Numerosity estimation in visual stimuli in the absence of luminance-based cues

Effect of 3-D depth structure, element size, and area containing elements on total-element overestimation phenomenon

The number of elements distributed in a three-dimensional stimulus is overestimated compared to a two-dimensional stimulus when both stimuli have the same number of elements. We examined the effect of the properties of a three-dimensional stimulus (the number of overlapping stereo surfaces, size of the elements, and size of the area containing elements, on the overestimation phenomenon in four experiments. The two stimuli were presented side-by-side with the same diameters. Observers judged which of the three-dimensional standard and two-dimensional comparison had more elements. The results showed that (a) the overestimation phenomenon occurred for the three-dimensional standard stimuli, (b) the size of the areas affected the amount of overestimation, while the number of overlapping stereo surfaces and size of elements did not, and (c) the amount of overestimation increased when the stimuli included more than 100 elements. Implications of these findings were discussed in the framework of back-surface bias, occlusion, and disparity-processing interference models.

Spatial number estimation has a higher linear range than temporal number estimation; differential affordances for subdivision might help to explain why

Estimation of visuospatial number typically has a limited linear range that goes well beyond the subitizing range but typically not beyond 20 items without calibration procedures. Three experiments involving a total of 104 undergraduate students, each tested once, sought to determine if the limit on the linear range represented a capacity limitation of a linear accumulator or might be the result of a strategy based on subdividing spatial displays into potentially subitizable subsets. For visual and auditory temporal numbers for a large range of numbers (2–58; Experiment 1), the (unbiased) linear range was found to be quite restricted (three or four items). Using matched linear spatial number stimuli (Experiment 2), the linear range observed extended to about nine or 10 items. Experiment 3 compared estimates when simultaneous two-dimensional spatial number displays were presented briefly, with estimates for identical displays that accumulated over time. The linear range of estimates for accumulating spatial displays reached only 11 items, whereas that for briefly presented displays extended to about 20 items. These results suggest that the limit on the linear range is not simply a capacity limitation in a linear accumulator. Rather, they support the idea that linear spatial number estimation for the range from five to 20 may be based on subdividing the display into a subitizable number of (potentially) subitizable groups, even if those groups are not outwardly marked.

Nonsymbolic numerosity in sets with illusory-contours exploits a context-sensitive, but contrast-insensitive, visual boundary formation process

The visual mechanisms underlying approximate numerical representation are still intensely debated because numerosity information is often confounded with continuous sensory cues (e.g., texture density, area, convex hull). However, numerosity is underestimated when a few items are connected by illusory contours (ICs) lines without changing other physical cues, suggesting in turn that numerosity processing may rely on discrete visual input. Yet, in these previous works, ICs were generated by black-on-gray inducers producing an illusory brightness enhancement, which could represent a further continuous sensory confound. To rule out this possibility, we tested participants in a numerical discrimination task in which we manipulated the alignment of 0, 2, or 4 pairs of open/closed inducers and their contrast polarity. In Experiment 1, aligned open inducers had only one polarity (all black or all white) generating ICs lines brighter or darker than the gray background. In Experiment 2, open inducers had always opposite contrast polarity (one black and one white inducer) generating ICs without strong brightness enhancement. In Experiment 3, reverse-contrast inducers were aligned but closed with a line preventing ICs completion. Results showed that underestimation triggered by ICs lines was independent of inducer contrast polarity in both Experiment 1 and Experiment 2, whereas no underestimation was found in Experiment 3. Taken together, these results suggest that mere brightness enhancement is not the primary cause of the numerosity underestimation induced by ICs lines. Rather, a boundary formation mechanism insensitive to contrast polarity may drive the effect, providing further support to the idea that numerosity processing exploits discrete inputs.

