Numerical Cognition

Numerical elaboration and the extension of numbers to non-tangible domains such as time have been linked to cultural complexity in several studies. However, the reasons for this phenomenon remain insufficiently explored. In the present analysis, Material Engagement Theory, an emerging perspective in cognitive archaeology, provides a new perspective from which to reinterpret the cultural nexus in which quantification develops. These insights are then applied to representative Neolithic, Upper Palaeolithic, and Middle Stone Age artifacts used for quantification: clay tokens from Neolithic Mesopotamia, notched tallies from the European Upper Palaeolithic, hand stencils with possible finger-counting patterns as documented at Cosquer and Gargas, and stringed beads from Blombos Cave in South Africa.

In this study, we investigated in school-age children the relationship among mathematical performance, the perception of numerosity (discrimination and mapping to number line), and sustained visual attention. The results (on 68 children between 8 and 11 years of age) show that attention and numerosity perception predict math scores but not reading performance. Even after controlling for several variables, including age, gender, nonverbal IQ, and reading accuracy, attention remained correlated with math skills and numerosity discrimination. These findings support previous reports showing the interrelationship between visual attention and both numerosity perception and math performance. It also suggests that attentional deficits may be implicated in disturbances such as developmental dyscalculia.

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This « genealogy of concepts », so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis by this too often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to the philosophical frame as well as to the some recent advances in Cognition that support our claim, the cognitive origin and the constitutive role of mathematical intuition. Now, as the keenest among the founding fathers would say, let's put aside this "heuristic" and rather focus our attention on the a posteriori reconstruction of the logical certitude of proof. An absolutely indispensable program, as we were saying, at the beginning of the XXth century, following the technical richness as well as confusion of the XIXth -a century having produced many results, among which several were false or unproven or badly stated -, a program, however, which excluded from foundational analysis any scientific examination of the constitutive process of mathematical concepts and structures. Such is the aim of the new project regarding the cognitive foundations of Mathematics, which is also an epistemological project. It is certainly not a question of discarding proof, with its logical and formal components, but simply of steering away from the formal (and computational) monomania, formerly justified and which dominated the previous century. It produced these marvelous logico-formal machines surrounding us, acting without meaning. The analysis of the constitution of sense and meaning in Mathematics, through cognition and history, is the purpose of the investigations in Mathematical Cognition, also in order to compensate the provable incompleteness of formalisms.

Even though mathematics is considered one of the most abstract domains of human cognition, recent work on embodiment of mathematics has shown that we make sense of mathematical concepts by using insights and skills acquired through bodily activity. Fingers play a significant role in many of these bodily interactions. Finger-based interactions provide the preliminary access to foundational mathematical constructs, such as one-to-one correspondence and whole-part relations in early development. In addition, children across cultures use their fingers to count and do simple arithmetic. There is also some evidence for an association between children’s ability to individuate fingers (finger gnosis) and mathematics ability. Paralleling these behavioral findings, there is accumulating evidence for overlapping neural correlates and functional associations between fingers and number processing. In this paper, we synthesize mathematics education and neurocognitive research on the relevance of ...

In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children's representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of 'number sense'. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children's mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children's performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring 'number sense' tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement.

& Numerical magnitude processing is an essential everyday skill. Functional brain imaging studies with human adults have repeatedly revealed that bilateral regions of the intraparietal sulcus are correlated with various numerical and mathematical skills. Surprisingly little, however, is known about the development of these brain representations. In the present study, we used functional neuroimaging to compare the neural correlates of nonsymbolic magnitude judgments between children and adults. Although behavioral performance was similar across groups, in comparison to the group of children the adult participants exhibited greater effects of numerical distance on the left intraparietal sulcus. Our findings are the first to reveal that even the most basic aspects of numerical cognition are subject to age-related changes in functional neuroanatomy. We propose that developmental impairments of number may be associated with atypical specialization of cortical regions underlying magnitude processing. & D

