Associations between Auditory Pitch and Visual Elevation Do Not Depend on Language: Evidence from a Remote Population

Associations between auditory pitch and visual elevation are widespread in many languages, and behavioral associations have been extensively documented between height and pitch among speakers of those languages. However, it remains unclear whether perceptual correspondences between auditory pitch and visual elevation inform these linguistic associations, or merely reflect them. We probed this cross-modal mapping in members of a remote Kreung hill tribe in northeastern Cambodia who do not use spatial language to describe pitch. Participants viewed shapes rising or falling in space while hearing sounds either rising or falling in pitch, and reported on the auditory change. Associations between pitch and vertical position in the Kreung were similar to those demonstrated in populations where pitch is described in terms of spatial height. These results suggest that associations between visual elevation and auditory pitch can arise independently of language. Thus, widespread linguistic associations between pitch and elevation may reflect universally predisposed perceptual correspondences.

Patterns in wireless phone estimation data from a cross-sectional survey: what are the implications for epidemiology?

OBJECTIVE

Self-reported recall data are often used in wireless phone epidemiological studies, which in turn are used to indicate relative risk of health outcomes from extended radiofrequency exposure. We sought to explain features commonly observed in wireless phone recall data and to improve analytical procedures.

SETTING

Wellington Region, New Zealand.

PARTICIPANTS

Each of the 16 schools selected a year 7 and/or 8 class to participate, providing a representative regional sample based on socioeconomic school ratings, school type and urban/rural balance. There was an 85% participation rate (N=373).

MAIN OUTCOME MEASURES

Planned: the distribution of participants' estimated extent of SMS-texting and cordless phone calls, and the extent of rounding to a final zero or five within the full set of recall data and within each order of magnitude. Unplanned: the distribution of the leading digits of these raw data, compared with that of billed data in each order of magnitude.

RESULTS

The nature and extent of number-rounding, and the distribution of data across each order in recall data indicated a logarithmic (ratio-based) mental process for assigning values. Responses became less specific as the leading-digit increased from 1 to 9, and 69% of responses for weekly texts sent were rounded by participants to a single non-zero digit (eg, 2, 20 and 200).

CONCLUSIONS

Adolescents' estimation of their cellphone use indicated that it was performed on a mental logarithmic scale. This is the first time this phenomenon has been observed in the estimation of recalled, as opposed to observed, numerical quantities. Our findings provide empirical justification for log-transforming data for analysis. We recommend the use of the geometric rather than arithmetic mean when a recalled numerical range is provided. A point of calibration may improve recall.

Developmental Changes in the Profiles of Dyscalculia: An Explanation Based on a Double Exact-and-Approximate Number Representation Model

Studies on developmental dyscalculia (DD) have tried to identify a basic numerical deficit that could account for this specific learning disability. The first proposition was that the number magnitude representation of these children was impaired. However, Rousselle and Noël (2007) brought data showing that this was not the case but rather that these children were impaired when processing the magnitude of symbolic numbers only. Since then, incongruent results have been published. In this paper, we will propose a developmental perspective on this issue. We will argue that the first deficit shown in DD regards the building of an exact representation of numerical value, thanks to the learning of symbolic numbers, and that the reduced acuity of the approximate number magnitude system appears only later and is secondary to the first deficit.

Comparing biological motion perception in two distinct human societies

Cross cultural studies have played a pivotal role in elucidating the extent to which behavioral and mental characteristics depend on specific environmental influences. Surprisingly, little field research has been carried out on a fundamentally important perceptual ability, namely the perception of biological motion. In this report, we present details of studies carried out with the help of volunteers from the Mundurucu indigene, a group of people native to Amazonian territories in Brazil. We employed standard biological motion perception tasks inspired by over 30 years of laboratory research, in which observers attempt to decipher the walking direction of point-light (PL) humans and animals. Do our effortless skills at perceiving biological activity from PL animations, as revealed in laboratory settings, generalize to people who have never before seen representational depictions of human and animal activity? The results of our studies provide a clear answer to this important, previously unanswered question. Mundurucu observers readily perceived the coherent, global shape depicted in PL walkers, and experienced the classic inversion effects that are typically found when such stimuli are turned upside down. In addition, their performance was in accord with important recent findings in the literature, in the abundant ease with which they extracted direction information from local motion invariants alone. We conclude that the effortless, veridical perception of PL biological motion is a spontaneous and universal perceptual ability, occurring both inside and outside traditional laboratory environments.

