Number sense across the lifespan as revealed by a massive Internet-based sample

Sharpening the number sense: Developmental trends in numerosity perception

Numerosity perception, the ability to process and estimate the number of objects in a set without explicitly counting, has been widely studied, and one well-established finding is that children become more accurate at perceiving numerosity with age. The question remains, however, what the underlying cognitive processes and mechanisms are that drive this improvement. Some authors have suggested that this is due to an increased numerical precision (i.e., the sharpening hypothesis), whereas others have proposed that the more accurate performance is due to the improved ability to inhibit non-numerical features of the display such as object size and spacing of items (i.e., the filtering hypothesis). The current study examined the developmental trajectory of numerosity perception across three age groups (M = 5.65, M = 11.03, and M = 20.10 years). As expected, more accurate performance was observed with age. Regression and analyses of variance revealing the contribution of numerical and non-numerical predictors in performance show that the performance in all age groups was primarily driven by numerical information and that its contribution increased with age. In addition, a consistent bias toward non-numerical features was observed in all age groups. These results support the sharpening hypothesis for children from 5 years of age to early adulthood, suggesting that from this age onward children increasingly focus on numerical information as they get older. These results have important implications for the understanding of the development and specific improvements of numerical perception.

Exploring the classical and numerical Delboeuf illusion: the impact of transcranial alternating current stimulation on magnitude processing

Understanding cognitive and neural mechanisms underlying quantity processing is crucial for unraveling human cognition. The existence of a single magnitude system, encompassing non-symbolic number estimation alongside other magnitudes like time and space, is still highly debated since clear evidence is limited. Recent research examined whether spatial biases also influence numerosity judgments, using visual illusions like the Delboeuf illusion. While findings support a generalized magnitude system, direct comparisons of spatial and numerical Delboeuf illusions are missing. This study explored whether perceptual biases similarly affect different magnitude processing and whether transcranial alternating current stimulation (tACS) modulates these processes. Participants underwent three tACS conditions (seven Hz, 18 Hz, placebo) while performing tasks involving the classic and numerical Delboeuf illusions. We hypothesized that theta-frequency tACS (seven Hz) would enhance visual integration and illusion strength, while beta tACS (18 Hz) would reduce it by promoting visual segregation. Results indicated higher discrimination accuracy in area-based tasks than numerical judgments. Nonetheless, a significant correlation between performances in spatial and numerical illusions supported the existence of a shared mechanism for magnitude processing. Contrary to expectations, seven Hz tACS reduced the perceptual illusion's strength. No significant interaction emerged between tACS frequency and discrimination abilities. These findings deepen our understanding of the cognitive processes involved in magnitude perception, potentially supporting the hypothesis of a generalized magnitude system. They also highlight the potential and limitations of non-invasive brain stimulation techniques, such as tACS, in modulating perceptual processes, offering insights into the neural underpinnings of quantity perception.

Separating neurocognitive mechanisms of maintenance and compensation to support financial ability in middle-aged and older adults: The role of language and the inferior frontal gyrus

This study investigated the role of brain regions involved in arithmetic processing in explaining individual differences in financial ability in 67 50-74-year-old cognitively normal adults. Structural integrity and resting-state functional connectivity measures were collected in the MRI scanner. Outside the scanner, participants performed financial ability and other cognitive tasks, and answered questionnaires to determine dementia risk, and financial risk and protective factors. Regions of interest involved in arithmetic processing were defined, focusing on language- and quantity-processing areas in temporo-frontal and parieto-frontal cortices, respectively. Our results showed that structural integrity and functional connectivity in brain regions associated with arithmetic retrieval were positively associated with financial ability, with language skill mediating left IFG structural integrity and financial ability. Connectivity patterns suggested that reliance on quantity mechanisms (i.e. calculation) was associated with poorer financial ability. Analyses revealed that reliance on these brain mechanisms did not depend on participants' age or risk of dementia and that protective factors such as household income or financial literacy supported the maintenance of connectivity related to financial abilities.

No Evidence of Reliability Across 36 Variations of the Emotional Dot-Probe Task in 9,600 Participants

The emotional dot-probe task is a widely used measure of attentional bias to threat. Recent work suggests, however, that subtraction-based behavioral measures of emotional dot-probe performance may not be appropriate for measuring such attentional biases due to poor reliability. In the two current studies, we systematically tested thirty-six versions of the emotional dot-probe that varied in stimuli (faces, scenes, snakes/spiders), timings (stimulus onset asynchrony of 100, 500, 900 milliseconds), stimulus orientations (horizontal, vertical), and trial types (e.g., threat congruent and threat incongruent). Across 9,600 participants, none of the 36 versions demonstrated internal reliability greater than zero. Reliability was similarly poor in anxious participants (based on Generalized Anxiety Disorder 7 Items or Brief Hypervigilance Scale). We conclude that the standard behavioral scores (reaction time- or accuracy-based difference scores) derived from the emotional dot-probe are not adequately reliable measures of attentional biases to threat in anxious or nonanxious populations.

Numerical cognition across the lifespan: A selective review of key developmental stages and neural, cognitive, and affective underpinnings

Numerical cognition constitutes a set of hierarchically related skills and abilities that develop-and may subsequently begin to decline-over developmental time. An innate "number sense" has long been argued to provide a foundation for the development of increasingly complex and applied numerical cognition, such as symbolic numerical reference, arithmetic, and financial literacy. However, evidence for a direct link between basic perceptual mechanisms that allow us to determine numerical magnitude (e.g., "how many" objects are in front of us and whether some of these are of a "greater" or "lesser" quantity), and later symbolic applications for counting and mathematics, has recently been challenged. Understanding how one develops an increasingly precise sense of number and which neurocognitive mechanisms support arithmetic development and achievement is crucial for developing successful mathematics curricula, supporting individual financial literacy and decision-making, and designing appropriate intervention and remediation programs for mathematical learning disabilities as well as mathematics anxiety. The purpose of this review is to provide a broad overview of the cognitive, neural, and affective underpinnings of numerical cognition-spanning the earliest hours of infancy to senior adulthood-and highlight gaps in our knowledge that remain to be addressed.

Neural Representation Precision of Distance Predicts Children's Arithmetic Performance

Focusing on the distance between magnitudes as the starting point to investigate the mechanism of relation detection and its contribution to mathematical thinking, this study explores the precision of neural representations of numerical distance and their impact on children's arithmetic performance. By employing neural decoding techniques and representational similarity analysis, the present study investigates how accurately the brain represents numerical distances and how this precision relates to arithmetic skills. Twenty-nine school-aged children participated, completing a dot number comparison task during fMRI scanning and an arithmetic fluency test. Results indicated that neural activation patterns in the intra-parietal sulcus decoded the distance between the presented pair of dots, and higher precision in neural distance representation correlates with better arithmetic performance. These findings suggest that the accuracy of neural decoding can serve as an index of neural representation precision and that the ability to precisely encode numerical distances in the brain is a key factor in mathematical abilities. This provides new insights into the neural basis of mathematical cognition and learning.

