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      You Can Count on Your Fingers: The Role of Fingers in Early Mathematical Development

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          Abstract

          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 fingers for early mathematics development. We delve into issues such as how the early multimodal (tactile, motor, visuospatial) experiences with fingers might be the gateway for later numerical skills, how finger gnosis, finger counting habits, and numerical abilities are associated at the behavioral and neural levels, and implications for mathematics education. We argue that, taken together, the two bodies of research can better inform how different finger skills support the development of numerical competencies, and we provide a road map for future interdisciplinary research that can yield to development of diagnostic tools and interventions for preschool and primary grade classrooms.

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          Perceptual symbol systems.

          Prior to the twentieth century, theories of knowledge were inherently perceptual. Since then, developments in logic, statistics, and programming languages have inspired amodal theories that rest on principles fundamentally different from those underlying perception. In addition, perceptual approaches have become widely viewed as untenable because they are assumed to implement recording systems, not conceptual systems. A perceptual theory of knowledge is developed here in the context of current cognitive science and neuroscience. During perceptual experience, association areas in the brain capture bottom-up patterns of activation in sensory-motor areas. Later, in a top-down manner, association areas partially reactivate sensory-motor areas to implement perceptual symbols. The storage and reactivation of perceptual symbols operates at the level of perceptual components--not at the level of holistic perceptual experiences. Through the use of selective attention, schematic representations of perceptual components are extracted from experience and stored in memory (e.g., individual memories of green, purr, hot). As memories of the same component become organized around a common frame, they implement a simulator that produces limitless simulations of the component (e.g., simulations of purr). Not only do such simulators develop for aspects of sensory experience, they also develop for aspects of proprioception (e.g., lift, run) and introspection (e.g., compare, memory, happy, hungry). Once established, these simulators implement a basic conceptual system that represents types, supports categorization, and produces categorical inferences. These simulators further support productivity, propositions, and abstract concepts, thereby implementing a fully functional conceptual system. Productivity results from integrating simulators combinatorially and recursively to produce complex simulations. Propositions result from binding simulators to perceived individuals to represent type-token relations. Abstract concepts are grounded in complex simulations of combined physical and introspective events. Thus, a perceptual theory of knowledge can implement a fully functional conceptual system while avoiding problems associated with amodal symbol systems. Implications for cognition, neuroscience, evolution, development, and artificial intelligence are explored.
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            Neural reuse: a fundamental organizational principle of the brain.

            An emerging class of theories concerning the functional structure of the brain takes the reuse of neural circuitry for various cognitive purposes to be a central organizational principle. According to these theories, it is quite common for neural circuits established for one purpose to be exapted (exploited, recycled, redeployed) during evolution or normal development, and be put to different uses, often without losing their original functions. Neural reuse theories thus differ from the usual understanding of the role of neural plasticity (which is, after all, a kind of reuse) in brain organization along the following lines: According to neural reuse, circuits can continue to acquire new uses after an initial or original function is established; the acquisition of new uses need not involve unusual circumstances such as injury or loss of established function; and the acquisition of a new use need not involve (much) local change to circuit structure (e.g., it might involve only the establishment of functional connections to new neural partners). Thus, neural reuse theories offer a distinct perspective on several topics of general interest, such as: the evolution and development of the brain, including (for instance) the evolutionary-developmental pathway supporting primate tool use and human language; the degree of modularity in brain organization; the degree of localization of cognitive function; and the cortical parcellation problem and the prospects (and proper methods to employ) for function to structure mapping. The idea also has some practical implications in the areas of rehabilitative medicine and machine interface design.
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              Developmental change in the acuity of the "Number Sense": The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults.

              Behavioral, neuropsychological, and brain imaging research points to a dedicated system for processing number that is shared across development and across species. This foundational Approximate Number System (ANS) operates over multiple modalities, forming representations of the number of objects, sounds, or events in a scene. This system is imprecise and hence differs from exact counting. Evidence suggests that the resolution of the ANS, as specified by a Weber fraction, increases with age such that adults can discriminate numerosities that infants cannot. However, the Weber fraction has yet to be determined for participants of any age between 9 months and adulthood, leaving its developmental trajectory unclear. Here we identify the Weber fraction of the ANS in 3-, 4-, 5-, and 6-year-old children and in adults. We show that the resolution of this system continues to increase throughout childhood, with adultlike levels of acuity attained surprisingly late in development.
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                Author and article information

                Contributors
                Journal
                JNC
                J Numer Cogn
                Journal of Numerical Cognition
                J. Numer. Cogn.
                PsychOpen
                2363-8761
                2018
                07 June 2018
                : 4
                : 1
                : 107-135
                Affiliations
                [a ] The University of Alabama , Tuscaloosa, AL, USA
                [b ] Indiana University , Bloomington, IN, USA
                [3]Department of Mathematics, Virginia Tech , Blacksburg, VA, USA
                [4]School of Teaching, Learning, and Curriculum Studies, Kent State University , Kent, OH, USA
                Author notes
                [* ]College of Education, Box 870231, Tuscaloosa, AL 35487, USA. Phone: (205) 348 6267. fsoylu@ 123456ua.edu
                Article
                jnc.v4i1.85
                10.5964/jnc.v4i1.85
                b9feceae-ebdf-4412-8910-3246063b8629
                Copyright @ 2018

                This is an open-access article distributed under the terms of the Creative Commons Attribution (CC BY) 4.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

                History
                : 31 October 2016
                : 21 May 2017
                Categories
                Theoretical Contributions

                Psychology
                mathematics education,finger counting,neuroscience,finger gnosis,cognitive development,embodied cognition,numerical cognition

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