Educational Studies in Mathematics

, Volume 72, Issue 1, pp 111–126 | Cite as

Approach to mathematical problem solving and students’ belief systems: two case studies



The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12–13 years old. In this study we identified different types of approaches to problems that determine the behavior of students in the problem-solving process. The research found two aspects that explain the students’ approaches to problem solving: (1) the presence of a dualistic belief system originating in the student’s school experience; and (2) motivation linked to beliefs regarding the difficulty of the task. Our results indicate that there is a complex relationship between students’ belief systems and approaches to problem solving, if we consider a wide variety of beliefs about the nature of mathematics and problem solving and motivational beliefs, but that it is not possible to establish relationships of causality between specific beliefs and problem-solving activity (or vice versa).


Approach to problem solving Patterns of action Problem solving Students’ belief systems 



The authors would like to express their gratitude for the help provided by Salvador Llinares (Universidad de Alicante) in preparing this manuscript, by Jody Doran (Washington University in St. Louis) in its translation and for the suggestions made by anonymous reviewers.


  1. Callejo, M. L. (1994). Les représentations graphiques dans la résolution de problèmes: Une expérience d’entraînement d’étudiantes dans un club mathématique. Educational Studies in Mathematics, 27(1), 1–33. doi: 10.1007/BF01284526.CrossRefGoogle Scholar
  2. De Corte, E., Verschaffel, L., & Op’t Eynde, P. (2000). Self-regulation: A characteristic and a goal of mathematics learning. In M. Boekaerts, P. R. Pintrich & M. Zeidner (Eds.), Handbook of self-regulation (pp. 687–726). San Diego: Academic Press.CrossRefGoogle Scholar
  3. Frank, M. L. (1988). Problem solving and mathematical beliefs. The Arithmetic Teacher, 35(5), 32–35.Google Scholar
  4. Furinghetti, F., & Morselli, F. (2009). Every unsuccessful problem solver is unsuccessful in his or her own way: Affective and cognitive factor in proving. Educational Studies in Mathematics, 70(1), 71–90. doi: 10.1007/s10649-008-9134-4.CrossRefGoogle Scholar
  5. Furinghetti, F., & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G. C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 39–57). Dordrecht, The Netherlands: KluwerGoogle Scholar
  6. Garofalo, J. (1989). Beliefs and their influence on mathematical performance. Mathematics Teacher, 82(7), 502–505.Google Scholar
  7. Gómez-Chacón, I. M. (2000). Affective influences in the knowledge of mathematics. Educational Studies in Mathematics, 43(2), 149–168. doi: 10.1023/A:1017518812079.CrossRefGoogle Scholar
  8. Green, T. F. (1971). The activities of teaching (Ch. 3). New York: McGraw Hill.Google Scholar
  9. Greer, B. (1993). The mathematical modelling perspective on wor(l)d problems. The Journal of Mathematical Behavior, 12(3), 239–250.Google Scholar
  10. Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7(4), 293–307. doi: 10.1016/S0959-4752(97)00006-6.CrossRefGoogle Scholar
  11. Hoyles, C. (1992). Mathematics teaching and mathematics teachers: A meta-case study. For the Learning of Mathematics, 12(3), 32–44.Google Scholar
  12. Kloosterman, P. (2002). Beliefs about mathematics and matematics learning in the secondary school: Measurement and implications for motivation. In G. C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 247–269). Dordrecht, The Netherlands: Kluwer.Google Scholar
  13. Kloosterman, P., & Stage, F. (1992). Measuring beliefs about mathematical problem solving. School Science and Mathematics, 92(3), 109–115.CrossRefGoogle Scholar
  14. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63.Google Scholar
  15. Leder, G. C. (2008). Beliefs: What lies behind the mirror. In B. Sriraman (Ed.), Beliefs and mathematics (pp. 39–54). Charlotte, NC: Information Age Publishing.Google Scholar
