Nonverbal estimation during numerosity judgements by adult humans

Performance of Asian elephants (Elephas maximus) on a quantity discrimination task is similar to that of African savanna elephants (Loxodonta africana)

Using an object-choice task, we measured the relative quantity discrimination ability of Asian elephants. Two zoo-housed elephants were given auditory cues of food being dropped into two containers (Nonvisible condition), and in one condition they could also see the food on top of the containers (Visible condition). Elephants received sets of varying ratios and magnitudes. We found that the elephants chose the greater quantity of food significantly above chance in both the Visible and Nonvisible conditions. Additionally, we found the elephants' ability to discriminate between quantities decreased as the ratio, and not the absolute difference, between the quantities increased, which is predicted by the accumulator model. We also compare our findings to those from a study using the same methods with African savanna elephants and found that the two species performed at similar levels, but given our small sample size it is difficult to make strong species-level conclusions. In discussing our results, we consider differences between the two species' wild environments as well as the types of sensory cues provided in human care, and we provide recommendations for extensions of this work.

The Enculturated Move From Proto-Arithmetic to Arithmetic

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation.

Capuchin monkeys (Cebus apella) treat small and large numbers of items similarly during a relative quantity judgment task

A key issue in understanding the evolutionary and developmental emergence of numerical cognition is to learn what mechanism(s) support perception and representation of quantitative information. Two such systems have been proposed, one for dealing with approximate representation of sets of items across an extended numerical range and another for highly precise representation of only small numbers of items. Evidence for the first system is abundant across species and in many tests with human adults and children, whereas the second system is primarily evident in research with children and in some tests with non-human animals. A recent paper (Choo & Franconeri, Psychonomic Bulletin & Review, 21, 93-99, 2014) with adult humans also reported "superprecise" representation of small sets of items in comparison to large sets of items, which would provide more support for the presence of a second system in human adults. We first presented capuchin monkeys with a test similar to that of Choo and Franconeri in which small or large sets with the same ratios had to be discriminated. We then presented the same monkeys with an expanded range of comparisons in the small number range (all comparisons of 1-9 items) and the large number range (all comparisons of 10-90 items in 10-item increments). Capuchin monkeys showed no increased precision for small over large sets in making these discriminations in either experiment. These data indicate a difference in the performance of monkeys to that of adult humans, and specifically that monkeys do not show improved discrimination performance for small sets relative to large sets when the relative numerical differences are held constant.

Ratio dependence in small number discrimination is affected by the experimental procedure

Adults, infants and some non-human animals share an approximate number system (ANS) to estimate numerical quantities, and are supposed to share a second, 'object-tracking,' system (OTS) that supports the precise representation of a small number of items (up to 3 or 4). In relative numerosity judgments, accuracy depends on the ratio of the two numerosities (Weber's Law) for numerosities >4 (the typical ANS range), while for numerosities ≤4 (OTS range) there is usually no ratio effect. However, recent studies have found evidence for ratio effects for small numerosities, challenging the idea that the OTS might be involved for small number discrimination. Here we tested the hypothesis that the lack of ratio effect in the numbers 1-4 is largely dependent on the type of stimulus presentation. We investigated relative numerosity judgments in college students using three different procedures: a simultaneous presentation of intermingled and separate groups of dots in separate experiments, and a further experiment with sequential presentation. As predicted, in the large number range, ratio dependence was observed in all tasks. By contrast, in the small number range, ratio insensitivity was found in one task (sequential presentation). In a fourth experiment, we showed that the presence of intermingled distractors elicited a ratio effect, while easily distinguishable distractors did not. As the different ratio sensitivity for small and large numbers has been often interpreted in terms of the activation of the OTS and ANS, our results suggest that numbers 1-4 may be represented by both numerical systems and that the experimental context, such as the presence/absence of task-irrelevant items in the visual field, would determine which system is activated.

