Learning to be an opportunistic word problem solver: going beyond informal solving strategies

Abstract

Informal strategies reflecting the representation of a situation described in an arithmetic word problem mediate students’ solving processes. When the informal strategies are inefficient, teaching students to make way for more efficient ways to find the solution is an important educational issue in mathematics. The current paper presents a pedagogical design for arithmetic word problem solving, which is part of a first-grade arithmetic intervention (ACE). The curriculum was designed to promote adaptive expertise among students through semantic analysis and recoding, which would lead students to favor the more adequate solving strategy when several options are available. We describe the ways in which students were taught to engage in a semantic analysis of the problem, and the representational tools used to favor this conceptual change. Within the word problem solving curriculum, informal and formal solving strategies corresponding to the different formats of the same arithmetic operation, were comparatively studied. The performance and strategies used by students were assessed, revealing a greater use of formal arithmetic strategies among ACE classes. Our findings illustrate a promising path for going past informal strategies on arithmetic word problem solving.

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Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Inversion is also a term used to describe the subtraction as addition relation (e.g., 7 − 3 = ? can be solved by considering 3 + ? = 7).

  2. 2.

    In order to display the distribution of the strategies among the correct answers in Fig. 2, we counted the number of occurrences of informal strategies and the number of occurrences where no strategy was written down.

References

  1. Alibali, M. W., & Rittle-Johnson, B. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology,91(1), 175–189.

    Article  Google Scholar 

  2. Baroody, A. J., Ginsburg, H. P., & Waxman, B. (1983). Children’s use of mathematical structure. Journal for Research in Mathematics Education,14(3), 156–168.

    Article  Google Scholar 

  3. Baroody, A. J., Torbeyns, J., & Verschaffel, L. (2009). Young children’s understanding and application of subtraction-related principles. Mathematical Thinking and Learning,11(1–2), 2–9.

    Article  Google Scholar 

  4. Blote, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology,93(3), 627–638.

    Article  Google Scholar 

  5. Brissiaud, R., & Sander, E. (2010). Arithmetic word problem solving: A situation strategy first framework. Developmental Science,13(1), 92–107.

    Article  Google Scholar 

  6. Campbell, J. I. D. (2008). Subtraction by addition. Memory & Cognition,36(6), 1094–1102.

    Article  Google Scholar 

  7. Carpenter, T. P., Hiebert, J., & Moser, J. M. (1981). Problem structure and first-grade children’s initial solution processes for simple addition and subtraction problems. Journal for Research in Mathematics Education,12(1), 27–39.

    Article  Google Scholar 

  8. Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review,34(4), 344–377.

    Article  Google Scholar 

  9. De Corte, E., & Verschaffel, L. (1987). The effect of semantic structure on first graders’ strategies for solving addition and subtraction word problems. Journal for Research in Mathematics Education,18(5), 363–381.

    Article  Google Scholar 

  10. Fischbein, E. (1987). Intuition in science and mathematics. An educational approach. Dordrecht: Reidel.

    Google Scholar 

  11. Fischer, J.-P., Sander, E., Sensevy, G., Vilette, B., & Richard, J.-F. (2018). Can young students understand the mathematical concept of equality? A whole-year arithmetical teaching experimentation in second grade. European Journal of Psychology of Education,34(2), 439–456.

    Article  Google Scholar 

  12. Gamo, S., Sander, E., & Richard, J. (2010). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction,20(5), 400–410.

    Article  Google Scholar 

  13. Goldin-Meadow, S., Alibali, M. W., & Church, R. B. (1993). Transitions in concept acquisition: Using the hand to read the mind. Psychological Review,100(2), 279–297.

    Article  Google Scholar 

  14. Gros, H., Sander, E., & Thibaut, J. (2019). When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems. Psychonomic Bulletin & Review,26(5), 1738–1746.

    Article  Google Scholar 

  15. Gvozdic, K., & Sander, E. (2017). Solving additive word problems: Intuitive strategies make the difference. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 39th annual conference of the Cognitive Science Society. London: Cognitive Science Society.

  16. Gvozdic, K., & Sander, E. (2018). When intuitive conceptions overshadow pedagogical content knowledge: Teachers’ conceptions of students’ arithmetic word problem solving strategies. Educational Studies in Mathematics,98(2), 157–175.

    Article  Google Scholar 

  17. Hattikudur, S., Sidney, P. G., & Alibali, M. W. (2016). Does comparing informal and formal procedures promote mathematics learning? The benefits of bridging depend on attitudes toward mathematics. The Journal of Problem Solving,9(1), 13–27.

    Article  Google Scholar 

  18. Hofstadter, D. R., & Sander, E. (2013). Surfaces and essences: Analogy as the fuel and fire of thinking. New York: Basic Books.

    Google Scholar 

  19. Joffredo-Le Brun, S., Morellato, M., Sensevy, G., & Quilio, S. (2018). Cooperative engineering as a joint action. European Educational Research Journal,17(1), 187–208.

    Article  Google Scholar 

  20. Kintsch, W., & van Dijk, A. (1978). Toward a model of text comprehension and production. Psychological Review,85(5), 363–394.

    Article  Google Scholar 

  21. Luwel, K., Schillemans, V., Onghena, P., & Verschaffel, L. (2009). Does switching between strategies within the same task involve a cost? British Journal of Psychology,100(4), 753–771.

