Semantic congruity affects numerical judgments similarly in monkeys and humans

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Neurobiology of numerical learning

Numerical abilities are complex cognitive skills essential for dealing with requirements of the modern world. Although the brain structures and functions underlying numerical cognition in different species have long been appreciated, genetic and molecular techniques have more recently expanded the knowledge about the mechanisms underlying numerical learning. In this review, we discuss the status of the research related to the neurobiological bases of numerical abilities. We consider how genetic factors have been associated with mathematical capacities and how these link to the current knowledge of brain regions underlying these capacities in human and non-human animals. We further discuss the extent to which significant variations in the levels of specific neurotransmitters may be used as potential markers of individual performance and learning difficulties and take into consideration the therapeutic potential of brain stimulation methods to modulate learning and improve interventional outcomes. The implications of this research for formulating a more comprehensive view of the neural basis of mathematical learning are discussed.

Convergent Circuit Computation for Categorization in the Brains of Primates and Songbirds

Categorization is crucial for behavioral flexibility because it enables animals to group stimuli into meaningful classes that can easily be generalized to new circumstances. A most abstract quantitative category is set size, the number of elements in a set. This review explores how categorical number representations are realized by the operations of excitatory and inhibitory neurons in associative telencephalic microcircuits in primates and songbirds. Despite the independent evolution of the primate prefrontal cortex and the avian nidopallium caudolaterale, the neuronal computations of these associative pallial circuits show surprising correspondence. Comparing cellular functions in distantly related taxa can inform about the evolutionary principles of circuit computations for cognition in distinctly but convergently realized brain structures.

Distinct Mechanisms in Number Comparison of Random and Regular Dots: An ERP Study

Numerosity comparison for regular patterns shows different features compared with that for random ones in previous studies, suggesting an underlying mechanism distinct from numerosity. In this study, we went further to compare the event-related potentials (ERP) components in numerosity processing of random and regular patterns, which are identical in all aspects of texture features except for the distribution. ERP components were recorded and analyzed while participants compared which of the two successively presented sets was more numerous. P2p amplitude was revealed to be significantly weaker for regular patterns compared with that for random patterns over right occipital-parietal cites, whereas no difference was found for P1 or N1 components. The difference in P2p amplitude, which is consistent with the behavior dissociation revealed in our previous studies, suggests that regular distribution can trigger distinct processing in numeral comparison tasks. Processing of continuous magnitudes or configuration cannot explain the decrease in P2p amplitude for regular distributed patterns. Therefore, this study further supports that P2p is mediated by numerosity processing.

No evidence for a core deficit in developmental dyscalculia or mathematical learning disabilities

BACKGROUND

Two hypotheses were tested regarding the characteristics of children with mathematical learning disabilities (MLD): (a) that children with MLD would have a 'core deficit' in basic number processing skills; and (b) that children with MLD would be at the end of a developmental continuum and have impairments in many cognitive skills.

METHODS

From a large sample (N = 1,303) of typically developing children, we selected a group definable as having MLD. The children were given measures of basic number processing and domain-general constructs. Differences between the observed sample and a simulated population were estimated using Cohen's d and Bayes factors. Receiver operating characteristic curves were plotted, and the area under the curve was computed to ascertain the diagnostic power of measures.

RESULTS

Results suggest that the differences between the MLD and control group can be defined along with general characteristics of the population rather than assuming single or multiple 'core deficits'. None of the measures of interest exceeded the diagnostic power that could be derived via simulation from the dimensional characteristics of the general population.

CONCLUSIONS

There is no evidence for core deficit(s) in MLD. We suggest that future research should focus on representative samples of typical populations and on carefully tested clinical samples confirming to the criteria of international diagnostic manuals. Clinical diagnoses require that MLD is persistent and resistant to intervention, so studies would deliver results less exposed to measurement fluctuations. Uniform diagnostic criteria would also allow for the easy cross-study comparison of samples overcoming a serious limitation of the current literature.

