New measures of number line estimation performance reveal children's ordinal understanding of numbers

Children's performance on the number line estimation task, often measured by the percentage of absolute error, predicts their later mathematics achievement. This task may also reveal (a) children's ordinal understanding of the target numbers in relation to each other and the benchmarks (e.g., endpoints, midpoint) and (b) the ordinal skills that are a necessary precursor to children's ability to understand the interval nature of a number line as measured by percentage of absolute error. Using data from 104 U.S. kindergartners, we measured whether children's estimates were correctly sequenced across trials and correctly positioned relative to given benchmarks within trials at two time points. For both time points, we found that each ordinal error measure revealed a distinct pattern of data distribution, providing opportunities to tap into different aspects of children's ordinal understanding. Furthermore, children who made fewer ordinal errors scored higher on the Test of Early Mathematics Ability and showed greater improvement on their interval understanding of numbers as reflected by a larger reduction of percentage of absolute error from Time 1 to Time 2. The findings suggest that our number line measures reveal individual differences in children's ordinal understanding of numbers, and that such understanding may be a precursor to their interval understanding and later mathematics performance.

Rational number representation by the approximate number system

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known "connectedness illusion" to provide evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers.

The relationship between numerical magnitude processing and math anxiety, and their joint effect on adult math performance, varied by indicators of numerical tasks

According to the hypothesis of Maloney et al. (Cognition 114(2):293-297, 2010. https://doi.org/10.1016/j.cognition.2009.09.013), math anxiety is related to deficits in numerical magnitude processing, which in turn compromises the development of advanced math skills. Because previous studies on this topic are contradictory, which may be due to methodological differences in the measurement of numerical magnitude processing, we tested Maloney et al.'s hypothesis using different tasks and their indicators: numerical magnitude processing (symbolic and non-symbolic comparison tasks: accuracy, reaction time, numerical ratio, distance and size effects, and Weber fraction; number line estimation task: estimation error), math anxiety (combined scores of learning, testing, math problem solving, and general math anxiety), and math performance. The results of our study conducted on 119 young adults mostly support the hypothesis proposed by Maloney et al. that deficiency in symbolic magnitude processing is related to math anxiety, but the relationship between non-symbolic processes and math anxiety was opposite to the assumptions. Moreover, the results indicate that estimation processes (but not comparison processes) and math anxiety are related to math performance in adults. Finally, high math anxiety moderated the relationship between reaction time in the symbolic comparison task, reaction time in the non-symbolic comparison task, numerical ratio effect in the symbolic comparison task, and math performance. Because the results of the joint effect of numerical magnitude processing and math anxiety on math performance were inconsistent, this part of the hypothesis is called into question.

What curves are parallel? The core feature of preschoolers' intuitive parallel category

Existing evidence has revealed that humans can spontaneously categorize many geometric shapes without formal education. Children around 4 years could distinguish between intersecting lines and parallel lines. Three features can be used to identify parallel lines, namely "translational congruence," "never meet," and "constant distance." This study separated them by using pairs of curves that possess only one of these features. Two experiments across 2021-2023, respectively, compared the relative priority of "translational congruence" with "constant distance," and "never meet" with "constant distance" among 3- to 5-year-old Chinese preschoolers (Ntotal = 314, 48% female). The results showed that preschoolers consistently grouped "constant distance" curves with parallel lines. This suggests that the core feature of intuitive parallel category is "constant distance" at this age.

Is core knowledge a natural subdivision of infant cognition?

We examine Spelke's core knowledge taxonomy and test its boundaries. We ask whether Spelke's core knowledge is a distinct type of cognition in the sense that the cognitive processes it includes and excludes are biologically and mechanically coherent.

Divisive language

What language devises, it might divide. By exploring the relations among the core geometries of the physical world, the abstract geometry of Euclid, and language, I give new insight into both the persistence of core knowledge into adulthood and our access to it through language. My extension of Spelke's language argument has implications for pedagogy, philosophy, and artificial intelligence.

Geometry in the brain optimized for sign language - A unique role of the anterior superior parietal lobule in deaf signers

Geometry has been identified as a cognitive domain where deaf individuals exhibit relative strength, yet the neural mechanisms underlying geometry processing in this population remain poorly understood. This fMRI study aimed to investigate the neural correlates of geometry processing in deaf and hearing individuals. Twenty-two adult deaf signers and 25 hearing non-signers completed a geometry decision task. We found no group differences in performance, while there were some differences in parietal activation. As expected, the posterior superior parietal lobule (SPL) was recruited for both groups. The anterior SPL was significantly more activated in the deaf group, and the inferior parietal lobule was significantly more deactivated in the hearing group. In conclusion, despite similar performance across groups, there were differences in the recruitment of parietal regions. These differences may reflect inherent differences in brain organization due to different early sensory and linguistic experiences.

