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stimuli were presented with 20 deviant numerosities of a 2.0 ratio both larger and smaller. For example, out of the 232 trials, 16 dots were presented in varying size and distance but 10 of those trials had 8 dots, and 10 of those trials had 32 dots, making up the 20 deviant stimuli. The same applied to the blocks with 32 as the base numerosity. To ensure the adults and children were attending to the stimuli, they put 3 fixation points throughout the trial where the participant had to move a joystick to move forward. Their findings indicated that the adults in the experiment had significant activation of the IPS when viewing the deviant number stimuli, aligning with what was previously found in the aforementioned paragraph. In the 4 year olds, they found significant activation of the IPS to the deviant number stimuli, resembling the activation found in adults. There were some differences in the activations, with adults displaying more robust bilateral activation, where the 4 year olds primarily showed activation in their right IPS and activated 112 less voxels than the adults. This suggests that at age 4, children have an established mechanism of neurons in the IPS tuned for processing non-symbolic numerosities. Other studies have gone deeper into this mechanism in children and discovered that children do also represent approximate numbers on a
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were 4vs.12, 8vs.16, and 4vs.8. The auditory stimuli consisted of tones in different frequencies with a set number of tones, with some deviant trials where the tones were shorter but more numerous or longer and less numerous to account for duration and its potential confounds. After the auditory stimuli was presented with 2 minutes of familiarization, the visual stimuli was presented with a congruent or incongruent array of colorful dots with facial features. they remained on the screen until the infant looked away. They found that infants looked longer at the stimuli that matched the auditory tones, suggesting that the system for approximating non-symbolic number, even across modalities, is present in infancy. What is important to note across these three particular human studies on nonsymbolic numerosities is that it is present in infancy and develops over the lifetime. The honing of their approximation and number sense abilities as indicated by the improving Weber fractions across time, and usage of the left IPS to provide a wider berth for processing of computations and enumerations lend support for the claims that are made for a nonsymbolic number processing mechanism in human brains.
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anchoring effect, the precision effect, and the ease of computation effect respectively. The left-digit effect refers to the observation that people tend to incorrectly judge the difference between $ 4.00 and $ 2.99 to be larger than that between $ 4.01 and $ 3.00 because of anchoring on left-most digits. The precision effect reflects the influence of the representativeness of digit patterns on magnitude judgments. Larger magnitudes are usually rounded and therefore have many zeros, whereas smaller magnitudes are usually expressed as precise numbers; so relying on the representativeness of digit patterns can make people incorrectly judge a price of $ 391,534 to be more attractive than a price of $ 390,000. The ease of computation effect shows that magnitude judgments are based not only on the output of a mental computation, but also on its experienced ease or difficulty. Usually it is easier to compare two dissimilar magnitudes than two similar magnitudes; overuse of this heuristic can make people incorrectly judge the difference to be larger for pairs with easier computations, e.g. $ 5.00 minus $ 4.00, than for pairs with difficult computations, e.g. $ 4.97 minus $ 3.96.
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the display is taken away. Then, after a delay period of several seconds, a second display is presented. If the number on the second display match that from the first, the monkey has to release a lever. If it is different, the monkey has to hold the lever. Neural activity recorded during the delay period showed that neurons in the intraparietal sulcus and the frontal cortex had a "preferred numerosity", exactly as predicted by behavioral studies. That is, a certain number might fire strongly for four, but less strongly for three or five, and even less for two or six. Thus, we say that these neurons were "tuned" for specific quantities. Note that these neuronal responses followed
452:. Additionally, the inferotemporal cortex is implicated in processing the numerical shapes and symbols, necessary for calculations with Arabic digits. More current research has highlighted the networks involved with multiplication and subtraction tasks. Multiplication is often learned through rote memorization and verbal repetitions, and neuroimaging studies have shown that multiplication uses a left lateralized network of the inferior frontal cortex and the superior-middle temporal gyri in addition to the IPL and IPS. Subtraction is taught more with quantity manipulation and strategy use, more reliant upon the right IPS and the posterior parietal lobule.
