Categorizing digits and the mental number line

  • Dennis ReikeEmail author
  • Wolf Schwarz
Short Report


Following the classical work of Moyer and Landauer (1967), experimental studies investigating the way in which humans process and compare symbolic numerical information regularly used one of two experimental designs. In selection tasks, two numbers are presented, and the task of the participant is to select (for example) the larger one. In classification tasks, a single number is presented, and the participant decides if it is smaller or larger than a predefined standard. Many findings obtained with these paradigms fit in well with the notion of a mental analog representation, or an Approximate Number System (ANS; e.g., Piazza 2010). The ANS is often conceptualized metaphorically as a mental number line, and data from both paradigms are well accounted for by diffusion models based on the stochastic accumulation of noisy partial numerical information over time. The present study investigated a categorization paradigm in which participants decided if a number presented falls into a numerically defined central category. We show that number categorization yields a highly regular, yet considerably more complex pattern of decision times and error rates as compared to the simple monotone relations obtained in traditional selection and classification tasks. We also show that (and how) standard diffusion models of number comparison can be adapted so as to account for mean and standard deviations of all RTs and for error rates in considerable quantitative detail. We conclude that just as traditional number comparison, the more complex process of categorizing numbers conforms well with basic notions of the ANS.


Categorization Numerical distance effect Mental number line Diffusion models 



We would like to thank Constantin G. Meyer-Grant for collecting the data.

The present work was supported by a research grant (SCHW 611/5-1) from the Deutsche Forschungsgemeinschaft (DFG).


