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“No! He starts walking backwards!”: interpreting motion graphs and the question of space, place and distance

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Abstract

This article deals with the interpretation of motion Cartesian graphs by Grade 8 students. Drawing on a sociocultural theoretical framework, it pays attention to the discursive and semiotic process through which the students attempt to make sense of graphs. The students’ interpretative processes are investigated through the theoretical construct of knowledge objectification and the configuration of mathematical signs, gestures, and words they resort to in order to achieve higher levels of conceptualization. Fine-grained video and discourse analyses offer an overview of the manner in which the students’ interpretations evolve into more condensed versions through the effect of what is called in the article “semiotic contractions” and “iconic orchestrations.”

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Notes

  1. The research team included Mélanie André, Serge Demers, Alain Girouard, Isaias Miranda, Andrew Sanderson, and Sonia Gonçalves .

  2. By utterance I do not mean something necessarily verbal. As Nemirovsky and Ferrara (2009) cogently argue, utterances can also incorporate actions, gestures and other sensorial elements.

  3. It was beyond the scope of this paper to investigate the question of time. Suffice it to say that the concept of time followed a similar historical developmental path as that of space (see, e.g., Radford 2008c, 2009b).

  4. “[T]he written sign is very frequently simply a fixed gesture” (Vygotsky 1997, p. 133).

  5. I have been using the term orchestration in the manner of Bakhtin (1981).

  6. For other examples of the role of iconicity and contraction in objectifying processes, see Radford (2008d).

References

  • Aristotle (1984). Physics. In J. Barnes (Ed.), Complete works of Aristotle (Vol. 1, pp. 315–446). Princeton: Princeton, University Press.

  • Arzarello, F. (2006). Semiosis as a multimodal process. In Revista Latinoamericana de Investigación en Matemática Educativa, Special Issue on Semiotics, Culture, and Mathematical Thinking (pp. 267–299).

  • Bakhtin, M. M. (1981). The dialogical imagination. Austin: University of Texas Press.

    Google Scholar 

  • Bartolini Bussi, M., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English (Ed.), Handbook of international research in mathematics education (2nd edn., pp. 746–783). New York: Routledge, Taylor and Francis.

  • Bühler, K. (1979). Teoría del lenguaje. Translated by Julián Marías. Madrid: Alianza Editorial.

    Google Scholar 

  • Burtt, E. A. (2003). The metaphysical foundations of modern science. New York: Dover (First published in 1932).

  • Cassirer, E. (1974). An essay on man. New Haven: Yale University Press.

    Google Scholar 

  • Descartes, R. (1644). The principles of philosophy. http://www.gutenberg.org/dirs/etext03/pnpph10.txt. Accessed August 13, 2008.

  • diSessa, A., Hammer, D., Sherrin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-reprensentational expertise in children. Journal of Mathematical Behavior, 10, 117–160.

    Google Scholar 

  • Edwards, L. (2009). Gestures and conceptual integration in mathematical talk. Educational Studies in Mathematics, 70(2), 127–141.

    Article  Google Scholar 

  • Elden, S. (2006). Speaking against number: Heidegger, language and the politics of calculation. Edinburgh: Edinburgh University Press.

    Google Scholar 

  • Ilyenkov, E. V. (1977). Dialectical logic. Moscow: Progress Publishers.

    Google Scholar 

  • Leont’ev, A. N. (1978). Activity, consciousness, and personality. New Jersey: Prentice-Hall.

    Google Scholar 

  • Lévy-Bruhl, L. (1922). La mentalité primitive. France: Presses Universitaires de France.

    Google Scholar 

  • Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159–174.

    Article  Google Scholar 

  • Nemirovsky, R., & Monk, S. (2000). “If you look at it the other way…”: An exploration into the nature of symbolizing. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms (pp. 177–221). Mahwah: Lawrence Erlbaum.

    Google Scholar 

  • Piaget, J. (1973). Introduction à l’épistémologie génétique [Introduction to genetic epistemology] (Vol. 2). Paris: Presses Universitaires de France.

    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 

  • 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.

