Numeracy, gist, literal thinking and the value of nothing in decision making

The onus on the average person is greater than ever before to make sense of large amounts of readily accessible quantitative information, but the ability and confidence to do so are frequently lacking. Many people lack practical mathematical skills that are essential for evaluating risks, probabilities and numerical outcomes such as survival rates for medical treatments, income from retirement savings plans or monetary damages in civil trials. In this Review, we integrate research on objective and subjective numeracy, focusing on cognitive and metacognitive factors that distort human perceptions and foment systematic biases in judgement and decision making. Paradoxically, an important implication of this research is that a literal focus on objective numbers and mechanical number crunching is misguided. Numbers can be a matter of life and death but a person who uses rote strategies (verbatim representations) cannot take advantage of the information contained in the numbers because 'rote' strategies are, by definition, processing without meaning. Verbatim representations (verbatim is only surface form, not meaning) treat numbers as data as opposed to information. We highlight a contrasting approach of gist extraction: organizing numbers meaningfully, interpreting them qualitatively and making meaningful inferences about them. Efforts to improve numerical cognition and its practical applications can benefit from emphasizing the qualitative meaning of numbers in context - the gist - building on the strengths of humans as intuitive mathematicians. Thus, we conclude by reviewing evidence that gist training facilitates transfer to new contexts and, because it is more durable, longer-lasting improvements in decision making.

Spontaneous supra-modal encoding of number in the infant brain

The core knowledge hypothesis postulates that infants automatically analyze their environment along abstract dimensions, including numbers. According to this view, approximate numbers should be encoded quickly, pre-attentively, and in a supra-modal manner by the infant brain. Here, we directly tested this idea by submitting the neural responses of sleeping 3-month-old infants, measured with high-density electroencephalography (EEG), to decoders designed to disentangle numerical and non-numerical information. The results show the emergence, in approximately 400 ms, of a decodable number representation, independent of physical parameters, that separates auditory sequences of 4 vs. 12 tones and generalizes to visual arrays of 4 vs. 12 objects. Thus, the infant brain contains a number code that transcends sensory modality, sequential or simultaneous presentation, and arousal state.

Information Theory Opens New Dimensions in Experimental Studies of Animal Behaviour and Communication

Over the last 40–50 years, ethology has become increasingly quantitative and computational. However, when analysing animal behavioural sequences, researchers often need help finding an adequate model to assess certain characteristics of these sequences while using a relatively small number of parameters. In this review, I demonstrate that the information theory approaches based on Shannon entropy and Kolmogorov complexity can furnish effective tools to analyse and compare animal natural behaviours. In addition to a comparative analysis of stereotypic behavioural sequences, information theory can provide ideas for particular experiments on sophisticated animal communications. In particular, it has made it possible to discover the existence of a developed symbolic “language” in leader-scouting ant species based on the ability of these ants to transfer abstract information about remote events.

Continual task learning in natural and artificial agents

How do humans and other animals learn new tasks? A wave of brain recording studies has investigated how neural representations change during task learning, with a focus on how tasks can be acquired and coded in ways that minimise mutual interference. We review recent work that has explored the geometry and dimensionality of neural task representations in neocortex, and computational models that have exploited these findings to understand how the brain may partition knowledge between tasks. We discuss how ideas from machine learning, including those that combine supervised and unsupervised learning, are helping neuroscientists understand how natural tasks are learned and coded in biological brains.

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.

A language compatibility effect in fraction processing

A language compatibility effect occurs when there is a match between what a language provides and what a mathematical task demands. Here, we investigated whether such an effect exists for fraction processing in English, which names the numerator first, versus Korean, which names the denominator first. We developed two new tasks: a fraction span task where participants view and then recall four fractions and a fraction identification task where they view one fraction and then another and judge whether the two fractions are the same or not. We generally found that English speakers were advantaged when the numerator drove task performance and Korean speakers were advantaged when the denominator was critical. These findings, particularly from the fraction identification task, were inconsistent with the attentional focus hypothesis, which proposes that the serialisation bias of a language guides which fraction component is attended to first. Rather, they were better explained by the verbal encoding hypothesis, which states that a necessary condition for observing language compatibility effects may be that the fraction components must be encoded in verbal working memory and rehearsed there.

