Calibrating the mental number line

Crossing the Boundary: No Catastrophic Limits on Infants' Capacity to Represent Linguistic Sequences

The boundary effect, namely the infants' failures to compare small and large numerosities, is well documented in studies using visual stimuli. The prevailing explanation is that the numerical system used to process sets up to 3 is incompatible with the system employed for numbers >3. This study investigates the boundary effect in 10-month-old infants presented with linguistic sequences. In Condition 1 (2 vs. 3), infants can differentiate small syllable sequences (2 vs. 3), with better performance for the 2-syllable sequence, which imposes a lower memory load. Condition 2 (2 vs. 4) revealed that infants are capable of discriminating across bounds, with relatively higher performance for the 4-syllable sequence, possibly encoded as one large ensemble. This study offers evidence that, when processing linguistic sounds, infants flexibly deal with small and large numerical representations with no boundaries or incompatibilities between them. Simultaneously encoding units of different magnitudes might aid early speech processing beyond memory limits.

Origins of numbers: a shared language-of-thought for arithmetic and geometry?

Concepts of exact number are often thought to originate from counting and the successor function, or from a refinement of the approximate number system (ANS). We argue here for a third origin: a shared language-of-thought (LoT) for geometry and arithmetic that involves primitives of repetition, concatenation, and recursive embedding. Applied to sets, those primitives engender concepts of exact integers through recursive applications of additions and multiplications. Links between geometry and arithmetic also explain the emergence of higher-level notions (squares, primes, etc.). Under our hypothesis, understanding a number means having one or several mental expressions for it, and their minimal description length (MDL) determines how easily they can be mentally manipulated. Several historical, developmental, linguistic, and brain imaging phenomena provide preliminary support for our proposal.

Money counts: effects of monetary vs. purely numerical values on the mental representation of quantities

It has been established that humans use different cognitive models to represent and process numerical quantities. In this study, we investigated whether the representation of monetary values fundamentally differs from the representation of numbers. We also examined the influence of both socio/economic factors and mathematical ability on such representation. A group of adults (N = 272) were tested anonymously with a variant of the number-to-position task (Siegler & Opfer, 2003). They were asked to position on a horizontal line quantities expressed either in numerical format (e.g., 50) in the "Number" conditions or as monetary values (e.g., 50€) in the "Money" conditions. The extremes of the line consisted either of specific values (i.e. "2 or 2€" and "503 or 503€") in the "Fixed" conditions or of unspecific concepts of quantity (e.g., "little" and "a lot") in the "Fuzzy" conditions. A linear model, as opposed to a logarithmic one, provided the best fit of group average data in all experimental conditions except for the "Money-Fuzzy" condition. The percentages of absolute error were significantly larger for Money stimuli than Number stimuli in both Fixed and Fuzzy conditions. This is consistent with the law of diminishing marginal utility, which entails that the value of monetary quantities is described by a concave curve rather than a linear relationship. As expected from previous research, participants who were more used to spending large quantities of money were closer to the linear representation model. Participants with higher mathematical abilities represented numerical values more closely to a linear model, but no such effect was found for monetary quantities.

Evidence for a Low Number Prior in Children's Intuitive Number Sense

Children rely on their Approximate Number System to intuitively perceive number. Such adaptations often exhibit sensitivity to real-world statistics. This study investigates a potential manifestation of the ANS's sensitivity to real-world statistics: a negative power-law distribution of objects in natural scenes should be reflected in children's expectations about number, or in more Bayesian terms, a low number prior distribution. Five- to eight-year-old children (n = 80; 39 girls, 41 boys) and adults (n = 20) in 2022 completed a number discrimination task in which one side was corrupted by perceptual noise. Children and adults demonstrate a low number prior. No age-related differences were observed, suggesting that the prior is formed by age five and does not strengthen with age.

Psychophysics over the counter: The effect of food preference on the perception of food quantity

Much research in food psychology has examined the influence of perceptual features - color, texture, packaging - on preference for that food. Here, we addressed the reverse question of the influence of food preference on the perception of its quantity. Does a portion of a loved food appear different than the same portion of a hated food? We probed this question by employing state-of-the-art tools of psychophysics, which allowed us the parallel examination of several long-standing issues of psychophysics. The latter included the difference between symbolic and non-symbolic number, the difference between the methods of Magnitude Estimation and Magnitude Production, the difference between under- or over-estimation and rate of growth measured by the (slope of) psychophysical function as well as the derivation of the Difference Threshold by the method of Constant Stimuli. We addressed the question of an effect on perception of food valence with four distinct experiments. Presenting real food items, we found that perceived quantity is a compressive function of objective quantity regardless of valence; both loved- and hated-food dynamics are governed by Stevens' power function with an exponent of 0.8. In absolute terms, applying Magnitude Estimation, we witnessed a gross underestimation for loved and hated food alike. In contrast, applying Magnitude Production, participants underproduced loved food, but overproduced hated food. For discrimination of food quantity, we found better resolving power for hated food. Collectively, our results show that, across diverse psychophysical evaluations, food valence affects its perception, especially when people actively regulate the to-be-consumed portion.

