Modularity, comparative cognition and human uniqueness

Darwin's claim 'that the difference in mind between man and the higher animals … is certainly one of degree and not of kind' is at the core of the comparative study of cognition. Recent research provides unprecedented support for Darwin's claim as well as new reasons to question it, stimulating new theories of human cognitive uniqueness. This article compares and evaluates approaches to such theories. Some prominent theories propose sweeping domain-general characterizations of the difference in cognitive capabilities and/or mechanisms between adult humans and other animals. Dual-process theories for some cognitive domains propose that adult human cognition shares simple basic processes with that of other animals while additionally including slower-developing and more explicit uniquely human processes. These theories are consistent with a modular account of cognition and the 'core knowledge' account of children's cognitive development. A complementary proposal is that human infants have unique social and/or cognitive adaptations for uniquely human learning. A view of human cognitive architecture as a mosaic of unique and species-general modular and domain-general processes together with a focus on uniquely human developmental mechanisms is consistent with modern evolutionary-developmental biology and suggests new questions for comparative research.

A number of options: rationalist, constructivist, and Bayesian insights into the development of exact-number concepts

The question of how human beings acquire exact-number concepts has interested cognitive developmentalists since the time of Piaget. The answer will owe something to both the rationalist and constructivist traditions. On the one hand, some aspects of numerical cognition (e.g. approximate number estimation and the ability to track small sets of one to four individuals) are innate or early-developing and are shared widely among species. On the other hand, only humans create representations of exact, large numbers such as 42, as distinct from both 41 and 43. These representations seem to be constructed slowly, over a period of months or years during early childhood. The task for researchers is to distinguish the innate representational resources from those that are constructed, and to characterize the construction process. Bayesian approaches can be useful to this project in at least three ways: (1) As a way to analyze data, which may have distinct advantages over more traditional methods (e.g. making it possible to find support for a nuli hypothesis); (2) as a way of modeling children's performance on specific tasks: Peculiarities of the task are captured as a prior; the child's knowledge is captured in the way the prior is updated; and behavior is captured as a posterior distribution; and (3) as a way of modeling learning itself, by providing a formal account of how learners might choose among alternative hypotheses.

The approximate number system and its relation to early math achievement: evidence from the preschool years

Humans rely on two main systems of quantification; one is nonsymbolic and involves approximate number representations (known as the approximate number system or ANS), and the other is symbolic and allows for exact calculations of number. Despite the pervasiveness of the ANS across development, recent studies with adolescents and school-aged children point to individual differences in the precision of these representations that, importantly, have been shown to relate to symbolic math competence even after controlling for general aspects of intelligence. Such findings suggest that the ANS, which humans share with nonhuman animals, interfaces specifically with a uniquely human system of formal mathematics. Other findings, however, point to a less straightforward picture, leaving open questions about the nature and ontogenetic origins of the relation between these two systems. Testing children across the preschool period, we found that ANS precision correlated with early math achievement but, critically, that this relation was nonlinear. More specifically, the correlation between ANS precision and math competence was stronger for children with lower math scores than for children with higher math scores. Taken together, our findings suggest that early-developing connections between the ANS and mathematics may be fundamentally discontinuous. Possible mechanisms underlying such nonlinearity are discussed.

A common rule for decision making in animal collectives across species

A diversity of decision-making systems has been observed in animal collectives. In some species, choices depend on the differences of the numbers of animals that have chosen each of the available options, whereas in other species on the relative differences (a behavior known as Weber's law), or follow more complex rules. We here show that this diversity of decision systems corresponds to a single rule of decision making in collectives. We first obtained a decision rule based on Bayesian estimation that uses the information provided by the behaviors of the other individuals to improve the estimation of the structure of the world. We then tested this rule in decision experiments using zebrafish (Danio rerio), and in existing rich datasets of argentine ants (Linepithema humile) and sticklebacks (Gasterosteus aculeatus), showing that a unified model across species can quantitatively explain the diversity of decision systems. Further, these results show that the different counting systems used by animals, including humans, can emerge from the common principle of using social information to make good decisions.