Graded human sensitivity to geometric and topological concepts

In a seminal study, Dehaene et al. (2006) found evidence that adults and children are sensitive to geometric and topological (GT) concepts using a novel odd-one-out task. However, performance on this task could reflect more general cognitive abilities than intuitive knowledge of GT concepts. Here, we developed a new 2-alternative forced choice (2-AFC) version of the original task where chance represents a higher bar to clear (50% vs. 16.67%) and where the role of general cognitive abilities is minimized. Replicating the original finding, American adult participants showed above-chance sensitivity to 41 of the 43 GT concepts tested. Moreover, their performance was not strongly driven by two general cognitive abilities, fluid intelligence and mental rotation, nor was it strongly associated with mathematical achievement as measured by ACT/SAT scores. The performance profile across the 43 concepts as measured by the new 2-AFC task was found to be highly correlated with the profiles as measured using the original odd-one-out task, as an analysis of data sets spanning populations and ages revealed. Most significantly, an aggregation of the 43 concepts into seven classes of GT concepts found evidence for graded sensitivity. Some classes, such as Euclidean geometry and Topology, were found to be more domain-specific: they "popped out" for participants and were judged very quickly and highly accurately. Others, notably Symmetry and Geometric transformations, were found to be more domain-general: better predicted by participants' general cognitive abilities and mathematical achievement. These results shed light on the graded nature of GT concepts in humans and challenge computational models that emphasize the role of induction.

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.

No intrinsic number bias: Evaluating the role of perceptual discriminability in magnitude categorization

Accumulating evidence suggests that there is a spontaneous preference for numerical, compared to non-numerical (e.g., cumulative surface area), information. However, given a paucity of research on the perception of non-numerical magnitudes, it is unclear whether this preference reflects a specific bias towards number, or a general bias towards the more perceptually discriminable dimension (i.e., number). Here, we found that when the number and area of visual dot displays were matched in mathematical ratio, number was more perceptually discriminable than area in both adults and children. Moreover, both adults and children preferentially categorized these ratio-matched stimuli based on number, consistent with previous work. However, when number and area were matched in perceptual discriminability, a different pattern of results emerged. In particular, children preferentially categorized stimuli based on area, suggesting that children's previously observed number bias may be due to a mismatch in the perceptual discriminability of number and area, not an intrinsic salience of number. Interestingly, adults continued to categorize the displays on the basis of number. Altogether, these findings suggest a dominant role for area during childhood, refuting the claim that number is inherently and uniquely salient. Yet they also reveal an increased salience of number that emerges over development. Potential explanations for this developmental shift are discussed. RESEARCH HIGHLIGHTS: Previous work found that children and adults spontaneously categorized dot array stimuli by number, over other magnitudes (e.g., area), suggesting number is uniquely salient. However, here we found that when number and area were matched by ratio, as in prior work, number was significantly more perceptually discriminable than area. When number and area were made equally discriminable ('perceptually-matched'), children, contra adults, spontaneously categorized stimuli by area over number (and other non-numerical magnitudes). These findings suggest that area may be uniquely salient early in childhood, with the previously-observed number bias not emerging until later in development.

Omit needless words: Sentence length perception

Short sentences improve readability. Short sentences also promote social justice through accessibility and inclusiveness. Despite this, much remains unknown about sentence length perception-an important factor in producing readable writing. Accordingly, we conducted a psychophysical study using procedures from Signal Detection Theory to examine sentence length perception in naive adults. Participants viewed real-world full-page text samples and judged whether a bolded target sentence contained more or fewer than 17 words. The experiment yielded four findings. First, naïve adults perceived sentence length in real-world text samples quickly (median = 300-400 ms) and precisely (median = ~90% correct). Second, flipping real-world text samples upside-down generated no reaction-time cost and nearly no loss in the precision of sentence length perception. This differs from the large inversion effects that characterize other highly practiced, real-world perceptual tasks involving canonically oriented stimuli, most notably face perception and reading. Third, participants significantly underestimated the length of mirror-reversed sentences-but not upside-down, nor standard sentences. This finding parallels participants' familiarity with commonly occurring left-justified right-ragged text, and suggests a novel demonstration of left-lateralized anchoring in scene syntax. Fourth, error patterns demonstrated that participants achieved their high speed, high precision sentence-length judgments by heuristically counting text lines, not by explicitly counting words. This suggests practical advice for writing instructors to offer students. When copy editing, students can quickly and precisely identify their long sentences via a line-counting heuristic, e.g., "a 17-word sentence spans about 1.5 text lines". Students can subsequently improve a long sentence's readability and inclusiveness by omitting needless words.

