Core knowledge of geometry in an Amazonian indigene group

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.

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.

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.

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.

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.

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.

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.

On the visual analytic intelligence of neural networks

Visual oddity task was conceived to study universal ethnic-independent analytic intelligence of humans from a perspective of comprehension of spatial concepts. Advancements in artificial intelligence led to important breakthroughs, yet excelling at such abstract tasks remains challenging. Current approaches typically resort to non-biologically-plausible architectures with ever-growing models consuming substantially more energy than the brain. Motivated by the brain's efficiency and reasoning capabilities, we present a biologically inspired system that receives inputs from synthetic eye movements - reminiscent of saccades, and processes them with neuronal units incorporating dynamics of neocortical neurons. We introduce a procedurally generated visual oddity dataset to train an architecture extending conventional relational networks and our proposed system. We demonstrate that both approaches are capable of abstract problem-solving at high accuracy, and we uncover that both share the same essential underlying mechanism of reasoning in seemingly unrelated aspects of their architectures. Finally, we show that the biologically inspired network achieves superior accuracy, learns faster and requires fewer parameters than the conventional network.

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

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

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.

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).

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.

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.

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.

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.

What does gender has to do with math? Complex questions require complex answers

Whether mathematics is a gendered domain or not is a long-lasting issue bringing along major social and educational implications. The females' underrepresentation in science, technology, engineering, and mathematics (STEM) has been considered one of the key signs of the math gender gap, although the current view largely attributes the origin of this phenomenon to sociocultural factors. Indeed, recent approaches to math gender differences reached the universal conclusion that nature and nurture exert reciprocal effects on each other, establishing the need for approaching the study of the math gender issue only once its intrinsic complexity has been accepted. Building upon a flourishing literature, this review provides an updated synthesis of the evidence for math gender equality at the start, and for math gender inequality on the go, challenging the role of biological factors. In particular, by combining recent findings from different research areas, the paper discusses the persistence of the "math male myth" and the associated "female are not good at math myth," drawing attention to the complex interplay of social and cultural forces that support such stereotypes. The suggestion is made that longevity of these myths results from the additive effects of two independent cognitive biases associated with gender stereotypes and with math stereotypes, respectively. Scholars' responsibility in amplifying these myths by pursuing some catching lines of research is also discussed.

Analyzing the misperception of exponential growth in graphs

Exponential growth is frequently underestimated, an error that can have a heavy social cost in the context of epidemics. To clarify its origins, we measured the human capacity (N = 521) to extrapolate linear and exponential trends in scatterplots. Four factors were manipulated: the function underlying the data (linear or exponential), the response modality (pointing or venturing a number), the scale on the y axis (linear or logarithmic), and the amount of noise in the data. While linear extrapolation was precise and largely unbiased, we observed a consistent underestimation of noisy exponential growth, present for both pointing and numerical responses. A biased ideal-observer model could explain these data as an occasional misperception of noisy exponential graphs as quadratic curves. Importantly, this underestimation bias was mitigated by participants' math knowledge, by using a logarithmic scale, and by presenting a noiseless exponential curve rather than a noisy data plot, thus suggesting concrete avenues for interventions.

Evaluating landscape knowledge and lithic resource selection at the French Middle Paleolithic site of the Bau de l'Aubesier

We report on the application of a novel approach to exploring the degree of landscape knowledge, wayfinding abilities, and the nature of decision-making processes reflected in the utilization of stone resources in the French Middle Paleolithic. Specifically, we use data from the site of the Bau de l'Aubesier to explore the reasons why a majority of the 350 raw material sources cataloged in the surrounding region appear not to have been utilized, including several located near the site and yielding high-quality lithic materials. To this end, we focus on the spatial relationships between sources as an explanatory variable, operationalized in terms of minimum travel times. Using geographic information system software and a generalized linear model of resource selection derived from the Bau assemblages, we compute source utilization probabilities from the perspective of hominins located off-site. We do so under three optimization scenarios, factoring in the intrinsic characteristics (e.g., quality) and time required to reach each source on the way to the Bau. More generally, we find that in slightly more than 50% of cases, seemingly viable sources may have been ignored simply because the minimum cost path leading back to the Bau passes through or requires only minimal deviations to reach, higher quality options. More generally, we found that throughout the entire region, a cost/benefit analysis of competing sources favors those from source areas known to have been utilized. Virtually all the available information on lithic procurement at the Bau is consistent with a model of landscape utilization premised on detailed knowledge of a very large area, an ability to accurately estimate travel times between locations, and a pragmatic strategy of stone resource exploitation based on minimizing costs (travel and search times) and maximizing utility.