Symbolic and non-symbolic number magnitude processing in children with developmental dyscalculia

The aim of this study was to evaluate if children with Developmental Dyscalculia (DD) exhibit a general deficit in magnitude representations or a specific deficit in the connection of symbolic representations with the corresponding analogous magnitudes. DD was diagnosed using a timed arithmetic task. The experimental magnitude comparison tasks were presented in non-symbolic and symbolic formats. DD and typically developing (TD) children showed similar numerical distance and size congruity effects. However, DD children performed significantly slower in the symbolic task. These results are consistent with the access deficit hypothesis, according to which DD children's deficits are caused by difficulties accessing magnitude information from numerical symbols rather than in processing numerosities per se.

Your space or mine? Mapping self in time

While humans are capable of mentally transcending the here and now, this faculty for mental time travel (MTT) is dependent upon an underlying cognitive representation of time. To this end, linguistic, cognitive and behavioral evidence has revealed that people understand abstract temporal constructs by mapping them to concrete spatial domains (e.g. past = backward, future = forward). However, very little research has investigated factors that may determine the topographical characteristics of these spatiotemporal maps. Guided by the imperative role of episodic content for retrospective and prospective thought (i.e., MTT), here we explored the possibility that the spatialization of time is influenced by the amount of episodic detail a temporal unit contains. In two experiments, participants mapped temporal events along mediolateral (Experiment 1) and anterioposterior (Experiment 2) spatial planes. Importantly, the temporal units varied in self-relevance as they pertained to temporally proximal or distal events in the participant’s own life, the life of a best friend or the life of an unfamiliar other. Converging evidence from both experiments revealed that the amount of space used to represent time varied as a function of target (self, best friend or unfamiliar other) and temporal distance. Specifically, self-time was represented as occupying more space than time pertaining to other targets, but only for temporally proximal events. These results demonstrate the malleability of space-time mapping and suggest that there is a self-specific conceptualization of time that may influence MTT as well as other temporally relevant cognitive phenomena.

Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence

Humans and nonhuman animals share the capacity to estimate, without counting, the number of objects in a set by relying on an approximate number system (ANS). Only humans, however, learn the concepts and operations of symbolic mathematics. Despite vast differences between these two systems of quantification, neural and behavioral findings suggest functional connections. Another line of research suggests that the ANS is part of a larger, more general system of magnitude representation. Reports of cognitive interactions and common neural coding for number and other magnitudes such as spatial extent led us to ask whether, and how, nonnumerical magnitude interfaces with mathematical competence. On two magnitude comparison tasks, college students estimated (without counting or explicit calculation) which of two arrays was greater in number or cumulative area. They also completed a battery of standardized math tests. Individual differences in both number and cumulative area precision (measured by accuracy on the magnitude comparison tasks) correlated with interindividual variability in math competence, particularly advanced arithmetic and geometry, even after accounting for general aspects of intelligence. Moreover, analyses revealed that whereas number precision contributed unique variance to advanced arithmetic, cumulative area precision contributed unique variance to geometry. Taken together, these results provide evidence for shared and unique contributions of nonsymbolic number and cumulative area representations to formally taught mathematics. More broadly, they suggest that uniquely human branches of mathematics interface with an evolutionarily primitive general magnitude system, which includes partially overlapping representations of numerical and nonnumerical magnitude.

Intuitive sense of number correlates with math scores on college-entrance examination

Many educated adults possess exact mathematical abilities in addition to an approximate, intuitive sense of number, often referred to as the Approximate Number System (ANS). Here we investigate the link between ANS precision and mathematics performance in adults by testing participants on an ANS-precision test and collecting their scores on the Scholastic Aptitude Test (SAT), a standardized college-entrance exam in the USA. In two correlational studies, we found that ANS precision correlated with SAT-Quantitative (i.e., mathematics) scores. This relationship remained robust even when controlling for SAT-Verbal scores, suggesting a small but specific relationship between our primitive sense for number and formal mathematical abilities.

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.