Estimation of factorial expressions and its improvement through calibration: A replication and extension of Tversky and Kahneman (1973)

Fifty years ago, Tversky and Kahneman (Cognitive Psychology, 5[2], 207-232, 1973) reported that people's speeded estimations of 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 were notably higher than their estimations for the equivalent expression in the opposite order, 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 (Median = 2,250 vs. 512, respectively). On top of this order effect, both groups grossly underestimated the correct value (40,320). The differential effect of the two orders on estimation has become famous as an early demonstration of the anchoring effect, where people's judgments under uncertainty are unduly influenced by an initial reference point (or "anchor"). Despite this fame, to the best of our knowledge, this effect has never been replicated. In a sample of 253 U.S. adults, the current study provides the first replication of this foundational example of anchoring. It extends this effect for the first time to a within-participants design, revealing its relative robustness even among participants who see the descending order first. Drawing on procedures from the mathematical cognition literature, it shows how the anchoring effect can be mitigated: calibrating to the correct value of 6! reduces this effect, and calibrating to 10! eliminates it altogether. An individual differences analysis measures the arithmetic fluency of participants and their accuracy on a new estimation assessment, and finds that higher estimation ability may be a "protective factor" against some anchoring effects. These findings affirm the anchoring effect of Tversky and Kahneman (1973, Study 6) while suggesting that calibration may be an effective strategy for helping to improve people's estimation of superlinear functions that are important in real-life contexts.

Distinct serial dependence between small and large numerosity processing

The serial dependence effect (SDE) is a perceptual bias where current stimuli are perceived as more similar to recently seen stimuli, possibly enhancing the stability and continuity of visual perception. Although SDE has been observed across many visual features, it remains unclear whether humans rely on a single mechanism of SDE to support numerosity processing across two distinct numerical ranges: subitizing (i.e., small numerosity processing, likely related to early object recognition) and estimation (i.e., large numerosity processing, likely related to ensemble numerosity extraction). Here, we show that subitizing and estimation exhibit distinct SDE patterns. Subitizing is characterized by an asymmetric SDE, whereas estimation demonstrates a symmetric SDE. Specifically, in subitizing, the SDE occurs only when the current magnitude is smaller than the previous magnitude but not when it is larger. In contrast, the SDE in estimation is present in both scenarios. We propose that these differences arise from distinct underlying mechanisms. A perceptual mechanism-namely, a 'temporal hysteresis' account, can explain the asymmetrical SDE in subitizing since object individuation resources are easily activated but resistant to deactivation. Conversely, a combination of perceptual and post-perceptual mechanisms can account for the SDEs in estimation, as both perceptual and post-perceptual interference can reduce the SDEs. Critically, a novel type of SDE characterized by reduced processing precision is found in subitizing only, implying that the continuity and stability of numerical processing can be dissociable in dynamic situations where numerical information is integrated over time. Our findings reveal the multifaceted nature of SDE mechanisms and suggest their engagement with cognitive modules likely subserving different functionalities.

Investigating acoustic numerosity illusions in professional musicians

Various studies have reported an association between musical expertise and enhanced visuospatial and mathematical abilities. A recent work tested the susceptibility of musicians and nonmusicians to the Solitaire numerosity illusion finding that also perceptual biases underlying numerical estimation are influenced by long-term music training. However, the potential link between musical expertise and different perceptual mechanisms of quantitative estimation may be either limited to the visual modality or universal (i.e., modality independent). We addressed this question by developing an acoustic version of the Solitaire illusion. Professional musicians and nonmusicians listened to audio file recordings of piano and trombone notes and were required to estimate the number of piano notes. The stimuli were arranged to form test trials, with piano and trombone notes arranged in a way to form the Solitaire pattern, and control trials, with randomly located notes to assess their quantitative abilities in the acoustic modality. In the control trials, musicians were more accurate in numerical estimation than nonmusicians. In the presence of illusory patterns, nonmusicians differed from musicians in the esteem of regularly arranged vs. randomly arranged notes. This suggests that the association between long-term musical training and different perceptual mechanisms underlying numerical estimation may not be confined to the visual modality. However, neither musicians nor nonmusicians seemed to be susceptible to the acoustic version of the Solitaire illusion, suggesting that the emergence of this illusion may be stimulus and task-dependent.

Anchoring bias in mental arithmetic

Mental arithmetic is widely studied, both with symbolic digits and with non-symbolic dot patterns that require operand estimation. Several studies reported surprising biases in adults' performance with both formats while their direction (over/underestimation in addition/subtraction) remains controversial (operational momentum effect or OM; Prado & Knops, Prado and Knops, Psychonomic Bulletin & Review, in Press., 2024). Theoretical accounts of OM make opposing predictions, thus enabling a decisive test: Using symbolic stimuli and responses, we enabled accurate operand encoding and result reporting, thus leaving mental calculation as only source of bias. Importantly, we manipulated operand order through calculation instructions (e.g., "29 + 19" vs. "add 19 to 29") to assess the crucial role of first operand size as cognitive anchor. With both auditory (Experiment 1, N = 30) and visual presentation (Experiment 2, N = 30), we observed reverse OM, i.e., overestimations in subtraction and underestimations in addition. Importantly, this instance of operation-based anchoring was independent of a second anchoring effect related to operand order: A large operand is a stronger anchor when mentioned first. Our discovery of both operation-based and order-based anchoring extends the well-known anchoring effect into mental arithmetic and eliminates several competing theories about the origin of OM.

