Knowledge of mathematics teachers in initial training regarding mathematical proofs: Logic-mathematical aspects in the evaluation of arguments

Keywords: mathematics teacher’s knowledge, mathematical proof, mathematics teachers in initial training

Abstract

The objective of this study is to characterize the knowledge of mathematics teachers in initial training (MTITs) at the Universidad Nacional (Costa Rica) on the logic-syntactic and mathematical aspects involved in proving, when evaluating mathematical arguments. The research is positioned in the interpretive paradigm and has a qualitative approach. It consists of two empirical phases: in the first, a questionnaire regarding logic-syntactic aspects was applied to 25 subjects, during the months of September and October 2018 and; in the second phase, a second questionnaire covering mathematical aspects was applied to 19 subjects, during the months of May and June 2019. For the analysis of the information, knowledge indicators were proposed.  Knowledge indicators are understood as phrases to determine evidence of knowledge in the responses of the subjects. It was appreciated that the vast majority of future mathematics teachers show knowledge to discriminate when a mathematical argument corresponds or not to a proof by virtue of the logic and syntactic aspects, and of mathematical elements associated with propositions with the structure of universal implication. In general, subjects displayed greater evidence of knowledge on the logic-syntactic aspects than on the mathematical aspects. Specifically, they evidenced that consideration of a particular case or the proof of the reciprocal proposition does not prove the result; likewise, subjects evidenced knowledge about the direct and indirect proof of the universal implication. In the case of the mathematical aspects considered as hypotheses, axioms, definitions and theorems, it was appreciated that subjects could have different levels of difficulties to understand a proof.

