Cognitive Processing

, Volume 17, Issue 3, pp 243–258 | Cite as

Training the equidistant principle of number line spacing

  • Tanja DackermannEmail author
  • Ursula Fischer
  • Stefan Huber
  • Hans-Christoph Nuerk
  • Korbinian Moeller
Research Report


The characteristics of effective numerical trainings are still under scientific debate. Given the importance of number line estimation due to the strong relation between task performance and arithmetic abilities, the current study aimed at training one important number line characteristic: the equidistant spacing of adjacent numbers. Following an embodied training approach, second-graders were trained using a randomized crossover design to divide a presented line into different numbers of equal segments by walking the line with equally spaced steps. Performance was recorded, and feedback as to the correct equidistant spacing was provided using the Kinect sensor system. Training effects were compared to a control training with no involvement of task-specific whole-body movements. Results indicated more pronounced specific training effects after the embodied training. Moreover, transfer effects to number line estimation and arithmetic performance were partially observed. In particular, differential training effects for bounded versus unbounded number line estimation corroborate the assumption that not only bodily experiences but also the need for a flexible adaption of the perspective on the training material might influence training success. Hence, more pronounced training effects of the embodied training might stem from different cognitive processes involved.


Mental number line Equidistance Number line estimation Bounded/unbounded number line estimation task Mathematical skills 



Tanja Dackermann, Korbinian Moeller and Hans-Christoph Nuerk were members of the “Cooperative Research Training Group” of the University of Education, Ludwigsburg, and the University of Tuebingen, which was supported by the Ministry of Science, Research and the Arts in Baden-Wuerttemberg and which served as a funding body of this study. Korbinian Moeller and Hans-Christoph Nuerk are principal investigators at the LEAD—Learning, Educational Achievement, and Life Course Development—graduate school which is supported by the German research foundation. We would like to thank primary school teachers for their cooperation and all children and their parents for participation. Furthermore, we are grateful to Leona Steinack for her help in data acquisition.


  1. Anelli F, Lugli L, Baroni G, Borghi AM, Nicoletti R (2014) Walking boosts your performance in making additions and subtractions. Front Psychol 5:1459. doi: 10.3389/fpsyg.2014.01459 CrossRefPubMedPubMedCentralGoogle Scholar
  2. Barsalou LW (2008) Grounded cognition. Annu Rev Psychol 59:617–645. doi: 10.1146/annurev.psych.59.103006.093639 CrossRefPubMedGoogle Scholar
  3. Barsalou LW (2010) Grounded cognition: past, present, and future. Top Cognit Sci 2:716–724. doi: 10.1111/j.1756-8765.2010.01115.x CrossRefGoogle Scholar
  4. Barth HC, Paladino AM (2011) The development of numerical estimation: evidence against a representational shift. Dev Sci 14:125–135. doi: 10.1111/j.1467-7687.2010.00962.x CrossRefPubMedGoogle Scholar
  5. Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B (Methodological) 57(1):289–300Google Scholar
