## 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.

### Similar content being viewed by others

## Notes

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

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

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.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.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.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.Brissiaud, R., & Sander, E. (2010). Arithmetic word problem solving: A situation strategy first framework.

*Developmental Science,**13*(1), 92–107.Campbell, J. I. D. (2008). Subtraction by addition.

*Memory & Cognition,**36*(6), 1094–1102.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.Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics.

*Developmental Review,**34*(4), 344–377.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.Fischbein, E. (1987).

*Intuition in science and mathematics. An educational approach*. Dordrecht: Reidel.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.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.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.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.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.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.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.Hofstadter, D. R., & Sander, E. (2013).

*Surfaces and essences: Analogy as the fuel and fire of thinking*. New York: Basic Books.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.Kintsch, W., & van Dijk, A. (1978). Toward a model of text comprehension and production.

*Psychological Review,**85*(5), 363–394.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.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.Resnick, L. B. (1989). Developing mathematical knowledge through microworlds.

*American Psychologist,**44*(2), 162–169.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.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.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.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.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.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.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.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.Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics.

*ZDM—The International Journal on Mathematics Education,**41*(5), 619–625.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.Sophian, C. (2008).

*The origins of mathematical knowledge in childhood*. New York: Routledge.Threlfall, J. (2009). Strategies and flexibility in mental calculation.

*ZDM—The International Journal on Mathematics Education,**41*(5), 541–555.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.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.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.Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.),

*Encyclopedia of mathematics education*(pp. 521–552). Dordrecht: Springer.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.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.Verschaffel, L., Greer, B., & De Corte, E. (2000).

*Making sense of word problems*. Lisse: Swets & Zeitlinger.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.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.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.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.Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching.

*Learning and Instruction,**14*(5), 445–451.Wolters, M. (1983). The part-whole schema and arithmetic problems.

*Educational Studies in Mathematics,**14*(2), 127–138.

## Funding

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

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

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

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11858-019-01114-z