Measuring the Approximate Number System

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.

Human nonverbal discrimination of relative and absolute number

The nonverbal discrimination of relative and absolute number of sequential visual stimuli was investigated with humans in bisection, reproduction, and report tasks. Participants viewed a sequence of 40 red and black objects on each trial, randomly intermixed, and had to identify the number of red objects, which varied from 1 to 20. To prevent the use of a verbal-counting strategy, participants were required to name the objects as they appeared. The characteristics of human performance resembled those of pigeons in analogous procedures (Tan & Grace Learning and Behavior 38:408-417, 2010; Tan, Grace, Holland, & McLean Journal of Experimental Psychology 33:409-427, 2007): Average response number increased systematically with sample number, and bisection points were located at the arithmetic, not the geometric, mean. Additionally, in both the reproduction and report tasks, coefficients of variation decreased for values less than 6 but increased or remained constant for larger values, suggesting that different representations were used for small and large numbers.

Numerical magnitude processing in children with mild intellectual disabilities

The present study investigated numerical magnitude processing in children with mild intellectual disabilities (MID) and examined whether these children have difficulties in the ability to represent numerical magnitudes and/or difficulties in the ability to access numerical magnitudes from formal symbols. We compared the performance of 26 children with MID on a symbolic (digits) and a non-symbolic (dot-arrays) comparison task with the performance of two control groups of typically developing children: one group matched on chronological age and one group matched on mathematical ability level. Findings revealed that children with MID performed more poorly than their typically developing chronological age-matched peers on both the symbolic and non-symbolic comparison tasks, while their performance did not substantially differ from the ability-matched control group. These findings suggest that the development of numerical magnitude representation in children with MID is marked by a delay. This performance pattern was observed for both symbolic and non-symbolic comparison tasks, although difficulties on the former task were more prominent. Interventions in children with MID should therefore foster both the development of magnitude representations and the connections between symbols and the magnitudes they represent.

Conformity to peer pressure in preschool children

Both adults and adolescents often conform their behavior and opinions to peer groups, even when they themselves know better. The current study investigated this phenomenon in 24 groups of 4 children between 4;2 and 4;9 years of age. Children often made their judgments conform to those of 3 peers, who had made obviously erroneous but unanimous public judgments right before them. A follow-up study with 18 groups of 4 children between 4;0 and 4;6 years of age revealed that children did not change their "real" judgment of the situation, but only their public expression of it. Preschool children are subject to peer pressure, indicating sensitivity to peers as a primary social reference group already during the preschool years.

False approximations of the approximate number system?

Prior research suggests that the acuity of the approximate number system (ANS) predicts future mathematical abilities. Modelling the development of the ANS might therefore allow monitoring of children's mathematical skills and instigate educational intervention if necessary. A major problem however, is that our knowledge of the development of the ANS is acquired using fundamentally different paradigms, namely detection in infants versus discrimination in children and adults. Here, we question whether such a comparison is justified, by testing the adult ANS with both a discrimination and a detection task. We show that adults perform markedly better in the discrimination compared to the detection task. Moreover, performance on discrimination but not detection, correlated with performance on mathematics. With a second similar experiment, in which the detection task was replaced by a same-different task, we show that the results of experiment 1 cannot be attributed to differences in chance level. As only task instruction differed, the discrimination and the detection task most likely reflect differences at the decisional level. Future studies intending to model the development of the ANS should therefore rely on data derived from a single paradigm for different age groups. The same-different task appears a viable candidate, due to its applicability across age groups.