  1. Banks, W.P., Fujii, M., & Kayra-Stuart, F. (1976). Semantic congruity effects in comparative judgments of magnitudes of digits. Journal of Experimental Psychology: Human Perception and Performance, 2, 435–447.Google Scholar
  2. Besner, D., & Coltheart, M. (1979). Ideographic and alphabetic processing in skilled reading of English. Neuropsychologia, 17, 467–472.Google Scholar
  3. Cartwright, D. (1941). Relation of decision-time to the categories of response. American Journal of Psychology, 54, 174–196.Google Scholar
  4. Dehaene, S. (1989). The psychophysics of numerical comparison: A reexamination of apparently incompatible data. Perception & Psychophysics, 45, 557–566.Google Scholar
  5. Dehaene, S., & Akhavein, R. (1995). Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 21, 314–326.Google Scholar
  6. Dehaene, S. (2003). The neural basis of the Weber–Fechner law: A logarithmic mental number line. Trends in Cognitive Sciences, 7, 145–147.Google Scholar
  7. Dehaene, S., & Brannon, E.M. (Eds.) (2011). Space, time and number in the brain. London: Academic Press.Google Scholar
  8. DeRosa, D.V., & Morin, R.E. (1970). Recognition reaction time for digits in consecutive and nonconsecutive memorized sets. Journal of Experimental Psychology, 83, 472–479.Google Scholar
  9. Gold, J.I., & Shadlen, M.N. (2007). The neural basis of decision making. Annual Review of Neuroscience, 30, 535–574.Google Scholar
  10. Henik, A., & Tzelgov, J. (1982). Is 3 greater than 5: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10, 389–395.Google Scholar
  11. Kamienkowski, J.E., Pashler, H., Dehaene, S., & Sigman, M. (2011). Effects of practice on task architecture: Combined evidence from interference experiments and random-walk models of decision making. Cognition, 119, 81–95.Google Scholar
  12. Lambrechts, A., Walsh, V., & van Wassenhove, V. (2013). Evidence accumulation in the magnitude system. PLoS ONE, 8, e82122.Google Scholar
  13. Luce, R.D. (1986) Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.Google Scholar
  14. Macé, M.J.-M., Joubert, O.R., Nespoulous, J.-L., & Fabre-Thorpe, M. (2009). The time-course of visual categorizations: You spot the animal faster than the bird. PLoS ONE, 4, e5927.Google Scholar
  15. Maxwell, S.E., & Delaney, H.D. (2004) Designing experiments and analyzing data, 2nd Edn. New York: Taylor & Francis.Google Scholar
  16. Miller, J.O. (1991). Reaction time analysis with outlier exclusion: Bias varies with sample size. The Quarterly Journal of Experimental Psychology, 13A, 907–912.Google Scholar
  17. Moyer, R.S., & Landauer, T.K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520.Google Scholar
  18. Nieder, A. (2005). Counting on neurons: The neurobiology of numerical competence. Nature Reviews Neuroscience, 6, 177–190.Google Scholar
  19. Piazza, M. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551.Google Scholar
  20. Ratcliff, R., & Smith, P.L. (2004). A comparison of sequential sampling models for two-choice reaction time. Psychological Review, 111, 333–367.Google Scholar
  21. Ratcliff, R., & McKoon, G. (2018). Modeling numerosity representation with an integrated diffusion model. Psychological Review, 125, 183–217.Google Scholar
  22. Reike, D., & Schwarz, W. (2016). One model fits all: Explaining many aspects of number comparison within a single coherent model – A random walk account. Journal of Experimental Psychology: Learning, Memory, and Cognition, 42, 1957–1971.Google Scholar
  23. Reike, D., & Schwarz, W. (2017). Exploring the origin of the number size congruency effect: Sensitivity or response bias? Attention, Perception, & Psychophysics, 79, 383–388.Google Scholar
  24. Reike, D., & Schwarz, W. (2019). Aging effects on symbolic number comparison: No deceleration of numerical information retrieval but more conservative decision-making. Psychology and Aging, 34, 4–16.Google Scholar
  25. Rosch, E. (1975). Cognitive representations of semantic categories. Journal of Experimental Psychology: General, 104, 192–233.Google Scholar
  26. Rugani, R., & de Hevia, M.-D. (2017). Number-space associations without language: Evidence from preverbal human infants and non-human animal species. Psychonomic Bulletin & Review, 24, 352–369.Google Scholar
  27. Schwarz, W., & Stein, F. (1998). On the temporal dynamics of digit comparison processes. Journal of Experimental Psychology: Learning, Memory, and Cognition, 24, 1275–1293.Google Scholar
  28. Schwarz, W., & Ischebeck, A. (2003). On the relative speed account of number—size interference effects in comparative judgments of numerals. Journal of Experimental Psychology: Human Perception and Performance, 29, 507–522.Google Scholar
  29. Schwarz, W., & Reike, D. (2019). The number–weight illusion. Accepted for publication in Psychonomic Bulletin and Review.Google Scholar
  30. Sigman, M., & Dehaene, S. (2005). Parsing a cognitive task: A characterization of the mind’s bottleneck. PLoS Biology, 3, 334–349.Google Scholar
  31. Smith, D.G., & Mewhort, D.J.K. (1998). The distribution of latencies constrains theories of decision time: A test of the random-walk model using numeric comparison. Australian Journal of Psychology, 50, 149–156.Google Scholar
  32. Sokal, R.R., & Rohlf, F.J. (1995) Biometry: The principles and practices of statistics in biological research, 3rd Edn. New York: W. H. Freeman.Google Scholar
  33. Thompson, C.A., Ratcliff, R., & McKoon, G. (2016). Individual differences in the components of children’s and adults’ information processing for simple symbolic and non-symbolic numeric decisions. Journal of Experimental Child Psychology, 150, 48–71.Google Scholar
  34. Walsh, V. (2003). A theory of magnitude: Common cortical metrics of time, space and quantity. Trends in Cognitive Sciences, 7, 483–488.Google Scholar
  35. Whalen, J., Gallistel, C.R., & Gelman, R. (1999). Nonverbal counting in humans: The psychophysics of number representation. Psychological Science, 10, 130–137.Google Scholar

Copyright information

© The Psychonomic Society, Inc. 2019

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of PotsdamPotsdamGermany

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