Subitizing: direct apprehension or serial processing?

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 Effects of Age and Set Size on the Fast Extraction of Egocentric Distance

Angular direction is a source of information about the distance to floor-level objects that can be extracted from brief glimpses (near one's threshold for detection). Age and set size are two factors known to impact the viewing time needed to directionally localize an object, and these were posited to similarly govern the extraction of distance. The question here was whether viewing durations sufficient to support object detection (controlled for age and set size) would also be sufficient to support well-constrained judgments of distance. Regardless of viewing duration, distance judgments were more accurate (less biased towards underestimation) when multiple potential targets were presented, suggesting that the relative angular declinations between the objects are an additional source of useful information. Distance judgments were more precise with additional viewing time, but the benefit did not depend on set size and accuracy did not improve with longer viewing durations. The overall pattern suggests that distance can be efficiently derived from direction for floor-level objects. Controlling for age-related differences in the viewing time needed to support detection was sufficient to support distal localization but only when brief and longer glimpse trials were interspersed. Information extracted from longer glimpse trials presumably supported performance on subsequent trials when viewing time was more limited. This outcome suggests a particularly important role for prior visual experience in distance judgments for older observers.

Bilateral and two-item advantage in subitizing

Subitizing refers to people's ability to enumerate small sets of items fast and accurately. The present study examined if the speed and scope of subitizing is improved when the items to be enumerated are presented bilaterally across hemifields rather than unilaterally in a single hemifield. Such an effect, known as the bilateral field advantage, has been observed in a number of other visual tasks. A second aim was to examine whether the speed of subitizing could be explained by the speed it takes to detect the items to be enumerated, as simple reaction times to multiple stimuli are known to be faster than responses to individual items (known as the redundant target effect, RTE). The results revealed a bilateral field advantage even for enumerating two items. Moreover, the two item condition was the optimal subitizing condition - even enumerating one single item took longer - but this effect was not due to the RTE. In fact, the RTE negatively correlated with the speed of enumerating the same stimuli.

Coarse guidance by numerosity in visual search

In five experiments, we examined whether the number of items can guide visual focal attention. Observers searched for the target area with the largest (or smallest) number of dots (squares in Experiment 4 and "checkerboards" in Experiment 5) among distractor areas with a smaller (or larger) number of dots. Results of Experiments 1 and 2 show that search efficiency is determined by target to distractor dot ratios. In searches where target items contained more dots than did distractor items, ratios over 1.5:1 yielded efficient search. Searches for targets where target items contained fewer dots than distractor items were harder. Here, ratios needed to be lower than 1:2 to yield efficient search. When the areas of the dots and of the squares containing them were fixed, as they were in Experiments 1 and 2, dot density and total dot area increased as dot number increased. Experiment 3 removed the density and area cues by allowing dot size and total dot area to vary. This produced a marked decline in search performance. Efficient search now required ratios of above 3:1 or below 1:3. By using more realistic and isoluminant stimuli, Experiments 4 and 5 show that guidance by numerosity is fragile. As is found with other features that guide focal attention (e.g., color, orientation, size), the numerosity differences that are able to guide attention by bottom-up signals are much coarser than the differences that can be detected in attended stimuli.

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 they could 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 results 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.

Global perception of small numerosities (subitizing) in cerebral-palsied children

The goal of this study was to investigate how global perception of small numerosities (subitizing) develops in cerebral-palsied children (CP), as compared to control children. Twenty-nine 4- to 8-year-old CP children were compared to controls matched on age and sex. Both groups were asked to quantify sets of one to six items displayed briefly on a screen (250 ms). The children were also assessed on counting and eye-hand coordination. CP children exhibited a lower subitizing limit than control children. In CP and control children, the subitizing limit increased significantly with age. In CP children, the subitizing limit was positively correlated with counting performance, and both were positively correlated with eye-hand coordination. In addition, the subitizing limit in CP children with no evidence of a right-hemisphere lesion tended to be higher than in children with a right or bilateral lesion, suggesting right-hemisphere involvement in subitizing. These results support the idea that subitizing and counting are not independent processes during development, and argue in favor of a model of subitizing that relies on a global process.

Quantification Judgement in High Functioning Autism: Superior or Different?

This study examined the hypothesis of superior quantification abilities of persons with high functioning autism (HFA). Fourteen HFA individuals (mean age: 15 years) individually matched with 14 typically developing (TD) participants (gender, chronological age, full-scale IQ) were asked to quantify as accurately and quickly as possible numerosities, represented by the number of squares (2-9) presented in random configurations. In addition, the visual angles of stimuli presentation were manipulated in order to induce a local (large visual angle) and a global (small visual angle) bias on participants' quantification performance (accuracy and naming time). Findings revealed no effect of local and global bias of stimuli presentation in the two groups' performance, and no superior quantification abilities in HFA participants. However, analyses of the naming time slopes for identification by HFA participants of small consecutive numerosities (2-5), suggested their use of counting processes instead of subitizing (or immediate apprehension of numerosities) as in TD participants. Possible explanations for these results are discussed with reference to models of locally-oriented information processing in autism.

