Numerical cognition
Cognitive psychology |
---|
Perception |
|
Attention |
Memory |
Metacognition |
Language |
Metalanguage |
Thinking |
Numerical cognition |
Numerical cognition is a subdiscipline of
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 infinity, the infinitesimal or the concept of the 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
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 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 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.[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 classic nature 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 an 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., Pica et al. (2004)); Butterworth & Reeve (2008), 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,[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 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.[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-unit
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
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 a logarithmic 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.
Relations between number and other cognitive processes
There is evidence that numerical cognition is intimately related to other aspects of thought – particularly spatial cognition.
Modification of the usual decimal representation was advocated by John Colson. The sense of 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, 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]
Ethnolinguistic variance
The numeracy of indigenous peoples is studied to identify universal aspects of numerical cognition in humans. Notable examples include the
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. Journal link
See also
- Addition – Arithmetic operation
- Approximate number system – Innate ability to detect differences in magnitude without counting
- Counting – Finding the number of elements of a finite set
- Estimation – Process of finding an approximation
- Numerosity adaptation effect – Phenomenon in numerical cognition
- Ordinal numerical competence – ability to count objects in order and to understand the greater than and less than relationships between numbers
- Parallel individuation system – non-symbolic cognitive system that supports the representation of numerical values from zero to three or four
- Plant arithmetic – Form of plant cognition
- The problem of the speckled hen– Type of epistemological problem
- Subitizing – Assessing the quantity of objects in a visual scene without individually counting each item
- Subtraction – One of the four basic arithmetic operations
Notes
- ^ Dehaene (1997), p. [page needed].
- ^ Agrillo (2012).
- ^ McComb, Packer & Pusey (1994).
- ^ Feigenson, Dehaene & Spelke (2004).
- ^ a b c Piazza et al. (2004).
- ^ Pinel et al. (2001).
- ^ Pinel et al. (2004).
- ^ Dehaene (1997).
- ^ Piazza & Eger (2016).
- ^ Campbell & Xue (2001).
- ^ Barrouillet, Mignon & Thevenot (2008).
- ^ Nieder (2005).
- ^ Nieder, Freedman & Miller (2002).
- ^ a b Nieder & Miller (2004).
- ^ a b Nieder & Miller (2003).
- ^ Berteletti et al. (2010).
- ^ Khanum et al. (2016).
- ^ Hubbard et al. (2005).
- ^ Galton (1880).
- ^ Dehaene, Bossini & Giraux (1993).
- ^ Fischer, Mills & Shaki (2010).
- ^ Núñez, Doan & Nikoulina (2011).
- ^ Walsh (2003).
- ^ Núñez (2009).
- ^ Dehaene (1992).
- ^ Thomas & Morwitz (2009), p. [page needed].
- ^ Pinker (2008), p. [page needed].
References
- Agrillo, C. (2012). "Evidence for Two Numerical Systems That Are Similar in Humans and Guppies". PLOS ONE. 7 (2). e31923. PMID 22355405.
- Barrouillet, P.; Mignon, M.; Thevenot, C. (2008). "Strategies in subtraction problem solving in children". Journal of Experimental Child Psychology. 99 (4): 233–251. PMID 18241880.
- Berteletti, I.; Lucangeli, D.; Piazza, M.; Dehaene, S.; Zorzi, M. (2010). "Numerical estimation in preschoolers". Developmental Psychology. 46 (2): 545–551. S2CID 8496112.
- S2CID 2662436.
- PMID 18757729.
- Campbell, J.I.D.; Xue, Q. (2001). "Cognitive Arithmetic Across Cultures" (PDF). Journal of Experimental Psychology: General. 130 (2): 299–315. PMID 11409105.
- PMID 16594732.
- Carey, S. (2004). "Bootstrapping and the origins of Concepts". Daedalus. 133: 59–68. S2CID 54493789.
- Carey, S. (2009). "Where our number concepts come from". Journal of Philosophy. 106 (4): 220–254. PMID 23136450.
