# Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics

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## Abstract

Many researchers have investigated flexibility of strategies in various mathematical domains. This study investigates strategy use and strategy flexibility, as well as their relations with performance in non-routine problem solving. In this context, we propose and investigate two types of strategy flexibility, namely inter-task flexibility (changing strategies across problems) and intra-task flexibility (changing strategies within problems). Data were collected on three non-routine problems from 152 Dutch students in grade 4 (age 9–10) with high mathematics scores. Findings showed that students rarely applied heuristic strategies in solving the problems. Among these strategies, the trial-and-error strategy was found to have a general potential to lead to success. The two types of flexibility were not displayed to a large extent in students’ strategic behavior. However, on the one hand, students who showed inter-task strategy flexibility were more successful than students who persevered with the same strategy. On the other hand, contrary to our expectations, intra-task strategy flexibility did not support the students in reaching the correct answer. This stemmed from the construction of an incomplete mental representation of the problems by the students. Findings are discussed and suggestions for further research are made.

## Keywords

Inter-task strategy flexibility Intra-task strategy flexibility Strategy use Non-routine problem solving## References

- Altun, M., & Sezgin-Memnun, D. (2008). Mathematics teacher trainees’ skills and opinions on solving non-routine mathematical problems.
*Journal of Theory and Practice in Education,**4*(2), 213–238.Google Scholar - Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.),
*The development of arithmetic concepts and skills*(pp. 1–34). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Beishuizen, M., Van Putten, C. M., & Van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred.
*Learning and Instruction,**7*(1), 87–106. doi: 10.1016/S0959-4752(96)00012-6.CrossRefGoogle Scholar - Bodin, A., Coutourier, R., & Gras, R. (2000).
*CHIC: Classification Hiérarchique Implicative et Cohésive-Version sous Windows—CHIC 1.2*. Rennes: Association pour la Recherche en Didactique des Mathématiques.Google Scholar - Cai, J. (2003). Singaporean students’ mathematical thinking in problem solving and problem posing: An exploratory study.
*International Journal of Mathematical Education in Science and Technology,**34*, 719–737. doi: 10.1080/00207390310001595401.CrossRefGoogle Scholar - Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework.
*Educational Studies in Mathematics,**58*, 45–75. doi: 10.1007/s10649-005-0808-x.CrossRefGoogle Scholar - Charles, R., Lester, F., & O’Daffer, P. (1992).
*How to evaluate progress in problem solving*. Reston, VA: NCTM.Google Scholar - De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and area of similar plane figures.
*Educational Studies in Mathematics,**35*, 65–83. doi: 10.1023/A:1003151011999.CrossRefGoogle Scholar - De Corte, E. (2007). Learning from instruction: The case of mathematics.
*Learning Inquiry,**1*, 19–30. doi: 10.1007/s11519-007-0002-4.CrossRefGoogle Scholar - Demetriou, A. (2004). Mind intelligence and development: A cognitive, differential, and developmental theory of intelligence. In A. Demetriou & A. Raftopoulos (Eds.),
*Developmental change: Theories, models and measurement*(pp. 21–73). Cambridge, UK: Cambridge University Press.Google Scholar - Doorman, M., Drijvers, P., Dekker, T., van den Heuvel-Panhuizen, M., de Lange, J., & Wijers, M. (2007). Problem solving as a challenge for mathematics education in the Netherlands.
*ZDM-The International Journal on Mathematics Education,**39*, 405–418. doi: 10.1007/s11858-007-0043-2.CrossRefGoogle Scholar - Elia, I., Panaoura, A., Gagatsis, A., Gravvani, K., & Spyrou, P. (2008). Exploring different aspects of the understanding of function: Toward a four-facet model.
*Canadian Journal of Science, Mathematics and Technology Education*,*8*(1), 49–69.CrossRefGoogle Scholar - Follmer, R. (2000).
*Reading, mathematics and problem solving: The effects of direct instruction in the development of fourth grade students’ strategic reading and problem solving approaches to textbased, nonroutine mathematical problems*. Unpublished doctoral thesis, University of Widener, Chester, PA.Google Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education*. Dordrecht: Reidel.Google Scholar - Gras, R., Suzuki, E., Guillet, F., & Spagnolo, F. (Eds.). (2008).