Anisotropy of perceived numerosity: Evidence for a horizontal-vertical numerosity illusion

Many studies have investigated whether numerical and spatial abilities share similar cognitive systems. A novel approach to this issue consists of investigating whether the same perceptual biases underlying size illusions can be identified in numerical estimation tasks. In this study, we required adult participants to estimate the number of white dots in arrays made of white and black dots displayed in such a way as to generate horizontal-vertical illusions with inverted T and L configurations. In agreement with previous literature, we found that participants tended to underestimate the target numbers. However, in the presence of the illusory patterns, participants were less inclined to underestimate the number of vertically aligned white dots. This reflects the perceptual biases underlying horizontal-vertical illusions. In addition, we identified an enhanced illusory effect when participants observed vertically aligned white dots in the T shape compared to the L shape, a result that resembles the length bisection bias reported in the spatial domain. Overall, we found the first evidence that numerical estimation differs as a function of the vertical or horizontal displacement of the stimuli. In addition, the involvement of the same perceptual biases observed in spatial tasks supports the idea that spatial and numerical abilities share similar cognitive processes.

Food density and preferred quantity: discrimination of small and large numbers in angelfish (Pterophyllum scalare)

Many animal species share the ability to discriminate between sets with different quantity of food items. In fish, this ability has rarely been investigated, although findings have been obtained do indicate a preference, as in other animals, for sets with large over small quantities. The role played by food item size has also been found to be important in the discrimination. However, another potentially important non-numerical variable, food density, has not been investigated. In this study, we examined the influence of density (inter-item distance) in the decision-making process of food discrimination in angelfish (Pterophyllum scalare). In a binary choice task, we kept the number and size of food items constant, but contrasted a set containing food items spaced further apart (sparse set) to another set with food items spaced more closely (dense set). We conducted this analysis with sets in the small (3 vs 3 food items) and in the large number range (5 vs 5 food items) and also varied the specific spatial arrangements of the food items in the sets. Contrary to expectations, angelfish showed a preference for the sparse sets over the dense sets in the five vs five contrasts irrespective of the specific spatial arrangement, but exhibited no preference in case of the three vs three contrasts. Subsequently, we slightly lengthened the inter-item distance in the dense sets, and found preference for the dense over the sparse sets. Last, we further examined the potential effect of spatial configuration of the items in the sets, but found no effect of this latter factor. Overall, these results indicate that higher density of the contrasted food item sets significantly influences choice in angelfish, which prefer denser sets if a clear discriminability of each individual item within the sets is provided.

The role of awareness of repetition during the development of automaticity in a dot-counting task

This study examined whether being aware of the repetition of stimuli in a simple numerosity task could aid the development of automaticity. The numerosity task used in this study was a simple counting task. Thirty-four participants were divided into two groups. One group was instructed that the stimuli would repeat many times throughout the experiment. The results showed no significant differences in the way automatic processing developed between the groups. Similarly, there was no correlation between the point at which automatic processing developed and the point at which participants felt they benefitted from the repetition of stimuli. These results suggest that extra-trial features of a task may have no effect on the development of automaticity, a finding consistent with the instance theory of automatisation.

Distinct mechanisms in the numerosity processing of random and regular dots

This study investigated the mechanisms of the numerosity coding of random and regular dot distribution patterns. Experiment 1 revealed that connectedness significantly affected the numerosity perception of randomly distributed dots, and two adjacent dots were considered to be one numeral unit when connected via lines. The connectedness effect was much weaker on the numerosity perception of regularly distributed dots in vertical or horizontal queues and was absent in the perception of dots in diagonal queues. Experiment 2 demonstrated that randomly distributed adaptors induced a stronger effect of adaptation compared with regular adaptors when random dots after adaptation were used to test participants' numerosity perception. Experiment 3 found that the change in stimulus orientation has no effect on adaptation for random patterns. However, for regular patterns, adapting stimuli with an orientation identical to the tests caused stronger aftereffects compared with those with a different orientation. In Experiment 4, when random adaptors were presented in one eye of a participant, the adaptation aftereffect was shown to exist in both the exposed and un-exposed eyes (binocular transfer), whereas the aftereffect of regular adaptors remained only in the exposed eye (monocular transfer). We interpret that distinct mechanisms might control the numerosity processing of randomly and regularly distributed dots. Generic numerosity processing seems to be automatically inhibited based on the coding of regular patterns. The absence of numeral unit individuation, which is coded at a higher visual-processing level, might play an important role in this inhibition.