W e examine two questions: Does the roundness or precision of prices bias magnitude judgments? If so, do these biased judgments affect buyer behavior? Results from five studies suggest that buyers underestimate the magnitudes of precise prices. We term this the precision effect. The first three studies are laboratory experiments designed to test the existence of the precision effect and examine the underlying psychological processes. In Study 1, we find that precise prices are judged to be smaller than round prices of similar magnitudes. For example, participants in this experiment incorrectly judged $395,425 to be smaller than $395,000. In Study 2, we show that precision is more likely to affect magnitude judgments under conditions of uncertainty. Study 3 demonstrates that manipulating prior experience with the pattern of roundness and precision in numbers can moderate the precision effect. Studies 4 and 5 examine whether the precision effect influences buyers' willingness to pay for negotiated purchases (e.g., houses). In Study 4, we conduct an experiment on a nationally representative sample of homeowners to demonstrate that participants are willing to pay more for houses when the sellers use precise (e.g., $364,578) instead of comparable round (e.g., $365,000) prices. In Study 5, we analyze data from residential real estate transactions in two separate markets and find that buyers pay higher sale prices when list prices are more precise. These empirical results enrich our understanding of the psychological processes that underlie price magnitude judgments and have substantive implications for buyer and seller behavior.

In this paper, I propose to sketch the main lines of some recent theoretical accounts of numerical cognition and evaluate their ability to explain the development of mathematically viable concepts of number (Dehaene 2011; Carey 2009; DeCruz 2008). I will try to show that, despite their differences, these all share a critical flaw in that they explain the development of number concepts by relying in part on numerical symbols. I will argue that explaining how number concepts develop in the brain by relying on the presence of mathematical symbols in the environment is akin to putting the cart before the ox, since such symbols cannot emerge without there being number concepts beforehand.

Title in English: 'Numbers in Simultaneous Interpreting: Objective and subjective difficulties.'
This experimental study investigates the causes for the high difficulty and error rate in the simultaneous interpretation of numbers. The book opens with a theoretical section, which proposes a framework for empirical inquiry including evaluation criteria and a model. It then shows the practical application of such framework to analysis of interpretations from English and from German into Italian by Master's degree students of Conference Interpreting. The study advances hypotheses on the dependency of quality upon training and lays the groundwork to develop targeted curriculum resources.

Numbers and prices can be processed and encoded in three different forms: 1) visual [based on their written form in Arabic numerals (e.g., 72)], 2) verbal [based on spoken word-sounds (e.g., "seventy" and "two"), and 3) analog (based on judgments of relative "size" or amount (e.g., more than 70 but less than 80)]. In this paper, we demonstrate that including commas (e.g., $1599 vs. $1599) and cents (e.g., $1599.85 vs. $1599) in a price's Arabic written form (i.e., how it is perceived visually) can change how the price is encoded and represented verbally in a consumer's memory. In turn, the verbal encoding of a written price can influence assessments of the numerical magnitude of the price. These effects occur because consumers non-consciously perceive that there is a positive relationship between syllabic length and numerical magnitude. Three experiments are presented demonstrating this important effect.

After the onset of formal schooling, little is known about the development of children's understanding of the arithmetic concepts of inversion and associativity. On problems of the form a + b À b (e.g., 3 + 26 À 26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a + b À c (e.g., 3 + 27 À 23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.

The study of numerical magnitude processing
provides a unique opportunity to examine interactions between phylogenetically ancient systems of semantic repre- sentations 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 process- ing of Hindu–Arabic numerals differs between literate and illiterate individuals. These findings reveal that nonsymbolic numerical magnitude processing is unaffected by encultura- tion, while the processing of numerical symbols is modulated by literacy

This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., three, seven, two hundred, etc.). I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey (2009). In this framework, the counting list (‘one,’ ‘two,’ ‘three,’ etc.) and the counting routine (i.e., reciting the list and pointing to objects, one at a time) form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers (e.g., There are 24 children in my class, so I need to bring 24 cupcakes for the party.) A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information (i.e., large, exact cardinal values) that they had no way of representing before.

The Material Origin of Numbers examines how number concepts are realized, represented, manipulated, and elaborated. Utilizing the cognitive archaeological framework of Material Engagement Theory and culling data from disciplines including neuroscience, ethnography, linguistics, and archaeology, the work offers a methodologically rich study of numbers and number concepts in the ancient Near East from the late Upper Paleolithic Period through the Bronze Age. This project has received funding from the Clarendon Fund at the University of Oxford, as well as the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793.

