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The role of intuition in the solving of optimization problems

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Abstract

This article presents the partial results obtained in the first stage of the research, which sought to answer the following questions: (a) What is the role of intuition in university students' solutions to optimization problems? (b) What is the role of rigor in university students' solutions to optimization problems? (c) How is the combination of intuition and rigor expressed in university students' solutions to optimization problems? (d) Is there really an optimizing intuition? In the first part, we provide reasons that make it plausible to consider intuition as a vector (metaphorically speaking) with three components: idealization, generalization, and argumentation. In the second part, we present the experimental design of the research and analyze the data to answer the questions previously asked. The experimental design does not allow us to falsify the hypothesis that some students have an optimizing intuition.

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References

  • Acevedo, J. I. (2008). Fenómenos relacionados con el uso de metáforas en el discurso del profesor. El caso de las gráficas de funciones. [Phenomena related with the use of metaphors in teachers' discourse.] Unpublished doctoral dissertation. Barcelona, Spain: University of Barcelona.

  • D'Amore, B., Font, V., & Godino, J. D. (2007). La dimensión metadidáctica en los procesos de enseñanza y aprendizaje de la matemática [The meta-didactical dimension of mathematics teaching and learning processes]. Paradigma, XXVIII(2), 49–77.

    Google Scholar 

  • Descartes, R. (1986). In J. Cottingham (Ed.), Meditations on first philosophy: with selections from the objections and replies. Cambridge: Cambridge University Press.

    Google Scholar 

  • Dubinsky, E. (2000). Meaning and formalism in mathematics. International Journal of Computers for Mathematical Learning, 5(3), 211–240.

    Article  Google Scholar 

  • Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht: Reidel.

    Google Scholar 

  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.

    Article  Google Scholar 

  • Fischbein, E., & Grossman, A. (1997). Schemata and intuitions in combinatorial reasoning. Educational Studies in Mathematics, 34, 27–47.

    Article  Google Scholar 

  • Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively-based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105.

    Article  Google Scholar 

  • Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10, 3–40.

    Article  Google Scholar 

  • Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, 69, 33–52.

    Article  Google Scholar 

  • Font, V., Godino, J. D., & D'Amore, B. (2007). An onto-semiotic approach to representations in mathematics education. For the Learning of Mathematics, 27(2), 1–7.

    Google Scholar 

  • Font, V., Godino, J. D., & Contreras, A. (2008). From representations to onto-semiotic configurations in analysing the mathematics teaching and learning processes. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: epistemology, historicity, classroom, and culture (pp. 157–173). Sense Publishers: The Netherlands.

    Google Scholar 

  • Font, V., Godino, J. D., Planas, N., & Acevedo, J. I. (2010). The object metaphor and synecdoche in mathematics classroom discourse. For the Learning of Mathematics, 30(1), 15–19.

    Google Scholar 

  • Font, V., Planas, N., & Godino, J. D. (2010). Modelo para el análisis didáctico en educación matemática [A model for didactic analysis in mathematics education]. Infancia y Aprendizaje, 33(1), 89–105.

    Article  Google Scholar 

  • Fujita, T., Jones, K., & Yamamoto, S. (2004). The role of intuition in geometry education: learning from the teaching practice in the early 20th century. 10th International Congress on Mathematical Education (ICME-10). Copenhagen, Denmark, 4–11 July 2004, pp 1–15.

  • Godino, J. D., Batanero, C., & Roa, R. (2005). An onto-semiotic analysis of combinatorial problems and the solving processes by university students. Educational Studies in Mathematics, 60(1), 3–36.

    Article  Google Scholar 

  • Godino, J. D., Contreras A., & Font, V. (2006). Análisis de procesos de instrucción basado en el enfoque ontológico-semiótico de la cognición matemática. [Analysis of teaching processes based on the onto-semiotic approach to mathematical cognition.] Recherches en Didactique des Mathématiques, 26(1), 39–88.

    Google Scholar 

  • Godino, J. D., Batanero, C., & Font, V. (2007). The onto semiotic approach to research in mathematics education. ZDM-The International Journal on Mathematics Education, 39(1–2), 127–135.

