View source for Numerical cognition

{{Short description|Study of numerical and mathematical abilities}}
{{use dmy dates|cs1-dates=ly|date=September 2021}}
{{Cognitive}}
'''Numerical cognition''' is a subdiscipline of [[cognitive science]] that studies the cognitive, developmental and neural bases of [[number]]s and [[mathematics]]. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in [[cognitive psychology]], [[developmental psychology]], [[neuroscience]] and [[cognitive linguistics]]. This discipline, although it may interact with questions in the [[philosophy of mathematics]], is primarily concerned with [[empirical]] questions.

Topics included in the domain of numerical cognition include:
*How do [[Number sense in animals|non-human animals process numerosity]]?
*How do infants acquire an understanding of numbers (and how much is inborn)?
*How do humans associate linguistic symbols with numerical quantities?
*How do these capacities underlie our ability to perform complex calculations?
*What are the neural bases of these abilities, both in humans and in non-humans?
*What metaphorical capacities and processes allow us to extend our numerical understanding into complex domains such as the concept of [[infinity]], the [[infinitesimal]] or the concept of the [[limit (mathematics)|limit]] in calculus?
* Heuristics in numerical cognition

== Comparative studies ==
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have an [[approximate number system|approximate sense of number]] (referred to as "[[Number sense in animals|numerosity]]").{{sfnp|Dehaene|1997|p={{pn|date=September 2021}}}} 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 [[Gaussian distribution|Gaussian]] 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 addition, in a few species the [[parallel individuation system]] has been shown, for example in the case of [[guppy|guppies]] which successfully discriminated between 1 and 4 other individuals.{{sfnp|Agrillo|2012}}

Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions.{{sfnp|McComb|Packer|Pusey|1994}} 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 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.

== Developmental studies ==
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" parameters such as total surface area, luminance, circumference, and so on. After the infants had been presented with many displays containing 16 items, they [[Habituation|habituated]], 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.

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 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.{{sfnp|Feigenson|Dehaene|Spelke|2004}}

In another series of studies, [[Karen Wynn]] 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 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).

There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the classic [[nature versus nurture]] debate. {{harvp|Gelman|Gallistel|1978}} suggested that a child innately has the concept of natural number, and only has to map this onto the words used in her language. {{harvs|txt|last=Carey|year1=2004|year2=2009}} disagreed, saying that these systems can only encode large numbers in an [[approximate number system|approximate way]], where language-based natural numbers can be exact. Without language, only numbers 1 to 4 are believed to have an exact representation, through the [[parallel individuation system]]. 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., {{harvp|Pica|Lemer|Izard|Dehaene|2004}}); {{harvp|Butterworth|Reeve|2008}}, {{harvp|Butterworth, Reeve, Reynolds & Lloyd|2008}}).

== Neuroimaging and neurophysiological studies ==
Human neuroimaging studies have demonstrated that regions of the [[parietal lobe]], including the [[intraparietal sulcus]] (IPS) and the [[inferior parietal lobule]] (IPL) are activated when subjects are asked to perform calculation tasks. Based on both human [[neuroimaging]] and [[neuropsychology]], 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,{{sfnp|Piazza|Izard|Pinel|Le Bihan|2004}} number comparison,{{sfnp|Pinel|Dehaene|Riviere|Le Bihan|2001}}{{sfnp|Pinel|Piazza|Le Bihan|Dehaene|2004}} 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.{{sfnp|Dehaene|1997}} 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 [[frontal lobe]] are also active in calculation tasks. These activations overlap with regions involved in language processing such as [[Broca's area]] and regions involved in [[working memory]] and [[attention]].  Additionally, the inferotemporal cortex is implicated in processing the numerical shapes and symbols, necessary for calculations with Arabic digits.{{sfnp|Piazza|Eger|2016}} 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.{{sfnp|Campbell|Xue|2001}} Subtraction is taught more with quantity manipulation and strategy use, more reliant upon the right IPS and the posterior parietal lobule.{{sfnp|Barrouillet|Mignon|Thevenot|2008}}

