Abstract
Studies have found that spatial-numerical associations could extend to arithmetic. Addition leads to rightward shift in spatial attention while subtraction leads to leftward shift (e.g., Knops et al. 2009; McCrink et al. 2007; Pinhas & Fischer 2008), which is consistent with the hypothesis of static mental number line (MNL) for arithmetic. The current investigation tested the hypothesis of dynamic mental number line which was shaped by the relative magnitudes of two operands in simple arithmetic. Horizontal and vertical electrooculograms (HEOG and VEOG) during simple arithmetic were recorded. Results showed that the direction of eye movements was dependent on the relative magnitudes of two operands. Subtraction was associated with larger rightward eye movements than addition (Experiment 1), and smaller-operand-first addition (e.g., 2+9) was associated with larger rightward eye movement than larger-operand-first addition (e.g., 9+2) only when the difference of two operands was large (Experiment 2). The results suggest that the direction of the mental number line could be dynamic during simple arithmetic, and that the eyes move along the dynamic mental number line to search for solutions.
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References
Bächtold, D., Baumüller, M., & Brugger, P. (1998). Stimulus-response compatibility in representational space. Neuropsychologia, 36(8), 731–735. doi:10.1016/S0028-3932(98)00002-5.
Butterworth, B., Zorzi, M., Girelli, L., & Jonckheere, A. R. (2001). Storage and retrieval of addition facts: the role of number comparison. Q. J. Exp. Psychol. Sect. A., 54(4), 1005–1029. doi:10.1080/713756007.
Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. J. Exp. Psychol. Gen., 122(3), 371–396. doi:10.1037/0096-3445.122.3.371.
Fischer, M. H., Castel, A. D., Dodd, M. D., & Pratt, J. (2003). Perceiving numbers causes spatial shifts of attention. Nat. Neurosci., 6(6), 555–556. doi:10.1038/nn1066.
Fischer, M. H., Fias, MH: Spatial representation of numbers. In: The handbook of mathematical cognition, Psychology Press, New York (2005).
Fischer, M. H., Mills, R. A., & Shaki, S. (2010). How to cook a SNARC: number placement in text rapidly changes spatial–numerical associations. Brain Cogn., 72(3), 333–336. doi:10.1016/j.bandc.2009.10.010.
Fischer, M. H., & Shaki, S. (2014). Spatial associations in numerical cognition–from single digits to arithmetic. Q. J. Exp. Psychol., 67(8), 1461–1483. doi:10.1080/17470218.2014.927515.
Galton, F. (1880a). Visualised numerals. Nature, 21, 252–256.
Galton, F. (1880b). Visualised numerals. Nature, 21, 494–495.
Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychol Rev, 79(4), 329–343. doi:10.1037/h0032950.
Hartmann, M. (2015). Numbers in the eye of the beholder: what do eye movements reveal about numerical cognition? Cogn. Process, 16, 245–248. doi:10.1007/s10339-015-0716-7.
Hartmann, M., Mast, F. W., & Fischer, M. H. (2015). Spatial biases during mental arithmetic: evidence from eye movements on a blank screen. Front Psychol., 6, 12. doi:10.3389/fpsyg.2015.00012.
Hartmann, M., Mast, F. W., Fischer, M. H: (in press). Counting is a spatial process: evidence from eye movements. Psychol. Res. doi:10.1007/s00426-015-0722-5.
Hung, Y., Hung, D. L., Tzeng, O. J.-L., & Wu, D. H. (2008). Flexible spatial mapping of different notations of numbers in Chinese readers. Cognition, 106(3), 1441–1450. doi:10.1016/j.cognition.2007.04.017.
Klein, E., Huber, S., Nuerk, H. C., & Moeller, K. (2014). Operational momentum affects eye fixation behaviour. Q. J. Exp. Psychol., 67(8), 1614–1625. doi:10.1080/17470218.2014.902976.
Knops, A., Dehaene, S., Berteletti, I., & Zorzi, M. (2014). Can approximate mental calculation account for operational momentum in addition and subtraction? Q. J. Exp. Psychol., 67(8), 1541–1556. doi:10.1080/17470218.2014.890234.
Knops, A., Thirion, B., Hubbard, E. M., Michel, V., & Dehaene, S. (2009). Recruitment of an area involved in eye movements during mental arithmetic. Science, 324(5934), 1583–1585. doi:10.1126/science.1171599.
Knops, A., Zitzmann, S., & McCrink, K. (2013). Examining the presence and determinants of operational momentum in childhood. Front. Psychol., 4, 325. doi:10.3389/fpsyg.2013.00325.
Lugli, L., Baroni, G., Anelli, F., Borghi, A. M., Nicoletti, R.: Counting is easier while experiencing a congruent motion. (2013) doi: 10.1371/journal.pone.0064500.
Masson, N., & Pesenti, M. (2014). Attentional bias induced by solving simple and complex addition and subtraction problems. Q. J Exp. Psychol., 67(8), 1514–1526. doi:10.1080/17470218.2014.903985.
McCrink, K., Dehaene, S., & Dehaene-Lambertz, G. (2007). Moving along the number line: operational momentum in nonsymbolic arithmetic. Percept. Psychophys., 69(8), 1324–1333. doi:10.3758/BF03192949.
