When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems
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.
KeywordsEncoding effects Mathematical cognition Mental models Semantics
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).
- “SCEI Statistics” (2017) Retrieved from http://www.scei-concours.fr/statistiques.php.
- 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
- 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
- Bhardwa, S. (2017). International Student Table 2017: Top 200 Universities. Times Higher Education.Google Scholar
- 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
- 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
- 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
- Davidson, J. E., & Sternberg, R. J. (Eds.). (2003). The psychology of problem solving. New York, NY: Cambridge University Press.Google Scholar
- Davis, P., Hersh, R., & Marchisotto, E. A. (2011). The mathematical experience, Study edition. Boston, MA: Birkhäuser.Google Scholar
- De Groot, A. D. (1965). Thought and choice in chess. The Hague, Netherlands: Mouton.Google Scholar
- Dehaene, S. (2011). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.Google Scholar
- 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
- 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
- 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
- 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
- Newell, A., & Simon, H. A. (1972). Human problem solving (Vol. 104, No. 9). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
- 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
- Russell, B. (1903). Principles of mathematics. Cambridge, UK: Cambridge University Press.Google Scholar
- 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
- 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