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

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

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.

Infants' sensitivity to shape changes in 2D visual forms.

Research in developmental cognitive science reveals that human infants perceive shape changes in 2D visual forms that are repeatedly presented over long durations. Nevertheless, infants' sensitivity to shape under the brief conditions of natural viewing has been little studied. Three experiments tested for this sensitivity by presenting 128 seven-month-old infants with shapes for the briefer durations under which they might see them in dynamic scenes. The experiments probed infants' sensitivity to two fundamental geometric properties of scale- and orientation-invariant shape: relative length and angle. Infants detected shape changes in closed figures, which presented changes in both geometric properties. Infants also detected shape changes in open figures differing in angle when figures were presented at limited orientations. In contrast, when open figures were presented at unlimited orientations, infants detected changes in relative length but not in angle. The present research therefore suggests that, as infants look around at the cluttered and changing visual world, relative length is the primary geometric property by which they perceive scale- and orientation-invariant shape.

Representations of space, time, and number in neonates.

A rich concept of magnitude--in its numerical, spatial, and temporal forms--is a central foundation of mathematics, science, and technology, but the origins and developmental relations among the abstract concepts of number, space, and time are debated. Are the representations of these dimensions and their links tuned by extensive experience, or are they readily available from birth? Here, we show that, at the beginning of postnatal life, 0- to 3-d-old neonates reacted to a simultaneous increase (or decrease) in spatial extent and in duration or numerical quantity, but they did not react when the magnitudes varied in opposite directions. The findings provide evidence that representations of space, time, and number are systematically interrelated at the start of postnatal life, before acquisition of language and cultural metaphors, and before extensive experience with the natural correlations between these dimensions.

Perceiving numerosity from birth.

Leibovich et al. opened up an important discussion on the nature and origins of numerosity perception. The authors rightly point out that non-numerical features of stimuli influence this ability. Despite these biases, there is evidence that from birth, humans perceive and represent numerosities, and not just non-numerical quantitative features such as item size, density, and convex hull.

Relative versus absolute numerical representation in fish: Can guppies represent "fourness"?

In recent years, the use of operant conditioning procedures has shown that species as diverse as chimpanzees, honeybees, and mosquitofish can be trained to discriminate between sets containing different numbers of objects. However, to succeed in this task, subjects can use two different strategies: either select the array containing a specific number of items (an absolute numerosity rule), or select the set containing the larger (or smaller) quantity of items (a relative numerosity rule). In the latter case, subjects need not only be able to judge whether two numerosities are equal or different but also be able to order numerosities. Here, in two experiments, we address whether fish can perform both kinds of judgment by training them with specific numerosities and testing their generalization to new numerosity contrasts. In Experiment 1, subjects were initially trained to select between visual arrays of 6 and 12 shapes, and were then tested with a contrast pairing the previously trained numerosity (either 6 or 12) with a novel numerosity (respectively, 3 or 24). Spontaneously, subjects selected the novel numerosity, in accordance with a relative numerosity rule. The second experiment tested whether guppies can also learn to select one specific number against all others, if appropriately trained. Fish trained to select an array of 4 shapes against several alternatives (4 vs. 1, 4 vs. 2, 4 vs. 8, 4 vs. 10) learned to recognize the number 4 against all alternatives and proved able to generalize their discrimination to novel, more difficult contrasts (4 vs. 3 and 4 vs. 6 items). In summary, although guppies preferentially opt for relative comparisons, they can flexibly learn either relative or absolute decision criteria on numerosity stimuli, depending on the context.

Toward exact number: young children use one-to-one correspondence to measure set identity but not numerical equality.

Exact integer concepts are fundamental to a wide array of human activities, but their origins are obscure. Some have proposed that children are endowed with a system of natural number concepts, whereas others have argued that children construct these concepts by mastering verbal counting or other numeric symbols. This debate remains unresolved, because it is difficult to test children's mastery of the logic of integer concepts without using symbols to enumerate large sets, and the symbols themselves could be a source of difficulty for children. Here, we introduce a new method, focusing on large quantities and avoiding the use of words or other symbols for numbers, to study children's understanding of an essential property underlying integer concepts: the relation of exact numerical equality. Children aged 32-36 months, who possessed no symbols for exact numbers beyond 4, were given one-to-one correspondence cues to help them track a set of puppets, and their enumeration of the set was assessed by a non-verbal manual search task. Children used one-to-one correspondence relations to reconstruct exact quantities in sets of 5 or 6 objects, as long as the elements forming the sets remained the same individuals. In contrast, they failed to track exact quantities when one element was added, removed, or substituted for another. These results suggest an alternative to both nativist and symbol-based constructivist theories of the development of natural number concepts: Before learning symbols for exact numbers, children have a partial understanding of the properties of exact numbers.