Children's estimates of equivalent rational number magnitudes are not equal: Evidence from fractions, decimals, percentages, and whole numbers

Integration of rational number knowledge with prior whole number knowledge has been theorized as critical for mathematical success. Fractions, decimals, and percentages are generally assumed to differ in difficulty based on the degree to which their structure is perceptually similar to whole numbers. Specifically, percentages are viewed as most similar to whole numbers with their fixed unstated denominator of 100. Decimals are often assumed to be easier than fractions because their place-value structure is an extension of the base-ten system for whole numbers, unlike fractions, which have a bipartite structure (i.e., a/b). However, there has been no comprehensive investigation of how fraction, decimal, and percentage knowledge compares with whole number knowledge. To assess understanding of the four notations, we measured within-participants number line estimation of equivalent fractions and decimals with shorter string lengths (e.g., 8/10 and 0.8) and longer string lengths (e.g., 80/100 and 0.80), percentages (e.g., 80%), and proportionally equivalent whole numbers on a 0-100 scale (e.g., 80.0). Middle school students (N = 65; 33 female) generally underestimated all formats relative to their actual values (whole numbers: 3% below; percentages: 2%; decimals: 17%; fractions: 5%). Shorter string-length decimals and fractions were estimated as smaller than equivalent longer string-length equivalents. Overall, percentages were estimated similarly to corresponding whole numbers, fractions had modest string-length effects, and decimals were the most underestimated, especially for single-digit decimals. These results highlight the strengths and weaknesses of children's understanding of each notation's magnitudes and challenge the assumption that decimals are easier than fractions.

The dark and bright side of the numbers: how emotions influence mental number line accuracy and bias

The traditional view of cognition as detached from emotions is recently being questioned. This study aimed to investigate the influence of emotional valence on the accuracy and bias in the representation of numbers on the mental number line (MNL). The study included 164 participants who were randomly assigned into two groups with induced positive and negative emotional valence using matched arousal film clips. Participants performed a computerised number-to-position (CNP) task to estimate the position of numbers on a horizontal line. The results showed that participants in the positive valence group exhibited a rightward bias, while those in the negative valence group showed an opposite pattern. The analysis of mean absolute error revealed that the negative valence group had higher error rates compared to the positive valence group. Furthermore, the MNL estimation pattern analysis indicated that a two-cycle cyclic power model (CPM) best explained the data for both groups. These findings suggest that emotional valence influences the spatial representation of numbers on the MNL and affects accuracy in numerical estimations. Our findings are finally discussed in terms of body-specificity and the Brain's Asymmetric Frequency Tuning (BAFT) theories. The study provides new insights into the interplay between emotions and numerical cognition.

The radial-tangential anisotropy of numerosity perception

Humans can estimate the number of visually presented items without counting. In most studies on numerosity perception, items are uniformly distributed across displays, with identical distributions in central and eccentric parts. However, the neural and perceptual representation of the human visual field differs between the fovea and the periphery. For example, in peripheral vision, there are strong asymmetries with regard to perceptual interferences between visual items. In particular, items arranged radially usually interfere more strongly with each other than items arranged tangentially (the radial-tangential anisotropy). This has been shown for crowding (the deleterious effect of clutter on target identification) and redundancy masking (the reduction of the number of perceived items in repeating patterns). In the present study, we tested how the radial-tangential anisotropy of peripheral vision impacts numerosity perception. In four experiments, we presented displays with varying numbers of discs that were predominantly arranged radially or tangentially, forming strong and weak interference conditions, respectively. Participants were asked to report the number of discs. We found that radial displays were reported as less numerous than tangential displays for all radial and tangential manipulations: weak (Experiment 1), strong (Experiment 2), and when using displays with mixed contrast polarity discs (Experiments 3 and 4). We propose that numerosity perception exhibits a significant radial-tangential anisotropy, resulting from local spatial interactions between items.

Motivating parents to protect their children from wildfire smoke: the impact of air quality index infographics

Background. Wildfire smoke events are increasing in frequency and intensity due to climate change. Children are especially vulnerable to health effects even at moderate smoke levels. However, it is unclear how parents respond to Air Quality Indices (AQIs) frequently used by agencies to communicate air pollution health risks. Methods. In an experiment (3 × 2 × 2 factorial design), 2,100 parents were randomly assigned to view one of twelve adapted AQI infographics that varied by visual (table, line, gauge), index type (AQI [0-500], AQHI [1-11+]), and risk level (moderate, high). Participants were told to imagine encountering the infographic in a short-term exposure scenario. They reported worry about wildfire smoke, intentions to take risk-mitigating actions (e.g., air purifier use), and support for various exposure reduction policies. Subsequently, participants were told to imagine encountering the same infographic daily during a school week in a long-term exposure scenario and again reported worry, action intentions, and policy support. Results. Parents' responses significantly differentiated between risk levels that both pose a threat to children's health; worry and action intentions were much higher in the high-risk group than the moderate-risk group in both short-exposure (F = 748.68 p<.001; F = 411.59, p<.001) and long-exposure scenarios (F = 470.51, p<.001; F = 212.01, p<.001). However, in the short-exposure scenario, when shown the AQHI [1-11+] with either the line or gauge visuals, parents' action intentions were more similar between moderate- and high-risk level groups (3-way interaction, F = 6.03, p = .002). Conclusions. These results suggest some index formats such as the AQHI-rather than the AQI-may better attune parents to moderate levels of wildfire smoke being dangerous to children's health. Our research offers insights for agencies and officials seeking to improve current public education efforts during wildfire smoke events and speaks to the critical need to educate parents and help them act short-term and long-term to protect children's health.

