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      Do We Have a Sense for Irrational Numbers?

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          Abstract

          Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

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          Most cited references41

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          Interactions between number and space in parietal cortex.

          Since the time of Pythagoras, numerical and spatial representations have been inextricably linked. We suggest that the relationship between the two is deeply rooted in the brain's organization for these capacities. Many behavioural and patient studies have shown that numerical-spatial interactions run far deeper than simply cultural constructions, and, instead, influence behaviour at several levels. By combining two previously independent lines of research, neuroimaging studies of numerical cognition in humans, and physiological studies of spatial cognition in monkeys, we propose that these numerical-spatial interactions arise from common parietal circuits for attention to external space and internal representations of numbers.
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            Core systems of number.

            What representations underlie the ability to think and reason about number? Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, other numerical abilities are widespread and can be observed in adults, infants and other animal species. We review recent behavioral and neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core systems for representing number. Performance signatures common across development and across species implicate one system for representing large, approximate numerical magnitudes, and a second system for the precise representation of small numbers of individual objects. These systems account for our basic numerical intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.
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              Time required for judgements of numerical inequality.

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                Author and article information

                Journal
                JNC
                J Numer Cogn
                Journal of Numerical Cognition
                J. Numer. Cogn.
                PsychOpen
                2363-8761
                10 February 2017
                : 2
                : 3
                : 170-189
                Affiliations
                [a ]TUM School of Education, Technical University of Munich, Munich, Germany
                [b ]University of Education Freiburg, Freiburg, Germany
                Author notes
                [* ]University of Education Freiburg, Kunzenweg 21, 79117 Freiburg, Germany. Phone: +49 761 682 430. andreas.obersteiner@ 123456ph-freiburg.de
                Article
                jnc.v2i3.43
                10.5964/jnc.v2i3.43
                9041b19b-074e-4eea-932b-8fac0d4827c9
                Copyright @ 2017

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

                History
                : 08 May 2016
                : 21 July 2016
                Categories
                Research Reports

                Psychology
                number comparison,natural number bias,numerical distance effect,number sense,magnitude representation

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