Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill1,2. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations3,4, and their performance suffers if this nonsymbolic system is impaired5. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is required6–10. Here (...) we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction11–15. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children’s difficulties learning symbolic arithmetic, and they suggest ways to enhance children’s engagement with formal mathematics. We presented children with approximate symbolic arithmetic problems in a format that parallels previous tests of non-symbolic arithmetic in preschool children8,9. In the first experiment, five- to six-year-old children were given problems such as ‘‘If you had twenty-four stickers and I gave you twenty-seven more, would you have more or less than thirty-five stickers?’’. Children performed well above chance (65.0%, t1952.77, P 5 0.012) without resorting to guessing or comparison strategies that could serve as alternatives to arithmetic. Children who have been taught no symbolic arithmetic therefore have some ability to perform symbolic addition problems. The children’s performance nevertheless fell short of performance on non-symbolic arithmetic tasks using equivalent addition problems with numbers presented as arrays of dots and with the addition operation conveyed by successive motions of the dots into a box (71.3% correct, F1,345 4.26, P 5 0.047)8.. (shrink)
& Behavioral and brain imaging research indicates that human infants, humans adults, and many nonhuman animals represent large nonsymbolic numbers approximately, discriminating between sets with a ratio limit on accuracy. Some behavioral evidence, especially with human infants, suggests that these representations differ from representations of small numbers of objects. To investigate neural signatures of this distinction, event-related potentials were recorded as adult humans passively viewed the sequential presentation of dot arrays in an adaptation paradigm. In two studies, subjects viewed successive (...) arrays of a single number of dots interspersed with test arrays presenting the same or a different number; numerical range (small numerical quantities 1–3 vs. large numerical.. (shrink)
Behavioral research suggests that two cognitive systems are at the foundations of numerical thinking: one for representing 1–3 objects in parallel and one for representing and comparing large, approximate numerical magnitudes. We tested for dissociable neural signatures of these systems in preverbal infants by recording event-related potentials (ERPs) as 6–7.5-month-old infants (n = 32) viewed dot arrays containing either small (1–3) or large (8–32) sets of objects in a number alternation paradigm. If small and large numbers are represented by the (...) same neural system, then the brain response to the arrays should scale with ratio for both number ranges, a behavioral and brain signature of the approximate numerical magnitude system obtained in animals and in human adults. Contrary to this prediction, a mid-latency positivity (P500) over parietal scalp sites was modulated by the ratio between successive large, but not small, numbers. Conversely, an earlier peaking positivity (P400) over occipital-temporal sites was modulated by the absolute cardinal value of small, but not large, numbers. These results provide evidence for two early developing systems of non-verbal numerical cognition: one that responds to small quantities as individual objects and a second that responds to large quantities as approximate numerical values. These brain signatures are functionally similar to those observed in previous studies of non-symbolic number with adults, suggesting that this dissociation may persist over vast differences in experience and formal training in mathematics. (shrink)
Previous research has shown that young children have difficulty searching for a hidden object whose location depends on the position of a partly visible physical barrier. Across four experiments, we tested whether children’s search errors are affected by two variables that influence adults’ object-directed attention: object boundaries and proximity relations. Toddlers searched for a car that rolled down a ramp behind an occluding panel and stopped on contact with a barrier. The car’s location on each trial depended on the placement (...) of the barrier behind one of two doors in the panel. In Experiment 1, when a part of the car (a pompom on an antenna) was visible at the same distance from the object as the barrier wall in past research, search performance was above chance but below ceiling. In Experiments 2 and 3, when the visible part was close to the hidden body of the car and could be seen through one of two windows in the doors of the occluding panel, performance was near ceiling. In Experiment 4, when only the barrier was visible through one of the same windows, performance was at chance. Toddlers’ search for a hidden object therefore is affected by the proximity of a visible part of the object, though not by the proximity of a separate visible landmark. These findings suggest a parallel between the object representations of young children and those of adults, whose attention is directed to objects and spreads in a gradient-like fashion within an object. (shrink)
Three experiments investigated changes from 15 to 30 months of age in children’s (N = 114) mastery of relations between an object and an aperture, supporting surface, or form. When choosing between objects to insert into an aperture, older children selected objects of an appropriate size and shape, but younger children showed little selectivity. Further experiments probed the sources of younger children’s difficulty by comparing children’s performance placing a target object in a hole, on a 2-dimensional form, or atop another (...) solid object. Together, the findings suggest that some factors limiting adults’ object representations, including the difficulty of comparing the shapes of positive and negative spaces and of representing shapes in 3 dimensions, contribute to young children’s errors in manipulating objects. (shrink)
