Trayectorias de aprendizaje del concepto de recta tangente en alumnos de Bachillerato

  1. Orts Muñoz, Abilio 1
  2. Llinares Ciscar, Salvador 2
  3. Boigues Planes, Francisco José 3
  1. 1 IES Tavernes Blanques
  2. 2 Universitat d'Alacant
    info

    Universitat d'Alacant

    Alicante, España

    ROR https://ror.org/05t8bcz72

  3. 3 Universidad Politécnica de Valencia
    info

    Universidad Politécnica de Valencia

    Valencia, España

    ROR https://ror.org/01460j859

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Revista:
Enseñanza de las ciencias: revista de investigación y experiencias didácticas

ISSN: 0212-4521,  2174-6486

Ano de publicación: 2018

Volume: 36

Número: 3

Páxinas: 121-140

Tipo: Artigo

DOI: 10.5565/REV/ENSCIENCIAS.2252 DIALNET GOOGLE SCHOLAR lock_openDDD editor

Outras publicacións en: Enseñanza de las ciencias: revista de investigación y experiencias didácticas

Resumo

The goal of this research is to characterize high school students’ learning trajectories of the concept of tangent line in a teaching experiment. The experiment considers a model of learning progression of the concept of tangent line using the local linearity of a function (Leibnizian model) to support the transition from the Euclidean conception to the Cartesian one. We identify three learning trajectories characterized by how high students manage both: i) the relation between the graphical and analytical registers to support the progression from the Euclidean conception to the Cartesian one through the Leibnizian model; and, ii) the approximation to the value of a function by the tangent line in the neighbourhood of the point of tangency. The findings suggest that the internalization of the Leibnizian conception is necessary to overcome the epistemological obstacle that the Euclidean model entails for the learning of the concept of tangent line.

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Información de financiamento

Esta investigación ha recibido el apoyo del Proyecto I+D+i, EDU2014-54526-R del Ministerio de Economía y Competitividad del Gobierno de España.Una versión previa de algunos de estos resultados fue presentada en el XX Simposio de la SEIEM, Málaga en septiembre de 2016.

