Lecture
Numerical cognition is a branch of cognitive science that studies the cognitive, developmental and neural foundations of numbers and mathematics . Like many other areas of cognitive science, it is a highly interdisciplinary topic that 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 field of numerical cognition include:
A variety of studies have shown that non-human animals, including rats, lions and various primate species, possess an approximate sense of number (called « numerosity ») (for a review see Dehaene 1997 ). For example, when a rat is trained to press a bar 8 or 16 times to obtain a food reward, the number of bar presses will approximately follow a Gaussian, or normal, distribution with a peak around 8 or 16 bar presses. When rats are hungrier, their bar-pressing behaviour occurs faster, so by showing that the peak number of bar presses is the same for sated or hungry rats it is possible to dissociate time and number of bar presses. In addition, in some species the parallel individuation system has been demonstrated, for instance in the case of guppies, which successfully discriminate between 1 and 4 other individuals.
Similarly, researchers set up hidden loudspeakers in the African savannah in order to test the natural (untrained) behaviour of lions ( McComb, Packer & Pusey, 1994 ). These loudspeakers could play back several lion roars, from 1 to 5. If a single lioness hears, for example, three roars of unknown lions, she will leave, whereas if she is with four of her sisters they will go to investigate. This suggests that lions can not only tell when they are «outnumbered», but that they can do so on the basis of signals from various sensory modalities, suggesting that numerosity is a multisensory concept.
Research in developmental psychology has shown that human infants, like non-human animals, possess an approximate sense of number. For example, in one study infants were repeatedly presented with arrays (in one block) of 16 dots. Careful controls were introduced to rule out information from «non-numerical» parameters such as total surface area, brightness, circumference and so on. After infants had been shown many displays containing 16 items, they habituated, or stopped looking at the display for so long. Infants were then presented with a display containing 8 items, and they looked at the new display for longer.
Because of the numerous controls that were used to rule out non-numerical factors, the experimenters concluded that six-month-old infants are sensitive to the difference between 8 and 16. Subsequent experiments using similar methodologies showed that six-month-old infants can discriminate between numbers differing by a ratio of 2:1 (8 versus 16 or 16 versus 32), but not by a ratio of 3:2 (8 versus 12 or 16 versus 24). However, 10-month-old infants succeed with both the 2:1 and the 3:2 ratio, suggesting an increasing sensitivity to differences in numerosity with age (for a review of this literature see Feigenson, Dehaene & Spelke 2004 ).
In another series of studies Karen Wynn showed that infants as young as five months can perform very simple addition (e.g., 1 + 1 = 2) and subtraction (3 - 1 = 2) operations. 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, the infants were presented with only one Mickey (the «impossible event»), they looked longer than if they were shown two Mickeys (the «possible» event). Further research by Karen Wynn and Koleen McCrink showed that although infants' ability to compute exact outcomes extends only to small numbers, infants can compute approximate outcomes of larger addition and subtraction events (e.g., «5 + 5»)
There is debate about how much these infant systems actually contain in terms of numerical concepts, proceeding from the classic nature versus nurture discussion. Gelman and Gallistel 1978 suggested that a child has the concept of natural number from birth, and needs only to map it onto the words used in his language. Carey 2004 , Carey 2009 disagreed, stating that these systems can encode large numbers only in an approximate manner , whereas language-based natural numbers can be exact. It is believed that without language only the numbers from 1 to 4 have an exact representation via the parallel individuation system.. One promising approach is to see whether cultures lacking number words can work with natural numbers. The results so far are mixed (e.g., Pica et al. 2004 ); Butterworth and Reeve 2008 , Butterworth, Reeve and Lloyd 2008 .