Differential roles of brain oscillations in numerical processing: evidence from resting-state EEG and mental number line

Recent works point to the importance of emotions in special-numerical associations. There remains a notable gap in understanding the electrophysiological underpinnings of such associations. Exploring resting-state (rs) EEG, particularly in frontal regions, could elucidate emotional aspects, while other EEG measures might offer insights into the cognitive dimensions correlating with behavioral performance. The present work investigated the relationship between rs-EEG measures (emotional and cognitive traits) and performance in the mental number line (MNL). EEG activity in theta (3-7 Hz), alpha (8-12 Hz, further subdivided into low-alpha and high-alpha), sensorimotor rhythm (SMR, 13-15 Hz), beta (16-25 Hz), and high-beta/gamma (28-40 Hz) bands was assessed. 76 university students participated in the study, undergoing EEG recordings at rest before engaging in a computerized number-to-position (CNP) task. Analysis revealed significant associations between frontal asymmetry, specific EEG frequencies, and MNL performance metrics (i.e., mean direction bias, mean absolute error, and mean reaction time). Notably, theta and beta asymmetries correlated with direction bias, while alpha peak frequency (APF) and beta activity related to absolute errors in numerical estimation. Moreover, the study identified significant correlations between relative amplitude indices (i.e., theta/beta ratio, theta/SMR ratio) and both absolute errors and reaction times (RTs). Our findings offer novel insights into the emotional and cognitive aspects of EEG patterns and their links to MNL performance.

The nature of anchor-biased estimates and its application to the wisdom of crowds

We propose a method to achieve better wisdom of crowds by utilizing anchoring effects. In this method, people are first asked to make a comparative judgment such as "Is the number of new COVID-19 infections one month later more or less than 10 (or 200,000)?" As in this example, two sufficiently different anchors (e.g., "10" or "200,000") are set in the comparative judgment. After this comparative judgment, people are asked to make their own estimates. These estimates are then aggregated. We hypothesized that the aggregated estimates using this method would be more accurate than those without anchor presentation. To examine the effectiveness of the proposed method, we conducted three studies: a computer simulation and two behavioral experiments (numerical estimation of perceptual stimuli and estimation of new COVID-19 infections by physicians). Through computer simulations, we could identify situations in which the proposed method is effective. Although the proposed method is not always effective (e.g., when a group can make fairly accurate estimations), on average, the proposed method is more likely to achieve better wisdom of crowds. In particular, when a group cannot make accurate estimations (i.e., shows biases such as overestimation or underestimation), the proposed method can achieve better wisdom of crowds. The results of the behavioral experiments were consistent with the computer simulation findings. The proposed method achieved better wisdom of crowds. We discuss new insights into anchoring effects and methods for inducing diverse opinions from group members.

A number sense as an emergent property of the manipulating brain

The ability to understand and manipulate numbers and quantities emerges during childhood, but the mechanism through which humans acquire and develop this ability is still poorly understood. We explore this question through a model, assuming that the learner is able to pick up and place small objects from, and to, locations of its choosing, and will spontaneously engage in such undirected manipulation. We further assume that the learner's visual system will monitor the changing arrangements of objects in the scene and will learn to predict the effects of each action by comparing perception with a supervisory signal from the motor system. We model perception using standard deep networks for feature extraction and classification. Our main finding is that, from learning the task of action prediction, an unexpected image representation emerges exhibiting regularities that foreshadow the perception and representation of numbers and quantity. These include distinct categories for zero and the first few natural numbers, a strict ordering of the numbers, and a one-dimensional signal that correlates with numerical quantity. As a result, our model acquires the ability to estimate numerosity, i.e. the number of objects in the scene, as well as subitization, i.e. the ability to recognize at a glance the exact number of objects in small scenes. Remarkably, subitization and numerosity estimation extrapolate to scenes containing many objects, far beyond the three objects used during training. We conclude that important aspects of a facility with numbers and quantities may be learned with supervision from a simple pre-training task. Our observations suggest that cross-modal learning is a powerful learning mechanism that may be harnessed in artificial intelligence.

Rationality, preferences, and emotions with biological constraints: it all starts from our senses

Is the role of our sensory systems to represent the physical world as accurately as possible? If so, are our preferences and emotions, often deemed irrational, decoupled from these 'ground-truth' sensory experiences? We show why the answer to both questions is 'no'. Brain function is metabolically costly, and the brain loses some fraction of the information that it encodes and transmits. Therefore, if brains maximize objective functions that increase the fitness of their species, they should adapt to the objective-maximizing rules of the environment at the earliest stages of sensory processing. Consequently, observed 'irrationalities', preferences, and emotions stem from the necessity for our early sensory systems to adapt and process information while considering the metabolic costs and internal states of the organism.

Children dynamically update and extend the interface between number words and perceptual magnitudes

As adults, we represent and think about number, space, and time in at least two ways: our intuitive-but imprecise-perceptual representations, and the slowly learned-but precise-number words. With development, these representational formats interface, allowing us to use precise number words to estimate imprecise perceptual experiences. We test two accounts of this developmental milestone. Either slowly learned associations are required for the interface to form, predicting that deviations from typical experiences (e.g., presentation of a novel unit or unpracticed dimension) will disrupt children's ability to map number words to their perceptual experiences or children's understanding of the logical similarity between number words and perceptual representations allows them to flexibly extend this interface to novel experiences (e.g., units and dimensions they have not yet learned how to formally measure). 5-11-year-olds completed verbal estimation and perceptual sensitivity tasks across three dimensions: Number, Length, and Area. For verbal estimation, they were given novel units (i.e., a three-dot unit called one "toma" for Number, a 44 px long line called one "blicket" for Length, a 111 px2 blob called one "modi" for Area) and asked to estimate how many tomas/blickets/modies they saw when shown a larger set of dots, lines, and blobs. Children could flexibly link number words to novel units across dimensions, demonstrating positive estimation slopes, even for Length and Area, which younger children had limited experience with. This suggests that the logic of structure mapping can be dynamically utilized across perceptual dimensions, even without extensive experience.