Are three zebras more than three frogs: examining conceptual and physical congruency in numerosity judgements of familiar objects

Researchers in numerical cognition have extensively studied the number sense-the innate human ability to extract numerical information from the environment quickly and effortlessly. Much of this research, however, uses abstract stimuli (e.g., dot configurations) that are also strictly controlled for their low-level visual confounds, such as size. Nonetheless, individuals rarely extract numerical information from abstract stimuli in everyday life. Yet, numerical judgments of familiar objects remain poorly understood and understudied. In the current study, we examined the cognitive mechanisms underlying the numerical decisions of familiar objects. In two experiments, we asked adult participants (Experiment 1) and two groups of children (aged 7-9 years and 11-12 years, Experiment 2) to perform an animal numerosity task (i.e., "Which animal is more numerous?"), while the conceptual congruency (i.e., the congruency between an object's real-life size and its numerosity) and physical congruency (the congruency between the number of items and the total space they occupy on the screen) were manipulated. Results showed that the conceptual congruency effect (i.e., better performance when the animal with a larger size in real life is more numerous) and a physical congruency effect (i.e., better performance when the physically larger animal is more numerous) were present in adults and children. However, the effects differed across the age groups and were also a subject of developmental change. To our knowledge, this study is the first one to demonstrate that conceptual knowledge can interfere with numerosity judgements in a top-down manner. This interference effect is distinct from the bottom-up interference effect, which comes from the physical properties of the set. Our results imply that the number sense is not a standalone core system for numbers but is embedded in a more extensive network where both low-level and higher-order influences are possible. We encourage numerical cognition researchers to consider employing not only abstract but also familiar objects when examining numerosity judgements across the lifespan.

Executive Function and young children's Cardinality Principle: the mediating role of the Approximate Number System and the moderating role of age

Background

Executive Function and the Approximate Number System are well-established as critical components in developing the Cardinality Principle in young children. However, most existing studies explore the relationship between these variables in isolation without examining whether Approximate Number System mediates the relationship between Executive Function and the Cardinality Principle and the role of age in this. This study aimed to address this gap by investigating the mediating role of the Approximate Number System in the relationship between Executive Function and the Cardinality Principle and the moderating role of age in young children.

Methods

This cross-sectional study was conducted in China from February to June 2024. A total of 203 young children (97 boys and 106 girls, Mean age = 68.93 ± 7.076 months) participated. Participants were assessed using a range of tests: the Day-Night Stroop Task, Digit Recall Task, Dimensional Change Card Sort Task, Panamath Test Software, How Many Task, and Give-N Task to measure Executive Function, Approximate Number System, and Cardinality Principle. Data were analyzed using SPSS 26.0 and PROCESS v4.1 (Model 4) to explore the relationships among Executive Function, the Approximate Number System, and the Cardinality Principle through Pearson correlations, multivariate regression, and mediation analysis with 5000 bootstrap samples.

Results

Correlation analysis revealed that the Cardinality Principle was significantly and positively correlated with Inhibitory Control, Working Memory, Cognitive Flexibility, Executive Function, and the Approximate Number System. Regression analyses indicated that Executive Function positively predicted young children's Cardinality Principle. Specifically, Working Memory and Cognitive Flexibility were positive predictors of the Cardinality Principle, while Inhibitory Control was not. Mediation analysis results demonstrated that the Approximate Number System mediated the relationships between Inhibitory Control and the Cardinality Principle, Working Memory and the Cardinality Principle, and Cognitive Flexibility and the Cardinality Principle, respectively. In addition, the study found that young children's age negatively moderated the relationship between the Approximate Number System and the Cardinality Principle.

Conclusions

The study emphasizes that in developing young children's Cardinality Principle, emphasis should be placed on improving their Executive Function and Approximate Number System while considering the age differences of young children and developing appropriate educational methods for different age groups.

What makes different number-space mappings interact?

Models of numerical cognition consider a visuo-spatial representation to be at the core of numerical processing, the 'mental number line'. Two main interference effects between number and space have been described: the SNARC effect reflects a small number/left side and large number/right side association (number-location mapping); the size-congruity effect (SCE) reflects a small number/small size and large number/large size association (number-size mapping). Critically, a thorough investigation on the representational source for these two number-space mappings is lacking, leaving open the question of whether the same representation underlies both phenomena. Here, we build on a recent study (Viarouge and de Hevia in Front Hum Neurosci 15:750964, 2021) in order to address this question in three experiments, by systematically manipulating the presence of the two conditions that might elicit an interaction between SNARC and SCE: (i) an implicit task whereby numerical and spatial information are task-irrelevant, (ii) a design in which the number-space congruency relative to both mappings vary at the same level -either both within or between blocks. Experiment 1 replicated the interaction between the two mappings when both factors were present. Experiments 2 and 3 dissociated the two factors by varying the two mappings at the same level but using an explicit comparison task (Experiment 2), or by using an implicit task but with mappings varying at different levels (Experiment 3). We found that both factors, either in combination or used in isolation, drive the interaction between the two number-space mappings. These findings are discussed in terms of the weight given to each mapping, suggesting that a single representation encompassing both number-space mappings is therefore activated whenever both mappings are given equal weight through task requirements.

Deficits in or Preservation of Basic Number Processing in Parkinson's Disease? A Registered Report

Neurodegenerative diseases such as Parkinson's disease (PD) have a huge impact on patients, caregivers, and the health care system. Until now, diagnosis of mild cognitive impairments in PD has been established based on domain-general functions such as executive functions, attention, or working memory. However, specific numerical deficits observed in clinical practice have not yet been systematically investigated. PD-immanent deterioration of domain-general functions and domain-specific numerical areas suggests mechanisms of both primary and secondary dyscalculia. The current study systematically investigated basic number processing performance in PD patients for the first time, targeting domain-specific cognitive representations of numerosity and the influence of domain-general factors. The overall sample consisted of patients with a diagnosis of PD, according to consensus guidelines, and healthy controls. PD patients were stratified into patients with normal cognition (PD-NC) or mild cognitive impairment (level I-PD-MCI based on cognitive screening). Basic number processing was assessed using transcoding, number line estimation, and (non-) symbolic number magnitude comparison tasks. Discriminant analysis was employed to assess whether basic number processing tasks can differentiate between a healthy control group and both PD groups. All participants were subjected to a comprehensive numerical and a neuropsychological test battery, as well as sociodemographic and clinical measures. Results indicate a profile of preserved (verbal representation) and impaired (magnitude representation, place × value activation) function in PD-MCI, hinting at basal ganglia dysfunction affecting numerical cognition in PD. Numerical deficits could not be explained by domain-general cognitive impairments, so that future research needs to incorporate domain-specific tasks of sufficient difficulty.

