The field of numerical cognition provides a fairly clear picture of the processes through which we learn basic arithmetical facts. This scientific picture, however, is rarely taken as providing a response to a much‐debated philosophical question, namely, the question of how we obtain number knowledge, since numbers are usually thought to be abstract entities located outside of space and time. In this paper, I take the scientific evidence on how we learn arithmetic as providing a response to the philosophical question (...) of how we obtain number knowledge. I reject the view that numbers are abstract entities located outside of space and time and, alternatively, derive from the scientific evidence a novel account of the nature of numbers. In this account, numbers are reifications of the counting procedure and arithmetic statements are seen as describing the functioning of counting and calculation techniques. (shrink)

This paper clarifies Merleau-Ponty’s original account of “higher-order” cognition as fundamentally embodied and enacted. Merleau-Ponty’s philosophy inspired theories that deemphasize overlaps between conceptual knowledge and motor intentionality or, on the contrary, focus exclusively on abstract thought. In contrast, this paper explores the link between Merleau-Ponty’s account of motor intentionality and his interpretations of our capacity to understand and interact productively with cultural symbolic systems. I develop my interpretation based on Merleau-Ponty’s analysis of two neuropathological modifications of motor intentionality, the case (...) of the brain-injured war veteran Schneider, and a neurological disorder known as Gerstmann’s syndrome. Building on my analysis of Schneider’s sensorimotor compensatory performances in relation to his limitations in the domains of algebra, geometry, and language usage, I demonstrate a strong continuity between the sense of embodiment and enaction at all these levels. Based on Merleau-Ponty’s interpretations, I argue that “higher-order” cognition is impaired in Schneider insofar as his injury limits his sensorimotor capacity to dynamically produce comparatively more complex differentiations of any given phenomenal structure. I then show how Merleau-Ponty develops and specifies his interpretation of Schneider’s intellectual difficulties in relation to the ambiguous role of the body, and in particular the hand, in Gerstmann’s syndrome. I explain how Merleau-Ponty defends the idea that sensorimotor and quasi-representational cognition are mutually irreducible, while maintaining that symbol-based cognition is a fundamentally enactive and embodied process. (shrink)

Mathematical cognition is one of the most important cognitive functions of human beings. The latest brain and cognitive research have shown that mathematical cognition is a system with multiple components and subsystems. It has phylogenetic root, also is related to ontogenetic development and learning, relying on a large-scale cerebral network including parietal, frontal and temporal regions. Especially, the parietal cortex plays an important role during mathematical cognitive processes. This indicates that language and visuospatial functions are both key to mathematical cognition. (...) All of those advances have important implications for basic mathematical education. (shrink)

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms ‘number,’ ‘numerosity,’ and ‘numerousness.’ Although the definitions he proposed were adopted by syeveral psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a “terminological chaos” in the field of numerical cognition. In this paper, I review Stevens’s distinctions to help bring order to this alleged chaos and to shed light on (...) two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I offer further support to Stevens’s distinction between numerosity and numerousness by showing that they are relational properties that emerge through different relationships between agents and their environments. The final conclusion is that, by adopting Stevens’s distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness. (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)

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

Current understanding of the development of quantity representations is based primarily on performance in the number‐line task. We posit that the data from number‐line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number‐line task: the production and estimation variations of the bounded and unbounded number‐line (...) tasks. Comparison of performance in the bounded and unbounded number‐line tasks provides a unique and direct way to assess the role of strategy in number‐line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number‐line task produce equivalent biases except for one: the estimation variation of the bounded number‐line task. We discuss the important implications of these findings. (shrink)

