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Subitizing
Assessing the quantity of objects in a visual scene without individually counting each item

Subitizing is the rapid and effortless ability to perceive small quantities, typically four or fewer items in a set, without counting or language. Larger sets often require counting or estimation, especially if unfamiliar. The term was coined in 1949 and derives from the Latin subitus meaning “sudden.” Judgments for one to four items are fast and accurate, but speed and confidence decline with more elements, as response times increase significantly per additional item. Research shows subitizing is distinct from mere apprehension (apprehension) and affects reaction time. Studies also reveal subitizing operates in tactile and auditory perception, supporting its role as a general perceptual mechanism beyond vision.

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Enumerating afterimages

As the derivation of the term "subitizing" suggests, the feeling associated with making a number judgment within the subitizing range is one of immediately being aware of the displayed elements.18 When the number of objects presented exceeds the subitizing range, this feeling is lost, and observers commonly report an impression of shifting their viewpoint around the display, until all the elements presented have been counted.19 The ability of observers to count the number of items within a display can be limited, either by the rapid presentation and subsequent masking of items,20 or by requiring observers to respond quickly.21 Both procedures have little, if any, effect on enumeration within the subitizing range. These techniques may restrict the ability of observers to count items by limiting the degree to which observers can shift their "zone of attention"22 successively to different elements within the display.

Atkinson, Campbell, and Francis23 demonstrated that visual afterimages could be employed in order to achieve similar results. Using a flashgun to illuminate a line of white disks, they were able to generate intense afterimages in dark-adapted observers. Observers were required to verbally report how many disks had been presented, both at 10 s and at 60 s after the flashgun exposure. Observers reported being able to see all the disks presented for at least 10 s, and being able to perceive at least some of the disks after 60 s. Unlike simply displaying the images for 10 and 60 second intervals, when presented in the form of afterimages, eye movement cannot be employed for the purpose of counting: when the subjects move their eyes, the images also move. Despite a long period of time to enumerate the number of disks presented when the number of disks presented fell outside the subitizing range (i.e., 5–12 disks), observers made consistent enumeration errors in both the 10 s and 60 s conditions. In contrast, no errors occurred within the subitizing range (i.e., 1–4 disks), in either the 10 s or 60 s conditions.24

Brain structures involved in subitizing and counting

The work on the enumeration of afterimages2526 supports the view that different cognitive processes operate for the enumeration of elements inside and outside the subitizing range, and as such raises the possibility that subitizing and counting involve different brain circuits. However, functional imaging research has been interpreted both to support different27 and shared processes.28

Bálint's syndrome

Social theory supporting the view that subitizing and counting may involve functionally and anatomically distinct brain areas comes from patients with simultanagnosia, one of the key components of Bálint's syndrome.29 Patients with this disorder suffer from an inability to perceive visual scenes properly, being unable to localize objects in space, either by looking at the objects, pointing to them, or by verbally reporting their position.30 Despite these dramatic symptoms, such patients are able to correctly recognize individual objects.31 Crucially, people with simultanagnosia are unable to enumerate objects outside the subitizing range, either failing to count certain objects, or alternatively counting the same object several times.32

However, people with simultanagnosia have no difficulty enumerating objects within the subitizing range.33 The disorder is associated with bilateral damage to the parietal lobe, an area of the brain linked with spatial shifts of attention.34 These neuropsychological results are consistent with the view that the process of counting, but not that of subitizing, requires active shifts of attention. However, recent research has questioned this conclusion by finding that attention also affects subitizing.35

Imaging enumeration

A further source of research on the neural processes of subitizing compared to counting comes from positron emission tomography (PET) research on normal observers. Such research compares the brain activity associated with enumeration processes inside (i.e., 1–4 items) for subitizing, and outside (i.e., 5–8 items) for counting.3637

Such research finds that within the subitizing and counting range activation occurs bilaterally in the occipital extrastriate cortex and superior parietal lobe/intraparietal sulcus. This has been interpreted as evidence that shared processes are involved.38 However, the existence of further activations during counting in the right inferior frontal regions, and the anterior cingulate have been interpreted as suggesting the existence of distinct processes during counting related to the activation of regions involved in the shifting of attention.39

Educational applications

Historically, many systems have attempted to use subitizing to identify full or partial quantities. In the twentieth century, mathematics educators started to adopt some of these systems, as reviewed in the examples below, but often switched to more abstract color-coding to represent quantities up to ten.