Visual nesting of stimuli affects rhesus monkeys' (Macaca mulatta) quantity judgments in a bisection task

Nonhuman animals are highly proficient at judging relative quantities presented in a variety of formats, including visual, auditory, and even cross-modal formats. Performance typically is constrained by the ratio between sets, as would be expected under Weber's law and as is described in the approximate number system (ANS) hypothesis. In most cases, tests are designed to avoid any perceptual confusion for animals regarding the stimulus sets, but despite this, animals show some of the perceptual biases that humans show based on organization of stimuli. Here, we demonstrate an additional perceptual bias that emerges from the illusion of nested sets. When arrays of circles were presented on a computer screen and were to be classified as larger than or smaller than an established central value, rhesus monkeys (Macaca mulatta) underestimated quantities when circles were nested within each other. This matched a previous report with adult humans (Chesney & Gelman, Attention, Perception, & Psychophysics 24:1104-1113, 2012), indicating that macaques, like humans, show the pattern of biased perception predicted by ANS estimation. Although some macaques overcame this perceptual bias, demonstrating that they could come to view nested stimuli as individual elements to be included in the estimates of quantity used for classifying arrays, the majority of the monkeys showed the bias of underestimating nested arrays throughout the experiment.

Inherently Analog Quantity Representations in Olive Baboons (Papio anubis)

Strong evidence indicates that non-human primates possess a numerical representation system, but the inherent nature of that system is still debated. Two cognitive mechanisms have been proposed to account for non-human primate numerical performance: (1) a discrete object-file system limited to quantities <4, and (2) an analog system which represents quantities comparatively but is limited by the ratio between two quantities. To test the underlying nature of non-human primate quantification, we asked eight experiment-naive olive baboons (Papio anubis) to discriminate between number pairs containing small (<4), large (>4), or span (small vs. large) numbers of food items presented simultaneously or sequentially. The prediction from the object-file hypothesis is that baboons will only accurately choose the larger quantity in small pairs, but not large or span pairs. Conversely, the analog system predicts that baboons will be successful with all numbers, and that success will be dependent on numerical ratio. We found that baboons successfully discriminated all pair types at above chance levels. In addition, performance significantly correlated with the ratio between the numerical values. Although performance was better for simultaneous trials than sequential trials, evidence favoring analog numerical representation emerged from both conditions, and was present even in the first exposure to number pairs. Together, these data favor the interpretation that a single, coherent analog representation system underlies spontaneous quantitative abilities in primates.

Evidence for two numerical systems that are similar in humans and guppies

BACKGROUND

Humans and non-human animals share an approximate non-verbal system for representing and comparing numerosities that has no upper limit and for which accuracy is dependent on the numerical ratio. Current evidence indicates that the mechanism for keeping track of individual objects can also be used for numerical purposes; if so, its accuracy will be independent of numerical ratio, but its capacity is limited to the number of items that can be tracked, about four. There is, however, growing controversy as to whether two separate number systems are present in other vertebrate species.

METHODOLOGY/PRINCIPAL FINDINGS

In this study, we compared the ability of undergraduate students and guppies to discriminate the same numerical ratios, both within and beyond the small number range. In both students and fish the performance was ratio-independent for the numbers 1-4, while it steadily increased with numerical distance when larger numbers were presented.

CONCLUSIONS/SIGNIFICANCE

Our results suggest that two distinct systems underlie quantity discrimination in both humans and fish, implying that the building blocks of uniquely human mathematical abilities may be evolutionarily ancient, dating back to before the divergence of bony fish and tetrapod lineages.

Quantity judgments of auditory and visual stimuli by chimpanzees (Pan troglodytes)

Many species can choose between two visual sets of stimuli on the basis of quantity. This is true when sets are both visible, or are presented one set at a time or even one item at a time. However, we know comparatively little about how well nonhuman animals can compare auditory quantities. Here, three chimpanzees (Pan troglodytes) chose between two sets of food items when they only heard each item fall into different containers rather than seeing those items. This method prevented the chimpanzees from summing the amount of visible food they saw because there were no visual cues. Chimpanzees performed well, and their performance matched that of previous experiments with regard to obeying Weber's law. They also performed well with comparisons between a sequentially presented auditory set and a fully visible set, demonstrating that duration of presentation was not being used as a cue. In addition, they accommodated empty sets into these judgments, although not perfectly. Thus, chimpanzees can judge auditory quantities in flexible ways that show many similarities to how they compare visual quantities.

Monkeys (macaca mulatta and cebus apella) and human adults and children (homo sapiens) compare subsets of moving stimuli based on numerosity

Two monkey species (Macaca mulatta and Cebus apella) and human children and adults judged the numerousness of two subsets of moving stimuli on a computer screen. Two sets of colored dots that varied in number and size were intermixed in an array in which all dots moved in random directions and speeds. Participants had to indicate which dot color was more numerous within the array. All species performed at high and comparable levels, including on trials in which the subset with the larger number of items had a smaller total area of coloration. This indicated a similarity across species to use the number of items in the subsets, and not dimensions such as area or volume, to guide decision making. Discrimination performance was constrained by the ratio between the subsets, consistent with other reports of numerousness judgments of stationary stimuli. These results indicate a similarity in numerical estimation ability for moving stimuli across primate species, and this capacity may be necessary for naturally occurring experiences in which moving stimuli must be summed.

