More Space, Better Mathematics: Is Space a Powerful Tool or a Cornerstone for Understanding Arithmetic?

  • Krzysztof CiporaEmail author
  • Philipp Alexander Schroeder
  • Mojtaba Soltanlou
  • Hans-Christoph NuerkEmail author
Part of the Research in Mathematics Education book series (RME)


Tight cognitive links between space and number processing exist. Usually, Spatial-Numerical Associations (SNAs) are interpreted causally: spatial capabilities are a cornerstone of math skill. We question this seemingly ubiquitous assumption. After presenting SNA taxonomy, we show that only some SNAs correlate with math skill. These correlations are not conclusive: (1) Their directions vary (stronger SNA relates sometimes to better, sometimes to poorer skill), (2) the correlations might be explained by mediator variables (e.g., SNA tasks involve cognitive control or reasoning), (3) the hypothetical course of causality is not resolved: For instance, contrary to conventional theories, arithmetic skills can underlie performance in some SNA tasks. However, benefits of SNA trainings on math skills seem to reinforce the claim of primary SNA role. Nevertheless, tasks used in such trainings may tap cognitive operations required in arithmetic, but not SNA representations themselves. Therefore, using space is a powerful tool rather than a cornerstone for math.


Spatial-Numerical Associations (SNA) Extension Spatial-Numerical Associations Directional Spatial-Numerical Associations SNARC effect Arithmetic skills Cognitive skills SNA Taxonomy Multi-digit number processing Compatibility effect Grounded cognition Embodied cognition Situated cognition Embodied math trainings Number Line Estimation (NLE) Cardinality Ordinality Place identification Place-value activation Place-value computation Place-value integration 



KC and MS are supported by a DFG grant [NU 265/3-1] to HCN. KC, MS, and HCN are further supported by the LEAD Graduate School & Research Network [GSC1028], which is funded within the framework of the Excellence Initiative of the German federal and state governments. We thank Julianne Skinner for proofreading the manuscript.


  1. Ashkenazi, S., Mark-Zigdon, N., & Henik, A. (2009). Numerical distance effect in developmental dyscalculia. Cognitive Development, 24(4), 387–400. Scholar
  2. Bachot, J., Gevers, W., Fias, W., & Roeyers, H. (2005). Number sense in children with visuospatial disabilities: Orientation of the mental number line. Psychology Science, 47(1), 172–183.Google Scholar
  3. Barth, H. C., & Paladino, A. M. (2011). The development of numerical estimation: Evidence against a representational shift. Developmental Science, 14(1), 125–135. Scholar
  4. Barth, H. C., Starr, A., & Sullivan, J. (2009). Children’s mappings of large number words to numerosities. Cognitive Development, 24(3), 248–264. Scholar
  5. Bloechle, J., Huber, S., & Moeller, K. (2015). In touch with numbers: Embodied and situated effects in number magnitude comparison. Journal of Cognitive Psychology, 27(4), 478–489. Scholar
  6. Bonato, M., Fabbri, S., Umiltà, C., & Zorzi, M. (2007). The mental representation of numerical fractions: Real or integer? Journal of Experimental Psychology: Human Perception and Performance, 33(6), 1410–1419. Scholar
  7. Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 41(6), 189–201. Scholar
  8. Brysbaert, M. (1995). Arabic number reading: On the nature of the numerical scale and the origin of phonological recoding. Journal of Experimental Psychology: General, 124(4), 434–452. Scholar
  9. Bueti, D., & Walsh, V. (2009). The parietal cortex and the representation of time, space, number and other magnitudes. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 364(1525), 1831–1840. Scholar