Cognitive processes of numerical estimation in children

We tested children in Grades 1 to 5, as well as college students, on a number line estimation task and examined latencies and errors to explore the cognitive processes involved in estimation. The developmental trends in estimation were more consistent with the hypothesized shift from logarithmic to linear representation than with an account based on a proportional judgment application of a power function model; increased linear responding across ages, as predicted by the log-to-lin shift position, yielded reasonable developmental patterns, whereas values derived from the cyclical power model were difficult to reconcile with expected developmental patterns. Neither theoretical position predicted the marked "M-shaped" pattern that was observed, beginning in third graders' errors and fourth graders' latencies. This pattern suggests that estimation comes to rely on a midpoint strategy based on children's growing number knowledge (i.e., knowledge that 50 is half of 100). As found elsewhere, strength of linear responding correlated significantly with children's performance on standardized math tests.

Memory for multiple visual ensembles in infancy

The number of individual items that can be maintained in working memory is limited. One solution to this problem is to store representations of ensembles that contain summary information about large numbers of items (e.g., the approximate number or cumulative area of a group of many items). Here we explored the developmental origins of ensemble representations by asking whether infants represent ensembles and, if so, how many at one time. We habituated 9-month-old infants to arrays containing 2, 3, or 4 spatially intermixed colored subsets of dots, then asked whether they detected a numerical change to one of the subsets or to the superset of all dots. Experiment Series 1 showed that infants detected a numerical change to 1 of the subsets when the array contained 2 subsets but not 3 or 4 subsets. Experiment Series 2 showed that infants detected a change to the superset of all dots no matter how many subsets were presented. Experiment 3 showed that infants represented both the approximate number and the cumulative surface area of these ensembles. Our results suggest that infants, like adults (Halberda, Sires, & Feigenson, 2006), can store quantitative information about 2 subsets plus the superset: a total of 3 ensembles. This converges with the known limit on the number of individual objects infants and adults can store and suggests that, throughout development, an ensemble functions much like an individual object for working memory.

Summation of large numerousness by newborn chicks

Newly hatched domestic chicks, reared with identical objects, when presented with sets of 3 vs. 2 objects disappearing one-by-one behind separate screens, spontaneously inspected the screen occluding the larger set; even when the continuous variables (area or perimeter) were controlled for (Rugani et al., 2009). Here, using a similar paradigm, we investigated the ability of chicks to perform addition on larger sets of objects. Chicks imprinted on five identical objects, were presented at test with 6 vs. 9 objects which disappeared one-by-one (Exp. 1). In Exp. 2, the same overall number of objects (15) was used, but employing an increased ratio, i.e., 5 vs. 10. In both experiments, when continuous variables were not made equal, chicks spontaneously inspected the screen occluding the larger set. However, when the size of the objects was adjusted so as to make the total surface area or perimeter equal for the two sets, chicks did not exhibit any preference. Lack of choice in the control conditions could be due to a combination of preferences; to rejoin the larger numerousness as well as the bigger objects (Rugani et al., 2010a). In Exp. 3, chicks were familiarized, during imprinting, with objects of various dimensions, in an attempt to reduce or suppress their tendency to approach objects larger than the familiar ones. Again chicks failed to choose at test between 5 vs. 10 objects when continuous variables were made equal. Results showed that chicks, after a one-by-one presentation of a large number of objects, rejoined the larger set. In order to choose the larger set, chicks estimated the objects in the two sets and then compared the outcomes. However, differently to what has been described for small numerousness, chicks succeeded only if non-numerical cues as well as numerical cues were available. This study suggests that continuous variables are computed by chicks for sets of objects that are not present at the same time and that are no longer visible at the time of choice.

Numerosity and number signs in deaf Nicaraguan adults

What abilities are entailed in being numerate? Certainly, one is the ability to hold the exact quantity of a set in mind, even as it changes, and even after its members can no longer be perceived. Is counting language necessary to track and reproduce exact quantities? Previous work with speakers of languages that lack number words involved participants only from non-numerate cultures. Deaf Nicaraguan adults all live in a richly numerate culture, but vary in counting ability, allowing us to experimentally differentiate the contribution of these two factors. Thirty deaf and 10 hearing participants performed 11 one-to-one matching and counting tasks. Results suggest that immersion in a numerate culture is not enough to make one fully numerate. A memorized sequence of number symbols is required, though even an unconventional, iconic system is sufficient. Additionally, we find that within a numerate culture, the ability to track precise quantities can be acquired in adulthood.