Geometry intuitions without vision? A study in blind children and adults

Geometry intuitions seem to be rooted in a non-verbal system that humans possess since early age. However, the mechanisms underlying the comprehension of basic geometric concepts remain elusive. Some authors have suggested that the starting point of geometry development could be found in the visual perception of specific features in our environment, thus conferring to vision a foundational role in the acquisition of geometric skills. To examine this assumption, a test probing intuitive understanding of basic geometric concepts was presented to congenitally blind children and adults. Participants had to detect the intruder among four different shapes, from which three instantiated a specific geometrical concept and one (the intruder) violated it. Although they performed above the chance level, the blind presented poorer performance than the sighted participants who did the task in the visual modality (i.e., with the eyes open), but performed equally well than the sighted who did the task in the tactile modality (i.e., with a blindfold). We therefore provide evidence that geometric abilities are impacted by the lack of vision.

What's in a question? Parents' question use in dyadic interactions and the relation to preschool-aged children's math abilities

The cognitive complexity of adults' questions, particularly during shared book reading, supports children's developing language skills. Questions can be described as having low cognitive demand (CD; e.g., labeling, matching) or high-CD (e.g., comparing, predicting). Little is known about the relation between different types of parental questioning and children's math abilities. The current study examined the quantity of low- and high-CD and domain-specific math questions that parents posed to their 4-year-old children in three structured activities and how the frequency of those questions relates to children's concurrent math and language skills. Parent-child dyads (n = 121) were observed interacting with a picture book, grocery store toys, and a puzzle for about 5 min each, and children completed math and spatial assessments. Although the frequency with which parents asked questions did not relate to children's outcomes, parents' use of high-CD questions was associated with children's spatial skills, standardized math scores, and vocabulary skills after controlling for parental utterances, child utterances, child age, and family socioeconomic status. However, domain-specific math questions were not related to any child outcomes above and beyond parents' total questions. This study suggests that domain-general questions that vary in CD (low and high) are differentially related to children's math and language abilities, which can inform the ways in which parents engage in early learning opportunities with their children.

Perception of geometric sequences and numerosity both predict formal geometric competence in primary school children

While most animals have a sense of number, only humans have developed symbolic systems to describe and organize mathematical knowledge. Some studies suggest that human arithmetical knowledge may be rooted in an ancient mechanism dedicated to perceiving numerosity, but it is not known if formal geometry also relies on basic, non-symbolic mechanisms. Here we show that primary-school children who spontaneously detect and predict geometrical sequences (non-symbolic geometry) perform better in school-based geometry tests indexing formal geometric knowledge. Interestingly, numerosity discrimination thresholds also predicted and explained a specific portion of variance of formal geometrical scores. The relation between these two non-symbolic systems and formal geometry was not explained by age or verbal reasoning skills. Overall, the results are in line with the hypothesis that some human-specific, symbolic systems are rooted in non-symbolic mechanisms.

Mental compression of spatial sequences in human working memory using numerical and geometrical primitives

How does the human brain store sequences of spatial locations? We propose that each sequence is internally compressed using an abstract, language-like code that captures its numerical and geometrical regularities. We exposed participants to spatial sequences of fixed length but variable regularity while their brain activity was recorded using magneto-encephalography. Using multivariate decoders, each successive location could be decoded from brain signals, and upcoming locations were anticipated prior to their actual onset. Crucially, sequences with lower complexity, defined as the minimal description length provided by the formal language, led to lower error rates and to increased anticipations. Furthermore, neural codes specific to the numerical and geometrical primitives of the postulated language could be detected, both in isolation and within the sequences. These results suggest that the human brain detects sequence regularities at multiple nested levels and uses them to compress long sequences in working memory.

Core knowledge of geometry can develop independently of visual experience

Geometrical intuitions spontaneously drive visuo-spatial reasoning in human adults, children and animals. Is their emergence intrinsically linked to visual experience, or does it reflect a core property of cognition shared across sensory modalities? To address this question, we tested the sensitivity of blind-from-birth adults to geometrical-invariants using a haptic deviant-figure detection task. Blind participants spontaneously used many geometric concepts such as parallelism, right angles and geometrical shapes to detect intruders in haptic displays, but experienced difficulties with symmetry and complex spatial transformations. Across items, their performance was highly correlated with that of sighted adults performing the same task in touch (blindfolded) and in vision, as well as with the performances of uneducated preschoolers and Amazonian adults. Our results support the existence of an amodal core-system of geometry that arises independently of visual experience. However, performance at selecting geometric intruders was generally higher in the visual compared to the haptic modality, suggesting that sensory-specific spatial experience may play a role in refining the properties of this core-system of geometry.

Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity

Among primates, humans are special in their ability to create and manipulate highly elaborate structures of language, mathematics, and music. Here we show that this sensitivity to abstract structure is already present in a much simpler domain: the visual perception of regular geometric shapes such as squares, rectangles, and parallelograms. We asked human subjects to detect an intruder shape among six quadrilaterals. Although the intruder was always defined by an identical amount of displacement of a single vertex, the results revealed a geometric regularity effect: detection was considerably easier when either the base shape or the intruder was a regular figure comprising right angles, parallelism, or symmetry rather than a more irregular shape. This effect was replicated in several tasks and in all human populations tested, including uneducated Himba adults and French kindergartners. Baboons, however, showed no such geometric regularity effect, even after extensive training. Baboon behavior was captured by convolutional neural networks (CNNs), but neither CNNs nor a variational autoencoder captured the human geometric regularity effect. However, a symbolic model, based on exact properties of Euclidean geometry, closely fitted human behavior. Our results indicate that the human propensity for symbolic abstraction permeates even elementary shape perception. They suggest a putative signature of human singularity and provide a challenge for nonsymbolic models of human shape perception.

Counterexample Search in Diagram-Based Geometric Reasoning

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object-wise, could underlie diagram-based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.

A three-channel fluorescent probe to image mitochondrial stress

Dual-recognition probes based on one reacting site inevitably encounter competition problems. Here, NPClA, a two-photon fluorescent probe based on a dual-site response for SO2/HOCl, was developed and applied in imaging mitochondrial stress.

Developmental differences in children's and adults' use of geometric information in map-reading tasks

Using maps effectively requires the ability to scale distances while preserving angle and orientation, the three properties of Euclidean geometry. The aim of the current study was twofold: first, to examine how the ability to represent and use these Euclidean properties changes with development when scaling maps in object-to-object relationships and, second, to explore the effects on the scaling performance of two variables of the array of objects, type of angular configuration and relative vector length. To this end, we tested seventy-five 4-, 6-, and 8-year-old children, as well as twenty-five adults, in a simple completion task with different linear and triangular configurations of objects. This study revealed important developmental changes between 4 and 6 years of age and between 8 years of age and adulthood for both distance and angle representation, while it also showed that the configuration variables affected younger and older children's performances in different ways when scaling distances and preserving angles and orientation. This study was instrumental in showing that, from an early age, children are able to exploit an intrinsic system of reference to scale geometrical configurations of objects.

Multi-Item Discriminability Pattern to Faces in Developmental Prosopagnosia Reveals Distinct Mechanisms of Face Processing

Previous studies have shown that individuals with developmental prosopagnosia (DP) show specific deficits in face processing. However, the mechanism underlying the deficits remains largely unknown. One hypothesis suggests that DP shares the same mechanism as normal population, though their faces processing is disproportionally impaired. An alternative hypothesis emphasizes a qualitatively different mechanism of DP processing faces. To test these hypotheses, we instructed DP and normal individuals to perceive faces and objects. Instead of calculating accuracy averaging across stimulus items, we used the discrimination accuracy for each item to construct a multi-item discriminability pattern. We found DP's discriminability pattern was less similar to that of normal individuals when perceiving faces than perceiving objects, suggesting that DP has qualitatively different mechanism in representing faces. A functional magnetic resonance imaging study was conducted to reveal the neural basis and found that multi-voxel activation patterns for faces in the right fusiform face area and occipital face area of DP were deviated away from the mean activation pattern of normal individuals. Further, the face representation was more heterogeneous in DP, suggesting that deficits of DP may come from multiple sources. In short, our study provides the first direct evidence that DP processes faces qualitatively different from normal population.

Training nonsymbolic proportional reasoning in children and its effects on their symbolic math abilities

Our understanding of proportions can be both symbolic, as when doing calculations in school mathematics, or intuitive, as when folding a bed sheet in half. While an understanding of symbolic proportions is crucial for school mathematics, the cognitive foundations of this ability remain unclear. Here we implemented a computerized training game to test a causal link from intuitive (nonsymbolic) to symbolic proportional reasoning and other math abilities in 4th grade children. An experimental group was trained in nonsymbolic proportional reasoning (PR) with continuous extents, and an active control group was trained on a remarkably similar nonsymbolic magnitude comparison. We found that the experimental group improved at nonsymbolic PR across training sessions, showed near transfer to a paper-and-pencil nonsymbolic PR test, transfer to symbolic proportions, and far transfer to geometry. The active control group showed only a predicted far transfer to geometry. In a second experiment, these results were replicated with an independent cohort of children. Overall this study extends previous correlational evidence, suggesting a functional link between nonsymbolic PR on one hand and symbolic PR and geometry on the other.