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However, in the realm of number, they share many similarities. As identified in monkeys, neurons selectively tuned to number were identified in the bilateral intraparietal sulci and prefrontal cortex in humans. Piazza and colleagues investigated this using fMRI, presenting participants with sets of dots where they either had to make same-different judgments or larger-smaller judgments. The sets of dots consisted of base numbers 16 and 32 dots with ratios in 1.25, 1.5, and 2. Deviant numbers were included in some trials in larger or smaller amounts than the base numbers. Participants displayed similar activation patterns as Neider found in the monkeys. The
436:, Stanislas Dehaene and colleagues have suggested that these two parietal structures play complementary roles. The IPS is thought to house the circuitry that is fundamentally involved in numerical estimation, number comparison, and on-line calculation, or quantity processing (often tested with subtraction) while the IPL is thought to be involved in rote memorization, such as multiplication. Thus, a patient with a lesion to the IPL may be able to subtract, but not multiply, and vice versa for a patient with a lesion to the IPS. In addition to these parietal regions, regions of the
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521:. Such individuals report that numbers are mentally represented with a particular spatial layout; others experience numbers as perceivable objects that can be visually manipulated to facilitate calculation. Behavioral studies further reinforce the connection between numerical and spatial cognition. For instance, participants respond quicker to larger numbers if they are responding on the right side of space, and quicker to smaller numbers when on the left—the so-called "Spatial-Numerical Association of Response Codes" or
33:
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another. If, when the screen was lowered, infants were presented with only one Mickey (the "impossible event") they looked longer than if they were shown two
Mickeys (the "possible" event). Further studies by Karen Wynn and Koleen McCrink found that although infants' ability to compute exact outcomes only holds over small numbers, infants can compute approximate outcomes of larger addition and subtraction events (e.g., "5+5" and "10-5" events).
476:, also implicated in number, communicate in approximating number and it was found in both species that the parietal neurons of the IPS had short firing latencies, whereas the frontal neurons had longer firing latencies. This supports the notion that number is first processed in the IPS and, if needed, is then transferred to the associated frontal neurons in the
529:. Moreover, neuroimaging studies reveal that the association between number and space also shows up in brain activity. Regions of the parietal cortex, for instance, show shared activation for both spatial and numerical processing. These various lines of research suggest a strong, but flexible, connection between numerical and spatial cognition.
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in monkeys has also found neurons in the frontal cortex and in the intraparietal sulcus that respond to numbers. Andreas Nieder trained monkeys to perform a "delayed match-to-sample" task. For example, a monkey might be presented with a field of four dots, and is required to keep that in memory after
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Developmental psychology studies have shown that human infants, like non-human animals, have an approximate sense of number. For example, in one study, infants were repeatedly presented with arrays of (in one block) 16 dots. Careful controls were in place to eliminate information from "non-numerical"
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for further numerations and applications. Humans displayed
Gaussian curves in the tuning curves of approximate magnitude. This aligned with monkeys, displaying a similarly structured mechanism in both species with classic Gaussian curves relative to the increasingly deviant numbers with 16 and 32 as
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investigated abstract number representations in infants using a different paradigm than the previous researchers because of the nature and developmental stage of the infants. For infants, they examined abstract number with both auditory and visual stimuli with a looking-time paradigm. The sets used
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With an established mechanism for approximating non-symbolic number in both humans and primates, a necessary further investigation is needed to determine if this mechanism is innate and present in children, which would suggest an inborn ability to process numerical stimuli much like humans are born
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It is important to note that while primates have remarkably similar brains to humans, there are differences in function, ability, and sophistication. They make for good preliminary test subjects, but do not show small differences that are the result of different evolutionary tracks and environment.
