Advertisement

Understanding Rational Numbers – Obstacles for Learners With and Without Mathematical Learning Difficulties

  • Andreas ObersteinerEmail author
  • Kristina Reiss
  • Wim Van Dooren
  • Jo Van Hoof
Chapter

Abstract

Many children have problems with learning rational numbers. Recent research has shed light on the cognitive mechanisms that may account for these difficulties. In this chapter, we first review theoretical frameworks and empirical evidence that help understanding learners’ difficulties with rational numbers. Next, we discuss whether these difficulties with rational numbers are the same for learners with and without mathematical learning difficulties. To identify effective teaching approaches, we briefly review recent intervention studies on rational number learning. Finally, we discuss implications for teaching and learning of rational numbers and desiderata for future research.

Keywords

Mathematical learning difficulties Rational numbers Fractions Conceptual change Dual processes 

References

  1. Alibali, M. W., & Sidney, P. G. (2015). Variability in the natural number bias: Who, when, how, and why. Learning and Instruction, 37, 56–61.  https://doi.org/10.1016/j.learninstruc.2015.01.003 CrossRefGoogle Scholar
  2. Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113, 447–455.  https://doi.org/10.1016/j.jecp.2012.06.004 CrossRefGoogle Scholar
  3. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal of Research in Mathematics Education, 15, 323–341.  https://doi.org/10.2307/748423 CrossRefGoogle Scholar
  4. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1985). Construct a sum: A measure of children’s understanding of fraction size. Journal for Research in Mathematics Education, 16, 120–131.  https://doi.org/10.2307/748369 CrossRefGoogle Scholar
  5. Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118.  https://doi.org/10.1016/j.jecp.2013.09.001 CrossRefGoogle Scholar
  6. Braithwaite, D. W., & Siegler, R. S. (2017). Developmental changes in the whole number bias. Developmental Sciences, 21.  https://doi.org/10.1111/desc.12541 CrossRefGoogle Scholar
  7. Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice, 18, 99–111.  https://doi.org/10.1111/1540-5826.00066 CrossRefGoogle Scholar
  8. Carraher, D. W. (1996). Learning about fractions. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 241–266). New Jersey: Lawrence Erlbaum Associates.Google Scholar
  9. Clarke, D. M., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72, 127–138 doi: 10.1007/ s10649-009-9198-9.CrossRefGoogle Scholar
  10. Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/wpcontent/uploads/Math_Standards.pdf Google Scholar
  11. Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth- and fifth- grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33, 111–144.CrossRefGoogle Scholar
  12. Desmet, L., Grégoire, J., & Mussolin, C. (2010). Developmental changes in the comparison of decimal fractions. Learning and Instruction, 20, 521–532.  https://doi.org/10.1016/j.learninstruc.2009.07.004 CrossRefGoogle Scholar
  13. DeWolf, M., Bassok, M., & Holyoak, K. J. (2015). Conceptual structure and the procedural affordances of rational numbers: Relational reasoning with fractions and decimals. Journal of Experimental Psychology: General, 144, 127–150.  https://doi.org/10.1037/xge0000034 CrossRefGoogle Scholar
  14. DeWolf, M., Chiang, J. N., Bassok, M., Holyoak, K. J., & Monti, M. M. (2016). Neural representations of magnitude for natural and rational numbers. NeuroImage, 141, 304–312.  https://doi.org/10.1016/j.neuroimage.2016.07.052 CrossRefGoogle Scholar
  15. DeWolf, M., Grounds, M. A., & Bassok, M. (2014). Magnitude comparison with different types of rational numbers. Journal of Experimental Psychology: Human Perception and Performance, 40, 71–82.  https://doi.org/10.1037/a0032916 CrossRefGoogle Scholar
  16. DeWolf, M., & Vosniadou, S. (2011). The whole number bias in fraction magnitude comparisons with adults. In L. Carlson, C. Hoelscher, & T. F. Shipley (Eds.), Proceedings of the 33rd annual conference of the cognitive science society (pp. 1751–1756). Austin, TX: Cognitive Science Society.Google Scholar
  17. DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, 39–49.  https://doi.org/10.1016/j.learninstruc.2014.07.002 CrossRefGoogle Scholar
  18. Fazio, L. K., Kennedy, C. A., & Siegler, R. S. (2016). Improving children’s knowledge of fraction magnitudes. PLoS One, 11, e0165243.  https://doi.org/10.1371/journal.pone.0165243 CrossRefGoogle Scholar
  19. Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., et al. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105, 683–700.  https://doi.org/10.1037/a0032446 CrossRefGoogle Scholar