  10. Bugden, S., & Ansari, D. (2011). Individual differences in children’s mathematical competence are related to the intentional but not automatic processing of Arabic numerals. Cognition, 118(1), 32–44. Scholar
  11. Bull, R., Cleland, A. A., & Mitchell, T. (2013). Sex differences in the spatial representation of number. Journal of Experimental Psychology: General, 142(1), 181–192. Scholar
  12. Cipora, K., Hohol, M., Nuerk, H.-C., Willmes, K., Brożek, B., Kucharzyk, B., & Nęcka, E. (2016). Professional mathematicians differ from controls in their spatial-numerical associations. Psychological Research, 80(4), 710–726. Scholar
  13. Cipora, K., & Nuerk, H.-C. (2013). Is the SNARC effect related to the level of mathematics? No systematic relationship observed despite more power, more repetitions, and more direct assessment of arithmetic skill. Quarterly Journal of Experimental Psychology, 66(10), 1974–1991. Scholar
  14. Cipora, K., Patro, K., & Nuerk, H.-C. (2018). Situated influences on spatial-numerical associations. In T. Hubbard (Ed.), Spatial biases in perception and cognition. (pp. 41–59). Cambridge, UK: Cambridge University Press.Google Scholar
  15. Cipora, K., Patro, K., & Nuerk, H.-C. (2015). Are Spatial-Numerical Associations a Cornerstone for Arithmetic Learning? The Lack of Genuine Correlations suggests: No. Mind, Brain, & Education, 9(4), 190–207. Scholar
  16. Cohen, D. J., & Blanc-Goldhammer, D. (2011). Numerical bias in bounded and unbounded number line tasks. Psychonomic Bulletin & Review, 18(2), 331–338. Scholar
  17. Cohen Kadosh, R., & Henik, A. (2007). Can synaesthesia research inform cognitive science? Trends in Cognitive Sciences, 11(4), 177–184. Scholar
  18. Cohen Kadosh, R., Lammertyn, J., & Izard, V. (2008). Are numbers special? An overview of chronometric, neuroimaging, developmental and comparative studies of magnitude representation. Progress in Neurobiology, 84(2), 132–147. Scholar
  19. Colome, A., Laka, I., & Sebastian-Galles, N. (2010). Language effects in addition: How you say it counts. Quarterly Journal of Experimental Psychology, 63(5), 965–983. Scholar
  20. Cragg, L., & Gilmore, C. (2014). Skills underlying mathematics: The role of executive function in the development of mathematics proficiency. Trends in Neuroscience and Education, 3(2), 63–68. Scholar
  21. Crollen, V., & Noël, M. P. (2015). Spatial and numerical processing in children with high and low visuospatial abilities. Journal of Experimental Child Psychology, 132, 84–98. Scholar
  22. Dackermann, T., Fischer, U., Nuerk, H.-C., & Cress, U. (2017). Applying embodied cognition: From useful interventions and their theoretical underpinnings to practical applications. ZDM, 49(4), 545–557. Scholar
  23. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371–396. Scholar
  24. de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110(2), 198–207. Scholar
  25. de Hevia, M. D., & Spelke, E. S. (2010). Number-space mapping in human infants. Psychological Science, 21(5), 653–660. Scholar
  26. De Smedt, B., Verschaffel, L., & Ghesquière, P. (2009). The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. Journal of Experimental Child Psychology, 103(4), 469–479. Scholar
  27. Dietrich, J. F., Huber, S., Dackermann, T., Moeller, K., & Fischer, U. (2016). Place-value understanding in number line estimation predicts future arithmetic performance. British Journal of Developmental Psychology, 34(4), 502–517. Scholar
  28. Dietrich, J. F., Huber, S., & Nuerk, H.-C. (2015). Methodological aspects to be considered when measuring the approximate number system (ANS)—A research review. Frontiers in Psychology, 6, 295.