Single-unit [[neurophysiology]] 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.{{sfnp|Nieder|2005}}{{sfnp|Nieder|Freedman|Miller|2002}}{{sfnp|Nieder|Miller|2004}} For example, a monkey might be presented with a field of four dots, and is required to keep that in memory after 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 [[Weber-Fechner law|Weber's law]], as has been demonstrated for other sensory dimensions, and consistent with the ratio dependence observed for non-human animals' and infants' numerical behavior.{{sfnp|Nieder|Miller|2003}}

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. 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{{sfnp|Piazza|Izard|Pinel|Le Bihan|2004}} 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 Nieder found in the monkeys.{{sfnp|Nieder|Miller|2003}} The [[intraparietal sulcus]] and the [[prefrontal cortex]], 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 [[prefrontal cortex]] 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 well as habituation. The results followed [[Weber's Law]], with accuracy decreasing as the ratio between numbers became smaller. This supports the findings made by Nieder in macaque monkeys{{sfnp|Nieder|Miller|2004}} and shows definitive evidence for an [[approximate number system|approximate number logarithmic scale]] in humans.{{sfnp|Berteletti|Lucangeli|Piazza|Dehaene|2010}}{{sfnp|Khanum|Hanif|Spelke|Berteletti|2016}}

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 ready to process language. {{harvp|Cantlon, Brannon, Carter & Pelphrey|2006}} set out to investigate this in 4 year old healthy, normally developing children in parallel with adults. A similar task to Piazza's{{sfnp|Piazza|Izard|Pinel|Le Bihan|2004}} 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 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 [[logarithmic scale]], aligning with the claims made by Piazza in adults.

{{harvp|Izard, Sann, Spelke & Streri|2009}} 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 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.

== Relations between number and other cognitive processes ==
There is evidence that numerical cognition is intimately related to other aspects of thought – particularly spatial cognition.{{sfnp|Hubbard|Piazza|Pinel|Dehaene|2005}} One line of evidence comes from studies performed on number-form [[Synaesthesia|synaesthetes]].{{sfnp|Galton|1880}} 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 [[Spatial-numerical association of response codes|SNARC effect]].{{sfnp|Dehaene|Bossini|Giraux|1993}} This effect varies across culture and context,{{sfnp|Fischer|Mills|Shaki|2010}} however, and some research has even begun to question whether the SNARC reflects an inherent number-space association,{{sfnp|Núñez|Doan|Nikoulina|2011}} instead invoking strategic problem solving or a more general cognitive mechanism like [[conceptual metaphor]].{{sfnp|Walsh|2003}}{{sfnp|Núñez|2009}} 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.{{sfnp|Dehaene|1992}} These various lines of research suggest a strong, but flexible, connection between numerical and spatial cognition.

Modification of the usual [[decimal representation]] was advocated by [[John Colson]]. The sense of [[complement (set theory)|complementation]], missing in the usual decimal system, is expressed by [[signed-digit representation]].

==Heuristics in numerical cognition==
Several consumer psychologists have also studied the heuristics that people use in numerical cognition. For example, {{harvp|Thomas|Morwitz|2009}} 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 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.{{sfnp|Thomas|Morwitz|2009|p={{pn|date=September 2021}}}}

==Ethnolinguistic variance==

The numeracy of indigenous peoples is studied to identify universal aspects of numerical cognition in humans. Notable examples include the [[Pirahã people]] who have no words for specific numbers and the [[Munduruku people]] who only have number words up to five. Pirahã adults are unable to mark an exact number of tallies for a pile of nuts containing fewer than ten items. Anthropologist [[Napoleon Chagnon]] spent several decades studying the [[Yanomami]] 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 language. The mental and lingual capabilities for numeracy are tied to the development of agriculture and with it large numbers of indistinguishable items.{{sfnp|Pinker|2008|p={{pn|date=September 2021}}}}

==Research outlet==
The ''Journal of Numerical Cognition'' is an open-access, free-to-publish, online-only Journal outlet specifically for research in the domain of numerical cognition. [http://jnc.psychopen.eu Journal link]