The longitudinal contribution of mapping to arithmetic: Do numeral knowledge, inhibition or analogical reasoning matter?

BACKGROUND

Recent studies have revealed the association between mapping and arithmetic (Ferres-Forga et al., J. Numer. Cogn., 8, 2022, 123; LeFevre et al., J. Numer. Cogn., 8, 2022, 1).

AIM

The underlying mechanism remains unclear.

MATERIALS & METHODS

The current study recruited 118 kindergarten children and followed up on them three times at 6-month intervals. They completed measures to assess mapping skills (T1), non-verbal IQ (T1), numeral knowledge (T2), inhibitory control (T2), analogical reasoning (T2) and arithmetic (T3).

RESULTS

The results showed that mapping accounted for significant variance in arithmetic ability over and above age, gender and non-verbal IQ. Furthermore, analogical reasoning played an important role in the relationship between mapping and mathematics ability.

DISCUSSION

The findings suggest the association between mapping and mathematics ability prior to formal schooling.

CONCLUSION

Analogical reasoning, rather than numeral knowledge or inhibitory control, may drive that association in young children.

Response boosts serial dependence in the numerosity estimation task

Perceptions of current stimuli are sometimes biased toward or away from past perceptions. This phenomenon is called serial dependence. However, the strength of the effect of past responses on serial dependence has not been fully elucidated. We conducted experiments with a task in which participants estimated the number of dot arrays (numerosity estimation task) and directly compared whether the strength of serial dependence changed in the numerosity estimation task when participants responded or did not respond in the immediately preceding trial. We also examined whether the strength of serial dependence affected the accuracy of the numerosity estimation. We found that attractive serial dependence was stronger when participants responded in the immediately preceding trial than when they only saw the stimulus. The results suggest that the information from the previous stimulus must reach the higher-level processes associated with perceptual decisions to influence the estimation of the current stimulus. However, it is possible that the results of this study are specific to tasks in which participants respond with numeric symbols. The magnitude of the serial dependence effect was not observed to affect numerosity estimation performance, and no evidence was found that serial dependence enhances accuracy in the numerosity estimation task.

The Interaction between Congruency and Numerical Ratio Effects in the Nonsymbolic Comparison Test

The nonsymbolic comparison task is used to investigate the precision of the Approximate Number Sense, the ability to process discrete numerosity without counting and symbols. There is an ongoing debate regarding the extent to which the ANS is influenced by the processing of non-numerical visual cues. To address this question, we assessed the congruency effect in a nonsymbolic comparison task, examining its variability across different stimulus presentation formats and numerical proportions. Additionally, we examined the variability of the numerical ratio effect with the format and congruency. Utilizing generalized linear mixed-effects models with a sample of 290 students (89% female, mean age 19.33 years), we estimated the congruency effect and numerical ratio effect for separated and intermixed formats of stimulus presentation, and for small and large numerical proportions. The findings indicated that the congruency effect increased in large numerical proportion conditions, but this pattern was observed only in the separated format. In the intermixed format, the congruency effect was insignificant for both types of numerical proportion. Notably, the numerical ratio effect varied for congruent and incongruent trials in different formats. The results may suggest that the processing of visual non-numerical parameters may be crucial when numerosity processing becomes noisier, specifically when numerical proportion becomes larger. The implications of these findings for refining the ANS theory are discussed.

Anatomical connectivity in children with developmental dyscalculia: A graph theory study

Current theories postulate that numerical processing depends upon a brain circuit formed by regions and their connections; specialized in the representation and manipulation of the numerical properties of stimuli. It has been suggested that the damage of these network may cause Developmental Dyscalculia (DD): a persistent neurodevelopmental disorder that significantly interferes with academic performance and daily life activities that require mastery of mathematical notions and operations. However, most of the studies on the brain foundations of DD have focused on regions of interest associated with numerical processing, and have not addressed numerical cognition as a complex network phenomenon. The present study explored DD using a Graph Theory network approach. We studied the association between topological measures of integration and segregation of information processing in the brain proposed by Graph Theory; and individual variability in numerical performance in a group of 11 school-aged children with DD (5 of which presented with comorbidity with Developmental Dyslexia, the specific learning disorder for reading) and 17 typically developing peers. A statistically significant correlation was found between the Weber fraction (a measure of numerical representations' precision) and the Clustering Index (a measure of segregation of information processing) in the whole sample. The DD group showed significantly lower Characteristic Path Length (average shortest path length among all pairs of regions in the brain network) compared to controls. Also, differences in critical regions for the brain network performance (hubs) were found between groups. The presence of limbic hubs characterized the DD brain network while right Temporal and Frontal hubs found in controls were absent in the DD group. Our results suggest that the DD may be associated with alterations in anatomical brain connectivity that hinder the capacity to integrate and segregate numerical information.

Conceptual foundations of early numeracy: Evidence from infant brain data

Understanding the conceptual resources that children bring to mathematics learning is crucial for developing effective instruction and interventions. Despite the considerable number of studies examining the neural underpinnings of number representations in adults and the growing number of reports in children, very few studies have examined the neural correlates of the link between foundational resources related to numerical information and symbolic number representations in infants. There is currently an active debate about which foundational resources are critical for symbolic mathematics. Is early numerical discrimination best explained by a holistic and generalized sense of magnitude rather than a number sense? Does early number sense provide the conceptual basis for mapping numerical symbols to their meaning? Are foundational number systems marginal while children learn to count and perform symbolic arithmetic, and only later children map non symbolic representations of numerical magnitudes onto symbols? After describing the mainstream theories of numerical cognition and the sources of controversy, we review recent studies of the neural bases of human infants' numerical performance with the aim of clarifying the link between early conceptual resources and symbolic number systems as children's mathematical minds develop.

