Evidence for distinct magnitude systems for symbolic and non-symbolic number

Abstract

Cognitive models of magnitude representation are mostly based on the results of studies that use a magnitude comparison task. These studies show similar distance or ratio effects in symbolic (Arabic numerals) and non-symbolic (dot arrays) variants of the comparison task, suggesting a common abstract magnitude representation system for processing both symbolic and non-symbolic numerosities. Recently, however, it has been questioned whether the comparison task really indexes a magnitude representation. Alternatively, it has been hypothesized that there might be different representations of magnitude: an exact representation for symbolic magnitudes and an approximate representation for non-symbolic numerosities. To address the question whether distinct magnitude systems exist, we used an audio–visual matching paradigm in two experiments to explore the relationship between symbolic and non-symbolic magnitude processing. In Experiment 1, participants had to match visually and auditory presented numerical stimuli in different formats (digits, number words, dot arrays, tone sequences). In Experiment 2, they were instructed only to match the stimuli after processing the magnitude first. The data of our experiments show different results for non-symbolic and symbolic number and are difficult to reconcile with the existence of one abstract magnitude representation. Rather, they suggest the existence of two different systems for processing magnitude, i.e., an exact symbolic system next to an approximate non-symbolic system.

Appendix: Experiment 1: Reaction time results

Control audio–visual matching tasks

Word–color matching task

Median reaction time (RT) on the correct responses of this task was 443.38 ms (SD = 79.28 ms).

Sound–letter matching task

Median RTs on the ‘non-match’ trials were submitted to a repeated measures analysis of variance (ANOVA) with distance (two levels: 1 and 4) as within-subject variable. There was no main effect of distance, F < 1.

Numerical audio–visual matching tasks

The median RTs of the numerical audio–visual matching tasks are shown per ratio and separately for the numbers within and outside the subitizing range in Table 1. The median RTs on the ‘non-match’ trials were submitted to a repeated measures analysis of variance (ANOVA) with task (four levels: number word–digit matching, number word–dots matching, digits–tones matching and tones–dots matching), ratio (two levels: small vs large) and number range (two levels: within vs outside subitizing range) as within-subject variables.²

All main effects were significant: there was a main effect of task, F(3,30) = 51.902, p < .0001, $\eta_{p}^{2}$ = .838, a main effect of ratio, F(1,32) = 29.038, p < .0001, $\eta_{p}^{2}$ = .476, and a main effect of number range, F(1,32) = 14.377, p < .0001, $\eta_{p}^{2}$ = .310. Moreover, there was a significant interaction between task and ratio, F(3,30) = 18.827, p < .0001, $\eta_{p}^{2}$ = .653, and between task and number range, F(3,30) = 16.103, p < .0001, $\eta_{p}^{2}$ = .617, which were in turn embedded in a three-way interaction between task, ratio and number range, F(3,30) = 6.146, p = .002, $\eta_{p}^{2}$ = .381. To disentangle these interactions, separate analyses per task were conducted (and post hoc analyses per number range if necessary).

Number word–digit matching task

There was no main effect of ratio, F < 1. There was a main effect of number range, F(1,33) = 8.671, p < .001, $\eta_{p}^{2}$ = .208, showing faster reaction times on the trials with numbers from within the subitizing range. There was no interaction between ratio and number range, F < 1.

Number word–dots matching task

There was a main effect of ratio, F(1,33) = 46.033, p < .0001, $\eta_{p}^{2}$ = .582, demonstrating faster reaction times on the small ratio’s than on the large ratio’s. There was also a main effect of number range, F(1,33) = 43.639, p < .0001, $\eta_{p}^{2}$ = .569, demonstrating much faster reaction times on the trials with the smallest numbers (i.e., within the subitizing range). There was also an interaction between ratio and number range, F(1,33) = 19.737, p < .0001, $\eta_{p}^{2}$ = .374. Post hoc paired t tests showed significant ratio effects in both number ranges: t(33) = 5.231, p < .0001, and t(33) = 5.649, p < .0001, for within and outside the subitizing range, respectively, although visual inspection of the data suggested a larger ratio effect in the larger number range.

Tones–digit matching task

There was a main effect of ratio, F(1,33) = 9.535, p < .001, $\eta_{p}^{2}$ = .224, demonstrating faster reaction times on the small ratio’s than on the large ratio’s. There was no main effect of number range and no ratio × number range interaction.

Tones–dots matching task

There was a main effect of ratio, F(1,32) = 19.986, p < .0001, $\eta_{p}^{2}$ = .384, demonstrating faster reaction times on the small ratio’s than on the large ratio’s. There was no effect of number range and no ratio × number range interaction.

Numerical ratio effects

The size of the ratio effect was computed for each participant by subtracting the average reaction time on trials with the most difficult ratios (i.e. 0.75 and 0.78) from the average reaction time on trials with the easiest ratios (i.e. 0.50 and 0.56). This difference was then divided by the average reaction time on trials with the easiest ratios to correct for individual differences in reaction times (see Holloway & Ansari, 2009; Sasanguie et al., 2013 for a similar method). We compared the size of the ratio effect in the number word–digit matching task with the weakest ratio effect in one of the other three numerical tasks, to statistically test the absence of a ratio effect in the number word–digit task. Paired t tests showed that both ratio effects significantly differed from each other, t(33) = 3.991, p < .0001.