Measuring temporal bias in sequential numerosity comparison

While several methods have been proposed to assess the influence of continuous visual cues in parallel numerosity estimation, the impact of temporal magnitudes on sequential numerosity judgments has been largely ignored. To overcome this issue, we extend a recently proposed framework that makes it possible to separate the contribution of numerical and non-numerical information in numerosity comparison by introducing a novel stimulus space designed for sequential tasks. Our method systematically varies the temporal magnitudes embedded into event sequences through the orthogonal manipulation of numerosity and two latent factors, which we designate as "duration" and "temporal spacing". This allows us to measure the contribution of finer-grained temporal features on numerosity judgments in several sensory modalities. We validate the proposed method on two different experiments in both visual and auditory modalities: results show that adult participants discriminated sequences primarily by relying on numerosity, with similar acuity in the visual and auditory modality. However, participants were similarly influenced by non-numerical cues, such as the total duration of the stimuli, suggesting that temporal cues can significantly bias numerical processing. Our findings highlight the need to carefully consider the continuous properties of numerical stimuli in a sequential mode of presentation as well, with particular relevance in multimodal and cross-modal investigations. We provide the complete code for creating sequential stimuli and analyzing participants' responses.

Face recognition's practical relevance: Social bonds, not social butterflies

Research on individual differences in face recognition has provided important foundational insights: their broad range, cognitive specificity, strong heritability, and resilience to change. Elusive, however, has been the key issue of practical relevance: do these individual differences correlate with aspects of life that go beyond the recognition of faces, per se? Though often assumed, especially in social realms, such correlates remain largely theoretical, without empirical support. Here, we investigate an array of potential social correlates of face recognition. We establish social relationship quality as a reproducible correlate. This link generalises across face recognition tasks and across independent samples. In contrast, we detect no robust association with the sheer quantity of social connections, whether measured directly via number of social contacts or indirectly via extraversion-related personality indices. These findings document the existence of a key social correlate of face recognition and provide some of the first evidence to support its practical relevance. At the same time, they challenge the naive assumption that face recognition relates equally to all social outcomes. In contrast, they suggest a focused link of face recognition to the quality, not quantity, of one's social connections.

Frontoparietal and salience network synchronizations during nonsymbolic magnitude processing predict brain age and mathematical performance in youth

The development and refinement of functional brain circuits crucial to human cognition is a continuous process that spans from childhood to adulthood. Research increasingly focuses on mapping these evolving configurations, with the aim to identify markers for functional impairments and atypical development. Among human cognitive systems, nonsymbolic magnitude representations serve as a foundational building block for future success in mathematical learning and achievement for individuals. Using task-based frontoparietal (FPN) and salience network (SN) features during nonsymbolic magnitude processing alongside machine learning algorithms, we developed a framework to construct brain age prediction models for participants aged 7-30. Our study revealed differential developmental profiles in the synchronization within and between FPN and SN networks. Specifically, we observed a linear increase in FPN connectivity, concomitant with a decline in SN connectivity across the age span. A nonlinear U-shaped trajectory in the connectivity between the FPN and SN was discerned, revealing reduced FPN-SN synchronization among adolescents compared to both pediatric and adult cohorts. Leveraging the Gradient Boosting machine learning algorithm and nested fivefold stratified cross-validation with independent training datasets, we demonstrated that functional connectivity measures of the FPN and SN nodes predict chronological age, with a correlation coefficient of .727 and a mean absolute error of 2.944 between actual and predicted ages. Notably, connectivity within the FPN emerged as the most contributing feature for age prediction. Critically, a more matured brain age estimate is associated with better arithmetic performance. Our findings shed light on the intricate developmental changes occurring in the neural networks supporting magnitude representations. We emphasize brain age estimation as a potent tool for understanding cognitive development and its relationship to mathematical abilities across the critical developmental period of youth. PRACTITIONER POINTS: This study investigated the prolonged changes in the brain's architecture across childhood, adolescence, and adulthood, with a focus on task-state frontoparietal and salience networks. Distinct developmental pathways were identified: frontoparietal synchronization strengthens consistently throughout development, while salience network connectivity diminishes with age. Furthermore, adolescents show a unique dip in connectivity between these networks. Leveraging advanced machine learning methods, we accurately predicted individuals' ages based on these brain circuits, with a more mature estimated brain age correlating with better math skills.

Are they equivalent? An examination of task variants of approximate number comparison

Nonverbal numerical ability supports individuals' numerical information processing in everyday life and is also correlated with their learning of mathematics. This ability is typically measured with an approximate number comparison paradigm, in which participants are presented with two sets of objects and instructed to choose the numerically larger set. This paradigm has multiple task variants, where the two sets are presented in different ways (e.g., two sets are presented either simultaneously or sequentially, or two sets are presented either intermixed or separately). Despite the fact that different task variants have often been used interchangeably, it remains unclear whether these variants measure the same aspects of nonverbal numerical ability. Using a latent variable modeling approach with 270 participants (Mage = 20.75 years, SDage = 2.03, 94 males), this study examined the degree to which three commonly used task variants tapped into the same construct. The results showed that a bi-factor model corresponding to the hypothesis that task variants had both commonalities and uniqueness was a better fit for the data than a single-factor model, corresponding to the hypothesis that task variants were construct equivalent. These findings suggested that task variants of approximate number comparison did not measure the same construct and cannot be used interchangeably. This study also quantified the extent to which general cognitive abilities were involved in both common and unique parts of these task variants.

Age-related effects in magnitude and place-value processing

While general cognitive skills decline during aging, numerical skills seem to be mainly preserved. Such skills are essential for an independent life up to old age, e.g., when dealing with money or time. Operating with numbers usually requires number magnitude and place-value processing. The question is whether these processes are negatively affected by aging due to the general cognitive decline or positively affected due to lifelong experience with numbers. Therefore, we investigated age-related changes in the distance and compatibility effects in single-digit, two-digit, and four-digit number comparison. On the one hand, older adults took longer for number processing and showed a smaller distance effect, indicating altered number magnitude representations. On the other hand, older adults were better in place-value processing as indicated by a smaller compatibility effect than in younger adults. We conclude that aging differentially affects basic numerical skills.