Visual number sense in untrained deep neural networks

Number sense, the ability to estimate numerosity, is observed in naïve animals, but how this cognitive function emerges in the brain remains unclear. Here, using an artificial deep neural network that models the ventral visual stream of the brain, we show that number-selective neurons can arise spontaneously, even in the complete absence of learning. We also show that the responses of these neurons can induce the abstract number sense, the ability to discriminate numerosity independent of low-level visual cues. We found number tuning in a randomly initialized network originating from a combination of monotonically decreasing and increasing neuronal activities, which emerges spontaneously from the statistical properties of bottom-up projections. We confirmed that the responses of these number-selective neurons show the single- and multineuron characteristics observed in the brain and enable the network to perform number comparison tasks. These findings provide insight into the origin of innate cognitive functions.

Application of an abstract concept across magnitude dimensions by fish

Mastering relational concepts and applying them to different contexts presupposes abstraction capacities and implies a high level of cognitive sophistication. One way to investigate extrapolative abilities is to assess cross-dimensional application of an abstract relational magnitude rule to new domains. Here we show that angelfish initially trained to choose either the shorter of two lines in a spatial task (line-length discrimination task) or the array with “fewer” items (numerical discrimination task) spontaneously transferred the learnt rule to novel stimuli belonging to the previously unseen dimension demonstrating knowledge of the abstract concept of “smaller”. Our finding challenges the idea that the ability to master abstract magnitude concepts across domains is unique to humans and suggests that the circuits involved in rule learning and magnitude processing might be evolutionary conserved.

Large as being on top of the world and small as hitting the roof: a common magnitude representation for the comparison of emotions and numbers

Previous work on the direct speed-intensity association (SIA) on comparative judgement tasks involved spatially distributed responses over spatially distributed stimuli with high motivational significance like facial expressions of emotions. This raises the possibility that the inferred stimulus-driven regulation of lateralized motor reactivity described by SIA, which was against the one expected on the basis of a valence-specific lateral bias, was entirely due to attentional capture from motivational significance (beyond numerical cognition). In order to establish the relevance of numerical cognition on the regulation of attentional capture we ran two complementary experiments. These involved the same direct comparison task on stimulus pairs that were fully comparable in terms of their analog representation of intensity but with different representational domain and motivational significance: symbolic magnitudes with low motivational significance in experiment 1 vs. emotions with rather high motivational significance in experiment 2. The results reveal a general SIA and point to a general mechanism regulating comparative judgements. This is based on the way spatial attention is captured toward locations that contain the stimulus which is closest in term of relative intensity to the extremal values of the series, regardless from its representational domain being it symbolic or emotional.

Limited evidence of number-space mapping in rhesus monkeys (Macaca mulatta) and capuchin monkeys (Sapajus apella)

Humans exhibit evidence of a mental number line that suggests a left-to-right, or sometimes right-to-left, representation of smaller to larger numbers. The Spatial Numerical Association of Response Codes (SNARC) effect is one example of this mental number line and has been investigated extensively in humans. Less research has been done with animals, and results have been inconclusive. Rugani, Vallortigara, Priftis, and Regolin (2015) found that young chicks showed a bias to respond to small quantities presented to their left and large quantities presented to their right when forced to move toward those stimuli to gain food reward. We replicated this design with rhesus macaques and capuchin monkeys using a computerized task, but we did not find this outcome. We also trained monkeys to choose between 2 arrays of dots, and then assessed biases in terms of choice location and response latency on trials with a numerical difference and on trials with equal numbers of items in both sets. There was no evidence of SNARC-like effects in equal trials, although when arrays differed in number, 12 of 19 monkeys showed differential performance depending on whether the smaller array was at the left or at the right onscreen. These results indicate that SNARC-like effects may not emerge in all contexts and may not be phylogenetically widespread. More effort is needed to broaden the number of species assessed and match other methods that are used with human participants so that we can better define the presence and extent of such effects. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Emotional Semantic Congruency based on stimulus driven comparative judgements