  29. Ebersbach, M., Luwel, K., Frick, A., Onghena, P., & Verschaffel, L. (2008). The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: Evidence for a segmented linear model. Journal of Experimental Child Psychology, 99(1), 1–17. Scholar
  30. Eerland, A., Guadalupe, T. M., & Zwaan, R. A. (2011). Leaning to the left makes the Eiffel Tower seem smaller: Posture-modulated estimation. Psychological Science, 22(12), 1511–1514. Scholar
  31. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. Scholar
  32. Fias, W., Lauwereyns, J., & Lammertyn, J. (2001). Irrelevant digits affect feature-based attention depending on the overlap of neural circuits. Cognitive Brain Research, 12(3), 415–423. Scholar
  33. Fischer, J. P. (2010). Numerical performance increased by finger training: A fallacy due to regression toward the mean? Cortex, 46(2), 272–273. Scholar
  34. Fischer, M. H. (2001). Number processing induces spatial performance biases. Neurology, 57(5), 822–826. Scholar
  35. Fischer, M. H. (2012). A hierarchical view of grounded, embodied, and situated numerical cognition. Cognitive Processing, 13(Suppl 1), S161–S164. Scholar
  36. Fischer, M. H., Mills, R. A., & Shaki, S. (2010). How to cook a SNARC: Number placement in text rapidly changes spatial-numerical associations. Brain and Cognition, 72(3), 333–336. Scholar
  37. Fischer, M. H., & Shaki, S. (2014). Spatial associations in numerical cognition—From single digits to arithmetic. Quarterly Journal of Experimental Psychology, 67(8), 1461–1483. Scholar
  38. Fischer, M. H., Shaki, S., & Cruise, A. (2009). It takes just one word to quash a SNARC. Experimental Psychology, 56(5), 361–366. Scholar
  39. Fischer, U., Moeller, K., Bientzle, M., Cress, U., & Nuerk, H.-C. (2011). Sensori-motor spatial training of number magnitude representation. Psychonomic Bulletin & Review, 18(1), 177–183. Scholar
  40. Fornaciai, M., Cicchini, G. M., & Burr, D. C. (2016). Adaptation to number operates on perceived rather than physical numerosity. Cognition, 151, 63–67. Scholar
  41. Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2), 59–65. Scholar
  42. Galton, F. (1880). Visualised numerals. Nature, 21(533), 252–256. Scholar
  43. Ganor-Stern, D., Tzelgov, J., & Ellenbogen, R. (2007). Automaticity and two-digit numbers. Journal of Experimental Psychology: Human Perception and Performance, 33(2), 483–496. Scholar
  44. Gebuis, T., & Reynvoet, B. (2012). The interplay between nonsymbolic number and its continuous visual properties. Journal of Experimental Psychology: General, 141(4), 642–648. Scholar
  45. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.Google Scholar
  46. Georges, C., Hoffmann, D., & Schiltz, C. (2017a). How and why do number-space associations co-vary in implicit and explicit magnitude processing tasks? Journal of Numerical Cognition, 3(2), 182–211. Scholar
  47. Georges, C., Hoffmann, D., & Schiltz, C. (2017b). Mathematical abilities in elementary school: Do they relate to number–space associations? Journal of Experimental Child Psychology, 161, 126–147. Scholar
  48. Gibson, L. C., & Maurer, D. (2016). Development of SNARC and distance effects and their relation to mathematical and visuospatial abilities. Journal of Experimental Child Psychology, 150, 301–313. Scholar
  49. Göbel, S. M. (2015). Up or down? Reading direction influences vertical counting direction in the horizontal plane—A cross-cultural comparison. Frontiers in Psychology, 6, 228. Scholar
  50. Göbel, S. M., Moeller, K., Pixner, S., Kaufmann, L., & Nuerk, H.-C. (2014). Language affects symbolic arithmetic in children: The case of number word inversion. Journal of Experimental Child Psychology, 119(1), 17–25. Scholar
  51. Gracia-Bafalluy, M., & Noël, M. P. (2008). Does finger training increase young children’s numerical performance? Cortex, 44(4), 368–375. Scholar
  52. Grant, E. (1972). Nicole Oresme and the medieval geometry of qualities and motions. A treatise on the uniformity and difformity of intensities known as “tractatus de configurationibus qualitatum et motuum”: Marshall Clagett (ed. and tr.), edited with an introduction (...). Studies in History and Philosophy of Science Part A, 3(2), 167–182. Scholar
  53. Henik, A., & Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10(4), 389–395. Scholar
  54. Ho, C. S.-H., & Cheng, F. S.-F. (1997). Training in place-value concepts improves children’s addition skills. Contemporary Educational Psychology, 22(4), 495–506. Scholar
  55. Hoffmann, D., Hornung, C., Martin, R., & Schiltz, C. (2013). Developing number-space associations: SNARC effects using a color discrimination task in 5-year-olds. Journal of Experimental Child Psychology, 116(4), 775–791. Scholar
  56. Hoffmann, D., Mussolin, C., Martin, R., & Schiltz, C. (2014). The impact of mathematical proficiency on the number-space association. PLoS One, 9(1), e85048. Scholar
  57. Hoffmann, D., Pigat, D., & Schiltz, C. (2014). The impact of inhibition capacities and age on number-space associations. Cognitive Processing, 15(3), 329–342. Scholar
  58. Hohol, M., Cipora, K., Willmes, K., & Nuerk, H.-C. (2017). Bringing back the balance: Domain-general processes are also important in numerical cognition. Frontiers in Psychology, 8, 499.