Associations Between Childhood Trauma Characteristics and Theory of Mind in Adults: Results From a Large, Diverse Sample

Theory of mind (ToM) is an essential social cognitive process encompassing abilities to represent and understand others' mental states. Although previous reports linked childhood trauma to social cognitive deficits, how characteristics of trauma exposure, such as subtype or timing, affect ToM remains unaddressed. Using data from a diverse adult sample (n = 2200), we tested whether exposure type and first exposure timing of common childhood trauma associated with ToM. Neither interpersonal loss (β = - 0.25, p = 0.170, [- 0.61, 0.10]) nor child maltreatment (β = - 0.21, p = 0.369, [- 0.66, 0.25]) was associated with lower ToM. There was no effect of timing of age at which trauma was experienced (F = 2.19, p = 0.087). While we did not identify age-dependent effects, future studies should examine links between timing or chronicity of prospectively reported childhood trauma and social cognition. Understanding of how childhood experiences shape ToM could reveal mechanisms underlying social cognition development and inform prevention efforts.

Developmental changes of the impact of visual cues on ANS acuity across grades 1-5: Different patterns of visual cues on numerosity processing

Numerous studies have consistently demonstrated the presence of the approximate number system (ANS) throughout development. Research has also revealed that visual cues may influence the ANS acuity, which may change with age. However, most studies have drawn conclusions based on performance differences between incongruent and congruent trials, which may be confounded by an individual's ability to inhibit interference. Therefore, to examine the developmental changes of the impact of visual cues on ANS acuity, we utilized congruent trials with varying visual cues. Our sample comprised Chinese children from grade one to grade five. We manipulated the salience of numerical cues (numerical ratio) and visual cues (dot size) in a non-symbolic numerosity comparison task. The results revealed a discernible leap in development from first to third grade and first to fifth grade; however, this upward trajectory did not persist into the transition from third to fifth grade, where no appreciable advancement was observed. Moreover, we observed different effects of visual cues on the dot-comparison task depending on the numerical cues and age. Specifically, visual cues (i.e., dot size) only facilitated ANS acuity in older school-aged children when numerical cues were weakened. The results indicate the presence of two distinct magnitude representational systems-one for the numerical dimension and another for the non-numerical dimension-during development.

Duration, numerosity and length processing in healthy ageing and Parkinson's disease

People constantly process temporal, numerical, and length information in everyday activities and interactions with the environment. However, it is unclear whether quantity perception changes during ageing. Previous studies have provided heterogeneous results, sometimes showing an age-related effect on a particular quantity, and other times reporting no differences between young and elderly samples. However, three dimensions were never compared within the same study. Here, we conducted two experiments with the aim of investigating the processing of duration, numerosity and length in both healthy and pathological ageing. The experimental paradigm consisted of three bisection tasks in which participants were asked to judge whether the presented stimulus (i.e. a time interval, a group of dots, or a line) was more similar to the short/few or long/many standards. The first study recruited healthy young and elderly participants, while the second recruited healthy elderly participants and patients with Parkinson's disease, a clinical condition commonly associated with temporal impairments. The results of both experiments showed that discrimination precision differed between domains in all groups, with higher precision in the numerosity task and lower sensitivity in judging duration. Furthermore, while discrimination abilities were affected in healthy elderly and, even more so, in Parkinson's disease group, no domain-specific impairments emerged. According to our research, reduced discrimination precision might be explained by an alteration of a single system for all quantities or by an age-related general cognitive decline.

Numerosity comparison, cognitive strategies, and general cognitive functioning in older people

Introduction

Studies have shown age-related differences in numerical cognition, for example, in the level of numerosity comparison ability. Moreover, some studies point out individual differences in the cognitive strategies employed during the performance of numerosity comparison tasks and reveal that they are related to the aging process. One probable cause of these differences is the level of cognitive functioning. The aim of our study was to determine the relationships among numerosity comparison ability, the cognitive strategies utilized in the performance of numerosity comparison tasks and the general cognitive functioning in older people.

Methods

Forty-seven elderly people participated in the study. The participants were examined using overall cognitive functioning scales and computerized numerosity comparison task.

Results

The results showed many correlations between the participants' level of cognitive functioning and the percent of correct responses (PCR) and response time (RT) during numerosity comparison, as well as with the cognitive strategies applied by the participants. Task correctness was positively related to the level of performance in the attention and executive function tasks. In contrast, the long-term memory resources index and visuospatial skills level were negatively correlated with RT regarding numerosity comparison task performance. The level of long-term memory resources was also positively associated with the frequency of use of more complex cognitive strategies. Series of regression analyses showed that both the level of general cognitive functioning and the cognitive strategies employed by participants in numerosity comparison can explain 9-21 percent of the variance in the obtained results.

Discussion

In summary, these results showed significant relationships between the level of cognitive functioning and proficiency in numerosity comparison measured in older people. Moreover, it has been shown that cognitive resources level is related to the strategies utilized by older people, which indicates the potential application for cognitive strategy examinations in the development of new diagnostic tools.

Magnitude representation of preschool children with autism spectrum condition

LAY ABSTRACT

The mathematical abilities of children with autism spectrum condition have been understudied. Magnitude representation (e.g. presenting the number of a collection of objects) is a fundamental numerical ability presented since early infancy and is correlated with children's later learning of formal mathematics. It remains unclear about whether children with autism spectrum condition differ from their peers without autism spectrum condition in precision of magnitude representations. This study compared preschool children with and without autism spectrum condition in their precision of magnitude representation with an approximate number comparison task, in which children compared two sets of dots without counting and chose the set with more dots. Children with autism spectrum condition exhibited the lower numerical comparison accuracy (i.e. the weaker magnitude representation) than their peers without autism spectrum condition. This difference existed even when multiple general cognitive abilities (working memory, inhibitory control, and nonverbal intelligence) and language abilities were statistically controlled. Moreover, the individual difference of the numerical comparison accuracy was larger in children with autism spectrum condition than without autism spectrum condition. These findings suggest that children with autism spectrum condition are at risk of weaker magnitude representation from an early age, emphasizing the need for specialized mathematics education or interventions to support their learning. In addition, the large variance in the precision of their magnitude representation suggests that individualized mathematics interventions are needed for children with autism spectrum condition.

Understanding the social-emotional components of our "number sense": insights from a novel non-symbolic numerical comparison task

Introduction

A large body of work has identified a core sense of number supported by the Approximate Number System (ANS) that is present in infancy and across species. Although it is commonly assumed that the ANS directly processes perceptual input and is relatively independent from affective factors, some evidence points at a correlation between ANS performance and math anxiety. However, the evidence is mixed. We tested whether giving participants active control in completing a numerical task would change the relationship between math anxiety on performance.

Methods

Adult participants (N = 103) completed a novel four-alternative-forced-choice non-symbolic numerical comparison task. In a repeated-measures design, participants either passively viewed different dot arrays or actively chose to view each array (i.e., active information-seeking) before deciding on the largest quantity. Participants also provided confidence judgments during the passive version of the task.