Relationships between number and space processing in adults with and without dyscalculia

A large body of evidence indicates clear relationships between number and space processing in healthy and brain-damaged adults, as well as in children. The present paper addressed this issue regarding atypical math development. Adults with a diagnosis of dyscalculia (DYS) during childhood were compared to adults with average or high abilities in mathematics across two bisection tasks. Participants were presented with Arabic number triplets and had to judge either the number magnitude or the spatial location of the middle number relative to the two outer numbers. For the numerical judgment, adults with DYS were slower than both groups of control peers. They were also more strongly affected by the factors related to number magnitude such as the range of the triplets or the distance between the middle number and the real arithmetical mean. By contrast, adults with DYS were as accurate and fast as adults who never experienced math disability when they had to make a spatial judgment. Moreover, number-space congruency affected performance similarly in the three experimental groups. These findings support the hypothesis of a deficit of number magnitude representation in DYS with a relative preservation of some spatial mechanisms in DYS. Results are discussed in terms of direct and indirect number-space interactions.

Comparison of quantities: core and format-dependent regions as revealed by fMRI

The perception and handling of numbers is central to education. Numerous imaging studies have focused on how quantities are encoded in the brain. Yet, only a few studies have touched upon number mining: the ability to extract the magnitude encoded in a visual stimulus. This article aims to characterize how analogue (i.e., disks and dots) and symbolic (i.e., positive and negative integers) formats influence number mining and the representation of quantities. Sixteen adult volunteers completed a comparison task while we recorded the blood oxygen level-dependent response using functional magnetic resonance imaging. The results revealed that a restricted set of specific subdivisions in the right intraparietal sulcus is activated in all conditions. With respect to magnitude assessment, the results show that 1) analogue stimuli are predominantly processed in the right hemisphere and that 2) symbolic stimuli encompass the analogue system and further recruit areas in the left hemisphere. Crucially, we found that polarity is encoded independently from magnitude. We refine the triple-code model by integrating our findings.

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.

Preschool acuity of the approximate number system correlates with school math ability

Previous research shows a correlation between individual differences in people's school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants' ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children's math ability and vocabulary size prior to the onset of formal math instruction. We found that children's ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.

Using plausible group sizes to communicate information about medical risks

OBJECTIVE

To make informed health decisions, patients must understand and recall risks, which often involve ratios with large denominators. Grasping the meaning of such numbers may be difficult, because of limited exposure to large groups of people in either our evolutionary history or daily life.

METHODS

In an experiment (n=98), we investigated whether medical risks are easier to understand and recall if their representation is based on small, evolutionarily plausible groups of people, and whether this representation especially helps patients with low numeracy.

RESULTS

Participants-especially those with low numeracy-often disregarded and incorrectly recalled denominators of ratios representing medical risks when the denominators involved were large. Risks were easier to understand and recall if their representation was based on smaller, evolutionarily plausible groups of people.

CONCLUSIONS

Our results extend previous literature on the role of numeracy in understanding health-relevant risk communications by showing the importance of using plausible group sizes to communicate these risks to people with low numeracy. Our results also support the notion that problems in risk perception occur because of inappropriate presentation formats rather than cognitive biases.

PRACTICE IMPLICATIONS

Our findings suggest suitable ways to communicate quantitative medical data-especially to people with low numeracy.

The powers of noise-fitting: reply to Barth and Paladino

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.

Swearing, euphemisms, and linguistic relativity

Participants read aloud swear words, euphemisms of the swear words, and neutral stimuli while their autonomic activity was measured by electrodermal activity. The key finding was that autonomic responses to swear words were larger than to euphemisms and neutral stimuli. It is argued that the heightened response to swear words reflects a form of verbal conditioning in which the phonological form of the word is directly associated with an affective response. Euphemisms are effective because they replace the trigger (the offending word form) by another word form that expresses a similar idea. That is, word forms exert some control on affect and cognition in turn. We relate these findings to the linguistic relativity hypothesis, and suggest a simple mechanistic account of how language may influence thinking in this context.