  59. Huber, S., Klein, E., Moeller, K., & Willmes, K. (2016). Spatial-numerical and ordinal positional associations coexist in parallel. Frontiers in Psychology, 7, 438. Scholar
  60. Huber, S., Moeller, K., & Nuerk, H.-C. (2014). Dissociating number line estimations from underlying numerical representations. Quarterly Journal of Experimental Psychology, 67(5), 991–1003. Scholar
  61. Huber, S., Nuerk, H.-C., Willmes, K., & Moeller, K. (2016). A general model framework for multisymbol number comparison. Psychological Review, 123(6), 667–695. Scholar
  62. Huber, S., Sury, D., Moeller, K., Rubinsten, O., & Nuerk, H.-C. (2015). A general number-to-space mapping deficit in developmental dyscalculia. Research in Developmental Disabilities, 43–44, 32–42. Scholar
  63. Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131(1), 92–107. Scholar
  64. Kallai, A. Y., & Tzelgov, J. (2012). The place-value of a digit in multi-digit numbers is processed automatically. Journal of Experimental Psychology: Learning, Memory, and Cognition, 38(5), 1221–1233. Scholar
  65. Kim, D., & Opfer, J. E. (2017). A unified framework for bounded and unbounded numerical estimation. Developmental Psychology, 53(6), 1088–1097. Scholar
  66. Klein, E., Huber, S., Nuerk, H.-C., & Moeller, K. (2014). Operational momentum affects eye fixation behaviour. Quarterly Journal of Experimental Psychology, 67(8), 1614–1625. Scholar
  67. Knops, A., Viarouge, A., & Dehaene, S. (2009). Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect. Attention, Perception & Psychophysics, 71(4), 803–821. Scholar
  68. Kornblum, S., Hasbroucq, T., & Osman, A. (1990). Dimensional overlap: cognitive basis for stimulus-response compatibility––a model and taxonomy. Psychological Review, 97(2), 253–270.