  6. Booth JL, Siegler RS (2006) Developmental and individual differences in pure numerical estimation. Dev Psychol 42:189–201. doi: 10.1037/0012-1649.41.6.189 CrossRefPubMedGoogle Scholar
  7. Booth JL, Siegler RS (2008) Numerical magnitude representations influence arithmetic learning. Child Dev 79:1016–1031. doi: 10.1111/j.1467-8624.2008.01173.x CrossRefPubMedGoogle Scholar
  8. Bueti D, Walsh V (2009) The parietal cortex and the representation of time, space, number and other magnitudes. Philos Trans R Soc B Biol Sci 364:1831–1840. doi: 10.1098/rstb.2009.0028 CrossRefGoogle Scholar
  9. Butterworth B, Varma S, Laurillard D (2011) Dyscalculia: from brain to education. Science. doi: 10.1126/science.1201536 PubMedGoogle Scholar
  10. Cohen DJ, Blanc-Goldhammer D (2011) Numerical bias in bounded and unbounded number line tasks. Psychon Bull Rev 18(2):331–338. doi: 10.3758/s13423-011-0059-z CrossRefPubMedPubMedCentralGoogle Scholar
  11. Cohen DJ, Sarnecka BW (2014) Children’s number-line estimation shows development of measurement skills (Not Number Representations). Dev Psychol 50:1640–1652. doi: 10.1037/a0035901 CrossRefPubMedGoogle Scholar
  12. de Hevia MD, Spelke ES (2010) Number-space mapping in human infants. Psychol Sci 21:653–660. doi: 10.1177/0956797610366091 CrossRefPubMedPubMedCentralGoogle Scholar
  13. de Hevia MD, Girelli L, Addabbo M, Cassia VM (2014) Human infants’ preference for left-to-right oriented increasing numerical sequences. PLoS One 9:e96412. doi: 10.1371/journal.pone.0096412 CrossRefPubMedPubMedCentralGoogle Scholar
  14. Dehaene S (2003) The neural basis of the Weber–Fechner law: a logarithmic mental number line. Trends in Cognit Sci 7(4):145–147. doi: 10.1016/S1364-6613(03)00055-X CrossRefGoogle Scholar
  15. Dehaene S (2005) Evolution of human cortical circuits for reading and arithmetic: the “neuronal recycling” hypothesis. In: Dehaene S, Duhamel JR, Hauser MD, Rizolatti G (eds) From monkey brain to human brain. MIT press, Cambridge, pp 133–157Google Scholar
  16. Dehaene S, Cohen L (2007) Cultural recycling of cortical maps. Neuron 56(2):384–398. doi: 10.1016/j.neuron.2007.10.004 CrossRefPubMedGoogle Scholar
  17. Dehaene S, Mehler J (1992) Cross-linguistic regularities in the frequency of number words. Cognition 43:1–29. doi: 10.1016/0010-0277(92)90030-L CrossRefPubMedGoogle Scholar
  18. Dehaene S, Dupoux E, Mehler J (1990) Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. J Exp Psychol Hum Percept Perform 16(3):626–641. doi: 10.1037/0096-1523.16.3.626 CrossRefPubMedGoogle Scholar
  19. Dehaene S, Bossini S, Giraux P (1993) The mental representation of parity and number magnitude. J Exp Psychol Gen 122:371–396. doi: 10.1037/0096-3445.122.3.371 CrossRefGoogle Scholar
  20. Dehaene S, Molko N, Cohen L, Wilson AJ (2004) Arithmetic and the brain. Curr Opin Neurobiol 14(2):218–224. doi: 10.1016/j.conb.2004.03.008 CrossRefPubMedGoogle Scholar
  21. Domahs F, Moeller K, Huber S, Willmes K, Nuerk HC (2010) Embodied numerosity: implicit hand-based representations influence symbolic number processing across cultures. Cognition 116:251–266. doi: 10.1016/j.cognition.2010.05.007 CrossRefPubMedGoogle Scholar
  22. Dowker A (2005) Early identification and intervention for students with mathematics difficulties. J Learn Disabil Austin 38:324–332. doi: 10.1177/00222194050380040801 CrossRefGoogle Scholar
