# Themes and Interplay of Beliefs in Mathematical Reasoning

- 509 Downloads
- 12 Citations

## Abstract

Upper secondary students’ task solving reasoning was analysed with a focus on arguments for strategy choices and conclusions. Passages in their arguments for reasoning that indicated the students’ beliefs were identified and, by using a thematic analysis, categorized. The results stress three themes of beliefs used as arguments for central decisions: safety, expectations and motivation. Arguments such as ‘I don’t trust my own reasoning’, ‘mathematical tasks should be solved in a specific way’ and ‘using this specific algorithm is the only way for me to solve this problem’ exemplify these three themes. These themes of beliefs seem to interplay with each other, for instance in students’ strategy choices when solving mathematical tasks.

## Key words

beliefs mathematical reasoning upper secondary school## Preview

Unable to display preview. Download preview PDF.

## References

- Amabile, T. M., Hill, K. G., Hennessey, B. A. & Tighe, E. M. (1994). The work preference inventory: Assessing intrinsic and extrinsic motivational orientations.
*Journal of Personality and Social Psychology, 66*, 950–967.CrossRefGoogle Scholar - Ball, D. & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, G. Martin & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 27–44). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Braun, V. & Clarke, V. (2006). Using thematic analysis in psychology.
*Qualitative Research in Psychology, 3*, 77–101.CrossRefGoogle Scholar - Callejo, M. L. & Vila, A. (2009). Approach to mathematical problem-solving and students’ belief systems.
*Educational Studies in Mathematics, 72*, 111–126.CrossRefGoogle Scholar - Carlson, M. P. (1999). The mathematical behavior of six successful mathematics graduate students: Influences leading to mathematical success.
*Educational Studies in Mathematics, 40*(3), 237–258.CrossRefGoogle Scholar - Cobb, P., Yackel, E. & Wood, T. (1989). Young children’s emotional acts while engaged in mathematical problem solving. In D. B. McLeod & V. M. Adams (Eds.),
*Affect and mathematical problem solving. A new perspective*(pp. 117–148). New York: Springer.CrossRefGoogle Scholar - Crawford, K., Gordon, S., Nicholas, J. & Prosser, M. (1994). Conceptions of mathematics and how it is learned: The perspectives of students entering university.
*Learning and Instruction, 4*, 331–345.CrossRefGoogle Scholar - Farmaki, V. & Paschos, T. (2007). The interaction between intuitive and formal mathematical thinking: A case study.
*International Journal of Mathematical Education in Science and Technology, 38*(3), 353–365.CrossRefGoogle Scholar - Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning.
*Educational Studies in Mathematics, 38*, 11–50.CrossRefGoogle Scholar - Frank, M. L. (1988). Problem solving and mathematical beliefs.
*Arithmetic Teacher, 35*(5), 32–34.Google Scholar - Fransisco, J. M. & Hähkiöniemi, M. (2011). Students’ ways of reasoning about nonlinear functions in guess-my-rule games.
*International Journal of Science and Mathematics Education*. doi: 10.1007/s10763-011-9310-3. - Furinghetti, F. & Morselli, F. (2009). Every unsuccessful problem solver in unsuccessful in his or her own way: Affective and cognitive factors in proving.
*Educational Studies in Mathematics, 70*, 71–90.CrossRefGoogle Scholar - Furinghetti, F. & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In E. Pehkonen, G. C. Leder & G. Törner (Eds.),
*Beliefs: A hidden variable in Mathematics education?*(pp. 39–57). Dordrecht: Kluwer.Google Scholar - Green, T. F. (1971).
*The activities of teaching*. New York: McGraw-Hill.Google Scholar - Hannula, M.S. (2004).
*Affect in mathematical thinking and learning*. PhD thesis, Finland: University of Turku.Google Scholar - Hannula, M. S. (2006). Affect in mathematical thinking and learning: Towards integration of emotion, motivation and cognition. In J. Maasz & W. Schloeglmann (Eds.),
*New Mathematics Education Research and Practice*(pp. 209–232). Rotterdam: Sense Publishers.Google Scholar - Hart, L. E. (1989). Describing the affective domain: Saying what we mean. In D. B. McLeod & V. M. Adams (Eds.),
*Affect and mathematical problem solving: A new perspective*(pp. 37–48). New York: Springer.CrossRefGoogle Scholar - Kloosterman, P. (2002). Beliefs about mathematics and mathematics learning in the secondary school. In E. Pehkonen, G. C. Leder & G. Törner (Eds.),