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Similarly, researchers have set up hidden speakers in the
African savannah to test natural (untrained) behavior in lions. These speakers can play a number of lion calls, from 1 to 5. If a single lioness hears, for example, three calls from unknown lions, she will leave, while if she is with four of
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reviewed several studies showing that the three heuristics that manifest in many everyday judgments and decisions – anchoring, representativeness, and availability – also influence numerical cognition. They identify the manifestations of these heuristics in numerical cognition as: the left-digit
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set out to investigate this in 4 year old healthy, normally developing children in parallel with adults. A similar task to Piazza's was used in this experiment, without the judgment tasks. Dot arrays of varying size and number were used, with 16 and 32 as the base numerosities. in each block, 232
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showed that infants as young as five months are able to do very simple additions (e.g., 1 + 1 = 2) and subtractions (3 - 1 = 2). To demonstrate this, Wynn used a "violation of expectation" paradigm, in which infants were shown (for example) one Mickey Mouse doll going behind a screen, followed by
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in the field. He concluded that they have no need for counting in their everyday lives. Their hunters keep track of individual arrows with the same mental faculties that they use to recognize their family members. There are no known hunter-gatherer cultures that have a counting system in their
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Because of the numerous controls that were in place to rule out non-numerical factors, the experimenters infer that six-month-old infants are sensitive to differences between 8 and 16. Subsequent experiments, using similar methodologies showed that 6-month-old infants can discriminate numbers
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or Normal distribution with peak around 8 or 16 bar presses. When rats are more hungry, their bar-pressing behavior is more rapid, so by showing that the peak number of bar presses is the same for either well-fed or hungry rats, it is possible to disentangle time and number of bar presses. In
525:. This effect varies across culture and context, however, and some research has even begun to question whether the SNARC reflects an inherent number-space association, instead invoking strategic problem solving or a more general cognitive mechanism like
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her sisters, they will go and explore. This suggests that not only can lions tell when they are "outnumbered" but that they can do this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept.
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differing by a 2:1 ratio (8 vs. 16 or 16 vs. 32) but not by a 3:2 ratio (8 vs. 12 or 16 vs. 24). However, 10-month-old infants succeed both at the 2:1 and the 3:2 ratio, suggesting an increased sensitivity to numerosity differences with age.
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There is evidence that numerical cognition is intimately related to other aspects of thought – particularly spatial cognition. One line of evidence comes from studies performed on number-form
485:, with accuracy decreasing as the ratio between numbers became smaller. This supports the findings made by Neider in macaque monkeys and shows definitive evidence for an
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who only have number words up to five. PirahĂŁ adults are unable to mark an exact number of tallies for a pile of nuts containing fewer than ten items. Anthropologist
464:, as has been demonstrated for other sensory dimensions, and consistent with the ratio dependence observed for non-human animals' and infants' numerical behavior.
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parameters such as total surface area, luminance, circumference, and so on. After the infants had been presented with many displays containing 16 items, they
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language. The mental and lingual capabilities for numeracy are tied to the development of agriculture and with it large numbers of indistinguishable items.
357:, or stopped looking as long at the display. Infants were then presented with a display containing 8 items, and they looked longer at the novel display.
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Fischer, M. H.; Mills, R. A.; Shaki, S. (April 2010). "How to cook a SNARC: Number placement in text rapidly changes spatial–numerical associations".
392:, where language-based natural numbers can be exact. Without language, only numbers 1 to 4 are believed to have an exact representation, through the
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suggested that a child innately has the concept of natural number, and only has to map this onto the words used in her language. Carey (
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The numeracy of indigenous peoples is studied to identify universal aspects of numerical cognition in humans. Notable examples include the
327:"). For example, when a rat is trained to press a bar 8 or 16 times to receive a food reward, the number of bar presses will approximate a
2055:
Núñez, R.; Doan, D.; Nikoulina, A. (August 2011). "Squeezing, striking, and vocalizing: Is number representation fundamentally spatial?".
396:. One promising approach is to see if cultures that lack number words can deal with natural numbers. The results so far are mixed (e.g.,
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Nieder, A.; Freedman, D. J.; Miller, E. K. (2002). "Representation of the quantity of visual items in the primate prefrontal cortex".
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What metaphorical capacities and processes allow us to extend our numerical understanding into complex domains such as the concept of
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Thomas, Manoj; Morwitz, Vicki (2009). "Heuristics in
Numerical Cognition: Implications for Pricing". In Rao, Vithala R. (ed.).
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is an open-access, free-to-publish, online-only
Journal outlet specifically for research in the domain of numerical cognition.
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A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have an
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Hubbard, E. M.; Piazza, M.; Pinel, P.; Dehaene, S. (June 2005). "Interactions between number and space in parietal cortex".
1164:; Reeve, R. (2008). "Verbal Counting and Spatial Strategies in Numerical Tasks : Evidence From Indigenous Australia".
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There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the classic
2242:; Riviere, D.; Le Bihan, D. (2001). "Modulation of parietal activation by semantic distance in a number comparison task".