In the 1990s, babies three weeks old were shown to differentiate between 1–3 objects, that is, to subitize.40 A more recent meta-study summarizing five different studies concluded that infants are born with an innate ability to differentiate quantities within a small range, which increases over time.41 By the age of seven that ability increases to 4–7 objects. Some practitioners claim that with training, children are capable of subitizing 15+ objects correctly.

Abacus

The hypothesized use of yupana, an Inca counting system, placed up to five counters in connected trays for calculations.

In each place value, the Chinese abacus uses four or five beads to represent units, which are subitized, and one or two separate beads, which symbolize fives. This allows multi-digit operations such as carrying and borrowing to occur without subitizing beyond five.

European abacuses use ten beads in each register, but usually separate them into fives by color.

Twentieth century teaching tools

The idea of instant recognition of quantities has been adopted by several pedagogical systems, such as Montessori, Cuisenaire and Dienes. However, these systems only partially use subitizing, attempting to make all quantities from 1 to 10 instantly recognizable. To achieve it, they code quantities by color and length of rods or bead strings representing them. Recognizing such visual or tactile representations and associating quantities with them involves different mental operations from subitizing.

Other applications

One of the most basic applications is in digit grouping in large numbers, which allow one to tell the size at a glance, rather than having to count. For example, writing one million (1000000) as 1,000,000 (or 1.000.000 or 1000000) or one (short) billion (1000000000) as 1,000,000,000 (or other forms, such as 1,00,00,00,000 in the Indian numbering system) makes it much easier to read. This is particularly important in accounting and finance, as an error of a single decimal digit changes the amount by a factor of ten. This is also found in computer programming languages for literal values, some of which use digit separators.

Dice, playing cards and other gaming devices traditionally split quantities into subitizable groups with recognizable patterns. The behavioural advantage of this grouping method has been scientifically investigated by Ciccione and Dehaene,42 who showed that counting performances are improved if the groups share the same amount of items and the same repeated pattern.

A comparable application is to split up binary and hexadecimal number representations, telephone numbers, bank account numbers (e.g., IBAN, social security numbers, number plates, etc.) into groups ranging from 2 to 5 digits separated by spaces, dots, dashes, or other separators. This is done to support overseeing completeness of a number when comparing or retyping. This practice of grouping characters also supports easier memorization of large numbers and character structures.

Self assessment

There is at least one game that can be played online to self assess one's ability to subitize.43

See also

References

  1. Kaufman, E.L.; Lord, M.W.; Reese, T.W. & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology. 62 (4). The American Journal of Psychology: 498–525. doi:10.2307/1418556. JSTOR 1418556. PMID 15392567. /wiki/American_Journal_of_Psychology ↩

  2. Kaufman, E.L.; Lord, M.W.; Reese, T.W. & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology. 62 (4). The American Journal of Psychology: 498–525. doi:10.2307/1418556. JSTOR 1418556. PMID 15392567. /wiki/American_Journal_of_Psychology ↩

  3. Saltzman, I.J. & Garner, W.R. (1948). "Reaction time as a measure of span of attention". The Journal of Psychology. 25 (2): 227–241. doi:10.1080/00223980.1948.9917373. PMID 18907281. /wiki/The_Journal_of_Psychology ↩

  4. Jevons, W.S. (1871). "The power of numerical discrimination". Nature. 3 (67): 281–282. Bibcode:1871Natur...3..281J. doi:10.1038/003281a0. https://doi.org/10.1038%2F003281a0 ↩