Spontaneous perception of numerosity in humans

Humans, including infants, and many other species have a capacity for rapid, nonverbal estimation of numerosity. However, the mechanisms for number perception are still not clear; some maintain that the system calculates numerosity via density estimates-similar to those involved in texture-while others maintain that more direct, dedicated mechanisms are involved. Here we show that provided that items are not packed too densely, human subjects are far more sensitive to numerosity than to either density or area. In a two-dimensional space spanning density, area and numerosity, subjects spontaneously react with far greater sensitivity to changes in numerosity, than either area or density. Even in tasks where they were explicitly instructed to make density or area judgments, they responded spontaneously to number. We conclude, that humans extract number information, directly and spontaneously, via dedicated mechanisms.

Number As a Primary Perceptual Attribute: A Review

Although humans are the only species to possess language-driven abstract mathematical capacities, we share with many other animals a nonverbal capacity for estimating quantities or numerosity. For some time, researchers have clearly differentiated between small numbers of items--less than about four--referred to as the subitizing range, and larger numbers, where counting or estimation is required. In this review, we examine more recent evidence suggesting a further division, between sets of items greater than the subitizing range, but sparse enough to be individuated as single items; and densely packed stimuli, where they crowd each other into what is better considered as a texture. These two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws and show different dependencies on eccentricity and on luminance levels. But provided the elements are not too crowded (less than about two items per square degree in central vision, less in the periphery), there is little evidence that estimation of numerosity depends on mechanisms responsive to texture. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

Modeling the approximate number system to quantify the contribution of visual stimulus features

The approximate number system (ANS) subserves estimation of the number of items in a set. Typically, ANS function is assessed by requiring participants to compare the number of dots in two arrays. Accuracy is determined by the numerical ratio of the sets being compared, and each participant's Weber fraction (w) provides a quantitative index of ANS acuity. When making numerical comparisons, however, performance is also influenced by non-numerical features of the stimuli, such as the size and spacing of dots. Current models of numerosity comparison do not account for these effects and consequently lead to different estimates of w depending on the methods used to control for non-numerical features. Here we proffer a new model that teases apart the effects of ANS acuity from the effects of non-numerical stimulus features. The result is an estimate of w that is a more theoretically valid representation of numerical acuity and novel terms that denote the degree to which a participant's perception of number is affected by non-numerical features. We tested this model in a sample of 20 adults and found that, by correctly attributing errors due to non-numerical stimulus features, the w obtained was more reliable across different stimulus conditions. We found that although non-numerical features biased numerosity discriminations in all participants, number was the primary feature driving discriminations in most of them. Our findings support the idea that, while numerosity is a distinct visual quantity, the internal representation of number is tightly bound to the representation of other magnitudes. This tool for identifying the different effects of the numerical and non-numerical features of a stimulus has important implications not only for the behavioral investigation of the ANS, but also for the collection and analyses of neural data sets associated with ANS function.

