Numerical representation in the parietal lobes: abstract or not abstract?

The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex.

Number-processing skills in adults with dyslexia

The present study investigated basic numerical skills and arithmetic in adults with developmental dyslexia. Participants performed exact and approximate calculation, basic numerical tasks (e.g., counting; symbolic number comparison; spatial-numerical association of response codes, SNARC), and visuospatial tasks (mental rotation and visual search tasks). The group with dyslexia showed a marginal impairment in counting compared to age- and IQ-matched controls, and they were impaired in exact addition, in particular with respect to speed. They were also significantly slower in multiplication. In basic number processing, however, there was no significant difference in performance between those with dyslexia and controls. Both groups performed similarly on subtraction and approximate addition tasks. These findings indicate that basic number processing in adults with dyslexia is intact. Their difficulties are restricted to the verbal code and are not associated with deficits in nonverbal magnitude representation, visual Arabic number form, or spatial cognition.

How numeracy influences risk comprehension and medical decision making

We review the growing literature on health numeracy, the ability to understand and use numerical information, and its relation to cognition, health behaviors, and medical outcomes. Despite the surfeit of health information from commercial and noncommercial sources, national and international surveys show that many people lack basic numerical skills that are essential to maintain their health and make informed medical decisions. Low numeracy distorts perceptions of risks and benefits of screening, reduces medication compliance, impedes access to treatments, impairs risk communication (limiting prevention efforts among the most vulnerable), and, based on the scant research conducted on outcomes, appears to adversely affect medical outcomes. Low numeracy is also associated with greater susceptibility to extraneous factors (i.e., factors that do not change the objective numerical information). That is, low numeracy increases susceptibility to effects of mood or how information is presented (e.g., as frequencies vs. percentages) and to biases in judgment and decision making (e.g., framing and ratio bias effects). Much of this research is not grounded in empirically supported theories of numeracy or mathematical cognition, which are crucial for designing evidence-based policies and interventions that are effective in reducing risk and improving medical decision making. To address this gap, we outline four theoretical approaches (psychophysical, computational, standard dual-process, and fuzzy trace theory), review their implications for numeracy, and point to avenues for future research.

Representation of number in the brain

Number symbols have allowed humans to develop superior mathematical skills that are a hallmark of technologically advanced cultures. Findings in animal cognition, developmental psychology, and anthropology indicate that these numerical skills are rooted in nonlinguistic biological primitives. Recent studies in human and nonhuman primates using a broad range of methodologies provide evidence that numerical information is represented and processed by regions of the prefrontal and posterior parietal lobes, with the intraparietal sulcus as a key node for the representation of the semantic aspect of numerical quantity.

Possible dendritic contribution to unimodal numerosity tuning and weber-fechner law-dependent numerical cognition

Humans and animals are known to share an ability to estimate or compare the numerosity of visual stimuli, and this ability is considered to be supported by the cortical neurons that have unimodal tuning for numerosity, referred to as the numerosity detector neurons. How such unimodal numerosity tuning is shaped through plasticity mechanisms is unknown. Here, I propose a testable hypothetical mechanism based on recently revealed features of the neuronal dendrite, namely, cooperative plasticity induction and nonlinear input integration at nearby dendritic sites, on the basis of the existing proposal that individual visual stimuli are represented as similar localized activities regardless of the size or the shape in a cortical region in the dorsal visual pathway. Intriguingly, the proposed mechanism naturally explains a prominent feature of the numerosity detector neurons, namely, the broadening of the tuning curve in proportion to the preferred numerosity, which is considered to underlie the known Weber-Fechner law-dependent accuracy of numerosity estimation and comparison. The simulated tuning curves are less sharp than reality, however, and together with the evidence from human imaging studies that numerical representation is a distributed phenomenon, it may not be likely that the proposed mechanism operates by itself. Rather, the proposed mechanism might facilitate the formation of hierarchical circuitry proposed in the previous studies, which includes neurons with monotonic numerosity tuning as well as those with sharp unimodal tuning, by serving as an efficient initial condition.

The case for a notation-independent representation of number

Cohen Kadosh & Walsh (CK&W) neglect the solid empirical evidence for a convergence of notation-specific representations onto a shared representation of numerical magnitude. Subliminal priming reveals cross-notation and cross-modality effects, contrary to CK&W's prediction that automatic activation is modality and notation-specific. Notation effects may, however, emerge in the precision, speed, automaticity, and means by which the central magnitude representation is accessed.

Symbolic, numeric, and magnitude representations in the parietal cortex

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.

Beyond format-specificity: Is analogue magnitude really the core abstract feature of the cultural number representation?

The issue of abstractness raises two distinct questions. First, is there a format-independent magnitude representation? Second, does analogue magnitude really play a crucial role in the development of human mathematics? We suggest that neither developmental nor cultural studies support this notion. The field needs to redefine the properties of the core number representation as used in human arithmetic.