The plural counts: Inconsistent grammatical number hinders numerical development in preschoolers - A cross-linguistic study

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiting differences between Polish, which has a complex grammatical number paradigm, leading to a partially inconsistent mapping between numerical quantities and grammatical number, and German, which has a comparatively easy verbal paradigm: 151 Polish-speaking and 123 German-speaking kindergarten children were tested using a symbolic numerical comparison task. Additionally, counting skills (Give-a-Number and count-list), and mapping between non-symbolic (dot sets) and symbolic representations of numbers, as well as working memory (Corsi blocks and Digit span) were assessed. Based on the Give-a-Number and mapping tasks, the children were divided into subset-knowers, CP-knowers-non-mappers, and CP-knowers-mappers. Linguistic background was related to performance in several ways: Polish-speaking children expectedly progressed to the CP-knowers stage later than German children, despite comparable non-numerical capabilities, and even after this stage was achieved, they fared worse in the numerical comparison task. There were also meaningful differences in spatial-numerical mapping between the Polish and German groups. Our findings are in line with the theory that grammatical number paradigms influence. the development of representations and processing of numbers, not only at the stage of acquiring the meaning of the first number-words but at later stages as well, when dealing with symbolic numbers.

Different impacts of long-term abacus training on symbolic and non-symbolic numerical magnitude processing in children

Abacus-based mental calculation (AMC) has been shown to be effective in promoting math ability in children. Given that AMC relies on a visuospatial strategy to perform rapid and precise arithmetic, previous studies mostly focused on the promotion of AMC training on arithmetic ability and mathematical visual-spatial ability, as well as its transfer of advanced cognitive ability. However, little attention has been given to its impact on basic numerical comparison ability. Here, we aim to examine whether and how long-term AMC training impacts symbolic and non-symbolic numerical comparisons. The distance effect (DE) was utilized as a marker, indicating that the comparison between two numbers becomes faster as their numerical distance enlarges. In the current study, forty-one children matched for age and sex were recruited at primary school entry and randomly assigned to the AMC group and the control group. After three years of training, the event-related potential (ERP) recording technique was used to explore the temporal dynamics of number comparison, of which tasks were given in symbolic (Arabic number) or non-symbolic (dot array) format. In the symbolic task, the children in the AMC group showed a smaller DE than those in the control group. Two ERP components, N1 and P2p, located in parietal areas (PO7, PO8) were selected as neural markers of numerical processing. Both groups showed DE in the P2p component in both tasks, but only the children in the AMC group showed DE in the N1 component in the non-symbolic task. In addition, the DE size calculated from reaction times and ERP amplitudes was correlated with higher cognitive capacities, such as coding ability. Taken together, the present results provide evidence that long-term AMC training may be beneficial for numerical processing in children, which may be associated with neurocognitive indices of parietal brain regions.

SymCog: An open-source toolkit for assessing human symbolic cognition

Symbol systems have a profound influence on human behavior, spanning countless modalities such as natural language, clothing styles, monetary systems, and gestural conventions (e.g., handshaking). Selective impairments in understanding and manipulating symbols are collectively known as asymbolia. Here we address open questions about the nature of asymbolia in the context of both historical and contemporary approaches to human symbolic cognition. We describe a tripartite perspective on symbolic cognition premised upon (1) mental representation of a concept, (2) a stored pool of symbols segregated from their respective referents, and (3) fast and accurate mapping between concepts and symbols. We present an open-source toolkit for assessing symbolic knowledge premised upon matching animated video depictions of abstract concepts to their corresponding verbal and nonverbal symbols. Animations include simple geometric shapes (e.g., filled circles, squares) moving in semantically meaningful ways. For example, a rectangle bending under the implied weight of a large square denotes "heaviness." We report normative data for matching words and images to these target animations. In a second norming study, participants rated target animations across a range of semantic dimensions (e.g., valence, dominance). In a third study, we normed a set of concepts familiar to American English speakers but lacking verbal labels (e.g., the feeling of a Sunday evening). We describe how these tools may be used to assess human symbolic processing and identify asymbolic deficits across the span of human development.