- Dehaene, Stanislas (1992). "Varieties of numerical abilities". Cognition. 44 (1–2): 1–42. S2CID 24382907.
- ISBN 978-0-19-513240-3.
- Dehaene, S.; Bossini, S.; Giraux, P. (September 1993). "The mental representation of parity and number magnitude". Journal of Experimental Psychology. 122 (3): 371–396. .
- Feigenson, L.; S2CID 17313189.
- Fischer, M. H.; Mills, R. A.; Shaki, S. (April 2010). "How to cook a SNARC: Number placement in text rapidly changes spatial–numerical associations". Brain and Cognition. 72 (3): 333–336. S2CID 19626981.
- Galton, Francis (25 March 1880). "Visualised Numerals". Nature. 21 (543): 494–495. S2CID 4074444.
- ISBN 9780674116368.
- Hubbard, E. M.; Piazza, M.; Pinel, P.; Dehaene, S. (June 2005). "Interactions between number and space in parietal cortex". Nature Reviews Neuroscience. 6 (1–2): 435–448. S2CID 1465072.
- Izard, V.; Sann, C.; Spelke, E. S.; Streri, A. (23 June 2009). "Newborn infants perceive abstract numbers". Proceedings of the National Academy of Sciences. 106 (25): 10382–10385. PMID 19520833.
- Khanum, S.; Hanif, R.; Spelke, E. S.; Berteletti, I.; Hyde, D. C. (20 October 2016). "Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children". PLOS ONE. 11 (10): e0164436. PMID 27764117.
- McComb, K.; Packer, C.; Pusey, A. (1994). "Roaring and numerical assessment in contests between groups of female lions, Panthera leo". Animal Behaviour. 47 (2): 379–387. S2CID 53183852.
- Nieder, A. (2005). "Counting on neurons: The neurobiology of numerical competence". Nature Reviews Neuroscience. 6 (3): 177–190. S2CID 14578049.
- Nieder, A.; Freedman, D. J.; Miller, E. K. (2002). "Representation of the quantity of visual items in the primate prefrontal cortex". Science. 297 (5587): 1708–1711. S2CID 20871267.
- Nieder, A.; Miller, E. K. (2003). "Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex". Neuron. 37 (1): 149–157. S2CID 5704850.
- Nieder, A.; Miller, E. K. (2004). "A parieto-frontal network for visual numerical information in the monkey". Proceedings of the National Academy of Sciences. 101 (19): 7457–7462. PMID 15123797.
- Núñez, R. (2009). "Numbers and Arithmetic: Neither Hardwired Nor Out There". Biological Theory. 4 (1): 68–83. S2CID 1707771.
- Núñez, R.; Doan, D.; Nikoulina, A. (August 2011). "Squeezing, striking, and vocalizing: Is number representation fundamentally spatial?". Cognition. 120 (2): 225–235. S2CID 16362508.
- Piazza, M.; Eger, E. (2016). "Neural foundations and functional specificity of number representations". Neuropsychologia. 83: 257–273. S2CID 22957569.
- Piazza, M.; Izard, V.; Pinel, P.; Le Bihan, D.; S2CID 6288232.
- S2CID 10653745.
- Pinel, P.; S2CID 17633857.
- Pinel, P.; Piazza, M.; Le Bihan, D.; S2CID 9372570.
- ISBN 978-0143114246. Retrieved 2012-11-08.
- Thomas, Manoj; Morwitz, Vicki (2009). "Heuristics in Numerical Cognition: Implications for Pricing". In Rao, Vithala R. (ed.). Handbook of Pricing Research in Marketing. Edward Elgar. pp. 132–. OCLC 807401627.
- Walsh, V. (November 2003). "A theory of magnitude: common cortical metrics of time, space and quantity". Trends in Cognitive Sciences. 7 (11): 483–488. S2CID 1761795.
Further reading
- ISBN 978-0-465-03770-4.