Experiments using a preferential looking method, a perceptual judgment method, and a predictive judgment method investigated the development, from 7 months to 6 years of age, of sensitivity to the effects of gravity and inertia on inanimate object motion. The experiments focused on a situation in which a ball rolled off a flat surface and either continued in linear motion (contrary to gravity), turned abruptly and moved downward (contrary to inertia), or underwent natural, parabolic motion. When children viewed the three (...) fully visible motions, both the preferential looking method and the perceptual judgment method provided evidence that sensitivity to inertia developed between 7 months and 2 years, and that sensitivity to gravity began to develop after 3 years. When children predicted the future location of the object without viewing the motions, the predictive judgment method provided evidence that sensitivity to gravity had developed by 2 years, whereas sensitivity to inertia began to develop only at 5±6 years. These findings suggest that knowledge of object motion develops slowly over childhood, in a piecemeal fashion. Moreover, the same system of knowledge appears to be tapped both in preferential looking tasks and in judgment tasks when children view fully visible events, but a different system may underlie children's inferences about unseen object motions. (shrink)
��Disoriented 4-year-old children use a distinctive container to locate a hidden object, but do they reorient by this information? We addressed this question by testing children’s search for objects in a circular room containing one distinctive and two identical containers. Children’s search patterns provided evidence that the distinctive container served as a direct cue to a hidden object’s location, but not as a directional signal guiding reorientation. The findings suggest that disoriented children’s search behavior depends on two distinct processes: a (...) modular reorientation process attuned to the geometry of the surface layout and an associative process linking landmarks to specific locations. (shrink)
Although disoriented young children reorient themselves in relation to the shape of the surrounding surface layout, cognitive accounts of this ability vary. The present paper tests three theories of reorientation: a snapshot theory based on visual image-matching computations, an adaptive combination theory proposing that diverse environmental cues to orientation are weighted according to their experienced reliability, and a modular theory centering on encapsulated computations of the shape of the extended surface layout. Seven experiments test these theories by manipulating four properties (...) of objects placed within a cylindrical space: their size, motion, dimensionality, and distance from the space’s borders. Their findings support the modular theory and suggest that disoriented search behavior centers on two processes: a reorientation process based on the geometry of the 3D surface layout, and a beacon-guidance process based on the local features of objects and surface markings. Ó 2010 Elsevier Inc. All rights reserved. (shrink)
Disoriented animals from ants to humans reorient in accord with the shape of the surrounding surface layout: a behavioral pattern long taken as evidence for sensitivity to layout geometry. Recent computational models suggest, however, that the reorientation process may not depend on geometrical analyses but instead on the matching of brightness contours in 2D images of the environment. Here we test this suggestion by investigating young children's reorientation in enclosed environments. Children reoriented by extremely subtle geometric properties of the 3D (...) layout: bumps and ridges that protruded only slightly off the floor, producing edges with low contrast. Moreover, children failed to reorient by prominent brightness contours in continuous layouts with no distinctive 3D structure. The findings provide evidence that geometric layout representations support children's reorientation. (shrink)
Although children take over a year to learn the meanings of the first three number words, they eventually master the logic of counting and the meanings of all the words in their count list. Here, we ask whether children’s knowledge applies to number words beyond those they have mastered: Does a child who can only count to 20 infer that number words above ‘twenty’ refer to exact cardinal values? Three experiments provide evidence for this understanding in preschool children. Before beginning (...) formal education or gaining counting skill, children possess a productive symbolic system for representing number. q 2005 Published by Elsevier B.V. (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)
Observations and experiments show that human adults preferentially share resources with close relations, with people who have shared with them (reciprocity), and with people who have shared with others (indirect reciprocity). These tendencies are consistent with evolutionary theory but could also reflect the shaping effects of experience or instruction in complex, cooperative, and competitive societies. Here, we report evidence for these three tendencies in 3.5-year-old children, despite their limited experience with complex cooperative networks. Three pillars of mature cooperative behavior therefore (...) appear to have roots extending deep into human development. Ó 2007 Elsevier B.V. All rights reserved. (shrink)