Specialization in the human brain: the case of numbers

How numerical representation is encoded in the adult human brain is important for a basic understanding of human brain organization, its typical and atypical development, its evolutionary precursors, cognitive architectures, education, and rehabilitation. Previous studies have shown that numerical processing activates the same intraparietal regions irrespective of the presentation format (e.g., symbolic digits or non-symbolic dot arrays). This has led to claims that there is a single format-independent, numerical representation. In the current study we used a functional magnetic resonance adaptation paradigm, and effective connectivity analysis to re-examine whether numerical processing in the intraparietal sulci is dependent or independent on the format of the stimuli. We obtained two novel results. First, the whole brain analysis revealed that format change (e.g., from dots to digits), in the absence of a change in magnitude, activated the same intraparietal regions as magnitude change, but to a greater degree. Second, using dynamic causal modeling as a tool to disentangle neuronal specialization across regions that are commonly activated, we found that the connectivity between the left and right intraparietal sulci is format-dependent. Together, this line of results supports the idea that numerical representation is subserved by multiple mechanisms within the same parietal regions.

Neural dissociation of number from letter recognition and its relationship to parietal numerical processing

The visual recognition of letters dissociates from the recognition of numbers at both the behavioral and neural level. In this article, using fMRI, we investigate whether the visual recognition of numbers dissociates from letters, thereby establishing a double dissociation. In Experiment 1, participants viewed strings of consonants and Arabic numerals. We found that letters activated the left midfusiform and inferior temporal gyri more than numbers, replicating previous studies, whereas numbers activated a right lateral occipital area more than letters at the group level. Because the distinction between letters and numbers is culturally defined and relatively arbitrary, this double dissociation provides some of the strongest evidence to date that a neural dissociation can emerge as a result of experience. We then investigated a potential source of the observed neural dissociation. Specifically, we tested the hypothesis that lateralization of visual number recognition depends on lateralization of higher-order numerical processing. In Experiment 2, the same participants performed addition, subtraction, and counting on arrays of nonsymbolic stimuli varying in numerosity, which produced neural activity in and around the intraparietal sulcus, a region associated with higher-order numerical processing. We found that individual differences in the lateralization of number activity in visual cortex could be explained by individual differences in the lateralization of numerical processing in parietal cortex, suggesting a functional relationship between the two regions. Together, these results demonstrate a neural double dissociation between letter and number recognition and suggest that higher-level numerical processing in parietal cortex may influence the neural organization of number processing in visual cortex.

Spatial reasoning in Tenejapan Mayans

Language communities differ in their stock of reference frames (coordinate systems for specifying locations and directions). English typically uses egocentrically-defined axes (e.g., "left-right"), especially when describing small-scale relationships. Other languages such as Tseltal Mayan prefer to use geocentrically-defined axes (e.g., "north-south") and do not use any type of projective body-defined axes. It has been argued that the availability of specific frames of reference in language determines the availability or salience of the corresponding spatial concepts. In four experiments, we explored this hypothesis by testing Tseltal speakers' spatial reasoning skills. Whereas most prior tasks in this domain were open-ended (allowing several correct solutions), the present tasks required a unique solution that favored adopting a frame-of-reference that was either congruent or incongruent with what is habitually lexicalized in the participants' language. In these tasks, Tseltal speakers easily solved the language-incongruent problems, and performance was generally more robust for these than for the language-congruent problems that favored geocentrically-defined coordinates. We suggest that listeners' probabilistic inferences when instruction is open to more than one interpretation account for why there are greater cross-linguistic differences in the solutions to open-ended spatial problems than to less ambiguous ones.

Dissociation of subtraction and multiplication in the right parietal cortex: evidence from intraoperative cortical electrostimulation

Previous research has consistently shown that the left parietal cortex is critical for numerical processing, but the role of the right parietal lobe has been much less clear. This study used the intraoperative cortical electrical stimulation approach to investigate neural dissociation in the right parietal cortex for subtraction and multiplication. Results showed that multiplication (as well as picture naming) was not affected by the cortical electrical stimulation on all the targeted sites of the right parietal cortex as well as those of the right temporal cortex. In contrast, stimulation at three right parietal sites (two sites in the right inferior parietal lobule and one in the right angular gyrus) impaired performance on simple subtraction problems. This study provided the first evidence from an intraoperative cortical electrical stimulation study to show the dissociation of arithmetic operations in the right parietal cortex. This dissociation between subtraction and multiplication suggests that the right parietal cortex plays a more significant role in quantity processing (subtraction) than in verbal processing (multiplication) in numerical processing.

