Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

Infants' auditory enumeration: evidence for analog magnitudes in the small number range

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

From numerical concepts to concepts of number

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas.

Beyond perceptual symbols: a call for representational pluralism

Recent evidence from cognitive neuroscience suggests that certain cognitive processes employ perceptual representations. Inspired by this evidence, a few researchers have proposed that cognition is inherently perceptual. They have developed an innovative theoretical approach that rests on the notion of perceptual simulation and marshaled several general arguments supporting the centrality of perceptual representations to concepts. In this article, I identify a number of weaknesses in these arguments and defend a multiple semantic code approach that posits both perceptual and non-perceptual representations.

Comment on "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures"

Dehaene et al. (Reports, 30 May 2008, p. 1217) argued that native speakers of Mundurucu, a language without a linguistic numerical system, inherently represent numerical values as a logarithmically spaced spatial continuum. However, their data do not rule out the alternative conclusion that Mundurucu speakers encode numbers linearly with scalar variability and psychologically construct space-number mappings by analogy.

Judgments of discrete and continuous quantity: an illusory Stroop effect

Evidence from human cognitive neuroscience, animal neurophysiology, and behavioral research demonstrates that human adults, infants, and children share a common nonverbal quantity processing system with nonhuman animals. This system appears to represent both discrete and continuous quantity, but the proper characterization of the relationship between judgments of discrete and continuous quantity remains controversial. Some researchers have suggested that both continuous and discrete quantity may be automatically extracted from a scene and represented internally, and that competition between these representations leads to Stroop interference. Here, four experiments provide evidence for a different explanation of adults' performance on the types of tasks that have been said to demonstrate Stroop interference between representations of discrete and continuous quantity. Our well-established tendency to underestimate individual two-dimensional areas can provide an alternative explanation (introduced here as the "illusory-Stroop" hypothesis). Though these experiments were constructed like Stroop tasks, and they produce patterns of performance that initially appear consistent with Stroop interference, Stroop interference effects are not involved. Implications for models of the construction of cumulative area representations and for theories of discrete and continuous quantity processing in large sets are discussed.

Automaticity for numerical magnitude of two-digit Arabic numbers in children

This paper examines the automatic processing of the numerical magnitude of two-digit Arabic numbers using a Stroop-like task in school-aged children. Second, third, and fourth graders performed physical size judgments on pairs of two-digit numbers varying on both physical and numerical dimensions. To investigate the importance of synchrony between the speed of processing of the numerical magnitude and the physical dimensions on the size congruity effect (SCE), we used masked priming: numerical magnitude was subliminally primed in half of the trials, while neutral priming was used in the other half. The results indicate a SCE in physical judgments, providing the evidence of automatic access to the magnitude of two-digit numbers in children. This effect was modulated by the priming type, as a SCE only appeared when the numerical magnitude was primed. This suggests that young children needed a relative synchronization of numerical and physical dimensions to access the magnitude of two-digit numbers automatically.

Delays without mistakes: response time and error distributions in dual-task

Background

When two tasks are presented within a short interval, a delay in the execution of the second task has been systematically observed. Psychological theorizing has argued that while sensory and motor operations can proceed in parallel, the coordination between these modules establishes a processing bottleneck. This model predicts that the timing but not the characteristics (duration, precision, variability…) of each processing stage are affected by interference. Thus, a critical test to this hypothesis is to explore whether the qualitiy of the decision is unaffected by a concurrent task.

Methodology/Principal Findings

In number comparison–as in most decision comparison tasks with a scalar measure of the evidence–the extent to which two stimuli can be discriminated is determined by their ratio, referred as the Weber fraction. We investigated performance in a rapid succession of two non-symbolic comparison tasks (number comparison and tone discrimination) in which error rates in both tasks could be manipulated parametrically from chance to almost perfect. We observed that dual-task interference has a massive effect on RT but does not affect the error rates, or the distribution of errors as a function of the evidence.

Conclusions/Significance

Our results imply that while the decision process itself is delayed during multiple task execution, its workings are unaffected by task interference, providing strong evidence in favor of a sequential model of task execution.