Methodological aspects to be considered when measuring the approximate number system (ANS) - a research review

According to a dominant view, the approximate number system (ANS) is the foundation of symbolic math abilities. Due to the importance of math abilities for education and career, a lot of research focuses on the investigation of the ANS and its relationship with math performance. However, the results are inconsistent. This might be caused by studies differing greatly regarding the operationalization of the ANS (i.e., tasks, dependent variables). Moreover, many methodological aspects vary from one study to the next. In the present review, we discuss commonly used ANS tasks and dependent variables regarding their theoretical foundation and psychometric features. We argue that the inconsistent findings concerning the relationship between ANS acuity and math performance may be partially explained by differences in reliability. Furthermore, this review summarizes methodological aspects of ANS tasks having important impacts on the results, including stimulus range, visual controls, presentation duration of the stimuli and feedback. Based on this review, we give methodological recommendations on how to assess the ANS most reliably and most validly. All important methodological aspects to be considered when designing an ANS task or comparing results of different studies are summarized in two practical checklists.

Effects of awareness on numerosity adaptation

Numerosity perception is a process involving several stages of visual processing. This study investigated whether distinct mechanisms exist in numerosity adaptation under different awareness conditions to characterize how numerosity perception occurs at each stage. The status of awareness was controlled by masking conditions, in which monoptic and dichoptic masking were proposed to influence different levels of processing. Numerosity adaptation showed significant aftereffects when the participants were aware (monoptic masking) and unaware (dichoptic masking) of adaptors. The interocular transfer for numerosity adaptation was distinct under the different awareness conditions. Adaptation was primarily binocular when participants were aware of stimuli and was purely monocular when participants were unaware of adaptors. Moreover, numerosity adaptation was significantly reduced when the adaptor dots were clustered into chunks with awareness, whereas clustering had no effect on unaware adaptation. These results show that distinct mechanisms exist in numerosity processing under different awareness conditions. It is suggested that awareness is crucial to numerosity cognition. With awareness, grouping (by clustering) influences numerosity coding through altered object representations, which involves higher-level cognitive processing.

Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement

Given the well-documented failings in mathematics education in many Western societies, there has been an increased interest in understanding the cognitive underpinnings of mathematical achievement. Recent research has proposed the existence of an Approximate Number System (ANS) which allows individuals to represent and manipulate non-verbal numerical information. Evidence has shown that performance on a measure of the ANS (a dot comparison task) is related to mathematics achievement, which has led researchers to suggest that the ANS plays a critical role in mathematics learning. Here we show that, rather than being driven by the nature of underlying numerical representations, this relationship may in fact be an artefact of the inhibitory control demands of some trials of the dot comparison task. This suggests that recent work basing mathematics assessments and interventions around dot comparison tasks may be inappropriate.

Activity counts: the effect of swimming activity on quantity discrimination in fish

Human infants and non-human animals can discriminate the larger of two sets of discrete items. This quantity discrimination may be based upon the number of items, or upon non-numerical variables of the sets that co-vary with number. We have demonstrated that angelfish select the larger of two shoals of conspecifics without using inter-fish distance or space occupied by the stimuli as cues. However, density appeared to influence the choice between large shoals. Here, we examine the role of another non-numerical cue, swimming activity of the stimulus fish, in quantity discrimination by angelfish. To control this variable, we varied the water temperature of the stimulus aquaria or restricted the space occupied by each fish in the stimulus shoals. We used the previously successfully discriminated contrasts consisting of large (10 vs. 5) and small (3 vs. 2) shoals. We also studied whether more active or less active shoals are preferred in case of equally sized shoals (10 vs. 10, 5 vs. 5, and 3 vs. 3). When differences in stimulus fish activity were minimized by temperature manipulation we found angelfish to prefer the larger shoal in the 3 vs. 2 comparison, but not in the 10 vs. 5 comparison. When activity was controlled by space restriction, angelfish preferred the larger shoal in both numerical contrasts. These results imply that the overall activity level of the contrasted shoals is not a necessary condition for small shoals discrimination in angelfish. On the other hand, the results obtained for the large shoals, together with results obtained in the control treatments (equal numerical contrasts and differing activity levels), suggest that activity is a sufficient condition for discrimination when large shoals are involved. Further experiments are needed to evaluate the influence of other continuous variables, and to assess whether the mechanisms underlying performance are comparable to those suggested for other animals.