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Educational Studies in Mathematics

, Volume 70, Issue 1, pp 71–90 | Cite as

Every unsuccessful problem solver is unsuccessful in his or her own way: affective and cognitive factors in proving

  • Fulvia FuringhettiEmail author
  • Francesca Morselli
Article

Abstract

It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors, such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect are present. The study is performed within a university course in mathematics education, where students were given a statement in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical activity shape the performance of our students.

Keywords

Affective factors Cognitive factors Beliefs Problem solving Proof University 

Notes

Acknowledgements

Research program supported by MIUR (PRIN 2005019721_002 ‘Meanings, conjectures, proofs: from basic research in mathematics education to curricular implications’).

References

  1. Barnard, T., & Tall, D. (1997). Cognitive units, connections and mathematical proof. In E. Pehkonen (Ed.) Proceedings of PME 21 (Vol. 2, pp. 41–48). Lahti - Helsinki: University of Helsinki – Lahti Research and Training Centre.Google Scholar
  2. Berger, P. (1998). Exploring mathematical beliefs – the naturalistic approach. In M. Ahtee, & E. Pehkonen (Eds.), Research methods in mathematics and science education (pp. 25–40). Helsinki: Department of Teacher Education, Research Report 185.Google Scholar
  3. Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex and setting. Buckingham, (Bucks): Open University Press. As quoted in (Burton, 1999).Google Scholar
  4. Boaler, J., Wiliam, D., & Brown, M. (1998). Students’ experiences of ability grouping-disaffection, polarisation and the construction of failure. Paper given to the Mathematics Education and Society Conference (Measi1), Nottingham, September. As quoted in (Burton, 1999).Google Scholar
  5. Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 99–119). Dordrecht: Kluwer.Google Scholar
  6. Brousseau, G. (1997). Theory of didactical situations in mathematics, Didactique des mathématiques 1970–1990. Dordrecht: Kluwer.Google Scholar
  7. Burton, L. (1999). Why is intuition so important to mathematicians but missing from mathematics education? For the Learning of Mathematics, 19(3), 27–32.Google Scholar
  8. Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: an emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45–75.CrossRefGoogle Scholar
  9. Clarkson, P., & Leder, G. C. (1984). Causal attributions for success and failure in mathematics: a cross cultural perspective. Educational Studies in Mathematics, 15, 413–422.CrossRefGoogle Scholar
  10. Crawford, K., Gordon S., Nicholas, J., & Prosser, M. (1994). Conceptions of mathematics and it is learning: The perspectives of students entering university. Learning and Instruction, 4, 331–345. As quoted in (Burton, 1999).CrossRefGoogle Scholar
  11. DeBellis, V. A., & Goldin, G. A. (1997). The affective domain in mathematical problem solving. In E. Pehkonen (Ed.) Proceedings of PME 21 (Vol. 2, pp. 209–216). Lahti - Helsinki: University of Helsinki–Lahti Research and Training Centre.Google Scholar
  12. DeBellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: a representational perspective. Educational Studies in Mathematics, 63, 131–147.CrossRefGoogle Scholar
  13. Evans, J., Hannula, M. S., Zan, R., & Brown, L. C. (Guests eds.) (2006a). Affect in Mathematics Education: exploring theoretical frameworks. A PME special issue. Educational Studies in Mathematics, 63(2).Google Scholar
  14. Evans, J., Morgan, C., & Tsatsaroni, A. (2006b). Discursive positioning and emotion in school mathematics practices. Educational Studies in Mathematics, 63, 209–226.CrossRefGoogle Scholar
  15. Frank, M. L. (1985). What myths about mathematics are held and conveyed by teachers? Arithmetic Teacher, 37, 10–12.Google Scholar
  16. Furinghetti, F., & Morselli, F. (2007). For whom the frog jumps: the case of a good problem solver. For the Learning of Mathematics, 27(2), 22–27.Google Scholar
  17. Garofalo, J. (1989). Beliefs and their influence on mathematical performance. Mathematics Teacher, 82, 502–505.Google Scholar
  18. Gholamazad, S., Liljedahl, & P., Zazkis, R. (2003). One line proof: what can go wrong? In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of PME 27 and PMENA (Vol. 2, pp. 429–435). Honolulu: University of Hawai’i.Google Scholar
  19. Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 59–72). Dordrecht: Kluwer.Google Scholar
  20. Gómez-Chacón, I. N. (2000). Affective influence in the knowledge of mathematics. Educational Studies in Mathematics, 43, 149–168.CrossRefGoogle Scholar
  21. Hadamard, J. (1954). An essay on the psychology of invention in the mathematical field. New York: Dover, (First published by PUP in 1945).Google Scholar
  22. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education (Vol. 3, pp. 234–283). Providence, RI: American Mathematical Society.Google Scholar
  23. Haylock, D. W. (1987). A framework for assessing mathematical creativity in schoolchildren. Educational Studies in Mathematics, 18, 59–74.CrossRefGoogle Scholar
