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.     

Implicit alternatives to a stimulus,difficulty of encoding,and schema-plus-correction representation

Garner has hypothesized that a S exposed to a stimulus infers alternatives to it. Data are reviewed which establish a close correlation between the number of implicit alternatives to a stimulus and the difficulty of encoding that stimulus. This correlation provides a datum to be explained and suggests that the identification of specific implicit alternatives may provide clues to the encoding process. For dot patterns of a certain kind, it has been suggested that the rotations and reflections of a stimulus are its only alternatives, but a close examination of certain data indicates that, for many of these patterns, the alternatives also include one or more other patterns which differ from the stimulus only in the position of a single dot and which locate that dot in a position where it is more predictable from the other dots. It is suggested that the one-dot-different alternatives represent schemas used in schema-plus-correction encoding.     

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.     

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.     

Simultaneous encoding of direction at a local and global scale

Human observers can simultaneously encode direction information at two different scales, one local (an individual dot) and one global (the coherent motion of a field of dots distrisbuted over a 10°-diameter display). We assessed whether encoding global motion would preclude the encoding of a local trajectory component and vice versa. In the present experiments, a large number (100–150) of dots were randomly assigned directions in each frame from a uniform distribution of directions spanning a range of 160° to create global motion in a single direction (Williams & Sekuler, 1984). Amidst these background dots, 1 dot moved in a consistent direction (trajectory) for the duration of the display. The direction of this “trajectory dot” was similar to the mean direction of the distribution of directions determining the movement of the background dots. Direction discrimination for both the global motion and the trajectory was measured, using the method of constant stimuli, under precued and postcued partial report conditions. A low- or high-frequency 85-msec tone signaled which motion the subject was to judge. In the precue condition, the tone was presented 200 msecbefore the onset of the stimulus, whereas in the postcue condition, the tone was presented immediatelyafter the offset of the stimulus. Direction discrimination thresholds for both global and local motion in the postcued condition were not significantly different from those obtained in the precued condition. These results suggest that direction information for both global and local motion is encoded simultaneously and that the observer has access to either motion signal after the presentation of a stimulus.     

Strength theories of disjunctive visual detection

Zero, 1, 2, or 3 black dots are tachistoscopically presented on a white field. There are two alternative tasks: (a) to decide on the presence of each of the left, middle, and right dots (multiple detection) or (b) to decide whether any of the dots was present (disjunctive detection), The results indicate that in disjunctive detection, Ss do not add together thestrengths of the three dot positions and compare this sum to a criterion. Rather they combine theirdecisions about each dot, responding “yes” to the array, if and only if they decide “yes” to any one dot. Strength distributions appear to be invariant with respect to irrelevant stimuli. Invariance with respect to report order holds approximately. However, dots reported on first are slightly more detectable. This suggests a successive scanning process, whose rate is independent of whether a stimulus is present or absent at the position scanned.     

Span and rate of apprehension in children and adults.

Children and adults quantified random patterns of dots, under unlimited exposure duration. For adults and children two distinct processes appear to operate. For adults the quantification of collections of from one to three dots is essentially errorless, and proceeds at the rate of 46 msec per item, while the quantification rate for from 4 to 10 dots is 307 msec per dot. For children the same operating ranges appear to hold, however children are much slower. The lower slope is 195 msec per dot, while the upper is 1049. Although the results for adults and children are similar except for the overall rates, the nature of the isomorphism between children and adults is unclear.     

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.     

Wholistic and analytic stimulus processing: The development of selective perceptual strategies

This experiment reports an investigation of the development of selective processing strategies as subjects become increasingly more practised at a serial self-paced RT task. It demonstrates the effect a preceding stimulus can have on the analysis of a current signal and the development of active analytic strategies in favour of passive wholistic processing, with practice. The stimuli used were letters with irrelevant visual noise dot patterns superimposed on them. The letter, or the dot pattern, or both, could be repeated on successive trials. Early in practice repetition of both stimulus components simultaneously produced short reponse latencies relative to repetition of the letter alone. The number of noise dots markedly affected RT. Late in practice, however, letter repetition RTs were small, irrespective of whether or not the noise dot pattern was repeated. Furthermore, the number of noise dots no longer had an effect on the RT to these stimulus transitions. The results suggest that subjects appear to be able to select relevant information to process as they become progressively more practised, even though there is evidence that they compare the current stimulus with an iconic representation of the immediately preceding one.     

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.     

Facilitation of retrieval by perceptual structure

Summary The effectiveness of a part of a stimulus as a retrieval cue for the whole is investigated in two experiments. Stimuli consisting of a series of eight colored dots were presented for 7s. In Experiment 1, the first three dots of a stimulus made up the cue. A structural compatibility between stimulus and cue was shown to enhance recall. Experiment 2 used only the first dot as a cue for the same stimuli as in Experiment 1 in order to control the possible alternative prediction based on Garner's work that the effect was due to a difference in complexity of stimuli, regardless of compatibility with the cue. The compatibility predictions are derived from a model based on a hierarchy principle by Buffart.     

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.     

