Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a (...) few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 (...) or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (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)

Only human beings have a rich conceptual repertoire with concepts like tort, entropy, Abelian group, mannerism, icon and deconstruction. How have humans constructed these concepts? And once they have been constructed by adults, how do children acquire them? While primarily focusing on the second question, in The Origin of Concepts , Susan Carey shows that the answers to both overlap substantially. Carey begins by characterizing the innate starting point for conceptual development, namely systems of core cognition. Representations of core cognition (...) are the output of dedicated input analyzers, as with perceptual representations, but these core representations differ from perceptual representations in having more abstract contents and richer functional roles. Carey argues that the key to understanding cognitive development lies in recognizing conceptual discontinuities in which new representational systems emerge that have more expressive power than core cognition and are also incommensurate with core cognition and other earlier representational systems. Finally, Carey fleshes out Quinian bootstrapping, a learning mechanism that has been repeatedly sketched in the literature on the history and philosophy of science. She demonstrates that Quinian bootstrapping is a major mechanism in the construction of new representational resources over the course of childrens cognitive development. Carey shows how developmental cognitive science resolves aspects of long-standing philosophical debates about the existence, nature, content, and format of innate knowledge. She also shows that understanding the processes of conceptual development in children illuminates the historical process by which concepts are constructed, and transforms the way we think about philosophical problems about the nature of concepts and the relations between language and thought. (shrink)

The current investigations coordinate math cognition and cultural approaches to numeric thinking to examine the linkages between numeric and spatial processes, and how these linkages are modified by the cultural artifact of writing. Previous research in the adult numeric cognition literature has shown that English monoliterates have a spatialised mental number line which is oriented from left-to-right with smaller magnitudes associated with the left side of space and larger magnitudes are associated with the right side of space. These associations between (...) number and space have been termed the Spatial Numeric Association Response Code Effect. The current study investigates the spatial orientation of the mental number line in the following groups: English monoliterates, Arabic monoliterates who use only the right-left writing system, Arabic-English biliterates, and illiterate Arabic speakers who only read numerals. Current results indicate, for the first time, a Reverse SNARC effect for Arabic monoliterates, such that the mental number line had a right-to-left directionality. Furthermore, a weakened Reverse SNARC was observed for Arabic-English biliterates, and no effect was observed among Illiterate Arabic speakers. These findings are especially notable since left-right biases are neurologically supported and are observed in pre-literate children regardless of which writing system is used by adults. The broader implications of how cultural artifacts affect basic numeric cognition will be discussed. (shrink)

This article described three heuristics that are employed in making judgements under uncertainty: representativeness, which is usually employed when people are asked to judge the probability that an object or event A belongs to class or process B; availability of instances or scenarios, which is often employed when people are asked to assess the frequency of a class or the plausibility of a particular development; and adjustment from an anchor, which is usually employed in numerical prediction when a relevant value (...) is available. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. A better understanding of these heuristics and of the biases to which they lead could improve judgements and decisions in situations of uncertainty. (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)

Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate f'mal questions to incomplete word problems, and (3) modeling performance pattems using a computer simulation. Solution performance was found to be systematically related to recall and question generation performance. (...) Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution "errors" were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error pattems were successfully simulated by manipulating a computer model’s language comprehension strategies, as opposed to its knowledge of logical set relations. o was.. (shrink)

Various defenses of amodal symbol systems are addressed, including amodal symbols in sensory-motor areas, the causal theory of concepts, supramodal concepts, latent semantic analysis, and abstracted amodal symbols. Various aspects of perceptual symbol systems are clarified and developed, including perception, features, simulators, category structure, frames, analogy, introspection, situated action, and development. Particular attention is given to abstract concepts, language, and computational mechanisms.

Cultural differences in the amount of time spent on homework and in beliefs and attitudes about homework were investigated through interviews with more than 3,500 elementary school children, their mothers, and their teachers. The children lived in 5 cities: Beijing, Chicago, Minneapolis, Sendai, and Taipei. Chinese children were assigned more homework and spent more time on homework than Japanese children, who in turn were assigned more and spent more time on homework than American children. Chinese children also received more help (...) from family members with their homework than American and Japanese children. Chinese children were found to have more positive attitudes about homework than American children; Japanese children's attitudes were between those of the Chinese and American children. Relations between amount of time spent on homework by children, amount of time parents spent assisting their children with homework, and children's achievement were also explored. The views of both parents and teachers about the value of homework are discussed. (shrink)

The scientist Richard Lovell Edgeworth, educated at Trinity College, Dublin, and Oxford, was a Member of the Lunar Society of Birmingham, where he exchanged ideas with other scientists, including James Watt, and was known for his significant mechanical inventions. However, Edgeworth's real interest was education: in this 1788 two-volume work, written with his daughter, the poet Maria Edgeworth, he draws on his own experience of raising twenty children, from which the work derives its authority and innovative character. The work was (...) very influential, and led to his Essays on Professional Education. The two volumes discuss the theories of philosophers and educationalists, while in general arguing for the importance and formative character of early childhood experiences. Volume 1 deals with different areas of childhood education, including play, learning, and obedience and good behaviour. (shrink)

Wales uses languages with both regular (Welsh) and irregular (English) counting systems. Three groups of 6- and 8-year-old Welsh children with varying degrees of exposure to the Welsh language—those who spoke Welsh at both home and school; those who spoke Welsh only at home; and those who spoke only English—were given standardized tests of arithmetic and a test of understanding representations of two-digit numbers. Groups did not differ on the arithmetic tests, but both groups of Welsh speakers read and compared (...) 2-digit numbers more accurately than monolingual English children. A similar study was carried out with Tamil/English bilingual children in England. The Tamil counting system is more transparent than English but less so than Welsh or Chinese. Tamil-speaking children performed better than monolingual English-speaking children on one of the standardized arithmetic tests but did not differ in their comparison of two-digit numbers. Reasons for the findings are discussed. (shrink)

Small numbers are spontaneously associated with left space and larger numbers with right space (the SNARC effect), for example when classifying numbers by parity. This effect is often attributed to reading habits but a causal link has so far never been documented. We report that bilingual Russian-Hebrew readers show a SNARC effect after reading Cyrillic script (from left-to-right) that is significantly reduced after reading Hebrew script (from right-to-left). In contrast, they have similar SNARC effects after listening to texts in either (...) language. These results support the view that spatially directional scanning habits contribute to the spatial association of numbers but also emphasize its flexibility. (shrink)

A decade of heightened emphasis in the United States on mathematics and science education has had little influence on academic achievement or parental attitudes. American elementary school children in 1990 lagged behind their Chinese and Japanese peers to as great a degree as they did in 1980. Comparison of the performance of elementary and secondary school students between 1980 and 1990 reveals a decline from first to eleventh grade in the relative position of American students in mathematics. Parental satisfaction with (...) American students' achievement and education remains high and standards remain low. Innate ability continues to be emphasized by Americans as a basis for achievement. American eleventh graders report more indications of stress than do their Chinese and Japanese counterparts. (shrink)

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)