Young children reorient by computing layout geometry, not by matching images of the environment

Disoriented animals from ants to humans reorient in accord with the shape of the surrounding surface layout: a behavioral pattern long taken as evidence for sensitivity to layout geometry. Recent computational models suggest, however, that the reorientation process may not depend on geometrical analyses but instead on the matching of brightness contours in 2D images of the environment. Here we test this suggestion by investigating young children's reorientation in enclosed environments. Children reoriented by extremely subtle geometric properties of the 3D layout: bumps and ridges that protruded only slightly off the floor, producing edges with low contrast. Moreover, children failed to reorient by prominent brightness contours in continuous layouts with no distinctive 3D structure. The findings provide evidence that geometric layout representations support children's reorientation.

Flexible intuitions of Euclidean geometry in an Amazonian indigene group

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ~180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics.

Adults' number-line estimation strategies: evidence from eye movements

Although the development of number-line estimation ability is well documented, little is known of the processes underlying successful estimators' mappings of numerical information onto spatial representations during these tasks. We tracked adults' eye movements during a number-line estimation task to investigate the processes underlying number-to-space translation, with three main results. First, eye movements were strongly related to the target number's location, and early processing measures directly predicted later estimation performance. Second, fixations and estimates were influenced by the size of the first number presented, indicating that adults calibrate their estimates online. Third, adults' number-line estimates demonstrated patterns of error consistent with the predictions of psychophysical models of proportion estimation, and eye movement data predicted the specific error patterns we observed. These results support proportion-based accounts of number-line estimation and suggest that adults' translation of numerical information into spatial representations is a rapid, online process.

Language and analogy in conceptual change

Carey proposes that the acquisition of the natural numbers relies on the interaction between language and analogical processes: specifically, on an analogical mapping from ordinal linguistic structure to ordinal conceptual structure. We suggest that this analogy in fact requires several steps. Further, we propose that additional analogical processes enter into the acquisition of number.

Concept innateness, concept continuity, and bootstrapping

The commentators raised issues relevant to all three important theses of The Origin of Concepts (henceforth TOOC). Some questioned the very existence of innate representational primitives, and others questioned my claims about their richness and whether they should be thought of as concepts. Some questioned the existence of conceptual discontinuity in the course of knowledge acquisition and others argued that discontinuity is much more common than was portrayed in TOOC. Some raised issues with my characterization of Quinian bootstrapping, and others questioned the dual factor theory of concepts motivated by my picture of conceptual development.

The Cultural Number Line: A Review of Cultural and Linguistic Influences on the Development of Number Processing

Approximate processing of numerosities is a universal and preverbal skill, while exact number processing above 4 involves the use of culturally acquired number words and symbols. The authors first review core concepts of numerical cognition, including number representation in the brain and the influential view that numbers are associated with space along a “mental number line.” Then, they discuss how cultural influences, such as reading direction, finger counting, and the transparency of the number word system, can influence the representation and processing of numbers. Spatial mapping of numbers emerges as a universal cognitive strategy. The authors trace the impact of cultural factors on the development of number skills and conclude that a cross-cultural perspective can reveal important constraints on numerical cognition.

How Numbers Bias Preschoolers’ Spatial Search

Numbers often bias adults’ spatial performance. Because the direction of this bias (left-to-right versus right-to-left) is culture-specific, it has been assumed that spatial-numeric associations develop with reading practice or schooling. The authors tested this assumption by examining spatial-numeric associations in pre-reading preschoolers. Preschoolers were shown two boxes (sample and matching boxes) subdivided into seven verbally numbered “rooms” (e.g., “the four room”). A “winner” card was revealed in the sample box, and children searched for the “winner” in the matching box (located in the same-numbered room). Preschoolers were faster and more accurate when rooms increased numerically from left-to-right versus right-to-left. This advantage was apparently caused by numbers influencing preschoolers’ encoding of spatial locations: Ordering of numbers in the sample box affected preschoolers’ search greatly, whereas ordering of numbers in the matching box did not. The authors conclude that numeric effects on spatial encoding develop far too early to be caused by reading practice or schooling.