The association between experience-based risky choice and mathematical ability

Mathematical ability has always been considered an important influencing factor in description-based risky choices. Experience-based risky choices, which occur frequently in daily life, are very different from description-based risky choices. The association between experience-based risky choice and mathematical ability remains unknown. This study adopts the feedback paradigm for experience-based risky choice to explore the association between multiple mathematical abilities and experience-based risky choice. The results show that, in experience-based risky choice, mathematical ability did not influence the decision to pursue higher expected value, but it did influence preference for risky. Thus, our study contributes to a more comprehensive view of mathematical ability and risky choice.

Biologically-Based Computation: How Neural Details and Dynamics Are Suited for Implementing a Variety of Algorithms

The Neural Engineering Framework (Eliasmith & Anderson, 2003) is a long-standing method for implementing high-level algorithms constrained by low-level neurobiological details. In recent years, this method has been expanded to incorporate more biological details and applied to new tasks. This paper brings together these ongoing research strands, presenting them in a common framework. We expand on the NEF's core principles of (a) specifying the desired tuning curves of neurons in different parts of the model, (b) defining the computational relationships between the values represented by the neurons in different parts of the model, and (c) finding the synaptic connection weights that will cause those computations and tuning curves. In particular, we show how to extend this to include complex spatiotemporal tuning curves, and then apply this approach to produce functional computational models of grid cells, time cells, path integration, sparse representations, probabilistic representations, and symbolic representations in the brain.

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.

Testing geometry and 3D perception in children following vision restoring cataract-removal surgery

As neuroscience and rehabilitative techniques advance, age-old questions concerning the visual experience of those who gain sight after blindness, once thought to be philosophical alone, take center stage and become the target for scientific inquiries. In this study, we employ a battery of visual perception tasks to study the unique experience of a small group of children who have undergone vision-restoring cataract removal surgery as part of the Himalayan Cataract Project. We tested their abilities to perceive in three dimensions (3D) using a binocular rivalry task and the Brock string task, perceive visual illusions, use cross-modal mappings between touch and vision, and spatially group based on geometric cues. Some of the children in this study gained a sense of sight for the first time in their lives, having been born with bilateral congenital cataracts, while others suffered late-onset blindness in one eye alone. This study simultaneously supports yet raises further questions concerning Hubel and Wiesel's critical periods theory and provides additional insight into Molyneux's problem, the ability to correlate vision with touch quickly. We suggest that our findings present a relatively unexplored intermediate stage of 3D vision development. Importantly, we spotlight some essential geometrical perception visual abilities that strengthen the idea that spontaneous geometry intuitions arise independently from visual experience (and education), thus replicating and extending previous studies. We incorporate a new model, not previously explored, of testing children with congenital cataract removal surgeries who perform the task via vision. In contrast, previous work has explored these abilities in the congenitally blind via touch. Taken together, our findings provide insight into the development of what is commonly known as the visual system in the visually deprived and highlight the need to further empirically explore an amodal, task-based interpretation of specializations in the development and structure of the brain. Moreover, we propose a novel objective method, based on a simple binocular rivalry task and the Brock string task, for determining congenital (early) vs. late blindness where medical history and records are partial or lacking (e.g., as is often the case in cataract removal cases).

Cortical representations of numbers and nonsymbolic quantities expand and segregate in children from 5 to 8 years of age

Number symbols, such as Arabic numerals, are cultural inventions that have transformed human mathematical skills. Although their acquisition is at the core of early elementary education in children, it remains unknown how the neural representations of numerals emerge during that period. It is also unclear whether these relate to an ontogenetically earlier sense of approximate quantity. Here, we used multivariate fMRI adaptation coupled with within- and between-format machine learning to probe the cortical representations of Arabic numerals and approximate nonsymbolic quantity in 89 children either at the beginning (age 5) or four years into formal education (age 8). Although the cortical representations of both numerals and nonsymbolic quantities expanded from age 5 to age 8, these representations also segregated with learning and development. Specifically, a format-independent neural representation of quantity was found in the right parietal cortex, but only for 5-year-olds. These results are consistent with the so-called symbolic estrangement hypothesis, which argues that the relation between symbolic and nonsymbolic quantity weakens with exposure to formal mathematics in children.