Auditory time perception impairment in children with developmental dyscalculia

Developmental dyscalculia (DD) is a specific learning disability which prevents children from acquiring adequate numerical and arithmetical competences. We investigated whether difficulties in children with DD spread beyond the numerical domain and impact also their ability to perceive time. A group of 37 children/adolescent with and without DD were tested with an auditory categorization task measuring time perception thresholds in the sub-second (0.25-1 s) and supra-second (0.75-3 s) ranges. Results showed that auditory time perception was strongly impaired in children with DD at both time scales. The impairment remained even when age, non-verbal reasoning, and gender were regressed out. Overall, our results show that the difficulties of DD can affect magnitudes other than numerical and contribute to the increasing evidence that frames dyscalculia as a disorder affecting multiple neurocognitive and perceptual systems.

Number sense-arithmetic link in Grade 1 and Grade 2: A case of fluency

BACKGROUND

Recent research suggested fluent processing as an explanation on why number sense contributes to simple arithmetic tasks-'Fluency hypothesis'.

AIMS

The current study investigates whether number sense contributes to such arithmetic tasks when other cognitive factors are controlled for (including those that mediate the link); and whether this contribution varies as a function of participants' individual maths fluency levels.

SAMPLE

Four hundred and thirty-seven Chinese schoolchildren (186 females; Mage = 83.49 months) completed a range of cognitive measures in Grade 1 (no previous classroom training) and in Grade 2 (a year later).

METHODS

Number sense, arithmetic (addition and subtraction), spatial ability, visuo-spatial working memory, perception, reaction time, character reading and general intelligence were measured.

RESULTS

Our data showed that the link between number sense and arithmetic was weaker in Grade 1 (Beta = .15 for addition and .06 (ns) for subtraction) compared to Grade 2 (.23-.28), but still persisted in children with no previous maths training. Further, math's performance in Grade 1 did not affect the link between number sense and maths performance in Grade 2.

CONCLUSION

Our data extended previous findings by showing that number sense is linked with simple maths task performance even after controlling for multiple cognitive factors. Our results brought some evidence that number sense-arithmetic link is somewhat sensitive to previous formal maths education. Further research is needed, as the differences in effects between grades were quite small, and arithmetic in Grade 1 did not moderate the link at question in Grade 2.

Young children intuitively organize numbers on straight, horizontal lines from left to right before the onset of formal instruction

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number-space mapping. Three- to six-year-old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non-symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left-to-right-oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space.

Diversity of spatial activities and parents' spatial talk complexity predict preschoolers' gains in spatial skills

Children's spatial activities and parental spatial talk were measured to examine their associations with variability in preschoolers' spatial skills (N = 113, Mage = 4 years, 4 months; 51% female; 80% White, 11% Black, and 9% other). Parents who reported more diversity in daily spatial activities and used longer spatial talk utterances during a spatial activity had children with greater gains in spatial skills from ages 4 to 5 (β = .17 and β = .40, respectively). Importantly, this study is the first to move beyond frequency counts of spatial input and investigate the links among the diversity of children's daily spatial activities, as well as the complexity of parents' spatial language across different contexts, and preschoolers' gains in spatial skills, an important predictor of later STEM success.

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

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

Caricaturing Shapes in Visual Memory

When representing high-level stimuli, such as faces and animals, we tend to emphasize salient features-such as a face's prominent cheekbones or a bird's pointed beak. Such mental caricaturing leaves traces in memory, which exaggerates these distinctive qualities. How broadly does this phenomenon extend? Here, in six experiments (N = 700 adults), we explored how memory automatically caricatures basic units of visual processing-simple geometric shapes-even without task-related demands to do so. Participants saw a novel shape and then immediately adjusted a copy of that shape to match what they had seen. Surprisingly, participants reconstructed shapes in exaggerated form, amplifying curvature, enlarging salient parts, and so on. Follow-up experiments generalized this bias to new parameters, ruled out strategic responding, and amplified the effects in serial transmission. Thus, even the most basic stimuli we encounter are remembered as caricatures of themselves.

Human-level few-shot concept induction through minimax entropy learning

Humans learn concepts both from labeled supervision and by unsupervised observation of patterns, a process machines are being taught to mimic by training on large annotated datasets-a method quite different from the human pathway, wherein few examples with no supervision suffice to induce an unfamiliar relational concept. We introduce a computational model designed to emulate human inductive reasoning on abstract reasoning tasks, such as those in IQ tests, using a minimax entropy approach. This method combines identifying the most effective constraints on data via minimum entropy with determining the best combination of them via maximum entropy. Our model, which applies this unsupervised technique, induces concepts from just one instance, reaching human-level performance on tasks of Raven's Progressive Matrices (RPM), Machine Number Sense (MNS), and Odd-One-Out (O3). These results demonstrate the potential of minimax entropy learning for enabling machines to learn relational concepts efficiently with minimal input.

What Makes Us Smart?