Bootstrapping the mind: analogical processes and symbol systems

Human cognition is striking in its brilliance and its adaptability. How do we get that way? How do we move from the nearly helpless state of infants to the cognitive proficiency that characterizes adults? In this paper I argue, first, that analogical ability is the key factor in our prodigious capacity, and, second, that possession of a symbol system is crucial to the full expression of analogical ability.

Dissociated neural correlates of quantity processing of quantifiers, numbers, and numerosities

Quantities can be represented using either mathematical language (i.e., numbers) or natural language (i.e., quantifiers). Previous studies have shown that numerical processing elicits greater activation in the brain regions around the intraparietal sulcus (IPS) relative to other semantic processes. However, little research has been conducted to investigate whether the IPS is also critical for the semantic processing of quantifiers in natural language. In this study, 20 adults were scanned with functional magnetic resonance imaging while they performed semantic distance judgment involving six types of materials (i.e., frequency adverbs, quantity pronouns and nouns, animal names, Arabic digits, number words, and dot arrays). Conjunction analyses of brain activation showed that numbers and dot arrays elicited greater activation in the right IPS than did words (i.e., animal names) or quantifiers (i.e., frequency adverbs and quantity pronouns and nouns). Quantifiers elicited more activation in left middle temporal gyrus and inferior frontal gyrus than did numbers and dot arrays. No differences were found between quantifiers and animal names. These findings suggest that, although quantity processing for numbers and dot arrays typically relies on the right IPS region, quantity processing for quantifiers typically relies on brain regions for general semantic processing. Thus, the IPS does not appear to be the only brain region for quantity processing.

The origins and structure of quantitative concepts

"Number" is the single most influential quantitative dimension in modern human society. It is our preferred dimension for keeping track of almost everything, including distance, weight, time, temperature, and value. How did "number" become psychologically affiliated with all of these different quantitative dimensions? Humans and other animals process a broad range of quantitative information across many psychophysical dimensions and sensory modalities. The fact that adults can rapidly translate one dimension (e.g., loudness) into any other (e.g., handgrip pressure) has been long established by psychophysics research (Stevens, 1975 ). Recent literature has attempted to account for the development of the computational and neural mechanisms that underlie interactions between quantitative dimensions. We review evidence that there are fundamental cognitive and neural relations among different quantitative dimensions (number, size, time, pitch, loudness, and brightness). Then, drawing on theoretical frameworks that explain phenomena from cross-modal perception, we outline some possible conceptualizations for how different quantitative dimensions could come to be related over both ontogenetic and phylogenetic time scales.

Vision and Sketching

This essay reflects on the revolution David Marr brought about in vision research, and in cognitive science more broadly. I start with an insider's view, then examine the methodological impact of his framework in cognitive science in general. My group's work on sketch understanding descends from Marr's approach to vision, a connection which I make to provide a concrete illustration. I close with a few thoughts about how research in vision and other areas of cognitive science might come together in the future.

The relation between spatial skill and early number knowledge: the role of the linear number line

Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers.

Developmental change in the acuity of approximate number and area representations

From very early in life, humans can approximate the number and surface area of objects in a scene. The ability to discriminate between 2 approximate quantities, whether number or area, critically depends on the ratio between the quantities, with the most difficult ratio that a participant can reliably discriminate known as the Weber fraction. While developmental improvements in the Weber fraction have been demonstrated for number, the developmental trajectory of improvement in area discrimination remains unknown. Here we investigated whether the development of area discrimination parallels that of number discrimination. We tested forty 3- to 6-year-old children and adults in both a number and an area discrimination task in which participants selected the greater of 2 quantities across a range of ratios. We used formal psychophysical models to derive, for each participant and each age group, the Weber fraction for both number and area discrimination. We found that, like number acuity, area acuity steadily improves during childhood. However, we also found area acuity to be consistently higher than number acuity, suggesting a potential difference in the underlying mechanisms that encode and/or represent approximate area and approximate number. We discuss these findings in the context of quantity processing and its development.

