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.     

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.     

Enumerating by pointing to locations: a new method for measuring the numerosity of visual object representations

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.     

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.     

Effects of age on searching for and enumerating targets that cannot be detected efficiently

We investigated the effects of old age on search, subitizing, and counting of difficult-to-find targets. In Experiment 1, young and older adults enumerated targets (Os) with and without distractors (Qs). Without distractors, the usual subitization-counting function occurred in both groups, with the same subitization span of 3.3 items. Subitization disappeared with distractors; older adults were slowed more overall by their presence but enumeration rates were not slowed by ageing either with or without distractors. In contrast, search rates for a single target (O among Qs) were twice as slow for older as for young adults. Experiment 2 tested and ruled out one account of age-equivalent serial enumeration based on the need to subvocalize numbers as items are enumerated. Alternative explanations based on the specific task differences between detecting and enumerating stimuli are discussed.     

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     

Spatial working memory load affects counting but not subitizing in enumeration

The present study investigated whether subitizing reflects capacity limitations associated with two types of working memory tasks. Under a dual-task situation, participants performed an enumeration task in conjunction with either a spatial (Experiment 1) or a nonspatial visual (Experiment 2) working memory task. Experiment 1 showed that spatial working memory load affected the slope of a counting function but did not affect subitizing performance or subitizing range. Experiment 2 showed that nonspatial visual working memory load affected neither enumeration efficiency nor subitizing range. Furthermore, in both spatial and nonspatial memory tasks, neither subitizing efficiency nor subitizing range was affected by amount of imposed memory load. In all the experiments, working memory load failed to influence slope, subitizing range, or overall reaction time. These findings suggest that subitizing is performed without either spatial or nonspatial working memory. A possible mechanism of subitizing with independent capacity of working memory is discussed.     

ERP研究反映感数与计数的不同脑机制

对于较小数目的快速准确认知的“感数”现象一直是一个引人兴趣的问题,这一感数过程到底本质是什么,它与计数过程究竟是同一种加工还是分属两种不同类别,对于这个问题多年来一直存在着争论。本实验的目的在于研究感数加工的本质,运用ERP手段来探索其与计数的不同机制;并且以分心物变量为指标,研究在有分心物呈现时,感数与计数过程将会有哪些变化以及其潜在的神经机制。对14名正常青年人记录感数与计数加工过程中的事件相关脑电位（ERP）。刺激图片由位于屏幕中心的靶(白色矩形)或者靶+分心物(白色圆形)组成,靶的数目分为感数（1－3个靶）及计数（4－6）;而分心物的数目有三种水平：零（无分心物）,与靶的数目相同及两倍于靶的数目。被试计算图片上白色矩形的个数并对所得数目进行奇偶判断,最后根据奇偶性用左右手进行按键反应。行为结果表明,感数与计数的反应时在靶数目之间有显著性差异,靶数目相同时,分心物越多,则被试所需要的反应时就越长。ERP测量表明P1波幅随着靶数目的增加而增大,随着分心物数目的增加而增加;N1波潜伏期随着靶数目的增加和分心物数目增加而减小,N1波幅随着靶数目和分心物的增加而增加;P3波幅随着靶数目的增加而减小,在某些记录点具有靶效应和干扰效应。研究结果提示感数加工具有明显的分心物效应,而计数加工则不然,支持感数与计数分属两种不同功能加工过程的观点,感数加工更易受到分心物出现的干扰。     

Do the magnocellular and parvocellular visual pathways contribute differentially to subitizing and counting?

We investigated the neurobiologies! basis of visual processes involved in object enumeration. Subitizing, the ability to rapidly and accurately enumerate four or fewer objects, is thought to depend on preattentive processing of visual stimuli, whereas counting of more numerous objects is thought to require serial shifts of attention. We attempted to distinguish between the hypothesis that the magnocellular (M) visual pathway is the preferential route for subitizing, and the alternative hypothesis that there is no selectivity for the M pathway or its counterpart, the parvocellular (P) visual pathway, in visual object enumeration. Green rectangles were presented on an equiluminant red background to impair M pathway processing. This slowed enumeration performance relative to a control condition in which object/background luminance differed, especially when the rectangles were relatively large and widely spaced and had constant retinal eccentricity. When low luminance contrast was used to impair processing along the P pathway, enumeration performance was slowed relative to a high-contrast control condition, especially when the rectangles were small and closely spaced. Overall, our manipulations affected enumeration performance without selectivity for subitizing or counting ranges and without altering the slope of the functions relating reaction time to numerosity. Thus, our results favor the hypothesis that visual enumeration does not depend preferentially on either the M or the P pathway.     