==See also==

* {{annotated link|Addition}}
* {{annotated link|Approximate number system}}
* {{annotated link|Counting}}
* {{annotated link|Estimation}}
* {{annotated link|Numerosity adaptation effect}}
* {{annotated link|Ordinal numerical competence}}
* {{annotated link|Parallel individuation system}}
* {{annotated link|Plant arithmetic}}
* {{annotated link|The problem of the speckled hen}}
* {{annotated link|Subitizing}}
* {{annotated link|Subtraction}}

== Notes ==
{{Reflist|24em}}

== References ==
{{refbegin|35em}}
*{{cite journal |last=Agrillo |first=C. |year=2012 |title=Evidence for Two Numerical Systems That Are Similar in Humans and Guppies |journal=PLOS ONE |volume=7 |issue=2 |doi=10.1371/journal.pone.0031923 |at=e31923 |pmid=22355405 |pmc=3280231 |bibcode=2012PLoSO...731923A|doi-access=free }}
*{{cite journal |last1=Barrouillet |first1=P. |last2=Mignon |first2=M. |last3=Thevenot |first3=C. |year=2008 |title=Strategies in subtraction problem solving in children |journal=Journal of Experimental Child Psychology |volume=99 |issue=4 |pages=233–251 |doi=10.1016/j.jecp.2007.12.001 |pmid=18241880 |url=https://archive-ouverte.unige.ch/unige:88295}}
*{{Cite journal |last1=Berteletti |first1=I. |last2=Lucangeli |first2=D. |last3=Piazza |first3=M. |last4=Dehaene |first4=S. |last5=Zorzi |first5=M. |year=2010 |title=Numerical estimation in preschoolers |journal=Developmental Psychology |volume=46 |issue=2 |pages=545–551 |doi=10.1037/a0017887 |pmid=20210512 |s2cid=8496112 }}
*{{Cite journal
 | last1 = Butterworth | first1 = B. | last2= Reeve | first2= R.
 | author-link = Brian Butterworth
 | year= 2008
 | title= Verbal Counting and Spatial Strategies in Numerical Tasks : Evidence From Indigenous Australia
 | journal=  Philosophical Psychology | issue=21 | volume= 4
 | pages= 443–457 | doi=10.1080/09515080802284597| s2cid = 2662436 }}
*{{Cite journal
| doi = 10.1073/pnas.0806045105
| volume = 105
| issue = 35
| pages = 13179–13184
| first1 = B.
|  last1 = Butterworth
| author-link = Brian Butterworth 
| first2 = R.
|  last2 = Reeve
| first3 = F.
|  last3 = Reynolds
| first4 = D.
|  last4 = Lloyd
| title = Numerical thought with and without words: Evidence from indigenous Australian children
| journal = Proceedings of the National Academy of Sciences
| year = 2008
| pmid=18757729
| pmc=2527348
| bibcode = 2008PNAS..10513179B
| ref = {{harvid|Butterworth, Reeve, Reynolds & Lloyd|2008}}
| doi-access = free
}}
*{{cite journal |last1=Campbell |first1=J.I.D. |last2=Xue |first2=Q. |year=2001 |title=Cognitive Arithmetic Across Cultures |journal=Journal of Experimental Psychology: General |volume=130 |issue=2 |pages=299–315 |doi=10.1037/0096-3445.130.2.299 |pmid=11409105 |url=http://www.apa.org/news/press/releases/xge1302299.pdf}}
*{{Cite journal |last1=Cantlon |first1=J. F. |author-link1=Jessica F. Cantlon |last2=Brannon |first2=E. M. |last3=Carter |first3=E. J. |last4=Pelphrey |first4=K. A. |date=2006-04-11 |title=Functional Imaging of Numerical Processing in Adults and 4-y-Old Children |journal=PLOS Biology |volume=4 |issue=5 |at=e125 |doi=10.1371/journal.pbio.0040125 |pmid=16594732 |pmc=1431577 |issn=1545-7885 |ref={{harvid|Cantlon, Brannon, Carter & Pelphrey|2006}} |doi-access=free }}
*{{Cite journal
 | last = Carey | first = S. | year= 2004
