Subitizing and counting

Subitizing, coined in 1949 by E.L. Kaufman et al.cite journal|author=Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J|year=1949|title=The discrimination of visual number|journal=American Journal of Psychology|volume=62|pages=498–525|doi=10.2307/1418556] refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective "subitus" (meaning "sudden") and captures a feeling of immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range. Number judgments for larger set-sizes were referred to either as counting or estimating, depending on the number of elements present within the display, and the time given to observers in which to respond (i.e., estimation occurs if insufficient time is available for observers to accurately count all the items present).

The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapidcite journal|author=Saltzman, I. J., & Garner, W. R|year=1948|title=Reaction time as a measure of span of attention|journal=The Journal of Psychology|volume=25|pages=227–241] , accuratecite journal|author=Jevons, W. S|year=1871|title=The power of numerical discrimination|journal=Nature|volume=3|pages=281–282|doi=10.1038/003281a0] and confident. [cite journal|author=Taves, E. H.|year=1941|title=Two mechanisms for the perception of visual numerousness|journal=Archives of Psychology|volume=37|pages=1–47] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence. In addition, response times rise in a dramatic fashion, with an extra 250 ms – 350 ms added for each additional item within the display beyond about four. [cite journal|author=Trick, L. M., & Pylyshyn, Z. W|year=1994|title=Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision|journal=Psychological Review|volume=101|issue=1|pages=80–102|doi=10.1037/0033-295X.101.1.80]

While the increase in response time for each additional element within a display is relatively large outside the subitizing range (i.e., 250 ms – 350 ms per item), there is still a significant, albeit smaller, increase within the subitizing range, for each additional element within the display (i.e., 40 ms – 100 ms per item). A similar pattern of reaction times is found in young children, although with steeper slopes for both the subitizing range and the enumeration range. . [cite journal|author= Chi, M.T.H. & Klahr, D. |year=1975|title= Span and rate of apprehension in children and adults |journal= Journal of Experimental Child Psychology |volume=19| pages=434- 439| ] This suggests there is no span of apprehension as such, if this is defined as the number of items which can be immediately apprehended by cognitive processes, since there is an extra cost associated with each additional item enumerated. However, the relative difference in costs associated with enumerating items within the subitizing range are small, whether measured in terms of accuracy, confidence, or speed of response. Furthermore, the values of all measures appear to differ markedly inside and outside the subitizing range. So, while there may be no span of apprehension, there appear to be real differences in the ways in which a small number of elements is processed by the visual system (i.e., approximately < 4 items), compared with larger numbers of elements (i.e., approximately > 4 items). Recent findings [cite journal|author=Riggs, K. J, Ferrand, L., Lancelin, D., Fryziel, L., Dumur, G., & Simpson, A.|year=2006|title=Subitizing in tactile perception|journal=Psychological Science|volume=17|issue=4|pages=271–272|doi=10.1111/j.1467-9280.2006.01696.x] demonstrated that subitizing and counting are not restricted to visual perception, but also extend to tactile perception (when observers had to name the number of stimulated fingertips).

=Enumerating after

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. 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. The ability of observers to count the number of items within a display can be limited, either by the rapid presentationand subsequent masking of items, [cite journal|author=Mandler, G., & Shebo, B. J|year=1982|title=Subitizing: An analysis of its component processes|journal=Journal of Experimental Psychology: General|volume=111|pages=1–22|doi=10.1037/0096-3445.111.1.1] or by requiring observers to respond quickly. 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" [cite journal|author=LaBerge, D., Carlson, R. L., Williams, J. K., & Bunney, B. G|year=1997|title=Shifting attention in visual space: Tests of moving-spotlight models versus an activity-distribution model|journal=Journal of Experimental Psychology: Human Perception and Performance|volume=23|pages=1380–1392|doi=10.1037/0096-1523.23.5.1380] successively to different elements within the display.

