Counting in everyday life: discrimination and enumeration

Spatial and chromatic properties of numerosity estimation in isolation and context

Numerosity estimation around the subitizing range is facilitated by a shape-template matching process and shape-coding mechanisms are selective to visual features such as colour and luminance contrast polarity. Objects in natural scenes are often embedded within other objects or textured surfaces. Numerosity estimation is improved when objects are grouped into small clusters of the same colour, a phenomenon termed groupitizing, which is thought to leverage on the subitizing system. Here we investigate whether numerosity mechanisms around the subitizing range are selective to colour, luminance contrast polarity and orientation, and how spatial organisation of context and target elements modulates target numerosity estimation. Stimuli consisted of a small number (3-to-6) of target elements presented either in isolation or embedded within context elements. To examine selectivity to colour, luminance polarity and orientation, we compared target-only conditions in which all elements were either the same or different along one of these feature dimensions. We found comparable performance in the same and different feature conditions, revealing that subitizing mechanism do not depend on ‘on-off’ luminance-polarity, colour or orientation channel interactions. We also measured the effect of varying spatial organisation of (i) context, by arranging the elements either in a grid, mirror-symmetric, translation-symmetric or random; (ii) target, by placing the elements either mirror-symmetric, on the vertices of simple shapes or random. Our results indicate higher accuracy and lower RTs in the grid compared to all other context types, with mirror symmetric, translation and random arrangements having comparable effects on target numerosity. We also found improved performance with shape-target followed by symmetric and random target arrangements in the absence and presence of context. These findings indicate that numerosity mechanisms around the subitizing range are not selective to colour, luminance polarity and orientation, and that symmetric, translation and random contexts organisations inhibit target-numerosity encoding stronger than regular/grid context.

The complexity of simple counting: ERP findings reveal early perceptual and late numerical processes in different arrangements

The counting process can only be fully understood when taking into account the visual characteristics of the sets counted. Comparing behavioral data as well as event-related brain potentials (ERPs) evoked by different task-irrelevant arrangements of dots during an exact enumeration task, we aimed to investigate the effect of illusory contour detection on the counting process while other grouping cues like proximity were controlled and dot sparsity did not provide a cue to the numerosity of sets. Adult participants (N = 37) enumerated dots (8-12) in irregular and two different types of regular arrangements which differed in the shape of their illusory dot lattices. Enumeration speed was affected by both arrangement and magnitude. The type of arrangement influenced an early ERP negativity peaking at about 270 ms after stimulus onset, whereas numerosity only affected later ERP components (> 300 ms). We also observed that without perceptual cues, magnitude was constructed at a later stage of cognitive processing. We suggest that chunking is a prerequisite for more fluent counting which influences automatic processing (< 300 ms) during enumeration. We conclude that the procedure of exact enumeration depends on the interaction of several perceptual and numerical processes that are influenced by magnitude and arrangement.

Developmental brain dynamics of numerical and arithmetic abilities

The development of numerical and arithmetic abilities constitutes a crucial cornerstone in our modern and educated societies. Difficulties to acquire these central skills can lead to severe consequences for an individual's well-being and nation's economy. In the present review, we describe our current broad understanding of the functional and structural brain organization that supports the development of numbers and arithmetic. The existing evidence points towards a complex interaction among multiple domain-specific (e.g., representation of quantities and number symbols) and domain-general (e.g., working memory, visual-spatial abilities) cognitive processes, as well as a dynamic integration of several brain regions into functional networks that support these processes. These networks are mainly, but not exclusively, located in regions of the frontal and parietal cortex, and the functional and structural dynamics of these networks differ as a function of age and performance level. Distinctive brain activation patterns have also been shown for children with dyscalculia, a specific learning disability in the domain of mathematics. Although our knowledge about the developmental brain dynamics of number and arithmetic has greatly improved over the past years, many questions about the interaction and the causal involvement of the abovementioned functional brain networks remain. This review provides a broad and critical overview of the known developmental processes and what is yet to be discovered.

