Tactile enumeration of small quantities using one hand

Our study explores various aspects of enumerating small quantities in the tactile modality. Fingertips of one hand were stimulated by a vibro-tactile apparatus (for 100/800 ms). Between 1 and 5 stimuli were presented to the right or the left hand and applied to neighboring (e.g., thumb–index–middle) or non-neighboring (e.g., thumb–middle–pinkie) fingers. The results showed a moderate increase in RT up to 4 stimuli and then a decrease for 5 stimuli. Right hand stimulation evoked more accurate performance than left hand stimulation only under short exposures (100 ms). Importantly, when the stimuli were presented to neighboring fingers, the accuracy rate was higher and the RT was faster than when presented to non-neighboring fingers. We discuss the results and suggest that when the stimuli are presented to one hand the subitizing range is 4 rather than 3. Furthermore, the right hand advantage and the efficiency for neighboring fingers are further support for the association between number and spatial arrangement of the fingers.     

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.     

Search, enumeration, and aging: eye movement requirements cause age-equivalent performance in enumeration but not in search tasks

Response time (RT) and fixation frequency were measured for young and older adults in search and enumeration tasks under high- and low-attentional competition conditions. There was an age-related decrement in search rates for single targets both for RT and fixation frequency, but there was no deficit in enumeration rates either with or without distractors even though serial enumeration rates were much slower than single target search rates. Fixation frequency per item in serial enumeration was much greater than in serial visual search for a single target. Enumerating targets with distractors produced an overall increase in RT and fixation frequency that was greater for older adults. The data are contrary to a generalized slowing account, and an alternative is proposed on the basis of the need to make eye movements in enumeration but not in search tasks.     

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.     

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 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.     

Parallel non-verbal enumeration is constrained by a set-based limit

Adults can represent approximate numbers of items independently of language. This approximate number system can discriminate and compare entities as varied as dots, sounds, or actions. But can multiple different types of entities be enumerated in parallel and stored as independent numerosities? Subjects who were prevented from verbally counting watched an experimenter hide sequences of objects in two locations. The number of object types, which contrasted in category membership, color, shape, and texture, varied from 1 to 5, and object types were completely temporally intermixed. Subjects were then asked how many objects of each type were in each location. In three experiments, subjects successfully enumerated the objects of each type in each location when 1-3 types were presented, but failed with 4 or 5 types, regardless of the total number of objects seen. Thus, adults can perform simultaneous enumeration of multiple sets that unfold in temporally intermixed fashion, but are limited to 3 such sets at a time. Furthermore, they perform these parallel enumerations in the absence of training or instruction, and can do so for sets of objects that are hidden in distinct locations. The convergence of this 3-set capacity limit with the 3-item capacity limit widely observed in studies of working memory suggests that each enumeration requires a single slot in memory, and that storage in memory is required before enumeration can occur.     

Infants' enumeration of actions: numerical discrimination and its signature limits

Are abstract representations of number – representations that are independent of the particular type of entities that are enumerated – a product of human language or culture, or do they trace back to human infancy? To address this question, four experiments investigated whether human infants discriminate between sequences of actions (jumps of a puppet) on the basis of numerosity. At 6 months, infants successfully discriminated four‐ versus eight‐jump sequences, when the continuous variables of sequence duration, jump duration, jump rate, jump interval and duration, and extent of motion were controlled, and rhythm was eliminated. In contrast, infants failed to discriminate two‐ versus four‐jump sequences, suggesting that infants fail to form cardinal number representations of small numbers of actions. Infants also failed to discriminate between sequences of four versus six jumps at 6 months, and succeeded at 9 months, suggesting that infants’ number representations are imprecise and increase in precision with age. All of these findings agree with those of studies using visual–spatial arrays and auditory sequences, providing evidence that a single, abstract system of number representation is present and functional in infancy.     

An implicit enumeration method for an exact test of weighted kappa

The kappa coefficient is one of the most widely used measures for evaluating the agreement between two raters asked to assign N objects to one of K nominal categories. Weighted versions of kappa enable partial credit to be awarded for near agreement, most notably in the case of ordinal categories. An exact significance test for weighted kappa can be conducted by enumerating all rater agreement tables with the same fixed marginal frequencies as the observed table, and accumulating the probabilities for all tables that produce a weighted kappa index that is greater than or equal to the observed measure. Unfortunately, complete enumeration of all tables is computationally unwieldy for modest values of N and K. We present an implicit enumeration algorithm for conducting an exact test of weighted kappa, which can be applied to tables of non‐trivial size. The algorithm is particularly efficient for ‘good’ to ‘excellent’ values of weighted kappa that typically have very small p‐values. Therefore, our method is beneficial for situations where resampling tests are of limited value because the number of trials needed to estimate the p‐value tends to be large.     