How did humans become clever enough to live in nearly every major ecosystem on earth, create vaccines against deadly plagues, explore the oceans depths, and routinely traverse the globe at 30,000 feet in aluminum tubes while nibbling on roasted almonds? Drawing on recent developments in our understanding of human evolution, we consider what makes us distinctively smarter than other animals. Contrary to conventional wisdom, human brilliance emerges not from our innate brainpower or raw computational capacities, but from the sharing of information in communities and networks over generations. We review how larger, more diverse, and more optimally interconnected networks of minds give rise to faster innovation and how the cognitive products of this cumulative cultural evolutionary process feedback to make us individually "smarter"-in the sense of being better at meeting the challenges and problems posed by our societies and socioecologies. Here, we consider not only how cultural evolution supplies us with "thinking tools" (like counting systems and fractions) but also how it has shaped our ontologies (e.g., do germs and witches exist?) and epistemologies, including our notions of what constitutes a "good reason" or "good evidence" (e.g., are dreams a source of evidence?). Building on this, we consider how cultural evolution has organized and distributed cultural knowledge and cognitive tasks among subpopulations, effectively shifting both thinking and production to the level of the community, population, or network, resulting in collective information processing and group decisions. Cultural evolution can turn mindless mobs into wise crowds by facilitating and constraining cognition through a wide variety of epistemic institutions-political, legal, and scientific. These institutions process information and aid better decision-making by suppressing or encouraging the use of different cultural epistemologies and ontologies.

Magnitude representation of preschool children with autism spectrum condition

LAY ABSTRACT

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

Recruitment of the premotor cortex during arithmetic operations by the monkey

Arithmetic operations are complex mental processes rooted in the abstract concept of numerosity. Despite the significance, the neural architecture responsible for these operations has remained largely uncharted. In this study, we explored the presence of specific neuronal activity in the dorsal premotor cortex of the monkey dedicated to numerical addition and subtraction. Our findings reveal that many of these neural activities undergo a transformation, shifting their coding from arithmetic to motor representations. These motor representations include information about which hand to use and the number of steps involved in the action. We consistently observed that cells related to the right-hand encoded addition, while those linked to the left-hand encoded subtraction, suggesting that arithmetic operations and motor commands are intertwining with each other. Furthermore, we used a multivariate decoding technique to predict the monkey's behaviour based on the activity of these arithmetic-related cells. The classifier trained to discern arithmetic operations, including addition and subtraction, not only predicted the arithmetic decisions but also the subsequent motor actions of the right and left-hand. These findings imply a cognitive extension of the motor cortex's function, where inherent neural systems are repurposed to facilitate arithmetic operations.

Exploring the Importance of Environmental Complexity for Newly Hatched Zebrafish

The effects of an early impoverished social or physical environment on vertebrate neural development and cognition has been known for decades. While existing studies have focused on the long-term effects, measuring adult cognitive phenotypes, studies on the effects of environmental complexity on the early stages of development are lacking. Zebrafish (Danio rerio) hatchlings are assumed to have minimal interaction with their environment and are routinely reared in small, bare containers. To investigate the effects of being raised under such conditions on development of behaviour and cognition, hatchlings housed for 10 days in either an enriched or a standard environment underwent two cognitive tasks. The results were mixed. Subjects of the two treatments did not differ in performance when required to discriminate two areas. Conversely, we found a significant effect in a number discrimination task, with subjects from impoverished condition performing significantly worse. In both experiments, larvae reared in impoverished environment showed a reduced locomotor activity. Given the effects that enrichment appears to exert on larvae, a third experiment explored whether hatchlings exhibit a spontaneous preference for more complex environments. When offered a choice between a bare setting and one with objects of different shapes and colors, larvae spent over 70% of time in the enriched sector. Deepening these effects of an early impoverished environment on cognitive development is crucial for the welfare of captive zebrafish populations and for enhancing the quality and reliability of studies on larval zebrafish.

The left digit effect in an unbounded number line task

The left digit effect in number line estimation refers to the phenomenon where numerals with similar magnitudes but different leftmost digits (e.g., 19 and 22) are estimated to be farther apart on a number line than is warranted. The effect has been studied using a bounded number line task, a task in which a line is bounded by two endpoints (e.g., 0 and 100), and where one must indicate the correct location of a target numeral on the line. The goal of the present work is to investigate the left digit effect in an unbounded number line task, a task that involves using the size of one unit to determine a target numeral's location, and that elicits strategies different from those used in the bounded number line task. In a preregistered study, participants (N = 58 college students) completed four blocks of 38 trials each of an unbounded number line task, with target numerals ranging between 0 and 100. We found a medium and statistically reliable left digit effect (d = 0.70). The study offers further evidence that the effect is not driven by response strategies specific to the bounded number line task. We discuss other possible sources of the effect including conversion of symbols to magnitudes in these and other contexts.

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.

Examining Profiles of U.S. Children's Screen Time and Associations with Academic Skills

Children's screen time (ST) increased in recent years, but investigations of the content and context (e.g., parental presence, and device type) of ST in predicting early academic skills remains understudied. In this study of 127 four- and five-year-olds, we examined whether the educational content and contextual information about children's ST from time diary interviews at age 4 predicted direct assessments of academic skills at age five. Cluster analyses detected three unique groups: Cluster One was "lowest ST and highest non-educational TV, with little parental presence," Cluster Two was "moderate ST and highest educational mobile devices, with moderate parental presence," and Cluster Three was "highest total ST and moderate educational ST and high parental presence." Children in Cluster Two scored significantly higher in literacy skills at age five than peers in Cluster One and Cluster Three. Children in Cluster One scored significantly higher in spatial skills compared to those in Cluster Three. No significant associations were detected between children's ST and number skills. Findings suggest that educational content and contextual features of ST combine in meaningful ways and appear to better predict children's literacy and spatial skills compared to variable-centered approaches. Methodological implications of this research and future directions are discussed.