For millennia, human beings have believed that it is morally wrong to judge others by the fortuitous or unfortunate events that befall them or by the actions of another person. Rather, an individual’s own intended, deliberate actions should be the basis of his or her evaluation, reward, and punishment. In a series of studies, the authors investigated whether such rules guide the judgments of children. The first 3 studies demonstrated that children view lucky others as more likely than unlucky others (...) to perform intentional good actions. Children similarly assess the siblings of lucky others as more likely to perform intentional good actions than the siblings of unlucky others. The next 3 studies demonstrated that children as young as 3 years believe that lucky people are nicer than unlucky people. The final 2 studies found that Japanese children also demonstrate a robust preference for the lucky and their associates. These findings are discussed in relation to M. J. Lerner’s (1980) just-world theory and J. Piaget’s (1932/1965) immanent-justice research and in relation to the development of intergroup attitudes. (shrink)
A series of experiments investigated the effect of speakers’ language, accent, and race on children’s social preferences. When presented with photographs and voice recordings of novel children, 5-year-old children chose to be friends with native speakers of their native language rather than foreign-language or foreign-accented speakers. These preferences were not exclusively due to the intelligibility of the speech, as children found the accented speech to be comprehensible, and did not make social distinctions between foreign-accented and foreign-language speakers. Finally, children chose (...) same-race children as friends when the target children were silent, but they chose other-race children with a native accent when accent was pitted against race. A control experiment provided evidence that children’s privileging of accent over race was not due to the relative familiarity of each dimension. The results, discussed in an evolutionary framework, suggest that children preferentially evaluate others along dimensions that distinguished social groups in prehistoric human societies. (shrink)
Human cognition is founded, in part, on four systems for representing objects, actions, number, and space. It may be based, as well, on a fifth system for representing social partners. Each system has deep roots in human phylogeny and ontogeny, and it guides and shapes the mental lives of adults. Converging research on human infants, non-human primates, children and adults in diverse cultures can aid both understanding of these systems and attempts to overcome their limits.
A dedicated, non-symbolic, system yielding imprecise representations of large quantities (Approximate Number System, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5-7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the (...) outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction. (shrink)
Six experiments investigated 7-month-old infants’ capacity to learn about the self-propelled motion of an object. After observing 1 wind-up toy animal move on its own and a second wind-up toy animal move passively by an experimenter’s hand, infants looked reliably longer at the former object during a subsequent stationary test, providing evidence that infants learned and remembered the mapping of objects and their motions. In further experiments, infants learned the mapping for different animals and retained it over a 15-min delay, (...) providing evidence that the learning is robust and infants’ expectations about self-propelled motion are enduring. Further experiments suggested that infants’ learning was less reliable when the self-propelled objects were novel or lacked faces, body parts, and articulated, biological motion. The findings are discussed in relation to infants’ developing knowledge of object categories and capacity to learn about objects in the first year of life. (shrink)
Three experiments investigated the role of a speci®c language in human representations of number. Russian±English bilingual college students were taught new numerical operations (Experiment 1), new arithmetic equations (Experiments 1 and 2), or new geographical or historical facts involving numerical or non-numerical information (Experiment 3). After learning a set of items in each of their two languages, subjects were tested for knowledge of those items, and new items, in both languages. In all the studies, subjects retrieved information about exact numbers (...) more effectively in the language of training, and they solved trained problems more effectively than untrained problems. In contrast, subjects retrieved information about approximate numbers and non-numerical facts with equal ef®ciency in their two languages, and their training on approximate number facts generalized to new facts of the same type. These ®ndings suggest that a speci®c, natural language contributes to the representation of large, exact numbers but not to the approximate number representations that humans share with other mammals. Language appears to play a role in learning about exact numbers in a variety of contexts, a ®nding with implications for practice in bilingual education. The ®ndings prompt more general speculations about the role of language in the development of speci®cally human cognitive abilities. q 2001 Elsevier Science B.V. All rights reserved. (shrink)
Under many circumstances, children and adult rats reorient themselves through a process which operates only on information about the shape of the environment (e.g., Cheng, 1986; Hermer & Spelke, 1996). In contrast, human adults relocate themselves more flexibly, by conjoining geometric and nongeometric information to specify their position (Hermer & Spelke, 1994). The present experiments used a dual-task method to investigate the processes that underlie the flexible conjunction of information. In Experiment 1, subjects reoriented themselves flexibly when they performed no (...) secondary task, but they reoriented themselves like children and adult rats when they engaged in verbal shadowing of continuous speech. In Experiment 2, subjects who engaged in nonverbal shadowing of a continuous rhythm reoriented like nonshadowing subjects, suggesting that the interference effect in Experiment 1 did not stem from general limits on working memory or attention but from processes more specific to language. In further experiments, verbally shadowing subjects detected and remembered both nongeometric information (Experiment 3) and geometric information (Experiments 1, 2, and 4), but they failed to conjoin the two types of information to specify the positions of objects (Experiment 4). Together. (shrink)