Typical development of quantity comparison in school-aged children

Although basic numerical skills have been widely studied in the last years, very few studies have undertaken a developmental approach. The present study evaluated the development of the magnitude comparison basic numerical ability, in children from first, third and sixth grades by means of the subject's response time in numerical tasks presented in symbolic and non-symbolic formats. The results showed a significant decrease on quantities processing speed as age increases, which suggests numerical skills tend to become automatic with instruction. The differences found, concerning the general achievement pattern in each school year, might express the maturational specificities of the numerical representation system through development.

No Innate Number Line in the Human Brain

Many authors in the field of numerical cognition have adopted a rather nativist view that all humans share the intuition that numbers map onto space and, more specifically, that an oriented left-to-right mental number line (MNL) is localized bilaterally in the intraparietal sulcus of the human brain. We review results from archaeological and historical (diachronic) studies as well as cross-cultural (synchronic) ones and contends that these claims are not well founded. The data actually suggest that the MNL is not innate. We argue that the MNL—and number-to-space mappings in general—emerges outside of natural selection proper requiring top-down dynamics that are culturally and historically mediated through high-order cognitive mechanisms such as fictive motion, conceptual mappings, and external representational media. These mechanisms, which are not intrinsically numerical and usually are acquired through education, are not genetically determined and are biologically realized through the systematic consolidation of specific brain phenotypes that support number-to-space mappings.

How social influence can undermine the wisdom of crowd effect

Social groups can be remarkably smart and knowledgeable when their averaged judgements are compared with the judgements of individuals. Already Galton [Galton F (1907) Nature 75:7] found evidence that the median estimate of a group can be more accurate than estimates of experts. This wisdom of crowd effect was recently supported by examples from stock markets, political elections, and quiz shows [Surowiecki J (2004) The Wisdom of Crowds]. In contrast, we demonstrate by experimental evidence (N = 144) that even mild social influence can undermine the wisdom of crowd effect in simple estimation tasks. In the experiment, subjects could reconsider their response to factual questions after having received average or full information of the responses of other subjects. We compare subjects' convergence of estimates and improvements in accuracy over five consecutive estimation periods with a control condition, in which no information about others' responses was provided. Although groups are initially "wise," knowledge about estimates of others narrows the diversity of opinions to such an extent that it undermines the wisdom of crowd effect in three different ways. The "social influence effect" diminishes the diversity of the crowd without improvements of its collective error. The "range reduction effect" moves the position of the truth to peripheral regions of the range of estimates so that the crowd becomes less reliable in providing expertise for external observers. The "confidence effect" boosts individuals' confidence after convergence of their estimates despite lack of improved accuracy. Examples of the revealed mechanism range from misled elites to the recent global financial crisis.

Exceptional abilities in the spatial representation of numbers and time: insights from synesthesia

In the study of basic and high-level cognitive functions, neuroscientists, psychologists, and philosophers have tended to focus on normal psychological processes and on deficits in these processes, whereas the study of exceptional abilities has been largely neglected. Here the authors emphasize the value of researching exceptional abilities. They make the case that studies of exceptional representations, such as of time, number, and space in synesthesia, can provide us with insights regarding the nature of the neurocognitive mechanisms of these dimensions, as well as their developmental, evolutionary, and cultural origins.

Cultural Variation in Numeration Systems and Their Mapping Onto the Mental Number Line

The ability to exactly assess large numbers hinges on cultural tools such as counting sequences and thus offers a great opportunity to study how culture interacts with cognition. To obtain a more comprehensive picture of the cultural variance in number representation, this article argues for the inclusion of cross-linguistic analyses. It scrutinizes the specific counting systems of Polynesian and Micronesian languages that were once derived from an abstract and regular system by extension in three dimensions. The linguistic origins, cognitive properties, and cultural context of these specific counting systems are analyzed, and their implications for the nature of a (putative) mental number line are discussed.

