More than superstition: differential effects of featural heterogeneity and change on subitizing and counting

This study investigates the effects of item heterogeneity (differences in color and shape) and moment-to-moment feature change as it relates to the issue of whether subitizing and counting involve different processes. Participants enumerated displays of up to eight items that were either homogeneous or heterogeneous. In situations where the heterogeneous displays always had approximately half of the items of one type and half of the other, heterogeneity significantly sped enumeration in the counting range (6-8 items) and significantly slowed enumeration in the subitizing range (1-3 items), a dissociation that suggests that subitizing and counting involve different operations. Moment-to-moment feature change had no effect on subitizing. However, feature change slowed counting, but only when participants were enumerating heterogeneous items that were half of one type and half of the other, as might be expected if participants were using differences in features to select items by type.     

The role of attention in subitizing

The process of rapidly and accurately enumerating small numbers of items without counting, i.e. subitizing, is often believed to rest on parallel preattentive processes. However, the possibility that enumeration of small numbers of items would also require attentional processes has remained an open question. The present study is the first that directly contrasts the preattentive and attentive models of subitizing. We used an inattentional blindness paradigm to manipulate the availability of attentional resources during enumeration. In the inattention condition, the items to be enumerated were presented unexpectedly while participants focused on a line length comparison task. Divided- and full-attention conditions were also included. The results showed that only numbers one and two could be enumerated when the effects of attention were minimized. Freeing attentional resources increased the enumeration accuracies considerably, including for number two. The results suggest that even for enumerating small numbers, the attentional demands increase as the number of objects increases.     

感数——它能告诉我们什么?

研究者在列举任务中发现,被试对3个以内项目的报告既快又准确（一般称之为“感数”）,而对3个以上项目的报告既慢又容易出错误（“计数”）,由此他们提出感数和计数属于两种不同性质的加工过程,一系列行为数据的反应时和正确率指标支持了这一假设。此外,最近的脑成像、电生理研究还发现,感数和计数在对注意的需求上同样存在着分离——感数无需注意,只有计数过程才需注意的参与。在此基础上,研究者从不同角度提出了一些解释感数现象的理论     

Evidence for a shared mechanism used in multiple-object tracking and subitizing

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.     

Effects of color heterogeneity on subitization

Small numbers of items can be enumerated rapidly and accurately via a process termedsubitizing. In two experiments, we examined the effect of target color heterogeneity on subitizing efficiency. In contrast to the findings of Puts and de Weert (1997), we found that observers were no less efficient at subitizing displays containing red and green items than they were at subitizing displays of a single color. We propose that these findings are consistent with subitization operating on items represented within a location master map that codes where objects are but not what they are. The data are discussed in relation to recent findings concerning the detection of single-feature targets and the functional architecture of early visual processing.     

Enumeration of small and large numerosities: the effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1-8) or by tens (10-80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

Enumeration of small and large numerosities: The effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1–8) or by tens (10–80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

What dot clusters and bar graphs reveal: subitizing is fast counting and subtraction

In two studies, we found that dot enumeration tasks resulted in shallow-sloped response time (RT) functions for displays of 1-4 dots and steep-sloped functions for displays of 5-8 dots, replicating results implicating subitizing and counting processes for low and high ranges of dots, respectively. Extracting number from a specific type of bar graph within the same numerical range produced a shallow-sloped but scallop-shaped RT function. Factor analysis confirmed two independent subranges for dots, but all bar graph values defined a unitary factor. Significantly, factor scores and asymmetries both showed correlations of bar graph recognition to dot subitizing but not to dot counting, strongly suggesting that subitizing was used in both enumeration of low numbers of dots and bar graph recognition. According to these results, subitizing appears to be a nonverbal process operating flexibly in either additive or subtractive fashion on analog quantities having spatial extent, a conclusion consistent with a fast-counting model of subitizing but not with other models of the subitizing process.     

Do subitizing deficits in developmental dyscalculia involve pattern recognition weakness?

The abilities of children diagnosed with developmental dyscalculia (DD) were examined in two types of object enumeration: subitizing, and small estimation (5–9 dots). Subitizing is usually defined as a fast and accurate assessment of a number of small dots (range 1 to 4 dots), and estimation is an imprecise process to assess a large number of items (range 5 dots or more). Based on reaction time (RT) and accuracy analysis, our results indicated a deficit in the subitizing and small estimation range among DD participants in relation to controls. There are indications that subitizing is based on pattern recognition, thus presenting dots in a canonical shape in the estimation range should result in a subitizing‐like pattern. In line with this theory, our control group presented a subitizing‐like pattern in the small estimation range for canonically arranged dots, whereas the DD participants presented a deficit in the estimation of canonically arranged dots. The present finding indicates that pattern recognition difficulties may play a significant role in both subitizing and subitizing deficits among those with DD.     