Financiadores

Referencias bibliográficas

  • Aranda, M. C., & Callejo, M. L. (2017). Formas de aproximar el área bajo una curva: un estudio con estudiantes de bachillerato. Enseñanza de las Ciencias, 35(1), 157-174. https://doi.org/10.5565/rev/ensciencias.2075.
  • Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa, S., Trigueros, M., & Weller, K. (2014). Apos Theory. A framework for research and curriculum development in Mathematics Education. Londres, Springer. https://doi.org/10.1007/978-1-4614-7966-6.
  • Battista, M. (2011). Conceptualizations and issues related to learning progressions, learning trajectories and levels of sophistication. The Mathematics Enthusiast, 8(3), 507-570.
  • Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis. Research in Mathematics Education, 10(1), 53-70. https://doi.org/10.1080/14794800801916457.
  • Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract on visualisation. For the learning of Mathematics, 29(3), 31-36.
  • Biza, I., & Zachariades, T. (2010). First year mathematics undergraduates’ settled images of tangent line. The Journal of Mathematical Behavior, 29(4), 218-229. https://doi.org/10.1016/j.jmathb.2010.11.001.
  • Canul, E., Dolores. C., & Martínez-Sierra, G. (2011). De la concepción global a la concepción local. El caso de la recta tangente en el marco de la convención matemática. Revista Latinoamericana de Investigación en Matemática Educativa, 14(2), 173-202.
  • Castela, C. (1995). Apprendre avec et contre ses connaissances antérieures: Un exemple concret, celui de la tangente. Recherches en Didactique des Mathématiques, 15(1), 7-47.
  • Clement, J. (2000). Analysis of clinical interviews: foundations and model viability. En A. E. Kelly & R. A. Lesh (eds.), Handbook of research design in mathematics and science education (547-590). Mahwah, NJ, Lawrence Erlbaum Associates.
  • Clements, D. G., & Sarama, J. (2004). Learning trajectories in Mathematics Education. Mathematical Thinking and Learning, 6(2), 81-89. https://doi.org/10.1207/s15327833mtl0602_1.
  • Mariotti, M. A. (2012). ICT as opportunities for teaching-learning in a mathematics classroom: The semiotic potential of artefacts. En T. Y. Tso (ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 25-40). Taipei, PME.
  • Maschietto, M. (2008). Graphic calculators and micro-straightness: analysis of a didactic engineering. International Journal of Computers for Mathematical Learning, 13, 207-230. https://doi.org/10.1007/s10758-008-9141-7.
  • Milani, R., & Baldino, R. (2002). The theory of limits as an obstacle to infinitesimal analysis. En A. D. Cockburn & E. Nardi (eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 345-352). Norwich, Universidad de East Anglia.
  • Orts, A., Llinares, S., & Boigues, F. J. (2016). Elementos para una Descomposición Genética del concepto recta tangente. Avances de Investigación en Educación Matemática, 10, 111-134.
  • Páez, R., & Vivier, L. (2013). Teachers’ conception of tangent line. The Journal of Mathematical Behavior, 32(2), 209-229. https://doi.org/10.1016/j.jmathb.2013.02.005.
  • Piaget, J., & García, R. (1989). Psychogenesis and the history of science. Nueva York, Columbia University Press. https://doi.org/10.1086/356015.
  • Robles, M. G., Del Castillo, A. G., & Font, V. (2010). La función derivada a partir de una visualización de la linealidad local. En M. M. Moreno, A. Estrada, J. Carrillo & T. A. Sierra (eds.), Investigación en Educación Matemática XIV (523-532). Lleida, SEIEM.
  • Roig, A. I., Llinares, S., & Penalva, M. C. (2010). Construcción del concepto de múltiplo común en el dominio de los números naturales. Enseñanza de las Ciencias, 28(2), 261-274.
  • Roig, A. I., Llinares, S., & Penalva, M. C. (2012). Different moments in the participatory stage of the secondary students’ abstraction of mathematical conceptions. BOLEMA, 26(44), 1345-1366. https://doi.org/10.1590/s0103-636x2012000400011.
  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. https://doi.org/10.2307/749205.
  • Simon, M. A. (2014). Hypothetical learning trajectories in Mathematics Education. En S. Lerman (ed.), Encyclopedia of Mathematics Education (pp. 272-275). Londres: Springer. https://doi.org/10.1007/978-94-007-4978-8_72.
  • Simon, M. A. & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91-104. https://doi.org/10.1207/s15327833mtl0602_2.
  • Simon, M. A., Tzur, R., Heinz, K., & Kinzel, M. (2004). Explicating a mechanism for conceptual learning: elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35(3), 305-329. https://doi.org/10.2307/30034818.
  • Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology. Underlying principles and essential elements. En A. E. Kelly & R. A. Lesh (eds.), Handbook of research design in mathematics and science education (267-306). Mahwah, NJ, Lawrence Erlbaum Associates.
  • Stylianides, A., & Stylianides, G. (2013). Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education. ZDM. Mathematics Education, 45, 333-341. https://doi.org/10.1007/s11858-013-0501-y.
  • Tall, D. (1985). Chords, tangents and the Leibniz notation. Mathematics Teaching, 112, 48-52.
  • Tzur, R., & Simon, M. A. (2004). Distinguishing two stages of mathematics conceptual learning. International Journal of Science and Mathematics Education, 2, 287-304. https://doi.org/10.1007/s10763-004-7479-4.
  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. En D. Tall (ed.), Advanced Mathematical Thinking (65-81). Dordrecht, Kluwer. https://doi.org/10.1007/0-306-47203-1_5.
  • Vivier, L. (2010). Un milieu théorique pour la notion de tangente dans l’enseignament secondaire. Annales de Didactique et de Sciencies Cognitives, 15, 173-199.
  • Weber, E., Walkington, C., & McGalliard, W. (2015). Expanding notions of learning trajectories in Mathematics Education. Mathematical Thinking and Learning, 17(4), 253-272. https://doi.org/10.1080/10986065.2015.1083836.
Fonte de datos: Dialnet