Subitizing

This proposal explores how the activity of subitizing – quickly apprehending the numerosity of a small set of items – changes with the development of number concepts. We describe how varying the orientations of items in teaching experiment sessions promoted one pre-schooler, Frank, to attend to subgroups of items and change his thinking about conjoining two numbers. The results illustrate how game-play oriented subitizing activities may support growth in number understandings.

Humans seem to be genetically endowed with a mathematical aptitude known as subitizing. It refers to the ability to correctly recognize very limited quantities (up to 3/4 items). Conversely, higher cardinalities require a different, non-innate process: counting. In keeping with the hypothesis that cognition plays a major role in structuring the linguistic complexity (cf. Lakoff and Johnson, 1980; Heine, 1997), here I suggest regarding subitizing as a significant shaping force in language. Our cognitive limitations may be reflected in the organization of linguistic structures, especially in the field of numerals. There are a number of clues which lead us to believe that certain fundamental linguistic features have autonomy and priority compared to others. These include: the diffusion of “1-2-(3)-many” numeral systems, the morpho-syntax of the lowest number words, the suppletion of the first ordinals, etc. Evidence of this in the IE. family was collected in §4. The split between the set of the first few elements and the rest is sometimes regarded as accidental or due to a chronological gap in historical evolution. However, I think that subitizing, as an innate and universal cognitive pattern, can provide a more satisfactory explanation. Since the first quantities are perceived more clearly and immediatly, the structure of some linguistic domains (e.g. numerals) reflects this cognitive hierarchy.

Descartes holds that the tell-tale sign of a solid proof is that its entailments appear clearly and distinctly. Yet, since there is a limit to what a subject can consciously fathom at any given moment, a mnemonic shortcoming threatens to render complex geometrical reasoning impossible. Thus, what enables us to recall earlier proofs, according to Descartes, is God’s benevolence: He is too good to pull a deceptive switch on us. Accordingly, Descartes concludes that geometry and belief in God must go hand in hand. However, I argue that, while theism adds a layer of psychological reassurance, the mind-independent reality of God would ensure the preservation of past demonstrations for atheists as well.

HUMAN infants can discriminate between different small numbers of items 1���4, and can determine numerical equivalence across perceptual modalities 5, 6. This may indicate the possession of true numerical concepts 1, 4���7. Alternatively, purely perceptual discriminations may underlie these abilities 8, 9. This debate addresses the nature of subitization, the ability to quantify small numbers of items without conscious counting 10, 11. Subitization may involve the holistic recognition of canonical perceptual patterns that do ...

This 22-session constructivist teaching experiment set out to investigate a preschool student’s number understanding relative to his subitizing activity. Subitizing, a quick apprehension of the numerosity of a small set of items, has been found to characterize perceptual and conceptual processes students rely on as their understanding of number develops. The purpose for this study is to investigate how a preschool student’s, Ben, perceptual subitizing activity changed relative to the density of items and the development of his number understanding. Findings indicated that early on in the teaching experiment, Ben’s perceptual subitizing activity was influenced by his primary reliance upon the perceived amount of space between items. Shifts in reasoning
when perceptually subitizing indicated physiological and experiential development in Ben’s number understanding, as Ben described the number of items increasing when the perceived amount of space between items decreased. Number conservation was considered as relevant to these findings because Ben’s explanation for why a number
could increase or decrease mirrored similar logic when unable to conserve number. Implications of this study suggest nuances in number understanding development which can explain preschool students’ reliance upon a more refined set of perceptual subitizing.

Subitizing and Mathematics Performance in Early Childhood Cathy Yun (Cathy.Yun@Vanderbilt.Edu) Peabody Research Institute, Box 0181 GPC, 230 Appleton Place Nashville, TN 37203 USA Amanda Havard (Amanda.Havard@Gmail.Com) Peabody Research Institute, Box 0181 GPC, 230 Appleton Place Nashville, TN 37203 USA Dale C. Farran (Dale. Farran@Vanderbilt.Edu) Peabody Research Institute, Box 0181 GPC, 230 Appleton Place Nashville, TN 37203 USA Mark W. Lipsey (Mark.Lipsey@Vanderbilt.Edu) Peabody Research Institute, Box 0181 GPC, 230 Appleton Place Nashville, TN 37203 USA Carol Bilbrey (Carol.Bilbrey@Vanderbilt.Edu) Peabody Research Institute, Box 0181 GPC, 230 Appleton Place Nashville, TN 37203 USA Kerry G. Hofer (Kerry.G.Hofer@Vanderbilt.Edu) Peabody Research Institute, Box 0181 GPC, 230 Appleton Place Nashville, TN 37203 USA but many studies agree on a range of 1-4 (Trick & Pylyshyn, Children as young as 2 can subitize, but both children’s subitizing speed and range may differ from that of adul...

Subitizing is a fast and accurate enumeration process of small sets of usually less than four objects. Several models were proposed in the literature. Critically, only pattern recognition theory suggests that subitizing performance is sensitive to the arrangement of the array. In our study, arrays of dots in random or canonical arrangements were enumerated. The subitizing range was larger and the reaction time slope was less steep in the canonical arrangements. When noise was added to the canonical pattern, the reaction time slope was proportional to the amount of noise. Moreover, arrangement has a stronger effect on sets with more than four objects. These results support the pattern recognition model of subitizing.

