Knowledge of mathematics teachers in initial training regarding mathematical proofsLogic-mathematical aspects in the evaluation of arguments

  1. Alfaro-Carvajal, Christian 1
  2. Flores-Martínez, Pablo 2
  3. Valverde-Soto, Gabriela 3
  1. 1 Universidad Nacional
  2. 2 Universidad de Granada
    info

    Universidad de Granada

    Granada, España

    ROR https://ror.org/04njjy449

  3. 3 Universidad de Costa Rica
    info

    Universidad de Costa Rica

    San José, Costa Rica

    ROR https://ror.org/02yzgww51

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Journal:
Uniciencia

ISSN: 2215-3470

Year of publication: 2022

Issue Title: Uniciencia. January-December, 2022

Volume: 36

Issue: 1

Type: Article

DOI: 10.15359/RU.36-1.9 DIALNET GOOGLE SCHOLAR lock_openDialnet editor

More publications in: Uniciencia

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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.

Bibliographic References

  • 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
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