Skip to main content
SpringerLink
Log in

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

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • 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.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Frank, M. L. (1988). Problem solving and mathematical beliefs. The Arithmetic Teacher, 35(5), 32–35.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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: Kluwer

    Google Scholar 

  • Garofalo, J. (1989). Beliefs and their influence on mathematical performance. Mathematics Teacher, 82(7), 502–505.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Green, T. F. (1971). The activities of teaching (Ch. 3). New York: McGraw Hill.

    Google Scholar 

  • Greer, B. (1993). The mathematical modelling perspective on wor(l)d problems. The Journal of Mathematical Behavior, 12(3), 239–250.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hoyles, C. (1992). Mathematics teaching and mathematics teachers: A meta-case study. For the Learning of Mathematics, 12(3), 32–44.

    Google Scholar 

  • 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 

  • Kloosterman, P., & Stage, F. (1992). Measuring beliefs about mathematical problem solving. School Science and Mathematics, 92(3), 109–115.

    Article  Google Scholar 

  • 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 

  • 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 

  • Leder, G. C., Pehkonen, E., & Törner, G. (eds). (2002). Beliefs: A hidden variable in mathematics education?. Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • Mason, J. (2004). Are beliefs believable? Mathematical Thinking and Learning, 6(3), 343–352. doi:10.1207/s15327833mtl0603_4.

    Article  Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: Addison-Wesley.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • McLeod, D. B., & Adams, V. M. (eds). (1989). Affect and mathematical problem solving. New York: Springer.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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: http://www.uni-duisburg.de/FB11/PROJECTS/MFO_abstracts.pdf.

  • Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Rokeach, M. (1968). Beliefs, attitudes and values. San Francisco: Jassey-Bass.

    Google Scholar 

  • Ruthven, K., & Coe, R. (1994). A structural analysis of students’ epistemic views. Educational Studies in Mathematics, 27(1), 101–109. doi:10.1007/BF01284530.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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: http://www.tdx.cesca.es/TDX-0925101-170122.

  • 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 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Departamento de Innovación y Formación Didáctica, Universidad de Alicante, 03690, San Vicente del Raspeig, España

    María Luz Callejo

  2. IES Gabriel Ferrater, 43206, Reus, España

    Antoni Vila

Authors
  1. María Luz Callejo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antoni Vila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to María Luz Callejo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Callejo, M.L., Vila, A. Approach to mathematical problem solving and students’ belief systems: two case studies. Educ Stud Math 72, 111–126 (2009). https://doi.org/10.1007/s10649-009-9195-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-009-9195-z

Keywords

Search

Navigation