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Abstract

The teaching-learning process is analyzed in a course for a group of professors who were taught subjects on Calculus, to study the episodes of problem-solving in them, focused on the identification of patterns and argumentation using counterexamples. The explanation and the argument in the classroom can be used together so that the argument (issued as a counterexample) supports the explanation (conjecture). Developing the mathematics class so that the above occurs is a form of interaction and how to encourage students to move from explanation to argumentation (placing a hybrid system). Furthermore, both forms of reasoning can influence dialogue protocols and strategies. In this work, the dialogue model is described as a tool to address the problem that arises when working with students.

Keywords

Argument Conjecture Counterexample Dialogue

Article Details

References

  1. Association of Mathematics Teacher Educators [AMTE]. (2017). Standards for mathematics teacher preparation. Association of Mathematics Teacher Educators. Retrieved from https://amte.net/standards

  2. Balacheff, N. (1999). Is Argumentation an Obstacle? Invitation to a Debate. Retrieved from: https://files.eric.ed.gov/fulltext/ED435644.pdf

  3. Beam, J., Belnap, J., Kuennen, E., Parrott, A., Seaman, C., & Szydlik, J. (2019). Big ideas in mathematics for future teachers: Big ideas in geometry and data analysis. Retrieved from https://www.uwosh.edu/mathematics/BigIdeas/BigIdeas

  4. Beam, J., Belnap, J., Kuennen, E., Parrott, A., Seaman, C., & Szydlik, J. (2019). Big ideas in mathematics for future teachers: big ideas in numbers and operations. Retrieved from https://www.uwosh.edu/mathematics/BigIdeas/BigIdeas

  5. Benitez, D. (2006). The use of the Internet to support research in educational mathematics. In Memoirs of the XIV Meeting of Professors of Mathematics (pp. 19-36). Educational Mathematics Area, Universidad Michoacana de San Nicolás de Hidalgo.

  6. Cañadas, M. C. (2002). Razonamiento inductivo puesto de manifiesto por alumnos de secundaria. Universidad de Granada.

  7. Canadas, M. C., Castro, E., & Castro, E. (2008). Patterns, generalization and inductive strategies of secondary students working on the tiles problem. PNA-REVISTA DE INVESTIGACION EN DIDACTICA DE LA MATEMATICA, 2(3), 137-151.

  8. Castro, F., Rodríguez, A. A. T., Campos Nava, M., & Morales Maure, L. (2021). La construcción científica del conocimiento de los estudiantes a partir de las gráficas con tracker. Revista Universidad y Sociedad, 13(1), 83-88.

  9. García, O., & Morales, L. (2013). El contraejemplo como recurso didáctico en la enseñanza del cálculo. UNIÓN. Revista Iberoamericana de Educación Matemática, 35, 161-175.

  10. Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. Zdm, 39(1), 127-135. https://doi.org/10.1007/s11858-006-0004-1

  11. Godino, J. D., Font, V., Wilhelmi, M. R., & Lurduy, O. (2011). Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. Educational Studies in Mathematics, 77(2), 247-265. https://doi.org/10.1007/s10649-010-9278-x

  12. Ibañes, M. (2001). Aspectos cognitivos del aprendizaje de la demostración matemática en alumnos de primer curso de bachillerato [Cognitive aspects of learning mathematical proofs in students in fifth year of secondary education]. Valladolid: Universidad de Valladolid.

  13. Johnson-Laird, P. N., & Byrne, R. M. J. (1993). Précis of Deduction. Behavioral and Brain Sciences, 16(2), 323-333. https://doi.org/10.1017/S0140525X00030260

  14. Klauer, K. J. (2001). Training des induktiven Denkens. Handbuch kognitives Training, 2, 165-209.

  15. Klauer, K. J., & Phye, G. D. (1994). Cognitive training for children: A developmental program of inductive reasoning and problem solving. Seattle; Toronto: Hogrefe & Huber.

  16. Mallart, A., Font, V., & Diez, J. (2018). Case study on mathematics pre-service teachers’ difficulties in problem posing. Eurasia Journal of Mathematics, Science and Technology Education, 14(4), 1465-1481. https://doi.org/10.29333/ejmste/83682

  17. Marrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1), 87-125. https://doi.org/10.1023/A:1012785106627

  18. Martin, M. O., von Davier, M., & Mullis, I. V. (2020). Methods and Procedures: TIMSS 2019 Technical Report. International Association for the Evaluation of Educational Achievement.

  19. Maure, L. M., Fábrega, D., Nava, M. C., & Marimón, O. G. (2018). Articulation of ethnomathematical knowledge in the intercultural bilingual education of the Guna people. Educational Research and Reviews, 13(8), 307-318. https://doi.org/10.5897/ERR2017.3438

  20. Morales-Maure, L., García-Marimónb, O., García-Vázquez, E., Campos-Navad, M., Gutiérrez, J., & Esbríe, M. Á. (2022). Leading teachers who promote math learning. Journal of Positive Psychology and Wellbeing, 6(1), 2098-2108.

  21. Neubert, G. A., & Binko, J. B. (1992). Inductive Reasoning in the Secondary Classroom. Washington DC: National Education Association.

  22. Osorio, V. L. (2002). Demostraciones y conjeturas en la escuela media. Revista electrónica de didáctica de las matemáticas, 2(3), 45-55.

  23. Plato. (2003). Dialogues. Volume V: Parmenides. Teeteto. Sophist. Political. Madrid: Editorial Gredos.

  24. Polya, G. (1967). La découverte des mathématiques (2 volumes). Paris: Wiley/Dunod.

  25. Sánchez, A., Font, V., & Breda, A. (2021). Significance of creativity and its development in mathematics classes for preservice teachers who are not trained to develop students’ creativity. Mathematics Education Research Journal, 1-23. https://doi.org/10.1007/s13394-021-00367-w

  26. Schleicher, A. (2019). PISA 2018: Insights and interpretations. OECD Publishing.

  27. Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (Reprint). Journal of Education, 196(2), 1-38. https://doi.org/10.1177/002205741619600202

  28. Stenning, K., & Monaghan, P. (2005). Strategies and knowledge representation. In J. P. Leighton & R. J. Sternberg (Eds.), The nature of reasoning (pp. 129-168). Cambridge: Cambridge University Press.

  29. Sternberg, R. J., & Gardner, M. K. (1983). Unities in inductive reasoning. Journal of Experimental Psychology: General, 112(1), 80-116. https://doi.org/10.1037/0096-3445.112.1.80

  30. Yáñez, J. C., Rojas, N., & Martínez, P. F. (2013). Caracterización del conocimiento matemático para la enseñanza de los números racionales. Avances de investigación en Educación Matemática(4), 47-64.