Results

We replicated the ratio-dependent signature in participants' accuracy in both the passive and active versions of the task using this novel paradigm, as well as in trial-level confidence judgments and information-seeking behavior. Participants' self-reported math anxiety significantly correlated with their accuracy on the passive version of the task. Critically, the correlation disappeared in the active version of the task. Gender also emerged as a predictor of confidence judgments and a moderator of the effect of task on overall accuracy and the effect of active information seeking on accuracy in the active version of the task. Exploratory analysis of estimated Weber Fraction suggests that these results may be driven by auxiliary factors instead of changes in ANS acuity.

Conclusion

These findings have implications for understanding the relationship between math anxiety and performance on numerical tasks.

Modeling Magnitude Discrimination: Effects of Internal Precision and Attentional Weighting of Feature Dimensions

Given a rich environment, how do we decide on what information to use? A view of a single entity (e.g., a group of birds) affords many distinct interpretations, including their number, average size, and spatial extent. An enduring challenge for cognition, therefore, is to focus resources on the most relevant evidence for any particular decision. In the present study, subjects completed three tasks-number discrimination, surface area discrimination, and convex hull discrimination-with the same stimulus set, where these three features were orthogonalized. Therefore, only the relevant feature provided consistent evidence for decisions in each task. This allowed us to determine how well humans discriminate each feature dimension and what evidence they relied on to do so. We introduce a novel computational approach that fits both feature precision and feature use. We found that the most relevant feature for each decision is extracted and relied on, with minor contributions from competing features. These results suggest that multiple feature dimensions are separately represented for each attended ensemble of many items and that cognition is efficient at selecting the appropriate evidence for a decision.

Neurocognitive Assessment of Mathematics-Related Capacities in Neurosurgical Patients

A precise neuropsychological assessment is of the utmost importance for neurosurgical patients undergoing the surgical excision of cerebral lesions. The assessment of mathematical abilities is usually limited to arithmetical operations while other fundamental visuo-spatial aspects closely linked to mathematics proficiency, such as the perception of numerical quantities and geometrical reasoning, are completely neglected. We evaluated these abilities with two objective and reproducible psychophysical tests, measuring numerosity perception and non-symbolic geometry, respectively. We tested sixteen neuro-oncological patients before the operation and six after the operation with classical neuropsychological tests and with two psychophysical tests. The scores of the classical neuropsychological tests were very heterogeneous, possibly due to the distinct location and histology of the tumors that might have spared (or not) brain areas subserving these abilities or allowed for plastic reorganization. Performance in the two non-symbolic tests reflected, on average, the presumed functional role of the lesioned areas, with participants with parietal and frontal lesions performing worse on these tests than patients with occipital and temporal lesions. Single-case analyses not only revealed some interesting exceptions to the group-level results (e.g., patients with parietal lesions performing well in the numerosity test), but also indicated that performance in the two tests was independent of non-verbal reasoning and visuo-spatial working memory. Our results highlight the importance of assessing non-symbolic numerical and geometrical abilities to complement typical neuropsychological batteries. However, they also suggest an avoidance of reliance on an excessively rigid localizationist approach when evaluating the neuropsychological profile of oncological patients.

Intuitive mapping between nonsymbolic quantity and observed action across development

Adults' concurrent processing of numerical and action information yields bidirectional interference effects consistent with a cognitive link between these two systems of representation. This link is in place early in life: infants create expectations of congruency across numerical and action-related stimuli (i.e., a small [large] hand aperture associated with a smaller [larger] numerosity). Although these studies point to a developmental continuity of this mapping, little is known about the later development and thus how experience shapes such relationships. We explored how number-action intuitions develop across early and later childhood using the same methodology as in adults. We asked 3-, 6-, and 8-year-old children, as well as adults, to relate the magnitude of an observed action (a static hand shape, open vs. closed, in Experiment 1; a dynamic hand movement, opening vs. closing, in Experiment 2) to either a small or large nonsymbolic quantity (numerosity in Experiment 1 and numerosity and/or object size in Experiment 2). From 6 years of age, children started performing in a systematic congruent way in some conditions, but only 8-year-olds (added in Experiment 2) and adults performed reliably above chance in this task. We provide initial evidence that early intuitions guiding infants' mapping between magnitude across nonsymbolic number and observed action are used in an explicit way only from late childhood, with a mapping between action and size possibly being the most intuitive. An initial coarse mapping between number and action is likely modulated with extensive experience with grasping and related actions directed to both arrays and individual objects.

Developmental psychologists should adopt citizen science to improve generalization and reproducibility

Widespread failures of replication and generalization are, ironically, a scientific triumph, in that they confirm the fundamental metascientific theory that underlies our field. Generalizable and replicable findings require testing large numbers of subjects from a wide range of demographics with a large, randomly-sampled stimulus set, and using a variety of experimental parameters. Because few studies accomplish any of this, meta-scientists predict that findings will frequently fail to replicate or generalize. We argue that to be more robust and replicable, developmental psychology needs to find a mechanism for collecting data at greater scale and from more diverse populations. Luckily, this mechanism already exists: Citizen science, in which large numbers of uncompensated volunteers provide data. While best-known for its contributions to astronomy and ecology, citizen science has also produced major findings in neuroscience and psychology, and increasingly in developmental psychology. We provide examples, address practical challenges, discuss limitations, and compare to other methods of obtaining large datasets. Ultimately, we argue that the range of studies where it makes sense *not* to use citizen science is steadily dwindling.

Development of precision of non-symbolic arithmetic operations in 4-6-year-old children

Children can represent the approximate quantity of sets of items using the Approximate Number System (ANS), and can perform arithmetic-like operations over ANS representations. Previous work has shown that the representational precision of the ANS develops substantially during childhood. However, less is known about the development of the operational precision of the ANS. We examined developmental change in the precision of the solutions to two non-symbolic arithmetic operations in 4-6-year-old U.S. children. We asked children to represent the quantity of an occluded set (Baseline condition), to compute the sum of two sequentially occluded arrays (Addition condition), or to infer the quantity of an addend after observing an initial array and then the array incremented by the unknown addend (Unknown-addend condition). We measured the precision of the solutions of these operations by asking children to compare their solutions to visible arrays, manipulating the ratio between the true quantity of the solution and the comparison array. We found that the precision of ANS representations that were not the result of operations (in the Baseline condition) was higher than the precision of solutions to ANS operations (in the Addition and Unknown-addend conditions). Further, we found that precision in the Baseline and Addition conditions improved significantly between 4 and 6 years, while precision in the Unknown-Addend condition did not. Our results suggest that ANS operations may inject "noise" into the representations they operate over, and that the development of the precision of different operations may follow different trajectories in childhood.

