Numerical distance effect size is a poor metric of approximate number system acuity

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Abstract

Individual differences in the ability to compare and evaluate nonsymbolic numerical magnitudes—approximate number system (ANS) acuity—are emerging as an important predictor in many research areas. Unfortunately, recent empirical studies have called into question whether a historically common ANS-acuity metric—the size of the numerical distance effect (NDE size)—is an effective measure of ANS acuity. NDE size has been shown to frequently yield divergent results from other ANS-acuity metrics. Given these concerns and the measure’s past popularity, it behooves us to question whether the use of NDE size as an ANS-acuity metric is theoretically supported. This study seeks to address this gap in the literature by using modeling to test the basic assumption underpinning use of NDE size as an ANS-acuity metric: that larger NDE size indicates poorer ANS acuity. This assumption did not hold up under test. Results demonstrate that the theoretically ideal relationship between NDE size and ANS acuity is not linear, but rather resembles an inverted J-shaped distribution, with the inflection points varying based on precise NDE task methodology. Thus, depending on specific methodology and the distribution of ANS acuity in the tested population, positive, negative, or null correlations between NDE size and ANS acuity could be predicted. Moreover, peak NDE sizes would be found for near-average ANS acuities on common NDE tasks. This indicates that NDE size has limited and inconsistent utility as an ANS-acuity metric. Past results should be interpreted on a case-by-case basis, considering both specifics of the NDE task and expected ANS acuity of the sampled population.

Keywords

Numerical distance effect Estimation Approximate number system Acuity Numerical magnitudes 