  20. Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Malone, A. S., Wang, A., et al. (2016). Effects of intervention to improve at-risk fourth graders’ understanding, calculations, and word problems with fractions. The Elementary School Journal, 116, 625–651.  https://doi.org/10.1086/686303 CrossRefGoogle Scholar
  21. Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2012). Developing children’s understanding of fractions: An intervention study. Mind, Brain, and Education, 6, 137–146.  https://doi.org/10.1111/j.1751-228X.2012.01149.x CrossRefGoogle Scholar
  22. Gillard, E., Van Dooren, W., Schaeken, W., & Verschaffel, L. (2009). Dual processes in the psychology of mathematics education and cognitive psychology. Human Development, 52, 95–108.  https://doi.org/10.1159/000202728 CrossRefGoogle Scholar
  23. Gómez, D. M., Jiménez, A., Bobadilla, R., Reyes, C., & Dartnell, P. (2015). The effect of inhibitory control on general mathematics achievement and fraction comparison in middle school children. ZDM Mathematics Education, 47, 801–811.  https://doi.org/10.1007/s11858-015-0685-4 CrossRefGoogle Scholar
  24. Hamdan, N., & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53, 587–596.  https://doi.org/10.1037/dev0000252 CrossRefGoogle Scholar
  25. Hart, K. (1981). Fractions. In K. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 66–81). London: John Murray Publishers.Google Scholar
  26. Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843–859.  https://doi.org/10.1037/a0019824 CrossRefGoogle Scholar
  27. Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing and representation of fractions within the brain. NeuroImage, 47, 403–413.  https://doi.org/10.1016/j.neuroimage.2009.03.041 CrossRefGoogle Scholar
  28. Leron, U., & Hazzan, O. (2009). Intuitive vs. analytical thinking: Four perspectives. Educational Studies in Mathematics, 71, 263–278.  https://doi.org/10.1007/sl0649-008-9175-8 CrossRefGoogle Scholar
  29. Mack, N. (1993). Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 85–105). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  30. Matthews, P. G., & Chesney, D. L. (2015). Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes. Cognitive Psychology, 78, 28–56.  https://doi.org/10.1016/j.cogpsych.2015.01.006 CrossRefGoogle Scholar
  31. Mazzocco, M. M. M., & Devlin, K. T. (2008). Parts and ‘holes’: Gaps in rational number sense among children with vs. without mathematical learning disabilities. Developmental Science, 11, 681–691.  https://doi.org/10.1111/j.1467-7687.2008.00717.x CrossRefGoogle Scholar
  32. Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limon & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233–258). Dordrecht, The Netherlands: Kluwer.Google Scholar
  33. Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30, 122–147.  https://doi.org/10.2307/749607 CrossRefGoogle Scholar
  34. Mou, Y., Li, Y., Hoard, M. K., Nugent, L. D., Chu, F. W., Rouder, J. N., & Geary, D. C. (2016). Developmental foundations of children’s fraction magnitude knowledge. Cognitive Development, 39, 141–153.  https://doi.org/10.1016/j.cogdev.2016.05.002 CrossRefGoogle Scholar
  35. Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40, 27–52.  https://doi.org/10.1207/s15326985ep4001_3 CrossRefGoogle Scholar
  36. Obersteiner, A., & Tumpek, C. (2016). Measuring fraction comparison strategies with eye-tracking. ZDM Mathematics Education, 48, 255–266.  https://doi.org/10.1007/s11858-015-0742-z CrossRefGoogle Scholar
  37. Obersteiner, A., Van Dooren, W., Van Hoof, J., & Verschaffel, L. (2013). The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learning and Instruction, 28, 64–72.  https://doi.org/10.1016/j.learninstruc.2013.05.003 CrossRefGoogle Scholar
  38. Obersteiner, A., Van Hoof, J., Van Dooren, W., & Verschaffel, L. (2016). Who can escape the natural number bias in rational number tasks? A study involving students and experts. British Journal of Psychology, 107, 537–555.  https://doi.org/10.1111/bjop.12161 CrossRefGoogle Scholar
  39. Prediger, S. (2008). The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions. Learning and Instruction, 18, 3–17.  https://doi.org/10.1016/j.learninstruc.2006.08.001 CrossRefGoogle Scholar
  40. Rau, M. A., Aleven, V., & Rummel, N. (2013). Interleaved practice in multi-dimensional learning tasks: Which dimension should we interleave? Learning and Instruction, 23, 98–114.  https://doi.org/10.1016/j.learninstruc.2012.07.003 CrossRefGoogle Scholar
  41. Resnick, I., Jordan, N. C., Hansen, N., Rajan, V., Rodrigues, J., Siegler, R. S., & Fuchs, L. S. (2016). Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Developmental Psychology, 52, 746–757.  https://doi.org/10.1037/dev0000102 CrossRefGoogle Scholar
  42. Rinne, L. F., Ye, A., & Jordan, N. C. (2017). Development of fraction comparison strategies: A latent transition analysis. Developmental Psychology, 53, 713–730.  https://doi.org/10.1037/dev0000275 CrossRefGoogle Scholar