Standard experimental paradigm designs and data exclusion practices in cognitive psychology can inadvertently introduce systematic "shadow" biases in participant samples

Standard cognitive psychology research practices can introduce inadvertent sampling biases that reduce the reliability and generalizability of the findings. Researchers commonly acknowledge and understand that any given study sample is not perfectly generalizable, especially when implementing typical experimental constraints (e.g., limiting recruitment to specific age ranges or to individuals with normal color vision). However, less obvious systematic sampling constraints, referred to here as "shadow" biases, can be unintentionally introduced and can easily go unnoticed. For example, many standard cognitive psychology study designs involve lengthy and tedious experiments with simple, repetitive stimuli. Such testing environments may 1) be aversive to some would-be participants (e.g., those high in certain neurodivergent symptoms) who may self-select not to enroll in such studies, or 2) contribute to participant attrition, both of which reduce the sample's representativeness. Likewise, standard performance-based data exclusion efforts (e.g., minimum accuracy or response time) or attention checks can systematically remove data from participants from subsets of the population (e.g., those low in conscientiousness). This commentary focuses on the theoretical and practical issues behind these non-obvious and often unacknowledged "shadow" biases, offers a simple illustration with real data as a proof of concept of how applying attention checks can systematically skew latent/hidden variables in the included population, and then discusses the broader implications with suggestions for how to manage and reduce, or at a minimum acknowledge, the problem.

A Shared Intuitive (Mis)understanding of Psychophysical Law Leads Both Novices and Educated Students to Believe in a Just Noticeable Difference (JND)

Humans are both the scientists who discover psychological laws and the thinkers who behave according to those laws. Oftentimes, when our natural behavior is in accord with those laws, this dual role serves us well: our intuitions about our own behavior can serve to inform our discovery of new laws. But, in cases where the laws that we discover through science do not agree with the intuitions and biases we carry into the lab, we may find it harder to believe in and adopt those laws. Here, we explore one such case. Since the founding of psychophysics, the notion of a Just Noticeable Difference (JND) in perceptual discrimination has been ubiquitous in experimental psychology-even in spite of theoretical advances since the 1950's that argue that there can be no such thing as a threshold in perceiving difference. We find that both novices and psychologically educated students alike misunderstand the JND to mean that, below a certain threshold, humans will be unable to tell which of two quantities is greater (e.g., that humans will be completely at chance when trying to judge which is heavier, a bag with 3000 grains of sand or 3001). This belief in chance performance below a threshold is inconsistent with psychophysical law. We argue that belief in a JND is part of our intuitive theory of psychology and is therefore very difficult to dispel.

The role of non-numerical information in the perception of temporal numerosity

Numerosity perception refers to the ability to make rapid but approximate estimates of the quantity of elements in a set (spatial numerosity) or presented sequentially (temporal numerosity). Whether numerosity is directly perceived or indirectly recomputed from non-numerical features is a highly debated issue. In the spatial domain, area and density have been suggested as the main parameters through which numerosity would be recomputed. In the temporal domain, stimuli duration and temporal frequency could be similarly exploited to retrieve numerosity. By adapting a psychophysical technique previously exploited in the spatial domain, we investigated whether temporal visual numerosity is directly perceived. Adult participants observed sequences of visual impulses sampled from a stimulus space spanning several levels of temporal frequency and duration (and hence numerosity), and then reproduced the sequence as accurately as possible via a series of keypresses. Crucially, participants were not asked to reproduce any particular property (such as number of impulses) but were free to choose any available cue (such as total duration, or temporal frequency). The results indicate that while the overall sequence duration was barely considered, numerosity and temporal frequency were both spontaneously used as the main cues to reproduce the sequences, with a slight but significant dominance of numerosity. Overall, the results are in line with previous literature suggesting that numerosity is directly encoded, even for temporal sequences, but a non-numerical feature (temporal frequency) is also used in reproducing sequences.

Trend judgment as a perceptual building block of graphicacy and mathematics, across age, education, and culture

Data plots are widely used in science, journalism and politics, since they efficiently allow to depict a large amount of information. Graphicacy, the ability to understand graphs, has thus become a fundamental cultural skill comparable to literacy or numeracy. Here, we introduce a measure of intuitive graphicacy that assesses the perceptual ability to detect a trend in noisy scatterplots ("does this graph go up or down?"). In 3943 educated participants, responses vary as a sigmoid function of the t-value that a statistician would compute to detect a significant trend. We find a minimum level of core intuitive graphicacy even in unschooled participants living in remote Namibian villages (N = 87) and 6-year-old 1st-graders who never read a graph (N = 27). The sigmoid slope that we propose as a proxy of intuitive graphicacy increases with education and tightly correlates with statistical and mathematical knowledge, showing that experience contributes to refining graphical intuitions. Our tool, publicly available online, allows to quickly evaluate and formally quantify a perceptual building block of graphicacy.

Effects of intelligence and approximate number system on the non-symbolic division ability in preschoolers

Recently, it has become evident that cognitive abilities such as the approximate number system (ANS), number knowledge, and intelligence affect individuals' fundamental mathematical ability. However, it is unclear which of these cognitive abilities have the greatest impact on the non-symbolic division ability in preschoolers. Therefore, in the present study, we included 4- to 6-year-old Korean preschoolers without prior formal education of division in order to test their ability to solve non-symbolic division problems, ANS acuity, and intelligence, and to determine the interrelationships among those functions (N = 38). We used the Panamath Dot Comparison Paradigm to measure the ANS acuity, employed non-symbolic division tasks to measure the ability to solve non-symbolic division problems, and measured the intelligence using the Korean version of the WPPSI-IV (Wechsler Preschool Primary Scale of Intelligence-IV). Our results showed that, in all conditions of the non-symbolic division tasks, the 4- to 6-years old children were able to perform better than chance level. Additionally, in a relatively easy condition, the children's performance showed a significant positive correlation with full-scale intelligence quotient (FSIQ) and ANS acuity; however, in a more complex condition, only FSIQ was significantly correlated with their performance. Overall, we found significant relationships between the children's performance in the non-symbolic division tasks and verbal comprehension, fluid reasoning, and processing speed index. Taken together, our findings demonstrate that preschoolers without formal education on the arithmetic problem solving can solve non-symbolic division problems. Moreover, we suggest that both FSIQ and ANS ability play essential roles in children's ability to solve non-symbolic division problems, highlighting the significance of intelligence on children's fundamental mathematical ability.

