Abstract
Assuming that dynamic features of Dynamic Geometry Software may provide a basic representation of both variation and functional dependency, and taking the Vygotskian perspective of semiotic mediation, a teaching experiment was designed with the aim of introducing students to the idea of function. This paper focuses on the use of the Trace tool and its potentialities for constructing the meaning of function. In particular, starting from a dynamic approach aimed at grounding the meaning of function in the experience of covariation, the Trace tool can be used to introduce the twofold meaning of trajectory, at the same time global and pointwise, and leads students to grasp the notion of function.
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References
Bartolini Bussi, M. G. (1998). Verbal interaction in mathematics classroom: a Vygotskian analysis. In H. Steinbring, M. G. Bartolini Bussi & A. Sierpinska (Eds.), Language and communication in mathematics classroom (pp. 65–84). Reston, VA: NCTM.
Bartolini Bussi, M. G., & Mariotti, M. A. (in press). Semiotic mediation in the mathematics classroom. In L. English & al. (Eds), Handbook of International Research in Mathematics Education. LEA.
Bottazzini, U. (1990). Storia della matematica. Torino, Italy: UTET.
Bourbaki, N. (1939). Théorie des ensembles (fascicule des résultats). Paris, France: Herman.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, The Netherlands: Kluwer.
Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. CBMS Issues in Mathematics Education, 7, 114–162.
Confrey, J., & Maloney, A. (1996). Function probe. Communications of the ACM archive, 39(8), 86–87. New York: ACM.
Dreyfus, T., & Eisenberg, T. (1983). The function concept in college students. Focus on the Learning Problems in Mathematics, 5, 119–132.
Dubinsky, E., & Harel, G. (1992). The concept of function: Aspects of epistemology and pedagogy, Mathematical Association of America, Vol. 25. Washington DC: MAA.
Euler, L. (1945). Introductio in analysin infinitorum 2nd part. In Opera Omnia: Birkhäuser, Series 1, Vol. 9 (Original title: Introductio in analysin Infinitorum Tomus secundus Theoriam Linearum curvarum. Lousannae, 1743).
Falcade, R. (2002). Cabri-géomètre outil de médiation sémiotique pour la notion de graphe de fonction. Petit x, 58, 47–81.
Goldenberg, E. P., Lewis, P., & O’Keefe, J. (1992). Dynamic representation and the development of an understanding of functions. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 235–260). Washington DC: MAA.
Hazzan, O., & Goldenberg, E. P. (1997). Student’s understanding of the notion of function. International Journal of Computers for Mathematical Learning, 1(3), 263–290.
Kieran, C., & Yerushalmy, M. (2004). Research on the role of technological environments in algebra learning and teaching. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra (pp. 99–154). Dordrecht, The Netherlands: Kluwer.
Jahn, A. P. (2002). “Locus” and “trace” in Cabri-géomètre: relationships between geometric and functional aspects in a study of transformations. Zentralblatt für Didaktik der Mathematik, 34(3), 78–84.
Laborde, C. (1999). Core geometrical knowledge for using the modelling power of Geometry with Cabri-geometry. Teaching Mathematics and its Applications, 18(4), 166–171.
Leinhardt, G., Zaslavky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64.
Malik, M. A. (1980). Historical and pedagogical aspects of the definition of function. International Journal of Mathematical Education in Science and Technology, 11(4), 489–492.
Mariotti, M. A. (2001). Justifying and proving in the Cabri environment. International Journal of Computers for Mathematical Learning, 6(3), 257–281.
Mariotti, M. A. (2002). Influence of technologies advances on students’ math learning. In L. English, M.G. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 695–723). LEA.
Mariotti, M. A., & Bartolini Bussi, M. G. (1998). From drawing to construction: Teachers mediation within the Cabri environment. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd International Conference of PME, vol. 1 (pp. 180–95). Stellenbosch, South Africa: University of Stellenbosch.
Mariotti, M. A., Laborde, C., & Falcade, R. (2003). Function and graph in a DGS environment. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA, vol. 3 (pp. 237–244). University of Hawai’i, Honolulu, HI, USA: CRDG, College of Education.
Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings. Dordrecht, The Netherlands: Kluwer.
Radford, L. (2002). The seen, the spoken and the written: A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23.
Schwartz, J., & Yerushalmy, M. (1992). Getting students to function in and with algebra. In E. Dubinsky & G. Harel (Eds.), The concept of function : Aspects of epistemology and pedagogy (pp. 261–289). Washington DC: MAA.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 3–22.
Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 25–58). Washington DC: MAA.
Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 3–21). Dordrecht, The Netherlands: Kluwer.
Tall, D. (1996). Function and calculus. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), International handbook of mathematics education, vol. 1 (pp. 289–325). Dordrecht, The Netherlands: Kluwer.
Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. CBMS Issues in Mathematics Education, 4, 21–44.
Trigueros, M., & Ursini, S. (1999). Does the understanding of variable evolve through schooling? In O.Zaslavsky (Ed.), Proceedings of the 23rd International Conference of PME, vol. 4 (pp. 273–280). Haifa, Israel: Israel Institute of Technology.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20, 356–366.
Vygotsky, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge, MA: Harvard University Press.
Wertsch, J. V., & Addison Stone, C. (1985). The concept of internalization in Vygotsky’s account of the genesis of higher mental functions. In J. V. Wertsch (Ed.), Culture, communication and cognition: Vygotskian perspectives (pp. 162–166). New York: Cambridge University Press.
Acknowledgments
We are grateful to the teachers, Brigitte Lacaze, at the High School “Pablo Neruda” in Grenoble, and Daniela Venturi, at the High School “Liceo Scientifico Michelangelo Buonarroti” in Forte dei Marmi, for their active contribution, and to the students.
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Falcade, R., Laborde, C. & Mariotti, M.A. Approaching functions: Cabri tools as instruments of semiotic mediation. Educ Stud Math 66, 317–333 (2007). https://doi.org/10.1007/s10649-006-9072-y
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DOI: https://doi.org/10.1007/s10649-006-9072-y