Abstract
In this paper, we analyze two episodes from an inquiry-based didactical research; the complete analysis of our research data is still ongoing. By taking into consideration various developments from the history of the geometry of space-time, our general aim is to explore high school students’ conceptions about measurement of length and time in relatively moving systems, and lead the students to reconsider these conceptions in an attempt of constructing a new metric for space-time. The episodes are extracted from long (focused) interviews with two couples of students, based on a carefully designed fictional scenario. Two main strategies have been identified and are analyzed in the paper: one of them relies on imagination and intuition; the other one makes use of preexisting school mathematical knowledge, in arriving to a simplified formula of a Minkowskian metric.
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Notes
The students were given a table with 24 lines. Here, we present an indicative part of this table.
References
Al-Balushi, S. (2013). The nature of anthropomorphic mental images created by low and high spatial ability students for different astronomical and microscopic scientific topics. International Journal of Science in Society, 4(4), 51–63.
Barbin, E. (1991). The reading of original texts: how and why to introduce a historical perspective. For the Learning of Mathematics, 11(2), 12–13.
van den Brink, F. J. (1993). Different aspects in designing mathematics education: three examples from the Freudenthal Institute. Educational Studies in Mathematics, 24, 35–64.
van den Brink, F. J. (1995). Geometry education in the midst of theories. For the Learning of Mathematics, 15(1), 21–28.
Chapman, O., & Heater, B. (2010). Understanding change through a high school mathematics teacher’s journey to inquiry-based teaching. Journal of Mathematics Teacher Education, 13(6), 445–458.
Cornford, F. M. (1937). Plato’s cosmology: the Timaeus of Plato with a running commentary. London: Routledge and Kegan Paul.
Crawford, B. A. (2000). Embracing the essence of inquiry: new roles for science teachers. Journal of Research in Science Teaching, 37, 916–937.
Driver, R., Guesne, E., & Tiberghien, A. (Eds.). (1985). Children’s ideas in science. London: Open University Press.
Durell, C. V. (1956). The theory of relativity. In J. Newman (Ed.), The world of mathematics (pp. 1107–1143). New York: Dover.
Eckstein, S. G., & Kozhevnikov, M. (1997). Parallelism in the development of children’s ideas and the historical development of projectile motion theories. International Journal of Science Education, 19(9), 1057–1073.
Eckstein, S. G., & Shemesh, M. (1993). Stage theory of the development of alternative conceptions. Journal of Research in Science Teaching, 30, 45–64.
Eddington, A. S. (1920). Space, time and gravitation. London: Cambridge University Press.
Einstein, A. (1962). Relativity: the special and general theory. London: Methuen & Co Ltd..
Einstein, A., & Infeld, L. (1938). The evolution of physics. Cambridge: Cambridge University Press.
Felsager, B. (2004a). Introducing Minkowski geometry using dynamic geometry programs, a short paper for the Topic Study Group 10 at ICME-10, Copenhagen, July 2004.
Felsager, B. (2004b). Through the looking glass: a glimpse of Euclid’s twin geometry, Supplementary Notes, ICME-10, Copenhagen, July 2004.
Gamow, G. (1939). Mr. Tompkins in wonderland. Cambridge: Cambridge University Press.
Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., da Silva, C. M. S., & Weeks, C. (2002). The use of original sources in the mathematics classroom. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education (Vol. 6, pp. 291–328). New York: Springer.
Kaisari, M., & Patronis, T. (2010). So we decided to call ‘straight line’ (…): mathematics students’ interaction and negotiation of meaning in constructing a model of elliptic geometry. Education Studies in Mathematics, 75, 253–269.
Klein, F. (1872). Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen: Verlag von Andreas Deichert (in German).
Klein, J. (1985). The world of physics and the “natural” world. In R. B. Williamson & E. Zuckerman (Eds.), Jacob Klein, lectures and essays (pp. 1–34). Annapolis: St John’s College Press.
Klein, F. (2004). Elementary mathematics from an advanced standpoint: geometry. New York: Dover.
Kuhn, T. (1962). The structure of scientific revolutions. Chicago: The University of Chicago Press.
Lappas, D., & Spyrou, P. (2006). A reading of Euclid’s elements as embodied mathematics and its educational implications. Mediterranean Journal for Research in Mathematics Education, 5(1), 1–16.
Lorentz, H. A. (1904). Electromagnetic phenomena in a system moving with any velocity smaller than that of light. Koninklijke Akademie van Wetenschappen te Amsterdam. Section of Sciences. Proceedings, 6(1903–1904): 809–831.
Matthews, M. R. (1994). Science teaching: the role of history and philosophy of science. New York: Routledge.
Niss, M. (Ed.). (2008). Proceedings of the 10th International Congress on Mathematical Education, 4–11 July, 2004. IMFUFA, Department of Science, Systems and Models, Roskilde University, Denmark.
Numbers, R. L., & Kampourakis, K. (Eds.). (2015). Newton’s apple and other myths about science. Cambridge: Harvard University Press.
Patronis, T. (1994). On students’ conception of axioms in school geometry. Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 9–16). Lisbon, Portugal.
Patronis, T., & Spanos, D. (2013). Exemplarity in mathematics education: from a romanticist viewpoint to a modern hermeneutical one. Science & Education, 22, 1993–2005.
Poincaré, H. (1952). Science and hypothesis. New York: Dover.
Sartori, L. (1996). Understanding relativity. California: University of California Press.
Stewart, D. W., & Shamdasani, P. N. (1990). Focus groups: theory and practice. Applied social research method series (volume 20). Newbury Park: Sage Publications.
Taylor, E. F., & Wheeler, J. A. (1963). Spacetime physics. San Francisco: Freeman & Co..
Verne, J. (1865). De la Terre à la Lune. Paris: P. J. Hetzel et Co (in French).
Wells, H. G. (1895). The time machine. London: William Heinemann.
Whitaker, R. J. (1983). Aristotle is not dead: student understanding of trajectory motion. American Journal of Physics, 51, 352–357.
Yaglom, I. M. (1979). A simple non-Euclidean geometry and its physical basis. New York: Springer-Verlag.
Zeichner, K. M. (1983). Alternative paradigms of teacher education. Journal of Teacher Education, 34, 3–9.
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Rizos, I., Patronis, A.(. & Lappas, D. “There is One Geometry and in Each Case There is a Different Formula”. Sci & Educ 26, 691–710 (2017). https://doi.org/10.1007/s11191-017-9915-1
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DOI: https://doi.org/10.1007/s11191-017-9915-1