Atkinson, Campbell, and Franciscite journal|author=Atkinson, J., Campbell, F. W., & Francis, M. R.|year=1976|title=The magic number 4+-0: A new look at visual numerosity judgements|journal=Perception|volume=5|pages=327–334|doi=10.1068/p050327] 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. 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. This result was replicated by Simon and Vaishnavi.cite journal|author=Simon, T. J., & Vaishnavi, S|year=1996|title=Subitizing and counting depend on different attentional mechanisms: Evidence from visual enumeration in afterimages|journal=Perception & Psychophysics|volume=58|issue=6|pages=915–926]

Brain structures involved in subitizing and counting

The work on the enumeration of afterimages 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.

Balint's syndrome

Clinical evidence 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 Balint's syndrome.cite journal|author=Balint, R|year=1909|title=Seelenlahmung des 'Schauens', optische Ataxie, raumliche Storung der Aufmerksamkeit|journal=Monatschrift für Psychiatrie und Neurologie|volume=25|pages=5–81] 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. Despite these dramatic symptoms, such patients are able to correctly recognize individual objects. [cite journal|author=Robertson, L., Treisman, A., Freidman-Hill, S., & Grabowecky, M.|year=1997|title=The interaction of spatial and object pathways: Evidence from Balint's Syndrome|journal=Journal of Cognitive Neuroscience|volume=9|issue=3|pages=295–317|doi=10.1162/jocn.1997.9.3.295] 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. [cite book|author=Dehaene, S|year=1997|title=The number sense: How the mind creates mathematics|location=New York | publisher=Oxford University Press]

However, people with simultanagnosia have no difficulty enumerating objects within the subitizing range. [cite journal|author=Dehaene, S., & Cohen, L|year=1994|title=Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients|journal=Journal of Experimental Psychology: Human Perception and Performance|volume=20|issue=5|pages=958–975|doi=10.1037/0096-1523.20.5.958] The disorder is associated with bilateral damage to the parietal lobe , an area of the brain linked with spatial shifts of attention.cite journal|author=Corbetta, M., Shulman, G. L., Miezin, F. M., & Petersen, S. E|year=1995|title=Superior parietal cortex activation during spatial attention shifts and visual feature conjunction|journal=Science|volume=270|pages=802–805|doi=10.1126/science.270.5237.802|pmid=7481770] These neuropsychological results are consistent with the view that the process of counting, but not that of subitizing, requires active shifts of attention.

Imaging enumeration

Further evidence for this view comes from a positron emission tomography (PET) study in normal observers. In it the relative brain activity associated with enumeration processes inside (i.e., 1&ndash;4 items), and outside (i.e., 5–8 items), the subitizing range was compared.Fact|date=February 2007

Observers were presented with visual arrays. Each array was composed of 16 bars, which were arranged on a 4 x 4 imaginary grid. The task of observers was to enumerate the number of vertical targets present. The performance of observers was consistent with previous enumeration experiments of this type.cite journal|author=Trick, L. M., & Pylyshyn, Z. W|year=1993|title=What enumeration studies can show us about spatial attention: Evidence for limited capacity preattentive processing|journal=Journal of Experimental Psychology: Human Perception and Performance|volume=19|pages=331–351|doi=10.1037/0096-1523.19.2.331] Reaction times were fast, and error rates low, within the subitizing range; outside the subitizing range both measures increased monotonically. Enumeration within the subitizing range activated bilateral sites in occipital extrastriate cortex, consistent with the hypothesis that subitizing is associated with early stages of visual processing. Enumeration outside the subitizing range (i.e., counting) activated the same extrastriate regions associated with subitizing, as well as additional sites in the superior parietal lobe/intraparietal sulcus bilaterally, the right inferior frontal regions, and the anterior cingulate. These patterns of activation support the view that subitizing and counting involve distinct cortical processes. Although both subitizing and counting are associated with activation of extrastriate regions, only counting is associated with activation of regions involved in the shifting of attention.

References