Contributions of Lower Structures to Higher Cognition: Towards a Dynamic Network Model

Researchers often attribute higher cognition to the enlargement of cortical regions throughout evolution, reflecting the belief that humans sit at the top of the cognitive pyramid. Implicitly, this approach assumes that the subcortex is of secondary importance for higher-order cognition. While it is now recognized that subcortical regions can be involved in various cognitive domains, it remains unclear how they contribute to computations essential for higher-level cognitive processes such as endogenous attention and numerical cognition. Herein, we identify three models of subcortical-cortical relations in these cognitive processes: (i) subcortical regions are not involved in higher cognition; (ii) subcortical computations support elemental forms of higher cognition mainly in species without a developed cortex; and (iii) higher cognition depends on a whole-brain dynamic network, requiring integrated cortical and subcortical computations. Based on evolutionary theories and recent data, we propose the SEED hypothesis: the Subcortex is Essential for the Early Development of higher cognition. According to the five principles of the SEED hypothesis, subcortical computations are essential for the emergence of cognitive abilities that enable organisms to adapt to an ever-changing environment. We examine the implications of the SEED hypothesis from a multidisciplinary perspective to understand how the subcortex contributes to various forms of higher cognition.

Learning artificial number symbols with ordinal and magnitude information

The question of how numerical symbols gain semantic meaning is a key focus of mathematical cognition research. Some have suggested that symbols gain meaning from magnitude information, by being mapped onto the approximate number system, whereas others have suggested symbols gain meaning from their ordinal relations to other symbols. Here we used an artificial symbol learning paradigm to investigate the effects of magnitude and ordinal information on number symbol learning. Across two experiments, we found that after either magnitude or ordinal training, adults successfully learned novel symbols and were able to infer their ordinal and magnitude meanings. Furthermore, adults were able to make relatively accurate judgements about, and map between, the novel symbols and non-symbolic quantities (dot arrays). Although both ordinal and magnitude training was sufficient to attach meaning to the symbols, we found beneficial effects on the ability to learn and make numerical judgements about novel symbols when combining small amounts of magnitude information for a symbol subset with ordinal information about the whole set. These results suggest that a combination of magnitude and ordinal information is a plausible account of the symbol learning process.

Précis of What Babies Know

Where does human knowledge begin? Research on human infants, children, adults, and nonhuman animals, using diverse methods from the cognitive, brain, and computational sciences, provides evidence for six early emerging, domain-specific systems of core knowledge. These automatic, unconscious systems are situated between perceptual systems and systems of explicit concepts and beliefs. They emerge early in infancy, guide children's learning, and function throughout life.

Quantity discrimination by kittens of the domestic cat (Felis silvestris catus)

Quantitative abilities are well described in many species and in diverse life situations, including in the adult domestic cat. However, such abilities have been much less studied during ontogeny. In the present study we examined spontaneous quantity discrimination by pre-weaning age kittens in two-way food choice experiments. In Experiment 1, 26 kittens performed 12 trials with different ratios between the number of same-size food items. In Experiment 2, 24 other kittens performed eight trials with different ratios between the size of two food items. We found, in general, that the kittens discriminated between the different amounts of food and spontaneously chose the larger one, but that their choice was influenced by the ratio of difference. The kittens in Experiment 1 chose the larger number of same-size food items if the ratio was smaller than 0.4 and in Experiment 2 they chose the larger pieces of food if the ratio between the items was smaller than 0.5. Because the kittens' choice was not influenced by the absolute number of food items or the numerical difference between them in Experiment 1, it suggests that their cognitive performance relied on an analog magnitude system rather than on an object file system during the quantity discrimination tasks. We discuss our results considering the ecological and social background of cats and compare it with the performance of previously studied species.

Effects of Robotics Education on Young Children's Cognitive Development: a Pilot Study with Eye-Tracking

The emerging field of robotics education (RE) is a new and rapidly growing subject area worldwide. It may provide a playful and novel learning environment for children to engage with all aspects of science, technology, engineering, and mathematics (STEM) learning. The purpose of this research is to examine how robotics learning activities may affect the cognitive abilities and cognitive processes of 6-8 years old children. The study adopted the mixed methods approach with a repeated measures design; three waves of data collection over 6 months, including quantitative data obtained from cognitive assessments and eye-tracking, and qualitative data from the interviews. A total of 31 children were recruited from an afterschool robotics program. To the best of our knowledge, this study is the first RE research that used a combination of eye-tracking, cognitive assessments, and interviews for examining the effect of RE on children. Using linear growth models, the results of cognitive assessments showed that children's visuospatial working memory as well as logical and abstract reasoning skills improved over time. The interview data were analyzed by a thematic analysis. The results revealed that children perceived RE activities as game play, which made children more engaged in their study; parents found their children to be more focused on activities comparing to six months ago. Additionally, the visualization of the eye-tracking data suggested that children became more focused on RE activities and got faster to process the information across six months in general, which echoed the findings in assessments and interviews. Our findings may help educators and policymakers better understand the benefits of RE for young children.

Neurophysiological signatures of approximate number system acuity in preschoolers

BACKGROUND

A hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers.

METHOD

ERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task.

RESULTS

Only children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks.

CONCLUSION

Results demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.

Are infants' preferences in the number change detection paradigm driven by sequence patterns?

Inter-individual differences in infants' numerosity processing have been assessed using a change detection paradigm, where participants were presented with two concurrent streams of images, one alternating between two numerosities and the other showing one constant numerosity. While most infants look longer at the changing stream in this paradigm, the reasons underlying these preferences have remained unclear. We suggest that, besides being attracted by numerosity changes, infants perhaps also respond to the alternating pattern of the changing stream. We conducted two experiments (N = 32) with 6-month-old infants to assess this hypothesis. In the first experiment, infants responded to changes in numerosity even when the changing stream showed numerosities in an unpredictable random order. In the second experiment, infants did not display any preference when an alternating stream was pitted against a random stream. These findings do not provide evidence that the alternating pattern of the changing stream contributes to drive infants' preferences. Instead, around the age of 6 months, infants' responses in the numerosity change detection paradigm appear to be mainly driven by changes in numerosity, with different levels of preference reflecting inter-individual difference in the acuity of numerosity perception.

Large-scale study of the precision of the approximate number system: Differences between formats, heterogeneity and congruency effects

The study used a large sample of elementary schoolchildren in Russia (N = 3,448, 51.6% were girls, with a mean age of 8.70 years, ranging 6-11 years) to investigate the congruency, format and heterogeneity effects in a nonsymbolic comparison test and between-individual differences in these effects with generalized linear mixed effects models (GLMMs). The participants were asked to compare two arrays of figures of different colours in spatially separated or spatially intermixed formats. In addition, the figures could be similar or different for the two arrays. The results revealed that congruency (difference between congruent and incongruent items), format (difference between mixed and separated formats) and heterogeneity (difference between homogeneous and heterogeneous conditions) interacted. The heterogeneity effect was higher in the separated format, while the format effect was higher for the homogeneous condition. The separated format produced a greater congruency effect than the mixed format. In addition, the congruency effect was lower in the heterogeneous condition than in the homogeneous condition. Analysis of between-individual differences revealed that there was significant between-individual variance in the format and congruency effects. Analysis of between-grade differences revealed that accuracy improved from grade 1 to grade 4 only for congruent trials in separated formats. Consequently, the congruency effect increased in separated/homogeneous and separated/heterogeneous conditions. In general, the study demonstrated that the test format and heterogeneity affected accuracy and that this effect varied for congruent and incongruent items.

