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The Acuity for Numerosity (but Not Continuous Magnitude) Discrimination Correlates with Quantitative Problem Solving but Not Routinized Arithmetic

Abstract

Approximate number sense (ANS) acuity refers to the ability to non-symbolically recognize, estimate and operate upon large numerosities. ANS acuity has been reported to be correlated with math achievement in children and adolescents. However, reports of this relationship in adults have been inconsistent. The present study aimed to resolve the inconsistency in the relationship between adults’ ANS acuity and math achievement by contrasting between different kinds of mathematical problem solving. We hypothesized that the correlation between ANS acuity mathematical performance would be stronger when deep quantitative processing is required during problem solving. In Experiment 1, ANS acuity was correlated with Mathematical Reasoning but not Directed Calculation performance. In Experiment 2, ANS acuity was correlated with Two-digit Subtraction (but not Addition) performance only when Regrouping (i.e., borrowing) was required. The results from two experiments demonstrated that ANS acuity was correlated with mathematical performance only when problem solving involved effortful, quantitative processing that goes beyond automatized, routinized arithmetic. In addition, ANS acuity was distinguishable from Area acuity regarding its unique relationship with math achievement, which was unconfounded by the influence of demographic variables and fluid intelligence. Overall, the present results help resolve the inconsistency in previous reports of the correlation between ANS acuity and math achievement in adults.

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Acknowledgments

This research was supported by grants from the National Research Foundation of Korea (2012R1A1A1011872, 2014R1A1A3051034) funded by the Korean government.

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Correspondence to Soohyun Cho.

Appendix

Appendix

Measuring the Acuity for Numerosity Discrimination

Individual differences in numerosity and length acuity was calculated the Weber fraction coefficient (w) which based on a psychophysical model for numerosity comparison (Halberda et al. 2008; Pica et al. 2004). The formula of the psychophysics model is as follows:

$$ \mathrm{Proportion}\ \mathrm{Correct} = 1\kern0.5em -\kern0.5em \frac{1}{2}erf\kern0.5em c\left(\frac{{\mathrm{n}}_1-{\mathrm{n}}_2}{w\sqrt{2\left({{\mathrm{n}}_1}^2+{{\mathrm{n}}_2}^2\right)}}\right) $$

The ”erfc” is a complementary Gaussian error function given by:

$$ \mathrm{erfc}\left(\mathrm{x}\right)\kern0.5em =\kern0.5em 1\kern0.5em -\kern0.5em \frac{2}{\sqrt{\uppi}}{\displaystyle {\int}_0^{\mathrm{x}}{e}^{-{t}^2}dt} $$

This model fits the proportion correct when comparing between a pair of quantities or lines (n1 and n2). The resulting value of w for each individual served as an estimator of the ratio of n1 and n2 that one can readily differentiate. Lower w indicates a higher sensitivity of discrimination.

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Jang, S., Cho, S. The Acuity for Numerosity (but Not Continuous Magnitude) Discrimination Correlates with Quantitative Problem Solving but Not Routinized Arithmetic. Curr Psychol 35, 44–56 (2016). https://doi.org/10.1007/s12144-015-9354-6

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Keywords

  • Approximate number system
  • Magnitude representation
  • Mathematical problem solving
  • Mathematical reasoning
  • Arithmetic fact retrieval