Research on numerical cognition with infants, children, adults, cross-cultural studies and animal studies converge on the conclusion that there are two distinct systems for the perception of numerical quantity: A small-number system (1~3) invoking parallel individuation, or “subitizing”, and a large-number system (4+) that is based on Weberian magnitude estimation. The present study employed high density EEG to investigate the neural basis of differentiation between small-number (1~3), and large-number (4~6) perception. During EEG data collection, participants were presented with dot pattern stimuli containing 1 to 6 dots of varying size. They were instructed to press a key when they detected a change in the number of dots presented and were rewarded for correctly detecting changes. ERP analyses were conducted over the left and right occipital-temporal-parietal junction. For the early ERP component (N1; 160~180 ms), we observed that there was separation of ERPs within the subitizing range (1~3) but not beyond (4~6). A second later positivity (P3) was found in Cz and neighboring electrodes, that was associated with change detection. Numerical changes were categorized as “within small” (1~2, 2~1, 2~3, 3~2), “within large” (4~5,5~4,5~6,6~5). Changes that crossed the small to large boundary were labeled “small-to-large” and “large-to- small”. The P3 was observed for all change categories except for “within large.” These data, taken together, suggest a neural basis for the differentiation of small vs. large number perception at early stages of processing, and a later stage that involves more semantic numerical processing that is employed in change detection task.

Developmental dyscalculia is a learning disability that affects the acquisition of knowledge about numbers and arithmetic. It is widely assumed that numeracy is rooted on the ''number sense", a core ability to grasp numerical quantities that humans share with other animals and deploy spontaneously at birth. To probe the links between number sense and dyscalculia, we used a psychophysical test to measure the Weber fraction for the numerosity of sets of dots, hereafter called number acuity. We show that number acuity improves with age in typically developing children. In dyscalculics, numerical acuity is severely impaired, with 10-year-old dyscalculics scoring at the level of 5-year-old normally developing children. Moreover, the severity of the number acuity impairment predicts the defective performance on tasks involving the manipulation of symbolic numbers. These results establish for the first time a clear association between dyscalculia and impaired ''number sense", and they may open up new horizons for the early diagnosis and rehabilitation of mathematical learning deficits.

The dataset contains Neolithic artifacts from the Ancient Near East identified as tokens. Data on 8,477 tokens were drawn from Schmandt-Besserat (1992), Before Writing, Vol. II: A Catalogue of Near Eastern Tokens. This catalogue was converted to electronic format with the permission of the author and her publisher for analyses conducted as part of a doctoral research project at the University of Oxford (2013–2016). With Schmandt-Besserat’s permission and assistance, her data were expanded with over 2,352 new finds sourced from 30 documents. The data were analyzed in:
Overmann, Karenleigh A. (2016). Materiality in numerical cognition: Material Engagement Theory and the counting technologies of the Ancient Near East. DPhil thesis, University of Oxford.
Overmann, Karenleigh A. (2019). The material origin of numbers: Insights from the archaeology of the Ancient Near East. Gorgias Studies in the Ancient Near East 14. Piscataway, NJ: Gorgias Press.

Number sense is critical to the development of higher order mathematic abilities. However, some children have difficulty acquiring these fundamental skills and the knowledge base of effective interventions/remediation is relatively limited. Based on emerging neuro-scientific research which has identified the association between finger movement/representation and numerical cognition, this one-year study examined the effectiveness of a finger calculation program for at-risk and non-at-risk school children. Specifically, the study sought to determine the impact of learning Chisanbop (Korean finger calculation method) on the number sense, computational, quantitative concepts, and problem solving skills, as well as attitudes towards mathematics, of children at the grade 2 and grade 5 levels. The findings indicate a significant impact on attitudes for math for at-risk grade 2 students.

How is number-concept acquisition related to overall language development? Experiments 1 and 2 measured number-word knowledge and general vocabulary in a total of 59 children, ages 30–60 months. A strong correlation was found between number-word knowledge and vocabulary, independent of the child’s age, contrary to previous results (D. Ansari et al., 2003). This result calls into question arguments that (a) the number-concept creation process is scaffolded mainly by visuo-spatial development and (b) that language only becomes integrated after the concepts are created (D. Ansari et al., 2003). Instead, this may suggest that having a larger nominal vocabulary helps children learn number words. Experiment 3 shows that the differences with previous results are likely due to changes in how the data were analyzed.