Although most deaf individuals could use sign language or sign/spoken language mix, hearing loss would still affect their language acquisition. Compensatory plasticity holds that the lack of auditory stimulation experienced by deaf individuals, such as congenital deafness, can be met by enhancements in visual cognition. And the studies of hearing individuals have showed that visual form perception is the cognitive mechanism that could explain the association between numerical magnitude processing and arithmetic computation. Therefore, we examined numerical magnitude processing and its (...) contribution to arithmetical ability in deaf adolescents, and explored the differences between the congenital and acquired deafness. 112 deaf adolescents and 58 hearing adolescents performed a series of cognitive and mathematical tests, and it was found there was no significant differences between the congenital group and the hearing group, but congenital group outperformed acquired group in numerical magnitude processing and arithmetic computation. It was also found there was a close association between numerical magnitude processing and arithmetic computation in all deaf adolescents, and after controlling for the demographic variables and general cognitive abilities, numerical magnitude processing could predict arithmetic computation in all deaf adolescents but not in congenital group. The role of numerical magnitude processing in deaf adolescents' mathematical performance should be paid attention in the training of arithmetical ability. (shrink)

Nonhuman primates represent the most significant extant species for comparative studies of cognition, including such complex phenomena as numerical competence, among others. Studies of numerical skills in monkeys and apes have a long, though somewhat sparse history, although questions for current empirical studies remain of great interest to several fields, including comparative, developmental, and cognitive psychology; anthropology; ethology; and philosophy, to name a few. In addition to demonstrated similarities in complex information processing, empirical studies of a variety of potential cognitive (...) limitations or constraints have provided insights into similarities and differences across the primate order, and continue to offer theoretical and pragmatic directions for future research. An historical overview of primate numerical studies is presented, as well as a summary of the 17‐year research history, including recent findings, of the Comparative Cognition Project at The Ohio State University Chimpanzee Center. Overall, the archival literature on number‐related skills and counting in nonhuman primates offers important implications for revising our thinking about comparative neuroanatomy, cross‐species (human/ape) cognitive similarities and differences, and the evolution of cognition represented by the primate continuum. (shrink)

Math and reading are related, and math problems are often accompanied by problems in reading. In the present study, we used a dimensional approach and we aimed to assess the relationship of reading and math with the cognitive skills assumed to underlie the development of math. The sample included 97 children from 4th and 5th grades of a primary school. Children were administered measures of reading and math, non-verbal IQ, and various underlying cognitive abilities of math. We also included measures (...) of phonological awareness and working memory. Two approaches were undertaken to elucidate the relations of the cognitive skills with math and reading. In the first approach, we examined the unique contributions of math and reading ability, as well as their interaction, to each cognitive ability. In the second approach, the cognitive abilities were taken to predict math and reading. Results from the first set of analyses showed specific effects of math on number sense and number system knowledge, whereas counting was affected by both math and reading. No math-by-reading interactions were observed. In contrast, for phonological awareness, an interaction of math and reading was found. Lower performing children on both math and reading performed disproportionately lower. Results with respect to the second approach confirmed the specific relation of counting, number sense, and number system knowledge to math and the relation of counting to reading but added that each math-related marker contributed independently to math. Following this approach, no unique effects of phonological awareness on math and reading were found. In all, the results show that math is specifically related to counting, number sense, and number system knowledge. The results also highlight what each approach can contribute to an understanding of the relations of the various cognitive correlates with reading and math. (shrink)

The present research study focuses on how the language of instruction has an impact on the mathematical thinking development as a consequence of using a language of instruction different from the students’ mother tongue. In CLIL (Content and Language Integrated Learning) academic content and a foreign language are leant at the same time, a methodology that is widely used in the schools in the present times. It is, therefore, our main aim to study if the language of instruction in second (...) language immersion programs influences the development of the first formal mathematical concepts. More specifically, if the learning of mathematical concepts in the early ages develops in a similar way if it is taught in the students’ mother tongue and is not influenced by the language used for teaching. Or else, if it can influence the development of the first skills only in the students’ general performance or in certain areas. The results of both the analysis of variance and multiple regression confirm how influencing the language of instruction is when mathematical thinking is developed teaching formal contents in a non-coincidence language. The second language is affecting the resolution of daily life problems, being more competent those students in 1st grades whose language of instruction matched with their mother tongue. (shrink)