Grey parrot number acquisition: the inference of cardinal value from ordinal position on the numeral list

A Grey parrot (Psittacus erithacus) had previously been taught to use English count words ("one" through "sih" [six]) to label sets of one to six individual items (Pepperberg, 1994). He had also been taught to use the same count words to label the Arabic numerals 1 through 6. Without training, he inferred the relationship between the Arabic numerals and the sets of objects (Pepperberg, 2006b). In the present study, he was then trained to label vocally the Arabic numerals 7 and 8 ("sih-none", "eight", respectively) and to order these Arabic numerals with respect to the numeral 6. He subsequently inferred the ordinality of 7 and 8 with respect to the smaller numerals and he inferred use of the appropriate label for the cardinal values of seven and eight items. These data suggest that he constructed the cardinal meanings of "seven" ("sih-none") and "eight" from his knowledge of the cardinal meanings of one through six, together with the place of "seven" ("sih-none") and "eight" in the ordered count list.

How do spatial representations enhance cognitive numerical processing?

Several philosophical theories attempt to explain how actions performed in the world enhance cognitive processing: internalism, active externalism, and cognitive integration. The aim of this paper is to examine whether the use of spatial representations in arithmetic can shed light on this debate. Relying on philosophical analysis, on a discussion of empirical work in the cognitive neuroscience of number, and on a historical case study, I will show that spatial representations of number indicate an integration between internal and external cognitive processes.

Large quantity discrimination by North Island robins (Petroica longipes)

While numerosity-representation and enumeration of different numbers of objects-and quantity discrimination in particular have been studied in a wide range of species, very little is known about the numerical abilities of animals in the wild. This study examined spontaneous relative quantity judgments (RQJs) by wild North Island robins (Petroica longipes) of New Zealand. In Experiment 1, robins were tested on a range of numerical values of up to 14 versus 16 items, which were sequentially presented and hidden. In Experiment 2, the same numerical contrasts were tested on a different group of subjects but quantities were presented as whole visible sets. Experiment 3 involved whole visible sets that comprised of exceedingly large quantities of up to 56 versus 64 items. While robins shared with other species a ratio-based representation system for representing very large values, they also appeared to have developed an object indexing system with an extended upper limit (well beyond 4) that may be an evolutionary response to ecological challenges faced by scatter-hoarding birds. These results suggest that cognitive mechanism influencing an understanding of physical quantity may be deployed more flexibly in some contexts than previously thought, and are discussed in light of findings across other mammalian and avian species.

Dyscalculia and vestibular function

BACKGROUND

A few studies in humans suggest that changes in stimulation of the balance organs of the inner ear (the 'vestibular system') can disrupt numerical cognition, resulting in 'dyscalculia', the inability to manipulate numbers. Many studies have also demonstrated that patients with vestibular dysfunction exhibit deficits in spatial memory.

OBJECTIVES

It is suggested that there may be a connection between spatial memory deficits resulting from vestibular dysfunction and the occurrence of dyscalculia, given the evidence that numerosity is coupled to the processing of spatial information (e.g., the 'spatial numerical association of response codes ('SNARC') effect').

RESULTS AND CONCLUSION

The evidence supporting this hypothesis is summarised and potential experiments to test it are proposed.

Over-estimation in numerosity estimation tasks: more than an attentional bias?

Over- and under-estimation have been observed in numerosity estimation and approximate arithmetic tasks. Two different models have been proposed to account for these reverse patterns of performance: 1) the bi-directional mapping account (Crollen, Castronovo, & Seron, 2011); 2) the operational momentum hypothesis (McCrink, Dehaene, & Dehaene-Lambertz, 2007). Our study was designed to examine whether the operational momentum could account for the over-estimation found in numerosity estimation tasks. To this aim, a series of 3 experiments involving a symbolic to non-symbolic numerical mapping and a rightward or leftward displacement along the mental number line were designed. Over-estimation was observed in these three tasks irrespective of the direction and size of the displacement to be done on the mental number line. These results thus clearly demonstrated that overestimation was not merely due to an attentional bias, but rather relied on the cognitive operation of mapping two differently scaled numerical representations.