Only recently has the complex anatomo-functional system underlying numerical cognition become accessible to evaluation in the living brain. We identified 27 studies investigating brain connectivity in numerical cognition. Despite considerable heterogeneity regarding methodological approaches, populations investigated, and assessment procedures implemented, the results provided largely converging evidence regarding the underlying brain connectivity involved in numerical cognition. Analyses of both functional/effective as well as structural connectivity have consistently corroborated the assumption that numerical cognition is subserved by a frontoparietal network including (intra)parietal as well as (pre)frontal cortex sites. Evaluation of structural connectivity has indicated the involvement of fronto-parietal association fibers encompassing the superior longitudinal fasciculus dorsally and the external capsule/extreme capsule system ventrally. Additionally, commissural fibers seem to connect the bilateral intraparietal sulci when number magnitude information is processed. Finally, the identification of projection fibers such as the superior corona radiata indicates connections between cortex and basal ganglia as well as the thalamus in numerical cognition. Studies on functional/effective connectivity further indicated a specific role of the hippocampus. These specifications of brain connectivity augment the triple-code model of number processing and calculation with respect to how gray matter areas associated with specific number-related representations may work together.

Research on the mental representation of numbers has focused on a horizontally aligned mental number line, but more and more findings have begun to implicate a vertical orientation as well. We investigate the relationship between these two orientations when people generate random numbers. In the horizontal condition, people generated larger numbers when they looked right as opposed to left. In the vertical condition, people generated larger numbers when they looked up as opposed to down. We present two main results based on analyses that compare the two spatial orientations. First, we show that the vertical effect was stronger than the horizontal one. Second, we show a weak correlation between the vertical and the horizontal effect, potentially suggesting a shared underlying mechanism.

Ellis et al. (2021, this issue; in the following abbreviated as EEA) conducted a conceptual replication and extension of a recent meta-analysis relating number line estimation to broader mathematical competence (Schneider et al., 2018; in the following: SEA). EEA pooled and analyzed data from seven new studies and compared the results to SEA's findings. As authors of SEA we briefly comment on three aspects of the conceptual replication study.

The study of numerical cognition has undergone tremendous progress in recent years, accumulating scores of data on cognitive systems that could be involved in the uniquely human ability to practice formal arithmetic. Among the important questions tackled by this burgeoning domain of research is what happens to limited cognitive systems that we share with many animal species to allow us to develop arithmetically-viable numerical content. While answers to this questions have varied, most have appealed to the presence of culturally-inherited extracranial cognitive support to explain how numerical content emerges from our innate cognitive machinery. In this paper I challenge this externalist approach and argue that we should favor explanations that focus on cognitive processing inside the head. To support my claim, I take a look at research into Spontaneous Focusing on Numerosity (SFON) to show that internalism can explain some aspects of the development of numerical cognition without needing to factor in the cultural setting in which this development takes place.

I respond to three responses to my 2015 Current Anthropology article, “Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number.” This study examined the categorical and geographical distribution of lexical numbers, also known as counting numbers, and grammatical number, the ability to linguistically distinguish singular and plural. Both these features of language conform to the perceptual experience of quantity, which consists of subitization, the ability to rapidly and unambiguously identify one, two, and three, and magnitude appreciation, the ability to appreciate bigger and smaller in the numerical quantity of groups when the difference lies above a threshold of noticeability. My reply to Sutliff disagrees with her contention that mathematical ideas are mentally innate on the grounds that this ignores their explicit construction through the interaction of human psychological, physiological, and behavioral abilities with materiality. My reply to Read expands on the idea that language obscures cross-cultural conceptual variability in number concepts because everything that translates as “three” does not necessarily have the same numerical properties. Finally, my reply to Everett notes that investigating numerical origins means discarding the deeply entrenched assumption of linguistic primacy on the grounds that material forms make numerical intuitions tangible, visible, and manipulable in ways that language cannot and, moreover, provide an alinguistic bootstrap mechanism that accounts for the emergence of both concepts of number and words for the concepts.

The Material Origin of Numbers examines how number concepts are realized, represented, manipulated, and elaborated. Utilizing the cognitive archaeological framework of Material Engagement Theory and culling data from disciplines including neuroscience, ethnography, linguistics, and archaeology, the work offers a methodologically rich study of numbers and number concepts in the ancient Near East from the late Upper Paleolithic Period through the Bronze Age. This project has received funding from the Clarendon Fund at the University of Oxford, as well as the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793.

