Skip to main content

Response bias in numerosity perception at early judgments and systematic underestimation

Abstract

Mental number representation relies on mapping numerosity based on nonsymbolic stimuli to symbolic magnitudes. It is known that mental number representation builds on a logarithmic scale, and thus numerosity decisions result in underestimation. In the current study, we investigated the temporal dynamics of numerosity perception in four experiments by employing the response-deadline SAT procedure. We presented random number of dots and required participants to make a numerosity judgment by comparing the perceived number of dots to 50. Using temporal dynamics in numerosity perception allowed us to observe a response bias at early decisions and a systematic underestimation at late decisions. In all three experiments, providing feedback diminished the magnitude of underestimation, whereas in Experiment 3 the absence of feedback resulted in greater underestimation errors. These results were in accordance with the findings that suggested feedback is necessary for the calibration of the mental number representation.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Note that the percentage of removed trials was comparable across deadline conditions (e.g., 60 ms versus 700 ms) when the delay after cue onset was set to 600 ms or less. This might indicate that the participants waited to fully process the screen in early cues, whereas they responded before receiving the cue in the latest deadline condition (3,500 ms). However, when the delay after cue onset was set to 500 ms or less, the percentage of the remaining trials dropped to 83%, 90%, 94%, 94%, 92%, 93%, 90% for each deadline condition, respectively. That said, the asymptotes of the best fitting model did not differ while the speed parameters showed faster processing in general. The results of the additional analysis are presented in the supplementary materials for all experiments.

  2. 2.

    We would like to thank the reviewers for pointing out these possible explanations.

  3. 3.

    We would like to thank Reviewer 2 for pointing out this possibility.

References

  1. Anobile, G., Cicchini, G. M., & Burr, D. C. (2016). Number as a primary perceptual attribute: A review. Perception, 45(1/2), 5–31. https://doi.org/10.1177/0301006615602599

    Article  PubMed  Google Scholar 

  2. Bevan, W., & Turner, E. D. (1964). Assimilation and contrast in the estimation of number. Journal of Experimental Psychology, 67(5), 458–462. https://doi.org/10.1037/h0041141

    Article  PubMed  Google Scholar 

  3. Burr, D. C., Anobile, G., & Arrighi, R. (2018). Psychophysical evidence for the number sense. Philosophical Transactions of the Royal Society B: Biological Sciences, 373(1740), 20170045. https://doi.org/10.1098/rstb.2017.0045

    Article  Google Scholar 

  4. Castronovo, J., & Seron, X. (2007). Numerical estimation in blind subjects: Evidence of the impact of blindness and its following experience. Journal of Experimental Psychology: Human Perception and Performance, 33(5), 1089. https://doi.org/10.1037/0096-1523.33.5.1089

    Article  PubMed  Google Scholar 

  5. Cheyette, S. J., & Piantadosi, S. T. (2019). A primarily serial, foveal accumulator underlies approximate numerical estimation. Proceedings of the National Academy of Sciences of the United States of America, 116, 17729–17734.

    Article  Google Scholar 

  6. Cheyette, S. J., & Piantadosi, S. T. (2020). A unified account of numerosity perception. Nature Human Behaviour, 4(12), 1265–1272.

    Article  Google Scholar 

  7. 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(4), 698–707. https://doi.org/10.3758/BF03196206

    Article  Google Scholar 

  8. Crollen, V., & Seron, X. (2012). Over-estimation in numerosity estimation tasks: More than an attentional bias? Acta Psychologica, 140(3), 246–251. https://doi.org/10.1016/j.actpsy.2012.05.003

    Article  PubMed  Google Scholar 

  9. Crollen, V., Castronovo, J., & Seron, X. (2011). Under- and over-estimation: A bi-directional mapping process between symbolic and nonsymbolic representations of number? Experimental Psychology, 58, 39–49. https://doi.org/10.1027/1618-3169/a000064

    Article  PubMed  Google Scholar 

  10. Crollen, V., Grade, S., Pesenti, M., & Dormal, V. (2013). A common metric magnitude system for the perception and production of numerosity, length, and duration. Frontiers in Psychology, 4, 449. https://doi.org/10.3389/fpsyg.2013.00449

    Article  PubMed  PubMed Central  Google Scholar 

  11. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1/2), 1–42. https://doi.org/10.1016/0010-0277(92)90049-N

    Article  PubMed  Google Scholar 

  12. Dehaene S, Spelke E, Pinel P, Stanescu R, Tsivkin S. Sources of mathematical thinking: behavioral and brain-imaging evidence. Science. 1999 May 7;284(5416):970-4