Number sense across the lifespan as revealed by a massive Internet-based sample

It has been difficult to determine how cognitive systems change over the grand time scale of an entire life, as few cognitive systems are well enough understood; observable in infants, adolescents, and adults; and simple enough to measure to empower comparisons across vastly different ages. Here we address this challenge with data from more than 10,000 participants ranging from 11 to 85 years of age and investigate the precision of basic numerical intuitions and their relation to students' performance in school mathematics across the lifespan. We all share a foundational number sense that has been observed in adults, infants, and nonhuman animals, and that, in humans, is generated by neurons in the intraparietal sulcus. Individual differences in the precision of this evolutionarily ancient number sense may impact school mathematics performance in children; however, we know little of its role beyond childhood. Here we find that population trends suggest that the precision of one's number sense improves throughout the school-age years, peaking quite late at ∼30 y. Despite this gradual developmental improvement, we find very large individual differences in number sense precision among people of the same age, and these differences relate to school mathematical performance throughout adolescence and the adult years. The large individual differences and prolonged development of number sense, paired with its consistent and specific link to mathematics ability across the age span, hold promise for the impact of educational interventions that target the number sense.

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.

Young children's understanding of "more" and discrimination of number and surface area

The psychology supporting the use of quantifier words (e.g., "some," "most," "more") is of interest to both scientists studying quantity representation (e.g., number, area) and to scientists and linguists studying the syntax and semantics of these terms. Understanding quantifiers requires both a mastery of the linguistic representations and a connection with cognitive representations of quantity. Some words (e.g., "many") refer to only a single dimension, whereas others, like the comparative "more," refer to comparison by numeric ("more dots") or nonnumeric dimensions ("more goo"). In the present work, we ask 2 questions. First, when do children begin to understand the word "more" as used to compare nonnumeric substances and collections of discrete objects? Second, what is the underlying psychophysical character of the cognitive representations children utilize to verify such sentences? We find that children can understand and verify sentences including "more goo" and "more dots" at around 3.3 years-younger than some previous studies have suggested-and that children employ the Approximate Number System and an Approximate Area System in verification. These systems share a common underlying format (i.e., Gaussian representations with scalar variability). The similarity in the age of onset we find for understanding "more" in number and area contexts, along with the similar psychophysical character we demonstrate for these underlying cognitive representations, suggests that children may learn "more" as a domain-neutral comparative term.

Evolution of optimal Lévy-flight strategies in human mental searches

Recent analysis of empirical data [Radicchi, Baronchelli, and Amaral, PloS ONE 7, e029910 (2012)] showed that humans adopt Lévy-flight strategies when exploring the bid space in online auctions. A game theoretical model proved that the observed Lévy exponents are nearly optimal, being close to the exponent value that guarantees the maximal economical return to players. Here, we rationalize these findings by adopting an evolutionary perspective. We show that a simple evolutionary process is able to account for the empirical measurements with the only assumption that the reproductive fitness of the players is proportional to their search ability. Contrary to previous modeling, our approach describes the emergence of the observed exponent without resorting to any strong assumptions on the initial searching strategies. Our results generalize earlier research, and open novel questions in cognitive, behavioral, and evolutionary sciences.

Representation of numerosity in posterior parietal cortex

Humans and animals appear to share a similar representation of number as an analog magnitude on an internal, subjective scale. Neurological and neurophysiological data suggest that posterior parietal cortex (PPC) is a critical component of the circuits that form the basis of numerical abilities in humans. Patients with parietal lesions are impaired in their ability to access the deep meaning of numbers. Acalculiac patients with inferior parietal damage often have difficulty performing arithmetic (2 + 4?) or number bisection (what is between 3 and 5?) tasks, but are able to recite multiplication tables and read or write numerals. Functional imaging studies of neurologically intact humans performing subtraction, number comparison, and non-verbal magnitude comparison tasks show activity in areas within the intraparietal sulcus (IPS). Taken together, clinical cases and imaging studies support a critical role for parietal cortex in the mental manipulation of numerical quantities. Further, responses of single PPC neurons in non-human primates are sensitive to the numerosity of visual stimuli independent of low-level stimulus qualities. When monkeys are trained to make explicit judgments about the numerical value of such stimuli, PPC neurons encode their cardinal numerical value; without such training PPC neurons appear to encode numerical magnitude in an analog fashion. Here we suggest that the spatial and integrative properties of PPC neurons contribute to their critical role in numerical cognition.