  43. Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. Oxford: Oxford University Press.Google Scholar
  44. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691–697.  https://doi.org/10.1177/0956797612440101 CrossRefGoogle Scholar
  45. Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17, 13–19.  https://doi.org/10.1016/j.tics.2012.11.004 CrossRefGoogle Scholar
  46. Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107, 909–918.  https://doi.org/10.1037/edu0000025 CrossRefGoogle Scholar
  47. Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994–20014.  https://doi.org/10.1037/a0031200 CrossRefGoogle Scholar
  48. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273–296.  https://doi.org/10.1016/j.cogpsych.2011.03.001 CrossRefGoogle Scholar
  49. Stacey, K., Helme, S., Steinle, V., Baturo, A., Irwin, K., & Bana, J. (2001). Preservice teachers’ knowledge of difficulties in decimal numeration. Journal of Mathematics Teacher Education, 4, 205–225.  https://doi.org/10.1023/A:1011463205491 CrossRefGoogle Scholar
  50. Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518.  https://doi.org/10.1016/j.learninstruc.2004.06.015 CrossRefGoogle Scholar
  51. Szucs, D., Devine, A., Soltesz, F., Nobes, A., & Gabriel, F. (2013). Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment. Cortex, 49, 2674–2688.  https://doi.org/10.1016/j.cortex.2013.06.007 CrossRefGoogle Scholar
  52. Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5–13.  https://doi.org/10.1016/j.learninstruc.2014.03.002 CrossRefGoogle Scholar
  53. Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2004). Strategy development in children with mathematical disabilities: Insights from the choice/no-choice method and the chronological-age/ability-level-match design. Journal of Learning Disabilities, 37(2), 119–131.  https://doi.org/10.1177/00222194040370020301 CrossRefGoogle Scholar
  54. Vamvakoussi, X. (2015). The development of rational number knowledge: Old topic, new insights. Learning and Instruction, 37, 50–55.  https://doi.org/10.1016/j.learninstruc.2015.01.002 CrossRefGoogle Scholar
  55. Vamvakoussi, X., Christou, K. P., Mertens, L., & Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density. Learning and Instruction, 21, 676–685.  https://doi.org/10.1016/j.learninstruc.2011.03.005 CrossRefGoogle Scholar
  56. Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behavior, 31, 344–355.  https://doi.org/10.1016/j.jmathb.2012.02.001 CrossRefGoogle Scholar
  57. Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2013). Educated adults are still affected by intuitions about the effect of arithmetical operations: Evidence from a reaction- time study. Educational Studies in Mathematics, 82, 323–330.  https://doi.org/10.1007/s10649-012-9432-8 CrossRefGoogle Scholar
  58. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453–467.  https://doi.org/10.1016/j.learninstruc.2004.06.013 CrossRefGoogle Scholar
  59. Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding about rational numbers and their notation. Cognition and Instruction, 28, 181–209.  https://doi.org/10.1080/07370001003676603 CrossRefGoogle Scholar
  60. Van Hoof, J., Janssen, R., Verschaffel, L., & Van Dooren, W. (2015). Inhibiting natural knowledge in fourth graders: Towards a comprehensive test instrument. ZDM Mathematics Education, 47, 849–857.  https://doi.org/10.1007/s11858-014-0650-7 CrossRefGoogle Scholar
  61. Van Hoof, J., Lijnen, T., Verschaffel, L., & Van Dooren, W. (2013). Are secondary school students still hampered by the natural number bias? – A reaction time study on fraction comparison tasks. Research in Mathematics Education, 15, 154–164.  https://doi.org/10.1080/14794802.2013.797747 CrossRefGoogle Scholar
  62. Van Hoof, J., Verschaffel, L., Ghesquière, P., & Van Dooren, W. (2017). The natural number bias and its role in rational number understanding in children with dyscalculia. Delay or deficit? Research in Developmental Disabilities, 71, 181–190.  https://doi.org/10.1016/j.ridd.2017.10.006 CrossRefGoogle Scholar
  63. Wang, L.-C., Tasi, H.-J., & Yang, H.-M. (2012). Cognitive inhibition in students with and without dyslexia and dyscalculia. Research in Developmental Disabilities, 33, 1453–1461.  https://doi.org/10.1016/j.ridd.2012.03.019 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Andreas Obersteiner
    • 1
    Email author
  • Kristina Reiss
    • 2
  • Wim Van Dooren
    • 3
  • Jo Van Hoof
    • 4
  1. 1.Institute for Mathematics EducationUniversity of Education FreiburgFreiburgGermany
  2. 2.School of EducationTechnical University of MunichMunichGermany
  3. 3.Centre for Instructional Psychology and TechnologyKU LeuvenLeuvenBelgium
  4. 4.Centre for Instructional Psychology and TechnologyFaculty of Psychology and Educational Sciences, KU LeuvenLeuvenBelgium

Personalised recommendations