## Abstract

Research on multi-digit number processing suggests that, in Arabic numerals, their place-value magnitude is automatically activated, whenever a magnitude-relevant task was employed. However, so far, it is unknown, whether place-value is also activated when the target task is magnitude-irrelevant. The current study examines this question using the parity congruency effect in two-digit numbers: It describes that responding to decade-digit parity congruent numbers (e.g., 35, 46; same parity of decades and units) is faster than to decade-digit parity incongruent numbers (e.g., 25; 36; different parities of decades and units). Here we investigate the (a-) symmetry of the parity congruency effect; i.e. whether it makes a difference whether participants are assessing the parity of the unit digit or the decade digit. We elaborate, how and why such an asymmetry is related to place-value processing, because the parity of the unit digit only interferes with the parity of the decade digit, while the parity of the decade digit interferes with both the parity of the unit digit and the integrated parity of the whole two-digit number. We observed a significantly larger parity congruency effect in the decade parity decision than in the unit parity decision. This suggests that automatic place-value processing also takes place in a typical parity judgment task, in which magnitude is irrelevant. Finally, because of the cross-lingual design of the study, we can show that these results and their implications were language-independent.

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## Notes

- 1.
Place-value activation could in principle be further distinguished in approximate and exact place-value activation. Approximate place-value activation would refer to the approximate value of number magnitude on a fuzzy mental number line (Dehaene, 2001; Dehaene, Dupoux, and Mehler, 1990). A person (e.g., a child in a number line estimation task) would know that 92 is somehow larger than 29, which is comprised of the same digits, but on different positions, but may not be able to locate it exactly on a number line. For exact place-value activation, the exact value as indexed by the Arabic number system is derived. For the number 72, one would know that it is an even number, a multiplication table result of 8*9, and not just some approximate magnitude between 70 and 80. The parity derived from such an exact integration of place-value activation of both digits (i.e., 90 and 2 for 92) is termed “integrated parity” or sometimes “place-value integration” in this manuscript.

- 2.
This setup allows examining the SNARC effect (Dehaene et al., 1993) as well. This was not the main objective of this experiment, but we report the results in Appendix 1.

- 3.
- 4.
We would like to thank Attila Krajcsi for pointing out this alternative explanation.

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## Acknowledgements

We would like to thank all the participants. This research was funded by a DFG grant [NU 265/3-1] to HCN supporting KC and MS. KC, MS, and HCN are further supported by the LEAD Graduate School and Research Network [GSC1028], which is funded within the framework of the Excellence Initiative of the German federal and state governments. MS is also supported by the Institutional Strategy of the University of Tübingen (Deutsche Forschungsgemeinschaft, ZUK 63). Finally, we thank our assistants Florine Winkler, Lia Heubner and Marie-Lene Schlenker, who helped with data collection and Julianne Skinner for proofreading the manuscript.

## Funding

This research was funded by a DFG grant [NU 265/3-1] to HCN supporting KC and MS.

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## Appendices

### Appendix 1: The SNARC effect

In this additional analysis, we tested for the presence of the SNARC effect (Spatial-Numerical Association of Response Codes; Dehaene et al., 1993). This term refers to the observation that small magnitude numbers are responded to faster with left-side responses, whereas large magnitude numbers are responded to faster with right-side responses (see Fischer & Shaki, 2014 for a current review; Wood, Willmes, Nuerk, & Fischer, 2008 for a meta-analysis). The SNARC effect is considered as a behavioural signature of the semantic processing of numerical magnitude (Fias et al., 1996). The presence of the SNARC effect was evaluated using the method proposed by Fias et al. (1996). Namely, for each participant separately we calculated regression slopes where dRT (differences in right-hand–left-hand reaction times) were regressed on number magnitude. Slopes that are more negative represent a stronger SNARC effect. In the case of two-digit numbers, three types of SNARC effects can be calculated (Huber et al., 2016): (a) an overall SNARC effect—whole numerical magnitude considered, (b) a decade SNARC effect—decade number considered (collapsed across unit numbers), (c) a unit SNARC effect—unit number considered (collapsed across decade numbers). Each type of the SNARC slope was calculated separately for decade parity and unit parity condition. None of the SNARC slopes differed between English and German speakers (*t*s < 0.5, *p*s ≥ 0.630), thus the data were collapsed across language groups. Results are summarized in Table 5.