Differences between literates and illiterates on symbolic but not nonsymbolic numerical magnitude processing

The study of numerical magnitude processing provides a unique opportunity to examine interactions between phylogenetically ancient systems of semantic representations and those that are the product of enculturation. While nonsymbolic representations of numerical magnitude are processed similarly by humans and nonhuman animals, symbolic representations of numerical magnitude (e.g., Hindu-Arabic numerals) are culturally invented symbols that are uniquely human. Here, we report a comparison of symbolic and nonsymbolic numerical magnitude processing in two groups of participants who differ substantially in their level of literacy. In this study, level of literacy is used as an index of level of school-based numeracy skill. The data from these groups demonstrate that while the processing of nonsymbolic numerical magnitude (numerical distance effect) is unaffected by an individual's level of literacy, the processing of Hindu-Arabic numerals differs between literate and illiterate individuals who live in a literature culture and have limited symbolic recognition skills. These findings reveal that nonsymbolic numerical magnitude processing is unaffected by enculturation, while the processing of numerical symbols is modulated by literacy.

Impact of high mathematics education on the number sense

In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.

Number concepts without number lines in an indigenous group of Papua New Guinea

BACKGROUND

The generic concept of number line, which maps numbers to unidimensional space, is a fundamental concept in mathematics, but its cognitive origins are uncertain. Two defining criteria of the number line are that (i) there is a mapping of each individual number (or numerosity) under consideration onto a specific location on the line, and (ii) that the mapping defines a unidimensional space representing numbers with a metric--a distance function. It has been proposed that the number line is based on a spontaneous universal human intuition, rooted directly in brain evolution, that maps number magnitude to linear space with a metric. To date, no culture lacking this intuition has been documented.

METHODOLOGY/PRINCIPAL FINDINGS

By means of a number line task, we investigated the universality proposal with the Yupno of Papua New Guinea. Unschooled adults did exhibit a number-to-space mapping (criterion i) but, strikingly, despite having precise cardinal number concepts, they located numbers only on the endpoints, thus failing to use the extent of the line. The produced mapping was bi-categorical and metric-free, in violation of criterion ii. In contrast, Yupnos with scholastic experience used the extent of the segment according to known standards, but they did so not as evenly as western controls, exhibiting a bias towards the endpoints.

CONCLUSIONS/SIGNIFICANCE

Results suggest that cardinal number concepts can exist independently from number line representations. They also suggest that the number line mapping, although ubiquitous in the modern world, is not universally spontaneous, but rather seems to be learned through--and continually reinforced by--specific cultural practices.

Cross-dimensional mapping of number, length and brightness by preschool children

Human adults in diverse cultures, children, infants, and non-human primates relate number to space, but it is not clear whether this ability reflects a specific and privileged number-space mapping. To investigate this possibility, we tested preschool children in matching tasks where the dimensions of number and length were mapped both to one another and to a third dimension, brightness. Children detected variation on all three dimensions, and they reliably performed mappings between number and length, and partially between brightness and length, but not between number and brightness. Moreover, children showed reliably better mapping of number onto the dimension of length than onto the dimension of brightness. These findings suggest that number establishes a privileged mapping with the dimension of length, and that other dimensions, including brightness, can be mapped onto length, although less efficiently. Children's adeptness at number-length mappings suggests that these two dimensions are intuitively related by the end of the preschool years.