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

Introduction

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

Methods

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

Results

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

Conclusion

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

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.

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

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

Perceptual addition of continuous magnitudes in an 'artificial algebra'

Although there is substantial evidence for an innate 'number sense' that scaffolds learning about mathematics, whether the underlying representations are based on discrete or continuous perceptual magnitudes has been controversial. Yet the nature of the computations supported by these representations has been neglected in this debate. While basic computation of discrete non-symbolic quantities has been reliably demonstrated in adults, infants, and non-humans, far less consideration has been given to the capacity for computation of continuous perceptual magnitudes. Here we used a novel experimental task to ask if humans can learn to add non-symbolic, continuous magnitudes in accord with the properties of an algebraic group, by feedback and without explicit instruction. Three pairs of experiments tested perceptual addition under the group properties of commutativity (Experiments 1a-b), identity and inverses (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback. Responding on transfer trials was accurate and provided strong support for addition under all of the group axioms with line length, and for all except associativity with brightness. Our results confirm that adult human subjects can implicitly add continuous quantities in a manner consistent with symbolic addition over the integers, and that an 'artificial algebra' task can be used to study implicit computation.

Neurobiology of numerical learning

Numerical abilities are complex cognitive skills essential for dealing with requirements of the modern world. Although the brain structures and functions underlying numerical cognition in different species have long been appreciated, genetic and molecular techniques have more recently expanded the knowledge about the mechanisms underlying numerical learning. In this review, we discuss the status of the research related to the neurobiological bases of numerical abilities. We consider how genetic factors have been associated with mathematical capacities and how these link to the current knowledge of brain regions underlying these capacities in human and non-human animals. We further discuss the extent to which significant variations in the levels of specific neurotransmitters may be used as potential markers of individual performance and learning difficulties and take into consideration the therapeutic potential of brain stimulation methods to modulate learning and improve interventional outcomes. The implications of this research for formulating a more comprehensive view of the neural basis of mathematical learning are discussed.

Compositional diversity in visual concept learning

Humans leverage compositionality to efficiently learn new concepts, understanding how familiar parts can combine together to form novel objects. In contrast, popular computer vision models struggle to make the same types of inferences, requiring more data and generalizing less flexibly than people do. Here, we study these distinctively human abilities across a range of different types of visual composition, examining how people classify and generate "alien figures" with rich relational structure. We also develop a Bayesian program induction model which searches for the best programs for generating the candidate visual figures, utilizing a large program space containing different compositional mechanisms and abstractions. In few shot classification tasks, we find that people and the program induction model can make a range of meaningful compositional generalizations, with the model providing a strong account of the experimental data as well as interpretable parameters that reveal human assumptions about the factors invariant to category membership (here, to rotation and changing part attachment). In few shot generation tasks, both people and the models are able to construct compelling novel examples, with people behaving in additional structured ways beyond the model capabilities, e.g. making choices that complete a set or reconfigure existing parts in new ways. To capture these additional behavioral patterns, we develop an alternative model based on neuro-symbolic program induction: this model also composes new concepts from existing parts yet, distinctively, it utilizes neural network modules to capture residual statistical structure. Together, our behavioral and computational findings show how people and models can produce a variety of compositional behavior when classifying and generating visual objects.

Profiles of mathematical deficits in children with dyslexia

Despite a high rate of concurrent mathematical difficulties among children with dyslexia, we still have limited information regarding the prevalence and severity of mathematical deficits in this population. To address this gap, we developed a comprehensive battery of cognitive tests, known as the UCSF Mathematical Cognition Battery (MCB), with the aim of identifying deficits in four distinct mathematical domains: number processing, arithmetical procedures, arithmetic facts retrieval, and geometrical abilities. The mathematical abilities of a cohort of 75 children referred to the UCSF Dyslexia Center with a diagnosis of dyslexia, along with 18 typically developing controls aged 7 to 16, were initially evaluated using a behavioral neurology approach. A team of professional clinicians classified the 75 children with dyslexia into five groups, based on parents' and teachers' reported symptoms and clinical history. These groups included children with no mathematical deficits and children with mathematical deficits in number processing, arithmetical procedures, arithmetic facts retrieval, or geometrical abilities. Subsequently, the children underwent evaluation using the MCB to determine concordance with the clinicians' impressions. Additionally, neuropsychological and cognitive standardized tests were administered. Our study reveals that within a cohort of children with dyslexia, 66% exhibit mathematical deficits, and among those with mathematical deficits, there is heterogeneity in the nature of these deficits. If these findings are confirmed in larger samples, they can potentially pave the way for new diagnostic approaches, consistent subtype classification, and, ultimately personalized interventions.