Non-abstract numerical representations in the IPS: Further support, challenges, and clarifications

The commentators have raised many pertinent points that allow us to refine and clarify our view. We classify our response comments into seven sections: automaticity; developmental and educational questions; priming; multiple representations or multiple access(?); terminology; methodological advances; and simulated cognition and numerical cognition. We conclude that the default numerical representations are not abstract.

A compact representation of drawing movements with sequences of parabolic primitives

Some studies suggest that complex arm movements in humans and monkeys may optimize several objective functions, while others claim that arm movements satisfy geometric constraints and are composed of elementary components. However, the ability to unify different constraints has remained an open question. The criterion for a maximally smooth (minimizing jerk) motion is satisfied for parabolic trajectories having constant equi-affine speed, which thus comply with the geometric constraint known as the two-thirds power law. Here we empirically test the hypothesis that parabolic segments provide a compact representation of spontaneous drawing movements. Monkey scribblings performed during a period of practice were recorded. Practiced hand paths could be approximated well by relatively long parabolic segments. Following practice, the orientations and spatial locations of the fitted parabolic segments could be drawn from only 2-4 clusters, and there was less discrepancy between the fitted parabolic segments and the executed paths. This enabled us to show that well-practiced spontaneous scribbling movements can be represented as sequences ("words") of a small number of elementary parabolic primitives ("letters"). A movement primitive can be defined as a movement entity that cannot be intentionally stopped before its completion. We found that in a well-trained monkey a movement was usually decelerated after receiving a reward, but it stopped only after the completion of a sequence composed of several parabolic segments. Piece-wise parabolic segments can be generated by applying affine geometric transformations to a single parabolic template. Thus, complex movements might be constructed by applying sequences of suitable geometric transformations to a few templates. Our findings therefore suggest that the motor system aims at achieving more parsimonious internal representations through practice, that parabolas serve as geometric primitives and that non-Euclidean variables are employed in internal movement representations (due to the special role of parabolas in equi-affine geometry).

Economic cognition in humans and animals: the search for core mechanisms

Over the past few decades, research in judgment and decision-making has revealed that decision-makers, though not always rational, are often quite predictable. Here, we attempt to explore the nature of this systematicity with a different approach to decision-making. Specifically, we propose that some of the systematicity of human decision-making may result from the operation of core knowledge mechanisms, domain-specific learning mechanisms with characteristic processing limitations. In this review, we describe the core knowledge approach and argue that at least some aspects of human decision-making have the signature characteristics of a core knowledge system, namely, such strategies develop early in ontogeny and are shared with closely related primate relatives.

Prefrontal cortex and the evolution of symbolic reference

Symbol systems such as numbers and language are of paramount importance to human cognition. In number theory, numbers are symbolic signs embedded in a system of higher-order sign-sign relations. During ontogeny, numerical competence passes through different referential sign relations with increasing complexity, from an iconic to an indexical and finally symbolic stage. Animals such as nonhuman primates are constrained to indexical reference. However, because symbolic reference emerges from indexical reference, behavioral and neuronal representations of semantic sign-numerosity associations in animals can elucidate the precursors of symbol systems. A neurobiological explanation of how numerical signs take their meaning is proposed by suggesting that neurons in the granular prefrontal cortex, a novel brain structure evolved in primates, enable high-order associations and establish links between nonsymbolic numerosities and arbitrary signs.

How Humans Count: Numerosity and the Parietal Cortex

Numerosity (the number of objects in a set), like color or movement, is a basic property of the environment. Animal and human brains have been endowed by evolution by mechanisms based on parietal circuitry for representing numerosity in an highly abstract, although approximate fashion. These mechanisms are functional at a very early age in humans and spontaneously deployed in the wild by animals of different species. The recent years have witnessed terrific advances in unveiling the neural code(s) underlying numerosity representations and showing similarities as well as differences across species. In humans, during development, with the introduction of symbols for numbers and the implementation of the counting routines, the parietal system undergoes profound (yet still largely mysterious) modifications, such that the neural machinery previously evolved to represent approximate numerosity gets partially “recycled” to support the representation of exact number.

Origins of mathematical intuitions: the case of arithmetic

Mathematicians frequently evoke their "intuition" when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of "core knowledge" associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.

