Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. CITO (Central Institute for the Development of Tests) provides Dutch schools with standardized tests for different subjects and grade levels. One of the CITO Tests is the Student Monitoring Tests for Mathematics. The DLE Test (Didactic Age Equivalent Test) is a different instrument published by Eduforce that teachers can use to measure their students’ development in a particular subject.

  2. The original versions of these problems have been developed for the World Class Tests. In 2004, Peter Pool and John Trelfall from the Assessment and Evaluation Unit, School of Education, University of Leeds who were involved in the development of these problems asked us to pilot them in the Netherlands.

  3. The coding scheme was developed by two of the authors, Marja van den Heuvel-Panhuizen and Angeliki Kolovou, and our Freudenthal Institute colleague Arthur Bakker.

  4. This control coding was done by Conny Bodin-Baarends who was involved in the data collection, but did not participate in the development of the coding scheme.

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.

    Article  Google 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.

    Article  Google 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.

    Article  Google 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.

    Article  Google Scholar 

  • De Corte, E. (2007). Learning from instruction: The case of mathematics. Learning Inquiry, 1, 19–30. doi:10.1007/s11519-007-0002-4.

    Article  Google 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.

    Article  Google 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.

    Article  Google 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.

  • 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.

    Article  Google 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.

    Article  Google Scholar 

  • Kantowski, M. G. (1977). Processes involved in mathematical problem solving. Journal for Research in Mathematics Education, 8, 163–180. doi:10.2307/748518.

    Article  Google 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.

    Article  Google 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.

  • 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.

    Article  Google 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.

    Article  Google 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.

  • 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.

    Article  Google 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.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Iliada Elia.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Elia, I., van den Heuvel-Panhuizen, M. & Kolovou, A. Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM Mathematics Education 41, 605–618 (2009). https://doi.org/10.1007/s11858-009-0184-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-009-0184-6

Keywords

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