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

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

Observers Efficiently Extract the Minimal and Maximal Element in Perceptual Magnitude Sets: Evidence for a Bipartite Format

The mind represents abstract magnitude information, including time, space, and number, but in what format is this information stored? We show support for the bipartite format of perceptual magnitudes, in which the measured value on a dimension is scaled to the dynamic range of the input, leading to a privileged status for values at the lowest and highest end of the range. In six experiments with college undergraduates, we show that observers are faster and more accurate to find the endpoints (i.e., the minimum and maximum) than any of the inner values, even as the number of items increases beyond visual short-term memory limits. Our results show that length, size, and number are represented in a dynamic format that allows for comparison-free sorting, with endpoints represented with an immediately accessible status, consistent with the bipartite model of perceptual magnitudes. We discuss the implications for theories of visual search and ensemble perception.

"Now I Get It!": Eureka Experiences During the Acquisition of Mathematical Concepts

Many famous scientists have reported anecdotes where a new understanding occurred to them suddenly, in an unexpected flash. Do people generally experience such "Eureka" moments when learning science concepts? And if so, do these episodes truly vehicle sudden insights, or is this impression illusory? To address these questions, we developed a paradigm where participants were taught the mathematical concept of geodesic, which generalizes the common notion of straight line to straight trajectories drawn on curved surfaces. After studying lessons introducing this concept on the sphere, participants (N = 56) were tested on their understanding of geodesics on the sphere and on other surfaces. Our findings indicate that Eureka experiences are common when learning mathematics, with reports by 34 (61%) participants. Moreover, Eureka experiences proved an accurate description of participants' learning, in two respects. First, Eureka experiences were associated with learning and generalization: the participants who reported experiencing Eurekas performed better at identifying counterintuitive geodesics on new surfaces. Second, and in line with the firstperson experience of a sudden insight, our findings suggest that the learning mechanisms responsible for Eureka experiences are inaccessible to reflective introspection. Specifically, reports of Eureka experiences and of participants' confidence in their own understanding were associated with different profiles of performance, indicating that the mechanisms bringing about Eureka experiences and those informing reflective confidence were at least partially dissociated. Learning mathematical concepts thus appears to involve mechanisms that operate unconsciously, except when a key computational step is reached and a sudden insight breaks into consciousness.

How learning influences non-symbolic numerical processing: effects of feedback in the dot comparison task

It has long been debated how humans estimate the numerosity of sets of elements and what role continuous visual properties play in this process. The dot comparison task, in which the more numerous of two dot arrays must be selected, is a dominant method to investigate this phenomenon. It has been shown that the visual properties of the two dot patterns strongly influence the comparison. This influence can be systematically investigated by manipulating visual properties congruently and incongruently with numerosity. However, it remains unclear how learning and prior experience affect the influence of the visual properties. To address this question, we introduced feedback into the classical dot comparison task: during the learning phase, participants in the experimental group received feedback after each trial indicating whether their answer was correct whereas participants in the control group did not. After the learning phase, neither group received feedback. The convex hull of the dot patterns and the average dot diameter were manipulated congruently and incongruently with numerosity. Our results show that feedback had no effect on overall performance. However, when manipulated separately, dot diameter no longer affected performance in the experimental group after the learning phase, but it did in the control group. Moreover, this effect remained visible even when diameter and convex hull were manipulated simultaneously. This pattern of results is consistent with the notion of sensory integration which proposes that weights are assigned to different visual cues and that numerical judgments depend on an additive combination of these weights. We also found a correlation between performance on an arithmetic task and performance on trials in which dot size was manipulated incongruently with numerosity. However, there were no correlations between an inhibition task and performance in the dot comparison task. Taken together, the current results suggest that learning with feedback may affect some visual properties but not others. Future studies should further investigate a wider range of visual properties to examine which of them can be influenced by learning and under what conditions learning occurs.

A single tDCS session can enhance numerical competence

While numerical skills are increasingly important in modern life, few interventions have been developed to support those with numeracy skills difficulties. Previous studies have demonstrated that applying transcranial Direct Current Stimulation (tDCS) can improve numerical skills. However, tDCS interventions designed to induce lasting changes typically involve reapplying brain-stimulation over several days. Repeated tDCS application can increase the risks associated with the procedure, as well as restricts the transferability of the method to a wider population, particularly those who may experience mobility issues, such as stroke survivors with acalculia. The current study investigated whether a single session of tDCS (anodal to right parietal lobe and cathodal to left parietal lobe), followed by four self-practice sessions without tDCS, could result in enhancement of numerical skills. Nineteen healthy adults (n = 10 tDCS, n = 9 sham control) implicitly learnt the magnitude association of nine arbitrary symbols, previously used by Cohen Kadosh et al. (2010). Numerical proficiency was assessed using number-to-space task, while automaticity was assessed with numerical Stroop. Results revealed that single-session tDCS had a significant effect on participants' accuracy on the number-to-space tasks, but not on the numerical Stroop task's congruity effect, implying automaticity may require longer practice. We conclude that a single session of tDCS should be considered as an avenue for interventions.