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.

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.

Enumeration of briefly presented items by the chimpanzee (Pan troglodytes) and humans (Homo sapiens)

In this study, we compared the performances on an enumeration task (numerical labeling task) of 1 chimpanzee (Pan troglodytes) and 4 humans. In this task, two types of trials, with different exposure durations of the sample that was to be enumerated, were used. In the unlimited-exposure trials, the sample remained on until the subject made a choice. In the brief-exposure trials, the sample was presented for 100 msec and then was masked. The results show clear differences between the different species. The main differences had to do with accuracy during the unlimited trials and response times during the brief trials. Detailed analyses of the pattern of response times for the chimpanzee and of looking-back behavior during the task suggests that the enumeration process underlying the subject's performance was not counting but estimation.

Aging and enumeration: A selective deficit for the subitization of targets among distractors

The term subitization has been used to refer to the fast and accurate enumeration of up to about 4 items. Beyond 4 items, enumeration becomes more serial and less accurate, with each additional item adding to the total enumeration time. The effects of aging on visual enumeration of targets (Os) were assessed with and without the presence of distractors (Xs). Young participants (n = 30; 18-27 years) showed the usual subitization-counting enumeration function, both with and without the presence of distractors. In contrast, for older participants (n = 35; 65-79 years), evidence for subitization was found only when distractors were absent. This occurred even though they could detect an individual target among distractors in parallel across the field. The results are discussed in relation to recent theories of visual selection and enumeration.

A systematic study of visual extinction. Between- and within-field deficits of attention in hemispatial neglect

Mechanisms of visual extinction were investigated in four patients with right hemisphere damage using a partial report paradigm. Different shapes (star or triangle) were displayed in one, two or four possible locations so that double simultaneous stimuli occurred either across the two hemifields or within the same hemifield. Patients attended either to the location (right, left or both), number (one, two or four) or shape (no, one or two stars among the shapes presented) of stimuli in three separate experiments using the same displays and exposure duration. Reporting the location (Experiment 1) produced marked contralesional extinction, although reaction time was delayed compared with unilateral right trials, indicating unconscious processing. Reaction time was also delayed on correct bilateral and unilateral left trials. In contrast, enumerating stimuli (Experiment 2) caused no significant contralesional extinction on bilateral displays and reaction time was similar on bilateral and unilateral right trials, suggesting that information from both fields was grouped in a single numerable percept in this task. However, patients often detected only one of two stimuli within the left field. Whereas similarity of shapes improved localization and did not affect enumeration, identifying stars among shapes (Experiment 3) revealed a severe inability to detect two similar targets between hemifields as well as within each of the hemifields. Distracting triangles were generally less detrimental to the perception of a concurrent target on either side, but slowed the reaction time regardless of whether they were in the same or the opposite field. Relative difficulty in ignoring distractors correlated with neglect severity on a cancellation task, and was most prominent in one patient with a large amount of frontal damage. These findings suggest that (i) allocation of attention to identical stimuli can be modulated by task demand; (ii) enumerating a small set of items across fields may not require attending to individual stimuli but relies on preattentive subitizing ability, as found in normal subjects; (iii) location information may be critical for attentional mechanisms subserved by the parietal cortex and pathological competition for awareness in extinction; (iv) extinction entails a bilateral deficit in attending to two concurrent similar targets when their features must be identified; and (v) the relevance of the stimuli can modulate the distribution of attention, possibly through frontal top-down control. These findings are consistent with recent neurophysiological evidence of parietal and frontal attentional influences on ventral visual pathways.

Assessment of calculation and number processing by adults: cognitive and neuropsychological issues

Calculation and number processing abilities were assessed in normal (n = 138) and traumatic brain-injured (n = 15) Brazilian literature subjects. The study aimed (i) to analyse the effects of demographic factors and to provide tentative norms adjusted for the relevant variables, (ii) to examine the factorial structure of the battery and to evaluate its clinical validity for diagnosis purposes, and (iii) to question the power of current models to account for effects and dissociations found for these groups. Analysis indicated a main effect of education on most subtests and of sex on three, but none for age. Cut-off scores for normality were defined at Percentile 10 with reference to education. The sensitivity of the battery to the presence of arithmetical impairments was considered satisfactory since 11 out of the 15 patients showed pathological scores. A principal component analysis indicated that the different sub-tests were grouped into three factors, which were tentatively interpreted with reference to current information-processing models. The multiple single-case analysis of dissociations in patients' performance suggested some limits with respect to anatomo-functional models of calculation and number processing.