Interference in geometry among people who are blind

BACKGROUND

Geometry, a central branch of mathematics, is challenging for schoolchildren. Studies have shown that, when comparing perimeters of geometrical shapes, many sighted participants experience interference from the area variable, possibly stemming from the visual differences between the geometrical shapes. Accordingly, we hypothesized that such interference would not be observed in participants who are blind, who use the tactile modality to detect the properties of shapes.

METHODS

Thirty participants, 15 who are blind and 15 with sight, explored pairs of geometrical shapes tactilely or visually, respectively, and compared areas and perimeters.

RESULTS AND CONCLUSIONS

Surprisingly, accuracy and response time findings suggested that the two groups had a similar pattern of performance, and hence that area also interferes in comparison of perimeters among people who are blind.

Places in the Brain: Bridging Layout and Object Geometry in Scene-Selective Cortex

Diverse animal species primarily rely on sense (left-right) and egocentric distance (proximal-distal) when navigating the environment. Recent neuroimaging studies with human adults show that this information is represented in 2 scene-selective cortical regions-the occipital place area (OPA) and retrosplenial complex (RSC)-but not in a third scene-selective region-the parahippocampal place area (PPA). What geometric properties, then, does the PPA represent, and what is its role in scene processing? Here we hypothesize that the PPA represents relative length and angle, the geometric properties classically associated with object recognition, but only in the context of large extended surfaces that compose the layout of a scene. Using functional magnetic resonance imaging adaptation, we found that the PPA is indeed sensitive to relative length and angle changes in pictures of scenes, but not pictures of objects that reliably elicited responses to the same geometric changes in object-selective cortical regions. Moreover, we found that the OPA is also sensitive to such changes, while the RSC is tolerant to such changes. Thus, the geometric information typically associated with object recognition is also used during some aspects of scene processing. These findings provide evidence that scene-selective cortex differentially represents the geometric properties guiding navigation versus scene categorization.

Reductive stress imaging in the endoplasmic reticulum by using living cells and zebrafish

Excessive accumulation of reducing agents in the ER leads to a constitutively high UPR. And the co-function of GSH, Cys and HOCl in biological processes is not well understood. To address this, a TP probe, NPCC, was developed for monitoring reductive stress in the ER. It can also distinguish cancer cells from normal cells.

Discrimination of Small Forms in a Deviant-Detection Paradigm by 10-month-old Infants

Using eye tracking, we investigated if 10-month-old infants could discriminate between members of a set of small forms based on geometric properties in a deviant-detection paradigm, as suggested by the idea of a core cognitive system for Euclidian geometry. We also investigated the precision of infants' ability to discriminate as well as how the discrimination process unfolds over time. Our results show that infants can discriminate between small forms based on geometrical properties, but only when the difference is sufficiently large. Furthermore, our results also show that it takes infants, on average, <3.5 s to detect a deviant form. Our findings extend previous research in three ways: by showing that infants can make similar discriminative judgments as children and adults with respect to geometric properties; by providing a first crude estimate on the limit of the discriminative abilities in infants, and finally; by providing a first demonstration of how the discrimination process unfolds over time.

The development of symmetry concept in preschool children

Young children are exposed to symmetrical figures frequently before they are taught the concept of symmetry, which is a valuable experience for the development of geometry; however, limited research has explored how this concept develops. This study investigated the developmental sequence of "general symmetry" concept and "specific symmetry" concepts (i.e., bilateral, rotational, and translational symmetry) with 106 4-6-year-old children using a symmetry deviant detection task. The test examined children's conception of general symmetry against asymmetry, specific symmetry against asymmetry, and discrimination of specific symmetries. The results suggested that the concept of symmetry develops as a differentiation process. The concept of general symmetry was acquired first, followed by specific symmetries which were acquired in sequential order.

Violations of Core Knowledge Shape Early Learning

Research on cognitive development has revealed that even the youngest minds detect and respond to events that adults find surprising. These surprise responses suggest that infants have a basic set of "core" expectations about the world that are shared with adults and other species. However, little work has asked what purpose these surprise responses serve. Here we discuss recent evidence that violations of core knowledge offer special opportunities for learning. Infants and young children make predictions about the world on the basis of their core knowledge of objects, quantities, and social entities. We argue that when these predictions fail to match the observed data, infants and children experience an enhanced drive to seek and retain new information. This impact of surprise on learning is not equipotent. Instead, it is directed to entities that are relevant to the surprise itself; this drive propels children-even infants-to form and test new hypotheses about surprising aspects of the world. We briefly consider similarities and differences between these recent findings with infants and children, on the one hand, and findings on prediction errors in humans and non-human animals, on the other. These comparisons raise open questions that require continued inquiry, but suggest that considering phenomena across species, ages, kinds of surprise, and types of learning will ultimately help to clarify how surprise shapes thought.

