Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context

  • Mariana Montiel
  • Miguel R. Wilhelmi
  • Draga Vidakovic
  • Iwan Elstak
Article

Abstract

The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students’ responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.

Keywords

Double and triple integration Multivariate functions Spherical and cylindrical coordinates Semiotic registers Onto-semiotic approach Personal-institutional duality 

References

  1. Alson, P. (1989). Path and graphs of functions. Focus on Learning Problems in Mathematics, 11(2), 99–106.Google Scholar
  2. Alson, P. (1991). A qualitative approach to sketch the graph of a function. School Science and Mathematics, 91(7), 231–236.Google Scholar
  3. Douady, R. (1987). Jeux de cadres et dialectique outil-objet. Recherche en didactique des mathématiques, 7(2), 5–31.Google Scholar
  4. Dray, T., & Manogue, C.(2002). Conventions for spherical coordinates. Retrieved from: http://www.math.oregonstate.edu/bridge/papers/spherical.pdf.
  5. Dubinsky, E., & MacDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, et al. (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273–280). Dordrecht: Kluwer.Google Scholar
  6. Duval, R. (2002). Proof understanding in mathematics. Proceedings of 2002 International Conference on Mathematics: Understanding Proving and Proving to Understand (pp. 23–44). Department of Mathematics, National Taiwan Normal University.Google Scholar
  7. 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.CrossRefGoogle Scholar
  8. Font, V., & Godino, J. D. (2006). La noción de configuración epistémica como herramienta de análisis de textos matemáticos: Su uso en la formación de profesores. (The notion of epistemic configuration as a tool for the analysis of mathematical texts: Its use in teacher preparation). Educaçao Matemática Pesquisa, 8(1), 67–98.Google Scholar
  9. 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, and culture (pp. 157–173). The Netherlands: Sense.Google Scholar
  10. Font, V., Godino, J., & D’Amore, B. (2007). An onto-semiotic approach to representations in mathematics education. For the Learning of Mathematics, 27, 2–14.Google Scholar
  11. Godino, J. D., & Batanero, C. (1994). Significado institucional y personal de los objetos matemáticos. (Institutional and personal meaning of mathematical objects). Recherches en Didactique des Mathématiques, 14(3), 325–355.Google Scholar
  12. Godino, J., & Batanero, C. (1997). Clarifying the meaning of mathematical objects as a priority area for research in mathematics education. In A. Sierpinska, & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity, An ICMI Study Book 1. The Netherlands: Kluwer Academic.Google Scholar
  13. Godino, J., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM. Zentralblatt für Didaktik der Mathematik, 39, 127–135. doi:10.1007/s11858-006-0004-1.CrossRefGoogle Scholar
  14. Godino, J., Batanero, C., & Roa, R. (2005). An onto-semiotic analysis of combinatorial problems and the solving processes by university students. Educational Studies in Mathematics, 60, 3–36. doi:10.1007/s10649-005-5893-3.CrossRefGoogle Scholar
  15. Godino, J. D., Bencomo, D., Font, V., & Wilhelmi, M. R. (2006). Análisis y valoración de la idoneidad didáctica de procesos de estudio de las matemáticas. (Analysis and evaluation of didactic suitability in mathematical study processes). Paradigma, XXVII(2), 221–252.Google Scholar
  16. 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 instruction processes based on the onto-semiotic approach to mathematical cognition). Recherches en Didactiques des Mathematiques, 26(1), 39–88.Google Scholar
  17. Godino, J. D., Font, V. & Wilhelmi, M. R. (2006). Análisis ontosemiótico de una lección sobre la suma y la resta. (An onto-semiotic analysis of a class presentation on addition and subtraction). Revista Latinoamericana de Investigación en Matemática Educativa, Special Issue on Semiotics, Culture and Mathematical Thinking, 131–155.Google Scholar
  18. Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. The Journal of Mathematical Behavior, 17(2), 137–165. doi:10.1016/S0364-0213(99)80056-1.CrossRefGoogle Scholar
  19. Janvier, C. (1987). Representation and understanding: The notion of function as an example. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 67–71). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  20. Larson, R., Hostetler, B., & Edwards, B. (2005). Multivariable calculus. Boston: Houghton Mifflin.Google Scholar
  21. Leathrum, T. (2002). Mathlets: Java applets for math exploration, Retrieved from: http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/.
  22. Montiel, M., Vidakovic, D., & Kabael, T. (2008). Relationship between students’ understanding of functions in Cartesian and polar coordinate systems. Investigations in Mathematics Learning, 1(2), 52–70.Google Scholar
  23. Noss, R., Bakker, A., Hoyles, C., & Kent, P. (2007). Situating graphs as workplace knowledge. Educational Studies in Mathematics, 65(3), 367–384. doi:10.1007/s10649-006-9058-9.CrossRefGoogle Scholar
  24. Pimm, D. (1987). Speaking mathematically. London: Routledge.Google Scholar
  25. Radford, L. (1997). On psychology, historical epistemology and the teaching of mathematics: towards a socio-cultural history of mathematics. For the Learning of Mathematics, 17(1), 26–33.Google Scholar
  26. Salas, S., Hille, E., & Etgen, G. (2007). Calculus one and several variables. USA: Wiley.Google Scholar
  27. Stewart, J. (2004). Calculus. USA: Thomson/Cole.Google Scholar
  28. Tall, D. (Ed.) (1991). Advanced mathematical thinking. Dordrech, Netherlands: KluwerGoogle Scholar
  29. Varberg, D., & Purcell, E. (2006). Calculus. USA: Prentice-Hall.Google Scholar
  30. Weller, K., Clark, J., Dubinsky, E., Loch, S., McDonald, M., & Merkovsky, R. (2003). Student performance and attitudes in courses based on APOS theory and the ACE teaching cycle. In A. Selden, E. Dubinsky, G. Harel, & F. Hitt (Eds.), Research in collegiate mathematics education V (pp. 97–131). Providence: American Mathematical Society.Google Scholar
  31. Wells, C. (2003). A handbook of mathematical discourse. PA: Infinity.Google Scholar
  32. Wilhelmi, M., Godino, J., & Lacasta, E. (2007a). Configuraciones epstémicas asociadas a la noción de igualdad de números reales, (Epistemic configurations associated with the notion of equality in real numbers). Recherches en Didactique des Mathématiques, 27(1), 77–120.Google Scholar
  33. Wilhelmi, M., Godino, J., & Lacasta, E.(2007b). Didactic effectiveness of mathematical definitions the case of the absolute value. International Electronic Journal of Mathematics Education, 2,2, Retrieved from: http://www.iejme.com/.
  34. Williams, J., & Wake, G. (2007). Metaphors and models in translation between college and workplace mathematics. Educational Studies in Mathematics, 64(3), 345–371. doi:10.1007/s10649-006-9040-6.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Mariana Montiel
    • 1
  • Miguel R. Wilhelmi
    • 2
  • Draga Vidakovic
    • 1
  • Iwan Elstak
    • 1
  1. 1.Georgia State UniversityAtlantaUSA
  2. 2.Public University of SpainPamplonaSpain

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