Monkeys match and tally quantities across senses

We report here that monkeys can actively match the number of sounds they hear to the number of shapes they see and present the first evidence that monkeys sum over sounds and sights. In Experiment 1, two monkeys were trained to choose a simultaneous array of 1-9 squares that numerically matched a sample sequence of shapes or sounds. Monkeys numerically matched across (audio-visual) and within (visual-visual) modalities with equal accuracy and transferred to novel numerical values. In Experiment 2, monkeys presented with sample sequences of randomly ordered shapes or tones were able to choose an array of 2-9 squares that was the numerical sum of the shapes and sounds in the sample sequence. In both experiments, accuracy and reaction time depended on the ratio between the correct numerical match and incorrect choice. These findings suggest monkeys and humans share an abstract numerical code that can be divorced from the modality in which stimuli are first experienced.

Numerical thought with and without words: Evidence from indigenous Australian children

Are thoughts impossible without the words to express them? It has been claimed that this is the case for thoughts about numbers: Children cannot have the concept of exact numbers until they know the words for them, and adults in cultures whose languages lack a counting vocabulary similarly cannot possess these concepts. Here, using classical methods of developmental psychology, we show that children who are monolingual speakers of two Australian languages with very restricted number vocabularies possess the same numerical concepts as a comparable group of English-speaking indigenous Australian children.

Does subitizing reflect numerical estimation?

Subitizing is the rapid and accurate enumeration of small sets (up to 3-4 items). Although subitizing has been studied extensively since its first description about 100 years ago, its underlying mechanisms remain debated. One hypothesis proposes that subitizing results from numerical estimation mechanisms that, according to Weber's law, operate with high precision for small numbers. Alternatively, subitizing might rely on a distinct process dedicated to small numerosities. In this study, we tested the hypothesis that there is a shared estimation system for small and large quantities in human adults, using a masked forced-choice paradigm in which participants named the numerosity of displays taken from sets matched for discrimination difficulty; one set ranged from 1 through 8 items, and the other ranged from 10 through 80 items. Results showed a clear violation of Weber's law (much higher precision over numerosities 1-4 than over numerosities 10-40), thus refuting the single-estimation-system hypothesis and supporting the notion of a dedicated mechanism for apprehending small numerosities.

Distinct cerebral pathways for object identity and number in human infants

All humans, regardless of their culture and education, possess an intuitive understanding of number. Behavioural evidence suggests that numerical competence may be present early on in infancy. Here, we present brain-imaging evidence for distinct cerebral coding of number and object identity in 3-mo-old infants. We compared the visual event-related potentials evoked by unforeseen changes either in the identity of objects forming a set, or in the cardinal of this set. In adults and 4-y-old children, number sense relies on a dorsal system of bilateral intraparietal areas, different from the ventral occipitotemporal system sensitive to object identity. Scalp voltage topographies and cortical source modelling revealed a similar distinction in 3-mo-olds, with changes in object identity activating ventral temporal areas, whereas changes in number involved an additional right parietoprefrontal network. These results underscore the developmental continuity of number sense by pointing to early functional biases in brain organization that may channel subsequent learning to restricted brain areas.

Behavioural experiments indicate that infants aged 4½ months or older possess an early “number sense” that, for instance, enables them to detect changes in the approximate number of objects in a set. However, the neural bases of this competence are unknown. We recorded the electrical activity evoked by the brain on the surface of the scalp as 3-mo-old infants were watching images of sets of objects. Most images depicted the same objects and contained the same number of objects, but occasionally the number or the identity of the objects changed. As indicated by the voltage potential at the surface of the scalp, the infants' brains reacted when either object identity or number changes were introduced. Using a 3-D model of the infant head, we reconstructed the cortical sources of these responses. Brain areas responding to object or number changes are distinct, and reveal a basic ventral/dorsal organization already in place in the infant brain. As in adults and children, object identity in infants is encoded along a ventral pathway in the temporal lobes, although number activates an additional right parietoprefrontral network. These results underscore the developmental continuity of number sense by pointing to early functional biases in brain organization.

Cerebral imaging reveals that human infants are sensitive to numerical quantity at a very early age and that the basic dorsal/ventral functional organization is already in place in the infant brain.

