Skip to main content
SpringerLink
Log in

On the linear representation of numbers: evidence from a new two-numbers-to-two positions task

  • Original Article
  • Published:
Psychological Research Aims and scope Submit manuscript

Abstract

In the number-to-position methodology, a number is presented on each trial and the observer places it on a straight line in a position that corresponds to its felt subjective magnitude. In the novel modification introduced in this study, the two-numbers-to-two-positions method, a pair of numbers rather than a single number is presented on each trial and the observer places them in appropriate positions on the same line. Responses in this method indicate not only the subjective magnitude of each single number but, simultaneously, provide a direct estimation of their subjective numerical distance. The results of four experiments provide strong evidence for a linear representation of numbers and, commensurately, for the linear representation of numerical distances. We attribute earlier results that indicate a logarithmic representation to the ordered nature of numbers and to the task used and not to a truly non-linear underlying representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. In order to control for confounds, an inclusive ANOVA was performed, including gender (male, female) and hand (left, right) as between-subjects variables, in addition to the main reported variables. For both number and distance estimation, the ANOVA did not yield main effects or interactions for gender and hand (p > 0.05). A similar analysis with respect to RT did not yield effects.

  2. We did not record any systematic differences between our German and Israeli participants.

References

  • Algom, D. (1992). Memory psychophysics: An examination of its perceptual and cognitive prospects. In D. Algom (Ed.), Psychophysical approaches to cognition (pp. 441–513). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Algom, D., & Marks, L. E. (1990). Range and regression, loudness scales, and loudness processing: Toward a context-bound psychophysics. Journal of Experimental Psychology Human Perception and Performance, 16(4), 706–727.

    Article  PubMed  Google Scholar 

  • Anderson, N. H. (1981). Foundations of information integration theory‏. New York: Academic Press.

    Google Scholar 

  • Anderson, N. H. (1982). Methods of information integration theory. New York: Academic Press.‏.

    Google Scholar 

  • Ashby, F. G., & Maddox, W. T. (1994). A response time theory of separability and integrality in speeded classification. Journal of Mathematical Psychology, 38(4), 423–466.‏.

    Article  Google Scholar 

  • Barth, H. C., & Paladino, A. M. (2011). The development of numerical estimation: Evidence against a representational shift. Developmental Science, 14(1), 125–135.

    Article  PubMed  Google Scholar 

  • Ben-Nathan, M., & Algom, D. (2007). The perceived magnitude of two-digit numbers: A functional measurement analysis. Teori and Modelli, 12, 87–96.

    Google Scholar 

  • Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79(4), 1016–1031‏.

    Article  PubMed  Google Scholar 

  • Brannon, E. M., Wusthoff, C. J., Gallistel, C. R., & Gibbon, J. (2001). Numerical subtraction in the pigeon: Evidence for a linear subjective number scale. Psychological Science, 12(3), 238–243.‏.

    Article  PubMed  Google Scholar 

  • Cohen, D. (2009). Integers do not automatically activate their magnitude representation. Psychonomic Bulletin and Review, 16, 332–336.

    Article  PubMed  Google Scholar 

  • Dehaene, S. (1997). The number sense. New York: Oxford University Press.

    Google Scholar 

  • Dehaene, S. (2001). Précis of the number sense. Mind and Language, 16(1), 16–36.‏.

    Article  Google Scholar 

  • Dehaene, S. (2003). The neural basis of the Weber–Fechner law: A logarithmic mental number line. Trends in Cognitive Sciences, 7(4), 145–147.‏.

    Article  PubMed  Google Scholar 

  • Fischer, M. H., & Adam, J. J. (2001). Distractor effects in pointing: The role of spatial layout. Experimental Brain Research, 136(4), 507–513.

    Article  PubMed  Google Scholar 

  • Fischer, M. H., & Campens, H. (2009). Pointing to numbers and grasping magnitudes. Experimental Brain Research, 192(1), 149–153.

    Article  PubMed  Google Scholar 

  • Fitousi, D. (2010). Dissociating between cardinal and ordinal and between the value and size magnitudes of coins. Psychonomic Bulletin and Review, 17(6), 889–894.‏.

    Article  PubMed  Google Scholar 

  • Fitousi, D., & Algom, D. (2018). A system factorial technology analysis is of the size-congruity effect: Implications for numerical cognition. Journal of Mathematical Psychology, 84, 57–73.

    Article  Google Scholar 

  • Gallistel, C. R., Gelman, R., & Cordes, S. (2006). The cultural and evolutionary history of the real numbers. In Levinson, S. & Jaisson, P (Ed.). Evolution and culture: A Fyssen Foundation symposium (17, pp. 247–274). Cambridge: MIT Press.

    Google Scholar 

  • Garner, W. R. (1952). An equal discriminability scale for loudness judgments. Journal of Experimental Psychology, 43, 232–238.