Language, gesture, and judgment: Children's paths to abstract geometry

As infants, children are sensitive to geometry when recognizing objects or navigating through rooms; however, explicit knowledge of geometry develops slowly and may be unstable even in adults. How can geometric concepts be both so accessible and so elusive? To examine how implicit and explicit geometric concepts develop, the current study assessed, in 132 children (3-8 years old) while they played a simple geometric judgment task, three distinctive channels: children's choices during the game as well as the language and gestures they used to justify and accompany their choices. Results showed that, for certain geometric properties, children chose the correct card even if they could not express with words (or gestures) why they had made this choice. Furthermore, other geometric concepts were expressed and supported by gestures prior to their articulation in either choices or speech. These findings reveal that gestures and behavioral choices may reflect implicit knowledge and serve as a foundation for the development of geometric reasoning. Altogether, our results suggest that language alone might not be enough for expressing and organizing geometric concepts and that children pursue multiple paths to overcome its limitations, a finding with potential implications for primary education in mathematics.

The statistical shape of geometric reasoning

Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect, idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric reasoning? Here, we address this question using a simple geometric task – planar triangle completion. An analysis of the distribution of participants’ errors in localizing a fragmented triangle’s missing corner reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles. By considering the statistical physics of the process characterized via a correlated random walk with a natural length scale, we explain these results and further predict participants’ estimates of the missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning task about changes in the triangle size and shape even when such completion strategies need not be invoked. Taken together, our findings suggest a critical role for noisy physical processes in our reasoning about elementary Euclidean geometry.

At 4.5 but not 5.5 years, children favor kin when the stakes are moderately high

Adults report more willingness to help siblings over close friends when the stakes are extremely high, such as when deciding whether to donate a kidney or risk injury to rescue someone in peril. When dividing plentiful, low-value resources, in contrast, children expect people to share equally with friends and siblings. Even when distributing limited resources-one instead of many-and distributing to their own social partners rather than fictional characters, children share more with kin and friends than with strangers but do not favor kin over friends until 5.5 years of age. However, no study has tested whether children would preferentially benefit kin if the rewards require that children incur a higher personal cost of their own time and effort. In the present experiment, therefore, we asked if children would work harder for kin over non-kin when playing a challenging geometry game that allowed them to earn rewards for others. We found that 4.5-year-old children calibrated their time and effort in the game differently according to who received the rewards-they played for more trials and answered more trials correctly for kin over non-kin, but 5.5-year-old children did not. The older children may have found the task easier and less costly or may have different social experiences affecting their efforts to benefit others. Nonetheless, 4.5-year-old children's social decisions favored kin as recipients of their generosity.

Language is more abstract than you think, or, why aren't languages more iconic?

How abstract is language? We show that abstractness pervades every corner of language, going far beyond the usual examples of freedom and justice In the light of the ubiquity of abstract words, the need to understand where abstract meanings come from becomes ever more acute. We argue that the best source of knowledge about abstract meanings may be language itself. We then consider a seemingly unrelated question: Why isn't language more iconic? Iconicity-a resemblance between the form of words and their meanings-can be immensely useful in language learning and communication. Languages could be much more iconic than they currently are. So why aren't they? We suggest that one reason is that iconicity is inimical to abstraction because iconic forms are too connected to specific contexts and sensory depictions. Form-meaning arbitrariness may allow language to better convey abstract meanings.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.

Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Bayesian validation of grammar productions for the language of thought

Probabilistic proposals of Language of Thoughts (LoTs) can explain learning across different domains as statistical inference over a compositionally structured hypothesis space. While frameworks may differ on how a LoT may be implemented computationally, they all share the property that they are built from a set of atomic symbols and rules by which these symbols can be combined. In this work we propose an extra validation step for the set of atomic productions defined by the experimenter. It starts by expanding the defined LoT grammar for the cognitive domain with a broader set of arbitrary productions and then uses Bayesian inference to prune the productions from the experimental data. The result allows the researcher to validate that the resulting grammar still matches the intuitive grammar chosen for the domain. We then test this method in the language of geometry, a specific LoT model for geometrical sequence learning. Finally, despite the fact of the geometrical LoT not being a universal (i.e. Turing-complete) language, we show an empirical relation between a sequence’s probability and its complexity consistent with the theoretical relationship for universal languages described by Levin’s Coding Theorem.