  23. Feigenson L, Dehaene S, Spelke E (2004) Core systems of number. Trends Cogn Sci. doi: 10.1016/j.tics.2004.05.002 PubMedGoogle Scholar
  24. Fischer MH (2012) A hierarchical view of grounded, embodied, and situated numerical cognition. Cogn Process 13:161–164. doi: 10.1007/s10339-012-0477-5 CrossRefGoogle Scholar
  25. Fischer MH, Brugger P (2011) When digits help digits: spatial–numerical associations point to finger counting as prime example of embodied cognition. Front Psychol 2:260. doi: 10.3389/fpsyg.2011.00260 CrossRefPubMedPubMedCentralGoogle Scholar
  26. Fischer MH, Shaki S (2015) Two steps to space for numbers. Front Psychol 6:612. doi: 10.3389/fpsyg.2015.00612 PubMedPubMedCentralGoogle Scholar
  27. Fischer U, Moeller K, Bientzle M, Cress U, Nuerk HC (2011) Sensori-motor spatial training of number magnitude representation. Psychon Bull Rev 18:177–183. doi: 10.3758/s13423-010-0031-3 CrossRefPubMedGoogle Scholar
  28. Fischer U, Moeller K, Huber S, Cress U, Nuerk H-C (2015) Full-body movement in numerical trainings: a pilot study with an interactive whiteboard. Int J Ser Games 2:23–35. doi: 10.17083/ijsg.v2i4.93 Google Scholar
  29. Fischer U, Moeller K, Class F, Huber S, Cress U, Nuerk HC (in press) Dancing with the SNARC: measuring spatial numerical associations on a digital dance mat. Can J Exp PsycholGoogle Scholar
  30. Gallistel CR, Gelman R (1992) Preverbal and verbal counting and computation. Cognition 44:43–74. doi: 10.1016/0010-0277(92)90050-R CrossRefPubMedGoogle Scholar
  31. Geary DC, Hoard MK, Nugent L, Bailey DH (2012) Mathematical cognition deficits in children with learning disabilities and persistent low achievement: a five-year prospective study. J Educ Psychol 104:206–223. doi: 10.1037/a0025398 CrossRefPubMedPubMedCentralGoogle Scholar
  32. Grossberg S, Repin DV (2003) A neural model of how the brain represents and compares multi-digit numbers: spatial and categorical processes. Neural Netw 16(8):1107–1140. doi: 10.1016/S0893-6080(03)00193-X CrossRefPubMedGoogle Scholar
  33. Haffner J, Baro K, Parzer P, Resch F (2005) HRT 1-4, Heidelberger rechentest. Hogrefe, GöttingenGoogle Scholar
  34. Hartmann M, Grabherr L, Mast FW (2012a) Moving along the mental number line: interactions between whole-body motion and numerical cognition. J Exp Psychol Hum Percept Perform 38:1416–1427. doi: 10.1037/a0026706 CrossRefPubMedGoogle Scholar
  35. Hartmann M, Farkas R, Mast FW (2012b) Self-motion perception influences number processing: evidence from a parity task. Cogn Process 13:189–192. doi: 10.1007/s10339-012-0484-6 CrossRefGoogle Scholar
  36. Hubbard EM, Piazza M, Pinel P, Dehaene S (2005) Interactions between number and space in parietal cortex. Nat Rev Neurosci 6(6):435–448. doi: 10.1038/nrn1684 CrossRefPubMedGoogle Scholar
  37. Huber S, Moeller K, Nuerk H-C (2014) Dissociating number line estimations from underlying numerical representations. Q J Exp Psychol 67:991–1003. doi: 10.1080/17470218.2013.838974 CrossRefGoogle Scholar
  38. Jordan NC, Kaplan D, Ramineni C, Locuniak MN (2009) Early math matters: kindergarten number competence and later mathematics outcomes. Dev Psychol 45:850–867. doi: 10.1037/a0014939 CrossRefPubMedPubMedCentralGoogle Scholar