    Article  PubMed  Google Scholar 

  • Gescheider, G. A. (1997). Psychophysics: The fundamentals. Mahwah: Erlbaum.

    Google Scholar 

  • Gilboa, I. (2009). Theory of decision under uncertainty. Cambridge: Cambridge University Press.‏.

    Book  Google Scholar 

  • Goldfarb, L., Henik, A., Rubinsten, O., Bloch-David, Y., & Gertner, L. (2011). The numerical distance effect is task dependent. Memory and Cognition, 39, 1508–1517.

    Article  PubMed  Google Scholar 

  • Guilford, J. P. (1954). Psychometric methods.‏. New York: McGraw-Hill.

    Google Scholar 

  • Helson, H. (1964). Adaptation-level theory.‏. Oxford: Harper & Row.

    Google Scholar 

  • Henik, A., & Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory and Cognition, 10(4), 389–395.‏.

    Article  PubMed  Google Scholar 

  • Izard, V., & Dehaene, S. (2008). Calibrating the mental number line. Cognition, 106(3), 1221–1247.‏.

    Article  PubMed  Google Scholar 

  • Kaufmann, L., Koppelstaetter, F., Delazer, M., Siedentopf, C., Rhomberg, P., Golaszewski, S., & Ischebeck, A. (2005). Neural correlates of distance and congruity effects in a numerical Stroop task: An event-related fMRI study. Neuroimage, 25(3), 888–898.‏.

    Article  PubMed  Google Scholar 

  • Krajcsi, A. (2017). Numerical distance and size effects dissociate in Indo-Arabic number comparison. Psychonomic Bulletin and Review, 24(3), 927–934.

    Article  PubMed  Google Scholar 

  • Krajcsi, A., Lengyel, G., & Kojouharova, P. (2018). Symbolic number comparison is not processed by the analogue number system: Different symbolic and nonsymbolic numerical distance and size effects. Frontiers in Psychology, 9, 124.

    Article  PubMed  PubMed Central  Google Scholar 

  • Krause, F., Bekkering, H., & Lindemann, O. (2013). A feeling for numbers: Shared metric for symbolic and tactile numerosities. Frontiers in psychology, 4(7), 1–8.

    Google Scholar 

  • Krause, F., & Lindemann, O. (2014). Expyriment: A Python library for cognitive and neuroscientific experiments. Behavior Research Methods, 46(2), 416–428.

    Article  PubMed  Google Scholar 

  • Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.

    Article  PubMed  Google Scholar 

  • Leth-Steensen, C., & Marley, A. A. J. (2000). A model of response time effects in symbolic comparison. Psychological Review, 107(1), 62–100.

    Article  PubMed  Google Scholar 

  • Link, T., Nuerk, H.-G., & Moeller, K. (2014). On the relation between the mental number line and arithmetic competencies. The Quarterly Journal of Experimental Psychology, 67, 1597–1613.

    Article  PubMed  Google Scholar 

  • Marks, L. E. (1974). Sensory processes: The new psychophysics. New York: Academic Press.

    Google Scholar 

  • Marks, L. E., & Algom, D. (1998). Psychophysical scaling. In H. M. Birnbaum (Ed.), Measurement, judgment, and decision making (pp. 81–178). New York: Academic Press.

    Chapter  Google Scholar 

  • Moyer, R. S. (1973). Comparing objects in memory: Evidence suggesting an internal psychophysics. Perception and Psychophysics, 13(2), 180–184.‏.

    Article  Google Scholar 

  • Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality.‏. Nature, 215, 1519–1520.

    Article  PubMed  Google Scholar 

  • Namdar, G., Ganel, T., & Algom, D. (2018). The size congruity effect vanishes in grasping: Implications for the processing of numerical information. Scientific Reports, 8, 2723.

    Article  PubMed  PubMed Central  Google Scholar 

  • Norwich, K. H. (1992). Information, sensation, and perception. New York: Academic Press.

    Google Scholar 

  • Núñez, R. E. (2017). Is there really an evolved capacity for number? Trends in Cognitive Sciences, 21(6), 409–424.

    Article  PubMed  Google Scholar 

  • Pansky, A., & Algom, D. (2002). Comparative judgment of numerosity and numerical magnitude: Attention preempts automaticity. Journal of Experimental Psychology Learning Memory and Cognition, 28(2), 259.

    Article  PubMed  Google Scholar 

  • Parducci, A. (1965). Category judgment: A range-frequency model. Psychological Review, 72(6), 407–418‏.

    Article  PubMed  Google Scholar 

  • Pinel, P., Piazza, M., Le Bihan, D., & Dehaene, S. (2004). Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron, 41(6), 983–993.