  69. Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schönmann, C., Plangger, F., … von Aster, M. (2011). Mental number line training in children with developmental dyscalculia. NeuroImage, 57(3), 782–795. Scholar
  70. Landy, D., Charlesworth, A., & Ottmar, E. (2014). Cutting in line: Discontinuities in the use of large numbers in adults. In Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp. 815–820).Google Scholar
  71. Landy, D., Charlesworth, A., & Ottmar, E. (2017). Categories of large numbers in line estimation. Cognitive Science, 41(2), 326–353. Scholar
  72. Landy, D., & Goldstone, R. L. (2010). Proximity and precedence in arithmetic. Quarterly Journal of Experimental Psychology, 63(10), 1953–1968. Scholar
  73. Landy, D., Silbert, N., & Goldin, A. (2013). Estimating large numbers. Cognitive Science, 37(5), 775–799. Scholar
  74. Laski, E. V., & Siegler, R. S. (2014). Learning from number board games: You learn what you encode. Developmental Psychology, 50(3), 853–864. Scholar
  75. Laurillard, D. (2016). Learning number sense through digital games with intrinsic feedback. Australasian Journal of Educational Technology, 32(6), 32–44. Scholar
  76. LeFevre, J. A., Lira, C. J., Sowinski, C., Cankaya, O., Kamawar, D., & Skwarchuk, S. L. (2013). Charting the role of the number line in mathematical development. Frontiers in Psychology, 4, 641. Scholar
  77. Leibovich, T., & Henik, A. (2013). Magnitude processing in non-symbolic stimuli. Frontiers in Psychology, 4, 375. Scholar
  78. Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2017). From “sense of number” to “sense of magnitude”—The role of continuous magnitudes in numerical cognition. Behavioral and Brain Sciences, 40, e164.
  79. Lindemann, O., Alipour, A., & Fischer, M. H. (2011). Finger counting habits in middle eastern and western individuals: An online survey. Journal of Cross-Cultural Psychology, 42(4), 566–578. Scholar
  80. Lindskog, M., Winman, A., & Poom, L. (2016). Arithmetic training does not improve approximate number system acuity. Frontiers in Psychology, 7, 1364. Scholar
  81. Link, T., Huber, S., Nuerk, H.-C., & Moeller, K. (2014). Unbounding the mental number line-new evidence on children’s spatial representation of numbers. Frontiers in Psychology, 4, 1021.
  82. Link, T., Moeller, K., Huber, S., Fischer, U., & Nuerk, H.-C. (2013). Walk the number line—An embodied training of numerical concepts. Trends in Neuroscience and Education, 2(2), 74–84. Scholar
  83. Link, T., Nuerk, H.-C., & Moeller, K. (2014). On the relation between the mental number line and arithmetic competencies. Quarterly Journal of Experimental Psychology, 67(8), 1597–1613. Scholar
  84. Loetscher, T., Schwarz, U., Schubiger, M., & Brugger, P. (2008). Head turns bias the brain’s internal random generator. Current Biology, 18(2), R60–R62. Scholar
  85. Lonnemann, J., Krinzinger, H., Knops, A., & Willmes, K. (2008). Spatial representations of numbers in children and their connection with calculation abilities. Cortex, 44(4), 420–428. Scholar
  86. Lyons, I. M., Nuerk, H.-C., & Ansari, D. (2015). Rethinking the implications of numerical ratio effects for understanding the development of representational precision and numerical processing across formats. Journal of Experimental Psychology: General, 144(5), 1021–1035. Scholar
  87. Maertens, B., De Smedt, B., Sasanguie, D., Elen, J., & Reynvoet, B. (2016). Enhancing arithmetic in pre-schoolers with comparison or number line estimation training: Does it matter? Learning and Instruction, 46, 1–11. Scholar
  88. Masson, N., & Pesenti, M. (2014). Attentional bias induced by solving simple and complex addition and subtraction problems. Quarterly Journal of Experimental Psychology, 67(8), 1514–1526. Scholar
  89. Masson, N., Letesson, C., & Pesenti, M. (2018). Time course of attentional shifts in mental arithmetic: Evidence from gaze metrics. The Quarterly Journal of Experimental Psychology, 71(4), 1009–1019. Scholar