Dissociation between small and large numerosities in newborn infants.

In the first year of life, infants possess two cognitive systems encoding numerical information: one for processing the numerosity of sets of 4 or more items, and the second for tracking up to 3 objects in parallel. While a previous study showed the former system to be already present a few hours after birth, it is unknown whether the latter system is functional at this age. Here, we adapt the auditory-visual matching paradigm that previously revealed sensitivity to large numerosities to test sensitivity to numerosities spanning the range from 2 to 12. Across studies, newborns discriminated pairs of large numerosities in a 3:1 ratio, even when the smaller numerosity was 3 (3 vs. 9). In contrast, newborn infants failed to discriminate pairs including the numerosity 2, even at the same ratio (2 vs. 6). These findings mirror the dissociation that has been reported with older infants, albeit with a discontinuity situated between numerosities 2 and 3. Two alternative explanations are compatible with our results: either newborn infants have a separate system for processing small sets, and the capacity of this system is limited to 2 objects; or newborn infants possess only one system to represent numerosities, and this system either is not functional or is extremely imprecise when it is applied to small numerosities.

Reading angles in maps.

Preschool children can navigate by simple geometric maps of the environment, but the nature of the geometric relations they use in map reading remains unclear. Here, children were tested specifically on their sensitivity to angle. Forty-eight children (age 47:15-53:30 months) were presented with fragments of geometric maps, in which angle sections appeared without any relevant length or distance information. Children were able to read these map fragments and compare two-dimensional to three-dimensional angles. However, this ability appeared both variable and fragile among the youngest children of the sample. These findings suggest that 4-year-old children begin to form an abstract concept of angle that applies both to two-dimensional and three-dimensional displays and that serves to interpret novel spatial symbols.

Flexible intuitions of Euclidean geometry in an Amazonian indigene group.

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ~180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics.

"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.

Education enhances the acuity of the nonverbal approximate number system.

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction.

Numerical discrimination in Drosophila melanogaster.

Sensitivity to numbers is a crucial cognitive ability. The lack of experimental models amenable to systematic genetic and neural manipulation has precluded discovering neural circuits required for numerical cognition. Here, we demonstrate that Drosophila flies spontaneously prefer sets containing larger numbers of objects. This preference is determined by the ratio between the two numerical quantities tested, a characteristic signature of numerical cognition across species. Individual flies maintained their numerical choice over consecutive days. Using a numerical visual conditioning paradigm, we found that flies are capable of associating sucrose with numerical quantities and can be trained to reverse their spontaneous preference for large quantities. Finally, we show that silencing lobula columnar neurons (LC11) reduces the preference for more objects, thus identifying a neuronal substrate for numerical cognition in invertebrates. This discovery paves the way for the systematic analysis of the behavioral and neural mechanisms underlying the evolutionary conserved sensitivity to numerosity.

Newborn infants perceive abstract numbers.

Although infants and animals respond to the approximate number of elements in visual, auditory, and tactile arrays, only human children and adults have been shown to possess abstract numerical representations that apply to entities of all kinds (e.g., 7 samurai, seas, or sins). Do abstract numerical concepts depend on language or culture, or do they form a part of humans' innate, core knowledge? Here we show that newborn infants spontaneously associate stationary, visual-spatial arrays of 4-18 objects with auditory sequences of events on the basis of number. Their performance provides evidence for abstract numerical representations at the start of postnatal experience.

Abstract representations of small sets in newborns.