The relationship between numerical mapping abilities, maths achievement and socioeconomic status in 4- and 5-year-old children

BACKGROUND

Early numeracy skills are associated with academic and life-long outcomes. Children from low-income backgrounds typically have poorer maths outcomes, and their learning can already be disadvantaged before they begin formal schooling. Understanding the relationship between the skills that support the acquisition of early maths skills could scaffold maths learning and improve life chances.

AIMS

The present study aimed to examine how the ability of children from different SES backgrounds to map between symbolic (Arabic numerals) and non-symbolic (dot arrays) at two difficulty ratios related to their math performance.

SAMPLE

Participants were 398 children in their first year of formal schooling (Mean age = 60 months), and 75% were from low SES backgrounds.

METHOD

The children completed symbolic to non-symbolic and non-symbolic to symbolic mapping tasks at two difficulty ratios (1:2; 2:3) plus standardized maths tasks.

RESULTS

The results showed that all the children performed better for symbolic to non-symbolic mapping and when the ratio was 1:2. Mapping task performance was significantly related to maths task achievement, but low-SES children showed significantly lower performance on all tasks.

CONCLUSION

The results suggest that mapping tasks could be a useful way to identify children at risk of low maths attainment.

Individual risk attitudes arise from noise in neurocognitive magnitude representations

Humans are generally risk averse, preferring smaller certain over larger uncertain outcomes. Economic theories usually explain this by assuming concave utility functions. Here, we provide evidence that risk aversion can also arise from relative underestimation of larger monetary payoffs, a perceptual bias rooted in the noisy logarithmic coding of numerical magnitudes. We confirmed this with psychophysics and functional magnetic resonance imaging, by measuring behavioural and neural acuity of magnitude representations during a magnitude perception task and relating these measures to risk attitudes during separate risky financial decisions. Computational modelling indicated that participants use similar mental magnitude representations in both tasks, with correlated precision across perceptual and risky choices. Participants with more precise magnitude representations in parietal cortex showed less variable behaviour and less risk aversion. Our results highlight that at least some individual characteristics of economic behaviour can reflect capacity limitations in perceptual processing rather than processes that assign subjective values to monetary outcomes.

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.

The developmental relationship between nonsymbolic and symbolic fraction abilities

A fundamental research question in quantitative cognition concerns the developmental relationship between nonsymbolic and symbolic quantitative abilities. This study examined this developmental relationship in abilities to process nonsymbolic and symbolic fractions. There were 99 6th graders (M_age = 11.86 years), 101 10th graders (M_age = 15.71 years), and 102 undergraduate and graduate students (M_age = 21.97 years) participating in this study, and their nonsymbolic and symbolic fraction abilities were measured with nonsymbolic and symbolic fraction comparison tasks, respectively. Nonsymbolic and symbolic fraction abilities were significantly correlated in all age groups even after controlling for the ability to process nonsymbolic absolute quantity and general cognitive abilities, including working memory and inhibitory control. Moreover, the strength of nonsymbolic-symbolic correlations was higher in 6th graders than in 10th graders and adults. These findings suggest a weakened association between nonsymbolic and symbolic fraction abilities during development, and this developmental pattern may be related with participants' increasing proficiency in symbolic fractions.

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.

A Perception Study for Unit Charts in the Context of Large-Magnitude Data Representation

Unit charts are a common type of chart for visualizing scientific data. A unit chart is a chart used to communicate quantities of things by making the number of symbols on the chart proportional to the number of items represented. An accurate perception of the order of magnitude is essential to evaluating whether a unit chart can effectively convey information. Previous studies have primarily focused on perceptual properties at small order-of-magnitude scales or the efficacy of pictographs in unit charts. However, few researchers have explored the perceptual effectiveness of unit charts when representing large orders of magnitude. In this study, we performed a series of sampling measurements to investigate the visual–perceptual characteristics of unit charts when representing asymmetric interactions such as large-scale numbers. The results showed that under the restriction of the current conventional display medium, unit charts still offer a significant advantage over bar charts in a single-scale visual overview. However, this comes at the cost of a longer response time. Although this study constitutes basic research, accumulating evidence about how people reason about magnitudes beyond human perception is critical to the field of information science. This study may contribute to understanding how viewers perceive unit charts and the factors that influence graphical perception. This article provides some specific guidelines for designing unit charts that may be useful to visualization designers.

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.

EXPRESS: How negative emotions affect young and older adults' numerosity estimation performance

In this study, we investigated the influence of negative emotions on numerosity estimation and whether this influence changes with aging during adulthood. Young and older adults were asked to estimate and compare the numerosity of collections of elements (cars or dots) with a two-digit number. Collections of elements were preceded by emotionally neutral (e.g., mushrooms) or emotionally negative (e.g., a corpse) pictures. Stimuli were easier (i.e., small-ratio) or harder (i.e., large-ratio) items. Young and older participants obtained similar numerosity estimation performance. Interestingly, participants were less accurate under negative emotions than under neutral emotions when they estimated numerosity of collections of abstract elements (i.e., dots). In contrast, participants improved their performance under negative emotions while estimating collections of non-abstract, daily-life elements (i.e., cars). These findings have important implications for furthering our understanding of the role of negative emotions in numerosity estimation and age-related differences therein.