    Google Scholar 

  • Radford, L. (2008a). The ethics of being and knowing: Towards a cultural theory of Learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 215–234). Rotterdam: Sense Publishers.

    Google Scholar 

  • Radford, L. (2008b). Culture and cognition: Towards an anthropology of mathematical thinking. In L. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 439–464). New York: Routledge, Taylor and Francis.

    Google Scholar 

  • Radford, L. (2008c). Semiotic reflections on medieval and contemporary graphic representations of motion. Working paper presented at meeting of the History and Pedagogy of Mathematics Conference (HPM 2008), 14–18 July 2008, Mexico City. http://www.laurentian.ca/educ/lradford/. Accessed July 14, 2008.

  • Radford, L. (2008d). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM: The International Journal on Mathematics Education, 40(1), 83–96.

    Article  Google Scholar 

  • Radford, L. (2009a). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126.

    Article  Google Scholar 

  • Radford, L. (2009b). Signifying relative motion: Time, space and the semiotics of Cartesian graphs. In W.-M. Roth (Ed.), Mathematical representations at the interface of the body and culture. Charlotte, NC: Information Age Publishers (in press).

  • Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general: The multisemiotic dimension of students’ algebraic activity. Journal for Research in Mathematics Education, 38, 507–530.

    Article  Google Scholar 

  • Radford, L., Cerulli, M., Demers, S., & Guzmán, J. (2004). The sensual and the conceptual: Artefact-mediated kinesthetic actions and semiotic activity. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28 conference of the international group for the psychology of mathematics education (PME 28) (Vol. 4, pp. 73–80). Norway: Bergen University College.

  • Radford, L., Demers, S., Guzmán, J., & Cerulli, M. (2003). Calculators, graphs, gestures, and the production meaning. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27 conference of the international group for the psychology of mathematics education (PME27–PMENA25) (Vol. 4, pp. 55–62). University of Hawaii.

  • Radford, L., Miranda, I., & Guzmán, J. (2008). Relative motion, graphs and the heteroglossic transformation of meanings: A semiotic analysis. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the Joint 32nd conference of the international group for the psychology of mathematics education and the 30th North American chapter (Vol. 4, pp. 161–168). Morelia: Cinvestav-UMSNH.

  • Robutti, O. (2006). Motion, technology, gesture in interpreting graphs. The International Journal for Technology in Mathematics Education, 13(3), 117–126.

    Google Scholar 

  • Roth, W.-M., & Lee, Y. J. (2004). Interpreting unfamiliar graphs: A generative activity theoretical model. Educational Studies in Mathematics, 57, 265–290.

    Article  Google Scholar 

  • Slowik, E. (1999). Descartes, spacetime, and relational motion. Philosophy of Science, 66(1), 117–139.

    Article  Google Scholar 

  • Vygotsky, L. S. (1986). In A. Kozulin (Ed.), Thought and language. Cambridge: MIT Press.

  • Vygotsky, L. S. (1997). In R. Rieber (Ed.), Collected works (Vol. 4). New York: Plenum Press.

  • Vygotsky, L. S., & Luria, A. (1994). Tool and symbol in child development. In R. van der Veer & J. Valsiner (Eds.) ,The Vygotsky reader (pp. 99–174). Oxford: Blackwell Publishers.

    Google Scholar 

  • Wartofsky, M. (1979). Models, representation and the scientific understanding. Dordrecht: D. Reidel.

    Google Scholar 

  • Woodward, D. (1985). Reality, symbolism, time, and space in medieval world maps. Annals of the Association of American Geographers, 75(5), 510–521.

    Article  Google Scholar 

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Acknowledgments

This article is a result of a research program funded by The Social Sciences and Humanities Research Council of Canada/Le Conseil de recherches en sciences humaines du Canada (SSHRC/CRSH).

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  1. École des sciences de l’éducation, Université Laurentienne, Sudbury, ON, Canada

    Luis Radford

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Radford, L. “No! He starts walking backwards!”: interpreting motion graphs and the question of space, place and distance. ZDM Mathematics Education 41, 467–480 (2009). https://doi.org/10.1007/s11858-009-0173-9

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