Arithmetic in newborn chicks

Newly hatched domestic chicks were reared with five identical objects. On days 3 or 4, chicks underwent free-choice tests in which sets of three and two of the five original objects disappeared (either simultaneously or one by one), each behind one of two opaque identical screens. Chicks spontaneously inspected the screen occluding the larger set (experiment 1). Results were confirmed under conditions controlling for continuous variables (total surface area or contour length; experiment 2). In the third experiment, after the initial disappearance of the two sets (first event, FE), some of the objects were visibly transferred, one by one, from one screen to the other (second event, SE). Thus, computation of a series of subsequent additions or subtractions of elements that appeared and disappeared, one by one, was needed in order to perform the task successfully. Chicks spontaneously chose the screen, hiding the larger number of elements at the end of the SE, irrespective of the directional cues provided by the initial (FE) and final (SE) displacements. Results suggest impressive proto-arithmetic capacities in the young and relatively inexperienced chicks of this precocial species.

Spatial representation across species: geometry, language, and maps

We review growing evidence that the reorientation system-shared by both humans and nonhuman species-privileges geometric representations of space and exhibits many of the characteristic features of modular systems. We also review evidence showing that humans can move beyond the limits of nonhuman species by using two cultural constructions, language and explicit maps. We argue that, although both of these constructions are uniquely human means of enriching the spatial system we share with other species, their representational formats, functions, and developmental trajectories are quite different, yielding distinctly different tools for empowering human spatial cognition.The capacity to reorient using geometry is present in humans by the age of 18 months.

Doing Socrates experiment right: controlled rearing studies of geometrical knowledge in animals

The issue of whether encoding of geometric information for navigational purposes crucially depends on environmental experience or whether it is innately predisposed in the brain has been recently addressed in controlled rearing studies. Non-human animals can make use of the geometric shape of an environment for spatial reorientation and in some circumstances reliance on purely geometric information (metric properties and sense) can overcome use of local featural information. Animals reared in home cages of different geometric shapes proved to be equally capable of learning and performing navigational tasks based on geometric information. The findings suggest that effective use of geometric information for spatial reorientation does not require experience in environments with right angles and metrically distinct surfaces.

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.

Spontaneous mapping of number and space in adults and young children

Mature representations of space and number are connected to one another in ways suggestive of a 'mental number line', but this mapping could either be a cultural construction or a reflection of a more fundamental link between the domains of number and geometry. Using a manual bisection paradigm, we tested for number line representations in adults, young school children, and preschool children. Non-symbolic numerical displays systematically distorted localization of the midpoint of a horizontal line at all three ages. Numerical and spatial representations therefore are linked prior to the onset of formal instruction, in a manner that suggests a privileged relation between spatial and numerical cognition.

Compressed Scaling of Abstract Numerosity Representations in Adult Humans and Monkeys

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

Multiple spatial representations of number: evidence for co-existing compressive and linear scales

Although the spatial representation of number (mental number line) is well documented, the scaling associated with this representation is less clear. Sometimes people appear to rely on compressive scaling, and sometimes on linear scaling. Here we provide evidence for both compressive and linear representations on the same numerical bisection task, in which adult participants estimate (without calculating) the midpoint between two numbers. The same leftward bias (pseudoneglect) shown on physical line bisection appears on this task, and was previously shown to increase with the magnitude of bisected numbers, consistent with compressive scaling (Longo and Lourenco in Neuropsychologia 45:1400-1407, 2007). In the present study, participants held either small (1-9) or large (101-109) number primes in memory during bisection. When participants remembered small primes, bisection responses were consistent with compressive scaling. However, when they remembered large primes, responses were more consistent with linear scaling. These results show that compressive and linear representations may be accessed flexibly on the same task, depending on the numerical context.

Beyond the number domain

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

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.

How does the mind work? Insights from biology

Cognitive scientists must understand not just what the mind does, but how it does what it does. In this paper, I consider four aspects of cognitive architecture: how the mind develops, the extent to which it is or is not modular, the extent to which it is or is not optimal, and the extent to which it should or should not be considered a symbol-manipulating device (as opposed to, say, an eliminative connectionist network). In each case, I argue that insights from developmental and evolutionary biology can lead to substantive and important compromises in historically vexed debates.

Look Ma, no fingers! Are children numerical solipsists?

I ask whether it is necessary that principles of number be mentally represented and point to the role of language in determining cultural variation. Some cultures possess extensive counting systems that are finite. I suggest that learning number principles is similar to learning conservation and, as such, might be derived from learning about the empirical properties of objects and other individuals in combinations.

Finger counting: The missing tool?

Rips et al. claim that the principles underlying the structure of natural numbers cannot be inferred from interactions with the physical world. However, in their target article they failed to consider an important source of interaction: finger counting. Here, we show that finger counting satisfies all the conditions required for allowing the concept of numbers to emerge from sensorimotor experience through a bottom-up process.

The innate schema of natural numbers does not explain historical, cultural, and developmental differences

Rips et al.'s proposition cannot account for the facts that (1) a historical look at the word number systems suggests that the concept of natural numbers has been progressively elaborated; (2) people from cultures without an elaborate counting system do not master the concept of natural numbers; (3) children take time to master natural numbers; and (4) the competing advantage of the postulated math schema in the natural selection process is not obvious.