The impact of musical training in symbolic and non-symbolic audiovisual judgements of magnitude

Quantity estimation can be represented in either an analog or symbolic manner and recent evidence now suggests that analog and symbolic representation of quantities interact. Nonetheless, those two representational forms of quantities may be enhanced by convergent multisensory information. Here, we elucidate those interactions using high-density electroencephalography (EEG) and an audiovisual oddball paradigm. Participants were presented simultaneous audiovisual tokens in which the co-varying pitch of tones was combined with the embedded cardinality of dot patterns. Incongruencies were elicited independently from symbolic and non-symbolic modality within the audio-visual percept, violating the newly acquired rule that “the higher the pitch of the tone, the larger the cardinality of the figure.” The effect of neural plasticity in symbolic and non-symbolic numerical representations of quantities was investigated through a cross-sectional design, comparing musicians to musically naïve controls. Individual’s cortical activity was reconstructed and statistically modeled for a predefined time-window of the evoked response (130–170 ms). To summarize, we show that symbolic and non-symbolic processing of magnitudes is re-organized in cortical space, with professional musicians showing altered activity in motor and temporal areas. Thus, we argue that the symbolic representation of quantities is altered through musical training.

On the usefulness of graph-theoretic properties in the study of perceived numerosity

Observers can quickly estimate the quantity of sets of visual elements. Many aspects of this ability have been studied and the underlying system has been called the Approximate Number Sense (Dehaene, 2011). Specific visual properties, such as size and clustering of the elements, can bias an estimate. For intermediate numerical quantities at low density (above five, but before texturization), human performance is predicted by a model based on the region of influence of elements (occupancy model: Allïk & Tuulmets, 1991). For random 2D configurations we computed ten indices based on graph theory, and we compared them with the occupancy model: independence number, domination, connected components, local clustering coefficient, global clustering coefficient, random walk, eigenvector centrality, maximum clique, total degree of connectivity, and total edge length. We made comparisons across a range of parameters, and we varied the size of the region of influence around each element. The analysis of the pattern of correlations suggests two main groups of graph-based measures. The first group is sensitive to the presence of local clustering of elements, the second seems more sensitive to density and the way information spreads in graphs. Empirical work on perception of numerosity may benefit from comparing, or controlling for, these properties.

Circling around number: People can accurately extract numeric values from circle area ratios

It has long been known that people have the ability to estimate numerical quantities without counting. A standard account is that people develop a sense of the size of symbolic numbers by learning to map symbolic numbers (e.g., 6) to their corresponding numerosities (e.g. :::) and concomitant approximate magnitude system (ANS) representations. However, we here demonstrate that adults are capable of extracting fractional numerical quantities from non-symbolic visual ratios (i.e., labeling a ratio of two circle areas with the appropriate symbolic fraction). Not only were adult participants able to perform this task, but they were remarkably accurate: linear regressions on median estimates yielded slopes near 1, and accounted for 97% of the variability. Participants also performed at least as well on line-estimation and ratio-estimation tasks using non-numeric circular stimuli as they did in earlier experiments using non-symbolic numerosities, which are frequently considered to be numeric stimuli. We discuss results as consistent with accounts suggesting that non-symbolic ratios have the potential to act as a reliable and stable ground for symbolic number, even when composed of non-numeric stimuli.

Development of the Mental Number Line Representation of Numbers 0–10 and Its Relationship to Mental Arithmetic

The internal representation of numbers on the mental number line (MNL) was demonstrated by performing the computerized version of the number-to-position (CNP) task on a touchscreen while restricting response time. We found that the estimation pattern is best fit by a sigmoid function, further denoted as the “sigmoidal model”. Two developmental leaps occurring during elementary school were recognized: (1) the division of the number line into two segments and (2) consistent use of different anchor points on the number line—the left endpoint in first grade, the right endpoint in second grade, and finally the midpoint in third grade. Additionally, when examining the differences between the breakpoints, we found that first graders demonstrated a breakpoint close to 6, which linearly decreased over the years until stabilizing close to 5. The relation between the ability to place individual numbers on a number line and performance of mental arithmetic showed that the consistent use of anchor points correlated significantly with faster responses in mental arithmetic.

Crowd control: Reducing individual estimation bias by sharing biased social information

Cognitive biases are widespread in humans and animals alike, and can sometimes be reinforced by social interactions. One prime bias in judgment and decision-making is the human tendency to underestimate large quantities. Previous research on social influence in estimation tasks has generally focused on the impact of single estimates on individual and collective accuracy, showing that randomly sharing estimates does not reduce the underestimation bias. Here, we test a method of social information sharing that exploits the known relationship between the true value and the level of underestimation, and study if it can counteract the underestimation bias. We performed estimation experiments in which participants had to estimate a series of quantities twice, before and after receiving estimates from one or several group members. Our purpose was threefold: to study (i) whether restructuring the sharing of social information can reduce the underestimation bias, (ii) how the number of estimates received affects the sensitivity to social influence and estimation accuracy, and (iii) the mechanisms underlying the integration of multiple estimates. Our restructuring of social interactions successfully countered the underestimation bias. Moreover, we find that sharing more than one estimate also reduces the underestimation bias. Underlying our results are a human tendency to herd, to trust larger estimates than one's own more than smaller estimates, and to follow disparate social information less. Using a computational modeling approach, we demonstrate that these effects are indeed key to explain the experimental results. Overall, our results show that existing knowledge on biases can be used to dampen their negative effects and boost judgment accuracy, paving the way for combating other cognitive biases threatening collective systems.