  90. McCrink, K., Dehaene, S., & Dehaene-Lambertz, G. (2007). Moving along the number line: Operational momentum in nonsymbolic arithmetic. Perception & Psychophysics, 69(8), 1324–1333. Scholar
  91. Miura, I. T., Okamoto, Y., Kim, C. C., Chang, C.-M., Steere, M., & Fayol, M. (1994). Comparisons of children’s cognitive representation of number: China, France, Japan, Korea, Sweden, and the United States. International Journal of Behavioral Development, 17(3), 401–411. Scholar
  92. Mix, K., Levine, S., Cheng, Y.-L., Young, C., Hambrick, D., Ping, R., & Konstantopoulos, S. (2016). Separate but correlated: The latent structure of space and mathematics across development. Journal of Experimental Psychology: General, 145(9), 1206–1227. Scholar
  93. Moeller, K., Pixner, S., Kaufmann, L., & Nuerk, H.-C. (2009). Children’s early mental number line: Logarithmic or decomposed linear? Journal of Experimental Child Psychology, 103(4), 503–515. Scholar
  94. Moeller, K., Pixner, S., Zuber, J., Kaufmann, L., & Nuerk, H.-C. (2011). Early place-value understanding as a precursor for later arithmetic performance—A longitudinal study on numerical development. Research in Developmental Disabilities, 32(5), 1837–1851. Scholar
  95. Moeller, K., Shaki, S., Göbel, S. M., & Nuerk, H.-C. (2015). Language influences number processing—A quadrilingual study. Cognition, 136, 150–155. Scholar
  96. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215(5109), 1519–1520. Scholar
  97. Nemati, P., Schmid, J., Soltanlou, M., Krimly, J.-T., Nuerk, H.-C., & Gawrilow, C. (2017). Planning and self-control, but not working memory, directly predict multiplication performance in adults. Journal of Numerical Cognition, 3(2), 441–467. Scholar
  98. Nuerk, H.-C., Moeller, K., Klein, E., Willmes, K., & Fischer, M. H. (2011). Extending the mental number line: A review of multi-digit number processing. Zeitschrift Für Psychologie/Journal of Psychology, 219(1), 3–22. Scholar
  99. Nuerk, H.-C., Weger, U., & Willmes, K. (2005). Language effects in magnitude comparison: Small, but not irrelevant. Brain and Language, 92(3), 262–277. Scholar
  100. Nuerk, H.-C., Moeller, K., & Willmes, K. (2015). Multi-digit number processing: Overview, conceptual clarifications, and language influences. In R. Cohen Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition. Oxford: Oxford University Press (pp. 106–139). Scholar
  101. Nuerk, H.-C., Weger, U., & Willmes, K. (2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82(1), B25–B33. Scholar
  102. Obersteiner, A., Reiss, K., & Ufer, S. (2013). How training on exact or approximate mental representations of number can enhance first-grade students’ basic number processing and arithmetic skills. Learning and Instruction, 23(1), 125–135. Scholar
  103. Opfer, J. E., & Siegler, R. S. (2007). Representational change and children’s numerical estimation. Cognitive Psychology, 55(3), 169–195. Scholar
  104. Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24(10), 2013–2019. Scholar
  105. Patro, K., Fischer, U., Nuerk, H.-C., & Cress, U. (2016). How to rapidly construct a spatial-numerical representation in preliterate children (at least temporarily). Developmental Science, 19(1), 126–144. Scholar
  106. Patro, K., Nuerk, H.-C., Cress, U., & Haman, M. (2014). How number-space relationships are assessed before formal schooling: A taxonomy proposal. Frontiers in Psychology, 5, 419. Scholar
  107. Penner-Wilger, M., & Anderson, M. L. (2013). The relation between finger gnosis and mathematical ability: Why redeployment of neural circuits best explains the finding. Frontiers in Psychology, 4, 877. Scholar
  108. Pesenti, M. (2005). Calculation abilities in expert calculators. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 413–430). New York, NY: Psychology Press.Google Scholar
  109. Petersen, S. E., & Posner, M. I. (2012). The attention system of the human brain: 20 years after. Annual review of neuroscience, 35, 73–89. Scholar
