Numerical cognition
Encyclopedia
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:
Similarly, researchers have set up hidden speakers in the African jungle to test natural (untrained) behavior in lions . These speakers can play a number of lion calls, from 1 to 5. If a single lionness 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.
, 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 (for a review of this literature see ).
In another series of studies, Karen Wynn and her colleagues showed that infants as young as four months are able to do very simple additions (e.g., 1 + 1 = 2) and subtractions (3 - 1 = 2). To demonstrate this, Wynn and colleagues 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). However, this is only true if the number of items does not exceed four (e.g., infants fail at 5 - 2 = 3).
There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the classic nature versus nurture
debate. Gelman and 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. Susan Carey (2004, 2009) disagreed, saying that these systems can only encoded large numbers in an approximate way, where language-based natural numbers can be exact. 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, Lemer, Izard & Dehaene, 2004; Butterworth, Reevet, Reynolds & Lloyd, 2008).
, 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 , number comparison and on-line calculation (often tested with subtraction) while the IPL is thought to be involved in overlearned tasks, such as multiplication (see ). 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
. Future research will be needed to disentangle the complex influences of language, working memory and attention on numerical processes.
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. 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 numerosities. Note that these neuronal responses followed 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 .
. 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.
Cognitive science
Cognitive science is the interdisciplinary scientific study of mind and its processes. It examines what cognition is, what it does and how it works. It includes research on how information is processed , represented, and transformed in behaviour, nervous system or machine...
that studies the cognitive, developmental and neural bases of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in cognitive psychology
Cognitive psychology
Cognitive psychology is a subdiscipline of psychology exploring internal mental processes.It is the study of how people perceive, remember, think, speak, and solve problems.Cognitive psychology differs from previous psychological approaches in two key ways....
, developmental psychology
Developmental psychology
Developmental psychology, also known as human development, is the scientific study of systematic psychological changes, emotional changes, and perception changes that occur in human beings over the course of their life span. Originally concerned with infants and children, the field has expanded to...
, neuroscience
Neuroscience
Neuroscience is the scientific study of the nervous system. Traditionally, neuroscience has been seen as a branch of biology. However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics,...
and cognitive linguistics
Cognitive linguistics
In linguistics, cognitive linguistics refers to the branch of linguistics that interprets language in terms of the concepts, sometimes universal, sometimes specific to a particular tongue, which underlie its forms...
. This discipline, although it may interact with questions in the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
is primarily concerned with empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
questions.
Topics included in the domain of numerical cognition include:
- How do 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 infinityInfinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
, the infinitesimalInfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
or the concept of the limitLimit (mathematics)In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
in calculus?
Comparative studies
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have an approximate sense of number (referred to as "numerosity") (for a review, see ). 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 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.Similarly, researchers have set up hidden speakers in the African jungle to test natural (untrained) behavior in lions . These speakers can play a number of lion calls, from 1 to 5. If a single lionness 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 habituatedHabituation
Habituation can be defined as a process or as a procedure. As a process it is defined as a decrease in an elicited behavior resulting from the repeated presentation of an eliciting stimulus...
, 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 (for a review of this literature see ).
In another series of studies, Karen Wynn and her colleagues showed that infants as young as four months are able to do very simple additions (e.g., 1 + 1 = 2) and subtractions (3 - 1 = 2). To demonstrate this, Wynn and colleagues 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). However, this is only true if the number of items does not exceed four (e.g., infants fail at 5 - 2 = 3).
There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the classic nature versus nurture
Nature versus nurture
The nature versus nurture debate concerns the relative importance of an individual's innate qualities versus personal experiences The nature versus nurture debate concerns the relative importance of an individual's innate qualities ("nature," i.e. nativism, or innatism) versus personal experiences...
debate. Gelman and 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. Susan Carey (2004, 2009) disagreed, saying that these systems can only encoded large numbers in an approximate way, where language-based natural numbers can be exact. 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, Lemer, Izard & Dehaene, 2004; Butterworth, Reevet, Reynolds & Lloyd, 2008).
Neuroimaging and neurophysiological studies
Human neuroimaging studies have demonstrated that regions of the parietal lobeParietal lobe
The parietal lobe is a part of the Brain positioned above the occipital lobe and behind the frontal lobe.The parietal lobe integrates sensory information from different modalities, particularly determining spatial sense and navigation. For example, it comprises somatosensory cortex and the...
, including the intraparietal sulcus
Intraparietal sulcus
The intraparietal sulcus is located on the lateral surface of the parietal lobe, and consists of an oblique and a horizontal portion. The IPS contains a series of functionally distinct subregions that have been intensively investigated using both single cell neurophysiology in primates and human...
(IPS) and the inferior parietal lobule (IPL) are activated when subjects are asked to perform calculation tasks. Based on both human neuroimaging
Neuroimaging
Neuroimaging includes the use of various techniques to either directly or indirectly image the structure, function/pharmacology of the brain...
and neuropsychology
Neuropsychology
Neuropsychology studies the structure and function of the brain related to specific psychological processes and behaviors. The term neuropsychology has been applied to lesion studies in humans and animals. It has also been applied to efforts to record electrical activity from individual cells in...
, 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 (often tested with subtraction) while the IPL is thought to be involved in overlearned tasks, such as multiplication (see ). 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
Frontal lobe
The frontal lobe is an area in the brain of humans and other mammals, located at the front of each cerebral hemisphere and positioned anterior to the parietal lobe and superior and anterior to the temporal lobes...
are also active in calculation tasks. These activations overlap with regions involved in language processing such as Broca's area
Broca's area
Broca's area is a region of the hominid brain with functions linked to speech production.The production of language has been linked to the Broca’s area since Pierre Paul Broca reported impairments in two patients. They had lost the ability to speak after injury to the posterior inferior frontal...
and regions involved in working memory
Working memory
Working memory has been defined as the system which actively holds information in the mind to do verbal and nonverbal tasks such as reasoning and comprehension, and to make it available for further information processing...
and attention
Attention
Attention is the cognitive process of paying attention to one aspect of the environment while ignoring others. Attention is one of the most intensely studied topics within psychology and cognitive neuroscience....
. Future research will be needed to disentangle the complex influences of language, working memory and attention on numerical processes.
Single-unit neurophysiology
Neurophysiology
Neurophysiology is a part of physiology. Neurophysiology is the study of nervous system function...
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 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 numerosities. Note that these neuronal responses followed 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 .
Relations between number and other cognitive processes
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 synaesthetes. 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 SNARC effect. 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 conceptual metaphorConceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another, for example, understanding quantity in terms of directionality . A conceptual domain can be any coherent organization of human experience...
. 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.
See also
- CountingCountingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
- Subitizing
- EstimationEstimationEstimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...
- AdditionAdditionAddition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
- SubtractionSubtractionIn arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
- Numerosity adaptation effectNumerosity adaptation effectThe numerosity adaptation effect is a perceptual phenomenon in numerical cognition which demonstrates non-symbolic numerical intuition and exemplifies how numerical percepts can impose themselves upon human brain automatically. This effect was first described in 2008.Presently, this effect is...
- Approximate number systemApproximate number systemThe approximate number system is a hypothetical physiological basis for the intuitive "number sense". As its name suggests, it only provides an approximate sense of number, and is not an accurate symbolic computation system...
- Pirahã peoplePirahã peopleThe Pirahã people are an indigenous hunter-gatherer tribe of Amazon natives, a subgroup of the Mura, who mainly live on the banks of the Maici River in Brazil's Amazonas state, in the territory on Humaitá and Manicoré municipality....