*Statistical implicative analysis: Theory and applications*. Heidelberg: Springer-Verlag.Google Scholar - Jonassen, J. (2000). Toward a design theory of problem solving.
*Educational Technology Research and Development,**48*(4), 63–85. doi: 10.1007/BF02300500.CrossRefGoogle Scholar - Kaizer, C., & Shore, B. (1995). Strategy flexibility in more and less competent students on mathematical word problems.
*Creativity Research Journal,**8*(1), 77–82. doi: 10.1207/s15326934crj0801_6.CrossRefGoogle Scholar - Kantowski, M. G. (1977). Processes involved in mathematical problem solving.
*Journal for Research in Mathematics Education,**8*, 163–180. doi: 10.2307/748518.CrossRefGoogle Scholar - Kolovou, A., Van den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks-A needle in a haystack.
*Mediterranean Journal for Research in Mathematics Education,**8*(2), 31–67.Google Scholar - Krems, J. F. (1995). Cognitive flexibility and complex problem solving. In P. A. Frensch & J. Funke (Eds.),
*Complex problem solving: The European perspective*(pp. 201–218). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Krutetskii, V. A. (1976).
*The psychology of mathematical abilities in school children*. Chicago: University of Chicago Press.Google Scholar - Martinsen, O., & Kaufmann, G. (1991). Effect of imagery, strategy and individual differences in solving insight problems.
*Scandinavian Journal of Educational Research,**35*(1), 69–76. doi: 10.1080/0031383910350105.CrossRefGoogle Scholar - Mayer, R. E. (1985). Implications of cognitive psychology for instruction in mathematical problem solving. In E. A. Silver (Ed.),
*Teaching and learning mathematical problem solving*(pp. 123–145). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Muir, T., & Beswick, K. (2005). Where did I go wrong? Students’ success at various stages of the problem-solving process. Accessed August 13, 2008 from http://www.merga.net.au/publications/counter.php?pub=pub_conf&id=143.
- Pantziara, M., Gagatsis, A., & Elia, I. (in press). Using diagrams as tools for the solution of non-routine mathematical problems.
*Educational Studies in Mathematics*.Google Scholar - Pape, S. J., & Wang, C. (2003). Middle school children’s strategic behavior: Classification and relation to academic achievement and mathematical problem solving.
*Instructional Science,**31*, 419–449. doi: 10.1023/A:1025710707285.CrossRefGoogle Scholar - Polya, G. (1957).
*How to solve it: A new aspect of mathematical method*(2nd ed.). Garden City: Doubleday.Google Scholar - Schoenfeld, A. (1985).
*Mathematical problem solving*. San Diego, CA: Academic Press.Google Scholar - Schoenfeld, A. H. (1983). The wild, wild, wild, wild, wild world of problem solving: A review of sorts.
*For the Learning of Mathematics,**3*, 40–47.Google Scholar - Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.),
*Handbook for research on mathematics teaching and learning*(pp. 334–370). New York: MacMillan.Google Scholar - Shore, B. M., & Kanevsky, L. S. (1993). Thinking processes: Being and becoming gifted. In K. A. Heller, F. J. Monks, & A. H. Passow (Eds.),
*International handbook of research on giftedness and talent*(pp. 131–145). Oxford, England: Pergamon.Google Scholar - Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing.
*ZDM. Zentralblatt für Didaktik der Mathematik,**29*(3), 75–80. doi: 10.1007/s11858-997-0003-x.CrossRefGoogle Scholar - Stacey, K. (1991). The effects on student’s problem solving behaviour of long-term teaching through a problem solving approach. In F. Furinghetti (Ed.),
*Proceedings of the 15th conference of the international group for the psychology of mathematics education*(Vol. 3, pp. 278–285). Assisi: PME.Google Scholar - Stanic, G., & Kilpatrick, J. (1988). Historical perspectives on problem solving in the mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.),
*The teaching and assessing of mathematical problem solving*(pp. 1–22). Reston, VA: NCTM.Google Scholar - Torbeyns, J., Desmedt, B., Ghesquière, P., & Verschaffel, L. (2009). Solving subtractions adaptively by means of indirect addition: Influence of task, subject, and instructional factors.
*Mediterranean Journal for Research in Mathematics Education*,*8*(2), 1–29.Google Scholar - Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999). Learning to solve mathematical application problems: A design experiment with fifth graders.
*Mathematical Thinking and Learning,**1*(3), 195–229. doi: 10.1207/s15327833mtl0103_2.CrossRefGoogle Scholar - Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (in press). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education.
*European Journal of Psychology of Education*.Google Scholar