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

BACKGROUND

Numerosity estimation is a basic preverbal ability that humans share with many animal species and that is believed to be foundational of numeracy skills. It is notoriously difficult, however, to establish whether numerosity estimation is based on numerosity itself, or on one or more non-numerical cues like-in visual stimuli-spatial extent and density. Frequently, different non-numerical cues are held constant on different trials. This strategy, however, still allows numerosity estimation to be based on a combination of non-numerical cues rather than on any particular one by itself.

METHODOLOGY/PRINCIPAL FINDINGS

Here we introduce a novel method, based on second-order (contrast-based) visual motion, to create stimuli that exclude all first-order (luminance-based) cues to numerosity. We show that numerosities can be estimated almost as well in second-order motion as in first-order motion.

CONCLUSIONS/SIGNIFICANCE

The results show that numerosity estimation need not be based on first-order spatial filtering, first-order density perception, or any other processing of luminance-based cues to numerosity. Our method can be used as an effective tool to control non-numerical variables in studies of numerosity estimation.

Numerical values leave a semantic imprint on associated signs in monkeys

Animals and humans share an evolutionary ancient quantity representation which is characterized by analog magnitude features: Discriminating magnitudes becomes more difficult with increasing set sizes (size effect) and with decreasing distance between two numerosities (distance effect). Humans show these effects even with number symbols. We wondered whether monkeys would show the same psychophysical effects with numerical signs and addressed this issue by training three monkeys to associate visual shapes with numerosities. We then confronted the monkeys with trials in which they had to match these visual signs with each other. The monkeys' performance in this shape versus shape protocol was positively correlated with the numerical distance and the magnitudes associated with the signs. Additionally, the monkeys responded significantly slower for signs with higher assigned numerical values. These findings suggest that the numerical values imprint their analog magnitudes characteristics onto the associated visual sign in monkeys, an effect that we also found reflected in the discharges of prefrontal neurons. This provides evidence for a precursor of the human number symbol knowledge.

Range dependent processing of visual numerosity: similarities across vision and haptics

'Subitizing' refers to fast and accurate judgement of small numerosities, whereas for larger numerosities either counting or estimation are used. Counting is slow and precise, whereas estimation is fast but imprecise. In this study consisting of five experiments we investigated if and how the numerosity judgement process is affected by the relative spacing between the presented numerosities. To this end we let subjects judge the number of dots presented on a screen and recorded their response times. Our results show that subjects switch from counting to estimation if the relative differences between subsequent numerosities are large (a factor of 2), but that numerosity judgement in the subitizing range was still faster. We also show this fast performance for small numerosities only occurred when numerosity information is present. This indicates this is typical for number processing and not magnitude estimation in general. Furthermore, comparison with a previous haptic study suggests similar processing in numerosity judgement through haptics and vision.

Understanding quantity in semantic dementia

Patients with semantic dementia (SD) show relative preservation of number skills, contrasting with their severe multimodal semantic impairment. Underpinning this preservation, it is argued, is spared understanding of numerical quantity, a competence likened to a spatial map and subserved by the parietal lobes. The study investigated quantity knowledge in 14 SD patients to determine whether it is consistently preserved irrespective of disease severity or whether there are constraints on this preserved knowledge domain. Performance was well preserved on Piagetian conservation tasks, estimating object numerosity, and understanding of basic numerical magnitude. However, patients showed impairment on real-world estimation tasks, increasing with semantic severity. More surprisingly, on an analogue scale task, they produced implausible responses, suggesting degraded knowledge of precise numerical relationships. The findings challenge the view that knowledge of quantity is totally preserved in SD and suggest that the temporal lobes have a contributory role in the conceptual understanding of quantity.