This microgenetic study investigated the discovery and development of the multiplication and division concept of inversion. Little is known about multiplicative concepts relative to additive concepts, including the inversion concept. Grade 6 participants (mean age ϭ 11 years 6 months) solved multiplication and division inversion problems (e.g., d ϫ e/e) for several weeks. In the final week they solved inversion, modified inversion (e.g., e ϫ d/e), and lure problems (e.g., d/e ϫ d) to investigate transfer of knowledge. Despite years of formal arithmetic instruction and repeated exposure to inversion problems, over a third of the participants failed to discover the inversion-based shortcut whereas another third used the shortcut almost exclusively. Almost all participants had difficulty appropriately generalising the inversion concept. Current theories of mathematical understanding may need to be modified to include the developmental complexities of multiplicative concepts.

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals 1; @0; !, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems.

Number systems constitute one of the major domains in which language has been invoked as a source of variation in thought or cognition. The notion that the features of a language's numeral system index cognitive complexity in mathematics has been pervasive in anthropological linguistics, from nineteenth century unilinear evolutionists to contemporary neo-Whorfians. In particular, the extreme case of languages with small numerical vocabularies have attracted enormous scholarly interest, but other features for which representational effects on cognition are claimed include systemic irregularity and the presence of multiple parallel numeral systems (numeral classifiers and object-specific counting). Quite independently but relatedly, the comparison of graphic numerical notations has inferred cognitive advantages directly from notation, such as the idea that the Roman numerals limited Western mathematical progress. The question of cognitive effects of language is interwoven with issues of social complexity; in place of a pure relativistic language-thought relationship, the discussion has been, and continues to be framed through a triad of language structure-cognition-social structure. Recognizing the enormous cross-cultural variation in lexical numerals and numerical notations, how can we best evaluate the extent to which, and more importantly, the processes by which, they affect numerical cognition? An activity-based explanatory model in which materiality, discourse, and practice mutually engage to constitute knowledge systems allows us to move past the presumption that language structure has direct cognitive effects, without denying that there are linguistic patterns of real interest for future inquiry into numerical cognition.

The idea the New Zealand Māori once counted by elevens has been viewed as a cultural misunderstanding originating with a mid-nineteenth-century dictionary of their language. Yet this “remarkable singularity” had an earlier, Continental origin, the details of which have been lost over a century of transmission in the literature. The affair is traced to a pair of scientific explorers, René-Primevère Lesson and Jules Poret de Blosseville, as reconstructed through their publications on the 1822–1825 circumnavigational voyage of the Coquille, a French corvette. Possible explanations for the affair are briefly examined, including whether it might have been a prank by the Polynesians or a misunderstanding or hoax on the part of the Europeans. Reasons why the idea of counting by elevens remains topical are discussed. First, its very oddity has obscured the counting method actually used—setting aside every tenth item as a tally. This “ephemeral abacus” is examined for its physical and mental efficiencies and its potential to explain aspects of numerical structure and vocabulary (e.g., Mangarevan binary counting; the Hawaiian number word for twenty, iwakalua), matters suggesting material forms have a critical if underappreciated role in realising concepts like exponential value. Second, it provides insight into why it can be difficult to appreciate highly elaborated but unwritten numbers like those found throughout Polynesia. Finally, the affair illuminates the difficulty of categorising number systems that use multiple units as the basis of enumeration, like Polynesian pair-counting; potential solutions are offered. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793.

Malgré l’imposant corpus soutenant l’existence du “Approximate Number System” (ANS), certains proposent d’expliquer notre comportement dans des études sur la cognition numérique en se fiant à un système dédié au traitement de grandeurs continues comme la durée, la taille, la luminosité, etc. Selon ces sceptiques de l’ANS, les méthodes expérimentales utilisées pour étudier l’ANS ne nous permettent pas d’identifier ce système comme étant responsable de notre comportement dans de telles études, puisque la quantité d’objets discrets varie toujours avec une autre grandeur continue. Dans cet article, je soutiens qu’un tel scepticisme n’est pas tenable pour des raisons conceptuelles, étant donné l’opposition fondamentale entre le contenu mental associé au discret et celui associé au continu. Pour soutenir ma thèse, je propose un bref résumé de la conception de la relation entre le continu et le discret dans l’histoire de la philosophie, en particulier telle que conçue par le mathématicien intutionniste L.E.J Brouwer, selon qui il est impossible de construire un continu à partir d’éléments discrets, et vice versa.

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children's mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6-to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.