Intuitive numbers guide decisions

Measuring reaction times to number comparisons is thought to reveal a processing stage in elementary numerical cognition linked to internal, imprecise representations of number magnitudes. These intuitive representations of the mental number line have been demonstrated across species and human development but have been little explored in decision making. This paper develops and tests hypotheses about the influence of such evolutionarily ancient, intuitive numbers on human decisions. We demonstrate that individuals with more precise mental-number-line representations are higher in numeracy (number skills) consistent with previous research with children. Individuals with more precise representations (compared to those with less precise representations) also were more likely to choose larger, later amounts over smaller, immediate amounts, particularly with a larger proportional difference between the two monetary outcomes. In addition, they were more likely to choose an option with a larger proportional but smaller absolute difference compared to those with less precise representations. These results are consistent with intuitive number representations underlying: a) perceived differences between numbers, b) the extent to which proportional differences are weighed in decisions, and, ultimately, c) the valuation of decision options. Human decision processes involving numbers important to health and financial matters may be rooted in elementary, biological processes shared with other species.

Behavioral and prefrontal representation of spatial proportions in the monkey

Primate brains are equipped with evolutionarily old and dedicated neural circuits so that they can grasp absolute quantities, such as the number of items or the length of a line. Absolute magnitude, however, is often not informative enough to guide decisions in conflicting social and foraging situations that require an assessment of quantity ratios. We report that rhesus monkeys can discriminate proportions (1:4, 2:4, 3:4, and 4:4) specified by bars differing in lengths and that they can do so at a precision comparable to that shown by humans; the monkeys thus demonstrate an abstract understanding of proportionality. Moreover, neurons in the lateral prefrontal cortex selectively responded to preferred proportions regardless of the exact physical appearance of the stimuli. These results support the hypothesis that nonhuman primates can judge proportions and utilize the underlying information in behaviorally relevant situations.

Neural correlates of approximate quantification strategies in young and older adults: an fMRI study

Young and older adults assessed the approximate number of dots in collections including between 20 and 50 dots, with two strategies. The benchmark strategy is based on retrieving memory representations of quantities after visually scanning stimulus. The anchoring strategy involves both enumeration and estimation processes. Brain activations and performance were analyzed as a function of strategies, size of collections and age. Executing the benchmark strategy produced faster performance. It was associated with increased activity of a bilateral parieto/occipital and insular cortical network, including the postcentral gyrus, the cuneus, the middle occipital gyrus, and the insula. In addition to these bilateral activations, the benchmark strategy activated right prefrontal area. The anchoring strategy activated right superior parietal lobule, bilateral subcortical structures (putamen), and left dorso-lateral prefrontal cortex. The effects of aging on these cortical networks depended on strategies. These results suggest dissociation between two numerosity estimation strategies underlying different cognitive estimation processes and help to clarify age differences in numerosity estimation.

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.

The evolution of comparative cognition: is the snark still a boojum?

In "The Snark is a Boojum", Beach [Beach, F.A., 1950. The snark was a boojum. American Psychologist. 5, 115-124] famously asserted that animal psychology embraced too few species and too few problems to deserve the name comparative. Later in the 20th century, others [e.g. Kamil, A.C., 1988. A synthetic approach to the study of animal intelligence. In: Leger, D.W. (Ed.), Comparative Perspectives in Modern Psychology. Nebraska Symposium on Motivation, vol. 35. University of Nebraska Press, Lincoln, NE, pp. 230-257; Shettleworth, S.J., 1993. Where is the comparison in comparative cognition? Alternative research programs. Psychological Science. 4, 179-184] expressed similar concerns about the new subfield of comparative cognition, suggesting that a more biological approach to choice of species and problems was needed to balance a dominant anthropocentrism. The last 10-15 years have seen many new developments, and a recent survey like Beach's reveals a very different picture. Not only are many more species being studied, contributions by researchers from different backgrounds are increasing, and research on comparative cognition is better connected with developmental psychology, behavioral neuroscience, primatology, behavioral ecology, and other fields. Contemporary research addresses three major aspects of cognition about equally: basic processes, physical cognition, and social cognition. This article describes a selected research program from each area, chosen to exemplify current trends and challenges for the field.

Nonsymbolic, approximate arithmetic in children: abstract addition prior to instruction

Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this task successfully with all the signatures of adults' nonsymbolic number representations: accuracy modulated by the ratio of the sum and the comparison quantity, equal performance for within- and cross-modality tasks and for addition and comparison tasks, and performance superior to that of a matched subtraction task. The findings provide clear evidence for nonsymbolic numerical operations on abstract numerical quantities in children who have not yet been taught formal arithmetic.