Studies using human neuroimaging have shown 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. On the basis of human neuroimaging and neuropsychology , Stanislas Dehaene and his colleagues have suggested that these two parietal structures play complementary roles. The IPS is thought to contain circuitry that is primarily involved in numerical estimation ( Piazza et al. 2004 ), number comparison ( Pinel et al. 2001 ; Pinel et al. 2004).) and online computation or quantity processing (often tested by subtraction), while the IPL is thought to be involved in rote memorization, such as multiplication (see Dehaene 1997 ). Thus a patient with a lesion of the IPL may be able to subtract but not to multiply, and vice versa for a patient with a lesion of the IPS. In addition to these parietal regions, areas of the frontal lobe are also engaged in calculation tasks. These activations overlap with regions involved in language processing, such as Broca's area, and with regions involved in working memory and attention.. Furthermore, the inferior temporal cortex is involved in processing the numerical shapes and symbols necessary for calculation with Arabic numerals. More recent research has identified networks associated with multiplication and subtraction tasks. Multiplication is often learned through rote memorization and verbal repetition, and neuroimaging studies have shown that multiplication recruits a left-lateralized network of the inferior frontal cortex and the superior-middle temporal gyri in addition to the IPL and IPS. Subtraction is accounted for more by quantity manipulation and strategy use, relying more on the right IPS and the posterior parietal lobe.
Single-unit neurophysiology in monkeys has also revealed neurons in the frontal cortex and within the intraparietal sulcus that respond to numbers. Andreas Nieder ( Nieder, 2005 ; Nieder, Freedman & Miller, 2002 ; Nieder & Miller, 2004)) trained monkeys to perform a «delayed match-to-sample» task. For example, a monkey may be presented with a field of four dots, which must be held in memory after the image is removed. Then, after a delay period of several seconds, a second display appears. If the number on the second display matches the number on the first, the monkey must release a lever. If it is different, the monkey must hold the lever. Neural activity recorded during the delay period showed that neurons within the intraparietal sulcus and the frontal cortex possessed a «preferred numerosity», exactly as had been predicted by behavioural research. That is, a particular number may 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» to particular magnitudes. Weber's law , as has been demonstrated for other sensory dimensions, is consistent with the ratio dependence observed for the numerical behaviour of non-human animals and infants ( Nieder & Miller 2003 ) .
It is important to note that, although primates have brains remarkably similar to the human brain, there are differences in function, capacities and complexity. They make good experimental subjects for preliminary testing, but they do not show the small differences that result from different evolutionary paths and environments. In numerical terms, however, they have a great deal in common. As has been found in monkeys, neurons selectively tuned to number have been identified in the bilateral intraparietal sulci and the prefrontal cortex in humans. Piazza and colleagues investigated this using fMRI, presenting participants with sets of dots on which they had to make either same-different judgements or larger-smaller judgements. The dot sets consisted of base numbers of 16 and 32 dots with ratios of 1.25, 1.5 and 2. Deviant numbers were included in some trials in larger or smaller quantities than the base numbers. Participants displayed patterns of activation similar to those found by Nieder in monkeys. Located in the parietal lobe of the brain, the sulci and the prefrontal cortex, also involved in number, communicate about approximate quantity, and it was found in both species that parietal IPS neurons had short firing latencies, whereas frontal neurons had longer firing latencies. This supports the idea that number is first processed in the IPS and, if necessary, is then relayed to the connected frontal neurons in the prefrontal cortex for further enumeration and application. Humans displayed Gaussian curves in their tuning curves for approximate magnitude. This matched the monkeys, demonstrating a similarly structured mechanism in both species, with classic Gaussian curves with respect to increasingly deviant numbers from 16 and 32, as well as habituation. The results were consistent with Weber's law., with decreasing precision as the ratio of the numbers decreased. This corroborates the conclusions drawn by Nieder in macaque monkeys, and demonstrates compelling evidence for an approximate numerical logarithmic scale in humans.