In this study adults performed numerical and physical size judgments on a symbolic (Arabic numerals) and non-symbolic (groups of dots) size congruity task. The outcomes would reveal whether a size congruity effect (SCE) can be obtained irrespective of notation. Subsequently, 5-year-old children performed a physical size judgment on both tasks. The outcomes will give a better insight in the ability of 5-year-olds to automatically process symbolic and non-symbolic numerosities. Adult performance on the symbolic and non-symbolic size congruity tasks revealed a SCE for numerical and physical size judgments, indicating that the non-symbolic size congruity task is a valid indicator for automatic processing of nonsymbolic numerosities. Physical size judgments on both tasks by children revealed a SCE only for non-symbolic notation, indicating that the lack of a symbolic SCE is not related to the mathematical or cognitive abilities required for the task but instead to an immature association between the number symbol and its meaning.

Individual differences in mathematics performance may stem from domain-general factors like working memory and intelligence. Parietal and frontal brain areas have been implicated in number processing, but the influence of such cognitive factors on brain activity during mathematics processing is not known. The relationship between brain mechanisms during symbolic and nonsymbolic number comparison and individual differences in working memory, nonverbal IQ, and mathematics achievement were investigated to determine possible associations with behavior and brain function. A number of brain areas showed correlations with working memory and number processing. For symbolic digits, working memory showed a positive relationship with brain activity in a network of bilateral parietal, temporal, and right frontal regions. For nonsymbolic dot arrays, working memory showed a negative relationship with several parietal and frontal brain areas. This relationship indicates differences between behavioral and brain function measures and points to the importance of working memory and basic number processing.

How should we evaluate the merit of written numeral systems? The
present ubiquity of the Hindu-Arabic (Western) numerals might
suggest that narrow considerations of efficiency have promoted the
convergence of numerical traditions on a single, superior solution.
Comparing the historical evolution of numerical notations to the
history of writing systems suggests, instead, that a host of social
factors influence the adoption, transmission, retention and
replacement of numeral systems. The wide range of contextual uses
and functions of written numerals belie any simple explanation of the
choices underlying their abandonment. Following the criteria
outlined by Coulmas, a sociolinguistic model is proposed in which a
wide variety of technical, graphic and cultural factors must be
considered in order to fully explain the historical record.

Contemporary comparative cognition has a large repertoire of animal models and methods, with concurrent theoretical advances that are providing initial answers to crucial questions about human cognition. What cognitive traits are uniquely human? What are the speciestypical inherited predispositions of the human mind? What is the human mind capable of without certain types of specific experiences with the surrounding environment? Here, we review recent findings from the domains of space, time and number cognition. These findings are produced using different comparative methodologies relying on different animal species, namely birds and non-human great apes. The study of these species not only reveals the range of cognitive abilities across vertebrates, but also increases our understanding of human cognition in crucial ways.

Do numbers have gender? Wilkie and Bodenhausen (2012) examined this issue in a series of experiments on perceived gender. They examined the perceived gender of baby faces and foreign names. Arbitrary numbers presented with these faces and names influenced their perceived gender. Specifically, odd numbers connoted masculinity, while even numbers connoted femininity. In two new studies (total N = 315), we further examined the gendering of numbers. The first study examined explicit ratings of 1-digit numbers. We confirmed that odd numbers seemed masculine while even numbers seemed feminine. Although both men and women showed this pattern, it was more pronounced among women. We also examined whether this pattern holds for automatic as well as deliberated reactions. Results of an Implicit Association Test showed that it did, but only among the women. The implicit and explicit patterns of numerical gender ascription were moderately correlated. The second study examined explicit perception...

Recent research reveals a link between individual differences in mathematics achievement and performance on tasks that activate the approximate number system (ANS): a primitive
cognitive system shared by diverse animal species and by humans of all ages. Here we used a brief experimental paradigm to test one causal hypothesis suggested by this relationship: activation of the ANS may enhance children’s performance of symbolic arithmetic. Over 2 experiments, children who briefly practiced tasks that engaged primitive approximate numerical quantities performed better on subsequent exact, symbolic arithmetic problems than did children given other tasks involving comparison and manipulation of non-numerical magnitudes (brightness and length). The practice effect appeared specific to mathematics,
as no differences between groups were observed on a comparable sentence completion task. These results move beyond correlational research and provide evidence that the exercise of non-symbolic numerical processes can enhance children’s performance of symbolic mathematics.