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ZDM

, Volume 48, Issue 3, pp 255–266 | Cite as

Measuring fraction comparison strategies with eye-tracking

  • Andreas Obersteiner
  • Christine Tumpek
Original Article

Abstract

Research suggests that people use a variety of strategies for comparing the numerical values of two fractions. They use holistic strategies that rely on the fraction magnitudes, componential strategies that rely on the fraction numerators or denominators, or a combination of both. We investigated how mathematically skilled adults adapt their strategies to the type of fraction pair. To extend previous research on simple fraction comparison, we used a highly controlled set of more complex fractions with two-digit components. In addition to response times, we recorded eye movements to assess how often the participants fixated on and alternated between specific fraction components. In line with previous studies, our data suggest that the participants preferred componential over holistic strategies for fraction pairs with common numerators or common denominators. Conversely, they preferred holistic over componential strategies for fraction pairs without common components. These results support the assumption that mathematically skilled adults adapt their strategies to the type of fraction pair even in complex fraction comparison. Our study also suggests that eye-tracking is a promising method for measuring strategy use in solving fraction problems.

Keywords

Fraction processing Holistic strategies Componential strategies Eye movements 

Notes

Acknowledgments

The authors would like to thank the university students for participating in our study. We also would like to thank the anonymous reviewers for their constructive comments on an earlier draft of this article.