    Article  PubMed  Google Scholar 

  • Pinheiro-Chagas, P., Dotan, D., Piazza, M., & Dehaene, S. (2017). Finger tracking reveals the covert stages of mental arithmetic. Open Mind, 1(1), 30–41‏.

    Article  PubMed  Google Scholar 

  • Restle, F., & Greeno, J. G. (1970). Introduction to mathematical psychology. Oxford: Addison-Wesley‏.

    Google Scholar 

  • Reynvoet, B., & Sasanguie, D. (2016). The symbol grounding problem revisited: A thorough evaluation of the ANS mapping account and the proposal of an alternative account based on symbol–symbol associations. Frontiers in Psychology, 7, 1581

    Article  PubMed  PubMed Central  Google Scholar 

  • Rips, L. J. (2013). How many is a zillion? Sources of number distortion. Journal of Experimental Psychology Learning Memory and Cognition, 39(4), 1257–1264‏.

    Article  PubMed  Google Scholar 

  • Rouder, J. N., & Geary, D. C. (2014). Children’s cognitive representation of the mathematical number line. Developmental Science, 17(4), 525–536‏.

    Article  PubMed  PubMed Central  Google Scholar 

  • Sasanguie, D., et al. (2014). The approximate number system is not predictive for symbolic number processing in kindergarteners. Quarterly Journal of Experimental Psychology, 67(2), 271–280.

    Article  Google Scholar 

  • Sasanguie, D., De Smedt, B., & Reynvoet, B. (2017). Evidence for distinct magnitude systems for symbolic and non-symbolic number. Psychological Research Psychologische Forschung, 81(1), 231–242.

    Article  PubMed  Google Scholar 

  • Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237–250‏.

    Article  PubMed  Google Scholar 

  • Siegler, R. S., Thompson, C. A., & Opfer, J. E. (2009). The logarithmic-to-linear shift: One learning sequence, many tasks, many time scales. Mind Brain and Education, 3(3), 143–150.

    Article  Google Scholar 

  • Slusser, E., & Barth, H. (2017). Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks. Journal of Experimental Child Psychology, 162, 181–198.

    Article  PubMed  Google Scholar 

  • Slusser, E. B., Santiago, R. T., & Barth, H. C. (2013). Developmental change in numerical estimation. Journal of Experimental Psychology General, 142(1), 193–208.

    Article  PubMed  Google Scholar 

  • Stevens, J. C. (1971). Psychophysics. In W. S. Cain & L. E. Marks (Eds.), Stimulus and sensation: Readings is sensory psychology. Boston: Little, Brown & Company.

    Google Scholar 

  • Stevens, S. S. (1975). Psychophysics. New York: Wiley‏.

    Google Scholar 

  • Van’t Noordende, J. E., Van Hoogmoed, A. H., Schot, W. D., & Kroesbergen, E. H. (2016). Number line estimation strategies in children with mathematical learning difficulties measured by eye tracking. Psychological Research Psychologische Forschung, 80(3), 368–378.

    Article  Google Scholar 

  • Verguts, T., Fias, W., & Stevens, M. (2005). A model of exact small-number representation. Psychonomic Bulletin and Review, 12(1), 66–80.‏.

    Article  PubMed  Google Scholar 

  • Verguts, T., & Van Opstal, F. (2005). Dissociation of the distance and size effects in one-digit numbers. Psychonomic Bulletin and Review, 12(5), 925–930.

    Article  PubMed  Google Scholar 

  • Verguts, T., & Van Opstal, F. (2014). A delta-rule model of numerical and non-numerical order processing. Journal of Experimental Psychology Human Perception and Performance, 40(3), 1092–1102.

    Article  PubMed  Google Scholar 

Download references

Acknowledgements

We thank James Townsend, Attila Krajcsi, and an anonymous referee for helpful comments on earlier versions.

Funding

No funding was received for this study.

Author information

Authors and Affiliations

  1. The School of Psychological Sciences, Tel-Aviv University, Tel Aviv, Israel

    Hofit Bar & Daniel Algom

  2. Department of Psychology, Achva Academic College, Arugot, Israel

    Hofit Bar & Daniel Algom

  3. Department of Psychology, Potsdam University, Potsdam, Germany

    Martin H. Fischer

Authors
  1. Hofit Bar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin H. Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Algom
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hofit Bar.

Ethics declarations

Conflict of interest

All authors declare that they do not have a conflict of interest.

Ethical standards

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Research involving human and animal participants

This article does not contain any studies with animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (XLSX 20 KB)

Supplementary material 2 (XLSX 80 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bar, H., Fischer, M.H. & Algom, D. On the linear representation of numbers: evidence from a new two-numbers-to-two positions task. Psychological Research 83, 48–63 (2019). https://doi.org/10.1007/s00426-018-1063-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00426-018-1063-y

Search

Navigation