A common cognitive process in everyday life consists in the comparative judgements of emotions given a pair of facial expressions and the choice of the most positive/negative among them. Results from three experiments on complete-facial expressions (happy/angry) and mixed-facial expressions (neutral/happy-or-angry) pairs viewed with (Experiment 1 and 3) or without (Experiment 2) foveation and performed in conditions in which valence was either task relevant (Experiment 1 and 2) or task irrelevant (Experiment 3), show that comparative judgements of emotions are stimulus driven. Judgements' speed increased as the target absolute emotion intensity grew larger together with the average emotion of the pair, irrespective of the compatibility between the valence and the side of motor response: a semantic congruency effect in the domain of emotion. This result undermines previous interpretation of results in the context of comparative judgements based on the lateralization of emotions (e.g., SNARC-like instructional flexibility), and is fully consistent with our formalization of emotional semantic congruency: the direct Speed-Intensity Association model.

Spatial Presentations, but Not Response Formats Influence Spatial-Numerical Associations in Adults

According to theories of embodied numerosity, processing of numerical magnitude is anchored in bodily experiences. In particular, spatial representations of number interact with movement in physical space, but it is still unclear whether the extent of the movement is relevant for this interaction. In this study, we compared spatial-numerical associations over response movements of differing spatial expansion. We expected spatial-numerical effects to increase with the extent of physical response movements. In addition, we hypothesized that these effects should be influenced by whether or not a spatial representation of numbers was presented. Adult participants performed two tasks: a magnitude classification (comparing numbers to the fixed standard 5), from which we calculated the Spatial Numerical Association of Response Codes (SNARC) effect; and a magnitude comparison task (comparing two numbers against each other), from which we calculated a relative numerical congruity effect (NCE), which describes that when two relatively small numbers are compared, responses to the smaller number are faster than responses to the larger number; and vice versa for large numbers. A SNARC effect was observed across all conditions and was not influenced by response movement extent but increased when a number line was presented. In contrast, an NCE was only observed when no number line was presented. This suggests that the SNARC effect and the NCE reflect two different processes. The SNARC effect seems to represent a highly automated classification of numbers as large or small, which is further emphasized by the presentation of a number line. In contrast, the NCE likely results from participants not only classifying numbers as small or large, but also processing their relative size within the relevant section of their mental number line representation. An additional external presentation of a number line might interfere with this process, resulting in overall slower responses. This study follows up on previous spatial-numerical training studies and has implications for future spatial-numerical trainings. Specifically, similar studies with children showed contrasting results, in that response format but not number line presentation influenced spatial-numerical associations. Accordingly, during development, the relative relevance of physical experiences and presentation format for spatial-numerical associations might change.

Numerical ordering of zero in honey bees

Some vertebrates demonstrate complex numerosity concepts-including addition, sequential ordering of numbers, or even the concept of zero-but whether an insect can develop an understanding for such concepts remains unknown. We trained individual honey bees to the numerical concepts of "greater than" or "less than" using stimuli containing one to six elemental features. Bees could subsequently extrapolate the concept of less than to order zero numerosity at the lower end of the numerical continuum. Bees demonstrated an understanding that parallels animals such as the African grey parrot, nonhuman primates, and even preschool children.

The problem with percentages

A great many students at a major research university make basic conceptual mistakes in responding to simple questions about two successive percentage changes. The mistakes they make follow a pattern already familiar from research on the difficulties that elementary school students have in coming to terms with fractions and decimals. The intuitive core knowledge of arithmetic with the natural numbers makes learning to count and do simple arithmetic relatively easy. Those same principles become obstacles to understanding how to operate with rational numbers.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review

It is widely accepted that human and nonhuman species possess a specialized system to process large approximate numerosities. The theory of an evolutionarily ancient approximate number system (ANS) has received converging support from developmental studies, comparative experiments, neuroimaging, and computational modelling, and it is one of the most dominant and influential theories in numerical cognition. The existence of an ANS system is significant, as it is believed to be the building block of numerical development in general. The acuity of the ANS is related to future arithmetic achievements, and intervention strategies therefore aim to improve the ANS. Here we critically review current evidence supporting the existence of an ANS. We show that important shortcomings and confounds exist in the empirical studies on human and non-human animals as well as the logic used to build computational models that support the ANS theory. We conclude that rather than taking the ANS theory for granted, a more comprehensive explanation might be provided by a sensory-integration system that compares or estimates large approximate numerosities by integrating the different sensory cues comprising number stimuli.

Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children

Current theories of numerical cognition posit that uniquely human symbolic number abilities connect to an early developing cognitive system for representing approximate numerical magnitudes, the approximate number system (ANS). In support of this proposal, recent laboratory-based training experiments with U.S. children show enhanced performance on symbolic addition after brief practice comparing or adding arrays of dots without counting: tasks that engage the ANS. Here we explore the nature and generality of this effect through two brief training experiments. In Experiment 1, elementary school children in Pakistan practiced either a non-symbolic numerical addition task or a line-length addition task with no numerical content, and then were tested on symbolic addition. After training, children in the numerical training group completed the symbolic addition test faster than children in the line length training group, suggesting a causal role of brief, non-symbolic numerical training on exact, symbolic addition. These findings replicate and extend the core findings of a recent U.S. laboratory-based study to non-Western children tested in a school setting, attesting to the robustness and generalizability of the observed training effects. Experiment 2 tested whether ANS training would also enhance the consistency of performance on a symbolic number line task. Over several analyses of the data there was some evidence that approximate number training enhanced symbolic number line placements relative to control conditions. Together, the findings suggest that engagement of the ANS through brief training procedures enhances children's immediate attention to number and engagement with symbolic number tasks.

The neuronal code for number

Humans and non-human primates share an elemental quantification system that resides in a dedicated neural network in the parietal and frontal lobes. In this cortical network, 'number neurons' encode the number of elements in a set, its cardinality or numerosity, irrespective of stimulus appearance across sensory motor systems, and from both spatial and temporal presentation arrays. After numbers have been extracted from sensory input, they need to be processed to support goal-directed behaviour. Studying number neurons provides insights into how information is maintained in working memory and transformed in tasks that require rule-based decisions. Beyond an understanding of how cardinal numbers are encoded, number processing provides a window into the neuronal mechanisms of high-level brain functions.

Monkeys display classic signatures of human symbolic arithmetic

Non-human primates compare quantities in a crude manner, by approximating their values. Less is known about the mental transformations that non-humans can perform over approximate quantities, such as arithmetic transformations. There is evidence that human symbolic arithmetic has a deep psychological connection with the primitive, approximate forms of quantification of non-human animals. Here, we ask whether the subtle performance signatures that humans exhibit during symbolic arithmetic also bear a connection to primitive arithmetic. Specifically, we examined the problem size effect, the tie effect, and the practice effect-effects which are commonly observed in children's math performance in school. We show that, like humans, monkeys exhibited the problem size and tie effects, indicating commonalities in arithmetic algorithms with humans. Unlike humans, however, monkeys did not exhibit a practice effect. Together, these findings provide new evidence for a cognitive relation between non-symbolic and symbolic arithmetic.

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.

Time to decide? Simplicity and congruity in comparative judgment

What is the relationship between magnitude judgments relying on directly available characteristics versus probabilistic cues? Question frame was manipulated in a comparative judgment task previously assumed to involve inference across a probabilistic mental model (e.g., "Which city is largest"--the "larger" question-vs. "Which city is smallest"--the "smaller" question). Participants identified either the largest or smallest city (Experiments 1a and 2) or the richest or poorest person (Experiment 1b) in a 3-alternative forced-choice (3-AFC) task (Experiment 1) or a 2-AFC task (Experiment 2). Response times revealed an interaction between question frame and the number of options recognized. When participants were asked the smaller question, response times were shorter when none of the options were recognized. The opposite pattern was found when participants were asked the larger question: response time was shorter when all options were recognized. These task-stimuli congruity results in judgment under uncertainty are consistent with, and predicted by, theories of magnitude comparison, which make use of deductive inferences from declarative knowledge.