Subitizing and counting in typical and atypical development

Enumeration performance in standard dot counting paradigms was investigated for different age groups with typical and atypically poor development of arithmetic skills. Experiment 1 showed a high correspondence between response times and saccadic frequencies for four age groups with typical development. Age differences were more marked for the counting than the subitizing range. In Experiment 2 we found a discontinuity between subitizing and counting for dyscalculic children; however, their subitizing slopes were steeper than those of typically developing control groups, indicating a dysfunctional subitizing mechanism. Across both experiments a number of factors could be identified that affect enumeration in the subitizing and the counting range differentially. These differential patterns further support the assumption of two qualitatively different enumeration processes.     

Intentional subitizing: exploring the role of automaticity in enumeration

Subitizing is traditionally described as the rapid, preattentive and automatic enumeration of up to four items. Counting, by contrast, describes the enumeration of larger sets of items and requires slower serial shifts of attention. Although recent research has called into question the preattentive nature of subitizing, whether or not numerosities in the subitizing range can be automatically accessed is yet to be empirically tested. In the current study, participants searched for two pre-defined digits in a circular visual-search array. Distractor dots of various set sizes were placed at the centre of the array. Despite the relevance of the distractor numerosities to the target detection task, the distractors did not influence target detection, thereby suggesting that their numerosities were not automatically accessed in Experiment 1. In Experiment 2, participants were explicitly instructed to enumerate the distractor dots. Here, congruent and incongruent distractor numerosities influenced the target detection task, thereby revealing that the distractor dots were capable of generating interference. Experiment 3 ensured that dots were attended by asking participants to detect the luminance of dots. Data confirmed that subitizing was not automatic. The present study also supported the alleged discontinuity between the subitizing and counting ranges because an examination of reaction time gradients in Experiment 2 found the counting gradient to be significantly steeper than the subitizing gradient. In sum, the results suggest that subitizing is a distinct but non-automatic style of enumeration.     

Small Subitizing Range in People with Williams syndrome

Recent evidence suggests that the rapid apprehension of small numbers of objects-- often called subitizing-- engages a system which allows representation of up to 4 objects but is distinct from other aspects of numerical processing. We examined subitizing by studying people with Williams syndrome (WS), a genetic deficit characterized by severe visuospatial impairments, and normally developing children (4-6.5 years old). In Experiment 1, participants first explicitly counted displays of 1 to 8 squares that appeared for 5 s and reported "how many". They then reported "how many" for the same displays shown for 250 ms, a duration too brief to allow explicit counting, but sufficient for subitizing. All groups were highly accurate up to 8 objects when they explicitly counted. With the brief duration, people with WS showed almost perfect accuracy up to a limit of 3 objects, comparable to 4 year-olds but fewer than either 5 or 6.5 year-old children. In Experiment 2, participants were asked to report "how many" for displays that were presented for an unlimited duration, as rapidly as they could while remaining accurate. Individuals with WS responded as rapidly as 6.5 year-olds, and more rapidly than 4 year-olds. However, their accuracy was as in Experiment 1, comparable to 4 year-olds, and lower than older children. These results are consistent with previous results indicating that people with WS can simultaneously represent multiple objects, but that they have a smaller capacity than older children, on par with 4 year-olds. This pattern is discussed in the context of normal and abnormal development of visuospatial skills, in particular those linked to the representation of numerosity.     

Age differences in enumerating things that move: Implications for the development of multiple-object tracking

The attentional theory of spatial enumeration (Trick & Pylyshyn, 1994) predicts that subitizing, the rapid process (40-120 msec/item) used to enumerate 1-4 items, employs the same mechanism that permits individuals to track 4-5 moving items simultaneously, whereas enumerating more items requires moving attentional focus from area to area in the display. To test this theory, enumeration of static and moving items was investigated in 8-, 10-, 12-, and 20-year-old participants using a number discrimination task. As was predicted, random independent item motion did not substantially impede enumeration of 1-4 items regardless of age. However, even movement within a 1.14 degree square area slowed enumeration of 6-9 items, although on average the interference decreased with age from 788 msec for the 8-year-olds to 136 msec for the 20-year-olds. The relevance of this finding for theories of enumeration, multiple-object tracking, visual working memory, and object-based attention is discussed.     

Attentional involvement in subitizing: Questioning the preattentive hypothesis

Subitizing, the rapid, effortless, and accurate enumeration of small numbers of items has been said to be carried out preattentively. If so, the preattentive processing could occur entirely in primary visual cortex, in which case it would be completed within about 50 ms, or during the feedforward sweep, in which case it would be completed within about 100 ms after the onset of an enumeration display which was followed by a mask. The stimulus–mask SOAs were 50 or 83 ms in Experiment 1, and 100 or 150 ms in Experiment 2. In both experiments subjects were more accurate and more sensitive to one-target than to two-target displays. These outcomes, in conjunction with the finding that accuracy and sensitivity were higher at the longer SOA, are inconsistent with the preattentive viewpoint and point to the necessity for attentional involvement in subitizing.     

Is subitizing a unique numerical ability?