  110. Pfister, R., Schroeder, P. A., & Kunde, W. (2013). SNARC struggles: Instant control over spatial-numerical associations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 39(6), 1953–1958. Scholar
  111. Piazza, M., & Izard, V. (2009). How humans count: Numerosity and the parietal cortex. The Neuroscientist, 15(3), 261–273. Scholar
  112. Piazza, M., Mechelli, A., Butterworth, B., & Price, C. J. (2002). Are subitizing and counting implemented as separate or functionally overlapping processes? NeuroImage, 15(2), 435–446. Scholar
  113. Pinhas, M., & Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109(3), 408–415. Scholar
  114. Pinhas, M., Shaki, S., & Fischer, M. H. (2014). Heed the signs: Operation signs have spatial associations. Quarterly Journal of Experimental Psychology, 67(8), 1527–1540. Scholar
  115. Pinheiro-Chagas, P., Dotan, D., Piazza, M., & Dehaene, S. (2017). Finger tracking reveals the covert stages of mental arithmetic pedro. Open Mind, 1(1), 30–41. Scholar
  116. Pixner, S., Moeller, K., Zuber, J., & Nuerk, H.-C. (2009). Decomposed but parallel processing of two-digit numbers in 1st graders. The Open Psychology Journal, 2, 40–48. Scholar
  117. Ramani, G. B., Siegler, R. S., & Hitti, A. (2012). Taking it to the classroom: Number board games as a small group learning activity. Journal of Educational Psychology, 104(3), 661–672. Scholar
  118. Raz, A., & Buhle, J. (2006). Typologies of attentional networks. Nature Reviews Neuroscience, 7(5), 367–379. Scholar
  119. Restle, F. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 83(2, Pt. 1), 274–278. Scholar
  120. Rodic, M., Zhou, X., Tikhomirova, T., Wei, W., Malykh, S., Ismatulina, V., … Kovas, Y. (2015). Cross-cultural investigation into cognitive underpinnings of individual differences in early arithmetic. Developmental Science, 18(1), 165–174. Scholar
  121. Rubinsten, O., & Sury, D. (2011). Processing ordinality and quantity: The case of developmental dyscalculia. PLoS One, 6(9), e24079. Scholar
  122. Sasanguie, D., & Reynvoet, B. (2014). Adults’ arithmetic builds on fast and automatic processing of Arabic digits: Evidence from an audiovisual matching paradigm. PLoS One, 9(2), e87739. Scholar
  123. Schneider, M., Beeres, K., Coban, L., Merz, S., Susan Schmidt, S., Stricker, J., & De Smedt, B. (2017). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: A meta-analysis. Developmental Science, 20(30), e12372. Scholar
  124. Schneider, M., Grabner, R. H., Zurich, E., & Paetsch, J. (2009). Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6. Journal of Educational Psychology, 101(2), 359–372. Scholar
  125. Schroeder, P. A., Nuerk, H.-C., & Plewnia, C. (2017a). Space in numerical and ordinal information: A common construct? Journal of Numerical Cognition, 3(2), 164–181. Scholar
  126. Schroeder, P. A., Nuerk, H.-C., & Plewnia, C. (2017b). Switching between Multiple Codes of SNARC-Like Associations: Two Conceptual Replication Attempts with Anodal tDCS in Sham-Controlled Cross-Over Design. Frontiers in Neuroscience, 11, 654. Scholar
  127. Schroeder, P. A., & Pfister, R. (2015). Arbitrary numbers counter fair decisions: Trails of markedness in card distribution. Frontiers in Psychology, 6, 240. Scholar
  128. Sella, F., Tressoldi, P., Lucangeli, D., & Zorzi, M. (2016). Training numerical skills with the adaptive videogame “The Number Race”: A randomized controlled trial on preschoolers. Trends in Neuroscience and Education, 5(1), 20–29. Scholar
  129. Shaki, S., & Fischer, M. H. (2014). Random walks on the mental number line. Experimental Brain Research, 232(1), 43–49. Scholar
  130. Shaki, S., Fischer, M. H., & Göbel, S. M. (2012). Direction counts: A comparative study of spatially directional counting biases in cultures with different reading directions. Journal of Experimental Child Psychology, 112(2), 275–281. Scholar