Object geometry serves humans' intuitive physics of stability

How do humans judge physical stability? A prevalent account emphasizes the mental simulation of physical events implemented by an intuitive physics engine in the mind. Here we test the extent to which the perceptual features of object geometry are sufficient for supporting judgments of falling direction. In all experiments, adults and children judged the falling direction of a tilted object and, across experiments, objects differed in the geometric features (i.e., geometric centroid, object height, base size and/or aspect ratio) relevant to the judgment. Participants' performance was compared to computational models trained on geometric features, as well as a deep convolutional neural network (ResNet-50), none of which incorporated mental simulation. Adult and child participants' performance was well fit by models of object geometry, particularly the geometric centroid. ResNet-50 also provided a good account of human performance. Altogether, our findings suggest that object geometry may be sufficient for judging the falling direction of tilted objects, independent of mental simulation.

Human brain representations of internally generated outcomes of approximate calculation revealed by ultra-high-field brain imaging

Much of human culture's advanced technology owes its existence to the ability to mentally manipulate quantities. Neuroscience has described the brain regions overall recruited by numerical tasks and the neuronal codes representing individual quantities during perceptual tasks. Nevertheless, it remains unknown how quantity representations are combined or transformed during mental computations and how specific quantities are coded in the brain when generated as the result of internal computations rather than evoked by a stimulus. Here, we imaged the brains of adult human subjects at 7 Tesla during an approximate calculation task designed to disentangle in- and outputs of the computation from the operation itself. While physically presented sample numerosities were distinguished in activity patterns along the dorsal visual pathway and within frontal and occipito-temporal regions, a representation of the internally generated result was most prominently detected in higher order regions such as angular gyrus and lateral prefrontal cortex. Behavioral precision in the task was related to cross-decoding performance between sample and result representations in medial IPS regions. This suggests the transformation of sample into result may be carried out within dorsal stream sensory-motor integration regions, and resulting outputs maintained for task purposes in higher-level regions in a format possibly detached from sensory-evoked inputs.

Understanding others' preferences: A comparison across primate species and human societies

We investigated children's and non-human great apes' ability to anticipate others' choices from their evident food preferences-regardless of whether these preferences deviate or align with one's own. We assessed children from three culturally-diverse societies (Namibia, Germany, and Samoa; N = 71; age range = 5-11) and four non-human great ape species (chimpanzees (Pan troglodytes), bonobos (Pan paniscus), gorillas (Gorilla gorilla), and orangutans (Pongo abelii); N = 25; age range = 7-29) regarding their choices in a dyadic food-retrieval task. Across conditions, participants' preferences were either aligned (same preference condition) or opposed (opposite preference condition) to those of their competitors. Children across societies altered their choices based on their competitor's preferences, indicating a cross-culturally recurrent capacity to anticipate others' choices relying on preferences-based inferences. In contrast to human children, all non-human great apes chose according to their own preferences but independent of those of their competitors. In sum, these results suggest that the tendency to anticipate others' choices based on their food preferences is cross-culturally robust and, among the great apes, most likely specific to humans.

Neurocognitive Assessment of Mathematics-Related Capacities in Neurosurgical Patients

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

Creating ad hoc graphical representations of number

The ability to communicate about exact number is critical to many modern human practices spanning science, industry, and politics. Although some early numeral systems used 1-to-1 correspondence (e.g., 'IIII' to represent 4), most systems provide compact representations via more arbitrary conventions (e.g., '7' and 'VII'). When people are unable to rely on conventional numerals, however, what strategies do they initially use to communicate number? Across three experiments, participants used pictures to communicate about visual arrays of objects containing 1-16 items, either by producing freehand drawings or combining sets of visual tokens. We analyzed how the pictures they produced varied as a function of communicative need (Experiment 1), spatial regularities in the arrays (Experiment 2), and visual properties of tokens (Experiment 3). In Experiment 1, we found that participants often expressed number in the form of 1-to-1 representations, but sometimes also exploited the configuration of sets. In Experiment 2, this strategy of using configural cues was exaggerated when sets were especially large, and when the cues were predictably correlated with number. Finally, in Experiment 3, participants readily adopted salient numerical features of objects (e.g., four-leaf clover) and generally combined them in a cumulative-additive manner. Taken together, these findings corroborate historical evidence that humans exploit correlates of number in the external environment - such as shape, configural cues, or 1-to-1 correspondence - as the basis for innovating more abstract number representations.

Combining and segmenting geometric shapes into parts depending on symmetry type: Evidence from children and adults

Symmetry is an important geometric feature that affects object segmentation into parts, though De Winter and Wagemans note that partly occluded objects can still be identified by the remaining visible parts. In two sets of experiments with children (n = 31, age 7-11, M = 8.8, SD = 1.4) and adults (n = 19, age 17-57, M = 30.4, SD = 12.6), we used 13 basic geometric figures distinguished by symmetry types to test how they are naturally segmented or combined and what the developmental impacts are on the segmentation and combination. In the first experiment, participants were asked to cut figures into two along a straight line; in the second experiment, participants had to create five sets of connected two-figure combinations where overlapping figures were allowed. The results confirmed the importance of the symmetry axis in both tasks. Other relevant criteria were dividing into half, maximal/minimal curvature, and use of edges or corners for reference. This study allows comparisons of the impact of symmetry type on the segmentation and combining of geometric figures and indicates developmental differences between children and adults.