Is my visualization better than yours? Analyzing factors modulating exponential growth bias in graphs

Humans tend to systematically underestimate exponential growth and perceive it in linear terms, which can have severe consequences in a variety of fields. Recent studies attempted to examine the origins of this bias and to mitigate it by using the logarithmic vs. the linear scale in graphical representations. However, they yielded conflicting results as to which scale induces more perceptual errors. In the current study, in an experiment with a short educational intervention, we further examine the factors modulating the exponential bias in graphs and suggest a theoretical explanation for our findings. Specifically, we test the hypothesis that each of the scales can induce misperceptions in a particular context. In addition to this, we explore the effect of mathematical education by testing two groups of participants (with a background in humanities vs. formal sciences). The results of this study confirm that when used in an inadequate context, these scales can have a dramatic effect on the interpretation of visualizations representing exponential growth. In particular, while the log scale leads to more errors in graph description tasks, the linear scale misleads people when they have to make predictions on the future trajectory of exponential growth. The second part of the study revealed that the difficulties with both scales can be reduced by means of a short educational intervention. Importantly, while no difference between participants groups was observed prior to the intervention, participants with a better mathematical education showed a stronger learning effect at posttest. The findings of this study are discussed in light of a dual-process model.

Modeling the left digit effect in adult number line estimation

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

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.

The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6484 Luxembourgish ninth-graders

The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.

Does auditory numerosity and non-numerical magnitude affect visual non-symbolic numerical representation?

In the present study, we investigated the effects of auditory numerosity and magnitude (loudness) on visual numerosity processing. Participants compared numerosities of two sequential dot arrays. The second dot array was paired with a tone array that was independent of visual comparison. The numerosity (One-tone vs. Multiple-tone) and the non-numerical magnitude of tones (loudness) were manipulated in Experiments 1 and 2, respectively. In Experiment 1, participants' inverse efficiency score (IES), that is, the quotient between response time and accuracy, was significantly smaller in the One-tone and Multiple-tone conditions than that in the No-tone condition, and linear trend analyses showed that the IES decreased with the number of tones. In Experiment 2, the IES in the Loud-tone condition was significantly smaller than that in the No-tone condition, and the IES decreased as the loudness of the tones increased. In Experiment 3, both auditory numerosity and magnitude were manipulated. For soft tones, the IES was smaller in the Multiple-tone condition than in the One-tone condition, whereas no significant difference was found between two conditions in loud tones. In sum, these findings suggest that the visual numerical representation can be spontaneously affected by the numerosity and non-numerical magnitude of stimuli from another modality.

Aprendizaje del conteo y los números naturales en preescolar: una revisión sistemática de la literatura

Aprender a contar cantidades discretas de forma exacta constituye uno de los primeros hitos del desarrollo del conocimiento matemático infantil. En los últimos años, ha habido un extenso debate en torno a cómo ocurre este proceso de aprendizaje en preescolar. La actual investigación tuvo como objetivo conocer las temáticas y preguntas de investigación generales desarrolladas en los últimos cinco años en cuanto al aprendizaje del conteo y los números naturales en preescolar. Para ello, se realizó una revisión sistemática en la que se hizo una indagación en las bases de datos ScienceDirect, EBSCO, Web of Science, SpringerLink, JSTOR y Sage. Se obtuvieron 98 artículos de investigación que fueron examinados mediante análisis de conglomerados y mapas jerárquicos a través de NVIVO 11.0.  Se encontraron cuatro núcleos temáticos (Ideas sobre los procesos cognitivos implicados en la comprensión del número, Representación de magnitudes numéricas, Intervenciones para favorecer el desarrollo de habilidades matemáticas y Aspectos estructurales del número), que muestran el panorama actual de investigación sobre aprendizaje del conteo. Los resultados de este estudio son importantes para delimitar posibles programas futuros de investigación, y pueden ser usados por docentes como insumo para enriquecer los ambientes de aprendizaje de sus aulas de clase.