Shape facilitates number: brain potentials and microstates reveal the interplay between shape and numerosity in human vision

Recognition of simple shapes and numerosity estimation for small quantities are often studied independently of each other, but we know that these processes are both rapid and accurate, suggesting that they may be mediated by common neural mechanisms. Here we address this issue by examining how spatial configuration, shape complexity, and luminance polarity of elements affect numerosity estimation. We directly compared the Event Related Potential (ERP) time-course for numerosity estimation under shape and random configurations and found a larger N2 component for shape over lateral-occipital electrodes (250–400 ms), which also increased with higher numbers. We identified a Left Mid Frontal (LMF; 400–650 ms) component over left-lateralised medial frontal sites that specifically separated low and high numbers of elements, irrespective of their spatial configuration. Different luminance-polarities increased N2 amplitude only, suggesting that shape but not numerosity is selective to polarity. Functional microstates confined numerosity to a strict topographic distribution occurring within the LMF time-window, while a microstate responding only to shape-configuration was evidenced earlier, in the N2 time-window. We conclude that shape-coding precedes numerosity estimation, which can be improved when the number of elements and shape vertices are matched. Thus, numerosity estimation around the subitizing range is facilitated by a shape-template matching process.

Early selection versus late correction: Age-related differences in controlling working memory contents

We examined whether a reduced ability to ignore irrelevant information is responsible for the age-related decline of working memory (WM) functions. By means of event-related brain potentials, we will show that filtering is not out of service in older adults but shifted to a later processing stage. Participants performed a visual short-term memory task (change-detection task) in which targets were presented along with distractors. To allow early selection, a cue was presented in advance of each display, indicating where the targets were to appear. Despite this relatively easy selection criterion, older adults' filtering was delayed as indicated by the amplitude pattern of the contralateral delay activity. Importantly, WM-equated younger adults did not show a delay indicating that the delay is specific to older adults and not a general phenomenon that comes with low WM capacity. Moreover, the analysis of early visual potentials revealed qualitatively different perceptual/attentional processing between the age groups. Young adults exhibited stronger distractor sensitivity that in turn facilitated filtering. Older adults, in contrast, seemed to initially store distractors and to suppress them after the fact. These early selection versus late-correction modes suggest an age-related shift in the strategy to control the contents of WM. (PsycINFO Database Record

Object individuation and compensation in healthy aging

Theories on neural compensation suggest that aged participants overactivate the brain areas involved in a task to compensate for the age-related decline. In this electrophysiological study, we investigated the temporal locus of neural overactivation in aging during multiple target processing. We measured performance and three event-related brain potential responses (N1, N2pc, and contralateral delay activity) in young and old adults, while they enumerated a variable number (1-4) of targets presented in an easy (distractor absent) or difficult (distractor present) condition. The main results indicated that although N2pc (∼200 ms) increased in amplitude in the distractor-present condition in the young group, no modulation occurred for the old group. Old participants were associated with larger N2pc amplitudes than young participants in the distractor-absent condition, where both groups had comparable levels of accuracy. These effects were not present for N1 and contralateral delay activity. Overall, the data suggest that in enumeration, aging is associated with compensatory effects that rely on the selection mechanism responsible for target individuation.

Multiple object individuation and subitizing in enumeration: a view from electrophysiology

What are the processes involved in determining that there are exactly n objects in the visual field? The core level of representation for this process is based on a mechanism that iteratively individuates each of the set of relevant objects for exact enumeration. In support of this proposal, we review recent electrophysiological findings on enumeration-at-a-glance and consider three temporally distinct responses of the EEG signal that are modulated by object numerosity, and which have been associated respectively with perceptual modulation, attention selection, and working memory. We argue that the neural response associated with attention selection shows the hallmarks of an object individuation mechanism, including the property of simultaneous individuation of a limited number of objects thought to underlie the behavioral subitizing effect. The findings support the view that the core component of exact enumeration is an attention-based individuation mechanism that binds specific features to locations and provides a stable representation of a limited set of relevant objects. The resulting representation is made available for further cognitive operations for exact enumeration.