  5. Taves, E.H. (1941). "Two mechanisms for the perception of visual numerousness". Archives of Psychology. 37: 1–47. ↩

  6. Kaufman, E.L.; Lord, M.W.; Reese, T.W. & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology. 62 (4). The American Journal of Psychology: 498–525. doi:10.2307/1418556. JSTOR 1418556. PMID 15392567. /wiki/American_Journal_of_Psychology ↩

  7. Trick, L.M. & Pylyshyn, Z.W. (1994). "Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision". Psychological Review. 101 (1): 80–102. doi:10.1037/0033-295X.101.1.80. PMID 8121961. /wiki/Doi_(identifier) ↩

  8. Saltzman, I.J. & Garner, W.R. (1948). "Reaction time as a measure of span of attention". The Journal of Psychology. 25 (2): 227–241. doi:10.1080/00223980.1948.9917373. PMID 18907281. /wiki/The_Journal_of_Psychology ↩

  9. Chi, M.T.H. & Klahr, D. (1975). "Span and rate of apprehension in children and adults". Journal of Experimental Child Psychology. 19 (3): 434–439. doi:10.1016/0022-0965(75)90072-7. PMID 1236928. /wiki/Doi_(identifier) ↩

  10. Kaufman, E.L.; Lord, M.W.; Reese, T.W. & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology. 62 (4). The American Journal of Psychology: 498–525. doi:10.2307/1418556. JSTOR 1418556. PMID 15392567. /wiki/American_Journal_of_Psychology ↩

  11. Riggs, K.J.; Ferrand, L.; Lancelin, D.; Fryziel, L.; Dumur, G. & Simpson, A. (2006). "Subitizing in tactile perception". Psychological Science. 17 (4): 271–272. doi:10.1111/j.1467-9280.2006.01696.x. PMID 16623680. S2CID 37333935. /wiki/Doi_(identifier) ↩

  12. Camos, V. & Tillmann, B. (2008). "Discontinuity in the enumeration of sequentially presented auditory and visual stimuli". Cognition. 107 (3): 1135–1143. doi:10.1016/j.cognition.2007.11.002. PMID 18068696. S2CID 14999504. /wiki/Doi_(identifier) ↩

  13. Gallace A.; Tan H.Z.; Spence C. (2008). "Can tactile stimuli be subitised? An unresolved controversy within the literature on numerosity judgments". Perception. 37 (5): 782–800. doi:10.1068/p5767. PMID 18605150. S2CID 2820818. /wiki/Hong_Z._Tan ↩

  14. Plaisier, M.A.; Bergmann Tiest, W.M. & Kappers, A.M.L. (2009). "One, two, three, many - Subitizing in active touch". Acta Psychologica. 131 (2): 163–170. doi:10.1016/j.actpsy.2009.04.003. hdl:1874/35195. PMID 19460685. /wiki/Doi_(identifier) ↩

  15. Plaisier, M.A.; Bergmann Tiest, W.M. & Kappers, A.M.L. (2010). "Range dependent processing of visual numerosity: similarities across vision and haptics". Experimental Brain Research. 204 (4): 525–537. doi:10.1007/s00221-010-2319-y. PMC 2903696. PMID 20549196. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2903696 ↩

  16. Plaisier, M.A. & Smeets, J.B.J. (2011). "Haptic subitizing across the fingers". Attention, Perception, & Psychophysics. 73 (5): 1579–1585. doi:10.3758/s13414-011-0124-8. PMC 3118010. PMID 21479724. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3118010 ↩

  17. Ferrand, L.; Riggs, K.J. & Castronovo, J. (2010). "Subitizing in congenitally blind adults". Psychonomic Bulletin & Review. 17 (6): 840–845. doi:10.3758/PBR.17.6.840. PMID 21169578. https://doi.org/10.3758%2FPBR.17.6.840 ↩

  18. Jevons, W.S. (1871). "The power of numerical discrimination". Nature. 3 (67): 281–282. Bibcode:1871Natur...3..281J. doi:10.1038/003281a0. https://doi.org/10.1038%2F003281a0 ↩

