Advertisement

When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems

  • Hippolyte GrosEmail author
  • Emmanuel Sander
  • Jean-Pierre Thibaut
Article

Abstract

Can our knowledge about apples, cars, or smurfs hinder our ability to solve mathematical problems involving these entities? We argue that such daily-life knowledge interferes with arithmetic word problem solving, to the extent that experts can be led to failure in problems involving trivial mathematical notions. We created problems evoking different aspects of our non-mathematical, general knowledge. They were solvable by one single subtraction involving small quantities, such as 14 – 2 = 12. A first experiment studied how university-educated adults dealt with seemingly simple arithmetic problems evoking knowledge that was either congruent or incongruent with the problems’ solving procedure. Results showed that in the latter case, the proportion of participants incorrectly deeming the problems “unsolvable” increased significantly, as did response times for correct answers. A second experiment showed that expert mathematicians were also subject to this bias. These results demonstrate that irrelevant non-mathematical knowledge interferes with the identification of basic, single-step solutions to arithmetic word problems, even among experts who have supposedly mastered abstract, context-independent reasoning.

Keywords

Encoding effects Mathematical cognition Mental models Semantics 

Notes

Acknowledgments

We sincerely thank Pernille Hemmer and two anonymous reviewers whose insightful feedback helped improve and clarify this manuscript. We also acknowledge gratefully Pierre Barrouillet, Katarina Gvozdic, and Maxime Maheu for helpful comments on previous versions of this work.

This research was supported by grants from the Regional Council of Burgundy, Pari Feder Grants (20159201AAO050S02982 & 20169201AAO050S01845, JPT), from the Experimental Fund for the Youth and French Ministry of Education (HAP10-CRE-EXPE-S1, ES), and from the French Ministry of Education and Future Investment Plan (CS-032-15-836-ARITHM-0, ES). HG was further supported by a doctoral fellowship from the Paris Descartes University.

Open practices statement

The data and materials for all experiments are available at (https://osf.io/fxgqh/?view_only=ed1374ef4d204c90a0cb03a30cb0a099).