Neural substrates of numerosity estimation in autism

Visual skills, including numerosity estimation are reported to be superior in autism spectrum disorders (ASD). This phenomenon is attributed to individuals with ASD processing local features, rather than the Gestalt. We examined the neural correlates of numerosity estimation in adults with and without ASD, to disentangle perceptual atypicalities from numerosity processing. Fourteen adults with ASD and matched typically developed (TD) controls estimated the number of dots (80-150) arranged either randomly (local information) or in meaningful patterns (global information) while brain activity was recorded with magnetoencephalography (MEG). Behavioral results showed no significant group difference in the errors of estimation. However, numerical estimation in ASD was more variable across numerosities than TD and was not affected by the global arrangement of the dots. At 80-120 ms, MEG analyses revealed early significant differences (TD > ASD) in source amplitudes in visual areas, followed from 120 to 400 ms by group differences in temporal, and then parietal regions. After 400 ms, a source was found in the superior frontal gyrus in TD only. Activation in temporal areas was differently sensitive to the global arrangement of dots in TD and ASD. MEG data show that individuals with autism exhibit widespread functional abnormalities. Differences in temporal regions could be linked to atypical global perception. Occipital followed by parietal and frontal differences might be driven by abnormalities in the processing and conversion of visual input into a number-selective neural code and complex cognitive decisional stages. These results suggest overlapping atypicalities in sensory, perceptual and number-related processing during numerosity estimation in ASD.

Reduced beta band connectivity during number estimation in autism

Recent evidence suggests that disruption of integrative processes in sensation and perception may play a critical role in cognitive and behavioural atypicalities characteristic of ASD. In line with this, ASD is associated with altered structural and functional brain connectivity and atypical patterns of inter-regional communication which have been proposed to contribute to cognitive difficulties prevalent in this group. The present MEG study used atlas-guided source space analysis of inter-regional phase synchronization in ASD participants, as well as matched typically developing controls, during a dot number estimation task. This task included stimuli with globally integrated forms (animal shapes) as well as randomly-shaped stimuli which lacked a coherent global pattern. Early task-dependent increases in inter-regional phase synchrony in theta, alpha and beta frequency bands were observed. Reduced long-range beta-band phase synchronization was found in participants with ASD at 70-145 ms during presentation of globally coherent dot patterns. This early reduction in task-dependent inter-regional connectivity encompassed numerous areas including occipital, parietal, temporal, and frontal lobe regions. These results provide the first evidence for inter-regional phase synchronization during numerosity estimation, as well as its alteration in ASD, and suggest that problems with communication among brain areas may contribute to difficulties with integrative processes relevant to extraction of meaningful 'Gestalt' features in this population.

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.

Multiple Object Individuation and Exact Enumeration

Exact computation of numerosity requires the selective individuation of the elements to be enumerated so that each element is counted once and only once. Such a mechanism should operate not only when the elements to be enumerated are presented in isolation but also when they are presented in cluttered scenes. To uncover the electrophysiological correlates of the level of object representation necessary for exact enumeration, we examined ERP measures during the execution of a target enumeration task. A variable number (1–4) of lateralized targets were presented with or without distracters on the target side. An early nonlateralized response (N1, 120–180 msec) was modulated by target numerosity only when presented without distracters. By contrast, the amplitudes of a lateralized and later response (N2pc, 180–300 msec) increased as a function of target numerosity both with and without distracters, reaching a plateau at three targets. We propose that the stage of processing reflected in the N2pc corresponds to the component of individuation that binds specific indexes to properties and locations and that this provides the representation type necessary for exact enumeration.

Cortical activation patterns during subitizing and counting

The exact amount of small number of items (1-4) can be detected fast and accurately (subitizing) while the enumeration of large number of items (over 4) is slower and error-prone (counting). Several counting-related cortical areas have been identified mainly in frontal and parietal regions, but cortical events associated with subitizing have remained unclear. Similarly, little is known about the temporal sequence of cortical activation during enumeration. In this study, we examined the temporal and spatial pattern of subitizing and counting using magnetoencephalography (MEG). During the MEG-recordings, black dots (2-8) in a visual display were shown to ten adults, who then responded with a button press as soon as they knew the number of items. The behavioural results showed a regularly reported dichotomy in enumeration of small (2-4) and large (5-8) numbers. In brain responses, pronounced activation peak during subitizing was detected around 250 ms in the bilateral posterior temporo-parietal area, which presumably reflects the function of ventral visual stream. During counting, pronounced activation was first detected in bilateral parietal areas, followed by a growing activation in the frontal cortices. The activation of frontal areas indicates the involvement of task guidance and attention, while the parietal areas activated earlier may have a key role in maintaining numerical representations and spatial attention. Brain functions during counting seem to consist of several constituent processes that reflect number processing, attention and task guidance. Our results demonstrated temporally and spatially specific brain activation for fast subitizing and effortful counting.

