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.     

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.     

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.     

Benefits of interhemispheric collaboration can be eliminated by mixing stimulus formats that involve different cortical access routes

Previous studies indicate that the benefits of dividing an information processing load across both cerebral hemispheres outweigh the costs of interhemispheric transfer as tasks become more difficult or cognitively complex. This is demonstrated as better performance when two stimuli to be compared are presented one to each visual field and hemisphere than when both stimuli are presented to the same single hemisphere (an across-hemisphere advantage). Two experiments indicate that this finding does not generalize to complex tasks that require matching numeric quantities represented by two very different visual formats whose processing involves somewhat different cortical areas: digits and dice-like dot patterns. In fact, mixing these stimulus formats consistently produces a within-hemisphere advantage. We propose that, when two simultaneously presented stimuli are presented in sufficiently different visual formats, identification of the two stimuli may take place in parallel, via different cortical access routes and with little or no interference, even when they are presented to the same cerebral hemisphere.     

小学生和成人不同注意条件下的感数与计数加工的比较研究

采用经典的空间线索范式,以成年人和小学二、四、六年级的学生为研究对象,考察了内、外源注意对感数和计数加工过程的影响。结果表明：（1）六年级学生和成人的感数范围为1-3或4,计数范围为4或5以上,二、四年级学生的感数范围在1-3,计数范围在4以上,支持感数和计数分离的观点。（2）对感数加工,成人和六年级学生均不受两种注意条件的影响,而对计数加工,外源性注意条件下的反应比内源性注意快;二年级学生1-6的数量加工受两种注意条件的影响,四年级学生1-5的加工不受两种注意条件的影响。（3）成人和小学生在感数和计数加工上都存在线索效应。以上结果说明：二-四年级是发展的关键期,随着年龄的增加,儿童逐渐有意识地有效利用内源和外源性线索进行数量加工,其注意指向性更明确;从发展的角度为选择注意加工理论提供了发展证据。     

Early visual tagging: effects of target-distractor similarity and old age on search, subitization, and counting

Three experiments examined the effects of target-distractor (T-D) similarity and old age on the efficiency of searching for single targets and enumerating multiple targets. Experiment 1 showed that increasing T-D similarity selectively reduced the efficiency of enumerating small (< 4) numerosities (subitizing) but had little effect on enumerating larger numerosities (counting) or searching for a single target. Experiment 2 provided converging evidence using fixation frequencies and a finer range of T-D similarities. Experiment 3 showed that T-D similarity had a greater impact on older than on young adults, but only for subitizing. The data are discussed in terms of the mechanisms and architecture of early visual tagging, dissociable effects in search and enumeration, and the effects of aging on visual processing.     

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.     

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.     

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     

Visual field differences in the processing of numerical stimuli

Twenty-four right-handed subjects received random presentations of the numbers 1-6 in the form of words, digits, and dot patterns, to the left and right visual fields. Accuracy and reaction time were recorded for an odd-even judgment requiring a manual response. A significant stimulus type of visual field interaction was obtained, with words showing a left-hemisphere advantage and digits and dot patterns showing a right-hemisphere advantage. This pattern supports Coltheart's (1980, Deep dyslexia: A right hemisphere hypothesis, In M. Coltheart, K. Patterson, & J.C. Marshall (Eds.), Deep dyslexia, London: Routledge & Kegan Paul) right hemisphere reading hypothesis, which suggests that the left hemisphere's general advantage in processing linguistic material may be specific to stimuli which involve phonological processing. When phonological processing is not possible (e.g., for arabic digits and other ideographic orthographies), the right hemisphere may have an advantage because of its superior visuospatial processing capabilities.     

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.     

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.     

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.     

Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

Humans appear to share with animals a nonverbal counting process. In a nonverbal counting condition, subjects pressed a key a numeral-specified number of times, while saying “the” at every press. The mean number of presses increased as a power function of the target number, with a constant coefficient of variation (c.v.), both within and beyond the proposed subitizing range (1–4 or 5), suggesting small numbers are represented on the same continuum as larger numbers and subject to the same noise process (scalar variability). By contrast, when subjects counted their presses out loud as fast as they could, the c.v. decreased as the inverse square root of the target value (binomial variability instead of scalar variability). The unexpected power-law relation between target value and mean number of presses in nonverbal counting suggests a new hypothesis about the development of the function relating number symbols to mental magnitudes.     

Some primitive mechanisms of spatial attention

Our approach to studying the architecture of mind has been to look for certain extremely simple mechanisms which we have good reason to suspect must exist, and to confirm these empirically. We have been concerned primarily with certain low-level mechanisms in vision which allow the visual system to simultaneously index items at multiple spatial locations, and have developed a provisional model (called the FINST model) of these mechanisms. Among the studies we have carried out to support these ideas are ones showing that subjects can track multiple independent moving targets in a field of identical distractors, and that their ability to track these targets and detect changes occuring on them does not generalize to non-targets or to items lying inside the convex polygon that they form (so that a zoom lens of attention does not fit the data). We have used a visual search paradigm to show that (serial or parallel) search can be confined to a subset of indexed items and the layout of these items is of little importance. We have also carried out a large number of studies on the phenomenon known as subitizing and have shown that subitizing occurs only when items can be preattentively individuated and in those cases location precuing has little effect, compared with when counting occurs, which suggests that subitizing may be carried out by counting active indexes rather than items in the visual field. And finally we have run studies showing that a certain motion effect which is sensitive to attention can occur at multiple precued loci. We believe that taken as a whole the evidence is most parsimoniously accounted for in terms of the hypothesis that there is an early preattentive stage in vision where a small number of salient items in the visual field are indexed and thereby made readily accessible for a variety of visual tasks.     

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.     

Further evidence on the effects of symbolic distance on Stroop-like interference

Pavese and Umiltà found that, in an enumeration task, Stroop-like interference is larger when the digit identity is symbolically close to the enumeration response than when it is symbolically far. In two experiments testing 49 undergraduates, we further explored this phenomenon using Francolini and Egeth's paradigm. We found that symbolic distance affected interference even when the stimulus was briefly presented and masked. In Exp. 2, which tested numerosities outside the subitizing range, individuals used a different enumeration strategy but showed the same symbolic distance effect. These results support the hypothesis that Stroop interference found in enumeration tasks depends on a rapid and automatic activation of digits' magnitude representation. Received: 10 November 1997 / Accepted: 23 June 1998     

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.     

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.