Malleability of the approximate number system: effects of feedback and training

Prior research demonstrates that animals and humans share an approximate number system (ANS), characterized by ratio dependence and that the precision of this system increases substantially over human development. The goal of the present research was to investigate the malleability of the ANS (as measured by Weber fraction) in adult subjects in response to feedback and to explore the relationship between ANS acuity and acuity on another magnitude comparison task. We tested each of 20 subjects over six 1-h sessions. The main findings were that (a) Weber fractions rapidly decreased when trial-by-trial feedback was introduced in the second session and remained stable over continued training, (b) Weber fractions remained steady when trial-by-trial feedback was removed in session 6, (c)Weber fractions from the number comparison task were positively correlated with Weber fractions from a line length comparison task, (d) improvement in Weber fractions in response to feedback for the number task did not transfer to the line length task, (e) finally, the precision of the ANS was positively correlated with math, but not verbal, standardized aptitude scores. Potential neural correlates of the perceptual information and decision processes are considered, and predictions regarding the neural correlates of ANS malleability are discussed.

Navigation as a source of geometric knowledge: young children's use of length, angle, distance, and direction in a reorientation task

Geometry is one of the highest achievements of our species, but its foundations are obscure. Consistent with longstanding suggestions that geometrical knowledge is rooted in processes guiding navigation, the present study examines potential sources of geometrical knowledge in the navigation processes by which young children establish their sense of orientation. Past research reveals that children reorient both by the shape of the surface layout and the shapes of distinctive landmarks, but it fails to clarify what shape properties children use. The present study explores 2-year-old children's sensitivity to angle, length, distance and direction by testing disoriented children's search in a variety of fragmented rhombic and rectangular environments. Children reoriented themselves in accord with surface distances and directions, but they failed to use surface lengths or corner angles either for directional reorientation or as local landmarks. Thus, navigating children navigate by some but not all of the abstract properties captured by formal Euclidean geometry. While navigation systems may contribute to children's developing geometric understanding, they likely are not the sole source of abstract geometric intuitions.

Generating nonsymbolic number stimuli

Studies investigating nonsymbolic numbers (e.g., dot arrays) are confronted with the problem that changes in numerosity are always accompanied by changes in the visual properties of the stimulus. It is therefore debated whether the visual properties of the stimulus rather than number can explain the results obtained in studies investigating nonsymbolic number processing. In this report, we present a program (available at http://titiagebuis.eu/Materials.html ; note that the program is designed to work with the Psychophysics Toolbox in MATLAB) that exports information about the visual properties of stimuli that co-vary with number (area extended, item size, total surface, density, and circumference). Consequently, insight into the relation between the visual properties of the stimulus and numerical distance can be achieved, and post hoc analyses can be conducted to directly reveal whether numerical distance or (some combinations of) the visual properties of a stimulus could be the most likely candidate underlying the results. Here, we report data that demonstrate the program's usefulness for research on nonsymbolic number stimuli.

Optimal encoding of interval timing in expert percussionists

We measured temporal reproduction in human subjects with various levels of musical expertise: expert drummers, string musicians, and non-musicians. While duration reproduction of the non-percussionists showed a characteristic central tendency or regression to the mean, drummers responded veridically. Furthermore, when the stimuli were auditory tones rather than flashes, all subjects responded veridically. The behavior of all three groups in both modalities is well explained by a Bayesian model that seeks to minimize reproduction errors by incorporating a central tendency prior, a probability density function centered at the mean duration of the sample. We measured separately temporal precision thresholds with a bisection task; thresholds were twice as low in drummers as in the other two groups. These estimates of temporal precision, together with an adaptable Bayesian prior, predict well the reproduction results and the central tendency strategy under all conditions and for all subject groups. These results highlight the efficiency and flexibility of sensorimotor mechanisms estimating temporal duration.

Rethinking the language of thought

In this piece, we overview the language of thought (LOT) program, a currently influential theory of the computational nature of thought. We focus on LOT's stance on concepts, computation in the central system, and mental symbols. We emphasize certain longstanding problems arising for the LOT approach, suggesting resolutions to these problems. Many of the solutions involve departures from the standard LOT program, i.e., the LOT program as developed by Jerry Fodor. We close by identifying avenues for future work. WIREs Cogn Sci 2012, 3:153-162. doi: 10.1002/wcs.1155 For further resources related to this article, please visit the WIREs website.