The Future of Decisions From Experience: Connecting Real-World Decision Problems to Cognitive Processes

In many important real-world decision domains, such as finance, the environment, and health, behavior is strongly influenced by experience. Renewed interest in studying this influence led to important advancements in the understanding of these decisions from experience (DfE) in the last 20 years. Building on this literature, we suggest ways the standard experimental design should be extended to better approach important real-world DfE. These extensions include, for example, introducing more complex choice situations, delaying feedback, and including social interactions. When acting upon experiences in these richer and more complicated environments, extensive cognitive processes go into making a decision. Therefore, we argue for integrating cognitive processes more explicitly into experimental research in DfE. These cognitive processes include attention to and perception of numeric and nonnumeric experiences, the influence of episodic and semantic memory, and the mental models involved in learning processes. Understanding these basic cognitive processes can advance the modeling, understanding and prediction of DfE in the laboratory and in the real world. We highlight the potential of experimental research in DfE for theory integration across the behavioral, decision, and cognitive sciences. Furthermore, this research could lead to new methodology that better informs decision-making and policy interventions.

Does instructional intervention reduce the left digit effect in number line estimation?

A robust left digit effect arises in number line estimation, whereby the leftmost digits of numerals have an undue influence on placements such that, for example, numbers like 298 are placed far to the left of numbers like 302. Past efforts to motivate more accurate performance using trial-by-trial and summary feedback have not led to a reduction in the left digit effect. In two experiments, we asked whether it is possible to reduce or eliminate the left digit effect in number line estimation through an instructional intervention in which one is explicitly taught about the left digit effect. In Experiment 1 (N = 134), participants completed two blocks (60 trials per block) of a self-paced 0-1,000 number line estimation task and were randomly assigned to either an instruction or a control condition. In Experiment 2 (N = 143), the procedure was enhanced with a learning check, and with additional measures to assess changes in behaviour as a result of instruction. In both experiments, a left digit effect was found in each block of each condition. Although there was evidence that instruction changed behaviour, these changes did not result in any reduction in the left digit effect relative to the control condition. These findings demonstrate that the left digit effect cannot be easily reduced by making people aware of it.

Efficient, continual, and generalized learning in the brain - neural mechanism of Mental Schema 2.0

There has been tremendous progress in artificial neural networks (ANNs) over the past decade; however, the gap between ANNs and the biological brain as a learning device remains large. With the goal of closing this gap, this paper reviews learning mechanisms in the brain by focusing on three important issues in ANN research: efficiency, continuity, and generalization. We first discuss the method by which the brain utilizes a variety of self-organizing mechanisms to maximize learning efficiency, with a focus on the role of spontaneous activity of the brain in shaping synaptic connections to facilitate spatiotemporal learning and numerical processing. Then, we examined the neuronal mechanisms that enable lifelong continual learning, with a focus on memory replay during sleep and its implementation in brain-inspired ANNs. Finally, we explored the method by which the brain generalizes learned knowledge in new situations, particularly from the mathematical generalization perspective of topology. Besides a systematic comparison in learning mechanisms between the brain and ANNs, we propose "Mental Schema 2.0," a new computational property underlying the brain's unique learning ability that can be implemented in ANNs.

Geometric experience sculpts the development and dynamics of hippocampal sequential cell assemblies

Euclidean space is the fabric of the world we live in. Whether and how geometric experience shapes our spatial-temporal representations of the world remained unknown. We deprived rats of experience with crucial features of Euclidean geometry by rearing them inside translucent spheres, and compared activity of large hippocampal neuronal ensembles during navigation and sleep with that of cuboid cage-reared controls. Sphere-rearing from birth permitted emergence of accurate neuronal ensemble spatial codes and preconfigured and plastic time-compressed neuronal sequences. However, sphere-rearing led to diminished individual place cell tuning, similar neuronal mapping of different track ends/corners, and impaired neuronal pattern separation and plasticity of multiple linear track experiences, partly driven by reduced preconfigured network repertoires. Subsequent experience with multiple linear environments over four days largely reversed these effects, substantiating the role of geometric experience on hippocampal neural development. Thus, early-life experience with Euclidean geometry enriches the hippocampal repertoire of preconfigured neuronal patterns selected toward unique representation and discrimination of multiple linear environments.

tDCS effects in basic symbolic number magnitude processing are not significantly lateralized

Functional lateralization was previously established for various cognitive domains-but not for number processing. Although numbers are considered to be bilaterally represented in the intraparietal sulcus (IPS), there are some indications of different functional roles of the left vs. right IPS in processing number pairs with small vs. large distance, respectively. This raises the question whether number size plays a distinct role in the lateralization within the IPS. In our preregistered study, we applied anodal transcranial direct current stimulation (tDCS) over the left vs. right IPS to investigate the effect of stimulation as compared to sham on small vs. large distance, in both single-digit and two-digit number comparison. We expected that anodal tDCS over the left IPS facilitates number comparison with small distance, while anodal tDCS over the right IPS facilitates number comparison with large distance. Results indicated no effect of stimulation; however, exploratory analyses revealed that tDCS over the right IPS slowed down single-digit number processing after controlling for the training effect. In conclusion, number magnitude processing might be bilaterally represented in the IPS, however, our exploratory analyses emphasise the need for further investigation on functional lateralization of number processing.