With the mechanism of non-symbolic number approximation established in both humans and primates, further investigation is required to determine whether this mechanism is innate and present in children, which would suggest an innate capacity to process numerical stimuli in the same way that humans are born ready to process language. Cantlon and colleagues set out to investigate this in 4-year-old healthy, normally developing children in parallel with adults. A task similar to Piazza's was used in this experiment, without the estimation tasks. Dot arrays of varying size and number were used, with base numbers of 16 and 32. In each block 232 stimuli were presented, with 20 deviant numerosities at a ratio of 2.0, both larger and smaller. For example, out of the 232 trials, 16 dots were presented at varying sizes and spacings, but in 10 of these trials there were 8 dots and in 10 of these trials 32 dots, making up the 20 deviant stimuli. The same applied to blocks with 32 as the base numerosity. To ensure that adults and children were attending to the stimuli, they placed 3 fixation points throughout the trial, at which the participant had to move a joystick in order to proceed. Their results showed that the adults in the experiment had significant IPS activation when viewing deviant-number stimuli, consistent with what was found earlier in the paragraph above. In the four-year-old children they found significant IPS activation to deviant-number stimuli, resembling the activation found in adults. There were some differences in the activations: adults displayed stronger bilateral activation, whereas the 4-year-olds mainly displayed activation in their right IPS and activated 112 fewer voxels than adults. This suggests that at the age of 4 children have an established mechanism of neurons in the IPS tuned to processing non-symbolic numerical values. Other studies have probed this mechanism further in children and found that children also represent approximate numbers on a chart. resembling the activation found in adults. There were some differences in the activations: adults displayed stronger bilateral activation, whereas the 4-year-olds mainly displayed activation in their right IPS and activated 112 fewer voxels than adults. This suggests that at the age of 4 children already have an established mechanism of neurons in the IPS tuned to processing non-symbolic numerical values. Other studies have probed this mechanism further in children and found that children also represent approximate numbers on a chart. resembling the activation found in adults. There were some differences in the activations: adults displayed stronger bilateral activation, whereas the 4-year-olds mainly displayed activation in their right IPS and activated 112 fewer voxels than adults. This suggests that at the age of 4 children already have an established mechanism of neurons in the IPS tuned to processing non-symbolic numerical values. Other studies have probed this mechanism further in children and found that children also represent approximate numbers on a chart. children have an established mechanism of neurons in the IPS tuned to processing non-symbolic numerical values. Other studies have probed this mechanism further in children and found that children also represent approximate numbers on a chart. children have an established mechanism of neurons in the IPS tuned to processing non-symbolic numerical values. Other studies have probed this mechanism further in children and found that children also represent approximate numbers on a chart.logarithmic scale , consistent with the claims made by Piazza for adults.
A study by Izard and colleagues investigated abstract representations of number in infants, using a different paradigm from previous researchers because of the nature and developmental stage of infants. For infants, they investigated abstract number with both auditory and visual stimuli, using a looking-time paradigm. Sets of 4 versus 12, 8 versus 16 and 4 versus 8 were used. The auditory stimuli consisted of tones of varying frequency with a given number of tones, with some deviations in trials where the tones were shorter but more numerous or longer and less numerous, in order to control for duration and its potential confounds. After the auditory stimuli had been presented for 2 minutes of familiarization, the visual stimuli were presented as a congruent or incongruent set of multicoloured 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 a system for approximating non-symbolic number, even across modalities, is present in infancy. What is important to note about these three particular studies of non-symbolic numerosity in humans is that it is present in infancy and develops across the lifespan. The honing of their abilities to approximate and recognize numbers, as indicated by the improvement of Weber fractions over time, and the recruitment of the left IPS to provide a broader base for processing calculation and enumeration, support the claims made for a mechanism of non-symbolic number processing. in the human brain. is present in infancy. What is important to note about these three particular studies of non-symbolic numerosity in humans is that it is present in infancy and develops across the lifespan. The honing of their abilities to approximate and recognize numbers, as indicated by the improvement of Weber fractions over time, and the recruitment of the left IPS to provide a broader base for processing calculation and enumeration, support the claims made for a mechanism of non-symbolic number processing. in the human brain. is present in infancy. What is important to note about these three particular studies of non-symbolic numerosity in humans is that it is present in infancy and develops across the lifespan. The honing of their abilities to approximate and recognize numbers, as indicated by the improvement of Weber fractions over time, and the recruitment of the left IPS to provide a broader base for processing calculation and enumeration, support the claims made for a mechanism of non-symbolic number processing. in the human brain.
There is evidence that numerical cognition is closely linked to other aspects of thinking, especially to spatial cognition. One piece of evidence comes from studies conducted on number-form synaesthetes . Such people report that numbers are mentally represented in a particular spatial layout; others experience numbers as perceptible objects that can be visually manipulated to facilitate calculation. Behavioural studies further strengthen the link between numerical and spatial cognition. For example, participants respond faster to large numbers if they answer on the right side of space, and faster to smaller numbers when on the left — the so-called «spatial-numerical association of response codes», or SNARC effect. This effect varies across culture and context, however, and some studies have even begun to question whether SNARC reflects an intrinsic association of number and space, appealing instead to strategic problem solving or to a more general cognitive mechanism such as conceptual metaphor . Moreover, neuroimaging studies show that the link between number and space also manifests itself in brain activity. For example, regions of the parietal cortex show shared activation for both spatial and numerical processing. These various lines of research suggest a strong but flexible link between numerical and spatial cognition.