Spatial intuition in elementary arithmetic: a neurocomputational account

Elementary arithmetic (e.g., addition, subtraction) in humans has been shown to exhibit spatial properties. Its exact nature has remained elusive, however. To address this issue, we combine two earlier models for parietal cortex: A model we recently proposed on number-space interactions and a modeling framework of parietal cortex that implements radial basis functions for performing spatial transformations. Together, they provide us with a framework in which elementary arithmetic is based on evolutionarily more basic spatial transformations, thus providing the first implemented instance of Dehaene and Cohen's recycling hypothesis.

Core knowledge of object, number, and geometry: a comparative and neural approach

Studies on the ontogenetic origins of human knowledge provide evidence for a small set of separable systems of core knowledge dealing with the representation of inanimate and animate objects, number, and geometry. Because core knowledge systems are evolutionarily ancient, they can be investigated from a comparative perspective, making use of various animal models. In this review, I discuss evidence showing precocious abilities in nonhuman species to represent (a) objects that move partly or fully out of view and their basic mechanical properties such as solidity, (b) the cardinal and ordinal/sequential aspects of numerical cognition and rudimentary arithmetic with small numerosities, and (c) the geometrical relationships among extended surfaces in the surrounding layout. Controlled rearing studies suggest that the abilities associated with core knowledge systems of objects, number, and geometry are observed in animals in the absence (or with very reduced) experience, supporting a nativistic foundation of such cognitive mechanisms. Animal models also promise a fresh approach to the issue of the neurobiological and genetic mechanisms underlying the expression of core knowledge systems.

Bootstrapping in a language of thought: a formal model of numerical concept learning

In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a rich system of interrelated numerical concepts. We present a computational framework for understanding this inductive leap as the consequence of statistical inference over a sufficiently powerful representational system. We provide an implemented model that is powerful enough to learn number word meanings and other related conceptual systems from naturalistic data. The model shows that bootstrapping can be made computationally and philosophically well-founded as a theory of number learning. Our approach demonstrates how learners may combine core cognitive operations to build sophisticated representations during the course of development, and how this process explains observed developmental patterns in number word learning.

Iterative Bayesian estimation as an explanation for range and regression effects: a study on human path integration

Systematic errors in human path integration were previously associated with processing deficits in the integration of space and time. In the present work, we hypothesized that these errors are de facto the result of a system that aims to optimize its performance by incorporating knowledge about prior experience into the current estimate of displacement. We tested human linear and angular displacement estimation behavior in a production-reproduction task under three different prior experience conditions where samples were drawn from different overlapping sample distributions. We found that (1) behavior was biased toward the center of the underlying sample distribution, (2) the amount of bias increased with increasing sample range, and (3) the standard deviation for all conditions was linearly dependent on the mean reproduced displacements. We propose a model of bayesian estimation on logarithmic scales that explains the observed behavior by optimal fusion of an experience-dependent prior expectation with the current noisy displacement measurement. The iterative update of prior experience is modeled by the formulation of a discrete Kalman filter. The model provides a direct link between Weber-Fechner and Stevens' power law, providing a mechanistic explanation for universal psychophysical effects in human magnitude estimation such as the regression to the mean and the range effect.

Multimodal semantic quantity representations: further evidence from korean sign language

Korean deaf signers performed a number comparison task on pairs of Arabic digits. In their response times profiles, the expected magnitude effect was systematically modified by properties of number signs in Korean sign language in a culture-specific way (not observed in hearing and deaf Germans or hearing Chinese). We conclude that finger-based quantity representations are automatically activated even in simple tasks with symbolic input although this may be irrelevant and even detrimental for task performance. These finger-based numerical representations are accessed in addition to another, more basic quantity system which is evidenced by the magnitude effect. In sum, these results are inconsistent with models assuming only one single amodal representation of numerical quantity.