A significant decade SNARC effect was found in the decade parity condition and a significant unit SNARC effect was found in the unit parity condition. The overall SNARC effect was only observed in the decade condition, but this might be due to a measurement artefact: decade digit largely determines the overall magnitude of a two-digit number. Note, however, that in principle, both decade and unit SNARC effects can be observed for multi-digit numbers in an appropriate setting (Weis et al., 2018).

Additionally, we calculated reliabilities of each SNARC effect using split-half method and adjusting for double test length with Spearman-Brown formula. Specifically, we adapted scripts described by (Cipora, van Dijck, et al., 2019), which are available at https://osf.io/n7szg/. In general, reliabilities for those conditions in which the SNARC effect was present were satisfactory and did not differ from values reported elsewhere in the literature. However, in those conditions in which SNARC was not significant, the reliabilities were very low or even negative. Please note that due to properties of Spearman-Brown formula, the negative values are more affected than positive ones: denominator of the formula is 1 + reliability. This is the reason why a relatively low negative correlation between halves (−0.30) was amplified to very low one of −0.86. This value is hardly interpretable (as any negative reliability estimate), and we present it for completeness only.

A 2 (Target digit: unit parity vs. decade parity) × 2 (SNARC type: unit SNARC vs. decade SNARC) repeated-measures ANOVA on mean SNARC slopes revealed no main effect of Target digit, *F*(1, 46) = 0.44, *p* = 0.510, *η*
^{2}_{p}
= 0.01, and no main effect of the SNARC type, *F*(1, 46) = 3.12, *p* = 0.084, *η*
^{2}_{p}
= 0.06. Crucially, there was a robust interaction between Target digit and SNARC type, *F*(1, 46) = 21.55, *p* < 0.001, *η*
^{2}_{p}
= 0.32. The SNARC was present only for target digits which were task-relevant (see Table 5 for descriptive results). An asymmetry was observed for the overall SNARC effect. It was only present in the decade parity condition. The overall SNARC slopes differed significantly between conditions, *t*(46) = −3.05, *p *= 0.004, *d* = 0.44.

These findings imply that spatial mapping can only be observed for numbers that were relevant in a given condition. The irrelevant numbers (unit number in the decade parity condition and decade number in the unit parity condition) did not evoke spatial mapping indexed by the SNARC effect.

In the last step of the analysis, we investigated correlations between slopes. In both conditions, we observed extremely strong correlations between decade SNARC slopes and overall SNARC slopes (*r*s > 0.92). This is due to the fact that decade magnitude plays a crucial role in overall number magnitude, thus these correlations are trivial. The overall SNARC slope in the unit condition correlated weakly with the unit SNARC slope from the unit condition (*r* = 0.32, *p* = 0.028). Interestingly, the two most robust SNARC effects, namely unit SNARC slope in the unit parity condition and the decade SNARC slope in the decade parity condition did not correlate with each other (*r* = 0.03, *p* = 0.859). None of the other correlations reached significance.

### Appendix 2: Complete ANOVA model

### Reaction times

Additionally, we conducted an ANOVA model considering all factors included in the design. Therefore, a 2 (Language: German vs. English, between subject) × 2 (Hand: right vs. left) × 2 (Parity of the target number: odd vs. even) × 2 (Congruency: congruent vs. incongruent) × 2 (Target digit: unit vs. decade) mixed-design ANOVA was conducted (cf. Table 6). Effects reported here reflect those reported in the main text, however, for the sake of completeness, we report them here again.

