On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical (...) confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

We consider connections between number sense—the ability to judge number—and the interpretation of natural language quantifiers. In particular, we present empirical evidence concerning the neuroanatomical underpinnings of number sense and quantifier interpretation. We show, further, that impairment of number sense in patients can result in the impairment of the ability to interpret sentences containing quantifiers. This result demonstrates that number sense supports some aspects of the language faculty.

In an interview setting, 21 6th‐graders from four public schools in south Taiwan were asked a series of questions, related to whole and decimal numbers, designed to assess their number sense. Eight students were randomly selected from both low and middle performance levels, and five were selected from a high‐performance level, according to their mathematical capabilities. Results indicated that, regardless of performance level, very few number sense strategies were used. The evidence also revealed that Taiwanese students (...) tended to apply rule‐based methods and standard written algorithms to explain their reasoning. Their inclination to use paper‐and‐pencil procedures narrowed their thinking and reasoning powers: this heavy reliance on written algorithms seemed to be a major impediment to the development of number sense. (shrink)

‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including (...) language; the existence of a homology between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

What Stanislas Debaene dubs "the number sense" is a natural ability humans share with other animals, enabling us to "count" to four virtually instantaneously. This so-called "accumulator" provides "a direct intuition of what numbers mean" (p. 5). Beyond four, our ability to perceive numbers becomes approximate, though concepts enable us to move beyond approximation. Because humans typically learn number concepts in early childhood, we easily forget that our brains retain the number sense throughout life. This (...) book examines the biological basis for this intuitive ability, with nine chapters organized into three readily graspable groups of three. Aside from its frustrating lack of a clear referencing system (making it difficult or impossible to trace Debaene's sources), the book is a pleasure to read. (shrink)

‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including (...) language; the existence of a homology between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation (the ‘number line’ ) and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

On a now orthodox view, humans and many other animals are endowed with a “number sense”, or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques, with critics maintaining either that numerical content is absent altogether, or else that some primitive analog of number (‘numerosity’) is represented as opposed to number itself. We distinguish three arguments for these claims – the arguments from congruency, confounds, and imprecision (...) – and show that none succeed. We then highlight positive reasons for thinking that the ANS genuinely represents numbers. The upshot is that proponents of the orthodox view should not feel troubled by recent critiques of their position. (shrink)

Against Clarke and Beck's proposal that the approximate number system represents natural and rational numbers, I suggest that the experimental evidence is better accommodated by the thesis that the ANS represents cardinality comparisons. Cardinality comparisons do not stand in arithmetical relations and being able to apply them does not involve basic arithmetical concepts and operations.

Clarke and Beck's view that numbers are both second-order and sensible is based on an empirically dubious claim, which is required to show that what they call the “weak sensitivity principle” is satisfied. The explanatory benefits that they say are gained by positing a sensory relation to numbers are also gained by positing such a relation to multitudes of objects.

‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including (...) language; the existence of a homology between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation (the ‘number line’ ) and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including (...) language; the existence of a homology between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation (the ‘number line’ ) and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

Although some cognitive studies provided reasons that children with low socioeconomic status (SES) showed poor mathematical achievements, there was no explicit evidence to directly explain the root of lagged performance in children with low SES. Therefore, the present study explored the differences in neural correlates in the process of symbolic magnitude comparison between children with different SES by the event-related potentials (ERP). A total of 16 second graders from low SES families and 16 from middle/high SES families participated in this (...) study. According to the results of anterior N1 (early attention) and P2 (extraction of numerical meaning) over frontal region, the differences among children with different SES were manifested as differences in general neural activities in terms of attention and top-down cognitive control. In the late stage of cognitive processing, there was no significant difference in the average amplitude of the LPC between the children with different SES, indicating that low SES did not influence the information encoding and memory updating of numerical representation responsible by the parietal lobe. The educational implications of this study were mentioned in the discussion. (shrink)

Generalized quantifiers are functions from pairs of properties to truth-values; these functions can be used to interpret natural language quantifiers. The space of such functions is vast and a great deal of research has sought to find natural constraints on the functions that interpret determiners and create quantifiers. These constraints have demonstrated that quantifiers rest on number and number sense. In the first part of the paper, we turn to developing this argument. In the remainder, we report (...) on work in neurobiology that test the empirical claims of the theory. In particular, we look at fMRI experiments and behavioral experiments with various patient populations that support the intimate connection between natural language quantification and number sense. (shrink)