References

  1. Cantrell, L. M., & Smith, L. B. (2013). Set size, individuation, and attention to shape. Cognition, 126, 258–267.  https://doi.org/10.1016/j.cognition.2012.10.007 CrossRefPubMedGoogle Scholar
  2. Chen, Q., & Li, J. (2014). Association between individual differences in non-symbolic number acuity and math performance: A meta-analysis. Acta Psychologica, 148, 163–172.  https://doi.org/10.1016/j.actpsy.2014.01.016 CrossRefPubMedGoogle Scholar
  3. Chesney, D. L, Bjälkebring, P., & Peters, E. (2015). How to estimate how well people estimate: Evaluating measures of individual differences in the approximate number system. Attention, Perception, & Psychophysics. 77, 2781–2802.  https://doi.org/10.3758/s13414-015-0974-6 CrossRefGoogle Scholar
  4. Chesney, D. L., & Haladjian, H. H. (2011). Evidence for a shared mechanism used in multiple-object tracking and subitizing. Attention, Perception, & Psychophysics, 73, 2457–2480. : https://doi.org/10.3758/s13414-011-0204-9 CrossRefGoogle Scholar
  5. Cordes, S., Gelman, R., Gallistel, C. R., & Whalen, J. (2001). Variability signatures distinguish verbal from nonverbal counting for both large and small numbers. Psychonomic Bulletin & Review, 8, 698–707.  https://doi.org/10.3758/BF03196206 CrossRefGoogle Scholar
  6. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–42.  https://doi.org/10.1016/0010-0277(92)90049-N CrossRefPubMedGoogle Scholar
  7. Dehaene, S., Bossini, S., & Pascal, G. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General. 122, 371–396.  https://doi.org/10.1037/0096-3445.122.3.371 CrossRefGoogle Scholar
  8. Dehaene, S., Izard, V., Spelke, E., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320, 1217–1220.  https://doi.org/10.1126/science.1156540 CrossRefPubMedPubMedCentralGoogle Scholar
  9. Feigenson, L., Dehaene, S., & Spelke, E. S. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314.  https://doi.org/10.1016/j.tics.2004.05.002 CrossRefPubMedGoogle Scholar
  10. Gilmore, C., Attridge, N., & Inglis, M. (2011). Measuring the approximate number system. The Quarterly Journal of Experimental Psychology, 64, 2009-2109.  https://doi.org/10.1080/17470218.2011.574710 CrossRefGoogle Scholar
  11. Gilmore, C.K., McCarthy, S.E., & Spelke, E.S. (2010). Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling. Cognition, 115, 394–406.  https://doi.org/10.1016/j.cognition.2010.02.002 CrossRefPubMedPubMedCentralGoogle Scholar
  12. Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the “number sense”: The approximate number system in 3-, 4-, 5-, 6-year-olds and adults. Developmental Psychology, 44, 1457–1465.  https://doi.org/10.1037/a0012682 CrossRefPubMedGoogle Scholar
  13. Halberda, J., Mazzocco, M., & Feigenson, L. (2008). Individual differences in nonverbal number acuity predict maths achievement. Nature, 455, 665–668.  https://doi.org/10.1038/nature07246 CrossRefPubMedGoogle Scholar
  14. Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103, 17–29.  https://doi.org/10.1016/j.jecp.2008.04.001 CrossRefPubMedGoogle Scholar
  15. Hurewitz, F., Gelman, R., & Schnitzer, B. (2006). Sometimes area counts more than number. Proceedings of the National Academy of Sciences, 103, 19599–19604.  https://doi.org/10.1073/pnas.0609485103 CrossRefGoogle Scholar
  16. Inglis, M., & Gilmore, C. (2014). Indexing the approximate number system. Acta Psychologica, 145, 147–155.  https://doi.org/10.1016/j.actpsy.2013.11.009 CrossRefPubMedGoogle Scholar
  17. Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525.  https://doi.org/10.2307/1418556 CrossRefPubMedGoogle Scholar
  18. Kingdom, F. A. A., & Prins, N. (2010). Psychophysics: A practical introduction. London, UK: Academic Press.Google Scholar
  19. Krajcsi, A., Lengyel, G., & Kojouharova, P. (2016). The source of the symbolic numerical distance and size effects. Frontiers in Psychology, 7.  https://doi.org/10.3389/fpsyg.2016.01795
  20. Maloney, E. A., Risko, E. F., Preston, F., Ansari, D., & Fugelsang, J. A. (2010). Challenging the reliability and validity of cognitive measures: The case of the numerical distance effect. Acta Psychologica 134, 154–161.  https://doi.org/10.1016/j.actpsy.2010.01.006 CrossRefPubMedGoogle Scholar
  21. Mechner, F. (1958). Probability relations within response sequence maintained under ratio reinforcement. Journal of the Experimental Analysis of Behavior, 1, 109–121.  https://doi.org/10.1901/jeab.1958.1-109 CrossRefPubMedPubMedCentralGoogle Scholar
  22. Meck, W. H., & Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320–334.  https://doi.org/10.1037/0097-7403.9.3.320 PubMedGoogle Scholar
  23. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520.  https://doi.org/10.1038/2151519a0 CrossRefPubMedGoogle Scholar
  24. Peters, E., Slovic, P., Västfjäll, D., & Mertz, C. K. (2008). Intuitive numbers guide decisions. Judgment and Decision Making, 3, 619–635. Retrieved from http://journal.sjdm.org/8827/jdm8827.pdf Google Scholar
  25. Price, G. R., Palmer, D., Battista, S., & Ansari, D. (2012). Nonsymbolic numerical magnitude comparison: Reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults. Acta Psychologica, 140, 50–57.  https://doi.org/10.1016/j.actpsy.2012.02.008 CrossRefPubMedGoogle Scholar
  26. Sasanguie, D., Defever, E., Van den Bussche, E., & Reynvoet, B. (2011). The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming. Acta Psychologica, 136, 73–80.  https://doi.org/10.1016/j.actpsy.2010.10.004 CrossRefPubMedGoogle Scholar
  27. Schley, D. R., & Peters, E. (2014). Assessing “economic value”: Symbolic number mappings predict risky and riskless valuations. Psychological Science, 25, 753–761.  https://doi.org/10.1177/0956797613515485 CrossRefPubMedGoogle Scholar
  28. Sekuler, R., & Mierkiewicz, D. (1977). Children’s Judgments of Numerical Inequality. Child Development, 48, 630–633.  https://doi.org/10.2307/1128664 CrossRefGoogle Scholar
  29. Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237–243.  https://doi.org/10.1111/1467-9280.02438 CrossRefPubMedGoogle Scholar
  30. Taves, E. H. (1941). Two mechanisms for the perception of visual numerousness. Archives of Psychology, 37(265), 1–47.Google Scholar
  31. Whalen, J., Gallistel, C. R., & Gelman, R. (1999). Non-verbal counting in humans: The psychophysics of number representation. Psychological Science, 10, 130–137.  https://doi.org/10.1111/1467-9280.00120 CrossRefGoogle Scholar
  32. Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74, B1–B11.  https://doi.org/10.1016/S0010-0277(99)00066-9 CrossRefPubMedGoogle Scholar

Copyright information

© The Psychonomic Society, Inc. 2018

Authors and Affiliations

  1. 1.Department of PsychologySt. John’s UniversityJamaicaUSA

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