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 processes in young children

Children attending the first grade in school were instructed to tell the number of dots (1–9) presented on a screen. The response latencies were related to the number of dots by two different linear relations for each subject. The first of these had a slope of about 0.1 sec/dot, was applicable for the encoding of 1–3 dots, and was taken as an indication of a subitizing process. The second linear relation was applicable for 5–9 dots and had a slope of about 1.0 sec/dot reflecting the speed of a counting process. The average intersection between the functions was located at 3.22 dots. The results were compared with earlier investigations of adult subjects who on the average subitize 6 dots and count 1 dot in about 0.4 sec. It was subitizing process being higher for adults. If the encoding of a stimulus has not been terminated within about 1.5 sec for children and adults the stimulus has instead to be identified in either a counting or an estimating process.     

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.     

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.     

Subitizing and similarity: Toward a pattern-matching theory of enumeration

Pattern-matching theories of subitizing claim that subjects enumerate displays with small numerosities by retrieving numerical responses associated with similar displays experienced in the past. Such retrieval implies that displays with small numerosities are similar to other displays of the same numerosity and dissimilar to other displays of different numerosities. These hypotheses were tested by having subjects rate the similarities of displays of dot patterns with numerosities in the range of 1-10. One group of subjects rated patterns of the same numerosity. Their ratings were higher for patterns in the subitizing range (numerosities of 1-3) than for patterns beyond the subitizing range (numerosities of 4-10). Another group rated patterns of different numerosities. Their ratings were lower in the subitizing range than beyond the subitizing range. An analysis based on multidimensional scaling suggested that numerosity could be retrieved accurately for displays of 1-3 dots, but not for displays of 4-10 dots.     

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.     

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     

Hemispheric specialization in quantification processes

Three experiments were carried out to study hemispheric specialization for subitizing (the rapid enumeration of small patterns) and counting (the serial quantification process based on some formal principles). The experiments consist of numerosity identification of dot patterns presented in one visual field, with a tachistoscopic technique, or eye movements monitored through glasses, and comparison between centrally presented dot patterns and lateralized tachistoscopically presented digits. Our experiments show left visual field advantage in the identification and comparison tasks in the subitizing range, whereas right visual field advantage has been found in the comparison task for the counting range. Received: 11 January 1999 / Accepted: 17 March 2000     

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.     

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.     

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.     

Where do we look when we count? The role of eye movements in enumeration

Two experiments addressed the coupling between eye movements and the cognitive processes underlying enumeration. Experiment 1 compared eye movements in a counting task with those in a “look” task, in which subjects were told to look at each dot in a pattern once and only once. Experiment 2 presented the same dot patterns to every subject twice, to measure the consistency with which dots were fixated between and within subjects. In both experiments, the number of fixations increased linearly with the number of objects to be enumerated, consistent with tight coupling between eye movements and enumeration. However, analyses of fixation locations showed that subjects tended to look at dots in dense, central regions of the display and tended not to look at dots in sparse, peripheral regions of the display, suggesting a looser coupling between eye movements and enumeration. Thus, the eyes do not mirror the enumeration process very directly.     

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.     

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.     

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.     

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.     

Subitizing and its subprocesses

 When people are presented with small sets of elements such as dots they can very easily determine their number. This ability has been called subitizing. The present paper reports results from four experiments. The discussion focuses on the question whether one or two psychological processes have to be assumed for this task of subitizing. Taken together our results support the two-process theories. In particular, a distinction between a process of separation and a process of enumeration is suggested. Experiments 3 and 4 support the canonical pattern hypothesis. Canonical patterns are processed substantially faster than noncanonical patterns. Furthermore, if a complex pattern can be broken down into small canonical patterns, then participants follow a partition and add strategy. Accepted: 22 December 1999     

Subitizing requires attention

While counting objects is typically a slow, serial process, enumerating about four or fewer objects has been considered to be a relatively effortless, parallel, and even preattentive process often referred to as subitizing. However, by combining a subitizing task with an attentional blink task, we show that subitizing is systematically affected by a closely preceding letter identification task. Vice versa, letter identification is also affected by a closely preceding subitizing task. Importantly, performance not only depended on the time between the two tasks, but also on the number of to-be-enumerated dots, even though this number fell within the subitizing range. The results imply that the processes underlying subitizing require attentional resources, suggesting that they are either serial in nature, or parallel, with capacity limited by the overall resources available.     

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.     

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。