Directionality in the interrelations between approximate number, verbal number, and mathematics in preschool-aged children

A fundamental question in numerical development concerns the directional relation between an early-emerging non-verbal approximate number system (ANS) and culturally acquired verbal number and mathematics knowledge. Using path models on longitudinal data collected in preschool children (M_age  = 3.86 years; N = 216; 99 males; 80.8% White; 10.8% Multiracial, 3.8% Latino; 1.9% Black; collected 2013-2017) over 1 year, this study showed that earlier verbal number knowledge was associated with later ANS precision (average β = .32), even after controlling for baseline differences in numerical, general cognitive, and language abilities. In contrast, earlier ANS precision was not associated with later verbal number knowledge (β = -.07) or mathematics abilities (average β = .10). These results suggest that learning about verbal numbers is associated with a sharpening of pre-existing non-verbal numerical abilities.

Leveraging Math Cognition to Combat Health Innumeracy

Rational numbers (i.e., fractions, percentages, decimals, and whole-number frequencies) are notoriously difficult mathematical constructs. Yet correctly interpreting rational numbers is imperative for understanding health statistics, such as gauging the likelihood of side effects from a medication. Several pernicious biases affect health decision-making involving rational numbers. In our novel developmental framework, the natural-number bias-a tendency to misapply knowledge about natural numbers to all numbers-is the mechanism underlying other biases that shape health decision-making. Natural-number bias occurs when people automatically process natural-number magnitudes and disregard ratio magnitudes. Math-cognition researchers have identified individual differences and environmental factors underlying natural-number bias and devised ways to teach people how to avoid these biases. Although effective interventions from other areas of research can help adults evaluate numerical health information, they circumvent the core issue: people's penchant to automatically process natural-number magnitudes and disregard ratio magnitudes. We describe the origins of natural-number bias and how researchers may harness the bias to improve rational-number understanding and ameliorate innumeracy in real-world contexts, including health. We recommend modifications to formal math education to help children learn the connections among natural and rational numbers. We also call on researchers to consider individual differences people bring to health decision-making contexts and how measures from math cognition might identify those who would benefit most from support when interpreting health statistics. Investigating innumeracy with an interdisciplinary lens could advance understanding of innumeracy in theoretically meaningful and practical ways.

Math difficulties in attention deficit hyperactivity disorder do not originate from the visual number sense

There is ample evidence from literature and clinical practice indicating mathematical difficulties in individuals with ADHD, even when there is no concomitant diagnosis of developmental dyscalculia. What factors underlie these difficulties is still an open question. Research on dyscalculia and neurotypical development suggests visual perception of numerosity (the number sense) as a building block for math learning. Participants with lower numerosity estimation thresholds (higher precision) are often those with higher math capabilities. Strangely, the role of numerosity perception in math skills in ADHD has been neglected, leaving open the question whether math difficulties in ADHD also originate from a deficitary visual number sense. In the current study we psychophysically measured numerosity thresholds and accuracy in a sample of children/adolescents with ADHD, but not concomitant dyscalculia (N = 20, 8-16 years). Math abilities were also measured by tasks indexing different mathematical competences. Numerosity performance and math scores were then compared to those obtained from an age-matched control group (N = 20). Bayesian statistics indicated no difference between ADHD and controls on numerosity perception, despite many of the symbolic math tasks being impaired in participants with ADHD. Moreover, the math deficits showed by the group with ADHD remained substantial even when numerosity thresholds were statistically regressed out. Overall, these results indicate that math difficulties in ADHD are unlikely to originate from an impaired visual number sense.

Learning with certainty in childhood

Learners use certainty to guide learning. They maintain existing beliefs when certain, but seek further information when they feel uninformed. Here, we review developmental evidence that this metacognitive strategy does not require reportable processing. Uncertainty prompts nonverbal human infants and nonhuman animals to engage in strategies like seeking help, searching for additional information, or opting out. Certainty directs children's attention and active learning strategies and provides a common metric for comparing and integrating conflicting beliefs across people. We conclude that certainty is a continuous, domain-general signal of belief quality even early in life.

The malleable impact of non-numeric features in visual number perception

Non-numeric stimulus features frequently influence observers' number judgments: when judging the number of items in a display, we will often (mis)perceive the set with a larger cumulative surface area as more numerous. These "congruency effects" are often used as evidence for how vision extracts numeric information and have been invoked in arguments surrounding whether non-numeric cues (e.g., cumulative area, density, etc.) are combined for number perception. We test whether congruency effects for one such cue - cumulative area - provide evidence that it is necessarily used and integrated in number perception, or if its influence on number is malleable. In Experiment 1, we replicate and extend prior work showing that the presence of feedback eliminates congruency effects between number and cumulative area, suggesting that the role of cumulative area in number perception is malleable rather than obligatory. In Experiment 2, we test whether this malleable influence is because of use of prior experiences about how number naturalistically correlates with cumulative area, or the result of response competition, with number and cumulative area actively competing for the same behavioral decision. We preserve cumulative area as a visual cue but eliminate response competition with number by replacing one side of the dot array with its corresponding Hindu-Arabic numeral. Independent of the presence or absence of feedback, we do not observe congruency effects in Experiment 2. These experiments suggest that cumulative area is not necessarily integrated in number perception nor a reflection of a rational use of naturalistic correlations, but rather congruency effects between cumulative area and number emerge as a consequence of response competition. Our findings help to elucidate the mechanism through which non-numeric cues and number interact, and provide an explanation for why congruency effects are only sometimes observed across studies.

Environmental influences on mathematics performance in early childhood

Math skills relate to lifelong career, health, and financial outcomes. Individuals' own cognitive abilities predict math performance and there is growing recognition that environmental influences including differences in culture and variability in math engagement also impact math skills. In this Review, we summarize evidence indicating that differences between languages, exposure to math-focused language, socioeconomic status, attitudes and beliefs about math, and engagement with math activities influence young children's math performance. These influences play out at the community and individual level. However, research on the role of these environmental influences for foundational number skills, including understanding of number words, is limited. Future research is needed to understand individual differences in the development of early emerging math skills such as number word skills, examining to what extent different types of environmental input are necessary and how children's cognitive abilities shape the impact of environmental input.