    Article  Google Scholar 

  22. Peters, G., De Smedt, B., Torbeyns, J., Ghesquière, P., & Verschaffel, L. (2013). Children’s use of addition to solve two-digit subtraction problems. British Journal of Psychology,104(4), 495–511.

    Google Scholar 

  23. Resnick, L. B. (1989). Developing mathematical knowledge through microworlds. American Psychologist,44(2), 162–169.

    Article  Google Scholar 

  24. Reusser, K. (1990). From text to situation to equation: Cognitive simulation of understanding and solving mathematical word problems. In H. Mandl, E. De Corte, N. Bennet, & H. F. Friedrich (Eds.), Learning and instruction: European research in an international context (Vol. II, pp. 477–498). New York: Pergamon Press.

    Google Scholar 

  25. Riley, M. S., & Greeno, J. G. (1988). Developmental analysis of understanding language about quantities and of solving problems. Cognition and Instruction,5(1), 49–101.

    Article  Google Scholar 

  26. Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s problem-solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). New York: Academic Press.

    Google Scholar 

  27. Rittle-Johnson, B., Fyfe, E. R., & Loehr, A. M. (2016). Improving conceptual and procedural knowledge: The impact of instructional content within a mathematics lesson. British Journal of Educational Psychology,86(4), 576–591.

    Article  Google Scholar 

  28. Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review,27(4), 587–597.

    Article  Google Scholar 

  29. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology,93(2), 346–362.

    Article  Google Scholar 

  30. Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. In B. Ross & J. Mestre (Eds.), Psychology of learning and motivation: Cognition in education (Vol. 55, pp. 199–226). San Diego: Elsevier.

    Google Scholar 

  31. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2017). The power of comparison in mathematics instruction: Experimental evidence from classrooms. In D. C. Geary, D. B. Berch, & K. M. Koepke (Eds.), Mathematical cognition and learning: Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 273–295). Waltham: Elsevier.

    Google Scholar 

  32. Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM—The International Journal on Mathematics Education,41(5), 619–625.

    Article  Google Scholar 

  33. Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference—A longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics,79(3), 389–408.

    Article  Google Scholar 

  34. Sophian, C. (2008). The origins of mathematical knowledge in childhood. New York: Routledge.

    Google Scholar 

  35. Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM—The International Journal on Mathematics Education,41(5), 541–555.

    Article  Google Scholar 

  36. Torbeyns, J., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2009a). Acquisition and use of shortcut strategies by traditionally schooled children. Educational Studies in Mathematics,71(1), 1–17.

    Article  Google Scholar 

  37. Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P., & Verschaffel, L. (2009b). Solving subtraction problems by means of indirect addition. Mathematical Thinking and Learning,11(1–2), 79–91.

    Article  Google Scholar 

  38. Torbeyns, J., Ghesquière, P., & Verschaffel, L. (2009c). Efficiency and flexibility of indirect addition in the domain of multi-digit subtraction. Learning and Instruction,19(1), 1–12.

    Article  Google Scholar 

  39. Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–552). Dordrecht: Springer.

    Google Scholar 

  40. Van den Heuvel-Panhuizen, M., & Treffers, A. (2009). Mathe-didactical reflections on young children’s understanding and application of subtraction-related principles. Mathematical Thinking and Learning,11(1–2), 102–112.

    Article  Google Scholar 

  41. Verschaffel, L., & De Corte, E. (1997). Word problems: A vehicle for promoting authentic mathematical understanding and problem solving in the primary school? In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 69–97). Hove: Psychology Press.

    Google Scholar 

  42. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.

    Google Scholar 

  43. Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester (Ed.), Second handbook on research on mathematics teaching and learning (pp. 557–628). Charlotte: Information Age Publishing.

    Google Scholar 

  44. Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education,24(3), 335–359.

    Article  Google Scholar 

  45. Vicente, S., Orrantia, J., & Verschaffel, L. (2007). Influence of situational and conceptual rewording on word problem solving. The British Journal of Educational Psychology,77(4), 829–848.

    Article  Google Scholar 

  46. Vilette, B., Fischer, J.-P., Sander, E., Sensevy, G., Quilio, S., & Richard, J.-F. (2017). Peut-on améliorer l’enseignement et l’apprentissage de l’arithmétique au CP? Le dispositif ACE [Can we improve the teaching and learning of arithmetic during first grade? The ACE program]. Revue Française de Pédagogie,201, 105–120.

    Article  Google Scholar 

  47. Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. Learning and Instruction,14(5), 445–451.

    Article  Google Scholar 

  48. Wolters, M. (1983). The part-whole schema and arithmetic problems. Educational Studies in Mathematics,14(2), 127–138.

    Article  Google Scholar 

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Funding

Funding was provided by Fonds d’expérimentation pour la jeunesse (Grant no. ACE_HAP_10).

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Correspondence to Katarina Gvozdic.

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Gvozdic, K., Sander, E. Learning to be an opportunistic word problem solver: going beyond informal solving strategies. ZDM Mathematics Education 52, 111–123 (2020). https://doi.org/10.1007/s11858-019-01114-z

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Keywords

  • Arithmetic word problem solving
  • Informal strategies
  • Arithmetic knowledge
  • Mathematics education
  • Adaptive expertise
  • Semantic recoding