The long and the short of it: rule-based relative length discrimination in carrion crows, Corvus corone

Birds and other nonhuman animals can choose the larger of two discrete or continuous quantities. However, whether birds possess the conceptual grasp and cognitive control to flexibly switch between relative more-or-less-than judgments remains elusive. We therefore tested carrion crows in a rule-based line-length discrimination task to flexibly select lines presented on a touchscreen according to their relative length. In the first experiment, the crows needed to discriminate a shorter from a longer line, and vice versa. In the second experiment, the crows were required to choose a medium long line among three lines of different length (intermediate-size task). The crows switched effortlessly between "longer than/shorter than" rules, showing no signs of trial history affecting switching performance. They reliably chose the relatively longer and shorter line length, thus demonstrating a concept of greater than/less than with a continuous magnitude. However, both crows failed to discriminate a line of 'medium' length embedded in longer and shorter lines. These results indicate that relational discrimination exhibits different cognitive demands. While a greater than/less than concept requires only one relational comparison (with the respectively greater or smaller magnitude), the discrimination of a 'medium' magnitude demands to relate two or more comparisons, which might overburden crows and maybe animals in general.

Quantifier processing can be dissociated from numerical processing: evidence from semantic dementia patients

Quantifiers such as frequency adverbs (e.g., "always", "never") and quantity pronouns (e.g., "many", "none") convey quantity information. Whether quantifiers are processed as numbers or as general semantics has been a matter of much debate. Some neuropsychological and fMRI studies have found that the processing of quantifiers depends on the numerical magnitude comprehension system, but others have found that quantifier processing is associated with semantic representation. The selective impairment of language in semantic dementia patients provides a way to examine the above controversy. We administered a series of neuropsychological tests (i.e., language processing, numerical processing and semantic distance judgment) to two patients with different levels of severity in semantic dementia (mild vs. severe). The results showed that the two patients had intact numerical knowledge, but impairments in semantic processing. Moreover, the patient with severe/late semantic dementia showed more impairment in quantifier and semantic processing than the patient with mild/early semantic dementia. We concluded that quantifier processing is associated with general semantic processing, not with numerical processing.

Representation of abstract quantitative rules applied to spatial and numerical magnitudes in primate prefrontal cortex

Processing quantity information based on abstract principles is central to intelligent behavior. Neural correlates of quantitative rule selectivity have been identified previously in the prefrontal cortex (PFC). However, whether individual neurons represent rules applied to multiple magnitude types is unknown. We recorded from PFC neurons while monkeys switched between "greater than/less than" rules applied to spatial and numerical magnitudes. A majority of rule-selective neurons responded only to the quantitative rules applied to one specific magnitude type. However, another population of neurons generalized the magnitude principle and represented the quantitative rules related to both magnitudes. This indicates that the primate brain uses rule-selective neurons specialized in guiding decisions related to a specific magnitude type only, as well as generalizing neurons that respond abstractly to the overarching concept "magnitude rules."

Goal-oriented, flexible use of numerical operations by monkeys

Previous studies have shown that elementary aspects of numerical abilities have developed in non-human primates. In the present study, we explored the potential for the development of a novel ability in the use of numerical operations by macaque monkeys (Macaca fuscata): adequate selection of a series of numerical actions toward achieving a behavioral goal. We trained monkeys to use a pair of devices to selectively add or subtract items to/from a digital array in order to match a previously viewed sample array. The monkeys determined whether to add or subtract on the basis of the feedback about numerosity given to the monkeys, which was displayed as an outcome of each step of the numerical operation. We also found that monkeys adapted flexibly to changes in the numerical rule that determined the relationship between device use and numerical operation. Our model analysis found that the numerosity-based model was a better fit for the monkeys' performance than was the reward-expectation-based model. Such a capacity for goal-oriented selection of numerical operations suggests a mechanism by which monkeys use numerical representations for purposeful behaviors.