McCrink, K., & Wynn, K. (2009). Operational momentum in large-number addition and subtraction by 9-month-olds. J. Exp. Child Psychol., 103(4), 400–408. doi:10.1016/j.jecp.2009.01.013.
Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520. doi:10.1038/2151519a0.
Myachykov, A., Ellis, R., Changelosi, A., Fischer, M. H.: (in press). Ocular drift along the mental number line. Psychol. Res.
Pinhas, M., & Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109(3), 408–415. doi:10.1016/j.cognition.2008.09.003.
Ranzini, M., Lisi, M., Zorzi, M.: (in press). Voluntary eye movements direct attention on the mental number space. Psychol. Res.
Restle, F. (1970). Speed of adding and comparing numbers. J. Exp. Psychol., 83(2p1), 274–278. doi:10.1037/h0028573.
Ristic, J., Wright, A., & Kingstone, A. (2006). The number line effect reflects top-down control. Psychon. Bull. Rev., 13(5), 862–868. doi:10.3758/BF03194010.
Schwarz, W., & Keus, I. M. (2004). Moving the eyes along the mental number line: comparing SNARC effects with saccadic and manual responses. Percept. Psychophys., 66(4), 651–664. doi:10.3758/BF03194909.
Shaki, S., & Fischer, M. H. (2008). Reading space into numbers—a cross-linguistic comparison of the SNARC effect. Cognition, 108(2), 590–599. doi:10.1016/j.cognition.2008.04.001.
Shaki, S., Fischer, M. H., & Petrusic, W. M. (2009). Reading habits for both words and numbers contribute to the SNARC effect. Psychon. Bull. Rev., 16(2), 328–331. doi:10.3758/PBR.16.2.328.
Shaki, S., Sery, N., & Fischer, M. H. (2015). 1+ 2 is more than 2+ 1: violations of commutativity and identity axioms in mental arithmetic. J. Cognit. Psychol., 27(4), 471–477. doi:10.1080/20445911.2014.973414.
Wiemers, M., Bekkering, H., & Lindemann, O. (2014). Spatial interferences in mental arithmetic: evidence from the motion-arithmetic compatibility effect. Q. J. Exp. Psychol., 67(8), 1557–1570. doi:10.1080/17470218.2014.889180.
Wood, G., Willmes, K., Nuerk, H., & Fischer, M. H. (2008). On the cognitive link between space and number: a meta-analysis of the SNARC effect. Psychol. Sci. Q., 50(4), 489–525.
Woods, S. S., Resnick, L. B., & Groen, G. J. (1975). An experimental test of five process models for subtraction. J. Educ. Psychol, 67(1), 17–21. doi:10.1037/h0078666.
Yang, T., Chen, C., Zhou, X., Xu, J., Dong, Q., & Chen, C. (2014). Development of spatial representation of numbers: a study of the SNARC effect in Chinese children. J. Exp. Child Psychol., 117, 1–11. doi:10.1016/j.jecp.2013.08.011.
Zebian, S. (2005). Linkages between number concepts, spatial thinking, and directionality of writing: the SNARC effect and the reverse SNARC effect in English and Arabic monoliterates, biliterates, and illiterate Arabic speakers. J. Cognit. Cult., 5(1), 165–190. doi:10.1163/1568537054068660.
Zhou, X., Chen, C., Dong, Q., Zhang, H., Zhou, R., Zhao, H., & Guo, Y. (2006). Event-related potentials of single-digit addition, subtraction, and multiplication. Neuropsychologia, 44(12), 2500–2507. doi:10.1016/j.neuropsychologia.2006.04.003.
Zhou, X., Chen, C., Qiao, S., Chen, C., Chen, L., Lu, N., & Dong, Q. (2009). Event-related potentials for simple arithmetic in Arabic digits and Chinese number words: a study of the mental representation of arithmetic facts through notation and operation effects. Brain Res., 1302, 212–224. doi:10.1016/j.brainres.2009.09.024.
Zhou, X., Chen, C., Zang, Y., Dong, Q., Chen, C., Qiao, S., & Gong, Q. (2007). Dissociated brain organization for single-digit addition and multiplication. Neuroimage, 35(2), 871–880. doi:10.1016/j.neuroimage.2006.12.017.
Zhou, F., Zhao, Q., Chen, C., & Zhou, X. (2012). Mental representations of arithmetic facts: evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis. Q. J. Exp. Psychol., 65(4), 661–674. doi:10.1080/17470218.2011.616213.
Zorzi, M., Priftis, K., & Umiltà, C. (2002). Brain damage: neglect disrupts the mental number line. Nature, 417(6885), 138–139. doi:10.1038/417138a.
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This research was supported by the State Key Basic Research Program of China 2014CB846100 and by two grants from the Natural Science Foundation of China (Project nos. 31221003 and 31271187).
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X. Yu and J. Liu contributed equally to this work.
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Yu, X., Liu, J., Li, D. et al. Dynamic mental number line in simple arithmetic. Psychological Research 80, 410–421 (2016). https://doi.org/10.1007/s00426-015-0730-5
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DOI: https://doi.org/10.1007/s00426-015-0730-5