Pre-service Mathematics Teachers’ Mathematical Connections in the Context of Problem-Solving About the Derivative

Year 2021, Volume: 12 Issue: 1, 202 - 220, 05.02.2021

Camılo Rodríguez Nıeto Flor Rodríguez Javier García García

https://doi.org/10.16949/turkbilmat.797182

Abstract

The mathematical connections made by five Pre-Service Mathematics Teachers (PSMTs) when solving problems on the derivative were analyzed. The conceptual framework used was the typology of intra-mathematical connections. For data collection, semi-structured interviews were conducted, and a questionnaire was designed, which included three tasks about of the derivative, which were analyzed through the thematic analysis method. The results showed that PSMTs made mathematical connections: meaning, different representations, procedural, part-whole, implication and feature. We identified that the PSMTs’ difficulties in establishing connections are caused by the meaning they have on the derivative concept, acquired in their received teaching and, if they are not attended, they can be reproduced in their future practice as in-service teachers.

Keywords

Mathematical connections, pre-service mathematics teachers, problem-solving, derivative

References

• Adams, W. C. (2015). Conducting semi-structured interviews. In K. E. Newcomer, H. P. Hatry, & J. S. Wholey (Eds.), Handbook of practical program evaluation (pp. 492–505). (Fourth ed.). New Jersey: Wiley.
• Antonio, R., Escudero, D. I., & Flores, E. (2019). Una introducción al concepto de derivada en estudiantes de bachillerato a través del análisis de situaciones de variación [An introduction to the concept of derivative in high school students]. Revista Educación matemática, 31(1), 258-280. https://doi.org/10.24844/EM3101.10
• Autonomous University of Guerrero [UAGro]. (2009). Plan de Estudios de la licenciatura en Matemáticas. México: Autor.
• Berry, J. y Nyman, M. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479–495. https://doi.org/10.1016/j.jmathb.2003.09.006
• Borji, V., Font, V., Alamolhodaei, H., & Sánchez, A. (2018). Application of the Complementarities of Two Theories, APOS and OSA, for the Analysis of the University Students’ Understanding on the Graph of the Function and its Derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(6), 2301-2315. https://doi.org/10.29333/ejmste/89514
• Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa
• Businskas, A. M. (2008). Conversations about connections: How secondary mathematics teachers conceptualize and contend with mathematical connections [Unpublished PhD Thesis]. Faculty of Education-Simon Fraser University, Canada.
• Byerley, C., & Thompson, P. W. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. The Journal of Mathematical Behavior, 48, 168-193. https://doi.org/10.1016/j.jmathb.2017.09.003
• Carter, N., Bryant-Lukosius, D., DiCenso, A., Blythe, J., & Neville, A., J. (2014). The use of triangulation in qualitative research. Oncology Nursing Forum, 41, 545–547. https://doi:10.1188/14.ONF.545-547
• Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education. London and New York: Routledge.
• Departament d’Ensenyament. (2017). Competències bàsiques de l’àmbit matemàtic. Recuperado de: http://ensenyament.gencat.cat/web/.content/home/departament/publicacions/colleccions/competenc ies-basiques/eso/eso-matematic.pdf.
• Dirección General de Bachillerado [DGB]. (2018). Cálculo diferencial. Recuperado el 06 de diciembre de 2019 https://www.dgb.sep.gob.mx/informacion-academica/programas-de-estudio/CFP/5to-Semestre/Calculo-Diferencial.pdf
• Dolores-Flores, C., & García-García, J. (2017). Conexiones Intramatemáticas y Extramatemáticas que se producen al Resolver Problemas de cálculo en Contexto: Un Estudio de Casos en el Nivel Superior. Bolema – Mathematics Education Bulletin, 31(57), 158–180. https://doi.org/10.1590/1980-4415v31n57a08
• Dolores-Flores, C., Rivera-López, M. I., & García-García, J. (2019). Exploring mathematical connections of pre-university students through tasks involving rates of change. International Journal of Mathematical Education in Science and Technology, 50(3), 369–389. https://doi.org/10.1080/0020739X.2018.1507050
• Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. https://doi.org/10.1007/s10649-006-0400-z
• Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2013). Mathematical Connections and Their Relationship to Mathematics Knowledge for Teaching Geometry. School Science and Mathematics, 113(3), 120–134. https://doi.org/10.1111/ssm.12009
• Evitts, T. (2004). Investigating the mathematical connections that preservice teachers use and develop while solving problems from reform curricula. [Unpublished dissertation], Pennsylvania State University College of Education. United States of America.