The processing of non-symbolic numerical magnitudes as indexed by ERPs

Previous research has shown that, in the context of event-related potential (ERP) prime-target experiments, processing meaningful stimuli such as words, phonemes, numbers, pictures of objects, and faces elicit negativities around 400 ms. However, there is little information on whether non-symbolic numerical magnitudes elicit this negative component. The present experiments recorded ERPs while adults made same/different judgments to serially presented prime-target pairs of non-symbolic numerical stimuli containing the same, close, or distant quantities. In Experiment 1, a negativity between 350 and 450 ms was elicited for targets preceded by primes of unequal quantity, and this was greater for close than for distant quantities. Change direction (decreasing or increasing) also modulated a similar negativity: a greater negativity was elicited by targets preceded by larger than by smaller quantities. Experiment 2 replicated the numerical distance and change direction effects for numerical judgments, but found no negative distance effect in a color comparison task when the same stimuli were used. Additionally, ERP effects of numerical distance were found under implicit conditions, and task proficiency in the number condition modulated implicit and explicit numerical distance ERP effects. These results suggest that the neural systems involved with processing numerical magnitudes contribute to the construction of meaningful, contextual representations, are partly automatic, and display marked individual differences.

Children's understanding of the relationship between addition and subtraction

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12+9-9 yields 12. Here, we investigate whether preschool children's approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.

Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.

Aging and numerosity estimation

In two experiments, young and older participants were asked to find the approximate number of dots in collections including between 40 and 460 dots. Experiment 1 showed that both age groups had comparable performance and no age-related differences in the power-function exponents for numerosity. Experiment 2 found that these age-related similarities were not due to speed-accuracy trade-offs or to compensation by older adults for potential age-related decline in numerosity estimation processes. Furthermore, young and older participants' estimation performance was influenced by physical features of stimuli only for very large numerosities, presumably because these are poorly represented in long-term memory. Implications of these findings for the further understanding of how participants accomplish numerosity estimation tasks and effects of aging in this domain are 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.

The multiple dimensions of sustained attention

Sustained counting (or temporal numerosity judgements) has been one of the key means of investigating anterior attentional processes. Forty-three patients with localised lesions to the frontal lobes were assessed on two tests of the ability to count the number (8-22) of stimuli presented at either a slow (roughly one per 3 sec) or fast (roughly three per sec) rate. Patients with lesions to the Superior Medial (SM) region (particularly Brodmann areas 24, 32, and 9) were impaired both in the Slow condition and also in the Fast condition, where they underestimated the number of stimuli. Patients with Right Lateral (RL) lesions (8, 45, and 46) also had difficulties in the Fast condition, especially when the number of targets was greater than 15. The results are considered from the perspectives of alternative positions on anterior attentional processes developed by Posner and Petersen (1990) and by Stuss et al. (1995). The most plausible interpretation is in terms of energising processes which involve the SM frontal cortex and monitoring processes which involve the RL frontal cortex.

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

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

Extracting number-selective responses from coherent oscillations in a computer model

Cortical neurons selective for numerosity may underlie an innate number sense in both animals and humans. We hypothesize that the number- selective responses of cortical neurons may in part be extracted from coherent, object-specific oscillations . Here, indirect evidence for this hypothesis is obtained by analyzing the numerosity information encoded by coherent oscillations in artificially generated spikes trains. Several experiments report that gamma-band oscillations evoked by the same object remain coherent, whereas oscillations evoked by separate objects are uncorrelated. Because the oscillations arising from separate objects would add in random phase to the total power summed across all stimulated neurons, we postulated that the total gamma activity, normalized by the number of spikes, should fall roughly as the square root of the number of objects in the scene, thereby implicitly encoding numerosity. To test the hypothesis, we examined the normalized gamma activity in multiunit spike trains, 50 to 1000 msec in duration, produced by a model feedback circuit previously shown to generate realistic coherent oscillations. In response to images containing different numbers of objects, regardless of their shape, size, or shading, the normalized gamma activity followed a square-root-of-n rule as long as the separation between objects was sufficiently large and their relative size and contrast differences were not too great. Arrays of winner-take-all numerosity detectors, each responding to normalized gamma activity within a particular band, exhibited tuning curves consistent with behavioral data. We conclude that coherent oscillations in principle could contribute to the number-selective responses of cortical neurons, although many critical issues await experimental resolution.