How does the human brain support abstract concepts such as seven or square? Studies of nonhuman animals, of human infants, and of children and adults in diverse cultures suggest these concepts arise from a set of cognitive systems that are phylogenetically ancient, innate, and universal across humans: systems of core knowledge. Two of these systems—for tracking small numbers of objects and for assessing, comparing and combining the approximate cardinal values of sets—capture the primary information in the system of positive integers. (...) Two other systems—for representing the shapes of small-scale forms and the distances and directions of surfaces in the large-scale navigable layout—capture the primary information in the system of Euclidean plane geometry. As children learn language and other symbol systems, they begin to combine their core numerical and geometrical representations productively, in uniquely human ways. These combinations may give rise to the first truly abstract concepts at the foundations of mathematics. (shrink)
To whom do children look when deciding on their own preferences? To address this question, 3-year-old children were asked to choose between objects or activities that were endorsed by unfamiliar people who differed in gender, race (White, Black), or age (child, adult). In Experiment 1, children demonstrated robust preferences for objects and activities endorsed by children of their own gender, but less consistent preferences for objects and activities endorsed by children of their own race. In Experiment 2, children (...) selected objects and activities favored by people of their own gender and age. In neither study did most children acknowledge the influence of these social categories. These findings suggest that gender and age categories are encoded spontaneously and influence children’s preferences and choices. For young children, gender and age may be more powerful guides to preferences than race. (shrink)
Infants were presented with an object that moved into reaching space on a path that was either continuously visible or interrupted by an occluder. Infants’ reaching was reduced sharply when an occluder was present, even though the occluder itself was out of reach and did not serve as a barrier to direct reaching for the object. We account for these findings and for the apparently contrasting findings of experiments using preferential looking methods to assess infants’ object representations, by proposing that (...) (a) object representations increase in precision over the infancy period, and (b) the precision of object representations varies in common ways at all ages as a function of object visibility and task demands. (shrink)
This article considers 3 claims that cognitive sex differ- ences account for the differential representation of men and women in high-level careers in mathematics and sci- ence: (a) males are more focused on objects from the beginning of life and therefore are predisposed to better learning about mechanical systems; (b) males have a pro- file of spatial and numerical abilities producing greater aptitude for mathematics; and (c) males are more variable in their cognitive abilities and therefore predominate at the upper (...) reaches of mathematical talent. Research on cogni- tive development in human infants, preschool children, and students at all levels fails to support these claims. Instead, it provides evidence that mathematical and scientific rea- soning develop from a set of biologically based cognitive capacities that males and females share. These capacities lead men and women to develop equal talent for mathe- matics and science. (shrink)
What leads humans to divide the social world into groups, preferring their own group and disfavoring others? Experiments with infants and young children suggest these tendencies are based on predispo- sitions that emerge early in life and depend, in part, on natural language. Young infants prefer to look at a person who previously spoke their native language. Older infants preferentially accept toys from native-language speakers, and preschool children preferentially select native-language speakers as friends. Variations in accent are sufficient to evoke (...) these social preferences, which are observed in infants before they produce or comprehend speech and are exhibited by children even when they comprehend the foreign-accented speech. Early-developing preferences for native-language speakers may serve as a foundation for later-developing preferences and conflicts among social groups. (shrink)
Infants learn from adults readily and cooperate with them spontaneously, but how do they select culturally appropriate teachers and collaborators? Building on evidence that children demonstrate social preferences for speakers of their native language, Experiment 1 presented 10- month-old infants with videotaped events in which a native and a foreign speaker introduced two different toys. When given a chance to choose between real exemplars of the objects, infants preferentially chose the toy modeled by the native speaker. In Experiment 2, 2.5-year-old (...) children were presented with the same videotaped native and foreign speakers, and played a game in which they could offer an object to one of two individuals. Children reliably gave to the native speaker. Together, the results suggest that infants and young children are selective social learners and cooperators, and that language provides one basis for this selectivity. (shrink)
& Visual object representation was studied in free-ranging rhesus monkeys. To facilitate comparison with humans, and to provide a new tool for neurophysiologists, we used a looking time procedure originally developed for studies of human infants. Monkeys’ looking times were measured to displays with one or two distinct objects, separated or together, stationary or moving. Results indicate that rhesus monkeys..