 | title= Bootstrapping and the origins of Concepts
 | journal= Daedalus  | volume = 133 | pages= 59–68| doi = 10.1162/001152604772746701 | s2cid = 54493789 | url = http://nrs.harvard.edu/urn-3:HUL.InstRepos:5109360 | doi-access = free }}
*{{Cite journal
 | last = Carey | first = S. | year= 2009
 | title= Where our number concepts come from
 | journal= Journal of Philosophy | issue = 4 | volume= 106 | pages= 220–254| doi = 10.5840/jphil2009106418 | pmid = 23136450 | pmc = 3489488 }}
*{{cite journal |last=Dehaene |first=Stanislas |title=Varieties of numerical abilities |journal=Cognition |year=1992 |volume=44 |issue=1–2 |pages=1–42 |doi=10.1016/0010-0277(92)90049-N|pmid=1511583|s2cid=24382907}}
*{{Cite book
 |last       = Dehaene
 |first      = Stanislas
 |author-link = Stanislas Dehaene
 |year       = 1997
 |title      = The number sense: How the mind creates mathematics
 |place      = New York
 |publisher  = Oxford University Press
 |isbn       = 978-0-19-513240-3
 |url        = https://archive.org/details/numbersensehowmi0000deha
 |url-access = registration
}}
*{{cite journal |last1=Dehaene |first1=S. |last2=Bossini |first2=S. |last3=Giraux |first3=P. |title=The mental representation of parity and number magnitude |journal=Journal of Experimental Psychology |date=September 1993 |volume=122 |issue=3 |pages=371–396 |doi=10.1037/0096-3445.122.3.371}}
*{{Cite journal
 | last1 = Feigenson | first1 = L. | last2= Dehaene | first2= S.
 | author-link2= Stanislas Dehaene
 | last3= Spelke | first3= E.
 | year= 2004
 | title= Core systems of number
 | journal= Trends in Cognitive Sciences | issue=7 | volume= 8
 | pages= 307–314 | doi=10.1016/j.tics.2004.05.002 | pmid=15242690| s2cid = 17313189 }}
*{{cite journal |last1=Fischer |first1=M. H. |last2=Mills |first2=R. A. |last3=Shaki |first3=S. |date=April 2010 |title=How to cook a SNARC: Number placement in text rapidly changes spatial–numerical associations |journal=Brain and Cognition |volume=72 |issue=3 |pages=333–336 |doi=10.1016/j.bandc.2009.10.010 |pmid=19917517 |s2cid=19626981}}
*{{cite journal |last=Galton |first=Francis |title=Visualised Numerals |journal=Nature |date=25 March 1880 |volume=21 |issue=543 |pages=494–495 |doi=10.1038/021494e0 |bibcode=1880Natur..21..494G |s2cid=4074444 |url=https://zenodo.org/record/1429243 |doi-access=free}}
*{{Cite book
 | last1=Gelman |first1=Rochel |last2=Gallistel |first2=Charles R.
 | author-link= Rochel Gelman
 | year=1978
 | title= The Child's Understanding of Number 
 | isbn=9780674116368
 | place=Cambridge Mass 
 | publisher= Harvard University Press}}
*{{cite journal |last1=Hubbard |first1=E. M. |last2=Piazza |first2=M. |last3=Pinel |first3=P. |last4=Dehaene |first4=S. |date=June 2005 |title=Interactions between number and space in parietal cortex |journal=Nature Reviews Neuroscience |volume=6 |issue=1–2 |pages=435–448 |doi=10.1038/nrn1684 |pmid=15928716 |s2cid=1465072}}
*{{Cite journal |last1=Izard |first1=V. |last2=Sann |first2=C. |last3=Spelke |first3=E. S. |last4=Streri |first4=A. |date=2009-06-23 |title=Newborn infants perceive abstract numbers |journal=Proceedings of the National Academy of Sciences |volume=106 |issue=25 |pages=10382–10385 |doi=10.1073/pnas.0812142106 |issn=0027-8424 |pmid=19520833 |pmc=2700913 |bibcode=2009PNAS..10610382I |ref={{harvid|Izard, Sann, Spelke & Streri|2009}}|doi-access=free }}