Right-hand responses (539 ms, SD = 64 ms) were significantly quicker than left-hand responses (544 ms, SD = 64 ms; i.e., a main effect of hand). Responses to even numbers were significantly faster than responses to odd numbers (535 ms, SD = 63 ms and 547 ms, SD = 64 ms, respectively; i.e., a main effect of parity). Congruent trials were responded to quicker (533 ms, SD = 62 ms) than incongruent trials (549 ms, SD = 64 ms; i.e., a main effect of congruency). Reaction times did not differ depending on Target digit (i.e., no main effect of Target digit). Thus, we replicated effects typically observed in such setups, that is, the dominant hand advantage, the odd effect (Hines, 1990) as well as the parity congruency effect.

There was no main effect of language and this factor did not interact with any other factor (cf. Table 6), thus not confirming our predictions on linguistic influences on place-value processing.

As concerns the automatic activation of integrated parity, there was a significant Congruency × Target digit interaction (cf. Fig. 4, panel A). Incongruent trials were responded to significantly slower in the decade condition compared to the unit condition. To disambiguate this interaction, the congruency effect was calculated separately for each target digit by subtracting RTs incongruent—RTs congruent. Subsequently, the congruency effect was tested against zero with one-sample *t*-tests. In both conditions, decade and unit parity judgement, the congruency effect was highly significant (*p*s < 0.001). However, the congruency effect differed significantly between conditions, *t*(46) = 2.54, *p* = 0.014, *d* = 0.37 with a larger effect size in the decade condition than in the unit condition (Cohen’s *d* of 1.54 and 1.29, respectively). This is in line with the idea of automatic processing of integrated parity.

The significant Hand × Congruency × Target digit interaction indicated that the effect of interest differed depending on the responding hand. To disambiguate this three-way interaction, the Target digit × Congruency interactions were tested for each hand separately. No interaction was present in the case of left-hand responses, *F*(1,46) = 0.66, *p* = 0.420, *η*
^{2}_{p}
= 0.01 (cf. Fig. 4, panel B). For right-hand responses, this interaction was robust, *F*(1,46) = 15.02, *p *< 0.001, *η*
^{2}_{p}
= 0.25. A follow-up one-sample *t* test indicated that the congruency effect was robust in both conditions (*p*s < 0.001) for right hand responses, but the congruency effect differed significantly between conditions, *t*(46) = 3.88, *p* < 0.001, *d* = 0.5 (cf. Fig. 4, panel C) with a larger effect size for the decade condition than for the unit condition (Cohen’s *d* of 1.52 and 1.07, respectively).

Interestingly, there was also a significant Hand × Congruency interaction. As already suggested by the results of the three-way interaction described above, the congruency effect was more pronounced in right-hand responses compared to left-hand responses (cf. Fig. 4, panels B and C). For both hands the congruency effect was robust (*p*s < 0.001), however, the effect was more pronounced for right-hand responses than for left-hand responses (Cohen’s *d* of 1.73 and 1.59, respectively). The difference between conditions was also significant, *t*(46) = 2.04, *p* = 0.048, *d* = 0.30.

There was also a significant Parity × Congruency interaction, whereby even numbers were more affected by the congruency effect than odd numbers. The congruency effect was robust in both conditions (*p*s < 0.001, Cohen’s *d* of 1.68 and 1.64 for even and odd numbers, respectively) and the difference between the congruency effects for even and odd numbers was also significant, *t*(46) = 2.25, *p* = 0.029, *d* = 0.33.

The Hand × Parity interaction was not significant, thus revealing no MARC effect (see Nuerk et al., 2004). This interaction was also not modulated by the Target digit (non-significant three-way Hand × Parity × Target digit interaction). No other interactions reached significance.

### Accuracies

Similar analysis for accuracies is summarized in Table 7.

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Cipora, K., Soltanlou, M., Smaczny, S. *et al.* Automatic place-value activation in magnitude-irrelevant parity judgement.
*Psychological Research* **85, **777–792 (2021). https://doi.org/10.1007/s00426-019-01268-1

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