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18-01-2020 | Original Article

# Numerals do not need numerosities: robust evidence for distinct numerical representations for symbolic and non-symbolic numbers

Auteurs: Mila Marinova, Delphine Sasanguie, Bert Reynvoet

Gepubliceerd in: Psychological Research | Uitgave 2/2021

• Optie A:
• Optie B:

## Abstract

In numerical cognition research, it has traditionally been argued that the processing of symbolic numerals (e.g., digits) is identical to the processing of the non-symbolic numerosities (e.g., dot arrays), because both number formats are represented in one common magnitude system—the Approximate Number System (ANS). In this study, we abandon this deeply rooted assumption and investigate whether the processing of numerals and numerosities can be dissociated, using an audio-visual paradigm in combination with various experimental manipulations. In Experiment 1, participants performed four comparison tasks with large symbolic and non-symbolic numbers: (1) number word–digit (2) tones–dots, (3) number word–dots, (4) tones–digit. In Experiment 2, we manipulated the number range (small vs. large) and the presentation modality (visual–auditory vs. auditory–visual). Results demonstrated ratio effects (i.e., the signature of ANS being addressed) in all tasks containing numerosities, but not in the task containing numerals only. Additionally, a cognitive cost was observed when participants had to integrate symbolic and non-symbolic numbers. Therefore, these results provide robust (i.e., independent of presentation modality or number range) evidence for distinct processing of numerals and numerosities, and argue for the existence of two independent number processing systems.
Voetnoten
1
With respect to the credibility and the scientific integrity of our research, we report how we determined our sample size, all data exclusions (if any), all manipulations, and all measures in the study (Simmons, Nelson, & Simonsohn, 2012).

2
Because our study focused on comparing both symbolic and non-symbolic number pairs, we manipulated the relative difference (i.e., ratio) between the numbers, but not the absolute distance between them. As we mentioned in “Introduction”, the ratio and the distance are two very strongly related metrics.

3
Because the duration of the silence is dependent on the amount of tones that have to be presented within the stimulus timeframe, for these experiments, the shortest and the longest inter-tone intervals registered by the program were 20 ms and 1373 ms, respectively.

4
The BF10 is the ratio of the likelihood of the alternative hypothesis and the likelihood of the null hypothesis (while BF01 is simply the inverse ratio of these two likelihoods). For the more complicated models involving a larger numbers of factors (e.g., repeated measures ANOVA), we reported the BFInclusion (see Wagenmakers et al., 2018 for the rationale). According to the interpretation of Jeffreys (1961), BF values between 1 and 3 are considered as anecdotal evidence (“not worth more than a bare mention”, Jeffreys, 1961) for the alternative hypothesis, BF values between 3 and 10 are considered as moderate evidence, BF values between 10 and 30 are considered as strong evidence, BF values between 30 and 100 are considered very strong evidence, and BF values above 100 are considered as extreme evidence.

5
The confidence intervals (CI) around the effect sizes were computed with an SPSS plug in calculator by Karl Wuensch, freely available from http://​core.​ecu.​edu/​psyc/​wuenschk/​SPSS/​SPSS-Programs.​htm. Following the author`s recommendations (see Wuensch, 2009), and Steigner (2004), we report the 90% CI for partial eta squared ($$\eta_{\text{p}}^{2}$$), and the 95% CI for Cohen’s d.

6
In JASP, the highest number that can be displayed as BF is 1e+305. Once this value has exceeded, the computer automatically changes this into an infinity (see Love, 2015; Wagenmakers, 2015).