Nothing to it: precursors to a zero concept in preschoolers

Do young children understand the numerical value of empty sets prior to developing a concept of symbolic zero? Are empty sets represented as mental magnitudes? In order to investigate these questions, we tested 4-year old children and adults with a numerical ordering task in which the goal was to select two stimuli in ascending numerical order with occasional empty set stimuli. Both children and adults showed distance effects for empty sets. Children who were unable to order the symbol zero (e.g., 0<1), but who successfully ordered countable integers (e.g., 2<4) nevertheless showed distance effects with empty sets. These results suggest that empty sets are represented on the same numerical continuum as non-empty sets and that children represent empty sets numerically prior to understanding symbolic zero.

Coding of abstract quantity by 'number neurons' of the primate brain

Humans share with nonhuman animals a quantification system for representing the number of items as nonverbal mental magnitudes. Over the past decade, the anatomical substrates and neuronal mechanisms of this quantification system have been unraveled down to the level of single neurons. Work with behaviorally trained nonhuman primates identified a parieto-frontal cortical network with individual neurons selectively tuned to the number of items. Such 'number neurons' can track items across space, time, and modality to encode numerosity in a most abstract, supramodal way. The physiological properties of these neurons can explain fundamental psychophysical phenomena during numerosity judgments. Functionally overlapping groups of parietal neurons represent not only numerable-discrete quantity (numerosity), but also innumerable-continuous quantity (extent) and relations between quantities (proportions), supporting the idea of a generalized magnitude system in the brain. These studies establish putative homologies between the monkey and human brain and demonstrate the suitability of nonhuman primates as model system to explore the neurobiological roots of the brain's nonverbal quantification system, which may constitute the evolutionary foundation of all further, more elaborate numerical skills in humans.

Bears "Count" Too: Quantity Estimation and Comparison in Black Bears (Ursus Americanus)

Studies of bear cognition are notably missing from the comparative record despite bears' large relative brain size and interesting status as generalist carnivores facing complex foraging challenges, but lacking complex social structures. We investigated the numerical abilities of three American black bears (Ursus Americanus) by presenting discrimination tasks on a touch-screen computer. One bear chose the larger of two arrays of dot stimuli, while two bears chose the smaller array of dots. On some trials the relative number of dots was congruent with the relative total area of the two arrays. On other trials number of dots was incongruent with area. All of the bears were above chance on trials of both types with static dots. Despite encountering greater difficulty with dots that moved within the arrays, one bear was able to discriminate numerically larger arrays of moving dots, and a subset of moving dots from within the larger array, even when area and number were incongruent. Thus, although the bears used area as a cue to guide responding, they were also able to use number as a cue. The pattern of performance was similar to that found previously with monkeys, and suggests that bears may also show other forms of sophisticated quantitative abilities.

Math, monkeys, and the developing brain

Thirty thousand years ago, humans kept track of numerical quantities by carving slashes on fragments of bone. It took approximately 25,000 y for the first iconic written numerals to emerge among human cultures (e.g., Sumerian cuneiform). Now, children acquire the meanings of verbal counting words, Arabic numerals, written number words, and the procedures of basic arithmetic operations, such as addition and subtraction, in just 6 y (between ages 2 and 8). What cognitive abilities enabled our ancestors to record tallies in the first place? Additionally, what cognitive abilities allow children to rapidly acquire the formal mathematics knowledge that took our ancestors many millennia to invent? Current research aims to discover the origins and organization of numerical information in humans using clues from child development, the organization of the human brain, and animal cognition.

Components representation of negative numbers: Evidence from auditory stimuli detection and number classification tasks

Past research suggested that negative numbers could be represented in terms of their components in the visual modality. The present study examined the processing of negative numbers in the auditory modality and whether it is affected by context. Experiment 1 employed a stimuli detection task where only negative numbers were presented binaurally. Experiment 2 employed the same task, but both positive and negative numbers were mixed as cues. A reverse attentional spatial–numerical association of response codes (SNARC) effect for negative numbers was obtained in these two experiments. Experiment 3 employed a number classification task where only negative numbers were presented binaurally. Experiment 4 employed the same task, but both positive and negative numbers were mixed. A reverse SNARC effect for negative numbers was obtained in these two experiments. These findings suggest that negative numbers in the auditory modality are generated from the set of positive numbers, thus supporting a components representation.