    Article  Google Scholar 

  • Godino, J. D., Font, V., Wilhelmi, M. R., & de Castro, C. (2009). Aproximación a la dimensión normativa en didáctica de las matemáticas desde un enfoque ontosemiótico [An approach to the normative dimension in mathematics teaching from the onto-semiotic perspective]. Enseñanza de las Ciencias, 27(1), 59–76.

    Google Scholar 

  • Johnson, M. (1987). The body in the mind. Chicago: Chicago University Press.

    Google Scholar 

  • Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in ‘filling’ the gap in children’s extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39, 131–147.

    Article  Google Scholar 

  • Malaspina, U. (2007). Intuición, rigor y resolución de problemas de optimización. [Intuition, rigour and solving of optimization problems.] Revista Latinoamericana de Investigación en Matemática Educativa, 10(3), 365–399.

    Google Scholar 

  • Malaspina, U. (2008). Intuición y rigor en la resolución de problemas de optimización. Un análisis desde el Enfoque Ontosemiótico de la Cognición e Instrucción Matemática. [Intuition and rigour in the solving of optimization problems. Analysis based on the ontosemiotic approach to mathematical cognition.] Unpublished doctoral dissertation. Pontificia Universidad Católica del Perú.

  • Malaspina, U., & Font, V. (2009). Optimizing intuition. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81–88). Thessaloniki: PME.

    Google Scholar 

  • Núñez, R. (2000). Mathematical idea analysis: what embodied cognitive science can say about the human nature of mathematics. In T. Nakaora & M. Koyama (Eds.), Proceedings of PME24 (Vol. 1, pp. 3–22). Hiroshima: Hiroshima University.

    Google Scholar 

  • Piaget, J., & Inhelder, B. (1963). The child's conception of space. London: Routledge and Kegan Paul.

    Google Scholar 

  • Raftopoulos, A. (2002). The spatial intuition of number and the number line. Mediterranean Journal for Research in Mathematics Education, 1(2), 17–36.

    Google Scholar 

  • Ramos, A. B., & Font, V. (2008). Criterios de idoneidad y valoración de cambios en el proceso de instrucción matemática [Suitability and assessment criteria of changes in the mathematics instruction process]. Revista Latinoamericana de Investigación en Matemática Educativa, 11(2), 233–265.

    Google Scholar 

  • Semandi, Z. (2008). Deep intuition as a level in the development of the concept image. Educational Studies in Mathematics, 68, 1–17.

    Article  Google Scholar 

  • Sirotic, N., & Zazkis, R. (2007). Irrational numbers: the gap between formal and intuitive knowledge. Educational Studies in Mathematics, 65, 49–76.

    Article  Google Scholar 

  • Tall, D. (2006). A theory of mathematical growth through embodiment, symbolism and proof. Annales de Didactique et de Sciences Cognitives, Irem de Strasbourg, 11, 195–215.

    Google Scholar 

  • Tirosh, D., & Stavy, R. (1999). Intuitive rules: a way to explain and predict students' reasoning. Educational Studies in Mathematics, 38, 51–66.

    Article  Google Scholar 

  • Tsamir, P. (2007). When intuition beats logic: prospective teachers' awareness of their same sides—same angles solutions. Educational Studies in Mathematics, 65, 255–279.

    Article  Google Scholar 

  • Tsamir, P., & Tirosh, D. (2006). PME 1 to 30—Summing up and looking ahead: A personal perspective on infinite sets. In J. Novotná, H. Moraová, M. Krátká & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, Prague (pp. 49–63).

  • Wittgenstein, L. (1953). Philosophical investigations. New York: The MacMillan Company.

    Google Scholar 

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Acknowledgment

The research reported in this article was carried out as part of the following project: EDU 2009-08120/EDUC.

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Authors and Affiliations

  1. Departamento de Ciencias – Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, Lima, 32, Perú

    Uldarico Malaspina

  2. Departament de Didàctica de les CCEE i la Matemàtica, Facultat de Formació del Professorat, Universitat de Barcelona, Passeig de la Vall d’Hebron, 171, Barcelona, 08035, Catalonia, Spain

    Vicenç Font

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  1. Uldarico Malaspina
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  2. Vicenç Font
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Correspondence to Uldarico Malaspina.

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Malaspina, U., Font, V. The role of intuition in the solving of optimization problems. Educ Stud Math 75, 107–130 (2010). https://doi.org/10.1007/s10649-010-9243-8

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