Full title: Objecthood Before Trees, Subitizing Before Forest: Does Perceptual Organization Affect Subitizing? Abstract: “Subitizing” refers to people's ability to rapidly and efficiently enumerate a small amount of items. According to a dominant explanation, this effect is based on pre-attentive individuation mechanisms that rely primarily on the processing of primitive features (e.g., color, orientation). However, other researchers claimed that other early perceptual processes are also likely to influence Subitizing. In order to pinpoint the locus of the effect, the present study tested whether and how perceptual organization affects Subitizing. Participants were asked to quickly determine the number of one to four red (blue) L-shaped targets, embedded among blue (red) distractors. The items were arranged in a 4 by 4 array and in some trials, four of the items were rotated such that they formed a square-like object. The results showed that this object had a strong influence on performance. When all the targets were part of the object responses were faster than when there was no object in the display. Conversely, responses were slower when there was an object, but it did not contain all the targets. This pattern was replicated in Experiment 2 in which the display was presented only briefly. Experiment 3 ruled out a proximity explanation by showing that the effects were eliminated when the same conditions were repeated but no object was present. These data clearly show that perceptual organization and Subitizing do not occur in parallel, but rather interact with one another. Furthermore, it demonstrates that even Subitizing can be improved by perceptual organization factors.

When seeing a jaguar, we can see all the spots on its mantle without seeing a determinate number, N, of spots on the mantle. How is this visual phenomenon possible? Philosophers have tried to provide a reliable answer to this question, by recruiting evidence from vision science about the way attention works. Here we push this idea forward, by suggesting that an alternative and less complex solution, with respect to the one proposed in the literature, is possible. In particular, we argue that the puzzling visual phenomenon of ‘seeing entities without seeing N-entities’ strictly depends on the specific number of entities we are simultaneously attending to: a problematic scenario concerning this visual process arises only when the number of entities in the visual field exceeds a specific quantity. This depends, as we argue, on the fact that the nature of our visual content is modulated, in this perceptual scenario, by the limited way we can exercise visual attention on the properties of the objects of our per- ception. Differently from other accounts, our proposal allows us to properly define when and why such a problem arises, and explains why such a situation has traditionally been found to be so puzzling. Our idea is also well motivated by experimental results that so far have not been taken into account in the debate.

Subitising, a quick apprehension of the numerosity of a small set of items, has been found to change from an individual's reliance on perceptual to conceptual processes. In this study, we utilised a constructivist teaching experiment methodology to investigate how the subitising activity of one preschool student, Amy, related to her construction of prenumerical units. Subitising and counting tasks were designed to assess and perturb Amy's thinking relative to her construction of units, and to observe changes in Amy's activity associated with the different tasks. Findings indicate that as Amy's subitising activity changed from perceptual to conceptual, she constructed subitised motor units and subitised figurative units. Implications of this study suggest that the construction of subitised units may support young children's later development of arithmetic units.

This study examines the encoding of multiple object locations into spatial memory by comparing localization accuracy for stimuli presented at different exposure durations. Participants in the longest duration condition viewed masked displays containing 1–10 discs for 1–10 s (durations typically used in simple span tasks), and then reported the locations of these discs on a blank screen. Compared to conditions that presented the same stimuli briefly for 50 or 200 ms (exposures more typical of simultaneous spatial arrays), localization accuracy did not improve significantly under longer viewing durations. Additionally, a clustering analysis found that responses were spread among different clusters of discs and not focused on individual clusters, regardless of viewing duration. A second experiment tested this performance for displays containing two distinct clusters of discs to determine if clearly grouped subsets of objects would improve performance, but there was no substantial improvement for these two-cluster displays when compared to displays with one cluster. Overall, the results indicate that spatial information for a set of objects is extracted globally and quickly, with little benefit from extended encoding durations that should have favored some deliberative form of grouping. Such results cast doubt on the validity of Corsi blocks or equivalent common neuropsychological tests purportedly designed to evaluate specifically spatial short-term memory spans.

The fast and accurate enumeration of a small set of objects, called subitizing, is thought to involve a different mechanism from other numerosity judgments, such as those based on estimation. In this report, we examine the subitizing limit using a novel enumeration task that obtained the perceived locations of enumerated objects. Observers were shown brief masked displays (50, 200, and 350 ms) of 2–9 small black discs randomly placed on a gray screen and then asked to place a marker where each disc had been located. The number of these markers provided an estimate of the number of items processed. This “pointing” methodology enabled observers to accurately “enumerate” displays containing up to six items in contrast with the four-item limit typically found when using standard reporting methods (and replicated here in Experiment 2). These results suggest a different account of the limits found in most subitizing and enumeration studies.

It has been proposed that the mechanism that supports the ability to keep track of multiple moving objects also supports subitizing—the ability to quickly and accurately enumerate a small set of objects. To test this hypothesis, we investigated the effects on subitizing when human observers were required to perform a multiple object tracking task and an enumeration task simultaneously. In three experiments, participants (Exp. 1, N = 24; Exp. 2, N = 11; Exp. 3, N = 37) enumerated sets of zero to nine squares that were flashed while they tracked zero, two, or four moving discs. The results indicated that the number of items participants could subitize decreased by one for each item they tracked. No such pattern was seen when the enumeration task was paired with an equally difficult, but nonvisual, working memory task. These results suggest that a shared visual mechanism supports multiple object tracking and subitizing.

A recent article argues that pure working memory in humans and chimpanzees have the same size. However, the data offered to support this claim show the opposite, namely that the size of pure working memory in chimpanzees is smaller than that of humans. In addition, extensive data show that the effective size of working memory in chimpanzees is much smaller than in humans. Altogether, there are quantitative differences in the size of working memory (pure or effective) between chimpanzees and humans leading to significant qualitative differences between them.