Extending the Mental Number Line—How Do Negative Numbers Contribute?

Previous studies suggest that there is an association between positive numbers and space; however, there is less agreement for negative numbers. The main purpose of the present study was to investigate the nature of the processing and representation of negative numbers, and the association between negative numbers and space. Results of the two experiments show that low-level processing (perception) of negative numbers can induce spatial shifts of attention. Whether this is caused by their numerical value or absolute value depends on the numerical context and task requirements, indicating that there are both components and holistic processing, and representation for negative numbers. The representation is automatically associated with leftward space; the coding and representation of the mental number line is adaptable to the specific numerical context and task requirements. The mental number line, therefore, can extend to the left side of zero, thus supporting the context-dependent view.

Comparison of the choice effect and the distance effect in a number-comparison task by FMRI

Behavioral and neurophysiological studies of numerical comparisons have shown a "distance effect," whereby smaller numerical distances between two digits are associated with longer response times and higher activity in the parietal region. In this experiment, we introduced a two-choice condition (between either the smaller/lower or the larger/higher of two digits) and examined its effect on brain activity by fMRI. We observed longer response times and greater activity with the choice of smaller numbers ("choice effect") in several brain regions including the right temporo-parietal region, (pre)cuneus, superior temporal sulcus, precentral gyrus, superior frontal gyrus, bilateral insula, and anterior cingulate cortex. These regions correspond to areas that have been suggested to play a role in attentional shift and response conflict. However, brain activity associated with the distance effect disappeared even though the behavioral distance effect remained. Despite the absence of the distance effect on brain activity, several areas changed activity in relation to response time, including regions that were reported to change activity in both a distance effect and a reaction-time-related manner. The result suggested that the level of task load may change the activity of regions that are responsible for magnitude detection.

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.

Can monkeys make investments based on maximized pay-off?

Animals can maximize benefits but it is not known if they adjust their investment according to expected pay-offs. We investigated whether monkeys can use different investment strategies in an exchange task. We tested eight capuchin monkeys (Cebus apella) and thirteen macaques (Macaca fascicularis, Macaca tonkeana) in an experiment where they could adapt their investment to the food amounts proposed by two different experimenters. One, the doubling partner, returned a reward that was twice the amount given by the subject, whereas the other, the fixed partner, always returned a constant amount regardless of the amount given. To maximize pay-offs, subjects should invest a maximal amount with the first partner and a minimal amount with the second. When tested with the fixed partner only, one third of monkeys learned to remove a maximal amount of food for immediate consumption before investing a minimal one. With both partners, most subjects failed to maximize pay-offs by using different decision rules with each partner' quality. A single Tonkean macaque succeeded in investing a maximal amount to one experimenter and a minimal amount to the other. The fact that only one of over 21 subjects learned to maximize benefits in adapting investment according to experimenters' quality indicates that such a task is difficult for monkeys, albeit not impossible.

Quantity discrimination in salamanders

We investigated discrimination of large quantities in salamanders of the genus Plethodon. Animals were challenged with two different quantities (8 vs 12 or 8 vs 16) in a two-alternative choice task. Stimuli were live crickets, videos of live crickets or images animated by a computer program. Salamanders reliably chose the larger of two quantities when the ratio between the sets was 1:2 and stimuli were live crickets or videos thereof. Magnitude discrimination was not successful when the ratio was 2:3, or when the ratio was 1:2 when stimuli were computer animated. Analysis of the salamanders' success and failure as well as analysis of stimulus features points towards movement as a dominant feature for quantity discrimination. The results are generally consistent with large quantity discrimination investigated in many other animals (e.g. primates, fish), current models of quantity representation (analogue magnitudes) and data on sensory aspects of amphibian prey-catching behaviour (neuronal motion processing).