217:
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McComb, K.; Packer, C.; Pusey, A. (1994). "Roaring and numerical assessment in contests between groups of female lions,
643: – ability to count objects in order and to understand the greater than and less than relationships between numbers
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652: – non-symbolic cognitive system that supports the representation of numerical values from zero to three or four
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Several consumer psychologists have also studied the heuristics that people use in numerical cognition. For example,
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are also active in calculation tasks. These activations overlap with regions involved in language processing such as
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Dehaene, S.; Bossini, S.; Giraux, P. (September 1993). "The mental representation of parity and number magnitude".
2295:"Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments"
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Berteletti, I.; Lucangeli, D.; Piazza, M.; Dehaene, S.; Zorzi, M. (2010). "Numerical estimation in preschoolers".
245:. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in
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1922:"Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex"
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1746:"Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children"
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Walsh, V. (November 2003). "A theory of magnitude: common cortical metrics of time, space and quantity".
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Piazza, M.; Eger, E. (2016). "Neural foundations and functional specificity of number representations".
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Khanum, S.; Hanif, R.; Spelke, E. S.; Berteletti, I.; Hyde, D. C. (20 October 2016).
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Nieder, A. (2005). "Counting on neurons: The neurobiology of numerical competence".
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What are the neural bases of these abilities, both in humans and in non-humans?
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Núñez, R. (2009). "Numbers and
Arithmetic: Neither Hardwired Nor Out There".
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How do these capacities underlie our ability to perform complex calculations?
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2145:"Tuning curves for approximate numerosity in the human intraparietal sulcus"
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How do infants acquire an understanding of numbers (and how much is inborn)?
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1290:"Functional Imaging of Numerical Processing in Adults and 4-y-Old Children"
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1040:"Evidence for Two Numerical Systems That Are Similar in Humans and Guppies"
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388:) disagreed, saying that these systems can only encode large numbers in an
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1965:"A parieto-frontal network for visual numerical information in the monkey"
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619: – Innate ability to detect differences in magnitude without counting
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How do humans associate linguistic symbols with numerical quantities?
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which successfully discriminated between 1 and 4 other individuals.
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1288:; Brannon, E. M.; Carter, E. J.; Pelphrey, K. A. (11 April 2006).
261:. This discipline, although it may interact with questions in the
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Human neuroimaging studies have demonstrated that regions of the
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2191:"Exact an Approximate Arithmetic in an Amazonian Indigene Group"
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Izard, V.; Sann, C.; Spelke, E. S.; Streri, A. (23 June 2009).
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Dehaene, Stanislas (1992). "Varieties of numerical abilities".
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Topics included in the domain of numerical cognition include:
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that studies the cognitive, developmental and neural bases of
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The Stuff of
Thought: Language as a Window Into Human Nature
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1093:"Strategies in subtraction problem solving in children"
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Pages displaying short descriptions of redirect targets
544:, missing in the usual decimal system, is expressed by
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625: – Finding the number of elements of a finite set
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Relations between number and other cognitive processes
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502:, aligning with the claims made by Piazza in adults.
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
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682: – One of the four basic arithmetic operations
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1463:The number sense: How the mind creates mathematics
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1091:Barrouillet, P.; Mignon, M.; Thevenot, C. (2008).
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1969:Proceedings of the National Academy of Sciences
1689:Proceedings of the National Academy of Sciences
1623:
1507:; Spelke, E. (2004). "Core systems of number".
1202:Proceedings of the National Academy of Sciences
573:who have no words for specific numbers and the
406:Butterworth, Reeve, Reynolds & Lloyd (2008)
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495:Cantlon, Brannon, Carter & Pelphrey (2006)
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1196:; Reeve, R.; Reynolds, F.; Lloyd, D. (2008).
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16:Study of numerical and mathematical abilities
2189:; Lemer, C.; Izard, V.; Dehaene, S. (2004).
1962:
1919:
1631:. Cambridge Mass: Harvard University Press.
877:
862:
2441:
1685:"Newborn infants perceive abstract numbers"
1343:"Bootstrapping and the origins of Concepts"
1258:Journal of Experimental Psychology: General
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631: – Process of finding an approximation
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412:Neuroimaging and neurophysiological studies
336:has been shown, for example in the case of
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481:well as habituation. The results followed
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506:Izard, Sann, Spelke & Streri (2009)
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1251:"Cognitive Arithmetic Across Cultures"
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2289:Pinel, P.; Piazza, M.; Le Bihan, D.;
2015:
1467:. New York: Oxford University Press.
