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Counting is a spatial process: evidence from eye movements

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Abstract

Spatial–numerical associations (small numbers—left/lower space and large numbers—right/upper space) are regularly found in simple number categorization tasks. These associations were taken as evidence for a spatially oriented mental number line. However, the role of spatial–numerical associations during more complex number processing, such as counting or mental arithmetic is less clear. Here, we investigated whether counting is associated with a movement along the mental number line. Participants counted aloud upward or downward in steps of 3 for 45 s while looking at a blank screen. Gaze position during upward counting shifted rightward and upward, while the pattern for downward counting was less clear. Our results, therefore, confirm the hypothesis of a movement along the mental number line for addition. We conclude that space is not only used to represent number magnitudes but also to actively operate on numbers in more complex tasks such as counting, and that the eyes reflect this spatial mental operation.

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Notes

  1. The algorithm checks the dispersion of consecutive data points in a moving window by summing the differences between the points’ maximum and minimum x and y values ([max(x) – min(x)] + [max(y) – min(y)]). If the sum is below 100 pixels, the window represents a fixation and expands until the sum exceeds 100 pixels. The final window is registered as fixation with a duration corresponding to the interval between the timestamps of the first and last included sample. The centroid of the included points determines the x and y coordinate of the fixation.

  2. Seven participants indeed performed their last few downward counting steps in the negative range (on average 4.2 counting steps, which cover approximately 15 % of their counting time). Counting downward in the negative range is similar to counting upwards and might potentially influence the hypothesized mental movement. However, visual inspection of the last part of the gaze path during downward counting for these seven participants showed no systematically different pattern when compared to the terminal gaze path of the other participants, and these cases were not treated differently. From the 11 participants who did not reach the negative range, only two had the number “2” as final counting result (which is the last number before reaching the negative range). These two participants reached “2” at the end of their counting time, as noted by the experimenter, suggesting that no participant stopped counting when reaching zero.

  3. For sum-coded predictors (addition = 1, subtraction = −1), two times the estimate (104 pixels) reflect the mean difference between upward and downward counting. Note that this value is slightly different from the value we reported above (111 pixels); this difference is mainly driven by the different averaging procedure (over 1-s time points).

  4. To validate the use of pupil size as indicator of cognitive effort, we first conducted another linear mixed model analysis with pupil size as fixed effect and participant as random intercept on response times, separately for addition and subtraction trials. Consistent with previous work (e.g., Kahneman et al., 1969; Nakayama et al., 2002), pupil size was a significant predictor of response times for both addition and subtraction trials (both ps < .001); response time increased with increasing pupil size, confirming that pupil size reliably reflects task difficulty.

  5. Even though the counting task still requires alternating between addition and subtraction trials (as in Anelli et al., 2014; Lugli et al., 2013), participants performed continuous additions and subtractions within one trial before they switched to the other operation.

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Acknowledgments

This research was funded by the Swiss National Science Foundation (P2BEP1_152104).

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  1. Division of Cognitive Sciences, University of Potsdam, Karl-Liebknecht-Strasse 24-25 House 14, 14476, Potsdam OT Golm, Germany

    Matthias Hartmann & Martin H. Fischer

  2. Department of Psychology, University of Bern, Bern, Switzerland

    Matthias Hartmann & Fred W. Mast

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Hartmann, M., Mast, F.W. & Fischer, M.H. Counting is a spatial process: evidence from eye movements. Psychological Research 80, 399–409 (2016). https://doi.org/10.1007/s00426-015-0722-5

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