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 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.     

Haptic subitizing across the fingers

Numerosity judgments of small sets of items (≤ 3) are generally fast and error free, while response times and error rates increase rapidly for larger numbers of items. We investigated an efficient process used for judging small numbers of items (known as subitizing) in active touch. We hypothesized that this efficient process for numerosity judgment might be related to stimulus properties that allow for efficient (parallel) search. Our results showed that subitizing was not possible for raised lines among flat surfaces, whereas this type of stimulus could be detected in parallel over the fingers. However, subitizing was possible when the number of fingers touching a surface had to be judged while the other fingers were lowered in mid-air. In the latter case, the lack of tactile input is essential, since subitizing was not enabled by differences in proprioceptive information from the fingers. Our results show that subitizing using haptic information from the fingers is possible only when some fingers receive tactile information while other fingers do not.     

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     

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.     

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。     

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.     

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.     

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     

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.     

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.     

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.     

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。     

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.     

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.     

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.     

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.     

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.     

自我参照和环境参照整合过程中的主方位判断

通过三个实验证明,自我参照和环境参照转化与整合过程中的主方位判断受拍摄方向和目标位置的影响。二者对主方位判断的作用模式不会随主方位判断任务的改变而改变,拍摄方向效表现为：朝北效应（匹配效应）和主方位效应,体现的是自我参照和环境参照整合过程中环境参照对主方位的影响;目标位置效应的认知加工时间模式可表示为：前-后轴<左-右轴<其他位置,体现的是自我参照和环境参照整合过程中自我参照对主方位的影响。本文以规范轴和规范方向为框架对数据结果进行了分析和解释。