  19. Kaufman, E.L.; Lord, M.W.; Reese, T.W. & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology. 62 (4). The American Journal of Psychology: 498–525. doi:10.2307/1418556. JSTOR 1418556. PMID 15392567. /wiki/American_Journal_of_Psychology ↩

  20. Mandler, G. & Shebo, B.J. (1982). "Subitizing: An analysis of its component processes". Journal of Experimental Psychology: General. 111 (1): 1–22. doi:10.1037/0096-3445.111.1.1. PMID 6460833. /wiki/Doi_(identifier) ↩

  21. Kaufman, E.L.; Lord, M.W.; Reese, T.W. & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology. 62 (4). The American Journal of Psychology: 498–525. doi:10.2307/1418556. JSTOR 1418556. PMID 15392567. /wiki/American_Journal_of_Psychology ↩

  22. LaBerge, D.; Carlson, R.L.; Williams, J.K. & Bunney, B.G. (1997). "Shifting attention in visual space: Tests of moving-spotlight models versus an activity-distribution model". Journal of Experimental Psychology: Human Perception and Performance. 23 (5): 1380–1392. doi:10.1037/0096-1523.23.5.1380. PMID 9336958. /wiki/Doi_(identifier) ↩

  23. Atkinson, J.; Campbell, F.W. & Francis, M.R. (1976). "The magic number 4±0: A new look at visual numerosity judgements". Perception. 5 (3): 327–334. doi:10.1068/p050327. PMID 980674. S2CID 26319756. /wiki/Doi_(identifier) ↩

  24. Simon, T.J. & Vaishnavi, S. (1996). "Subitizing and counting depend on different attentional mechanisms: Evidence from visual enumeration in afterimages". Perception & Psychophysics. 58 (6): 915–926. doi:10.3758/BF03205493. PMID 8768186. https://doi.org/10.3758%2FBF03205493 ↩

  25. Atkinson, J.; Campbell, F.W. & Francis, M.R. (1976). "The magic number 4±0: A new look at visual numerosity judgements". Perception. 5 (3): 327–334. doi:10.1068/p050327. PMID 980674. S2CID 26319756. /wiki/Doi_(identifier) ↩

  26. Simon, T.J. & Vaishnavi, S. (1996). "Subitizing and counting depend on different attentional mechanisms: Evidence from visual enumeration in afterimages". Perception & Psychophysics. 58 (6): 915–926. doi:10.3758/BF03205493. PMID 8768186. https://doi.org/10.3758%2FBF03205493 ↩

  27. Corbetta, M.; Shulman, G.L.; Miezin, F.M. & Petersen, S.E. (1995). "Superior parietal cortex activation during spatial attention shifts and visual feature conjunction". Science. 270 (5237): 802–805. Bibcode:1995Sci...270..802C. doi:10.1126/science.270.5237.802. PMID 7481770. S2CID 22131790. /wiki/Bibcode_(identifier) ↩

  28. Piazza, M; Mechelli, A; Butterworth, B & Price, CJ (2002). "Are subitizing and counting implemented as separate or functionally overlapping processes?". NeuroImage. 15 (2): 435–46. doi:10.1006/nimg.2001.0980. PMID 11798277. S2CID 13959500. /wiki/Doi_(identifier) ↩

  29. Balint, R. (1909). "Seelenlahmung des 'Schauens', optische Ataxie, raumliche Storung der Aufmerksamkeit". Monatsschr Psychiatr Neurol (in German). 25 (1): 51–66. doi:10.1159/000210464. https://zenodo.org/record/1448605 ↩

  30. Balint, R. (1909). "Seelenlahmung des 'Schauens', optische Ataxie, raumliche Storung der Aufmerksamkeit". Monatsschr Psychiatr Neurol (in German). 25 (1): 51–66. doi:10.1159/000210464. https://zenodo.org/record/1448605 ↩