  13. Dehaene, S. (2003). The neural basis of the Weber–Fechner law: A logarithmic mental number line. Trends in Cognitive Sciences, 7, 145147. https://doi.org/10.1016/S1364-6613(03)00055-X

    Article  Google Scholar 

  14. Dehaene, S. (2011). The number sense: How the mind creates mathematics (2nd ed.). Oxford University Press.

    Google Scholar 

  15. Dehaene, S., Izard, V., Spelke, E.S., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320(5880), 1217–1220. https://doi.org/10.1126/science.1156540

    Article  PubMed  PubMed Central  Google Scholar 

  16. DeWind, N. K., Park, J., Woldorff, M. G., & Brannon, E. M. (2019). Numerical encoding in early visual cortex. Cortex, 114, 76–89. https://doi.org/10.1016/j.cortex.2018.03.027

    Article  PubMed  Google Scholar 

  17. Dietrich, J. F., Huber, S., & Nuerk, H. C. (2015). Methodological aspects to be considered when measuring the approximate number system (ANS)–A research review. Frontiers in Psychology, 6, 295. https://doi.org/10.3389/fpsyg.2015.00295

    Article  PubMed  PubMed Central  Google Scholar 

  18. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314.

    Article  Google Scholar 

  19. Fornaciai, M., Cicchini, G. M., & Burr, D. C. (2016). Adaptation to number operates on perceived rather than physical numerosity. Cognition, 151, 63–67. https://doi.org/10.1016/j.cognition.2016.03.006

    Article  PubMed  PubMed Central  Google Scholar 

  20. Guillaume, M., & Gevers, W. (2016). Assessing the Approximate Number System: No relation between numerical comparison and estimation tasks. Psychological Research, 80(2), 248–258. https://doi.org/10.1007/s00426-015-0657-x

    Article  PubMed  Google Scholar 

  21. Guillaume, M., & Van Rinsveld, A. (2018). Comparing numerical comparison tasks: A meta-analysis of the variability of the weber fraction relative to the generation algorithm. Frontiers in Psychology, 9, 1694. https://doi.org/10.3389/fpsyg.2018.01694

    Article  PubMed  PubMed Central  Google Scholar 

  22. Indow, T., & Ida, M. (1977). Scaling of dot numerosity. Perception & Psychophysics, 22(3), 265–276. https://doi.org/10.3758/BF03199689

    Article  Google Scholar 

  23. Izard, V., & Dehaene, S. (2008). Calibrating the mental number line. Cognition, 106, 1221–1247. https://doi.org/10.1016/j.cognition.2007.06.004

    Article  PubMed  Google Scholar 

  24. Kılıç, A., & Öztekin, I. (2014). Retrieval dynamics of the strength based mirror effect in recognition memory. Journal of Memory and Language, 76, 158–173. https://doi.org/10.1016/j.jml.2014.06.009

    Article  Google Scholar 

  25. Kim, R. S., Seitz, A. R., & Shams, L. (2008). Benefits of stimulus congruency for multisensory facilitation of visual learning. PLoS One, 3(1), e1532

  26. Krueger, L. E. (1972). Perceived numerosity. Perception, & Psychophysics, 11(1), 5–9. https://doi.org/10.3758/BF03212674

    Article  Google Scholar 

  27. Krueger, L. E. (1982). Single judgments of numerosity. Perception & Psychophysics, 31, 175–182. https://doi.org/10.3758/BF03206218

    Article  Google Scholar 

  28. Krueger, L. E. (1984). Perceived numerosity: A comparison of magnitude production, magnitude estimation, and discrimination judgments. Perception & Psychophysics, 35(6), 536–542. https://doi.org/10.3758/BF03205949

    Article  Google Scholar 

  29. Mejias, S., & Schiltz, C. (2013). Estimation abilities of large numerosities in kindergartners. Frontiers in Psychology, 4, 518. https://doi.org/10.3389/fpsyg.2013.00518

    Article  PubMed  PubMed Central  Google Scholar 

  30. Mundy, E., & Gilmore, C. K. (2009). Children’s mapping between symbolic and nonsymbolic representations of number. Journal of Experimental Child Psychology, 103, 490–502. https://doi.org/10.1016/j.jecp.2009.02.003

    Article  PubMed  Google Scholar 

  31. Nieder, A. (2016). The neuronal code for number. Nature Reviews Neuroscience, 17(6), 366–382. https://doi.org/10.1038/nrn.2016.40

    Article  PubMed  Google Scholar 

  32. Norris, J. E., & Castronovo, J. (2016). Dot display affects approximate number system acuity and relationships with mathematical achievement and inhibitory control. PLOS ONE, 11(5). https://doi.org/10.1371/journal.pone.0155543