Continuous visual properties explain neural responses to nonsymbolic number

Nonsymbolic number and its continuous visual properties are confounded in everyday life: When number changes, its continuous visual properties also change. It could therefore be efficient to rely on the visual properties to judge number. The current consensus, however, holds that number is processed independent of its visual properties. In this study, we pitched these two opposing theories against each other. We used electroencephalography to look at the components suggested to process number. The first experiment showed that number and visual cues affect the N1 and/or the P2 component. To disentangle number and visual processes, we controlled the visual cues in the second experiment. Now, no number-related effects were present. When the data were reorganized according to visual cue instead of number size, N1 and P2 effects emerged. These results argue against the idea that number is processed independent of its continuous visual variables.

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.

Aging and visual counting

Background

Much previous work on how normal aging affects visual enumeration has been focused on the response time required to enumerate, with unlimited stimulus duration. There is a fundamental question, not yet addressed, of how many visual items the aging visual system can enumerate in a “single glance”, without the confounding influence of eye movements.

Methodology/Principal Findings

We recruited 104 observers with normal vision across the age span (age 21–85). They were briefly (200 ms) presented with a number of well- separated black dots against a gray background on a monitor screen, and were asked to judge the number of dots. By limiting the stimulus presentation time, we can determine the maximum number of visual items an observer can correctly enumerate at a criterion level of performance (counting threshold, defined as the number of visual items at which ≈63% correct rate on a psychometric curve), without confounding by eye movements. Our findings reveal a 30% decrease in the mean counting threshold of the oldest group (age 61–85: ∼5 dots) when compared with the youngest groups (age 21–40: 7 dots). Surprisingly, despite decreased counting threshold, on average counting accuracy function (defined as the mean number of dots reported for each number tested) is largely unaffected by age, reflecting that the threshold loss can be primarily attributed to increased random errors. We further expanded this interesting finding to show that both young and old adults tend to over-count small numbers, but older observers over-count more.

Conclusion/Significance

Here we show that age reduces the ability to correctly enumerate in a glance, but the accuracy (veridicality), on average, remains unchanged with advancing age. Control experiments indicate that the degraded performance cannot be explained by optical, retinal or other perceptual factors, but is cortical in origin.

Detecting children with arithmetic disabilities from kindergarten: evidence from a 3-year longitudinal study on the role of preparatory arithmetic abilities

In a 3-year longitudinal study, 471 children were classified, based on their performances on arithmetic tests in first and second grade, as having persistent arithmetic disabilities (AD), persistent low achieving (LA), persistent typical achieving, inconsistent arithmetic disabilities (DF1), or inconsistent low achieving in arithmetic. Significant differences in the performances on the magnitude comparison in kindergarten (at age 5-6) were found between the AD and LA and between the AD and DF1 groups. Furthermore, the percentage of true-positive AD children (at age 7-8) correctly diagnosed in kindergarten by combination of procedural counting, conceptual counting, and magnitude comparison tasks was 87.50%. When composing clinical samples, researchers should pay attention when stipulating restrictive or lenient cutoffs for arithmetic disabilities and select children based on their scores in 2 consecutive years, because the results of studies on persistent low achievers or children with inconsistent disabilities cannot be generalized to children with persistent arithmetic disabilities.

Can subitizing survive the attentional blink? An ERP study

This study was done to test whether subitizing versus counting are attention demanding based on whether they can be performed during the attentional blink (AB). ERPs were recorded while participants performed a task requiring them to judge the number of dots presented and this judgment task either followed the presentation of a task-relevant item in a rapid stimulus presentation stream (dual-task) or the potential target was task irrelevant (single-task). The behavioral data demonstrated that T2 accuracies decreased as a function of the number of dots not only in counting range, but also in subitizing range. The ERP results showed a delayed P3 component in the dual-task condition, and this was equally true for both subitizing and counting conditions. Furthermore, the P3 amplitude was reduced during the AB, and this was still equally true for both the subitizing and counting conditions. The present results suggest that both subitizing and counting require attention, and that subitizing is not a purely pre-attentive process.

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.