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Relating Computational Cartesian Graphs to a Real Motion: An Analysis of High School Students’ Activity

  • Ulises Salinas-HernándezEmail author
  • Isaias Miranda
Chapter
Part of the ICME-13 Monographs book series (ICME13Mo)

Abstract

In this chapter we apply the recent findings of the Theory of Objectification (TO) to analyze grade 12 students’ processes of interpretation of Cartesian graphs that are displayed by software—that manages real time video and reproduces it frame by frame in a computer—and are related to an experiment of a free falling tennis ball’s motion across an inclined plane. We specifically are interested in analyzing how students’ understanding of the mathematical relationships of the physical variables (space and time) precedes students’ understanding of how real motion occurs. Our data analysis shows that students firstly associated the Cartesian graphs with the linear trajectory of the motion; secondly, with the researcher’s intervention, students focused their attention on the software to deal with the shape of the Cartesian graph and the physical phenomena; thirdly, by simulating the experiment with artifacts and gestures, they established a functional relationship between the spatial (vertical and horizontal) and time variables. By doing so, they were able to describe the trajectory of the tennis ball in terms of various mathematical relations (vertical position and time, horizontal position and time, vertical position and horizontal position). We conclude that the opportunity the students have to manipulate (more or less on a whim) the software, is what affords students’ processes of interpretation of a specific characteristic.

Keywords

Theory of objectification Activity Computational cartesian graphs Real motion 

Notes

Acknowledgements

We thank the teacher and the students of this research for allowing us to video record their class. We also want to thank Luis Radford and the reviewers for providing us with critical reading of an earlier draft of this chapter.

References

  1. Aristotle. (2004). Metaphysics (H. Lawson-Tancred, Trans.). New York: Penguin Group.Google Scholar
  2. Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70(2), 97–109.CrossRefGoogle Scholar
  3. Bakthin, M. M. (1981). The dialogic imagination (C. Emerson & M. Holquirt, Trans). Texas, USA: University of Texas Press.Google Scholar
  4. Burke, J. (2010). Fourth grade student conceptions of piecewise linear position functions. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 227–233). Columbus, Ohio, USA.: The Ohio State University.Google Scholar
  5. Cicero, M. L. L., & Spagnolo, F. (2009). The use of motion sensor can lead the students to understanding the Cartesian graph. CERME 6–WORKING GROUP 11, 2106.Google Scholar
  6. Clement, J. (1994). Use of physical intuition and imagistic simulation in expert problema solving. In D. Tirosh (Ed.), Implicit and explicit knowledge (pp. 204–244). Hillsdale, NJ: Ablex.Google Scholar
  7. Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41, 143–163.CrossRefGoogle Scholar
  8. Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2003). Perspectives on technology mediated learning in secondary school mathematics classrooms. The Journal of Mathematical Behavior, 22(1), 73–89.CrossRefGoogle Scholar
  9. Hegel, G. W. F. (1977). Phenomenology of the spirit (A. V. Miller, Trans.). Oxford, UK: Oxford University Press.Google Scholar
  10. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  11. Leont’ev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  12. Miranda, I., Radford, L., & Guzmán, J. (2007). Interpretación de gráficas cartesianas sobre el movimiento desde el punto de vista de la teoría de la objetivación. Educación Matemática, 19(3), 5–30.Google Scholar
  13. Monaghan, J. M., & Clement, J. (1999). Use of computer simulation to develop mental simulations for understanding relative motion concepts. International Journal of Science Education, 21(9), 921–944.CrossRefGoogle Scholar
  14. Monaghan, J. M. & Clement, J. (2000). Algorithms, visualization, and mental models: High school students’ interactions with relative motion simulation. Journal of Science Education and Technology, 9(4), 311–325. ISSN: 1059-0145/00/1200-0311Google Scholar
  15. Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and the velocity sign. Journal of Mathematical Behavior, 13, 389–422. https://doi.org/10.1016/0732-3123(94)90002-7
  16. Nemirovsky, R., Tierney, C. & Wright, T. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119–172. https://doi.org/10.1207/s1532690xci1602_1
  17. Radford, L. (2009). “No! He starts walking backwards!”: Interpreting motion graphs and the question of space, place and distance. ZDM Mathematics Education, 41, 467–480. https://doi.org/10.1007/s11858-009-0173-9.
  18. Radford, L. (2013). Three key concepts of the theory of objectification: Knowledge, knowing, and learning. Journal of Research in Mathematics Education, 2(1), 7–44.Google Scholar
  19. Radford, L. (2014a). De la teoría de la objetivación. Revista Latinoamericana de Etnomatemática, 7(2), 132–150.Google Scholar
  20. Radford, L. (2014b). On teachers and students: An ethical cultural-historical perspective. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Plenary paper) (Vol. 1, pp. 1–20). Vancouver (PME).Google Scholar
  21. Radford, L. (2014c). On the role of representations and artefacts in knowing and learning. Educational Studies in Mathematics, 85(3), 405–422.CrossRefGoogle Scholar
  22. Radford, L. (2015a). Methodological aspects of the theory of objectification. Perspectivas Da Educação Matemática, 8(18), 547–567.
Google Scholar
  23. Radford, L. (2015b). The epistemological foundations of the theory of objectification. Isonomia, 127–149.Google Scholar
  24. Roth, W.-M., & Lee, Y. J. (2003). Interpreting unfamiliar graphs: A generative, activity theoretical model. Educational Studies in Mathematics, 57, 265–290.CrossRefGoogle Scholar
  25. So, W. C., Kita, S., & Goldin-Meadow, S. (2013). When do speakers use gestures to specify who does what to whom? The role of language proficiency and type of gestures in narratives. Journal of Psycholinguistic Research, 42(6), 581–594.CrossRefGoogle Scholar
  26. Speiser, B., Walter, C., & Maher, C. A. (2003). Representing motion: An experiment in learning. Journal of Mathematical Behavior, 22, 1–35.CrossRefGoogle Scholar
  27. Tao, P.-K., & Gunstone, R. (1999). The process of conceptual change in force and motion during computer-supported physics education. Journal of Research in Science Teaching, 36(7), 859–882.CrossRefGoogle Scholar
  28. Thornton, R. K., & Sokoloff, D. R. (1990). Learning motion concepts using real-time microcomputer-based laboratory tools. American Journal of Physics, 58, 858–867.CrossRefGoogle Scholar
  29. Urban-Woldron, H. (2015). Motion sensors in mathematics teaching: Learning tools for understanding general math concepts? International Journal of Mathematical Education in Science and Technology, 46(4), 584–598.CrossRefGoogle Scholar
  30. Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press.Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Cinvestav-IPNMexico CityMexico
  2. 2.Instituto Politécnico Nacional, CICATA-LegariaMexico CityMexico

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