Although debates continue, studies of cognition in infancy suggest that knowledge begins to emerge early in life and constitutes part of humans' innate endowment. Early-developing knowledge appears to be both domain-specific and task-specific, it appears to capture fundamental constraints on ecologically important classes of entities in the child's environment, and it appears to remain central to the commonsense knowledge systems of adults.

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic (...) mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

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. (shrink)

While endorsing Gopnik's proposal that studies of the emergence and modification of scientific theories and studies of cognitive development in children are mutually illuminating, we offer a different picture of the beginning points of cognitive development from Gopnik's picture of "theories all the way down." Human infants are endowed with several distinct core systems of knowledge which are theory-like in some, but not all, important ways. The existence of these core systems of knowledge has implications for the joint research program (...) between philosophers and psychologists that Gopnik advocates and we endorse. A few lessons already gained from this program of research are sketched. (shrink)

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for (...) representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems. (shrink)

Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate with these properties. Experiment 1 extended the standard habituation/dishabituation paradigm to a 1 vs 2 comparison with three-dimensional objects and confirmed that when number and total front surface area are confounded, infants discriminate the arrays. Experiment 2 revealed that (...) infants dishabituated to a change in front surface area but not to a change in number when the two variables were pitted against each other. Experiments 3 through 5 revealed no sensitivity to number when front surface area was controlled, and Experiments 6 and 7 extended this pattern of findings to the Wynn (1992) transformation task. Infants’ lack of a response to number, combined with their demonstrated sensitivity to one or more dimensions of continuous extent, supports the hypothesis that the representations subserving object-based attention, rather than those subserving enumeration, underlie performance in the above tasks. (shrink)

Research on human infants, adult nonhuman primates, and children and adults in diverse cultures provides converging evidence for four systems at the foundations of human knowledge. These systems are domain specific and serve to represent both entities in the perceptible world (inanimate manipulable objects and animate agents) and entities that are more abstract (numbers and geometrical forms). Human cognition may be based, as well, on a fifth system for representing social partners and for categorizing the social world into groups. Research (...) on infants and children may contribute both to understanding of these systems and to attempts to overcome misconceptions that they may foster. (shrink)

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. (shrink)

What are the brain and cognitive systems that allow humans to play baseball, compute square roots, cook soufflés, or navigate the Tokyo subways? It may seem that studies of human infants and of non-human animals will tell us little about these abilities, because only educated, enculturated human adults engage in organized games, formal mathematics, gourmet cooking, or map-reading. In this chapter, we argue against this seemingly sensible conclusion. When human adults exhibit complex, uniquely human, culture-specific skills, they draw on a (...) set of psychological and neural mechanisms with two distinctive properties: they evolved before humanity and thus are shared with other animals, and they emerge early in human development and thus are common to infants, children, and adults. These core knowledge systems form the building blocks for uniquely human skills. Without them we wouldn’t be able to learn about different kinds of games, mathematics, cooking, or maps. To understand what is special about human intelligence, therefore, we must study both the core knowledge systems on which it rests and the mechanisms by which these systems are orchestrated to permit new kinds of concepts and cognitive processes. What is core knowledge? A wealth of research on non-human primates and on human infants suggests that a system of core knowledge is characterized by four properties (Hauser, 2000; Spelke, 2000). First, it is domain-specific: each system functions to represent particular kinds of entities such as conspecific agents, manipulable objects, places in the environmental layout, and numerosities. Second, it is task-specific: each system uses its representations to address specific questions about the world, such as “who is this?” [face recognition], “what does this do?” [categorization of artifacts], “where am I?” [spatial orientation], and “how many are here?” [enumeration]. Third, it is relatively encapsulated: each uses only a subset of the information delivered by an animal’s input systems and sends information only to a subset of the animal’s output systems. (shrink)

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. (shrink)

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on (...) this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably. (shrink)

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The spatial content of the (...) formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (shrink)

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event (...) depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS. (shrink)

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 - 9 yields 12. Here, we investigate whether preschool children’s approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation (...) than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding. (shrink)

Des branches entières des mathématiques sont fondées sur des liens posés entre les nombres et l’espace : mesure de longueurs, définition de repères et de coordonnées, projection des nombres complexes sur le plan… Si les nombres complexes, comme l’utilisation de repères, sont apparus relativement récemment (vers le XVIIe siècle), la mesure des longueurs est en revanche un procédé très ancien, qui remonte au moins au 3e ou 4e millénaire av. J-C. Loin d’être fortuits, ces liens entre les nombres et l’espace (...) reflèteraient une intuition fondamentale, universelle, façonnée au cours des millénaires par la sélection naturelle, et qui aurait servi de guide et d’inspiration aux mathématiciens au fil des siècles. (shrink)

La connaissances noyau de la géométrie euclidienne est liée au raisonnement déductif et non à la reconnaissance de motifs perceptuels.

The performance of the Mundurucu on the number-space task may exemplify a general competence for drawing analogies between space and other linear dimensions, but Mundurucu participants spontaneously chose number when other dimensions were available. Response placement may not reflect the subjective scale for numbers, but Cantlon et al.'s proposal of a linear scale with scalar variability requires additional hypotheses that are problematic.

We agree with Nuñez that the Mundurucu do not master the formal propreties of number lines and logarithms, but as the term "intuition" implies, they spontaneously experience a logarithmic mapping of number to space as natural and "feeling right.".

Thirty-six 1-year-old middle-class children with fathers who spent differential time with them at home were observed in two experimental contexts separated by 2 weeks. In the first, each infant was shown six to eight repetitions of three different nonsocial events followed by a change in..

Four-month-old infants were allowed to manipulate, without vision, two rings attached to a bar that permitted each ring to undergo rotary motion against a fixed surface. In different conditions, the relative motions of the rings were rigid, independent, or opposite, and they circled either the same fixed point outside the zone of manipulation or spatially separated points. Infants’ perception of the ring assemblies were affected by the nature of the rotary motion in two ways. First, infants perceived a unitary object (...) when the felt ends of the object underwent a common, rigid rotary motion; perception of object unity was stronger in this condition than when the ends underwent either independent or opposite rotary motions. Second, infants perceived two distinct objects when the felt ends of the objects underwent independent rotary motions that centred on distinct fixed points. Perception of the distinctness of the objects was less clear when the ends underwent opposite or independent rotary motions that centred on a common fixed point. These findings provide the first evidence that infants are sensitive to rotary motion patterns and can extrapolate a global pattern of rigid motion from the distinct, local velocities that they produce and experience at their two hands. (shrink)

The results of two experiments examining infants at 6, 9, 12, 15, 18, and 21 months 0f age and varying levels of father interaction are summarized to show that separation protest is more a function of a strange person remaining in an unfamiliar laboratory situation with the infant than the temporary loss of a specific parent. The use of protest as an index of infant-parent attachment seems undesirable.