  39. Kaufmann L, Handl P, Thöny B (2003) Evaluation of a numeracy intervention program focusing on basic numerical knowledge and conceptual knowledge. A pilot study. J Learn Disabil 36:564–573. doi: 10.1177/00222194030360060701 CrossRefPubMedGoogle Scholar
  40. Keulen RF, Adam JJ, Fischer MH, Kuipers H, Jolles J (2002) Selective reaching: evidence for multiple frames of reference. J Exp Psychol Hum Percept Perform 28:515–526. doi: 10.1037/0096-1523.28.3.515 CrossRefPubMedGoogle Scholar
  41. Klein E, Moeller K, Willmes K, Nuerk HC, Domahs F (2011) The influence of implicit hand-based representations on mental arithmetic. Front Psychol 2:197. doi: 10.3389/fpsyg.2011.00197 CrossRefPubMedPubMedCentralGoogle Scholar
  42. Kroesbergen EH, Van Luit JE (2003) Mathematics interventions for children with special educational needs. A meta-analysis. Remedial Spec Educ 24:97–114. doi: 10.1177/07419325030240020501 CrossRefGoogle Scholar
  43. 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 57:782–795. doi: 10.1016/j.neuroimage.2011.01.070 CrossRefPubMedGoogle Scholar
  44. Kulik CLC, Kulik JA (1991) Effectiveness of computer-based instruction: an updated analysis. Comput Hum Behav 7:75–94. doi: 10.1016/0747-5632(91)90030-5 CrossRefGoogle Scholar
  45. Laski EV, Siegler RS (2007) Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Dev 78:1723–1743. doi: 10.1111/j.1467-8624.2007.01087.x CrossRefPubMedGoogle Scholar
  46. Leslie AM, Gelman R, Gallistel CR (2008) The generative basis of natural number concepts. Trends Cogn Sci 12(6):213–218. doi: 10.1016/j.tics.2008.03.004 CrossRefPubMedGoogle Scholar
  47. Li Q, Ma X (2010) A meta-analysis of the effects of computer technology on school students’ mathematics learning. Educ Psychol Rev 22:215–243. doi: 10.1007/s10648-010-9125-8 CrossRefGoogle Scholar
  48. Link T, Moeller K, Huber S, Fischer U, Nuerk HC (2013) Walk the number line—an embodied training of numerical concepts. Trends Neurosci Educ 2 74–84. doi: 10.1016/j.tine.2013.06.005 [Corrigendum, Trends in Neuroscience and Education, in press. doi: 10.1016/j.tine.2015.11.003]
  49. Link T, Huber S, Nuerk HC, Moeller K (2014a) Unbounding the mental number line—new evidence on children’s spatial representation of numbers. Front Psychol 4:1021. doi: 10.3389/fpsyg.2013.01021 CrossRefPubMedPubMedCentralGoogle Scholar
  50. Link T, Nuerk HC, Moeller K (2014b) On the relation between the mental number line and arithmetic competencies. Q J Exp Psychol 67:1597–1613. doi: 10.1080/17470218.2014.892517 CrossRefGoogle Scholar
  51. Link T, Schwarz EJ, Huber S, Fischer U, Nuerk HC, Cress U, Moeller K (2014c) Mathe mit der Matte-Verkörperlichtes training basisnumerischer kompetenzen. Zeitschrift für Erziehungswissenschaft 17(2):257–277. doi: 10.1007/s11618-014-0533-2 CrossRefGoogle Scholar
  52. Loetscher T, Schwarz U, Schubiger M, Brugger P (2008) Head turns bias the brain’s internal random generator. Curr Biol 18:R60–R62. doi: 10.1016/j.cub.2007.11.015 CrossRefPubMedGoogle Scholar
  53. Melnyk B, Morrison-Beedy D (2012) Intervention research: designing, conducting, analyzing, and funding. Springer, BerlinGoogle Scholar
  54. Mihulowicz U, Klein E, Nuerk HC, Willmes K, Karnath HO (2015) Spatial displacement of numbers on a vertical number line in spatial neglect. Front Hum Neurosci 9:240. doi: 10.3389/fnhum.2015.00240 CrossRefPubMedPubMedCentralGoogle Scholar