Because human languages vary in sound and meaning, children must learn which distinctions their language uses. For speech perception, this learning is selective: initially infants are sensitive to most acoustic distinctions used in any language1–3, and this sensitivity reflects basic properties of the auditory system rather than mechanisms specific to language4–7; however, infants’ sensitivity to non-native sound distinctions declines over the course of the first year8. Here we ask whether a similar process governs learning of word meanings. We investigated the (...) sensitivity of 5-month-old infants in an English-speaking environment to a conceptual distinction that is marked in Korean but not English; that is, the distinction between ‘tight’ and ‘loose’ fit of one object to another9,10. Like adult Korean speakers but unlike adult English speakers, these infants detected this distinction and divided a continuum of motion-into-contact actions into tightand loose-fit categories. Infants’ sensitivity to this distinction is linked to representations of object mechanics11that are shared by non-human animals12–14. Language learning therefore seems to develop by linking linguistic forms to universal, pre-existing representations of sound and meaning. Our research focuses on the crosscutting conceptual distinctions between actions producing loose- and tight-fitting contact relationships (compare left and right columns in Fig. 1a) and actions producing containment versus support relationships (compare first and second rows in Fig. 1a). As early as Korean and English children begin to talk about such actions, they categorize them differently from one another and similarly to Korean- and Englishspeaking adults9,15. Moreover, English and Korean adults differ in their performance on non-linguistic categorization tasks involving heterogeneous examples of these actions, in accord with the differing semantics of their languages16,17, whereas the performance of young children on such tasks has been mixed9,10,18,19.. (shrink)
Developmental research suggests that some of the mechanisms that underlie numerical cognition are present and functional in human infancy. To investigate these mechanisms and their developmental course, psychologists have turned to behavioral and electrophysiological methods using briefly presented displays. These methods, however, depend on the assumption that young infants can extract numerical information rapidly. Here we test this assumption and begin to investigate the speed of numerical processing in five-month-old infants. Infants successfully discriminated between arrays of 4 vs. 8 dots (...) on the basis of number when a new array appeared every 2 s, but not when a new array appeared every 1.0 or 1.5 s. These results suggest alternative interpretations of past findings, provide constraints on the design of future experiments, and introduce a new method for probing infants’ enumeration process. Further experiments using this method provide initial evidence that infants’ enumeration mechanism operates in parallel and yields increasingly accurate numerical representations over time, as does the enumeration mechanism used by adults in symbolic and non-symbolic tasks. q 2004 Elsevier B.V. All rights reserved. (shrink)
A preference method probed infants` perception of object motion on an inclined plane. Infants viewed videotaped events in which a ball rolled downward (or upward) while speeding up (or slowing down). Then infants were tested with events in which the ball moved in the opposite direction with appropriate or inappropriate acceleration. Infants aged 7 months, but not 5 months, looked longer at the test event with inappropriate acceleration, suggesting emerging sensitivity to gravity. A further study tested whether infants appreciate that (...) a stationary object released on an incline moves downward rather than upward; findings again were positive at 7 months and negative at 5 months. A final study provided evidence, nevertheless, that 5-monthold infants discriminate downward from upward motion and relate downward motion in videotaped events to downward motion in live events. Sensitivity to certain effects of gravity appears to develop in infancy. (shrink)
A new method was devised to test object permanence in young infants. Fivemonth-old infants were habituated to a screen that moved back and forth through a 180-degree arc, in the manner of a drawbridge. After infants reached habituation, a box was centered behind the screen. Infants were shown two test events: a possible event and an impossible event. In the possible event, the screen stopped when it reached the occluded box; in the impossible event, the screen moved through the space (...) occupied by the box. The results indicated that infants looked reliably longer at the impossible than at the possible event. This hnding suggested that infants (1) understood that the box continued to exist, in its same location, after it was occluded by the screen, and (2) expected the screen to stop against the occluded box and were surprised, or puzzled, when it failed to do so. A control experiment in which the box was placed next to the screen provided support for this interpretation of the results. Together, the results of these experiments indicate that, contrary to Piaget’s (1954) claims, infants as young as 5 months of age understand that objects continue to exist when occluded. The results, also indicate that 5-month-old infants realize that solid objects do not move through the space occupied by other solid objects. (shrink)