Mathematical skills in 3- and 5-year-olds with spina bifida and their typically developing peers: a longitudinal approach

Preschoolers with spina bifida (SB) were compared to typically developing (TD) children on tasks tapping mathematical knowledge at 36 months (n = 102) and 60 months of age (n = 98). The group with SB had difficulty compared to TD peers on all mathematical tasks except for transformation on quantities in the subitizable range. At 36 months, vocabulary knowledge, visual-spatial, and fine motor abilities predicted achievement on a measure of informal math knowledge in both groups. At 60 months of age, phonological awareness, visual-spatial ability, and fine motor skill were uniquely and differentially related to counting knowledge, oral counting, object-based arithmetic skills, and quantitative concepts. Importantly, the patterns of association between these predictors and mathematical performance were similar across the groups. A novel finding is that fine motor skill uniquely predicted object-based arithmetic abilities in both groups, suggesting developmental continuity in the neurocognitive correlates of early object-based and later symbolic arithmetic problem solving. Models combining 36-month mathematical ability and these language-based, visual-spatial, and fine motor abilities at 60 months accounted for considerable variance on 60-month informal mathematical outcomes. Results are discussed with reference to models of mathematical development and early identification of risk in preschoolers with neurodevelopmental disorder.

Under- and Over-Estimation

Over the last 30 years, numerical estimation has been largely studied. Recently, Castronovo and Seron (2007) proposed the bi-directional mapping hypothesis in order to account for the finding that dependent on the type of estimation task (perception vs. production of numerosities), reverse patterns of performance are found (i.e., under- and over-estimation, respectively). Here, we further investigated this hypothesis by submitting adult participants to three types of numerical estimation task: (1) a perception task, in which participants had to estimate the numerosity of a non-symbolic collection; (2) a production task, in which participants had to approximately produce the numerosity of a symbolic numerical input; and (3) a reproduction task, in which participants had to reproduce the numerosity of a non-symbolic numerical input. Our results gave further support to the finding that different patterns of performance are found according to the type of estimation task: (1) under-estimation in the perception task; (2) over-estimation in the production task; and (3) accurate estimation in the reproduction task. Moreover, correlation analyses revealed that the more a participant under-estimated in the perception task, the more he/she over-estimated in the production task. We discussed these empirical data by showing how they can be accounted by the bi-directional mapping hypothesis ( Castronovo & Seron, 2007 ).

Using Mental Representations of Space When Words Are Unavailable: Studies of Enumeration and Arithmetic in Indigenous Australia

Here the authors describe the nature and use of spatial strategies in a standard nonverbal addition task in two groups of children who speak only languages in which counting words are not available, as compared with children who were raised speaking English. They tested speakers of Warlpiri and Anindilyakwa aged between 4 and 7 years at two remote sites in the Northern Territory of Australia. These children used spatial strategies extensively and were significantly more accurate when they did so. English-speaking children used spatial strategies very infrequently but relied on an enumeration strategy supported by counting words to do the addition task. The main spatial strategy exploited the known visual memory strengths of Aboriginals and involved matching the spatial patterns of the augend and addend sets. These findings suggest that counting words, far from being necessary for exact arithmetic, offer one strategy among others. They also suggest that spatial models for numbers do not need to be one-dimensional vectors, as in a mental number line (MNL), but can be at least two-dimensional.

The cultural constitution of cognition: taking the anthropological perspective

To what extent is cognition affected by culture? And how might cognitive science profit from an intensified collaboration with anthropology in exploring this issue? In order to answer these questions, we will first give a brief description of different perspectives on cognition, one that prevails in most cognitive sciences – particularly in cognitive psychology – and one in anthropology. Three basic assumptions of cognitive science regarding the separability of content and process, the context-independence of processing, and the culture-independence of processing will then be discussed. We argue that these assumptions need to be questioned and scrutinized cross-culturally. A thorough examination of these issues would profit considerably from collaboration with anthropologists, not only by enabling deeper insight into the cultures under scrutiny, but also by synergistic effects that would allow for a more comprehensive understanding of human cognition.