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

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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.

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Notes

  1. The duality personal-institutional will be defined in the Section 2. Informally, we are referring to the shared criteria within an institution (university, mathematics department, classroom, business) of what should be previously known, or what should be taught, learned, and understood in some didactical situation or evaluation process.

  2. In single variable calculus, we treat real-variable and real-valued functions \(\left( {f:R \to R} \right)\), whose graph is in R 2 and we look at the “exponents” in terms of R n, in this case “1”, and point out that \(1 + 1 = 2\). When moving to multivariate calculus, we introduce R n. When our domain is two dimensional, we work with functions \(f:R^2 \to R\), whose graph can be represented in R 3 (2 + 1 = 3). Now, in terms of spatial dimensions, a function \(f:R^3 \to R\) would have to be graphed in \(R^4 \left( {3 + 1 = 4} \right)\), which is impossible. Triple integrals deal with functions of the type \(f\left( {x,y,z} \right) = w\). That is the reason for the relation with hyperspace.

  3. As was mentioned, his actions as well as the properties and arguments used were impeccable, if the words function and region had been exchanged in the two cases. On the other hand, the pragmatic nature of the meanings of the terms ‘function’ and ‘region’ can themselves cause the mistake, and can be explained in terms of institutional use (including the professor’s explanations). The identification of a function with its graph and, consequently, the area (definite integral) with the region, shows a semantic and syntactic coherence, in spite of the mistaken use of the terms.

References

  • Alson, P. (1989). Path and graphs of functions. Focus on Learning Problems in Mathematics, 11(2), 99–106.

    Google Scholar 

  • Alson, P. (1991). A qualitative approach to sketch the graph of a function. School Science and Mathematics, 91(7), 231–236.

    Google Scholar 

  • Douady, R. (1987). Jeux de cadres et dialectique outil-objet. Recherche en didactique des mathématiques, 7(2), 5–31.

    Google Scholar 

  • Dray, T., & Manogue, C.(2002). Conventions for spherical coordinates. Retrieved from: http://www.math.oregonstate.edu/bridge/papers/spherical.pdf.

  • 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 

  • 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.

  • 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. (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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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 

  • Larson, R., Hostetler, B., & Edwards, B. (2005). Multivariable calculus. Boston: Houghton Mifflin.

    Google Scholar 

  • Leathrum, T. (2002). Mathlets: Java applets for math exploration, Retrieved from: http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/.

  • 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 

  • 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.

    Article  Google Scholar 

  • Pimm, D. (1987). Speaking mathematically. London: Routledge.

    Google Scholar 

  • 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 

  • Salas, S., Hille, E., & Etgen, G. (2007). Calculus one and several variables. USA: Wiley.

    Google Scholar 

  • Stewart, J. (2004). Calculus. USA: Thomson/Cole.

    Google Scholar 

  • Tall, D. (Ed.) (1991). Advanced mathematical thinking. Dordrech, Netherlands: Kluwer

  • Varberg, D., & Purcell, E. (2006). Calculus. USA: Prentice-Hall.

    Google Scholar 

  • 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 

  • Wells, C. (2003). A handbook of mathematical discourse. PA: Infinity.

    Google Scholar 

  • 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 

  • 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/.

  • 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.

    Article  Google Scholar 

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Correspondence to Mariana Montiel.

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Fig. 4
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Expected answers and actual student answers of question 1

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Expected answers and actual student answers of question 2

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a Expected answers of question 3. b Actual student answers of question 3

Fig. 7
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a Expected answers of question 4. b Actual student answers of question 4

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Montiel, M., Wilhelmi, M.R., Vidakovic, D. et al. Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context. Educ Stud Math 72, 139–160 (2009). https://doi.org/10.1007/s10649-009-9184-2

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