Relative numerosity discrimination by chimpanzees (Pan troglodytes): evidence for approximate numerical representations

Two adult chimpanzees were trained on a relative "numerosity" discrimination task. In each trial, two arrays containing different numbers of red dots were presented on a CRT monitor. The subjects were required to choose the array containing the larger number of dots. In Experiment 1, using numerosities between 1 and 8, 28 different pairs were presented repeatedly, and accuracy scores were analyzed to explore which cues the chimpanzee subjects utilized to perform the task. Multiple regression analyses revealed that the subjects' performance was (1) not simply controlled by the "numerical" difference between arrays, but that it was (2) best described by Fechner's Law-that is accuracy increased linearly with the logarithmic value of the numerical difference between arrays divided by the number in the larger of the two arrays. This relationship was maintained when using much larger numerosities (Experiment 3). In Experiment 2, the chimpanzees were tested on the effects of total area and density by manipulating dot size and presentation area. The results revealed that these factors clearly affected the subjects' performance but that they could not alone explain the results, suggesting that the chimpanzees did use relative numerosity difference as a discriminative cue.

Semantic numerical representation in blind subjects: the role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.

Nonverbal representation of time and number in adults

A wealth of human and animal research supports common neural processing of numerical and temporal information. Here we test whether adult humans spontaneously encode number and time in a paradigm similar to those previously used to test the mode-control model in animals. Subjects were trained to classify visual stimuli that varied in both number and duration as few/short or many/long. Subsequently subjects were tested with novel stimuli that varied time and held number constant (eight flashes in 0.8-3.2s) or varied number and held time constant (4-16 flashes in 1.6s). Adult humans classified novel stimuli as many/long as monotonic functions of both number and duration, consistent with simultaneous, nonverbal, analog encoding. Numerical sensitivity, however, was finer than temporal sensitivity, suggesting differential salience of time and number. These results support the notion that adults simultaneously represent the number and duration of stimuli but suggest a possible asymmetry in their representations.

Nonverbal estimation during numerosity judgements by adult humans

On an automated task, humans selected the larger of two sets of items, each created through the one-by-one addition of items. Participants repeated the alphabet out loud during trials so that they could not count the items. This manipulation disrupted counting without producing major effects on other cognitive capacities such as memory or attention, and performance of this experimental group was poorer than that of participants who counted the items. In Experiment 2, the size of individual items was varied, and performance remained stable when the larger numerical set contained a smaller total amount than the smaller numerical set (i.e., participants used numerical rather than nonnumerical quantity cues in making judgements). In Experiment 3, reports of the number of items in a single set showed scalar variability as accuracy decreased, and variability in responses increased with increases in true set size. These data indicate a mechanism for the approximate representation of numerosity in adult humans that might be shared with nonhuman animals.

Cross-domain transfer of quantitative discriminations: is it all a matter of proportion?

Meck and Church (1983) estimated a 5:1 scale factor relating the mental magnitudes representing number to the mental magnitudes representing duration. We repeated their experiment with human subjects. We obtained transfer regardless of the objective scaling between the ranges; a 5:1 scaling for number versus duration (measured in seconds) was not necessary. We obtained transfer even when the proportions between the endpoints of the number range were different. We conclude that, at least in human subjects, transfer from a discrimination based on continuous quantity (duration) to a discrimination based on discrete quantity (number) is mediated by the cross-domain comparability of within-domain proportions. The results of our second and third experiments also suggest that the subjects compare a probe with a criterion determined by the range of stimuli tested rather than by trial-specific referents, in accordance with the pseudologistic model of Killeen, Fetterman, and Bizo (1997).