  16. Leder, G. C., Pehkonen, E., & Törner, G. (eds). (2002). Beliefs: A hidden variable in mathematics education?. Dordrecht, The Netherlands: Kluwer.Google Scholar
  17. Lester, F. K. (1994). Musings about mathematical problem-solving research: 1970–1994. Journal for Research in Mathematics Education, 25(6), 660–675. doi: 10.2307/749578.CrossRefGoogle Scholar
  18. Lester, F. K. (2002). Implications of research on students’ beliefs for classroom practice. In G. C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 345–353). Dordrecht, The Netherlands: Kluwer.Google Scholar
  19. Llinares, S. (2002). Participation and reification in learning to teach: The role of knowledge and beliefs. In G. C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 195–209). Dordrecht, The Netherlands: Kluwer.Google Scholar
  20. Mason, J. (2004). Are beliefs believable? Mathematical Thinking and Learning, 6(3), 343–352. doi: 10.1207/s15327833mtl0603_4.CrossRefGoogle Scholar
  21. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: Addison-Wesley.Google Scholar
  22. McLeod, D. B. (1994). Research on affect and mathematics learning in the jrme. 1970 to the present. Journal for Research in Mathematics Education, 25(6), 637–647. doi: 10.2307/749576.CrossRefGoogle Scholar
  23. McLeod, D. B., & Adams, V. M. (eds). (1989). Affect and mathematical problem solving. New York: Springer.Google Scholar
  24. Nesher, P., Hershkovitz, S., & Novotna, J. (2003). Situation model, text base and what else? Factors affecting problem solving. Educational Studies in Mathematics, 52(2), 151–176. doi: 10.1023/A:1024028430965.CrossRefGoogle Scholar
  25. Pehkonen, E., & Törner, G. (1999). Introduction to the abstract book for the Oberwolfach meeting on belief research. Abstract of the “Mathematical beliefs and their impact on teaching and learning of mathematics” (pp. 3–10). Conference at Mathematisches Forschungsinstitut Oberwolfach (MF0). Retrieved October 15, 2008, from:
  26. Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.Google Scholar
  27. Rokeach, M. (1968). Beliefs, attitudes and values. San Francisco: Jassey-Bass.Google Scholar
  28. Ruthven, K., & Coe, R. (1994). A structural analysis of students’ epistemic views. Educational Studies in Mathematics, 27(1), 101–109. doi: 10.1007/BF01284530.CrossRefGoogle Scholar
  29. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press.Google Scholar
  30. Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Hillsdale NJ: Lawrence Erlbaum.Google Scholar
  31. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of ‘well taught’ mathematics classes. Educational Psychologist, 23(2), 145–166. doi: 10.1207/s15326985ep2302_5.CrossRefGoogle Scholar
  32. Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338–355. doi: 10.2307/749440.CrossRefGoogle Scholar
  33. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and sense-making in mathematics. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 334–389). New York: MacMillan.Google Scholar
  34. Törner, G., Schoenfeld, A. H., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. ZDM Mathematics Education, 39(5–6), 353–473. doi: 10.1007/s11858-007-0053-0.CrossRefGoogle Scholar
  35. Verschaffel, L., Greer, B., & De Corte, E. (2002). Everyday knowledge and mathematical modeling of school word problems. In K. Gravemeijer, R. Lehrer, B. van Oers & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 257–276). Dordrecht, The Netherlands: Kluwer.Google Scholar
  36. Vila, A. (2001). Resolució de problemes de matemàtiques: identificació, origen i formació dels sistemes de creences en l’alumnat. Alguns efectes sobre l’abordatge dels problemes, Doctoral Dissertation, Universitat Autònoma de Barcelona, Spain. Retrieved October 15, 2008, from:
  37. Vila, A., & Callejo, M. L. (2004). Matemáticas para aprender a pensar. El papel de las creencias en la resolución de problemas. Madrid: Narcea.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Departamento de Innovación y Formación DidácticaUniversidad de AlicanteSan Vicente del RaspeigEspaña
  2. 2.IES Gabriel FerraterReusEspaña

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