Configurational effects on the enumeration of dots: Counting by groups

The processing time for quantifying numerosity of two-dimensional dot patterns was investigated as a function of both number of dots and relative proximity between dots. A cluster algorithm (CODE) was first developed as a formal model of how human subjects organize neighboring dots into groups. CODE-based predictions of grouping effects on number processing latencies were then tested with patterns consisting of n dots (range n = 13–23). The results largely confirmed CODE-based predictions and thereby indicated that large collections of dots are preferably counted by groups. Small (n ≤ 5) groups are subitized and their partial results are summed to a running total. Based on criteria other than dot proximity, large (n > 5), proximity-based groups are subdivided into smaller groups of two or three dots, which are again subitized.     

Discontinuity in the enumeration of sequentially presented auditory and visual stimuli

The seeking of discontinuity in enumeration was recently renewed because Cowan [Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24, 87–185; Cowan, N. (2005). Working memory capacity. Hove: Psychology Press] suggested that it allows evaluating the limit of the focus of attention, currently estimated at four items. A strong argument in favour of a general constraint of the cognitive system is that similar discontinuities should be observed in modalities different from the classic simultaneous presentation of visual objects. Recently, data were provided on tactile stimuli, but the authors diverged in their conclusion about the existence of such discontinuity [Gallace, A., Tan, H. Z., & Spence, C. (2006). Numerosity judgments for tactile stimuli distributed over the body surface. Perception, 35(2), 247–266; Riggs, K. J., Ferrand, L., Lancelin, D., Fryziel, L., Dumur, G., & Simpson, A. (2006). Subitizing in tactile perception. Psychological Science, 17(4), 271–272]. Following a similar rationale, our study aimed at evaluating discontinuity in the enumeration of auditory and visual stimuli presented sequentially. The clear and similar discontinuity observed in error rates, response times and given responses for both modalities favours the general capacity limit view, but also questions the size of this capacity, because the discontinuity occurred here at size 2. However, the masking of stimuli in sensory memory could not be entirely discarded.     

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.     

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.     

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.     

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.     

"Constructive" enumeration by chimpanzees (Pan troglodytes) on a computerized task

Two chimpanzees used a joystick to collect dots, one at a time, on a computer monitor (see video-clip in the electronic supplementary material), and then ended a trial when the number of dots collected was equal to the Arabic numeral presented for the trial. Both chimpanzees performed substantially and reliably above chance in collecting a quantity of dots equal to the target numeral, one chimpanzee for the numerals 1–7, and the second chimpanzee for the numerals 1–6. Errors that were made were seldom discrepant from the target by more than one dot quantity, and the perceptual process subitization was ruled out as an explanation for the performance. Additionally, analyses of trial duration data indicated that the chimpanzees were responding based on the numerosity of the constructed set rather than on the basis of temporal cues. The chimpanzees' decreasing performance with successively larger target numerals, however, appeared to be based on a continuous representation of magnitude rather than a discrete representation of number. Therefore, chimpanzee counting in this type of experimental task may be a process that represents magnitudes with scalar variability in that the memory for magnitudes associated with each numeral is imperfect and the variability of responses increases as a function of the numeral's value. Accepted after revision: 11 June 2001 Electronic Publication     

Infants’ auditory enumeration: Evidence for analog magnitudes in the small number range

Vigorous debate surrounds the issue of whether infants use different representational mechanisms to discriminate small and large numbers. We report evidence for ratio-dependent performance in infants’ discrimination of small numbers of auditory events, suggesting that infants can use analog magnitudes to represent small values, at least in the auditory domain. Seven-month-old infants in the present study reliably discriminated two from four tones (a 1:2 ratio) in Experiment 1, when melodic and continuous temporal properties of the sequences were controlled, but failed to discriminate two from three tones (a 2:3 ratio) under the same conditions in Experiment 2. A third experiment ruled out the possibility that infants in Experiment 1 were responding to greater melodic variety in the four-tone sequences. The discrimination function obtained here is the same as that found for infants’ discrimination of large numbers of visual and auditory items at a similar age, as well as for that obtained for similar-aged infants’ duration discriminations, and thus adds to a growing body of evidence suggesting that human infants may share with adults and nonhuman animals a mechanism for representing quantities as “noisy” mental magnitudes.