The best game in town: The reemergence of the language-of-thought hypothesis across the cognitive sciences

Mental representations remain the central posits of psychology after many decades of scrutiny. However, there is no consensus about the representational format(s) of biological cognition. This paper provides a survey of evidence from computational cognitive psychology, perceptual psychology, developmental psychology, comparative psychology, and social psychology, and concludes that one type of format that routinely crops up is the language-of-thought (LoT). We outline six core properties of LoTs: (i) discrete constituents; (ii) role-filler independence; (iii) predicate-argument structure; (iv) logical operators; (v) inferential promiscuity; and (vi) abstract content. These properties cluster together throughout cognitive science. Bayesian computational modeling, compositional features of object perception, complex infant and animal reasoning, and automatic, intuitive cognition in adults all implicate LoT-like structures. Instead of regarding LoT as a relic of the previous century, researchers in cognitive science and philosophy-of-mind must take seriously the explanatory breadth of LoT-based architectures. We grant that the mind may harbor many formats and architectures, including iconic and associative structures as well as deep-neural-network-like architectures. However, as computational/representational approaches to the mind continue to advance, classical compositional symbolic structures - that is, LoTs - only prove more flexible and well-supported over time.

A language of thought for the mental representation of geometric shapes

In various cultures and at all spatial scales, humans produce a rich complexity of geometric shapes such as lines, circles or spirals. Here, we propose that humans possess a language of thought for geometric shapes that can produce line drawings as recursive combinations of a minimal set of geometric primitives. We present a programming language, similar to Logo, that combines discrete numbers and continuous integration to form higher-level structures based on repetition, concatenation and embedding, and we show that the simplest programs in this language generate the fundamental geometric shapes observed in human cultures. On the perceptual side, we propose that shape perception in humans involves searching for the shortest program that correctly draws the image (program induction). A consequence of this framework is that the mental difficulty of remembering a shape should depend on its minimum description length (MDL) in the proposed language. In two experiments, we show that encoding and processing of geometric shapes is well predicted by MDL. Furthermore, our hypotheses predict additive laws for the psychological complexity of repeated, concatenated or embedded shapes, which we confirm experimentally.

Dissociation of categorical and coordinate spatial relations on dynamic network organization states

Humans can flexibly represent both categorical and coordinate spatial relations. Previous research has mainly focused on hemisphere lateralization in representing these two types of spatial relations, but little is known about how distinct network organization states support representations of the two. Here we used dynamic resting-state functional connectivity (FC) to explore this question. To do this, we separated a meta-identified navigation network into a ventral and two other subnetworks. We revealed a Weak State and a Strong State within the ventral subnetwork and a Negative State and a Positive State between the ventral and other subnetworks. Further, we found the Weak State (i.e., weak but positive FC) within the ventral subnetwork was related to the ability of categorical relation recognition, suggesting that the representation of categorical spatial relations was related to weak integration among focal regions in the navigation network. In contrast, the Negative State (i.e., negative FC) between the ventral and other subnetworks was associated with the ability of coordinate relation processing, suggesting that the representation of coordinate spatial relations may require competitive interactions among widely distributed regions. In sum, our study provides the first empirical evidence revealing different focal and distributed organizations of the navigation network in representing different types of spatial information.

Time spent playing predicts early reading and math skills through associations with self-regulation

Children's play time has declined in recent decades, which could negatively impact early self-regulation-a vital component of school readiness. To date, studies have not fully explored how the time spent playing relates to children's self-regulatory skills, and in turn, their early reading and math competencies. Using data from time diaries and direct assessments of self-regulation, prereading, and math skills, this study examined how minutes spent playing at home predict these skills in a sample of 128 children followed from age four to five. Additionally, it considered whether self-regulation explained links between play time and prereading and math. Results showed that the time spent playing positively related to children's self-regulation. Moreover, through its association with self-regulation, play time had indirect effects on prereading and math skills measured one year later. Results suggest that fostering opportunities for play time during the preschool years may help to boost school readiness skills.

The cultural origins of symbolic number

It is popular in psychology to hypothesize that representations of exact number are innately determined-in particular, that biology has endowed humans with a system for manipulating quantities which forms the primary representational substrate for our numerical and mathematical concepts. While this perspective has been important for advancing empirical work in animal and child cognition, here we examine six natural predictions of strong numerical nativism from a multidisciplinary perspective, and find each to be at odds with evidence from anthropology and developmental science. In particular, the history of number reveals characteristics that are inconsistent with biological determinism of numerical concepts, including a lack of number systems across some human groups and remarkable variability in the form of numerical systems that do emerge. Instead, this literature highlights the importance of economic and social factors in constructing fundamentally new cognitive systems to achieve culturally specific goals. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