  131. Shaki, S., Fischer, M. H., & Petrusic, W. M. (2009). Reading habits for both words and numbers contribute to the SNARC effect. Psychonomic Bulletin & Review, 16(2), 328–331. Scholar
  132. Shaki, S., & Gevers, W. (2011). Cultural characteristics dissociate magnitude and ordinal information processing. Journal of Cross-Cultural Psychology, 42(4), 639–650. Scholar
  133. Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118–124. Scholar
  134. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444. Scholar
  135. Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237–243. Scholar
  136. Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games—But not circular ones—Improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology, 101(3), 545–560. Scholar
  137. Siegler, R. S., Thompson, C. A., & Opfer, J. E. (2009). The logarithmic-to-linear shift: One learning sequence, many tasks, many time scales. Mind, Brain, and Education, 3(3), 143–150. Scholar
  138. Simner, J., Mayo, N., & Spiller, M. J. (2009). A foundation for savantism? Visuo-spatial synaesthetes present with cognitive benefits. Cortex, 45(10), 1246–1260. Scholar
  139. Stavy, R., & Tirosh, D. (2000). How students (mis-) understand science and mathematics: Intuitive rules. New York: Teachers College Press.Google Scholar
  140. Szűcs, D., Nobes, A., Devine, A., Gabriel, F. C., & Gebuis, T. (2013). Visual stimulus parameters seriously compromise the measurement of approximate number system acuity and comparative effects between adults and children. Frontiers in Psychology, 4, 444. Scholar
  141. Tudusciuc, O., & Nieder, A. (2007). Neuronal population coding of continuous and discrete quantity in the primate posterior parietal cortex. PNAS, 104(36), 14513–14518. Scholar
  142. van Dijck, J.-P., & Fias, W. (2011). A working memory account for spatial-numerical associations. Cognition, 119(1), 114–119. Scholar
  143. Verbruggen, F., Liefooghe, B., Notebaert, W., & Vandierendonck, A. (2005). Effects of stimulus-stimulus compatibility and stimulus-response compatibility on response inhibition. Acta Psychologica, 120(3), 307–326. Scholar
  144. Wasner, M., Moeller, K., Fischer, M. H., & Nuerk, H.-C. (2014). Aspects of situated cognition in embodied numerosity: The case of finger counting. Cognitive Processing, 15(3), 317–328. Scholar
  145. Wiemers, M., Bekkering, H., & Lindemann, O. (2014). Spatial interferences in mental arithmetic: Evidence from the motion-arithmetic compatibility effect. Quarterly Journal of Experimental Psychology, 67(8), 1557–1570. Scholar
  146. Wiemers, M., Bekkering, H., & Lindemann, O. (2017). Is more always up? Evidence for a preference of hand-based associations over vertical number mappings. Journal of Cognitive Psychology, 29(5), 642–652. Scholar
  147. Wilson, A. J., Dehaene, S., Pinel, P., Revkin, S. K., Cohen, L., & Cohen, D. (2006). Principles underlying the design of “The Number Race”, an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2, 19. Scholar
  148. Wood, G., Willmes, K., Nuerk, H.-C., & Fischer, R. (2008). On the cognitive link between space and number: A meta-analysis of the SNARC effect. Psychology Science Quarterly, 50(4), 489–525.Google Scholar
  149. Zohar-Shai, B., Tzelgov, J., Karni, A., & Rubinsten, O. (2017). It does exist! A left-to-right spatial–numerical association of response codes (SNARC) effect among native Hebrew speakers. Journal of Experimental Psychology: Human Perception and Performance, 43(4), 719–728. Scholar
  150. Zuber, J., Pixner, S., Moeller, K., & Nuerk, H.-C. (2009). On the language specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity. Journal of Experimental Child Psychology, 102(1), 60–77. Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of TuebingenTuebingenGermany
  2. 2.LEAD Graduate School and Research NetworkUniversity of TuebingenTuebingenGermany
  3. 3.Department of Psychiatry and PsychotherapyUniversity of TuebingenTuebingenGermany
  4. 4.Leibniz-Institut für WissensmedienTuebingenGermany

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