Comparing Calculus Communication across Static and Dynamic Environments Using a Multimodal Approach
Tóm tắt
In this article, a thinking-as-communicating approach is used to analyse calculus students’ thinking in two environments. The first is a ‘static’ environment in the sense of static visual representations, such as those found in textbook diagrams, while the second is a dynamic environment as exploited by the use of dynamic geometry environments (DGEs). The purpose of the article is to compare calculus students’ communication as it is facilitated by each of these two environments, and to explore the role of paper- and digital-mediated representations for positioning certain ways of thinking about calculus. The analysis provides evidence that the participants employed different modes of communication – utterances, gestures and touchscreen-dragging – and they communicated about fundamental calculus ideas differently when prompted by different types of representations. The study presents implications for teaching dynamic aspects of functions and calculus, and argues for a multimodal view of communication to capture the use of gestures and dragging for communicating dynamic and temporal mathematical relationships.
Tài liệu tham khảo
Arzarello, F. (2006). Semiosis as a multilingual process. Revista Latinoamericana de Investigación en matemática educativa, 9(special issue 1), 267–299.
Arzarello, F. & Paola, D. (2003). Mathematical objects and proofs within technological environments: an embodied analysis. In Proceedings of CERME-3, Bellaria, IT: CERME. (http://www.mathematik.uni-dortmund.de/~erme/CERME3/Groups/TG9/TG9_Arzarello_cerme3.pdf).
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. ZDM – The International Journal on Mathematics Education, 34(3), 66–72.
Berry, J., & Nyman, M. (2003). Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behavior, 22(4), 479–495.
Chen, C.-L., & Herbst, P. (2013). The interplay among gestures, discourse, and diagrams in students’ geometrical reasoning. Educational Studies in Mathematics, 83(2), 285–307.
Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 135–164.
Cuoco, A. (1994). Multiple representations for functions. In J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning (pp. 121–140). Washington: The Mathematical Association of America.
Falcade, R., Laborde, C., & Mariotti, M. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317–333.
Ferrara, F., Pratt, D., & Robutti, O. (2006). The role and uses of technologies for the teaching of algebra and calculus: Ideas discussed at PME over the last 30 years. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: past, present and future (pp. 237–273). Rotterdam: Sense Publishers.
Graham, K. & Ferrini-Mundy, J. (1989). An exploration of student understanding of central concepts in calculus. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco.
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: PME.
Jackiw, N. (2001). The Geometer’s Sketchpad. Emeryville: Key Curriculum Press.
Jackiw, N. (2011). SketchExplorer. Emeryville: Key Curriculum Press.
Lave, J., & Wenger, E. (1991). Situated learning: legitimate peripheral participation. Cambridge: Cambridge University Press.
McNeil, D. (1992). Hand and mind: what gestures reveal about thought. Chicago: University of Chicago Press.
NCTM. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.
Ng, O. (2014). The interplay between language, gestures, and dragging and diagrams in bilingual learners’ mathematical communications. In P. Liljedahl, S. Oesterle, C. Nicol, D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (vol. 4, pp. 289–296). Vancouver: PME.
Ng, O. & Sinclair, N. (2013). Gestures and temporality: Children’s use of gestures on spatial transformation tasks. In A. Lindmeier, A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 361–368). Kiel: PME.
Núñez, R. (2006). Do real numbers really move? Language, thought, and gesture: the embodied cognitive foundations of mathematics. In R. Hersh (Ed.), 18 unconventional essays on the nature of mathematics (pp. 160–181). New York: Springer.
Robutti, O., & Ferrara, F. (2002). Approaching algebra through motion experiments. In A. Cockburn, E. Nardi (Eds.), Proceedings of the 26th PME International Conference (vol. 1, pp. 111–118). Norwich, UK: PME.
Schwarz, B., & Bruckheimer, M. (1990). The function concept with microcomputers: multiple strategies in problem solving. School Science and Mathematics, 90(7), 597–614.
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.
Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70(2), 191–200.
Sinclair, N., & de Freitas, E. (2014). The haptic nature of gesture: rethinking gesture with new multitouch digital technologies. Gesture, 14(3), 351–374.
Sinclair, N., & Gol Tabaghi, S. (2010). Drawing space: mathematicians’ kinetic conceptions of eigenvectors. Educational Studies in Mathematics, 74(3), 223–240.
Sinclair, N., & Yurita, V. (2008). To be or to become: how dynamic geometry changes discourse. Research in Mathematics Education, 10(2), 135–150.
Stewart, J. (2008). Calculus: early transcendentals (6th ed.). Belmont: Brooks Cole.
Tall, D. (1980). Mathematical intuition, with special reference to limiting processes. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 170–176). Berkeley: PME.
Tall, D. (1986). Building and testing a cognitive approach to the calculus using computer graphics (unpublished doctoral dissertation). Coventry: Mathematics Education Research Centre, University of Warwick.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.
Thompson, P. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2/3), 229–274.
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.
Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: stability and change in students’ conceptions. International Journal of Mathematics Education in Science and Technology, 38(5), 609–637.
Vygotsky, L. (1978). Mind in society. Cambridge: Harvard University Press.
Weber, E., Tallman, M., Byerley, C., & Thompson, P. (2012). Understanding the derivative through the calculus triangle. The Mathematics Teacher, 106(4), 274–278.
Wenger, E. (1998). Communities of practice: learning, meaning and identity. Cambridge: Cambridge University Press.
Williams, S. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22(3), 219–236.
Wittgenstein, L. (1953). Philosophical investigations. Oxford: Basil Blackwell.
Yerushalmy, M., & Swidan, O. (2012). Signifying the accumulation graph in a dynamic and multi-representation environment. Educational Studies in Mathematics, 80(3), 287–306.