Time left in the mouse

Evidence suggests that the online combination of non-verbal magnitudes (durations, numerosities) is central to learning in both human and non-human animals [Gallistel, C.R., 1990. The Organization of Learning. MIT Press, Cambridge, MA]. The molecular basis of these computations, however, is an open question at this point. The current study provides the first direct test of temporal subtraction in a species in which the genetic code is available. In two experiments, mice were run in an adaptation of Gibbon and Church's [Gibbon, J., Church, R.M., 1981. Time left: linear versus logarithmic subjective time. J. Exp. Anal. Behav. 7, 87-107] time left paradigm in order to characterize typical responding in this task. Both experiments suggest that mice engaged in online subtraction of temporal values, although the generalization of a learned response rule to novel stimulus values resulted in slightly less systematic responding. Potential explanations for this pattern of results are discussed.

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.

Sometimes area counts more than number

Using an interference paradigm, we show across three experiments that adults' order judgments of numbers, sizes, or combined area of dots in pairs of arrays occur spontaneously and automatically, but at different speeds and levels of accuracy. Experiment 1 used circles whose sizes varied between but not within arrays. Variation in circle size interfered with judgments of which array had more circles. Experiment 2 used displays in which circle size varied within and between arrays. Between-array differences in the amount of "circle stuff" (area occupied by circles) interfered with judgments of number. Experiment 3 examined whether variation in number also interferes with judgments of area. Interference between discrete and continuous stimulus dimensions occurred in both directions, although it was stronger from the continuous to the discrete than vice versa. These results bear on interpretations of studies with infants and preschoolers wherein subjects respond on the basis of continuous quantity rather than discrete quantity. In light of our results with adults, these findings do not license the conclusion that young children cannot represent discrete quantity. Absent data on attentional hierarchies and speed of processing, it is premature to conclude that infant and child quantity processes are fundamentally different from that of adults.

Temporal and Spatial Enumeration Processes in the Primate Parietal Cortex

Humans and animals can nonverbally enumerate visual items across time in a sequence or rapidly estimate the set size of spatial dot patterns at a single glance. We found that temporal and spatial enumeration processes engaged different populations of neurons in the intraparietal sulcus of behaving monkeys. Once the enumeration process was completed, however, another neuronal population represented the cardinality of a set irrespective of whether it had been cued in a spatial layout or across time. These data suggest distinct neural processing stages for different numerical formats, but also a final convergence of the segregated information to form most abstract quantity representations.

Quantity-based judgments in the domestic dog (Canis lupus familiaris)

We examined the ability of domestic dogs to choose the larger versus smaller quantity of food in two experiments. In experiment 1, we investigated the ability of 29 dogs (results from 18 dogs were used in the data analysis) to discriminate between two quantities of food presented in eight different combinations. Choices were simultaneously presented and visually available at the time of choice. Overall, subjects chose the larger quantity more often than the smaller quantity, but they found numerically close comparisons more difficult. In experiment 2, we tested two dogs from experiment 1 under three conditions. In condition 1, we used similar methods from experiment 1 and tested the dogs multiple times on the eight combinations from experiment 1 plus one additional combination. In conditions 2 and 3, the food was visually unavailable to the subjects at the time of choice, but in condition 2, food choices were viewed simultaneously before being made visually unavailable, and in condition 3, they were viewed successively. In these last two conditions, and especially in condition 3, the dogs had to keep track of quantities mentally in order to choose optimally. Subjects still chose the larger quantity more often than the smaller quantity when the food was not simultaneously visible at the time of choice. Olfactory cues and inadvertent cuing by the experimenter were excluded as mechanisms for choosing larger quantities. The results suggest that, like apes tested on similar tasks, some dogs can form internal representations and make mental comparisons of quantity.

A competitive neural model of small number detection

The ability to represent numbers is a key attribute for both humans and animals. Recent developments in the understanding of numerical processing has led to the proposal that humans utilise two independent representations of number, one for real numbers and another for integers. We describe a computational model of small number detection to explore the relationship between these core systems of number. We use a combination of unsupervised and supervised neural networks to simulate the interaction between the real and integer representations. For real values we use a self-organised spatial representation of number. For integer values we use a supervised network motivated by linguistic processing. During training and testing, the networks exhibit behavioural characteristics such as the number size and numerical distance effects. Each representation is combined using the mixture-of-experts architecture that allows us to model the subitization limit (the maximum number of visual stimuli that can be accurately quantified almost immediately) as the competitive allocation of representations for number detection, where the crossover point between deploying the real and integer representations of number is obtained through a process of learning. Our results suggest that the existence of two core systems of number is at least computationally plausible and further suggests that the subitization limit emerges through the interaction of spatial and linguistic numerical processing. This provides computational evidence for one way in which small and large numbers are related in humans.