Response bias in numerosity perception at early judgments and systematic underestimation

Mental number representation relies on mapping numerosity based on nonsymbolic stimuli to symbolic magnitudes. It is known that mental number representation builds on a logarithmic scale, and thus numerosity decisions result in underestimation. In the current study, we investigated the temporal dynamics of numerosity perception in four experiments by employing the response-deadline SAT procedure. We presented random number of dots and required participants to make a numerosity judgment by comparing the perceived number of dots to 50. Using temporal dynamics in numerosity perception allowed us to observe a response bias at early decisions and a systematic underestimation at late decisions. In all three experiments, providing feedback diminished the magnitude of underestimation, whereas in Experiment 3 the absence of feedback resulted in greater underestimation errors. These results were in accordance with the findings that suggested feedback is necessary for the calibration of the mental number representation.

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

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

"BreaThink": breathing affects production and perception of quantities

Cognition is shaped by signals from outside and within the body. Following recent evidence of interoceptive signals modulating higher-level cognition, we examined whether breathing changes the production and perception of quantities. In Experiment 1, 22 adults verbally produced on average larger random numbers after inhaling than after exhaling. In Experiment 2, 24 further adults estimated the numerosity of dot patterns that were briefly shown after either inhaling or exhaling. Again, we obtained on average larger responses following inhalation than exhalation. These converging results extend models of situated cognition according to which higher-level cognition is sensitive to transient interoceptive states.

Impact of sharing full versus averaged social information on social influence and estimation accuracy

The recent developments of social networks and recommender systems have dramatically increased the amount of social information shared in human communities, challenging the human ability to process it. As a result, sharing aggregated forms of social information is becoming increasingly popular. However, it is unknown whether sharing aggregated information improves people's judgments more than sharing the full available information. Here, we compare the performance of groups in estimation tasks when social information is fully shared versus when it is first averaged and then shared. We find that improvements in estimation accuracy are comparable in both cases. However, our results reveal important differences in subjects' behaviour: (i) subjects follow the social information more when receiving an average than when receiving all estimates, and this effect increases with the number of estimates underlying the average; (ii) subjects follow the social information more when it is higher than their personal estimate than when it is lower. This effect is stronger when receiving all estimates than when receiving an average. We introduce a model that sheds light on these effects, and confirms their importance for explaining improvements in estimation accuracy in all treatments.

Do estimates of numerosity really adhere to Weber's law? A reexamination of two case studies

Both humans and nonhuman animals can exhibit sensitivity to the approximate number of items in a visual array or events in a sequence, and across various paradigms, uncertainty in numerosity judgments increases with the number estimated or produced. The pattern of increase is usually described as exhibiting approximate adherence to Weber’s law, such that uncertainty increases proportionally to the mean estimate, resulting in a constant coefficient of variation. Such a pattern has been proposed to be a signature characteristic of an innate “number sense.” We reexamine published behavioral data from two studies that have been cited as prototypical evidence of adherence to Weber’s law and observe that in both cases variability increases less than this account would predict, as indicated by a decreasing coefficient of variation with an increase in number. We also consider evidence from numerosity discrimination studies that show deviations from the constant coefficient of variation pattern. Though behavioral data can sometimes exhibit approximate adherence to Weber’s law, our findings suggest that such adherence is not a fixed characteristic of the mechanisms whereby humans and animals estimate numerosity. We suggest instead that the observed pattern of increase in variability with number depends on the circumstances of the task and stimuli, and reflects an adaptive ensemble of mechanisms composed to optimize performance under these circumstances.

Representation of visual numerosity information during working memory in humans: An fMRI decoding study

Both animal and human studies on numerosity have shown the importance of the parietal cortex for numerosity processing. However, most studies have focused on the perceptual processing of numerosity. Still, it is unclear how and where numerosity information is coded when this information is retained during a working memory delay phase. Such temporal storage could be realized by the same structures as perceptual processes, or be transformed to a more abstract representation, potentially involving prefrontal regions. FMRI decoding studies allow the identification of brain areas that exhibit multi‐voxel activation patterns specific to the content of working memory. Here, we used an assumption‐free searchlight‐decoding approach to test where numerosity‐specific codes can be found during a 12 s retention period. Participants (n = 24) performed a retro‐cue delayed match‐to‐sample task, in which numerosity information was presented as visual dot arrays. We found mnemonic numerosity‐specific activation in the right lateral portion of the intraparietal sulcus; an area well‐known for perceptual processing of numerosity. The applied retro‐cue design dissociated working memory delay activity from perceptual processes and showed that the intraparietal sulcus also maintained working memory representation independent of perception.

Approximate multiplication in young children prior to multiplication instruction

Prior work indicates that children have an untrained ability to approximately calculate using their approximate number system (ANS). For example, children can mentally double or halve a large array of discrete objects. Here, we asked whether children can perform a true multiplication operation, flexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruction. We presented 5- to 8-year-olds with nonsymbolic multiplicands (dot arrays) or symbolic multiplicands (Arabic numerals) ranging from 2 to 12 and with nonsymbolic multipliers ranging from 2 to 8. Children compared each imagined product with a visible comparison quantity. Children performed with above-chance accuracy on both nonsymbolic and symbolic approximate multiplication, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any single-digit symbolic multiplication equations (e.g., 2 × 3) on a basic math test were nevertheless successful on both our approximate multiplication tasks, indicating that children have an intuitive sense of multiplication that emerges independent of formal instruction about symbolic multiplication. Nonsymbolic multiplication performance mediated the relation between children's Weber fraction and symbolic math abilities, suggesting a pathway by which the ANS contributes to children's emerging symbolic math competence. These findings may inform future educational interventions that allow children to use their basic arithmetic intuition as a scaffold to facilitate symbolic math learning.