An illustrated tutorial for logarithmic scales and decibels in acoustics

Acoustics is generally defined as the science that deals with the production, transmission, and reception of sound and the understanding and control of its effects. In fact, the fields of acoustics cover an especially broad range of subjects and domains, and comprehensive acoustics textbooks are usually quite thick as a consequence. While they are valuable resources for researchers, these books might appear a little daunting for a young audience or for people who are new to acoustics. This paper is an example of how educational comics can be designed and used to introduce one of the most commonly discussed topics when the basics of acoustics are taught: decibel level. Seven drawn pages constitute a visual support to explain the origin and history of the decibel, together with examples from acoustics and other domains on the use of logarithmic scales and classical decibel calculations. Several comments and comprehensive bibliographical references are also provided for each drawn page to enlarge the range of subjects or exercises that can be discussed in courses and foster further readings.

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

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.

The distribution of initial estimates moderates the effect of social influence on the wisdom of the crowd

Whether, and under what conditions, groups exhibit “crowd wisdom” has been a major focus of research across the social and computational sciences. Much of this work has focused on the role of social influence in promoting the wisdom of the crowd versus leading the crowd astray and has resulted in conflicting conclusions about how social network structure determines the impact of social influence. Here, we demonstrate that it is not enough to consider the network structure in isolation. Using theoretical analysis, numerical simulation, and reanalysis of four experimental datasets (totaling 2885 human subjects), we find that the wisdom of crowds critically depends on the interaction between (i) the centralization of the social influence network and (ii) the distribution of the initial individual estimates. By adopting a framework that integrates both the structure of the social influence and the distribution of the initial estimates, we bring previously conflicting results under one theoretical framework and clarify the effects of social influence on the wisdom of crowds.

Reduced choice-confidence in negative numerals

Negative numbers are central in math. However, they are abstract, hard to learn, and manipulated slower than positive numbers regardless of math ability. It suggests that confidence, namely the post-decision estimate of being correct, should be lower than positives. We asked participants to pick the larger single-digit numeral in a pair and collected their implicit confidence with button pressure (button pressure was validated with three empirical signatures of confidence). We also modeled their choices with a drift-diffusion decision model to compute the post-decision estimate of being correct. We found that participants had relatively low confidence with negative numerals. Given that participants compared with high accuracy the basic base-10 symbols (0–9), reduced confidence may be a general feature of manipulating abstract negative numerals as they produce more uncertainty than positive numerals per unit of time.

Associations among socioeconomic status and preschool-aged children's, number skills, and spatial skills: The role of executive function

Extensive literature has documented socioeconomic status (SES) disparities in young children's standardized math achievement, which primarily reflect differences in basic number and arithmetic skills. In addition, growing evidence indicates that direct assessments of executive function (EF) both predict standardized math achievement and mediate SES differences in standardized math tests. However, early spatial skills and children's approximate number system (ANS) acuity, critical components of later math competence, have been largely absent in this past research. The current study examined SES associations with multiple direct assessments of early ANS, cardinality, and spatial skills, as well as standardized math achievement, in a socioeconomically diverse sample of 4-year-old children (N = 149). Structural equation modeling revealed SES effect sizes of .21 for geometric sensitivity skills, .23 for ANS acuity, .39 for cardinality skills, and .28 for standardized math achievement. Furthermore, relations between SES and children's spatial skills, ANS acuity, cardinality, and standardized math skills were mediated by a composite measure of children's EF skills. Implications of pervasive SES disparities across multiple domains of early math development, as well as the mitigating role of EF, are discussed.

EmBody/EmFace as a new open tool to assess emotion recognition from body and face expressions