Four-month-old infants can perceive bimodally speciiied events. They respond to relationships between the optic and acoustic stimulation that carries information about an object. Infants can do this by detecting the temporal synchrony of an object’s sounds and its optically specified impacts. They are sensitive both to the common tempo and to the simultaneity of such sounds and visible impacts. These findings support the view that intermodal perception depends at least in part on the detection of invariant relationships in patterns of (...) light and sound. (shrink)
3-; much of this book attests, a wealth of research provides evidence that human infants have a core capacity for representing objects and their zotions. The environment contains a diversity of objects, however, with varied properties and behaviors. Objects such as pebbles and blocks are inert; tev move or change only in response to an external force. Objects such as utterflies and cars have internal mechanisms generating forces that can propel them. Self-propelled objects can be further differentiated, according to ine (...) nature and characteristic pattern of their motions and the circumstances fiat evoke them. To navigate successfully in this diverse and changing envitonment, perceivers and actors must categorize the objects around them appropriately, determining what kind of thing each object is and how it is Llcely to behave. (shrink)
Mature representations of number are built on a core system of numerical representation that connects to spatial representations in the form of a ‘mental number line’. The core number system is functional in early infancy, but little is known about the origins of the mapping of numbers onto space. Here we show that preverbal infants transfer the discrimination of an ordered series of numerosities to the discrimination of an ordered series of line lengths. Moreover, infants construct relationships between individual numbers (...) and line lengths that vary positively, but not between numbers and lengths that vary inversely. These findings provide evidence for an early developing predisposition to relate representations of numerical magnitude and spatial length. A central foundation of mathematics, science and technology therefore emerges prior to experience with language, symbol systems, or measurement devices. (shrink)
Previous research with adults suggests that a catalog of minimally counterintuitive concepts, which underlies supernatural or religious concepts, may constitute a cognitive optimum and is therefore cognitively encoded and culturally transmitted more successfully than either entirely intuitive concepts or maximally counterintuitive concepts. This study examines whether children's concept recall similarly is sensitive to the degree of conceptual counterintuitiveness (operationalized as a concept's number of ontological domain violations) for items presented in the context of a fictional narrative. Seven- to nine-year-old children (...) who listened to a story including both intuitive and counterintuitive concepts recalled the counterintuitive concepts containing one (Experiment 1) or two (Experiment 2), but not three (Experiment 3), violations of intuitive ontological expectations significantly more and in greater detail than the intuitive concepts, both immediately after hearing the story and 1 week later. We conclude that one or two violations of expectation may be a cognitive optimum for children: They are more inferentially rich and therefore more memorable, whereas three or more violations diminish memorability for target concepts. These results suggest that the cognitive bias for minimally counterintuitive ideas is present and active early in human development, near the start of formal religious instruction. This finding supports a growing literature suggesting that diverse, early-emerging, evolved psychological biases predispose humans to hold and perform religious beliefs and practices whose primary form and content is not derived from arbitrary custom or the social environment alone. (shrink)
Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)
Four-month-old infants sometimes can perceive the unity of a partly hidden object. In each of a series of experiments, infants were habituated to one object whose top and bottom were visible but whose center was occluded by a nearer object. They were then tested with a fully visible continuous object and with two fully visible object pieces with a gap where the occluder had been. Pattems of dishabituation suggested that infants perceive the boundaries of a partly hidden object by analyzing (...) the movements of its surfaces: infants perceived a connected object when its ends moved in a common translation behind the occluder. Infants do not appear to perceive a connected object by analyzing the colors and forms of surfaces: they did not perceive a connected object when its visible parts were stationary, its color was homogeneous, its edges were aligned, and its shape was simple and regular. These findings do not support the thesis, from gestalt psychology, that object perception first arises as a consequence of a tendency to perceive the simplest, most regular configuration, or the Piagetian thesis that object perception depends on the prior coordination of action. Perception of objects may depend on an inherent conception of what an object is. (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 À 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. Ó 2008 Elsevier B.V. All rights reserved. (shrink)