From the very first days of life, newborns are not tied to represent narrow, modality- and object-specific aspects of their environment. Rather, they sometimes react to abstract properties shared by stimuli of very different nature, such as approximate numerosity or magnitude. As of now, however, there is no evidence that newborns possess abstract representations that apply to small sets: in particular, while newborns can match large approximate numerosities across senses, this ability does not extend to small numerosities. In two experiments, we presented newborn infants (N = 64, age 17 to 98 h) with patterned sets AB or ABB simultaneously in the auditory and visual modalities. Auditory patterns were presented as periodic sequences of sounds (AB: triangle-drum-triangle-drum-triangle-drum …; ABB: triangle-drum-drum-triangle-drum-drum-triangle-drum-drum …), and visual patterns as arrays of 2 or 3 shapes (AB: circle-diamond; ABB: circle-diamond-diamond). In both experiments, we found that participants reacted and looked longer when the patterns matched across the auditory and visual modalities - provided that the first stimulus they received was congruent. These findings uncover the existence of yet another type of abstract representations at birth, applying to small sets. As such, they bolster the hypothesis that newborns are endowed with the capacity to represent their environment in broad strokes, in terms of its most abstract properties. This capacity for abstraction could later serve as a scaffold for infants to learn about the particular entities surrounding them.

IntPhys 2019: A Benchmark for Visual Intuitive Physics Understanding.

In order to reach human performance on complex visual tasks, artificial systems need to incorporate a significant amount of understanding of the world in terms of macroscopic objects, movements, forces, etc. Inspired by work on intuitive physics in infants, we propose an evaluation benchmark which diagnoses how much a given system understands about physics by testing whether it can tell apart well matched videos of possible versus impossible events constructed with a game engine. The test requires systems to compute a physical plausibility score over an entire video. To prevent perceptual biases, the dataset is made of pixel matched quadruplets of videos, enforcing systems to focus on high level temporal dependencies between frames rather than pixel-level details. We then describe two Deep Neural Networks systems aimed at learning intuitive physics in an unsupervised way, using only physically possible videos. The systems are trained with a future semantic mask prediction objective and tested on the possible versus impossible discrimination task. The analysis of their results compared to human data gives novel insights in the potentials and limitations of next frame prediction architectures.

Distinct cerebral pathways for object identity and number in human infants.

All humans, regardless of their culture and education, possess an intuitive understanding of number. Behavioural evidence suggests that numerical competence may be present early on in infancy. Here, we present brain-imaging evidence for distinct cerebral coding of number and object identity in 3-mo-old infants. We compared the visual event-related potentials evoked by unforeseen changes either in the identity of objects forming a set, or in the cardinal of this set. In adults and 4-y-old children, number sense relies on a dorsal system of bilateral intraparietal areas, different from the ventral occipitotemporal system sensitive to object identity. Scalp voltage topographies and cortical source modelling revealed a similar distinction in 3-mo-olds, with changes in object identity activating ventral temporal areas, whereas changes in number involved an additional right parietoprefrontal network. These results underscore the developmental continuity of number sense by pointing to early functional biases in brain organization that may channel subsequent learning to restricted brain areas.

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.

Are numbers special? An overview of chronometric, neuroimaging, developmental and comparative studies of magnitude representation.

There is a current debate whether the human brain possesses a shared representation for various types of magnitude such as numerical quantities, physical size, or loudness. Here, we critically review evidence from chronometric, neuroimaging, developmental and comparative fields, and supplement it with a meta-analysis of the neuroimaging data. Together, based on such an integrative overview, we discuss limitations inherent in each approach, and the possibility whether shared, or distinct magnitude representation, or both representations exist.

Core cognition in adult vision: A surprising discrepancy between the principles of object continuity and solidity.

From an early age, humans intuitively expect physical objects to obey core principles, including continuity (objects follow spatiotemporally continuous paths) and solidity (two solid objects cannot occupy the same space at the same time). These 2 principles are sometimes viewed as deriving from a single overarching "persistence" principle. Indeed, violations of solidity where one solid object seemingly passes through another could theoretically be interpreted as a violation of continuity, with an object "teleporting" to switch places rather than passing through a solid obstacle. However, it is an empirical issue whether the two principles are processed distinctly or identically to one another. Here, adult participants tracked objects during dynamic events in a novel location detection task, which sometimes involved violations of the principles of continuity or solidity. Although participants explicitly noticed both types of violations and reported being equally surprised at both, they made more errors and answered more slowly after continuity violations than after solidity violations. Our results demonstrate that the two principles show different signature patterns and are thus represented distinctly in the mind. (PsycInfo Database Record (c) 2020 APA, all rights reserved).