References

  1. “SCEI Statistics” (2017) Retrieved from http://www.scei-concours.fr/statistiques.php.
  2. Anderson, S. F., Kelley, K., & Maxwell, S. E. (2017). Sample-size planning for more accurate statistical power: A method adjusting sample effect sizes for publication bias and uncertainty. Psychological Science, 28(11), 1547–1562.CrossRefGoogle Scholar
  3. Bassok, M. (2001). Semantic alignments in mathematical word problems. In D. Gentner, K. J. Holyoak, & B. Kokinov (Eds.), The analogical mind: Perspectives from cognitive science (pp. 401–433). Cambridge, MA: MIT Press.Google Scholar
  4. Bassok, M., Chase, V. M., & Martin, S. A. (1998). Adding apples and oranges: Alignment of semantic and formal knowledge. Cognitive Psychology, 35(2), 99–134.CrossRefGoogle Scholar
  5. Bassok, M., Pedigo, S. F., & Oskarsson, A. T. (2008). Priming addition facts with semantic relations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 34(2), 343–352.Google Scholar
  6. Bassok, M., Wu, L. L., & Olseth, K. L. (1995). Judging a book by its cover: Interpretative effects of content on problem-solving transfer. Memory and Cognition, 23, 354–367.CrossRefGoogle Scholar
  7. Bhardwa, S. (2017). International Student Table 2017: Top 200 Universities. Times Higher Education.Google Scholar
  8. Blessing, S. B., & Ross, B. H. (1996). Content effects in problem categorization and problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22(3), 792–810.Google Scholar
  9. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179–202.CrossRefGoogle Scholar
  10. Chi, M. T., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5(2), 121–152.CrossRefGoogle Scholar
  11. Chi, M. T. H. (1978). Knowledge structures and memory development. In R.S. Siegler (Ed.) Children’s thinking: What develops?, (pp. 73–96). Hillsdale, NJ: Lawrence Erlabaum Associates.Google Scholar
  12. Chi, M. T. H. (2006). Two approaches to the study of experts’ characteristics. In K. A. Ericsson, N. Charness, P. Feltovich, & R. Hoffman (Eds.), Cambridge handbook of expertise and expert performance (pp. 121–130). Cambridge: Cambridge University Press.Google Scholar
  13. Davidson, J. E., & Sternberg, R. J. (Eds.). (2003). The psychology of problem solving. New York, NY: Cambridge University Press.Google Scholar
  14. Davis, P., Hersh, R., & Marchisotto, E. A. (2011). The mathematical experience, Study edition. Boston, MA: Birkhäuser.Google Scholar
  15. De Groot, A. D. (1965). Thought and choice in chess. The Hague, Netherlands: Mouton.Google Scholar
  16. Dehaene, S. (2011). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.Google Scholar
  17. Ericsson, K. A., & Lehmann, A. C. (1996). Expert and exceptional performance: Evidence of maximal adaptation to task constraints. Annual Review of Psychology, 47(1), 273–305.CrossRefGoogle Scholar
  18. Gamo, S., Sander, E., & Richard, J. F. (2010). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction, 20(5), 400–410.CrossRefGoogle Scholar
  19. Goldberg, R. F., & Thompson-Schill, S. L. (2009). Developmental “roots” in mature biological knowledge. Psychological Science, 20(4), 480–487.CrossRefGoogle Scholar
  20. Gros, H., Sander, E., & Thibaut, J. P. (2016). “This problem has no solution”: When closing one of two doors results in failure to access any. In A. Papafragou, D. Grodner, D. Mirman, & J. C. Trueswell (Eds.), Proceedings of the 38th Annual Conference of the Cognitive Science Society (pp. 1271–1276). Austin, TX: Cognitive Science Society.Google Scholar
  21. Gros, H., Thibaut, J. P., & Sander, E. (2015). Robustness of semantic encoding effects in a transfer task for multiple-strategy arithmetic problems. In D. C. Noelle, R. Dale, A. S. Warlaumont, J. Yoshimi, T. Matlock, C. D. Jennings, & P. P. Maglio (Eds.), Proceedings of the 37th Annual Conference of the Cognitive Science Society (pp. 818–823). Austin, TX: Cognitive Science Society.Google Scholar
  22. Gros, H., Thibaut, J. P., & Sander, E. (2017). The nature of quantities influences the representation of arithmetic problems: Evidence from drawings and solving procedures in children and adults. In R. Granger, U. Hahn, & R. Sutton (Eds.), Proceedings of the 39th Annual Meeting of the Cognitive Science Society (pp 439–444). Austin, TX: Cognitive Science Society.Google Scholar
  23. Hartigan, J. A., & Hartigan, P. M. (1985). The dip test of unimodality. The Annals of Statistics, 13(1), 70–84.CrossRefGoogle Scholar
  24. Hector, A. (2015). The new statistics with R: An introduction for biologists. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
  25. Lesgold, A., Rubinson, H., Feltovich, P., Glaser, R., Klopfer, D., & Wang, Y. (1988). Expertise in a complex skill: Diagnosing x-ray pictures. In M. T. H. Chi, R. Glaser, M. J. Farr (Eds.), The nature of expertise (pp. 311–342). Hillsdale, NJ: ErlbaumGoogle Scholar
  26. Newell, A., & Simon, H. A. (1972). Human problem solving (Vol. 104, No. 9). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  27. 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.CrossRefGoogle Scholar
  28. Ross, B. H. (1987). This is like that: The use of earlier problems and the separation of similarity effects. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13(4), 629–639.Google Scholar
  29. Russell, B. (1903). Principles of mathematics. Cambridge, UK: Cambridge University Press.Google Scholar
  30. Sternberg, R. J., & Frensch, P. A. (1992). On being an expert: A cost-benefit analysis. In The psychology of expertise (pp. 191–203). New York, NY: Springer.CrossRefGoogle Scholar
  31. Thevenot, C., & Barrouillet, P. (2015). Arithmetic word problem solving and mental representations. In R. Cohen Kadosh, & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 158–179). Oxford, UK: Oxford University Press.Google Scholar
  32. Verschaffel, L., De Corte, E., & Vierstraete, H. (1999). Upper elementary school pupils' difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, 30(3), 265–285.CrossRefGoogle Scholar
  33. Vicente, S., Orrantia, J., & Verschaffel, L. (2007). Influence of situational and conceptual rewording on word problem solving. British Journal of Educational Psychology, 77(4), 829–848.CrossRefGoogle Scholar
  34. Voss, J. F., Greene, T. R., Post, T. A., & Penner, B. C. (1983). Problem-solving skill in the social sciences. In Psychology of learning and motivation (Vol. 17, pp. 165–213). New York, NY: Academic Press.Google Scholar
  35. Voss, J. F., Vesonder, G. T., & Spilich, G. J. (1980). Text generation and recall by high-knowledge and low-knowledge individuals. Journal of Verbal Learning and Verbal Behavior, 19(6), 651–667.CrossRefGoogle Scholar
  36. Westfall, J., Kenny, D. A., & Judd, C. M. (2014). Statistical power and optimal design in experiments in which samples of participants respond to samples of stimuli. Journal of Experimental Psychology: General, 143, 2020–2045.CrossRefGoogle Scholar

Copyright information

© The Psychonomic Society, Inc. 2019

Authors and Affiliations

  • Hippolyte Gros
    • 1
    • 2
    Email author
  • Emmanuel Sander
    • 2
  • Jean-Pierre Thibaut
    • 3
  1. 1.Center for Research and InterdisciplinarityParis Descartes UniversityParisFrance
  2. 2.IDEA Lab, Faculty of Psychology and Educational SciencesUniversity of GenevaGenevaSwitzerland
  3. 3.Lead, CNRS UMR 5022, University of Bourgogne Franche-ComtéBourgogneFrance

Personalised recommendations