References

Alfaro, C.; Flores, P. & Valverde, G. (2019). La demostración matemática: Significado, tipos, funciones atribuidas y relevancia en el conocimiento profesional de los profesores de matemáticas. Uniciencia, 33(2), 55-75. https://doi.org/10.15359/ru.33-2.5.
Ayalon, M. & Even, R. (2008). Deductive reasoning: In the eye of the beholder. Educational Studies in Mathematics, 69(3), 235-247. https://doi.org/10.1007/s10649-008-9136-2
Bryman, A. (2012). Social research methods. Oxford University Press.
Buchbinder, O. & McCrone, S. (2018). Taking proof into secondary classrooms–supporting future mathematics teachers. In T.E. Hodges, G. J. Roy, & A. M. Tyminski (Eds.), Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. University of South Carolina & Clemson University.
Cabassut, R., Conner, A., İşçimen, F. A, Furinghetti, F., Jahnke, H. N. & Morselli, F. (2012). Conceptions of proof–In research and teaching. En G. Hanna y M. De Villiers (Eds.), Proof and proving in mathematics education (pp. 169-190). Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_7
Carrillo, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D. & Muñoz-Catalán, C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236-253. https://doi.org/10.1080/14794802.2018.1479981
Cohen, L., Manion, L. & Morrison, K. (2007). Research Methods in Education. Routledge. https://doi.org/10.4324/9780203029053
Crespo, C. & Ponteville, C. (2005). Las funciones de la demostración en el aula de matemática. Acta Latinoamericana de Matemática Educativa, 18, 307-312. http://funes.uniandes.edu.co/5954/1/CrespoFuncionesAlme2005.pdf.
Durand-Guerrier, V., Boero, P., Douek, N., Epp, S. S. & Tanguay, D. (2012a). Argumentation and proof in the mathematics classroom. En G. Hanna y M. De Villiers (Eds.), Proof and proving in mathematics education (pp. 349-367). Springer. https://doi.org/10.1007/978-94-007-2129-6_15
Durand-Guerrier, V., Boero, P.; Douek, N., Epp, S. S. & Tanguay, D. (2012b). Examining the role of logic in teaching proof. En G. Hanna y M. De Villiers (Eds.), Proof and proving in mathematics education (pp. 369-389). Springer. https://doi.org/10.1007/978-94-007-2129-6_16
Elbaz, F. (1983). Teacher Thinking. A Study of Practica! Knowledge. Croom Helm.
Flores, Á. (2007). Esquemas de argumentación en profesores de matemáticas del bachillerato. Educación Matemática, 19(1), 63-98.
Flores-Medrano, E., Montes, M., Carrillo, J., Contreras, L., Muñoz-Catalán, M. & Liñán, M. (2016). El papel del MTSK como modelo de conocimiento del profesor en las interrelaciones entre los espacios de trabajo matemático. Bolema: Boletim de Educação Matemática, 30(54), 204-221. https://doi.org/10.1590/1980-4415v30n54a10.
Garrido, M. (1991). Lógica simbólica. Editorial Tecnos.
Hanna, G. & De Villiers, M. (2012). Aspects of proof in mathematics education. En G. Hanna y M. De Villiers (Eds.), Proof and proving in mathematics education (pp. 1-10). Springer. doi: https://doi.org/10.1007/978-94-007-2129-6_1
Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for research in mathematics education, 33(5), 379-405. https://doi.org/10.2307/4149959
Krippendorff, K. (2004). Content analysis: An introduction to its methodology. Sage publications.
Lin, F. L., Yang, K. L., Lo, J. J., Tsamir, P., Tirosh, D. & Stylianides, G. (2012). Teachers’ professional learning of teaching proof and proving. En G. Hanna y M. De Villiers (Eds.), Proof and proving in mathematics education (pp. 327-346). Springer. https://doi.org/10.1007/978-94-007-2129-6_14
Lo, J. & McCrory, R. (2009). Proof and proving in mathematics for prospective elementary teachers. En F.L. Lin, F.J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education Vol. 2 (pp. 41-46). Springer.
Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez y P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 173-204).: Sense Publisher. https://doi.org/10.1163/9789087901127_008
Martínez-Recio, A. (1999). Una aproximación epistemológica a la enseñanza y aprendizaje de la demostración matemática [Tesis doctoral]. Universidad de Granada. Documento físico.
Ministerio de Educación Pública. (2012). Programas de estudio de matemáticas I, II y III ciclos de la educación general básica y ciclo diversificado. https://mep.go.cr/sites/default/files/pro-gramadeestudio/programas/matematica.pdf.
Montoro, V. (2007). Concepciones de estudiantes de profesorado acerca del aprendizaje de la demostración. Revista electrónica de investigación en educación en ciencias, 2(1), 101-121.
National Council of Teachers of Mathematics (NCTM). (2003). Principios y estándares para la educación matemática. SAEM Thales.
Patterson, C. (1950). Los principios del pensamiento correcto: lógica. Editorial Americalee.
Pietropaolo, R. & Campos, T. (2009). Considerations about Proof in School Mathematics and in Teacher Development Programmes. En F.L. Lin, F.J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education Vol. 2 (pp. 142-147). Springer.
Ponte, J. P. & Chapman, O. (2006). Mathematics teachers’ knowledge and practices. In A. Gutierrez & P. Boero (Eds.), Handbook of reaserch on the psychology of mathematics education: Past, present and future (pp. 461-494). Sense Publisher. https://doi.org/10.1163/9789087901127_017
Ramos, M., Moreno, G. & Marmolejo, E. (2015). Concepciones de profesores de bachillerato sobre la demostración matemática en contexto escolar. XIV Conferencia Interamericana de Educación Matemática, 14. http://xiv.ciaem-redumate.org/index.php/xiv_ciaem/xiv_ciaem/paper/viewFile/1046/428.
Roberts, C. (2010). Introduction to mathematical proofs: a transition.:Chapman y Hall/CRC. https://doi.org/10.1201/b17173
Rodríguez, J. (2003). Paradigmas, enfoques y métodos en la investigación educativa. Revista del Instituto de Investigaciones Educativas, 7(12), 23-40.
Sandín, M. (2003). Investigación cualitativa en educación: Fundamentos y tradiciones. McGraw-Hill.
Shulman, L. (1986). Those who understand: Knowledge ghrotw in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for research in Mathematics Education, 38(3), 289-321.
Stylianides, G. J. & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40, 314-352.
Stylianides, G. J., Stylianides, A. J. & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 237–266).
Sullivan, P. & Woods, T. (Eds.). (2008). The International Handbook of Mathematics Teacher Education. Sense Publisher.
Tabach, M., Levenson, E., Barkai, R., Tsamir, P., Tirosh, D. & Dreyfus, T. (2009). Teachers' Knowledge of Students' Correct and Incorrect Proof Constructions. En F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education Vol. 2 (pp. 214-219). Springer.
Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M. & Cheng, Y. H. (2012). Cognitive development of proof. In Proof and proving in mathematics education (pp. 13-49). Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_2
Vicario, V. & Carrillo, J. (2005). Concepciones del profesor de secundaria sobre la demostración matemática: El caso de la irracionalidad de la raíz cuadrada de dos y las funciones de la demostración. En A. Maz, B. Gómez y M. Torralbo (Eds.), Investigación en Educación Matemática. Noveno Simposio de la SEIEM (pp. 145-152). Universidad de Córdoba.
Viseu, F., Menezes, L., Fernandes, J. A., Gomes, A., & Martins, P. M. (2017). Conceções de Professores do Ensino Básico sobre a Prova Matemática: influência da experiência profissional. Bolema, 31(57), 430-453. https://doi.org/10.1590/1980-4415v31n57a21.
Winicki-Landman, G. (1998). On Proofs and Their Performance as Works of Art. The Mathematics Teacher, 91(8), 722-725. http://www.jstor.org/stable/27970759
Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I. & Winicki-Landman, G. (2012). The need for proof and proving: Mathematical and pedagogical perspectives. In Proof and proving in mathematics education (pp. 215-229). Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_9
Published
2022-01-31
How to Cite
Alfaro-Carvajal, C., Flores-Martínez, P., & Valverde-Soto, G. (2022). Knowledge of mathematics teachers in initial training regarding mathematical proofs: Logic-mathematical aspects in the evaluation of arguments. Uniciencia, 36(1), 1-25. https://doi.org/10.15359/ru.36-1.9
Section
Original scientific papers (evaluated by academic peers)

Comentarios (ver términos de uso)