Exploring the developmental changes in automatic two-digit number processing

Even when two-digit numbers are irrelevant to the task at hand, adults process them. Do children process numbers automatically, and if so, what kind of information is activated? In a novel dot-number Stroop task, children (Grades 1-5) and adults were shown two different two-digit numbers made up of dots. Participants were asked to select the number that contained the larger dots. If numbers are processed automatically, reaction time for dot size judgment should be affected by numerical characteristics. The results suggest that, like adults, children process two-digit numbers automatically. Based on the current findings, we propose a developmental trend for automatic two-digit number processing that goes from decomposed sequential (activation of decade digit followed by that of unit digit) to decomposed parallel processing (simultaneous activation of decade and unit digits).     

Connectedness affects dot numerosity judgment: Implications for configural processing

Participants judged the number of dots in visual displays with brief presentations (200 msec), such that the numerosity judgment was based on an instantaneous impression without counting. In some displays, pairs of adjacent dots were connected by line segments, whereas, in others, line segments were freely hanging without touching the dots. In Experiments 1, 2A, and 2B, connecting pairs of dots by line segments led to underestimation of dot numbers in those patterns. In Experiment 3, we controlled for the number of freely hanging line segments, whereas Experiment 4 showed that line segments that were merely attached to dots without actually connecting them did not produce a considerable underestimation effect. Experiment 5 showed that a connectedness effect existed when stimulus duration was reduced (50 msec) or extended (1,000 msec). We conclude that connectivity affects dot numerosity judgments, consistent with earlier findings of a configural effect in numerosity processing. Implications of the role of connectedness in object representation are discussed.     

Vernier acuity as affected by target length and separation

Vernier acuity was measured for vertical lines of different lengths; the threshold (about 4 sec of arc) was almost as good for the shortest stimuli (1 min 20 sec squares) as for the longest (21 min 20 sec × 1 min 20 sec rectangles) and did not change when two round dots were shown in positions corresponding to the squares. The threshold for the two dots measured in terms of minimum detectable lateral offset increased when the vertical separation between the dots increased, but, when replotted in terms of the angle of tilt between them with respect to vertical, the threshold improved with dot separation; moreover, at asymptote, the threshold was comparable to that obtained for detecting that an actual line was tilted out of vertical. Our data suggest that, in performing the vernier task, Ss do not extrapolate the edges of the vernier elements; instead, they judge the deviation of the inner ends of the stimuli from verticality. This hypothesis explains the effect of increasing separation between vernier elements and also accounts for other types of acuity, such as the detection of curvature.     

Basic number processing deficits in developmental dyscalculia: Evidence from eye tracking

Recent research suggests that developmental dyscalculia is associated with a subitizing deficit (i.e., the inability to quickly enumerate small sets of up to 3 objects). However, the nature of this deficit has not previously been investigated. In the present study the eye-tracking methodology was employed to clarify whether (a) the subitizing deficit of two boys with dyscalculia resulted from a general slowing in the access to magnitude representation, or (b) children with dyscalculia resort to a back-up counting strategy even for small object sets. In a dot-counting task, a standard problem size effect for the number of fixations required to encode the presented numerosity within the subitizing range was observed. Together with the finding that problem size had no impact on the average fixation duration, this result suggested that children with dyscalculia may indeed have to count, while typically developing controls are able to enumerate the number of dots in parallel, i.e., subitize. Implications for the understanding of developmental dyscalculia are considered.     

On the detectability of straight lines in visual noise: An extension of French’s paradigm into the millisecond domain

We report the results of a study that investigated the effects of four experimental parameters on the tachistoscopic detectability of dotted lines in a noisy background consisting of dynamic patterns of identical dots. Characteristics of the lines that were explored included dot spacing, dot numerosity, and line orientation. Dot spacing was shown to be the most powerful parameter in determining the masking effect: the closer the dots, the more easily the line was detected. Increasing the number of the dots was also effective in increasing the detectability of a line, but only up to a limit of five dots. The orientation of the line, on the other hand, did not alter its detectability. Increasing the density of the visual masking noise monotonically reduced the detectability of the line. These studies are an extension of French’s (1953, 1954) work, differing mainly in the fact that the stimulus material was presented tachistoscopically rather than in a mode that allowed prolonged (up to 5 sec) examination of the figures. Thus, they put the results in a domain in which physiological interpretations may be more critically evaluated. A comparison of the results suggests that essentially the same sorts of cognitive mechanisms are operating in the millisecond domain as in the longer exposures. This suggests that simple neurological models utilizing the concept of feature filters or detectors may be being prematurely applied to relatively complex perceptual processes.     

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.     

Span of attention,backward masking,and reaction time

A test pattern consisting of 0 to 15 dots and a following random dot masking pattern were presented for 5 msec each with SOAs varying between 30 and 200 msec. The subject was asked to report the perceived number of dots in the test pattern as soon as possible and to assign a confidence rating to each report. The span of attention (upper limit for 50% correct numerosity judgments) increased from 2.4 to 9.5 as the SOA increased. Backward masking reduced the reported number of dots from the actual number in the test pattern, especially with small SOAs. Reaction time increased linearly at a low rate (approximately 40 msec/dot) up to 4 dots in the test pattern and then increased linearly at a high rate (approximately 370 msec/dot) as thereported, orperceived, number of dots increased. The two different branches of the reaction time curve were considered to represent two separate processes,subitizing andcounting, as suggested by Klahr (1973), who found similar dual increase rates as a function of the actual number of dots. These findings, as well as causal inference based on partial correlations and path analysis, indicated that the reported (perceived) number of dots and confidence rating were both determined by the number of stimulus dots and the SOA and that the reaction time was determined by the so-determined perceived number of dots and level of confidence. A multistage model is proposed.