Fostering Dialogue in the Calculus Classroom Using Dynamic Digital Technology
Tóm tắt
In this article, we discuss the dynamic digital software SimCalc MathWorlds and its potential to promote dialogue in the first calculus course for engineering students at Tecnológico de Monterrey, México. Sixty students participated in a pedagogical sequence of tasks that had been designed to help them appropriate the relations between a function and its derivative. In the classroom, the software provided a visual scenario supporting the various tasks. The simulation of cartoon motion over a straight line was included during the interaction. Corresponding graphs of position and velocity gave meaning to the function and its derivative. Active and exploratory visual perception allowed the interpretation of mathematical relations being sought as affordances provided by the software. Co-action between students and mathematical knowledge through software use promoted dialogue in order to identify those relations as invariants. A qualitative method, predominantly ethnographic, was applied during the 2 weeks of the classroom experience. The results revealed the students’ appropriation of the relations by means of mathematical language. With this experience, we propose the term ‘dialogic ecosystem’ as a way to emphasize the design and performance of the pedagogical sequence, where the teacher, students and software cohabit in an environment resulting in dialogue as an important component for the acquisition of mathematical knowledge.
Tài liệu tham khảo
Alanís, J., & Salinas, P. (2010). Cálculo de una variable: Acercamientos newtoniano y leibniziano integrados didácticamente. El Cálculo y su Enseñanza, 2, 1–14.
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM: The International Journal on Mathematics Education, 34(3), 66–72.
Bakhtin, M. (1982). The dialogic imagination. Austin, TX: University of Texas Press.
Bakhtin, M. (1986). Speech genres and other late essays. Austin, TX: University of Texas Press.
Burke, J., Hegedus, S., & Robidoux, R. (2013). Reflections on significant developments in designing SimCalc software. In S. Hegedus & J. Roschelle (Eds.), The SimCalc vision and contributions (pp. 65–83). Dordrecht, NL: Springer.
Fernández–Cárdenas, J. (2015). Dialogism: sequentiality, positioning, plurality and historicity in the analysis of educational practice. Sinéctica, 43, 1–20.
Goodwin, C. (2007). Participation, stance and affect in the organization of activities. Discourse & Society, 18(1), 53–73.
Hegedus, S., & Moreno-Armella, L. (2010). Accommodating the instrumental genesis framework within dynamic technological environments. For the Learning of Mathematics, 30(1), 26–31.
Heritage, J. (2001). Goffman, Garfinkel and conversation analysis. In M. Wetherell, S. Taylor, & S. Yates (Eds.), Discourse theory and practice: a reader (pp. 47–56). London, UK: Sage.
Hong, Y., & Thomas, M. (2013). Graphical construction of a local perspective. In A. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 81–90). Kiel, DE: PME.
Levinson, S. (1983). Pragmatics. Cambridge, UK: Cambridge University Press.
Mercer, N., & Howe, C. (2012). Explaining the dialogic processes of teaching and learning: the value and potential of sociocultural theory. Learning, Culture and Social Interaction, 1(1), 12–21.
Moreno-Armella, L., & Hegedus, S. (2009). Co-action with digital technologies. ZDM – The International Journal on Mathematics Education, 41(4), 505–519.
Moreno-Armella, L., & Sriraman, B. (2005). The articulation of symbol and mediation in mathematics education. ZDM – The International Journal on Mathematics Education, 37(6), 476–486.
Moreno-Armella, L., & Sriraman, B. (2010). Symbols and mediation in mathematics education. In B. Sriraman & L. English (Eds.), Theories of mathematics education: seeking new frontiers (pp. 213–232). Berlin, DE: Springer.
Moreno-Armella, L., Hegedus, S., & Kaput, J. (2008). From static to dynamic mathematics: historical and representational perspectives. Educational Studies in Mathematics, 68(2), 99–111.
Sacks, H., Schegloff, E., & Jefferson, G. (1974). A simplest systematics for the organization of turn-taking for conversation. Language, 50(4), 696–735.
Salinas, P. (2013). Approaching calculus with SimCalc: linking derivative and antiderivative. In S. Hegedus & J. Roschelle (Eds.), The SimCalc vision and contributions (pp. 383–399). Dordrecht, NL: Springer.
Salinas, P., & Alanís, J. (2009). Hacia un nuevo paradigma en la enseñanza del Cálculo dentro de una institución educativa. Revista Latinoamericana de Investigación En Matemática Educativa, 12(3), 355–382.
Salinas, P., & Quintero, E. (2013). Integrating digital technology for the innovation of calculus curriculum. In proceedings of the ASEE annual conference. Atlanta, GA: American Society for Engineering Education ( http://www.asee.org/public/conferences/20/papers/8153/view ).
Salinas, P., Alanís, J., & Pulido, R. (2011). Cálculo de una variable: reconstrucción para el aprendizaje y la enseñanza. Didac, 56–57, 62–69.
Salinas, P., Alanís, J., Pulido, R., Santos, F., Escobedo, J. & Garza, J. (2012a). Cálculo aplicado: Competencias matemáticas a través de contextos (Tomo I). D.F., MX: Cengage Learning.
Salinas, P., Alanís, J., Pulido, R., Santos, F., Escobedo, J. & Garza, J. (2012b). Cálculo aplicado: Competencias matemáticas a través de contextos (Tomo II). D.F., MX: Cengage Learning.
Salinas, P., Alanís, J., Pulido, R., Santos, F., Escobedo, J. & Garza, J. (2013). Cálculo aplicado: Competencias matemáticas a través de contextos (Tomo III). D.F., MX: Cengage Learning.
Salinas, P., Quintero, E., & González-Mendívil, E. (2014). An environment to promote a visual learning of calculus. In H. Arabnia, A. Bahrami, L. Deligiannidis, & G. Jandieri (Eds.), Proceedings of the 2014 international conference on frontiers in education: computer science and computer engineering (pp. 425–429). Las Vegas, NV: CSREA Press.
Schegloff, E. (1992). Repair after next turn: the last structurally provided defense of intersubjectivity in conversation. American Journal of Sociology, 97(5), 1295–1345.
Schegloff, E., Jefferson, G., & Sacks, H. (1977). The preference for self-correction in the organization of repair in conversation. Language, 53(2), 361–382.
Sinclair, N., & Robutti, O. (2014). Teaching practices in digital environments. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 598–601). Dordrecht, NL: Springer.
Thompson, P., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30(1–2), 124–147.
Vahey, P., Knudsen, J., Rafanan, K., & Lara-Meloy, T. (2013). Curricular activity systems supporting the use of dynamic representations to foster students’ deep understanding of mathematics. In C. Mouza & N. Lavigne (Eds.), Emerging technologies for the classroom (pp. 15–30). New York, NY: Springer.
Wegerif, R. (2004). The role of educational software as a support for teaching and learning conversations. Computers & Education, 43(1–2), 179–191.
Yerushalmy, M., & Swidan, O. (2012). Signifying the accumulation graph in a dynamic and multi-representation environment. Educational Studies in Mathematics, 80(3), 287–306.