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)

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)

The study has two objectives: to introduce grip force recording as a new technique for studying embodied numerical processing; and to demonstrate how three competing accounts of numerical magnitude representation can be tested by using this new technique: the Mental Number Line, A Theory of Magnitude and Embodied Cognition account. While 26 healthy adults processed visually presented single digits in a go/no-go n-back paradigm, their passive holding forces for two small sensors were recorded in both hands. Spontaneous and unconscious grip (...) force changes related to number magnitude occurred in the left hand already 100–140 ms after stimulus presentation and continued systematically. Our results support a two-step model of number processing where an initial stage is related to the automatic activation of all stimulus properties whereas a later stage consists of deeper conscious processing of the stimulus. This interpretation generalizes previous work with linguistic stimuli and elaborates the timeline of embodied cognition. We hope that the use of grip force recording will advance the field of numerical cognition research. (shrink)

Conscious experience is the direct observation of conscious events. Human conscious experience is four-dimensional. Conscious events are linked (associated) by spacetime intervals to produce a coherent conscious experience. This explains why conscious experience appears to us the way it does. Conscious experience is an orientation in space and time, an understanding of the position of the observer in space and time. Causality, past-future relations, learning, memory, cognitive processing, and goal-directed actions all evolve from four-dimensional conscious experience. A neural correlate for (...) four-dimensional conscious experience can be found in the human brain and is modelled by Einstein's special theory of relativity. The relativistic concept of spacetime interval is central for understanding conscious experience and cognition. (shrink)

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

Using a novel enumeration task, we examined the encoding of spatial information during subitizing. Observers were shown masked presentations of randomly-placed discs on a screen and were required to mark the perceived locations of these discs on a subsequent blank screen. This provided a measure of recall for object locations and an indirect measure of display numerosity. Observers were tested on three stimulus durations and eight numerosities. Enumeration performance was high for displays containing up to six discs—a higher subitizing range (...) than reported in previous studies. Error in the location data was measured as the distance between corresponding stimulus and response discs. Overall, location errors increased in magnitude with larger numerosities and shorter display durations. When errors were computed as disc distance from display centroid, results suggest a compressed representation by observers. Additionally, enumeration and localization accuracy increased with display regularity. (shrink)

A computer can come to understand natural language the same way Helen Keller did: by using “syntactic semantics”—a theory of how syntax can suffice for semantics, i.e., how semantics for natural language can be provided by means of computational symbol manipulation. This essay considers real-life approximations of Chinese Rooms, focusing on Helen Keller’s experiences growing up deaf and blind, locked in a sort of Chinese Room yet learning how to communicate with the outside world. Using the SNePS computational knowledge-representation system, (...) the essay analyzes Keller’s belief that learning that “everything has a name” was the key to her success, enabling her to “partition” her mental concepts into mental representations of: words, objects, and the naming relations between them. It next looks at Herbert Terrace’s theory of naming, which is akin to Keller’s, and which only humans are supposed to be capable of. The essay suggests that computers at least, and perhaps non-human primates, are also capable of this kind of naming. (shrink)

Number-form synesthetes consciously experience numbers in spatially-defined locations. For non-synesthete individuals, a similar association of numbers and space appears in the form of an implicit mental number line as signified by the distance effect–reaction time decreases as the numerical distance between compared numbers increases. In the current experiment, three number-form synesthetes and two different non-synesthete control groups performed a number comparison task. Synesthete participants exhibited a sizeable distance effect only when presented numbers were congruent with their number-form. In contrast, the (...) controls exhibited a distance effect regardless of congruency or presentation type. The findings suggest that: number-form synesthesia impairs the ability to represent numbers in a flexible manner according to task demands; number-form synesthesia is a genuine tangible experience, triggered involuntarily; and the classic mental number line can be more pliable than previously thought and appears to be independent of cultural-lingo direction. (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)

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.

People frequently report that, at times, their thought has a vocal character. Thinking commonly appears to be accompanied or constituted by silently ‘talking’ to oneself in inner speech. In this paper, we explore the specifically epistemic role of inner speech in conscious reasoning. A plausible position—but one I argue is ultimately wrong—is that inner speech plays asolelyfacilitative role that is exhausted by (i) serving as the vehicle of representation for conscious reasoning, and/or (ii) allowing one to focus on certain types (...) of objects or relations, e.g., causal relations, abstracta, counterfactuals, etc., or to consciously entertain structured propositional contents that it would be hard (or impossible) to focus on or entertain with representations in other (e.g., imagistic) formats. According to this position, inner speech doesn't figure as a justificatory element in our reasoning or as the partial epistemic basis of our conclusions—it merely facilitates reasoning through (i) and/or (ii). In contrast to the view that inner speech is amerefacilitator, I establish that (outside of potentially playing roles (i) and/or (ii)) the language we use itself serves as a crucial source of information in reasoning. In other words, we reason from propositions about the language we use in inner speech as opposed to exclusively reasoning from the semantic contents of the speech. My conclusion follows from how we use language as a cognitive tool to keep track of information, e.g., the contents of premises, lemmas, previous reasoning results, etc., in reasoning. (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)