A modification of the ordinary decimal notation was proposed by John Colson . The sense of the complement, absent in the ordinary decimal system, is expressed by the representation of signed digits .
Several consumer psychologists have also studied the heuristics that people use in numerical cognition. For example, Thomas and Morwitz (2009) reviewed several studies showing that three heuristics that manifest themselves in many everyday judgements 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 mistakenly judge the difference between $4.00 and $2.99 as larger than the difference between $4.01 and $3.00, because of anchoring on the leftmost digits. The precision effect reflects the influence of the representativeness of digit patterns on magnitude judgements. Large magnitudes are usually rounded and therefore have many zeros, whereas smaller magnitudes are usually expressed by precise numbers; therefore, relying on the representativeness of digit patterns, people may mistakenly consider a price of $391,534 more attractive than a price of $390,000. The ease-of-computation effect shows that magnitude judgements are based not only on the results of mental calculation, but also on their ease or difficulty as subjectively experienced. It is generally easier to compare two different magnitudes than two similar magnitudes; overuse of this heuristic can lead people to incorrectly judge the difference as larger for pairs with easier computations, e.g., $5.00 minus $4, than for pairs with difficult computations, e.g., $4.97 minus $3.96. whereas smaller magnitudes are usually expressed by precise numbers; therefore, relying on the representativeness of digit patterns, people may mistakenly consider a price of $391,534 more attractive than a price of $390,000. The ease-of-computation effect shows that magnitude judgements are based not only on the results of mental calculation, but also on their ease or difficulty as subjectively experienced. It is generally easier to compare two different magnitudes than two similar magnitudes; overuse of this heuristic can lead people to incorrectly judge the difference as larger for pairs with easier computations, e.g., $5.00 minus $4, than for pairs with difficult computations, e.g., $4.97 minus $3.96. whereas smaller magnitudes are usually expressed by precise numbers; therefore, relying on the representativeness of digit patterns, people may mistakenly consider a price of $391,534 more attractive than a price of $390,000. The ease-of-computation effect shows that magnitude judgements are based not only on the results of mental calculation, but also on their ease or difficulty as subjectively experienced. It is generally easier to compare two different magnitudes than two similar magnitudes; overuse of this heuristic can lead people to incorrectly judge the difference as larger for pairs with easier computations, e.g., $5.00 minus $4, than for pairs with difficult computations, e.g., $4.97 minus $3.96. 534 would be more attractive than a price of $390,000. The ease-of-computation effect shows that magnitude judgements are based not only on the results of mental calculation, but also on their ease or difficulty as subjectively experienced. It is generally easier to compare two different magnitudes than two similar magnitudes; overuse of this heuristic can lead people to incorrectly judge the difference as larger for pairs with easier computations, e.g., $5.00 minus $4, than for pairs with difficult computations, e.g., $4.97 minus $3.96. 534 would be more attractive than a price of $390,000. The ease-of-computation effect shows that magnitude judgements are based not only on the results of mental calculation, but also on their ease or difficulty as subjectively experienced. It is generally easier to compare two different magnitudes than two similar magnitudes; overuse of this heuristic can lead people to incorrectly judge the difference as larger for pairs with easier computations, e.g., $5.00 minus $4, than for pairs with difficult computations, e.g., $4.97 minus $3.96. [20]
The counting ability of indigenous peoples is studied in order to identify universal aspects of human numerical cognition. Well-known examples include the Pirahã people , who have no words for specific numbers, and the Mundurukú people, who have number words only up to five. Pirahã adults are unable to mark the exact number of counters for a pile of nuts containing fewer than ten items. The anthropologist Napoleon Chagnon spent several decades studying the Yanomami.in the field. He concluded that they have no need to count in everyday life. Their hunters keep track of individual arrows with the same mental faculties that they use to recognize members of their family. There are no known hunter-gatherer cultures that have a counting system in their language. Mental and linguistic capacities for mathematics are associated with the development of agriculture, and with it large numbers of indistinguishable objects.
The Journal of Numerical Cognition is an open access, free-to-publish, online-only journal devoted specifically to research in the field of numerical cognition. Link to the journal
Comments