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| Numerical cognition |
Numerical cognition is a subdiscipline ofcognitive science that studies the cognitive, developmental and neural bases ofnumbers andmathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers incognitive psychology,developmental psychology,neuroscience andcognitive linguistics. This discipline, although it may interact with questions in thephilosophy of mathematics, is primarily concerned withempirical questions.
Topics included in the domain of numerical cognition include:
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have anapproximate sense of number (referred to as "numerosity").[1] 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 aGaussian 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 theparallel individuation system has been shown, for example in the case ofguppies which successfully discriminated between 1 and 4 other individuals.[2]
Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions.[3] 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 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, theyhabituated, 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.[4]
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 classicnature versus nurture debate.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. Carey (2004,2009) disagreed, saying that these systems can only encode large numbers in anapproximate way, where language-based natural numbers can be exact. Without language, only numbers 1 to 4 are believed to have an exact representation, through theparallel 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.,Pica et al. (2004));Butterworth & Reeve (2008),Butterworth, Reeve, Reynolds & Lloyd (2008)).
Human neuroimaging studies have demonstrated that regions of theparietal lobe, including theintraparietal sulcus (IPS) and theinferior parietal lobule (IPL) are activated when subjects are asked to perform calculation tasks. Based on both humanneuroimaging andneuropsychology, 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,[5] number comparison,[6][7] 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.[8] 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 thefrontal lobe are also active in calculation tasks. These activations overlap with regions involved in language processing such asBroca's area and regions involved inworking memory andattention. Additionally, the inferotemporal cortex is implicated in processing the numerical shapes and symbols, necessary for calculations with Arabic digits.[9] 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.[10] Subtraction is taught more with quantity manipulation and strategy use, more reliant upon the right IPS and the posterior parietal lobule.[11]
Single-unitneurophysiology 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.[12][13][14] 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 followedWeber'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.[15]
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[5] 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.[15] Theintraparietal sulcus and theprefrontal 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 theprefrontal 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 followedWeber's Law, with accuracy decreasing as the ratio between numbers became smaller. This supports the findings made by Nieder in macaque monkeys[14] and shows definitive evidence for anapproximate number logarithmic scale in humans.[16][17]
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.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[5] 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 alogarithmic scale, aligning with the claims made by Piazza in adults.
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.
There is evidence that numerical cognition is intimately related to other aspects of thought – particularly spatial cognition.[18] One line of evidence comes from studies performed on number-formsynaesthetes.[19] 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" orSNARC effect.[20] This effect varies across culture and context,[21] however, and some research has even begun to question whether the SNARC reflects an inherent number-space association,[22] instead invoking strategic problem solving or a more general cognitive mechanism likeconceptual metaphor.[23][24] 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.[25] These various lines of research suggest a strong, but flexible, connection between numerical and spatial cognition.
Modification of the usualdecimal representation was advocated byJohn Colson. The sense ofcomplementation, missing in the usual decimal system, is expressed bysigned-digit representation.
Several consumer psychologists have also studied the heuristics that people use in numerical cognition. For example,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.[26]
The numeracy of indigenous peoples is studied to identify universal aspects of numerical cognition in humans. Notable examples include thePirahã people who have no words for specific numbers and theMunduruku 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. AnthropologistNapoleon Chagnon spent several decades studying theYanomami 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.[27]
TheJournal of Numerical Cognition is an open-access, free-to-publish, online-only journal outlet specifically for research in the domain of numerical cognition.