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)

This study presents the evaluation of a computer-based learning program for children with developmental dyscalculia and focuses on factors affecting individual responsiveness. The adaptive training program Calcularis 2.0 has been developed according to current neuro-cognitive theory of numerical cognition. It aims to automatize number representations, supports the formation and access to the mental number line and trains arithmetic operations as well as arithmetic fact knowledge in expanding number ranges. Sixty-seven children with developmental dyscalculia from second to fifth grade (mean age (...) 8.96 years) were randomly assigned to one of two groups (Calcularis group, waiting control group). Training duration comprised a minimum of 42 training sessions à 20 min within a maximum period of 13 weeks. Compared to the waiting control group, children of the Calcularis group demonstrated a higher benefit in arithmetic operations and number line estimation. These improvements were shown to be stable after a 3-months post training interval. In addition, this study examines which predictors accounted for training improvements. Results indicate that this self-directed training was especially beneficial for children with low math anxiety scores and without an additional reading and/or spelling disorder. In conclusion, Calcularis 2.0 supports children with developmental dyscalculia to improve their arithmetical abilities and their mental number line representation. However, it is relevant to further adapt the setting to the individual circumstances. (shrink)

Based on the notion that time, space, and number are part of a generalized magnitude system, we assume that the dual-systems approach to temporal cognition also applies to numerical cognition. Referring to theoretical models of the development of numerical concepts, we propose that children's early skills in processing numbers can be described analogously to temporal updating and temporal reasoning.

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A third hypothesis, (...) in line with dual-systems theory (e.g., Kahneman and Frederick, 2002 ) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others. (shrink)

Cohen Kadosh & Walsh (CK&W) argue that recent findings challenge the hypothesis of abstract numerical representations. Here we show that because, like many other authors in the field, they rely on inaccurate definitions of abstract and non-abstract representations, CK&W fail to provide compelling evidence against the abstract view.

Number processing abilities are important for academic and personal development. The course of initial specialization of ventral occipito-temporal cortex sensitivity to visual number processing is crucial for the acquisition of numeric and arithmetic skills. We examined the visual N1, the electrophysiological correlate of vOTC activation across five time points in kindergarten, middle and end of first grade, second grade, and fifth grade. A combination of cross-sectional and longitudinal EEG data of a total of 62 children at varying familial risk for (...) dyslexia were available to form groups of 23, 22, 27, 27, and 42 participants for each of the five time points. The children performed a target detection task which included visual presentation of single digits, false fonts, and letters to derive measures for coarse and fine digit sensitive processing across development. The N1 amplitude analyses indicated coarse and fine sensitivity characterized by a stronger N1 to digits than false fonts across all five time points, and stronger N1 to digits than letters at all but the second time point. In addition, lower arithmetic skills were associated with stronger coarse N1 digit sensitivity over the left hemisphere in second grade, possibly reflecting allocation of more attentional resources or stronger reliance on the verbal system in children with poorer arithmetic skills. To summarize, our results show persistent visual N1 sensitivity to digits that is already present early on in pre-school and remains stable until fifth grade. This pattern of digit sensitivity development clearly differs from the relatively sharp rise and fall of the visual N1 sensitivity to words or letters between kindergarten and middle of elementary school and suggests unique developmental trajectories for visual processing of written characters that are relevant to numeracy and literacy. (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)

Early mathematical abilities matter for later formal arithmetical performances, school and professional success. Accordingly, it seems central to accurately assess numerical school readiness at school entrance. This is a prerequisite for identifying school-starters who are at risk to encounter difficulties in mathematics and stimulate their acquisition of mathematical fundamentals as soon as possible. In the present study, we present a new test which allows professionals working with children (e.g., teachers, school psychologists, speech therapists, school doctors) to assess children’s numerical school (...) readiness when they enter formal schooling in a simple, rapid and efficient manner. 346 children were assessed at the entrance of 1st grade (6-to-7-year-olds) with the novel numerical school readiness test (T1). In addition, children’s numerical skill levels were evaluated with classical arithmetical tests at T1 and a year later in 2nd grade (T2, 7-to-8-year-olds). Performance in these classical tests in 1st and 2nd grade systematically related to their early mathematical abilities assessed with the numerical school readiness test. By using the present test to evaluate numerical school readiness it is possible to identify pupils at risk for developing poor mathematical skills right from the start of formal schooling. Such a tool is needed, as a child’s scholastic level in mathematic at school entrance (or school readiness) is known to be critical for his or her future school and professional carrier (Currie & Thomas, 1999; Duncan et al., 2007; Pagani, Fitzpatrick, Archambault, & Janosz, 2010; Rivera-Batiz, 1992; Romano, Babchishin, Pagani, & Kohen, 2010). (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)

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)