Searching for unknown feature targets on more than one dimension: investigating a "dimension-weighting" account

Search for odd-one-out feature targets takes longer when the target can be present in one of several dimensions as opposed to only one dimension (Müller, Heller, & Ziegler, 1995; Treisman, 1988). Müller et al. attributed this cost to the need to discern the target dimension. They proposed a dimension-weighting account, in which master map units compute, in parallel, the weighted sum of dimension-specific saliency signals. If the target dimension is known in advance, signals from that dimension are amplified. But if the target dimension is unknown, it is determined in a process that shifts weight from the nontarget to the target dimension. The weight pattern thus generated persists across trials, producing intertrial facilitation for a target (trial n + 1) dimensionally identical to the preceding target (trial n). In the present study, we employed a set of new tasks in order to reexamine and extend this account. Targets were defined along two possible dimensions (color or orientation) and could take on one of two feature values (e.g., red or blue). Experiments 1 and 2 required absent/present and color/orientation discrimination of a single target, respectively. They showed that (1) both tasks involve weight shifting, though (explicitly) discerning the dimension of a target requires some process additional to simply detecting its presence; and (2) the intertrial facilitation is indeed (largely) dimension specific rather than feature specific in nature. In Experiment 3, the task was to count the number of targets in a display (either three or four), which could be either dimensionally the same (all color or all orientation) or mixed (some color and some orientation). As predicted by the dimension-weighting account, enumerating four targets all defined within the same dimension was faster than counting three such targets or mixed targets defined in two dimensions.

Age and the ability to inhibit distractor information in visual selective attention

Adult age differences in the effects of different types of distractor interference on visual search were examined. Young adults (mean age = 18.5 years) and older adults (mean age = 69.5 years) performed a target-counting task that required a complete search of a visual display in each trial. Varying numbers of targets were presented alone in displays or were interspersed among eight distractor items that were either categorically related (letters) or conceptually related (numbers representing either the correct number or the incorrect number of targets in the display) to the target item (letter Q). An adult age difference in the speed of target enumeration was observed when targets were presented alone in the display. In addition, when targets appeared with distractors, both younger and older adults were penalized more by conceptually interfering distracters than by categorically related distractors. Results did not suggest an age-related decline in inhibitory processes.

Calculation and number processing: assessment battery; role of demographic factors

This paper describes the structure and contents of EC301, a standardized testing battery for the evaluation of brain-damaged adults in the area of calculation and number processing. The battery was administered to 180 normal subjects stratified by education (3 levels), age (3) and gender. EC301 is composed of a large variety of tasks dealing with basic arithmetic skills, and their linguistic, spatial, and mnesic dimensions. The three main notational systems for numbers--Arabic digits, written verbal, and spoken verbal number forms--are explored. Analysis of error rates indicated the effect of some demographic factors (principally, education; incidentally, gender) on normal performance in some tasks.

Coordination of what and where in visual attention

Performance often suffers when two visual discriminations must be made concurrently ('divided attention'). In the modular primate visual system, different cortical areas analyse different kinds of visual information. Especially important is a distinction between an occipitoparietal 'where?' system, analysing spatial relations, and an occipitotemporal 'what?' system responsible for object recognition. Though such visual subsystems are anatomically parallel, their functional relationship when 'what?' and 'where?' discriminations are made concurrently is unknown. In the present experiments, human subjects made concurrent discriminations concerning a brief visual display. Discriminations were either similar (two 'what?' or two 'where?' discriminations) or dissimilar (one of each), and concerned the same or different objects. When discriminations concerned different objects, there was strong interference between them. This was equally severe whether discriminations were similar--and therefore dependent on the same cortical system--or dissimilar. When concurrent 'what?' and 'where?' discriminations concerned the same object, however, all interference disappeared. Such results suggest that 'what?' and 'where?' systems are coordinated in visual attention: their separate outputs can be used simultaneously without cost, but only when they concern one object.

Varieties of numerical abilities

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.

Single-parameter power law psychophysics of auditory numerosity and the psychological moment hypothesis

In the present study, numerosity estimation was investigated. A two-parameter Stevens power law analysis was performed on a total of 944 subjects in six experiments. Two pulse ranges (2-17 or 17-253 pulses) and six pulse rates (either constant or randomly varied within trial blocks) were used, variously, in an unsuccessful attempt to find evidence for a psychological moment, under the supposition that the exponent (or, possibly, the measure constant) would become smaller as increasing numbers of pulses fell within the interval determined by each psychological moment. A single-parameter reanalysis of these six experiments under the initial value condition that a (standard) stimulus of one pulse be assigned a theoretical response (modulus) of one yielded single-parameter equations whose exponents were reliably less varied than those for conventional two-parameter equations in Experiments 1-4 (with randomly varying pulse rates from trial to trial) but not less varied in Experiments 5 and 6 (in which pulse rates were constant within trial blocks). It was concluded that the variable pulse rate condition, with its reduced exponent variability and presumed reduced temporal confounding, provides a more valid estimate of the single-parameter power law exponent for numerosity, which was found to be 0.80.

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 single-element 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 capacity-limited process, but not 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.