Compressed Scaling of Abstract Numerosity Representations in Adult Humans and Monkeys

There is general agreement that nonverbal animals and humans endowed with language possess an evolutionary precursor system for representing and comparing numerical values. However, whether nonverbal numerical representations in human and nonhuman primates are quantitatively similar and whether linear or logarithmic coding underlies such magnitude judgments in both species remain elusive. To resolve these issues, we tested the numerical discrimination performance of human subjects and two rhesus monkeys (Macaca mulatta) in an identical delayed match-to-numerosity task for a broad range of numerosities from 1 to 30. The results demonstrate a noisy nonverbal estimation system obeying Weber's Law in both species. With average Weber fractions in the range of 0.51 and 0.60, nonverbal numerosity discriminations in humans and monkeys showed similar precision. Moreover, the detailed analysis of the performance distributions exhibited nonlinearly compressed numerosity representations in both primate species. However, the difference between linear and logarithmic scaling was less pronounced in humans. This may indicate a gradual transformation of a logarithmic to linear magnitude scale in human adults as the result of a cultural transformation process during the course of mathematical education.

Basic math in monkeys and college students

Adult humans possess a sophisticated repertoire of mathematical faculties. Many of these capacities are rooted in symbolic language and are therefore unlikely to be shared with nonhuman animals. However, a subset of these skills is shared with other animals, and this set is considered a cognitive vestige of our common evolutionary history. Current evidence indicates that humans and nonhuman animals share a core set of abilities for representing and comparing approximate numerosities nonverbally; however, it remains unclear whether nonhuman animals can perform approximate mental arithmetic. Here we show that monkeys can mentally add the numerical values of two sets of objects and choose a visual array that roughly corresponds to the arithmetic sum of these two sets. Furthermore, monkeys' performance during these calculations adheres to the same pattern as humans tested on the same nonverbal addition task. Our data demonstrate that nonverbal arithmetic is not unique to humans but is instead part of an evolutionarily primitive system for mathematical thinking shared by monkeys.

Adult humans possess mathematical abilities that are unmatched by any other member of the animal kingdom. Yet, there is increasing evidence that the ability to enumerate sets of objects nonverbally is a capacity that humans share with other animal species. That is, like humans, nonhuman animals possess the ability to estimate and compare numerical values nonverbally. We asked whether humans and nonhuman animals also share a capacity for nonverbal arithmetic. We tested monkeys and college students on a nonverbal arithmetic task in which they had to add the numerical values of two sets of dots together and choose a stimulus from two options that reflected the arithmetic sum of the two sets. Our results indicate that monkeys perform approximate mental addition in a manner that is remarkably similar to the performance of the college students. These findings support the argument that humans and nonhuman primates share a cognitive system for nonverbal arithmetic, which likely reflects an evolutionary link in their cognitive abilities.

Monkeys have an ability to represent numerical values even though they lack linguistic abilities. The authors show that monkeys can also perform addition on numerical values and that they perform similarly to college students who are asked to add without counting.

Quantity representation in children and rhesus monkeys: linear versus logarithmic scales

The performances of 4- and 5-year-olds and rhesus monkeys were compared using a computerized task for quantity assessment. Participants first learned two quantity anchor values and then responded to intermediate values by classifying them as similar to either the large anchor or the small anchor. Of primary interest was an assessment of where the point of subjective equality (PSE) occurred for each species across four different sets of anchors to determine whether the PSE occurred at the arithmetic mean or the geometric mean. Both species produced PSEs that were closer to the geometric mean for three of four anchor sets. This indicates that monkeys and children access either a logarithmic scale for quantity representation or a linear scale that is subject to scalar variability, both of which are consistent with Weber's law and representation of quantity that takes the form of analog magnitudes.

Rhesus monkeys (Macaca mulatta) enumerate large and small sequentially presented sets of items using analog numerical representations

Two rhesus monkeys selected the larger of two sequentially presented sets of items on a computer monitor. In Experiment 1, performance was related to the ratio of set sizes, and the monkeys discriminated between sets with up to 10 items. Performance was not disrupted when 1 set had fewer than 4 items and 1 set had more than 4 items, a critical trial type for differentiating object file and analog models of numerical representation. Experiment 2 controlled the interitem rate of presentation. Experiment 3 included some trials on which number and amount (visual surface area) offered conflicting cues. Experiment 4 varied the total duration of set presentation and the duration of item visibility. In all of the experiments, performance remained high, although total set presentation duration also acted as a partial cue for the monkeys. Overall, the data indicated that rhesus monkeys estimate the approximate number of items in sequentially presented sets and that they are not relying solely on nonnumerical cues such as rate, duration, or cumulative amount.