Literatuur
Barth, H., Kanwisher, N., & Spelke, E. (2003). The construction of large number representations in adults. Cognition, 86(3), 201–221. https://​doi.​org/​10.​1016/​S0010-0277(02)00178-6.
Brysbaert, M. (2007). The language-as-fixed-effect-fallacy: Some simple SPSS solutions to a complex problem (Version 2.0). London: Royal Holloway, University of London.
Brysbaert, M., & Stevens, M. (2018). Power analysis and effect size in mixed effects models: A tutorial. Journal of Cognition, 1(1), 1–20. https://​doi.​org/​10.​5334/​joc.​10.
Bulthé, J., De Smedt, B., & de Beeck, H. O. (2014). Format-dependent representations of symbolic and non-symbolic numbers in the human cortex as revealed by multi-voxel pattern analyses. Neuroimage, 87, 311–322. https://​doi.​org/​10.​1016/​j.​neuroimage.​2013.​10.​049.
Bulthé, J., De Smedt, B., & Op de Beeck, H. P. (2015). Visual number beats abstract numerical magnitude: Format-dependent representation of Arabic digits and dot patterns in human parietal cortex. Journal of Cognitive Neuroscience, 27(7), 1376–1387. https://​doi.​org/​10.​1162/​jocn_​a_​00787.
Cantlon, J. F., Libertus, M. E., Pinel, P., Dehaene, S., Brannon, E. M., & Pelphrey, K. A. (2009). The neural development of an abstract concept of number. Journal of Cognitive Neuroscience, 21(11), 2217–2229. https://​doi.​org/​10.​1162/​jocn.​2008.​21159.
Defever, E., Sasanguie, D., Gebuis, T., & Reynvoet, B. (2011). Children’s representation of symbolic and nonsymbolic magnitude examined with the priming paradigm. Journal of Experimental Child Psychology, 109(2), 174–186. https://​doi.​org/​10.​1016/​j.​jecp.​2011.​01.​002.
Dehaene, S. (2001). Précis of the number sense. Mind & Language, 16(1), 16–36. https://​doi.​org/​10.​1111/​1468-0017.​00154. CrossRef
Dehaene, S. (2007). Symbols and quantities in parietal cortex: Elements of a mathematical theory of number representation and manipulation. In P. Haggard & Y. Rossetti (Eds.), Attention and performance XXII. Sensori-motor foundations of higher cognition (pp. 527–574). Cambridge: Harvard University Press.
Dehaene, S., & Akhavein, R. (1995). Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology. Learning, Memory, and Cognition, 21(2), 314.
Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1(1), 83–120.
Dehaene, S., & Mehler, J. (1992). Cross-linguistic regularities in the frequency of number words. Cognition, 43(1), 1–29.
Faul, F., Erdfelder, E., Lang, A. G., & Buchner, A. (2007). G* Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39(2), 175–191. https://​doi.​org/​10.​3758/​BF03193146.
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. https://​doi.​org/​10.​1016/​j.​tics.​2004.​05.​002.
Finke, S., Kemény, F., Perchtold, C., Göbel., S., & Landerl, K. (2018). Same or different? The ERP signatures of uni- and cross- modal integration of number words and Arabic digits. Poster presented at the First Mathematical Cognition and Learning Society Conference, Oxford, UK.
Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44(1–2), 43–74.
Gebuis, T., & Reynvoet, B. (2011). Generating nonsymbolic number stimuli. Behavior Research Methods, 43(4), 981–986. https://​doi.​org/​10.​3758/​s13428-011-0097-5.
Gielen, I., Brysbaert, M., & Dhondt, A. (1991). The syllable-length effect in number processing is task-dependent. Perception & Psychophysics, 50(5), 449–458. CrossRef
Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the “Number Sense”: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44(5), 1457. https://​doi.​org/​10.​1037/​a0012682.