The domain of numbers provides a paradigmatic case for investigating interactions of culture, language, and cognition: Numerical competencies are considered a core domain of knowledge, and yet the development of specifically human abilities presupposes cultural and linguistic input by way of counting sequences. These sequences constitute systems with distinct structural properties, the cross-linguistic variability of which has implications for number representation and processing. Such representational effects are scrutinized for two types of verbal numeration systems—general and object-specific ones—that were in parallel (...) use in several Oceanic languages. The analysis indicates that the object-specific systems outperform the general systems with respect to counting and mental arithmetic, largely due to their regular and more compact representation. What these findings reveal on cognitive diversity, how the conjectures involved speak to more general issues in cognitive science, and how the approach taken here might help to bridge the gap between anthropology and other cognitive sciences is discussed in the conclusion. (shrink)

To support the claim that the approximate number system represents rational numbers, Clarke and Beck argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

There is a relationship between reading and math skills, as well as comorbidity between reading and math disorders. A mutual foundation for this comorbidity could be that the quality of phonological representations is important for both early reading and arithmetic. In this study, we examine this hypothesis in a sample traced longitudinally from preschool to first grade. The results show that phonological awareness does not explain development in arithmetic, but that there is an indirect effect between phoneme awareness in preschool (...) and arithmetic in first grade via phoneme awareness in first grade. This effect is, however, weak and restricted to verbal arithmetic and not arithmetic fluency. This finding is only partly in line with other studies, and a reason could be that this study more strongly controls for confounders and previous skills than other studies. (shrink)

The pigeonhole principle is a well-known mathematical principle and is quite simple to understand. It goes as follows: If n items are placed into m containers, and if m (...) an identifier (nominal). They suggest quantities first need to be turned into categories. We believe this difference can instead be explained by pragmatic factors not taken into account in their study. We disambiguate the problems given to participants and study the effects of their disambiguation and the influence of different kinds of quantities: countable and mass. We show that using a problem with countable quantities instead of hair, usually understood as non-countable, increase the performance of participants, no longer distinguishable from that of the nominal problem. (shrink)

The study of everyday numeric thinking in adults directs our attention to several aspects of number cognition that have received almost no attention in the experimental cognitive science literature, namely the influences of socially situated artifact use on numeric processing. The current studies explore numeral recognition and conceptualisation processes in business people who engage in different types of numeracy practices; orally based numeracy practices which involve very little use of written records compared to paper-based numeracy practices. Ethnographic observations of Lebanese (...) business people were conducted to gain a detailed understanding of the socio-cognitive demands in orally-based paperless and paper-based business settings. These observations were in turn used to design experimental reaction time studies which investigated currency based numeral recognition and conceptualisation processes. The results of the numeral recognition and priming studies clearly illustrate that the use of artifacts in everyday numeracy practices influences numeral recognition and conceptualisation in a way that suggests tight linkages between the visio-spatial processes involve in recognizing numerals embedded in cultural artifacts and the semantically based processes involved in the conception of these numerals. The relevance of the current findings for the main models of adult numeric cognition and for research on everyday numeracy will be discussed. (shrink)

Some evolutionary psychologists contend that the best way to discover the functions of our present psychological systems is by appealing to the notion of functional mesh, that is, the assumed tight fit between a trait's design and the adaptive problem it is supposed to solve. In this paper, I argue that there exist theoretical considerations and empirical evidence that undermine this assumption of optimal design. Instead, I suggest that cognitive systems are constrained by what I call bounded functionality. This proposal (...) makes use of Jacob's (1977) notion of evolution as a bricoleur and Simon's (1981) idea that problems can have ``satisficing'' solutions. Functional mesh will thus be shown to neglect constraints that are necessary to explain the evolution of psychological mechanisms. (shrink)