Effects of distraction on visual enumeration in children and adults

Speeded enumeration of visual stimuli typically produces a bilinear function, with a shallow subitizing rate (<100 ms/item) up to 3-4 items (subitizing span) and a steeper counting rate (~300 ms/item) thereafter. FINST theory (L. M. Trick & Z. W. Pylyshyn, 1993, 1994) suggests that subitizing of targets is possible in the presence of distractors if attention is not required for target detection, but this has not been tested in children. The present study explored enumeration without distractors (Os alone) and with distractors (Os among Xs) in 35 children aged 6-11 years and 17 adults. Subitizing span increased significantly from childhood to adulthood, and counting rate increased significantly with age. Bilinear functions were significantly better than linear fits to the data for most children and adults both without distractors (97% and 100%, respectively) and with distractors (89% and 94%), consistent with their efficient visual search for a single O among multiple Xs. These findings are discussed in comparison with those from new modeling of earlier enumeration data from young and older adults, revealing striking asymmetries in subitizing with distractors between development and aging.     

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.     

The role of pattern recognition in children's exact enumeration of small numbers

Enumeration can be accomplished by subitizing, counting, estimation, and combinations of these processes. We investigated whether the dissociation between subitizing and counting can be observed in 4‐ to 6‐year‐olds and studied whether the maximum number of elements that can be subitized changes with age. To detect a dissociation between subitizing and counting, it is tested whether task manipulations have different effects in the subitizing than in the counting range. Task manipulations concerned duration of presentation of elements (limited, unlimited) and configuration of elements (random, line, dice). In Study 1, forty‐nine 4‐ and 5‐year‐olds were tested with a computerized enumeration task. Study 2 concerned data from 4‐, 5‐, and 6‐year‐olds, collected with Math Garden, a computer‐adaptive application to practice math. Both task manipulations affected performance in the counting, but not the subitizing range, supporting the conclusion that children use two distinct enumeration processes in the two ranges. In all age groups, the maximum number of elements that could be subitized was three. The strong effect of configuration of elements suggests that subitizing might be based on a general ability of pattern recognition.     

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.     

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.     

Subitizing speed,subitizing range,counting speed,the Stroop effect,and aging: capacity differences and speed equivalence

Thirty younger and 29 older adults were tested on reaction times for set size of a display of 1 to 9 digits. On half of the trials, the nominal value of the digits was equal to the size of the set displayed; on the other half, the value differed by 1 from the set size (Stroop interference). We found evidence for age differences in subitizing span (2.83 vs. 2.07). Once individual differences in subitizing range were taken into account, no age differences were found in the rate of either subitizing or counting, and no individual differences were apparent in subitizing speed. There was no age difference in the susceptibility to the Stroop effect. The results suggest that, with advancing age, the size of the focus of attention may shrink, but speed of access to elements in the focus of attention may remain constant.     

Subitizing in congenitally blind adults

We investigated the role of vision in tactile enumeration within and outside the subitizing range. Congenitally blind and sighted (blindfolded) participants were asked to enumerate quickly and accurately the number of fingers stimulated. Both groups of participants enumerated one to three fingers quickly and accurately but were much slower and less accurate with four to nine fingers. Within the subitizing range, blind participants performed no differently from both sighted (blindfolded) and sighted-seeing participants. Outside of the subitizing range, blind and sighted-seeing participants showed better performance than did sighted-blindfolded participants, suggesting that lack of access to the predominant sensory modality does affect performance. Together, these findings further support the claim that subitizing is a general perceptual mechanism and demonstrate that vision is not necessary for the development of the subitizing mechanism.     

Mathematics anxiety affects counting but not subitizing during visual enumeration

Individuals with mathematics anxiety have been found to differ from their non-anxious peers on measures of higher-level mathematical processes, but not simple arithmetic. The current paper examines differences between mathematics anxious and non-mathematics anxious individuals in more basic numerical processing using a visual enumeration task. This task allows for the assessment of two systems of basic number processing: subitizing and counting. Mathematics anxious individuals, relative to non-mathematics anxious individuals, showed a deficit in the counting but not in the subitizing range. Furthermore, working memory was found to mediate this group difference. These findings demonstrate that the problems associated with mathematics anxiety exist at a level more basic than would be predicted from the extant literature.     

Individual differences in working memory capacity and enumeration

Two experiments are reported in which subjects performed working memory and enumeration tasks. In the first experiment, subjects scoring low on the working memory task also performed poorly on the attention-demanding "counting" portion of the enumeration task. Yet no span differences were found for the non-attention-demanding "subitizing" portion. In Experiment 2, conjunctive and disjunctive distractors were added to the enumeration task. Although both high and low working memory span subjects were adversely affected by the addition of conjunctive distractors, the effect was much greater for the low-span subjects. Implications from these findings are that differences in working memory capacity correspond to differences in capability for controlled attention.