Jarosz, A. F., & Wiley, J. (2014). What are the odds? A practical guide to computing and reporting Bayes factors. The Journal of Problem Solving, 7(1), 2. https://​doi.​org/​10.​7771/​1932-6246.​1167. CrossRef
Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford: Oxford University Press.
Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. The American journal of psychology, 62(4), 498–525.
Koechlin, E., Naccache, L., Block, E., & Dehaene, S. (1999). Primed numbers: Exploring the modularity of numerical representations with masked and unmasked semantic priming. Journal of Experimental Psychology: Human Perception and Performance, 25(6), 1882–1905.
Krajcsi, A., Lengyel, G., & Kojouharova, P. (2016). The source of the symbolic numerical distance and size effects. Frontiers in Psychology, 7, 1795. https://​doi.​org/​10.​3389/​fpsyg.​2016.​01795.
Krajcsi, A., Lengyel, G., & Kojouharova, P. (2018). Symbolic number comparison is not processed by the analog number system: Different symbolic and non-symbolic numerical distance and size effects. Frontiers in Psychology, 9, 124. https://​doi.​org/​10.​3389/​fpsyg.​2018.​00124.
Kutter, E. F., Bostroem, J., Elger, C. E., Mormann, F., & Nieder, A. (2018). Single neurons in the human brain encode numbers. Neuron, 100(3), 753–761. https://​doi.​org/​10.​1016/​j.​neuron.​2018.​08.​036.
Love, J. (2015). Infinite number for BF Inclusion [Online discussion forum]. https://​github.​com/​jasp-stats/​jasp-desktop/​issues/​1039. Accessed 1 Feb 2019.
Lyons, I. M., Ansari, D., & Beilock, S. L. (2012). Symbolic estrangement: Evidence against a strong association between numerical symbols and the quantities they represent. Journal of Experimental Psychology: General, 141(4), 635. https://​doi.​org/​10.​1037/​a0027248. CrossRef
Marinova, M., Sasanguie, D., & Reynvoet, B. (2018). Symbolic estrangement or symbolic integration of numerals with quantities: Methodological pitfalls and a possible solution. PLoS One. https://​doi.​org/​10.​1371/​journal.​pone.​0200808.
Moeller, K., Huber, S., Nuerk, H. C., & Willmes, K. (2011). Two-digit number processing: Holistic, decomposed or hybrid? A computational modelling approach. Psychological Research, 75(4), 290–306. https://​doi.​org/​10.​1007/​s00426-010-0307-2.
Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520. https://​doi.​org/​10.​1038/​2151519a0.
Nieder, A. (2016). The neuronal code for number. Nature Reviews Neuroscience, 17(6), 366. https://​doi.​org/​10.​1038/​nrn.​2016.​40.
Nieder, A., & Dehaene, S. (2009). Representation of number in the brain. Annual Review of Neuroscience, 32, 185–208. https://​doi.​org/​10.​04122/​annurev.​neuro.​051508.​135550.
Nuerk, H. C., & Willmes, K. (2005). On the magnitude representations of two-digit numbers. Psychology Science, 47(1), 52–72.
Núñez, R. E. (2017). Is there really an evolved capacity for number? Trends in Cognitive Sciences, 21(6), 409–424. https://​doi.​org/​10.​1016/​j.​tics.​2017.​03.​005.
Philippi, T. G., van Erp, J. B. F., & Werkhoven, P. J. (2008). Multisensory temporal numerosity judgment. Brain Research, 1242, 116–125. https://​doi.​org/​10.​1016/​j.​brainres.​2008.​05.​056.
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.
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, 293–305. https://​doi.​org/​10.​1016/​j.​neuron.​2006.​11.​022.
Reynvoet, B., Notebaert, K., & Van den Bussche, E. (2011). The processing of two-digit numbers depends on task instructions. Zeitschrift für Psychologie/Journal of Psychology, 219(1), 37–41. https://​doi.​org/​10.​1027/​2151-2604/​a000044. CrossRef
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. https://​doi.​org/​10.​3389/​fpsyg.​2016.​01581.