Twenty classes in ten schools with 627 sixth?grade students in five cities in Taiwan participated in this study. The research provides information on the performance differences among written computation, pictorial representation, symbolic representation and number sense. The results of One?way ANOVA analysis indicate that significant difference was found among WCT, PRT, SRT and NST tests, with F=536.327, p=0.000. The a posteriori comparisons show for each pair (WCT vs PRT, WCT vs SRT, WCT vs NST, PRT vs SRT and (...) SRT vs NST) significant difference at the 0.001 level (p=0.000), except for the pair comparison between PRT and NST. This implies that these Taiwanese students were highly skilled in written computation but their written skills were not equally transferred to use of non?computational paths that depended on symbolic representation, pictorial representation and number sense to solve similar problems. (shrink)

The main purpose of this study was to investigate the number sense performance of 3rd?graders in Taiwan, and to diagnose areas of weakness or deficiency in number sense development. A total of 808 3rd?graders participated in this study. The results indicated that these students did not perform well on each of the five number sense components (correct rates approx. 34%), and they appeared worst on the performance of ?Judging the reasonableness of computational results?. Boys (...) and girls did not show any appreciable difference in their ability to solve number sense problems. The importance of number sense should be highlighted both by teachers and in textbooks and more time and opportunity provided for students to work on this type of exercise at lower grade levels. This would require that ?drill and practice? exercises in mathematics should not indeed be over? taught, and the teaching of number sense to children should begin as early as possible. (shrink)

ABSTRACTThis study examined the performance of sixth graders in Hong Kong on a number sense three-tier test and identified their possible misconceptions about number sense. The NSTTT compri...

There is ample evidence from literature and clinical practice indicating mathematical difficulties in individuals with ADHD, even when there is no concomitant diagnosis of developmental dyscalculia. What factors underlie these difficulties is still an open question. Research on dyscalculia and neurotypical development suggests visual perception of numerosity (the number sense) as a building block for math learning. Participants with lower numerosity estimation thresholds (higher precision) are often those with higher math capabilities. Strangely, the role of numerosity perception in (...) math skills in ADHD has been neglected, leaving open the question whether math difficulties in ADHD also originate from a deficitary visual number sense. In the current study we psychophysically measured numerosity thresholds and accuracy in a sample of children/adolescents with ADHD, but not concomitant dyscalculia (N = 20, 8–16 years). Math abilities were also measured by tasks indexing different mathematical competences. Numerosity performance and math scores were then compared to those obtained from an age-matched control group (N = 20). Bayesian statistics indicated no difference between ADHD and controls on numerosity perception, despite many of the symbolic math tasks being impaired in participants with ADHD. Moreover, the math deficits showed by the group with ADHD remained substantial even when numerosity thresholds were statistically regressed out. Overall, these results indicate that math difficulties in ADHD are unlikely to originate from an impaired visual number sense. (shrink)

This book announces a new academic discipline, the “cognitive science of mathematics” (p. XI), by demonstrating how an empirical examination of the ideas underlying our use of mathematical symbols and calculations must employ metaphors grounded in the “embodied mind.” The authors ruthlessly attack what they dub “The Romance of Mathematics” (p. XV), their metaphor for any approach that treats mathematics as grounded in an abstract, disembodied yet objective reality that mysteriously provides the essential structure to the natural, human world.

Cognitive Science, Volume 46, Issue 7, July 2022.

What Stanislas Debaene dubs "the number sense" is a natural ability humans share with other animals, enabling us to "count" to four virtually instantaneously. This so-called "accumulator" provides "a direct intuition of what numbers mean". Beyond four, our ability to perceive numbers becomes approximate, though concepts enable us to move beyond approximation. Because humans typically learn number concepts in early childhood, we easily forget that our brains retain the number sense throughout life. This book examines (...) the biological basis for this intuitive ability, with nine chapters organized into three readily graspable groups of three. Aside from its frustrating lack of a clear referencing system, the book is a pleasure to read. (shrink)

Many philosophers still countenance senses or meanings in the broadly Fregean vein. However, it is difficult to posit the existence of senses without positing quite a lot of them, including at least one presenting every entity in existence. I discuss a number of Cantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and nontraditional understanding of senses, and to what extent they fare better in solving the problems, are (...) also discussed. In the end, it is concluded that one must divide senses into various ramified-orders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic. (shrink)

A concise introduction to personal and professional numeracy skills, helping readers to become more mathematically competent.

Journal Name: Apeiron Issue: Ahead of print.

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence (...) that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)