  31. Robertson, L.; Treisman, A.; Freidman-Hill, S. & Grabowecky, M. (1997). "The interaction of spatial and object pathways: Evidence from Balint's Syndrome". Journal of Cognitive Neuroscience. 9 (3): 295–317. doi:10.1162/jocn.1997.9.3.295. PMID 23965009. S2CID 27076617. /wiki/Doi_(identifier) ↩

  32. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. ISBN 978-0195110043. 978-0195110043 ↩

  33. Dehaene, S. & Cohen, L. (1994). "Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients". Journal of Experimental Psychology: Human Perception and Performance. 20 (5): 958–975. doi:10.1037/0096-1523.20.5.958. PMID 7964531. /wiki/Doi_(identifier) ↩

  34. Corbetta, M.; Shulman, G.L.; Miezin, F.M. & Petersen, S.E. (1995). "Superior parietal cortex activation during spatial attention shifts and visual feature conjunction". Science. 270 (5237): 802–805. Bibcode:1995Sci...270..802C. doi:10.1126/science.270.5237.802. PMID 7481770. S2CID 22131790. /wiki/Bibcode_(identifier) ↩

  35. Vetter, P; Butterworth, B & Bahrami, B (2008). Warrant, Eric (ed.). "Modulating attentional load affects numerosity estimation: Evidence against a pre-attentive subitizing mechanism". PLOS ONE. 3 (9): e3269. Bibcode:2008PLoSO...3.3269V. doi:10.1371/journal.pone.0003269. PMC 2533400. PMID 18813345. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2533400 ↩

  36. Corbetta, M.; Shulman, G.L.; Miezin, F.M. & Petersen, S.E. (1995). "Superior parietal cortex activation during spatial attention shifts and visual feature conjunction". Science. 270 (5237): 802–805. Bibcode:1995Sci...270..802C. doi:10.1126/science.270.5237.802. PMID 7481770. S2CID 22131790. /wiki/Bibcode_(identifier) ↩

  37. Piazza, M; Mechelli, A; Butterworth, B & Price, CJ (2002). "Are subitizing and counting implemented as separate or functionally overlapping processes?". NeuroImage. 15 (2): 435–46. doi:10.1006/nimg.2001.0980. PMID 11798277. S2CID 13959500. /wiki/Doi_(identifier) ↩

  38. Piazza, M; Mechelli, A; Butterworth, B & Price, CJ (2002). "Are subitizing and counting implemented as separate or functionally overlapping processes?". NeuroImage. 15 (2): 435–46. doi:10.1006/nimg.2001.0980. PMID 11798277. S2CID 13959500. /wiki/Doi_(identifier) ↩

  39. Corbetta, M.; Shulman, G.L.; Miezin, F.M. & Petersen, S.E. (1995). "Superior parietal cortex activation during spatial attention shifts and visual feature conjunction". Science. 270 (5237): 802–805. Bibcode:1995Sci...270..802C. doi:10.1126/science.270.5237.802. PMID 7481770. S2CID 22131790. /wiki/Bibcode_(identifier) ↩

  40. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. ISBN 978-0195110043. 978-0195110043 ↩

  41. Rouselle, L. & Noël, M.P. (2008). "The development of automatic numerosity processes in preschoolers: Evidence for numerosity-perceptual interference". Developmental Psychology. 44 (2): 544–560. doi:10.1037/0012-1649.44.2.544. hdl:2078.1/23088. PMID 18331143. /wiki/Doi_(identifier) ↩

  42. Ciccione, L. & Dehaene, S. (2020). "Grouping mechanisms in numerosity perception". Open Mind. 4 (1): 102–118. doi:10.1162/opmi_a_00037. PMC 8412191. PMID 34485793. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8412191 ↩

  43. Moylan, Andrew. "Test Your "Subitizing" Ability". Wolfram Blog. Wolfram Research. Retrieved 28 December 2020. https://blog.wolfram.com/2011/06/01/test-your-subitizing-ability/ ↩