Shared system for ordering small and large numbers in monkeys and humans

There is increasing evidence that animals share with adult humans and perhaps human infants a system for representing objective number as psychological magnitudes that are an analogue of the quantities they represent. Here we show that rhesus monkeys can extend a numerical rule learned with the values 1 through 9 to the values 10, 15, 20, and 30, which suggests that there is no upper limit on a monkey's numerical capacity. Instead, throughout the numerical range tested, both accuracy and latency in ordering two numerical values were systematically controlled by the ratio of the values compared. In a second experiment, we directly compared humans' and monkeys' performance in the same ordinal comparison task. The qualitative and quantitative similarity in their performance provides the strongest evidence to date of a single nonverbal, evolutionarily primitive mechanism for representing and comparing numerical values.

Scalar effects in the visual discrimination of numerosity by pigeons

Pigeons trained in a conditional discrimination procedure to respond to a visual array made a left or right choice, depending on which of two numbers of elements (i.e., anchor numerosities) the array contained. They were then tested with novel arrays at these anchor numerosities, as well as at interpolated and extrapolated numerosities. Various control conditions showed that the birds' discrimination performance was primarily based on stimulus numerosity, and not on other factors, such as brightness or area. Results from a series of tests, spanning a wide range of numerosities, conformed to scalar principles. Psychometric functions showed superposition, indicating that Weber's law applies to numerosity discrimination. The subjective midpoint between anchor values was at the geometric mean. Variability about this bisection point increased in proportion to the numerical value of the mean.

The failure of Weber's law in time perception and production

Weber's law--constancy of the coefficient of variation--is an apparently ubiquitous feature of time perception, and forms the foundation of several theories of timing. We sought evidence for Weber's law in temporal production and categorization experiments. The production task required pigeons to switch between keys within a specified temporal window. The categorization task required them to classify a stimulus duration as either short or long. Weber fractions did not descend to a horizontal asymptote, but were U-shaped: they decreased as a function of target duration, and increased again at intermediate and long durations. This pattern conforms neither to Weber's law, nor to its generalized form (Getty, D.J., 1975. Discrimination of short temporal intervals: a comparison of two models. Percept. Psychophys. 18, 1-8). A model of counter failure accommodated the U-shaped pattern.

Preschool children's mapping of number words to nonsymbolic numerosities

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20-120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably.

The role of working memory in spatial enumeration: Patterns of selective interference in subitizing and counting

Articulatory suppression (repeatedly pronouncing a syllable or word while carrying out another task) is thought to interfere selectively with the phonological store in working memory. Although suppression interferes with temporal enumeration (enumerating successive light flashes), to date there has been little evidence of such interference in spatial enumeration (enumerating units laid out in space at one time)--a finding with serious ramifications for theories of enumeration. Participants carried out a spatial enumeration task, enumerating 1-8 dots while listening to a metronome (baseline condition) or while carrying out a secondary task to the rhythm of the metronome (dual-task condition). There were four secondary tasks: simple articulation (saying a letter), complex articulation (alternating between two letters), simple tapping (tapping a finger), and complex tapping (alternating between two fingers). Interference varied with number of items, but the pattern differed from that observed with temporal enumeration.

Counting on neurons: the neurobiology of numerical competence

Numbers are an integral part of our everyday life - we use them to quantify, rank and identify objects. The verbal number concept allows humans to develop superior mathematical and logic skills that define technologically advanced cultures. However, basic numerical competence is rooted in biological primitives that can be explored in animals, infants and human adults alike. We are now beginning to unravel its anatomical basis and neuronal mechanisms on many levels, down to its single neuron correlate. Neural representations of numerical information can engage extensive cerebral networks, but the posterior parietal cortex and the prefrontal cortex are the key structures in primates.