1378:"Where our number concepts come from"
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385:
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848:Nieder, Freedman & Miller (2002)
487:approximate number logarithmic scale
278:non-human animals process numerosity
1629:The Child's Understanding of Number
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332:addition, in a few species the
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402:Butterworth & Reeve (2008)
364:In another series of studies,
265:, is primarily concerned with
1:
2312:10.1016/s0896-6273(04)00107-2
1939:10.1016/s0896-6273(02)01144-3
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650:Parallel individuation system
394:parallel individuation system
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334:parallel individuation system
192:Parallel individuation system
2453:Where mathematics comes from
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604:
558:Thomas & Morwitz (2009)
546:signed-digit representation
493:ready to process language.
321:approximate sense of number
10:
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1521:10.1016/j.tics.2004.05.002
1360:10.1162/001152604772746701
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532:Modification of the usual
2457:. New York: Basic Books.
1178:10.1080/09515080802284597
812:Campbell & Xue (2001)
617:Approximate number system
263:philosophy of mathematics
187:Approximate number system
2502:Developmental psychology
2040:10.1162/biot.2009.4.1.68
1166:Philosophical Psychology
1126:Developmental Psychology
890:Berteletti et al. (2010)
800:Piazza & Eger (2016)
686:
565:Ethnolinguistic variance
444:and regions involved in
426:inferior parietal lobule
251:developmental psychology
2215:10.1126/science.1102085
1990:10.1073/pnas.0402239101
1897:10.1126/science.1072493
1710:10.1073/pnas.0812142106
1394:10.5840/jphil2009106418
1223:10.1073/pnas.0806045105
2492:Cognitive neuroscience
2266:10.1006/nimg.2001.0913
1823:10.1006/anbe.1994.1052
534:decimal representation
303:or the concept of the
233:is a subdiscipline of
2487:Cognitive linguistics
1583:"Visualised Numerals"
1382:Journal of Philosophy
914:Hubbard et al. (2005)
374:nature versus nurture
348:Developmental studies
259:cognitive linguistics
2497:Cognitive psychology
1038:Agrillo, C. (2012).
902:Khanum et al. (2016)
752:Piazza et al. (2004)
470:intraparietal sulcus
422:intraparietal sulcus
247:cognitive psychology
25:Cognitive psychology
2207:2004Sci...306..499P
1981:2004PNAS..101.7457N
1889:2002Sci...297.1708N
1883:(5587): 1708–1711.
1762:2016PLoSO..1164436K
1701:2009PNAS..10610382I
1695:(25): 10382–10385.
1599:1880Natur..21..494G
1546:Brain and Cognition
1214:2008PNAS..10513179B
1208:(35): 13179–13184.
1056:2012PLoSO...731923A
776:Pinel et al. (2004)
764:Pinel et al. (2001)
527:conceptual metaphor
315:Comparative studies
231:Numerical cognition
173:Numerical cognition
65:Pattern recognition
1457:Dehaene, Stanislas
1376:Carey, S. (2009).
1341:Carey, S. (2004).
398:Pica et al. (2004)
55:Object recognition
2464:978-0-465-03770-4
2201:(5695): 499–503.
2018:Biological Theory
1975:(19): 7457–7462.
1474:978-0-19-513240-3
536:was advocated by
500:logarithmic scale
478:prefrontal cortex
474:prefrontal cortex
323:(referred to as "
235:cognitive science
228:
227:
2514:
2468:
2456:
2447:Nuñez, Rafael E.
2430:
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2332:
2314:
2285:
2259:
2250:(5): 1013–1026.
2234:
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2117:
2094:Neuropsychologia
2088:
2051:
2033:
2012:
2002:
1992:
1959:
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1916:
1871:
1834:
1811:Animal Behaviour
1801:
1791:
1773:
1756:(10): e0164436.
1740:
1730:
1712:
1679:
1654:(1–2): 435–448.
1642:
1620:
1610:
1608:10.1038/021494e0
1593:(543): 494–495.