  24. Hazin, I., & da Rocha Falcâo, T. (2001). Self-esteem and performance in school mathematics: a contribution to the debate about the relationship between cognition and affect. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of PME 25 (Vol. 3, pp. 121–128). Utrecht: Freudenthal Institute.Google Scholar
  25. Heath, T. L. (1956). The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary. New York, NY: Dover Publications.Google Scholar
  26. Imai, T. (2000). The influence of overcoming fixation in mathematics towards divergent thinking in open-ended mathematics problems on Japanese junior high school students. International Journal of Mathematical Education in Science and Technology, 31, 187–193.CrossRefGoogle Scholar
  27. 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, 29–63.Google Scholar
  28. Leder, G. C., Pehkonen, E., & Törner, G. (2002). Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer.Google Scholar
  29. Leron, U., & Hazzan, O. (1997). The world according to Johnny: a coping perspective in mathematics education. Educational Studies in Mathematics, 32, 265–292.CrossRefGoogle Scholar
  30. Lithner, J. (2000). Mathematical reasoning in school tasks. Educational Studies in Mathematics, 41, 165–190.CrossRefGoogle Scholar
  31. Mason, J., & Pimm, D. (1984). Generic examples: seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289.CrossRefGoogle Scholar
  32. McLeod, D. B. (1992). Research on affect in mathematics education: a reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics learning and teaching (pp. 575–596). New York, NY: Macmillan.Google Scholar
  33. McLeod, D. B. (1994). Research on affect and mathematics learning in the JRME: 1970 to the present. Journal for Research in Mathematics Education, 25, 637–647.CrossRefGoogle Scholar
  34. Morselli, F. (2002). Analisi di processi dimostrativi in ambito algebrico, Unpublished Dissertation, Dipartimento di Matematica dell’Università di Genova.Google Scholar
  35. Morselli, F. (2006). Use of examples in conjecturing and proving: an exploratory study. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of PME 30 (Vol. 4, pp. 185–192). Prague: Charles University.Google Scholar
  36. Op’T Eynde, P., & Hannula, M. (2006). The case study of Frank. Educational Studies in Mathematics, 63, 123–129.CrossRefGoogle Scholar
  37. Pehkonen, E., & Pietilä, A. (2003). On relationships between beliefs and knowledge in mathematics education. In M.A.Mariotti (Ed.), Proceedings of CERME 3, CD-Rom.Google Scholar
  38. Poincaré, H. (1908). L’invention mathématique. L’Enseignement Mathématique, 10, 357–371.Google Scholar
  39. Polya, G. (1931). Comment chercher la solution d’un problème de mathématiques? L’Enseignement Mathématique, 30, 275–276.Google Scholar
  40. Polya, G. (1945). How to solve it, A new aspect of mathematical method. Princeton: PUP.Google Scholar
  41. Radford, L. (1996). La résolution des problèmes: comprendre puis résoudre. Bulletin AMQ, XXXVI(3), 19–30.Google Scholar
  42. Schoenfeld, A. H. (1983). Beyond the purely cognitive: Beliefs systems, social cognitions, and metacognitions as driving forces in intellectual performance. Cognitive Science, 7, 329–363.CrossRefGoogle Scholar
  43. Schoenfeld, A. H. (1987). Confessions of an accidental theorist. For the Learning of Mathematics, 7(1), 30–38.Google Scholar
  44. Schoenfeld, A. H. (1989). Exploration of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20, 338–355.CrossRefGoogle Scholar
  45. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York, NY: Macmillan.Google Scholar
  46. Simon, H. A. (1982). Comments. In M. S. Clark, & S. T. Fiske (Eds.), Affect and cognition (pp. 333–342). Hillsdale, NJ: L. Erlbaum, as quoted in (McLeod, 1992).Google Scholar
  47. Simon, M. A. (1996). Beyond inductive and deductive reasoning: the search for a sense of knowing. Educational Studies in Mathematics, 30, 197–210.CrossRefGoogle Scholar
  48. Walen, S. B., & Williams, S. R. (2002). A matter of time: Emotional responses to timed mathematics tests. Educational Studies in Mathematics, 49, 361–378.CrossRefGoogle Scholar
  49. Weber, K. (2001). Student difficulty in constructing proof: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119.CrossRefGoogle Scholar
  50. Weber, K. S., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.CrossRefGoogle Scholar
  51. Weiner, B. (1972). Theories of motivation. Chicago, IL: Rand McNally.Google Scholar
  52. Weiner, B. (1980). The order of affect in rational (attributional) approaches to human motivation. Educational Researcher, 19, 4–11.Google Scholar
  53. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.CrossRefGoogle Scholar
  54. Zan, R., Brown, L., Evans, J., & Hannula, M. S. (2006). Affect in mathematics education: an introduction. Educational Studies in Mathematics, 60, 113–121.CrossRefGoogle Scholar
  55. Zazkis, R., & Liljedahl, P. (2004). Understanding primes: The role of representation. International Journal for Research in Mathematics Education, 35, 164–186.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Dipartimento di Matematica dell’Università di GenovaGenoaItaly

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