Number sense in human infants

Four experiments used a preferential looking method to investigate 6-month-old infants' capacity to represent numerosity in visual-spatial displays. Building on previous findings that such infants discriminate between arrays of eight versus 16 discs, but not eight versus 12 discs (Xu & Spelke, 2000), Experiments 1 and 2 investigated whether infants' numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 versus 32 discs, but not 16 versus 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children and many non-human animals. Experiments 3 and 4 addressed a controversy concerning infants' ability to discriminate large numerosities (observed under conditions that control for total filled area, array size and density, item size and correlated properties such as brightness: Brannon, 2002; Xu, 2003b; Xu & Spelke, 2000) versus small numerosities (not observed under conditions that control for total contour length: Clearfield & Mix, 1999). To investigate the sources of these differing findings, Experiment 3 tested infants' large-number discrimination with controls for contour length, and Experiment 4 tested small-number discrimination with controls for total filled area. Infants successfully discriminated the large-number displays but showed no evidence of discriminating the small-number displays. These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large versus small numerosities.

Tuning Curves for Approximate Numerosity in the Human Intraparietal Sulcus

Number, like color or movement, is a basic property of the environment. Recently, single neurons tuned to number have been observed in animals. We used both psychophysics and neuroimaging to examine whether a similar neural coding scheme is present in humans. When participants viewed sets of items with a variable number, the bilateral intraparietal sulci responded selectively to number change. Functionally, the shape of this response indicates that humans, like other animal species, encode approximate number on a compressed internal scale. Anatomically, the intraparietal site coding for number in humans is compatible with that observed in macaque monkeys. Our results therefore suggest an evolutionary basis for human elementary arithmetic.

A neural model of quantity discrimination

A neural network model is proposed with the ability to extract abstract numerical representation from visual input. It simulates properties of a number detection system which is hypothesized to underlie simple language-independent numerical abilities. The network has three layers where the first layer computes the sum of the nearest neighbour inputs. The first layer is also augmented with multiplicative gating and gradient tonic activation which prevents interference. The second layer implements local lateral inhibition which enables a single node to represent a single object. The third layer exhibits number-tuning similar to recently described responses of neurons in the prefrontal cortex. Computer simulations showed that network response does not depend on visual attributes like the object's size, position or shape. The model is based on several biophysical mechanisms such as multiplicative interaction on dendrites, independent processing on different dendritic branches and disinhibition by glutamate spill-over on kainate receptors on inhibitory axons.

Numerical cognition without words: evidence from Amazonia

Members of the Pirahã tribe use a "one-two-many" system of counting. I ask whether speakers of this innumerate language can appreciate larger numerosities without the benefit of words to encode them. This addresses the classic Whorfian question about whether language can determine thought. Results of numerical tasks with varying cognitive demands show that numerical cognition is clearly affected by the lack of a counting system in the language. Performance with quantities greater than three was remarkably poor, but showed a constant coefficient of variation, which is suggestive of an analog estimation process.

Analog numerical representations in rhesus monkeys: evidence for parallel processing

Monkeys have been introduced as model organisms to study neural correlates of numerical competence, but many of the behavioral characteristics of numerical judgments remain speculative. Thus, we analyzed the behavioral performance of two rhesus monkeys judging the numerosities 1 to 7 during a delayed match-to-sample task. The monkeys showed similar discrimination performance irrespective of the exact physical appearance of the stimuli, confirming that performance was based on numerical information. Performance declined smoothly with larger numerosities, and reached discrimination threshold at numerosity "4." The nonverbal numerical representations in monkeys were based on analog magnitudes, object tracking process ("subitizing") could not account for the findings because the continuum of small and large numbers shows a clear Weber fraction signature. The lack of additional scanning eye movements with increasing set sizes, together with indistinguishable neuronal response latencies for neurons with different preferred numerosities, argues for parallel encoding of numerical information. The slight but significant increase in reaction time with increasing numerosities can be explained by task difficulty and consequently time-consuming decision processes. The behavioral results are compared to single-cell recordings from the prefrontal cortex in the same subjects. Models for numerosity discrimination that may account for these results are discussed.