*{{Cite journal |last1=Khanum |first1=S. |last2=Hanif |first2=R. |last3=Spelke |first3=E. S. |last4=Berteletti |first4=I. |last5=Hyde |first5=D. C. |date=2016-10-20 |title=Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children |journal=PLOS ONE |volume=11 |issue=10 |pages=e0164436 |doi=10.1371/journal.pone.0164436 |pmid=27764117 |pmc=5072670 |issn=1932-6203 |bibcode=2016PLoSO..1164436K|doi-access=free }}
*{{Cite journal
 | last1 = McComb | first1 = K. | last2= Packer | first2= C.
 | last3= Pusey | first3= A.
 | year= 1994
 | title= Roaring and numerical assessment in contests between groups of female lions, ''Panthera leo''
 | journal= Animal Behaviour | issue= 2| volume= 47
 | pages= 379–387 | doi=10.1006/anbe.1994.1052| s2cid = 53183852 }}
*{{Cite journal
 | last = Nieder | first = A.
 | year= 2005
 | title= Counting on neurons: The neurobiology of numerical competence
 | journal= Nature Reviews Neuroscience
 | issue=  3| volume= 6
 | pages= 177–190
 | doi=10.1038/nrn1626
 | pmid=15711599| s2cid = 14578049
 }}
*{{Cite journal
 | last1 = Nieder | first1 = A. | last2= Freedman | first2= D. J.
 | last3= Miller | first3= E. K.
 | year= 2002
 | title= Representation of the quantity of visual items in the primate prefrontal cortex
 | journal= Science | issue=  5587| volume= 297
 | pages= 1708–1711 | doi=10.1126/science.1072493| pmid = 12215649 | bibcode = 2002Sci...297.1708N | s2cid = 20871267 }}
*{{Cite journal
 | last1 = Nieder | first1 = A. | last2= Miller | first2= E. K.
 | year= 2003
 | title= Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex
 | journal= Neuron | issue=  1| volume= 37
 | pages= 149–157 | doi=10.1016/s0896-6273(02)01144-3| pmid = 12526780 | s2cid = 5704850 | doi-access= free }}
*{{Cite journal
 | last1 = Nieder | first1 = A. | last2= Miller | first2= E. K.
 | year= 2004
 | title= A parieto-frontal network for visual numerical information in the monkey
 | journal= Proceedings of the National Academy of Sciences
 | issue=  19| volume= 101
 | pages= 7457–7462 | doi=10.1073/pnas.0402239101| pmid = 15123797 | pmc= 409940| bibcode = 2004PNAS..101.7457N | doi-access = free }}
*{{cite journal |last=Núñez |first=R. |year=2009 |title=Numbers and Arithmetic: Neither Hardwired Nor Out There |journal=Biological Theory |volume=4 |issue=1 |pages=68–83 |doi=10.1162/biot.2009.4.1.68 |s2cid=1707771 |citeseerx=10.1.1.610.6016}}
*{{cite journal |last1=Núñez |first1=R. |last2=Doan |first2=D. |last3=Nikoulina |first3=A. |date=August 2011 |title=Squeezing, striking, and vocalizing: Is number representation fundamentally spatial? |journal=Cognition |volume=120 |issue=2 |pages=225–235 |pmid=21640338 |doi=10.1016/j.cognition.2011.05.001 |s2cid=16362508}}
* {{Cite journal |last1=Piazza |first1=M. |last2=Eger |first2=E. |year=2016 |title=Neural foundations and functional specificity of number representations |journal=Neuropsychologia |volume=83 |pages=257–273 |doi=10.1016/j.neuropsychologia.2015.09.025 |pmid=26403660 |hdl=11572/114302 |s2cid=22957569|hdl-access=free }}
*{{Cite journal
 | last1 = Piazza | first1 = M. | last2= Izard | first2= V.
 | last3= Pinel  | first3= P. | last4= Le Bihan | first4= D.
 | last5= Dehaene | first5= S. | author-link5= Stanislas Dehaene
 | year= 2004