Two models that predict the relation between mean enumeration time and numerosity in a speeded enumeration experiment are tested. The first is a bilinear two-process model, and the second is a log-linear single-process model. Previously, support for the bilinear model has provided evidence for the existence of a unique numerical ability called “subitizing.” Both models are shown to yield close approximations to the empirical data, but at the same time to consistently violate the robust shape of these data. Two fundamental discrepancies exist: (1) Enumeration of singleelement displays is unpredictably fast, both in the data reported here and elsewhere, and (2) the response-time function for multiple elements is continuously convex upward, with a significant log-quadratic component. The findings support the contention that enumeration is a capacitylimited process, but no statistically reliable change in processing character, that is, from subitizing to some other process, is evident in enumeration of displays of up to six elements.     

Small subitizing range in people with Williams syndrome

Recent evidence suggests that the rapid apprehension of small numbers of objects—often called subitizing—engages a system which allows representation of up to 4 objects but is distinct from other aspects of numerical processing. We examined subitizing by studying people with Williams syndrome (WS), a genetic deficit characterized by severe visuospatial impairments, and normally developing children (4–6.5 years old). In Experiment 1, participants first explicitly counted displays of 1 to 8 squares that appeared for 5 s and reported “how many”. They then reported “how many” for the same displays shown for 250 ms, a duration too brief to allow explicit counting, but sufficient for subitizing. All groups were highly accurate up to 8 objects when they explicitly counted. With the brief duration, people with WS showed almost perfect accuracy up to a limit of 3 objects, comparable to 4-year-olds but fewer than either 5- or 6.5-year-old children. In Experiment 2, participants were asked to report “how many” for displays that were presented for an unlimited duration, as rapidly as possible while remaining accurate. Individuals with WS responded as rapidly as 6.5-year-olds, and more rapidly than 4-year-olds. However, their accuracy was as in Experiment 1, comparable to 4-year-olds and lower than older children. These results are consistent with previous findings, indicating that people with WS can simultaneously represent multiple objects, but that they have a smaller capacity than older children, on par with 4-year-olds. This pattern is discussed in the context of normal and abnormal development of visuospatial skills, in particular those linked to the representation of numerosity.     

Subitizing and counting depend on different attentional mechanisms: Evidence from visual enumeration in afterimages

Two experiments showed that, when selective eye movements were disabled by the presentation of stimuli in the form of afterimages, increased inspection time and facilitative stimulus configurations failed to increase the subitizing limit of 4 objects. Afterimages of two to eight dots induced by a photographic flashgun were shown to 3 adult subjects. For more than 4 objects, enumeration errors occurred at a rate of 20%–30%. Enumeration was effectively perfect for 2–4 linearly configured dots, with occasional errors surprisingly occurring in that range when dots appeared in groups of up to 3 items. No errors occurred in nonafterimage control conditions. Enumeration errors were attributed to failures of individuating dots to be counted due to the deactivation of selective eye movements in afterimages. A third experiment supported this interpretation by disabling eye movements with briefly presented stimuli and producing results much like those of the afterimage conditions.     

The role of attention in subitizing: Is the magical number 1?

Subitizing, the fast and accurate enumeration of up to about 3 or 4 objects, has often been thought to be dependent on limited-capacity preattentive mechanisms. We used an attentional blink paradigm to investigate the extent to which subitizing requires attentional resources. On each trial, subjects identified a target letter in an RSVP stream and then enumerated dots presented in the stream that were either simultaneous with the target letter or followed it by up to 400 ms. For numerosities from 2 to 9, evidence of an attentional blink was observed; only enumeration of 0 or 1 elements was independent of lag. Thus, even enumeration of 2–3 objects, which is within the traditional subitizing range, appears to require attentional resources. The relation of this work to studies on the attentional requirements of detecting a unique item among distractors, a supposedly preattentive discrimination, is briefly discussed.     

Cross-modal attention modulates tactile subitizing but not tactile numerosity estimation

Debate remains about whether the same attentional mechanism subserves subitizing (with number of items less than or equal to 4) and numerosity estimation (with number of items equal to or larger than 5), and evidence is scarce from the tactile modality. Here, we examined tactile numerosity perception. Using tactile Braille displays, participants completed the following three main tasks: (1) Unisensory task with focused attention: Participants reported the number (1~12) of the tactile pins. (2) Unisensory task with divided attention: Participants compared the numbers of pins across the upper and lower area of their left index fingers, in addition to reporting the number of tactile pins on their right index fingers. (3) Cross-modal task with divided attention: Participants reported the number of tactile pins and compared the numbers of visual dots across the upper and lower part of a (illusory) rectangle that overlaid the tactile stimuli. We found that performance of subitizing rather than estimation was interfered with in dual tasks, regardless of whether distractor events were from the same modality (tactile modality) or from a different modality (visual modality). Moreover, a further test of visual/tactile working memory capacity revealed that the precision of tactile subitizing, in the presence of a visual distractor, was correlated with the capacity of visual working memory, not of tactile working memory. Overall, our study revealed that tactile numerosity perception is accounted for by amodal attentional modulation yet by differential attentional mechanisms in terms of subitizing and estimation.