  55. Moeller K, Pixner S, Kaufmann L, Nuerk H-C (2009) Children’s early mental number line: Logarithmic or rather decomposed linear? J Exp Child Psychol 103:503–515. doi: 10.1016/j.jecp.2009.02.006 CrossRefPubMedGoogle Scholar
  56. Moeller K, Pixner S, Zuber J, Kaufmann L, Nuerk HC (2011) Early place-value understanding as a precursor for later arithmetic performance—a longitudinal study on numerical development. Res Dev Disabil 32:1837–1851. doi: 10.1016/j.ridd.2011.03.012 CrossRefPubMedGoogle Scholar
  57. Moeller K, Fischer U, Link T, Wasner M, Huber S, Cress U, Nuerk HC (2012) Learning and development of embodied numerosity. Cogn Process 13:271–274. doi: 10.1007/s10339-012-0457-9 CrossRefGoogle Scholar
  58. Myachykov A, Scheepers C, Fischer MH, Kessler K (2013) TEST: a tropic, embodied, and situated theory of cognition. Topics Cogn Sci 63:442–460. doi: 10.1111/tops.12024 Google Scholar
  59. Nieder A (2015) Neuronal correlates of non-verbal numerical competence in primates. The Oxford handbook of numerical cognition. University Press, OxfordGoogle Scholar
  60. Nieder A, Dehaene S (2009) Representation of number in the brain. Annu Rev Neurosci 32:185–208. doi: 10.1146/annurev.neuro.051508.135550 CrossRefPubMedGoogle Scholar
  61. Nuerk H-C, Moeller K, Willmes K (2015) Multi-digit number processing-overview, conceptual clarifications, and language influences. The Oxford handbook of numerical cognition. University Press, OxfordGoogle Scholar
  62. Parsons S, Bynner J (2005) Does numeracy matter more?. National Research and Development Centre for Adult Literacy and Numeracy, LondonGoogle Scholar
  63. Patro K, Nuerk HC, Cress U, Haman M (2014) How number-space relationships are assessed before formal schooling: a taxonomy proposal. Front Psychol 5:419. doi: 10.3389/fpsyg.2014.00419 CrossRefPubMedPubMedCentralGoogle Scholar
  64. Pica P, Lemer C, Izard V, Dehaene S (2004) Exact and approximate arithmetic in an Amazonian indigene group. Science 306(5695):499-503. doi: 10.1126/science.1102085 CrossRefPubMedGoogle Scholar
  65. Pinel P, Piazza M, Le Bihan D, Dehaene S (2004) Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron 41(6):983–993. doi: 10.1016/S0896-6273(04)00107-2 CrossRefPubMedGoogle Scholar
  66. Ramani GB, Siegler RS (2011) Reducing the gap in numerical knowledge between low-and middle-income preschoolers. J Appl Dev Psychol 32:146–159. doi: 10.1016/j.appdev.2011.02.005 CrossRefGoogle Scholar
  67. Ramani GB, Siegler RS, Hitti A (2012) Taking it to the classroom: number board games as a small group learning activity. J Educ Psychol 104:661–672. doi: 10.1037/a0028995 CrossRefGoogle Scholar
  68. Reinert RM, Huber S, Nuerk HC, Moeller K (2014) Multiplication facts and the mental number line: evidence from unbounded number line estimation. Psychol Res 79:95–103. doi: 10.1007/s00426-013-0538-0 CrossRefPubMedGoogle Scholar
  69. Schneider M, Heine A, Thaler V, Torbeyns J, De Smedt B, Verschaffel L, Jacobs AM, Stern E (2008) A validation of eye movements as a measure of elementary school children’s developing number sense. Cogn Dev 23:409–422. doi: 10.1016/j.cogdev.2008.07.002 CrossRefGoogle Scholar
  70. Schwarz W, Müller D (2006) Spatial associations in number-related tasks. Exp Psychol 53:4–15. doi: 10.1027/1618-3169.53.1.4 CrossRefPubMedGoogle Scholar
  71. Shaki S, Fischer MH (2014) Random walks on the mental number line. Exp Brain Res 232:43–49. doi: 10.1007/s00221-013-3718-7 CrossRefPubMedGoogle Scholar