Context affects the numerical semantic congruity effect in rhesus monkeys (Macaca mulatta)

Do monkeys anchor their numerical judgments based on the context in which their choices are presented? We addressed this question by varying the numerical range across sessions while macaque monkeys made ordinal judgments. Monkeys were trained to make a conditional discrimination whereby they were reinforced for ordering arrays of dots in ascending or descending numerical order, dependent on a color cue. Monkeys were tested using two ranges of numerosities that converged on a single pair. Similar to the findings of Cantlon and Brannon (2005), we found a semantic congruity effect whereby decision time was systematically influenced by the congruity between the cue (ascending or descending) and the relative Numerical Magnitude of the stimuli within each range. Furthermore, monkeys showed a context effect, such that decision time for a given pair was dependent on whether it was a relatively small or large set of values compared to the other values presented in that session. This finding suggests that, similar to humans, the semantic congruity effect observed in monkeys is anchored by the context. Thus our data provide further evidence for the existence of a shared numerical comparison process in monkeys and humans.

Empty sets as part of the numerical continuum: conceptual precursors to the zero concept in rhesus monkeys

The goal of the current research was to explore whether monkeys possess conceptual precursors necessary for understanding zero. We trained rhesus monkeys on a nonsymbolic numerical matching-to-sample task, and on a numerical ordering task. We then introduced nondifferentially reinforced trials that contained empty sets to determine whether monkeys would treat empty sets as numerical values. All monkeys successfully matched and ordered the empty sets without any training. Accuracy showed distance effects, indicating that they treated empty sets as values on a numerical continuum.

The Representation of Negative Numbers: Exploring the Effects of Mode of Processing and Notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.

Beyond the number domain

In a world without numbers, we would be unable to build a skyscraper, hold a national election, plan a wedding or pay for a chicken at the market. The numerical symbols used in all these behaviors build on the approximate number system (ANS) which represents the number of discrete objects or events as a continuous mental magnitude. Here, we first discuss evidence that the ANS bears a set of behavioral and brain signatures that are universally displayed across animal species, human cultures and development. We then turn to the question of whether the ANS constitutes a specialized cognitive and neural domain - a question central to understanding how this system works, the nature of its evolutionary and developmental trajectory and its physical instantiation in the brain.

Effects of development and enculturation on number representation in the brain

A striking way in which humans differ from non-human primates is in their ability to represent numerical quantity using abstract symbols and to use these 'mental tools' to perform skills such as exact calculations. How do functional brain circuits for the symbolic representation of numerical magnitude emerge? Do neural representations of numerical magnitude change as a function of development and the learning of mental arithmetic? Current theories suggest that cultural number symbols acquire their meaning by being mapped onto non-symbolic representations of numerical magnitude. This Review provides an evaluation of this contention and proposes hypotheses to guide investigations into the neural mechanisms that constrain the acquisition of cultural representations of numerical magnitude.

Behavioral and near-infrared spectroscopy study of the effects of distance and choice in a number comparison task

Extensive behavioral and neurophysiological numerical comparison studies have shown that response times are longer and parietal activities are stronger when the numerical distance between two digits is smaller (the distance effect). However, only a few behavioral studies have considered the effect of the choice of larger or smaller numerals in numerical comparisons. Using near-infrared spectroscopy (NIRS), we investigated the neural basis of choosing larger/smaller numerals in number comparison tasks in which subjects were required to choose a larger or smaller digit. Our results showed that choosing a smaller digit induced significantly longer response times (the choice effect) and stronger parietal activities. We also obtained significantly longer response times as the distance effect in accordance with previous works. However, NIRS data did not show any significant difference corresponding to distance effect. Our results and previous studies suggest that the parietal cortex is involved not only in measuring numerical quantities, but also in choosing a numerically larger/smaller quantity among the categories of choice. Potentials and limitations of NIRS were discussed.

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.