  33. Peirce, J., Gray, J. R., Simpson, S., MacAskill, M., Höchenberger, R., Sogo, H., Kastman. E., & Lindeløv, J. K. (2019) PsychoPy2: Experiments in behavior made easy. Behavioral Research Methods, 51, 195–203. https://doi.org/10.3758/s13428-018-01193-y

  34. Piazza, M. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551. https://doi.org/10.1016/j.tics.2010.09.008

    Article  PubMed  Google Scholar 

  35. Piazza, M., Pinel, P., Le Bihan, D., & Dehaene, S. (2007). A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron, 53(2), 293–305. https://doi.org/10.1016/j.neuron.2006.11.022

    Article  PubMed  Google Scholar 

  36. Price, J., Clement, L. M., & Wright, B. J. (2014). The role of feedback and dot presentation format in younger and older adults’ number estimation. Aging, Neuropsychology, and Cognition, 21(1), 68–98. https://doi.org/10.1080/13825585.2013.786015

    Article  Google Scholar 

  37. R Core Team. (2019). R: A language and environment for statistical computing [Computer software]. R Foundation for Statistical Computing. https://www.R-project.org/

  38. Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85(2), 59

    Article  Google Scholar 

  39. Ratcliff, R. (2006). Modeling response signal and response time data. Cognitive Psychology, 53(3), 195–237. https://doi.org/10.1016/j.cogpsych.2005.10.002

    Article  PubMed  PubMed Central  Google Scholar 

  40. Ratcliff, R., & McKoon, G. (2018). Modeling numerosity representation with an integrated diffusion model. Psychological Review, 125(2), 183–217. https://doi.org/10.1037/rev0000085

    Article  PubMed  Google Scholar 

  41. Reed, A. V. (1973). Speed-accuracy trade-off in recognition memory. Science, 181(4099), 574-576

    Article  Google Scholar 

  42. Reinert, R. M., Hartmann, M., Huber, S., & Moeller, K. (2019). Unbounded number line estimation as a measure of numerical estimation. PLOS ONE, 14(3). https://doi.org/10.1371/journal.pone.0213102

  43. Stevens S. S. (1959). Cross-modality validation of subjective scales for loudness, vibration, and electric shock. Journal of Experimental Psychology, 57(4), 201–209

    Article  Google Scholar 

  44. Stevens, S. S. (1966). Matching functions between loudness and ten other continua1. Perception & Psychophysics, 1(1), 5–8. https://doi.org/10.3758/BF03207813

    Article  Google Scholar 

  45. Teghtsoonian, R., & Teghtsoonian, M. (1978). Range and regression effects in magnitude scaling. Perception & Psychophysics, 24(4), 305–314. https://doi.org/10.3758/BF03204247

    Article  Google Scholar 

  46. Trick, L. M., & Pylyshyn, Z. W. (1994). Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision. Psychological Review, 101(1), 80.

    Article  Google Scholar 

  47. Van den Berg, R., Lindskog, M., Poom, L., & Winman, A. (2017). Recent is more: A negative time-order effect in nonsymbolic numerical judgment. Journal of Experimental Psychology: Human Perception and Performance, 43(6), 1084–1097. https://doi.org/10.1037/xhp0000387

    Article  PubMed  Google Scholar 

  48. Verguts, T., & Fias, W. (2004). Representation of number in animals and humans: A neural model. Journal of cognitive neuroscience, 16(9), 1493–1504. https://doi.org/10.1162/0898929042568497

    Article  PubMed  Google Scholar 

  49. 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

    Article  Google Scholar 

  50. Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1–B11. https://doi.org/10.1016/S0010-0277(99)00066-9

    Article  PubMed  Google Scholar 

Download references

Author note

This work has been supported by Middle East Technical University Scientific Research Projects Coordination Unit under grant number BAP-08-11-2017-036.

We would like to thank Sinem Aytaç, Hatice Dedetaş, Zeynep Erbaş and Sema Betül Türk for their assistance in data collection.

Parts of this work were presented in the International Meeting of the Psychonomic Society in 2018 and the 51st Annual Meeting of the Society for Mathematical Psychology.

Data link (https://osf.io/u3yz5/).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Aslı Kılıç.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

ESM 1

(PDF 97 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kılıç, A., İnan, A.B. Response bias in numerosity perception at early judgments and systematic underestimation. Atten Percept Psychophys (2021). https://doi.org/10.3758/s13414-021-02365-3

Download citation

Keywords

  • Numerosity perception
  • Mental number line
  • Speed–accuracy trade-off
  • Response deadline procedure