Nonverbal expressions contribute substantially to social interaction by providing information on another person’s intentions and feelings. While emotion recognition from dynamic facial expressions has been widely studied, dynamic body expressions and the interplay of emotion recognition from facial and body expressions have attracted less attention, as suitable diagnostic tools are scarce. Here, we provide validation data on a new open source paradigm enabling the assessment of emotion recognition from both 3D-animated emotional body expressions (Task 1: EmBody) and emotionally corresponding dynamic faces (Task 2: EmFace). Both tasks use visually standardized items depicting three emotional states (angry, happy, neutral), and can be used alone or together. We here demonstrate successful psychometric matching of the EmBody/EmFace items in a sample of 217 healthy subjects with excellent retest reliability and validity (correlations with the Reading-the-Mind-in-the-Eyes-Test and Autism-Spectrum Quotient, no correlations with intelligence, and given factorial validity). Taken together, the EmBody/EmFace is a novel, effective (< 5 min per task), highly standardized and reliably precise tool to sensitively assess and compare emotion recognition from body and face stimuli. The EmBody/EmFace has a wide range of potential applications in affective, cognitive and social neuroscience, and in clinical research studying face- and body-specific emotion recognition in patient populations suffering from social interaction deficits such as autism, schizophrenia, or social anxiety.

Symbols and mental programs: a hypothesis about human singularity

Natural language is often seen as the single factor that explains the cognitive singularity of the human species. Instead, we propose that humans possess multiple internal languages of thought, akin to computer languages, which encode and compress structures in various domains (mathematics, music, shape…). These languages rely on cortical circuits distinct from classical language areas. Each is characterized by: (i) the discretization of a domain using a small set of symbols, and (ii) their recursive composition into mental programs that encode nested repetitions with variations. In various tasks of elementary shape or sequence perception, minimum description length in the proposed languages captures human behavior and brain activity, whereas non-human primate data are captured by simpler nonsymbolic models. Our research argues in favor of discrete symbolic models of human thought.

Visual foundations of Euclidean geometry

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as "natural". We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry - i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3-34 years), and 25 participants from the Amazon (age 5-67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in "sense"). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans' knowledge in Euclidean geometry could possibly be grounded.

Verbal counting and the timing of number acquisition in an indigenous Amazonian group

Children in industrialized cultures typically succeed on Give-N, a test of counting ability, by age 4. On the other hand, counting appears to be learned much later in the Tsimane’, an indigenous group in the Bolivian Amazon. This study tests three hypotheses for what may cause this difference in timing: (a) Tsimane’ children may be shy in providing behavioral responses to number tasks, (b) Tsimane’ children may not memorize the verbal list of number words early in acquisition, and/or (c) home environments may not support mathematical learning in the same way as in US samples, leading Tsimane’ children to primarily acquire mathematics through formalized schooling. Our results suggest that most of our subjects are not inhibited by shyness in responding to experimental tasks. We also find that Tsimane’ children (N = 100, ages 4-11) learn the verbal list later than US children, but even upon acquiring this list, still take time to pass Give-N tasks. We find that performance in counting varies across tasks and is related to formal schooling. These results highlight the importance of formal education, including instruction in the count list, in learning the meanings of the number words.

The dopamine circuit as a reward-taxis navigation system

Studying the brain circuits that control behavior is challenging, since in addition to their structural complexity there are continuous feedback interactions between actions and sensed inputs from the environment. It is therefore important to identify mathematical principles that can be used to develop testable hypotheses. In this study, we use ideas and concepts from systems biology to study the dopamine system, which controls learning, motivation, and movement. Using data from neuronal recordings in behavioral experiments, we developed a mathematical model for dopamine responses and the effect of dopamine on movement. We show that the dopamine system shares core functional analogies with bacterial chemotaxis. Just as chemotaxis robustly climbs chemical attractant gradients, the dopamine circuit performs ‘reward-taxis’ where the attractant is the expected value of reward. The reward-taxis mechanism provides a simple explanation for scale-invariant dopaminergic responses and for matching in free operant settings, and makes testable quantitative predictions. We propose that reward-taxis is a simple and robust navigation strategy that complements other, more goal-directed navigation mechanisms.

Large numbers cause magnitude neglect: The case of government expenditures

Four studies demonstrate that the public's understanding of government budgetary expenditures is hampered by difficulty in representing large numerical magnitudes. Despite orders of magnitude difference between millions and billions, study participants struggle with the budgetary magnitudes of government programs. When numerical values are rescaled as smaller magnitudes (in the thousands or lower), lay understanding improves, as indicated by greater sensitivity to numerical ratios and more accurate rank ordering of expenses. A robust benefit of numerical rescaling is demonstrated across a variety of experimental designs, including policy relevant choices and incentive-compatible accuracy measures. This improved sensitivity ultimately impacts funding choices and public perception of respective budgets, indicating the importance of numerical cognition for good citizenship.

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.