 | title= Tuning curves for approximate numerosity in the human intraparietal sulcus
 | journal= Neuron | issue= 3| volume= 44
 | pages= 547–555 | doi=10.1016/j.neuron.2004.10.014| pmid = 15504333 | s2cid = 6288232 | doi-access= free }}
*{{Cite journal
 | last1 = Pica | first1 = P. | last2= Lemer | first2= C.
 | author-link1=Pierre Pica
 | last3= Izard | first3= V. | last4= Dehaene| first4= S.
 | year= 2004
 | title= Exact an Approximate Arithmetic in an Amazonian Indigene Group
 | journal= Science | issue= 5695| volume= 306
 | pages= 499–503 | doi=10.1126/science.1102085 | pmid=15486303| url = https://hal.archives-ouvertes.fr/hal-00206219 | bibcode = 2004Sci...306..499P | s2cid = 10653745 }}
*{{Cite journal
 | last1 = Pinel | first1 = P. | last2= Dehaene | first2= S.
 | author-link2=Stanislas Dehaene
 | last3= Riviere | first3= D. | last4= Le Bihan | first4= D.
 | year= 2001
 | title= Modulation of parietal activation by semantic distance in a number comparison task
 | journal= NeuroImage | issue= 5 | volume= 14
 | pages= 1013–1026 | doi=10.1006/nimg.2001.0913 | pmid=11697933| citeseerx = 10.1.1.5.6247 | s2cid = 17633857 }}
*{{Cite journal
 | last1 = Pinel  | first1 = P. | last2= Piazza | first2= M.
 | last3= Le Bihan | first3= D. | last4= Dehaene | first4= S.
 | author-link4=Stanislas Dehaene
 | year= 2004
 | title= Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments
 | journal= Neuron | issue= 6 | volume= 41
 | pages= 983–993 | doi=10.1016/s0896-6273(04)00107-2| pmid = 15046729 | s2cid = 9372570 | doi-access= free }}
*{{cite book |last=Pinker |first=Steven |author-link=Steven Pinker |year=2008 |title=The Stuff of Thought: Language as a Window Into Human Nature |publisher=[[Penguin Books]] |isbn=978-0143114246 |access-date=November 8, 2012 |url=https://books.google.com/books?id=vCDX54DlYBYC&q=counting+system}}
*{{cite book |last1=Thomas |first1=Manoj |last2=Morwitz |first2=Vicki |year=2009 |editor-last=Rao |editor-first=Vithala R. |chapter=Heuristics in Numerical Cognition: Implications for Pricing |title=Handbook of Pricing Research in Marketing |pages=132– |publisher=Edward Elgar |oclc=807401627 |isbn=9781847202406}}
*{{cite journal |last=Walsh |first=V. |date=November 2003 |title=A theory of magnitude: common cortical metrics of time, space and quantity |journal=Trends in Cognitive Sciences |volume=7 |issue=11 |pages=483–488 |doi=10.1016/j.tics.2003.09.002 |pmid=14585444 |s2cid=1761795}}
{{refend}}

== Further reading ==
* {{cite book|last=Geary|first=David C.|author-link=David C. Geary|year=1994|title=Children's Mathematical Development: Research and Practical Applications|place=Washington, DC|publisher=American Psychological Association|isbn=978-1557982582}}
* {{Cite book
 | last1= Lakoff
 | first1= George 
 | author-link= George Lakoff
 | last2= Nuñez
 | first2= Rafael E.
 | author-link2= Rafael E. Núñez
 | year= 2000
 | title= Where mathematics comes from
 | place= New York
 | publisher= Basic Books.
 | isbn= 978-0-465-03770-4
 | url= https://archive.org/details/wheremathematics00lako
 | url-access= registration
 }}

{{Evolutionary psychology}}

[[Category:Cognition]]
[[Category:Cognitive linguistics]]
[[Category:Cognitive neuroscience]]
[[Category:Cognitive psychology]]
[[Category:Developmental psychology]]
[[Category:Quantity]]