Sasanguie, D., De Smedt, B., Defever, E., & Reynvoet, B. (2012). Association between basic numerical abilities and mathematics achievement. British Journal of Developmental Psychology, 30(2), 344–357. https://​doi.​org/​10.​1111/​j.​2044-835X.​2011.​02048.​x. CrossRef
Sasanguie, D., De Smedt, B., & Reynvoet, B. (2017). Evidence for distinct magnitude systems for symbolic and non-symbolic number. Psychological Research, 81(1), 231–242. https://​doi.​org/​10.​1007/​s00426-015-0734-1.
Simmons, J. P., Nelson, L. D., & Simonsohn, U. (2012). A 21 Word solution. SSRN Electronic Journal, 1–4. https://​doi.​org/​10.​2139/​ssrn.​2160588.
Steiger, J. H. (2004). Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological Methods, 9(2), 164–182. https://​doi.​org/​10.​1037/​1082-989X.​9.​2.​164.
Tokita, M., & Ishiguchi, A. (2012). Behavioral evidence for format-dependent processes in approximate numerosity representation. Psychonomic Bulletin and Review, 19(2), 285–293. https://​doi.​org/​10.​3758/​s13423-011-0206-6.
Tokita, M., & Ishiguchi, A. (2016). Precision and bias in approximate numerical judgment in auditory, tactile, and cross-modal presentation. Perception, 45(1–2), 56–70. https://​doi.​org/​10.​1177/​030100661559688.
Tokita, M., Ashitani, Y., & Ishiguchi, A. (2013). Is approximate numerical judgment truly modality-independent? Visual, auditory, and cross-modal comparisons. Attention, Perception, and Psychophysics, 75(8), 1852–1861. https://​doi.​org/​10.​3758/​s13414-013-0526-x. CrossRef
Van Hoogmoed, A. H., & Kroesbergen, E. H. (2018). On the difference between numerosity processing and number processing. Frontiers in Psychology, 9, 1650. https://​doi.​org/​10.​3389/​fpsyg.​2018.​01650.
Verguts, T., Fias, W., & Stevens, M. (2005). A model of exact small-number representation. Psychonomic Bulletin & Review, 12(1), 66–80. https://​doi.​org/​10.​3758/​BF03196349. CrossRef
Vos, H., Sasanguie, D., Gevers, W., & Reynvoet, B. (2017). The role of general and number-specific order processing in adults’ arithmetic performance. Journal of Cognitive Psychology, 29(4), 469–482. https://​doi.​org/​10.​1080/​20445911.​2017.​1282490. CrossRef
Wagenmakers, E. J. (2015). Zero-inclusion Probabilities When Multiple Bf10s Are Infinite [Online discussion forum]. https://​github.​com/​jasp-stats/​jasp-desktop/​issues/​771. Accessed 1 Feb 2019.
Wagenmakers, E. J., Love, J., Marsman, M., Jamil, T., Ly, A., Verhagen, A. J., & Morey, R. D. (2018a). Bayesian statistical inference for psychological science. Part II: Example applications with JASP. Psychonomic Bulletin & Review, 25(1), 58–76. https://​doi.​org/​10.​3758/​s13423-017-1323-7. CrossRef
Wagenmakers, E. J., Marsman, M., Jamil, T., Ly, A., Verhagen, J., Love, J., & Matzke, D. (2018b). Bayesian inference for psychology. Part I: Theoretical advantages and practical ramifications. Psychonomic Bulletin & Review, 25(1), 35–57. https://​doi.​org/​10.​3758/​s13423-017-1343-3. CrossRef
Wuensch, K. (2009). Confidence intervals for Eta-squared and RMSSE. http://​core.​ecu.​edu/​psyc/​wuenschk/​StatsLessons.​htm. Accessed 1 Feb 2019.
Metagegevens
Titel
Numerals do not need numerosities: robust evidence for distinct numerical representations for symbolic and non-symbolic numbers
Auteurs
Mila Marinova
Delphine Sasanguie
Bert Reynvoet
Publicatiedatum
18-01-2020
Uitgeverij
Springer Berlin Heidelberg
Gepubliceerd in
Psychological Research / Uitgave 2/2021
Print ISSN: 0340-0727
Elektronisch ISSN: 1430-2772
DOI
https://doi.org/10.1007/s00426-019-01286-z

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