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. doi: 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. doi: 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. doi: 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. doi: 10.2307/748369.CrossRefGoogle Scholar
  5. 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, 1410–1419. doi: 10.1037/0096-1523.33.6.1410.Google Scholar
  6. Booth, J. L., & Newton, K. J. (2012). Fractions: could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37, 247–253. doi: 10.1016/j.cedpsych.2012.07.001.CrossRefGoogle Scholar
  7. Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. (1981). Results from the second mathematics assessment of the National Assessment of Educational Progress. Washington, DC: National Council of Teachers of Mathematics.Google 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. 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. doi: 10.2307/749646.CrossRefGoogle Scholar
  11. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506. doi: 10.1080/02643290244000239.CrossRefGoogle Scholar
  12. 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. doi: 10.1037/a0032916.Google Scholar
  13. Ericsson, K. A., & Simon, H. A. (1980). Verbal reports as data. Psychological Review, 87, 215–251. doi: 10.1037/0033-295X.87.3.215.
  14. Faulkenberry, T. J., & Pierce, B. H. (2011). Mental representations in fraction comparison. Holistic versus component-based strategies. Experimental Psychology, 58, 480–489. doi: 10.1027/1618-3169/a000116.CrossRefGoogle Scholar
  15. Ganor-Stern, D., Karasik-Rivkin, I., & Tzelgov, J. (2011). Holistic representation of unit fractions. Experimental Psychology, 58, 201–206. doi: 10.1027/1618-3169/a000086.CrossRefGoogle Scholar
  16. 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,. doi: 10.1007/s11858-015-0685-4.Google Scholar
  17. Grant, E. R., & Spivey, M. J. (2003). Eye movements and problem solving: guiding attention guides thought. Psychological Science, 14, 462–466. doi: 10.1111/1467-9280.02454.CrossRefGoogle Scholar
  18. Green, H. J., Lemaire, P., & Dufau, S. (2007). Eye movement correlates of younger and older adults’ strategies for complex addition. Acta Psychologica, 125, 257–278. doi: 10.1016/j.actpsy.2006.08.001.
  19. Huber, S., Klein, E., Willmes, K., Nuerk, H.-C., & Moeller, K. (2014a). Decimal fraction representations are not distinct from natural number representations—evidence from a combined eye-tracking and computational modeling approach. Frontiers in Human Neuroscience, 8, 172. doi: 10.3389/fnhum.2014.00172.CrossRefGoogle Scholar
  20. Huber, S., Moeller, K., & Nuerk, H.-C. (2014b). Adaptive processing of fractions—evidence from eye-tracking. Acta Psychologica, 148, 37–48. doi: 10.1016/j.actpsy.2013.12.010.CrossRefGoogle Scholar
  21. Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing and representation of fractions within the brain. Neuroimage, 47, 403–413. doi: 10.1016/j.neuroimage.2009.03.041.CrossRefGoogle Scholar
  22. Ischebeck, A., Weilharter, M., & Körner, C. (2015). Eye movements reflect and shape strategies in fraction comparison. The Quarterly Journal of Experimental Psychology. Advance online publication. doi: 10.1080/17470218.2015.1046464.
  23. Jacob, S. N., & Nieder, A. (2009a). Notation-independent representation of fractions in the human parietal cortex. The Journal of Neuroscience, 29, 4652–4657. doi: 10.1523/JNEUROSCI.0651-09.2009.CrossRefGoogle Scholar
  24. Jacob, S. N., & Nieder, A. (2009b). Tuning to non-symbolic proportions in the human frontoparietal cortex. European Journal of Neuroscience, 30, 1432–1442. doi: 10.1111/j.1460-9568.2009.06932.x.CrossRefGoogle Scholar
  25. Liang, K. Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13–22. doi: 10.1093/biomet/73.1.13.CrossRefGoogle Scholar
  26. Matthews, P. G., & Chesney, D. L. (2015). Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes. Cognitive Psychology, 78, 28–56. doi: 10.1016/j.cogpsych.2015.01.006.CrossRefGoogle Scholar
  27. Meert, G., Grégoire, J., & Noël, M.-P. (2009). Rational numbers: componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology, 62, 1598–1616. doi: 10.1080/17470210802511162.CrossRefGoogle Scholar
  28. Meert, G., Grégoire, J., & Noël, M.-P. (2010a). Comparing 5/7 and 2/9: adults can do it by accessing the magnitude of the whole fractions. Acta Psychologica, 135, 284–292. doi: 10.1016/j.actpsy.2010.07.014.CrossRefGoogle Scholar
  29. Meert, G., Grégoire, J., & Noël, M.-P. (2010b). Comparing the magnitude of two fractions with common components: which representations are used by 10- and 12-year-olds? Journal of Experimental Child Psychology, 107, 244–259. doi: 10.1016/j.jecp.2010.04.008.CrossRefGoogle Scholar
  30. Merkley, R., & Ansari, D. (2010). Using eye tracking to study numerical cognition: the case of the ratio effect. Experimental Brain Research, 206, 455–460. doi: 10.1007/s00221-010-2419-8.CrossRefGoogle Scholar
  31. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520. doi: 10.1038/2151519a0.CrossRefGoogle Scholar
  32. 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. doi: 10.1207/s15326985ep4001_3.CrossRefGoogle Scholar
  33. Obersteiner, A., Dresler, T., Reiss, K., Vogel, C. M., Pekrun, R., & Fallgatter, A. J. (2010). Bringing brain imaging to the school to assess arithmetic problem solving. Chances and limitations in combining educational and neuroscientific research. ZDMThe International Journal on Mathematics Education, 42, 541–554. doi: 10.1007/s11858-010-0256-7.
  34. Obersteiner, A., Moll, G., Beitlich, J. T., Cui, C., Schmidt, M., Khmelivska, T., & Reiss, K. (2014). Expert mathematicians’ strategies for comparing the numerical values of fractions—evidence from eye movements. In S. Oesterle, C. Nicol, P. Liljedahl, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 4, pp. 338–345). Vancouver: PME.Google Scholar
  35. 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. doi: 10.1016/j.learninstruc.2013.05.003.CrossRefGoogle Scholar
  36. Padberg, F. (2009). Didaktik der Bruchrechnung (4th ed.). Heidelberg: Spektrum Akademischer Verlag.CrossRefGoogle Scholar
  37. Robinson, K. M. (2001). The validity of verbal reports in children’s subtraction. Journal of Educational Psychology, 93, 211–222. doi: 10.1037/0022-0663.93.1.211.CrossRefGoogle Scholar
  38. Schneider, M., & Siegler, R. S. (2010). Representations of the magnitudes of fractions. Journal of Experimental Psychology: Human Perception and Performance, 36, 1227–1238. doi: 10.1037/a0018170.Google Scholar
  39. Schneider, M., Heine, A., Thaler, V., Torbeyns, J., De Smedt, B., Verschaffel, L., Jacobs, A. M., & Stern, E. (2008). A validation of eye movements as a measure of elementary school children’s developing number sense. Cognitive Development, 23, 409–422. doi: 10.1016/j.cogdev.2008.07.002.
  40. Sekuler, R., & Mierkiewicz, D. (1977). Children’s judgments of numerical inequality. Child Development, 48, 630–633.CrossRefGoogle Scholar
  41. Siegler, R. S. (2013). Fractions: the new frontier for theories of numerical development. Trends in Cognitive Sciences, 17, 13–19. doi: 10.1016/j.tics.2012.11.004.CrossRefGoogle Scholar
  42. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691–697. doi: 10.1177/0956797612440101.CrossRefGoogle Scholar
  43. Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994–20014. doi: 10.1037/a0031200.CrossRefGoogle Scholar
  44. Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518. doi: 10.1016/j.learninstruc.2004.06.015.CrossRefGoogle Scholar
  45. Sullivan, J. L., Juhasz, B. J., Slattery, T. J, & Barth, H. C. (2011). Adults’ number-line estimation strategies: Evidence from eye movements. Psychonomic Bulletin and Review, 18, 557–563. doi: 10.3758/s13423-011-0081-1.
  46. Szücs, D., & Goswami, U. (2007). Educational neuroscience: defining a discipline for the study of mental representations. Mind, Brain, and Education, 1, 114–127. doi: 10.1111/j.1751-228X.2007.00012.x.CrossRefGoogle Scholar
  47. Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2014). Bridging the gap: fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction,. doi: 10.1016/j.learninstruc.2014.03.002.Google Scholar
  48. Tzelgov, J., Ganor-Stern, D., Kallai, A., & Pinhas, M. (2014). Primitives and non-primitives of numerical representations. Oxford Handbooks Online. Retrieved 24 July 2015. http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199642342.001.0001/oxfordhb-9780199642342-e-019.
  49. Vamvakoussi, X. (2015). The development of rational number knowledge: old topics, new insights. Learning and Instruction, 37, 50–55. doi: 10.1016/j.learninstruc.2015.01.002.
  50. 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. doi: 10.1007/s10649-012-9432-8.CrossRefGoogle Scholar
  51. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: a conceptual change approach. Learning and Instruction, 14, 453–467. doi: 10.1016/j.learninstruc.2004.06.013.CrossRefGoogle Scholar
  52. Van Hoof, J., Vandewalle, J., Verschaffel, L., & Van Dooren, W. (2015). In search for the natural number bias in secondary school students’ interpretation of the effect of arithmetical operations. Learning and Instruction, 37, 30–38. doi: 10.1016/j.learninstruc.2014.03.004.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2015

Authors and Affiliations

  1. 1.Heinz Nixdorf-Stiftungslehrstuhl für Didaktik der MathematikTUM School of Education, Technische Universität MünchenMunichGermany

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