*Beliefs: A hidden variable in Mathematics education?*(pp. 39–57). Dordrecht: Kluwer.Google Scholar - Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions.
*The Journal of Mathematical Behavior, 26*, 60–82.CrossRefGoogle Scholar - Lerch, C. M. (2004). Control decisions and personal beliefs: Their effect on solving mathematical problems.
*The Journal of Mathematical Behavior, 23*, 361–372.CrossRefGoogle Scholar - Lester, F. K., Garofalo, J. & Kroll, D. L. (1989). Self-confidence, interest, beliefs, and metacognition: Key influences on problem-solving behavior. In D. B. McLeod & V. M. Adams (Eds.),
*Affect and mathematical problem solving. A new perspective*(pp. 75–88). New York: Springer.CrossRefGoogle Scholar - Lithner, J. (2008). A research framework for creative and imitative reasoning.
*Educational Studies in Mathematics, 67*(3), 255–276.CrossRefGoogle Scholar - Liu, P.-H. (2010). Are beliefs believable? An investigation of college students’ epistemological beliefs and behavior in mathematics.
*The Journal of Mathematical Behavior, 29*(2), 86–98.CrossRefGoogle Scholar - McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. Grouws (Ed.),
*Handbook of Research in Mathematics Teaching and Learning*(pp. 575–596). New York: Macmillan Publishing Company.Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston: The Council.Google Scholar - Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. In
*Third Mediterranean conference on mathematics education*(pp. 115–124).Google Scholar - Op’t Eyende, P., De Corte, E. & Verschaffel, L. (2006). “Accepting emotional complexity”: A socio-constructivist perspective on the role of emotions in the mathematics classroom.
*Educational Studies in Mathematics, 35*(2), 189–206.Google Scholar - Philippou, G. N. & Christou, C. (1998). The effects of a preparatory mathematics program in changing prospective teachers’ attitudes towards mathematics.
*Educational Studies in Mathematics, 35*(2), 189–206.CrossRefGoogle Scholar - Pólya, G. (1945).
*How to solve it*. Princeton: Princeton University Press.Google Scholar - Prat-Sala, M. & Redford, P. (2010). The interplay between motivation, self-efficacy, and approaches to studying.
*British Journal of Educational Psychology, 80*, 283–305.CrossRefGoogle Scholar - Presmeg, N. C. (1993). Mathematics—‘A bunch of formulas’? Interplay of beliefs and problem solving styles. In I. Hirabayashi, N. Nohda, K. Shigematsu & F.-L. Lin (Eds.),
*Proceedings of the 17th PME International Conference, 3*(pp. 57–64)Google Scholar - Ryan, R. M. & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being.
*American Psychologist, 55*(1), 68–78.CrossRefGoogle Scholar - Schoenfeld, A. (1985).
*Mathematical problem solving*. Orlando: Academic.Google Scholar - Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition and sense-making in mathematics. In D. Grouws (Ed.),
*Handbook of Research in Mathematics Teaching and Learning*(pp. 334–370). New York: Macmillan Publishing Company.Google Scholar - Sfard, A. (2001). There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning.
*Educational Studies in Mathematics, 46*, 13–57.CrossRefGoogle Scholar - Skemp, R. (1978). Relational understanding and instrumental understanding.
*Arithmetic Teacher, 26*(3), 9–15.Google Scholar - Speer, N. (2005). Issues of methods and theory in the study of mathematics teachers’ professed and attributed beliefs.
*Educational Studies in Mathematics, 58*(3), 361–391.CrossRefGoogle Scholar - Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. Grouws (Ed.),
*Handbook of Research in Mathematics Teaching and Learning*(pp. 127–146). New York: Macmillan Publishing Company.Google Scholar - Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning.
*Educational Studies in Mathematics, 34*(2), 97–129.CrossRefGoogle Scholar - Voigt, J. (1994). Negotiation of Mathematical Meaning and Learning Mathematics.
*Educational Studies in Mathematics, 26*(2/3), 275–298.CrossRefGoogle Scholar - Wong, N.-Y., Marton, F., Wong, K.-M. & Lam, C.-C. (2002). The lived space of mathematics learning.
*The Journal of Mathematical Behavior, 21*, 25–47.CrossRefGoogle Scholar - Wyndhamn, J. & Säljö, R. (1997). Word problems and mathematical reasoning—a study of children’s mastery of reference and meaning in textual realities.
*Learning and Instruction, 7*(4), 361–382.CrossRefGoogle Scholar