Mathematical justifications motivated by the modeling of a physical phenomenon
Article Sidebar
Main Article Content
Abstract
The implementation of a modelling activity for engineering students that allows a curve fit to be made as a representation of the behaviour of a damped harmonic oscillator will be discussed. The discussion is centred on the relevance of the application of a modelling activity that uses digital tools for cell phones and computers, with the intention of recognizing elements that show skills to be developed, necessary to propose mathematical justifications to validate the mathematical model that describes the physical phenomenon. Some opportunities for pedagogical intervention are discussed.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
Apostol, T. M. (1969). Calculus, vol. 1 y 2. Reverté.
Baird, D. C. (1991). Experimentación. Una introducción a la teoría de mediciones y al diseño de experimentos. Prentice Hall.
Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18(2), 147–176. http://doi.org/10.1007/BF00314724
Balacheff, N. (1991). The Benefits and Limits of Social Interaction: The Case of Mathematical Proof. En A. J. Bishop, et al. (Eds.), Mathematical Knowledge: Its Growth Through Teaching. Mathematics Education Library, vol 10. Springer. https://doi.org/10.1007/978-94-017-2195-0_9
Balacheff, N. (2019) Contrôle, preuve et démonstration. Trois régimes de la validation. In: Pilet J., Vendeira C. (Eds.) Actes du séminaire national de didactique des mathématiques 2018 (pp.423-456). ARDM et IREM de Pris - Université de Pari Diderot.
Barbosa, J. C. (2019). Commentary on Affect, Cognition and Metacognition in Mathematical Modelling. In Scott¬ A.¬Chamberlin & Bharath¬ Sriraman (Eds.) Affect in Mathematical Modeling (pp. 3-13). Springer. https://doi.org/10.1007/978-3-030-04432-9_1
Barwell, R. (2013). Formal and informal language in mathematics classroom interaction: a dialogic perspective. In A. M. Lindmeier y A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the PME, Vol. 2 (pp. 73–80). http://www.lettredelapreuve.org/pdf/PME37/Barwell.pdf
Bell, W. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1), 23–40. https://doi.org/10.1007/BF00144356
Blum W. (2011). ¿Can modelling be taught and learnt? Some answers from empirical research. In Kaiser G., Blum W., Borromeo Ferri R. & Stillman G. (Eds) Trends in Teaching and Learning of Mathematical Modelling. International Perspectives on the Teaching and Learning of Mathematical Modelling, Vol 1 (pp. 15-30). Springer. https://doi.org/10.1007/978-94-007-0910-2_3
Boero, P. (1999). Argumentación y demostración: una relación compleja, productiva, e inevitable en las matemáticas y en la educación matemática. Prueba. http://www.lettredelapreuve.org/OldPreuve/Newsletter/990708Theme/990708ThemeES.html
Cirillo, M., Pelesko, J. A., Felton-Koestler, M. D., & Rubel, L. (2016). Perspectives on modeling in school mathematics. In Christian R. Hirsch & Amy Roth McDuffie (Eds). Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics (pp. 3-16). National Council of Teachers of Mathematics.
Duval, R. (1993). Registres de représentation sémiotique et le fonctionnement cognitif de la pensé. Annales de Didactique et de Sciences Cognitives, (5), 37- 65.
Duval, R. (1999). Semiosis y pensamiento humano. Universidad del Valle.
Fischbein, E. (1982). Intuition and Proof. For the Learning of Mathematics, 3(2), 9–18.
Halliday, D., Resnick, R., & Walker, J. (2001). Fundamentos de Física, volúmenes 1 y 2. CECSA
Halloun, I. A. (2007). Mediated modeling in science education. Science & Education, 16(7), 653-697. https://doi.org/10.1007/s11191-006-9004-3
Hanna, G., & Barbeau, E. (2002). What Is a Proof? In B. Baigrie (Ed.), History of Modern Science and Mathematics, Vol. 1, (pp. 36–48). Charles Scribner’s Sons.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Shoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education. Vol 3 (pp. 234–283). American Mathematical Society.
Hemmi, K., Lepik, M., & Viholainen, A. (2013). Analysing proof-related competences in Estonian, Finnish and Swedish mathematics curricula -- towards a framework of developmental proof. Journal of Curriculum Studies, 45(3), 354-378. https://doi.org/10.1080/00220272.2012.754055
Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3), 302-310. https://doi.org/10.1007/BF02652813
Kelly, G., Druker, S., & Chen, C. (1998). Students’ reasoning about electricity: Combining performance assessments with argumentation analysis. International Journal of Science Education, 20(7), 849–871. https://doi.org/10.1080/0950069980200707
Maaß, K. (2006). What are modelling competencies? Zentralblatt für Didaktik der Mathematik, 38(2), 113-142. https://doi.org/10.1007/BF02655885
Malafosse D., Lerouge A. & Dusseau J.M. (2000), Étude en inter-didactique des mathématiques et de la physique de l’acquisition de la loi d’Ohm au collège : espace de réalité, Didaskalia,16, 81–106
Marrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1–2), 87–125. http://doi:10.1023/A:1012785106627
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66, 23-41. https://doi.org/10.1007/s10649-006-9057-x
Protter, M., & Morrey, C. (1964). Modern Mathematical Analysis. Addison-Wesley
Sevinc, S., & Lesh, R. (2018). Training mathematics teachers for realistic math problems: a case of modeling-based teacher education courses. ZDM, 50(1), 301-314. https://doi.org/10.1007/s11858-017-0898-9
Toulmin, S. E. (1969). The uses of argument. Cambridge University Press
Touma, G. (2009). Une étude sémiotique sur l'activité cognitive d'interprétation. Annales de didactique et de sciences cognitives. (14): 79-101
Umland, K., & Sriraman, B. (2014). Argumentation in Mathematics. In Stephen Lerman (Ed.) Encyclopedia of Mathematics Education (pp. 44–46). Springer. https://doi.org/10.1007/978-94-007-4978-8
Watson, F. (1980). The role of proof and conjecture in mathematics and mathematics teaching. International Journal of Mathematical Education in Science and Technology, 11(2), 163-167. https://doi.org/10.1080/0020739800110202