Competing numerical magnitude codes in decimal comparison: Whole number and rational number distance both impact performance

A critical difference between decimal and whole numbers is that among whole numbers the number of digits provides reliable information about the size of the number, e.g., double-digit numbers are larger than single-digit numbers. However, for decimals, fewer digits can sometimes denote a larger number (i.e., 0.8 > 0.27). Accordingly, children and adults perform worse when comparing such Inconsistent decimal pairs relative to Consistent pairs, where the larger number also has more digits (i.e., 0.87 > 0.2). Two explanations have been posited for this effect. The string length congruity account proposes that participants compare each position in the place value system, and they additionally compare the number of digits. The semantic interference account suggests that participants additionally activate the whole number referents of numbers - the numbers unadorned with decimal points (e.g., 8 < 27) - and compare these. The semantic interference account uniquely predicts that for Inconsistent problems with the same actual rational distance, those with larger whole number distances should be harder, e.g., 0.9 vs. 0.81 should be harder than 0.3 vs. 0.21 because 9 < < 81 whereas 3 < 21. Here we test this prediction in two experiments with college students (Study 1: n = 58 participants, Study 2: n = 78). Across both, we find a main effect of consistency, demonstrating string length effects, and also that whole number distance interferes with processing conflicting decimals, demonstrating semantic interference effects. Evidence for both effects supports the semantic interference account, highlighting that decimal comparison difficulties arise from multiple competing numerical codes. Finally, for accuracy we found no relationship between whole number distance sensitivity and math achievement, indicating that whole number magnitude interference affects participants similarly across the spectrum of math achievement.

Numerical Representation for Action in Crows Obeys the Weber-Fechner Law

The psychophysical laws governing the judgment of perceived numbers of objects or events, called the number sense, have been studied in detail. However, the behavioral principles of equally important numerical representations for action are largely unexplored in both humans and animals. We trained two male carrion crows (Corvus corone) to judge numerical values of instruction stimuli from one to five and to flexibly perform a matching number of pecks. Our quantitative analysis of the crows' number production performance shows the same behavioral regularities that have previously been demonstrated for the judgment of sensory numerosity, such as the numerical distance effect, the numerical magnitude effect, and the logarithmical compression of the number line. The presence of these psychophysical phenomena in crows producing number of pecks suggests a unified sensorimotor number representation system underlying the judgment of the number of external stimuli and internally generated actions.

Effects of a 3-factor field intervention on numerical and geometric knowledge in preschool children

The main aim of this study was to develop and test the effects of a field math intervention program on both number and geometry knowledge. The intervention was developed based on three basic skills previously associated with mathematical performance: symbolic number knowledge, mapping processes and spatial reasoning. The participants were 117 preschoolers from six schools in Cali and Bogotá. The children were assigned to an intervention group (N = 55) or a control group (N = 62). The intervention lasted 11 weeks with 3 sessions per week where the children participated in different game-based activities. Tests of numerical and geometric knowledge were administered before and after the intervention. The effects of the intervention were tested twice, immediately after the program ended and six months later. The results show that the children in the intervention group improved more than the control group in both number and geometry. The second posttest revealed a significant intervention effect for geometry, but not for numerical knowledge. The implications of these mixed patterns of results are discussed in the paper.

The algorithmic origins of counting

The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics.

Number sense at 12 months predicts 4-year-olds' maths skills

Preverbal infants spontaneously represent the number of objects in collections. Is this 'sense of number' (also referred to as Approximate Number System, ANS) part of the cognitive foundations of mathematical skills? Multiple studies reported a correlation between the ANS and mathematical achievement in children. However, some have suggested that such correlation might be mediated by general-purpose inhibitory skills. We addressed the question using a longitudinal approach: we tested the ANS of 60 12 months old infants and, when they were 4 years old (final N = 40), their symbolic math achievement as well as general intelligence and inhibitory skills. Results showed that the ANS at 12 months is a specific predictor of later maths skills independent from general intelligence or inhibitory skills. The correlation between ANS and maths persists when both abilities are measured at four years. These results confirm that the ANS has an early, specific and longstanding relation with mathematical abilities in childhood. RESEARCH HIGHLIGHTS: In the literature there is a lively debate about the correlation between the ANS and maths skills. We longitudinally tested a sample of 60 preverbal infants at 12 months and rested them at 4 years (final sample of 40 infants). The ANS tested at 12 months predicted later symbolic mathematical skills at 4 years, even when controlling for inhibition, general intelligence and perceptual skills. The ANS tested at 4 years remained linked with symbolic maths skills, confirming this early and longstanding relation in childhood.