  72. Shaki S, Fischer MH, Petrusic WM (2009) Reading habits for both words and numbers contribute to the SNARC effect. Psychon Bull Rev 16:328–331. doi: 10.3758/PBR.16.2.328 CrossRefPubMedGoogle Scholar
  73. Siegler RS (2009) Improving the numerical understanding of children from low-income families. Child Dev Perspect 3:118–124. doi: 10.1111/j.1750-8606.2009.00090.x CrossRefGoogle Scholar
  74. Siegler RS, Booth JL (2004) Development of numerical estimation in young children. Child Dev 75:428–444. doi: 10.1111/j.1467-8624.2004.00684.x CrossRefPubMedGoogle Scholar
  75. Siegler RS, Opfer JE (2003) The development of numerical estimation evidence for multiple representations of numerical quantity. Psychol Sci 14:237–250. doi: 10.1111/1467-9280.02438 CrossRefPubMedGoogle Scholar
  76. Siegler RS, Ramani GB (2008) Playing linear numerical board games promotes low-income children’s numerical development. Dev Sci 11:655–661. doi: 10.1111/j.1467-7687.2008.00714.x CrossRefPubMedGoogle Scholar
  77. Siegler RS, Ramani GB (2009) Playing linear number board games—but not circular ones—improves low-income preschoolers’ numerical understanding. J Educ Psychol 101(3):545–560. doi: 10.1037/a0014239 CrossRefGoogle Scholar
  78. Simon O, Mangin JF, Cohen L, Le Bihan D, Dehaene S (2002) Topographical layout of hand, eye, calculation, and language-related areas in the human parietal lobe. Neuron 33(3):475–487. doi: 10.1016/S0896-6273(02)00575-5 CrossRefPubMedGoogle Scholar
  79. Simon O, Kherif F, Flandin G, Poline JB, Rivière D, Mangin JF, Dehaene S (2004) Automatized clustering and functional geometry of human parietofrontal networks for language, space, and number. Neuroimage 23(3):1192–1202. doi: 10.1016/j.neuroimage.2004.09.023 CrossRefPubMedGoogle Scholar
  80. Slusser EB, Santiago RT, Barth HC (2013) Developmental change in numerical estimation. J Exp Psychol Gen 142:193–208. doi: 10.1037/a0028560 CrossRefPubMedGoogle Scholar
  81. Tipper SP, Lortie C, Baylis GC (1992) Selective reaching: evidence for action-centered attention. J Exp Psychol Hum Percept Perform 18:891–905. doi: 10.1037/0096-1523.18.4.891 CrossRefPubMedGoogle Scholar
  82. Von Aster MG, Shalev RS (2007) Number development and developmental dyscalculia. Dev Med Child Neurol 49:868–873. doi: 10.1111/j.1469-8749.2007.00868.x CrossRefGoogle Scholar
  83. Walsh V (2003) A theory of magnitude: common cortical metrics of time, space and quantity. Trends Cogn Sci 7:483–488. doi: 10.1016/j.tics.2003.09.002 CrossRefPubMedGoogle Scholar
  84. Weiß RH, Osterland J (2013) Grundintelligenztest Skala 1- Revision (CFT 1-R). Hogrefe, GöttingenGoogle Scholar
  85. Wilson M (2002) Six views of embodied cognition. Psychon Bull Rev 9:625–636. doi: 10.3758/BF03196322 CrossRefPubMedGoogle Scholar

Copyright information

© Marta Olivetti Belardinelli and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Tanja Dackermann
    • 1
    Email author
  • Ursula Fischer
    • 1
    • 2
  • Stefan Huber
    • 1
  • Hans-Christoph Nuerk
    • 1
    • 3
  • Korbinian Moeller
    • 1
    • 3
  1. 1.Leibniz-Institut für Wissensmedien TübingenTuebingenGermany
  2. 2.University of RegensburgRegensburgGermany
  3. 3.Eberhard Karls University TuebingenTuebingenGermany

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