remarkably few research studies addressing this issue. This study, therefore, sought to examine autistic children's number estimation skills and whether variation in these skills can explain at least in part strengths and weaknesses in children's mathematical achievement. Thirty-two cognitively able children with autism (range = 8-13 years) and 32 typical children of similar age and ability were administered a standardized test of mathematical achievement and two estimation tasks, one psychophysical nonsymbolic estimation (numerosity discrimination) task and one symbolic estimation (numberline) task. Children with autism performed worse than typical children on the numerosity task, on the numberline task, which required mapping numerical values onto space, and on the test of mathematical achievement. These findings question the widespread belief that mathematical skills are generally enhanced in autism. For both groups of children, variation in performance on the numberline task was also uniquely related to their academic achievement, over and above variation in intellectual ability; better number-to-space mapping skills went hand-in-hand with better arithmetic skills. Future research should further determine the extent and underlying causes of some autistic children's difficulties with regards to number.

Improving arithmetic performance with number sense training: an investigation of underlying mechanism

A nonverbal primitive number sense allows approximate estimation and mental manipulations on numerical quantities without the use of numerical symbols. In a recent randomized controlled intervention study in adults, we demonstrated that repeated training on a non-symbolic approximate arithmetic task resulted in improved exact symbolic arithmetic performance, suggesting a causal relationship between the primitive number sense and arithmetic competence. Here, we investigate the potential mechanisms underlying this causal relationship. We constructed multiple training conditions designed to isolate distinct cognitive components of the approximate arithmetic task. We then assessed the effectiveness of these training conditions in improving exact symbolic arithmetic in adults. We found that training on approximate arithmetic, but not on numerical comparison, numerical matching, or visuo-spatial short-term memory, improves symbolic arithmetic performance. In addition, a second experiment revealed that our approximate arithmetic task does not require verbal encoding of number, ruling out an alternative explanation that participants use exact symbolic strategies during approximate arithmetic training. Based on these results, we propose that nonverbal numerical quantity manipulation is one key factor that drives the link between the primitive number sense and symbolic arithmetic competence. Future work should investigate whether training young children on approximate arithmetic tasks even before they solidify their symbolic number understanding is fruitful for improving readiness for math education.

Compressive mapping of number to space reflects dynamic encoding mechanisms, not static logarithmic transform

The mapping of number onto space is fundamental to measurement and mathematics. However, the mapping of young children, unschooled adults, and adults under attentional load shows strong compressive nonlinearities, thought to reflect intrinsic logarithmic encoding mechanisms, which are later "linearized" by education. Here we advance and test an alternative explanation: that the nonlinearity results from adaptive mechanisms incorporating the statistics of recent stimuli. This theory predicts that the response to the current trial should depend on the magnitude of the previous trial, whereas a static logarithmic nonlinearity predicts trialwise independence. We found a strong and highly significant relationship between numberline mapping of the current trial and the magnitude of the previous trial, in both adults and school children, with the current response influenced by up to 15% of the previous trial value. The dependency is sufficient to account for the shape of the numberline, without requiring logarithmic transform. We show that this dynamic strategy results in a reduction of reproduction error, and hence improvement in accuracy.

Association between individual differences in non-symbolic number acuity and math performance: a meta-analysis

Many recent studies have examined the association between number acuity, which is the ability to rapidly and non-symbolically estimate the quantity of items appearing in a scene, and symbolic math performance. However, various contradictory results have been reported. To comprehensively evaluate the association between number acuity and symbolic math performance, we conduct a meta-analysis to synthesize the results observed in previous studies. First, a meta-analysis of cross-sectional studies (36 samples, N = 4705) revealed a significant positive correlation between these skills (r = 0.20, 95% CI = [0.14, 0.26]); the association remained after considering other potential moderators (e.g., whether general cognitive abilities were controlled). Moreover, a meta-analysis of longitudinal studies revealed 1) that number acuity may prospectively predict later math performance (r = 0.24, 95% CI = [0.11, 0.37]; 6 samples) and 2) that number acuity is retrospectively correlated to early math performance as well (r = 0.17, 95% CI = [0.07, 0.26]; 5 samples). In summary, these pieces of evidence demonstrate a moderate but statistically significant association between number acuity and math performance. Based on the estimated effect sizes, power analyses were conducted, which suggested that many previous studies were underpowered due to small sample sizes. This may account for the disparity between findings in the literature, at least in part. Finally, the theoretical and practical implications of our meta-analytic findings are presented, and future research questions are discussed.