1577:
1540:
1499:
1478:
1466:
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1415:
1405:
1372:
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1337:
1327:
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1138:10.1037/a0017887
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659:Plant arithmetic
655:
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579:Napoleon Chagnon
575:Munduruku people
420:, including the
220:
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60:Face recognition
35:
21:
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2522:
2521:
2517:
2516:
2515:
2513:
2512:
2511:
2472:
2471:
2465:
2438:
2436:Further reading
2433:
2405:(11): 483–488.
2382:
2361:
2359:
2357:
2031:10.1.1.610.6016
1852:10.1038/nrn1626
1660:10.1038/nrn1684
1639:
1503:Feigenson, L.;
1475:
1253:
1194:Butterworth, B.
1172:(21): 443–457.
1162:Butterworth, B.
1033:
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542:complementation
540:. The sense of
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434:neuropsychology
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164:Problem solving
159:Decision making
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2443:Lakoff, George
2437:
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2432:
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2380:
2367:
2356:978-0143114246
2355:
2337:Pinker, Steven
2333:
2305:(6): 983–993.
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2235:
2183:
2155:(3): 547–555.
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2063:(2): 225–235.
2052:
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1373:
1338:
1286:Cantlon, J. F.
1282:
1264:(2): 299–315.
1246:
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1132:(2): 545–551.
1121:
1103:(4): 233–251.
1088:
1034:
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1002:
998:Dehaene (1992)
990:
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711:Agrillo (2012)
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446:working memory
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2347:Penguin Books
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2247:
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2198:
2194:
2152:
2148:
2115:11572/114302
2097:
2093:
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1807:Panthera leo
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600:Journal link
595:
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523:SNARC effect
519:synaesthetes
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455:Single-unit
454:
442:Broca's area
438:frontal lobe
430:neuroimaging
415:
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318:
307:in calculus?
271:
255:neuroscience
230:
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128:Metalanguage
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2291:Dehaene, S.
2240:Dehaene, S.
2238:Pinel, P.;
2141:Dehaene, S.
2100:: 257–273.
1505:Dehaene, S.
1300:(5). e125.
1024:, p. .
1012:, p. .
701:, p. .
680:Subtraction
538:John Colson
489:in humans.
483:Weber's Law
462:Weber's law
269:questions.
243:mathematics
2476:Categories
2362:2012-11-08
2244:NeuroImage
1031:References
674:Subitizing
629:Estimation
366:Karen Wynn
355:habituated
325:numerosity
41:Perception
2482:Cognition
2390:807401627
2252:CiteSeerX
2057:Cognition
2026:CiteSeerX
1780:1932-6203
1719:0027-8424
1421:Cognition
1353:: 59–68.
1316:1545-7885
450:attention
267:empirical
154:Reasoning
144:Cognition
105:Long-term
95:Emotional
74:Attention
2507:Quantity
2449:(2000).
2419:14585444
2339:(2008).
2321:15046729
2293:(2004).
2282:17633857
2274:11697933
2231:10653745
2223:15486303
2187:Pica, P.
2171:15504333
2143:(2004).
2132:22957569
2124:26403660
2085:16362508
2077:21640338
2009:15123797
1948:12526780
1913:20871267
1905:12215649
1868:14578049
1860:15711599
1831:53183852
1798:27764117
1750:PLOS ONE
1737:19520833
1668:15928716
1574:19626981
1566:19917517
1537:17313189
1529:15242690
1459:(1997).
1449:24382907
1412:23136450
1369:54493789
1347:Daedalus
1334:16594732
1278:11409105
1242:18757729
1146:20210512
1117:18241880
1084:22355405
1044:PLOS ONE
623:Counting
611:Addition
605:See also
583:Yanomami
472:and the
376:debate.
329:Gaussian
297:infinity
135:Thinking
121:Language
100:Learning
2427:1761795
2329:9372570
2203:Bibcode
2195:Science
2179:6288232
2048:1707771
1977:Bibcode
1956:5704850
1885:Bibcode
1877:Science
1789:5072670
1758:Bibcode
1728:2700913
1697:Bibcode
1676:1465072
1617:4074444
1595:Bibcode
1441:1511583
1403:3489488
1325:1431577
1233:2527348
1210:Bibcode
1186:2662436
1154:8496112
1075:3280231
1052:Bibcode
338:guppies
276:How do
239:numbers
149:Concept
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299:, the
81:Memory
50:Visual
2423:S2CID
2325:S2CID
2278:S2CID
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