• Fuentealba, C., Badillo, E., & Sánchez-Matamoros, G. (2018). Puntos de no-derivabilidad de una función y su importancia en la comprensión del concepto de derivada [The non-derivability points of a function and their importance in the understanding of the derivative concept]. Educação e Pesquisa, 44, 1-20. https://doi.org/10.1590/s1678-4634201844181974
• Fuentealba, C., Badillo, E., & Sánchez-Matamoros, G. (2019). Identificación y caracterización de los subniveles de desarrollo del esquema de derivada. Enseñanza de las Ciencias, 37(2), 63-84. https://doi.org/10.5565/rev/ensciencias.2518
• Fuentealba, C., Badillo, E., Sánchez-Matamoros, G., & Cárcamo, A. (2018). The understanding of the derivative concept in higher education. EURASIA Journal of Mathematics, Science and Technology Education, 15(2), 1-15. https://doi.org/10.29333/ejmste/100640
• Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645-657. https://doi.org/10.1080/0144341042000262953
• García-García, J. (2018). Conexiones matemáticas y concepciones alternativas asociadas a la derivada y a la integral en estudiantes del preuniversitario [Mathematical connections and alternative conceptions associated with the derivative and the integral in pre-university students]. (Unpublished doctoral dissertation). Autonomous University of Guerrero, Mexico.
• García-García, J. G. (2019). Escenarios de exploración de conexiones matemáticas [mathematical connection exploration scenarios]. Números: Revista de Didáctica de las Matemáticas, 4 (100), 129–133.
• García-García, J., & Dolores-Flores, F. (2020). Exploring pre-university students’ mathematical connections when solving Calculus application problems. International Journal of Mathematical Education in Science and Technology, https://doi.org/10.1080/0020739X.2020.1729429
• García-García, J., & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing Calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227–252. https://doi.org/10.1080/0020739X.2017.1355994
• García-García, J., & Dolores-Flores, C. (2019). Pre-university students’ mathematical connections when sketching the graph of derivative and antiderivative functions. Mathematics Education Research Journal, https://doi.org/10.1007/s13394-019-00286-x
• Hashemi, N., Abu, M. S., Kashefi, H., & Rahimi, K. (2014). Undergraduate students’ difficulties in conceptual understanding of derivation. Procedia-Social and Behavioral Sciences, 143, 358-366.
• Herbert, S., & Pierce, R. (2011). What is rate? Does context or representation matter?. Mathematics education research journal, 23(4), 455-477. https://doi.org/10.1007/s13394-011-0026-z
• Kenedi, A. K., Helsa, Y., Ariani, Y., Zainil, M., & Hendri, S. (2019). Mathematical connection of elementary school students to solve mathematical problems. Journal on Mathematics Education, 10(1), 69-80.
• Kula-Ünver, S. (2020). How do pre-service mathematics teachers respond to students’ unexpected questions related to the second derivative?. Journal of Pedagogical Research, 4(3), 359-374.
• Mwakapenda, W. (2008). Understanding connections in the school mathematics curriculum. South African Journal of Education, 28(2), 189-202.
• Mhlolo, M. K. (2012). Mathematical connections of a higher cognitive level: A tool we may use to identify these in practice. African Journal of Research in Mathematics, Science and Technology Education, 16(2), 176–191. https://doi.org/10.1080/10288457.2012.10740738 Ministerio de Educación Nacional (Ministry of National Education) [MEN]. (2006). Estándares básicos de competencias en lenguaje, Matemáticas, ciencia y ciudadanas. Bogotá, Colombia: MEN.
• Ministry of Education [MOE] (2006a). Mathematics syllabuses – Primary. Singapore: Author.
• Ministry of Education [MOE] (2006b). Mathematics syllabuses – Lower secondary. Singapore: Author.
• Moru, E. K. (2020). An APOS analysis of university students’ understanding of derivatives: A Lesotho case Study. African Journal of Research in Mathematics, Science and Technology Education, 24(2), 279-292. https://doi.org/10.1080/18117295.2020.1821500
• National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics. Nowell, L. S., Norris, J. M., White, D. E., & Moules, N. J. (2017). Thematic analysis: Striving to meet the trustworthiness criteria. International Journal of Qualitative Methods, 16(1), 1-13. https://doi.org/10.1177/1609406917733847
• Özgen, K. (2013). Self-efficacy beliefs in mathematical literacy and connections between mathematics and real world: The case of high school students. Journal of International Education Research, 9(4), 305–316. https://doi.org/10.19030/jier.v9i4.8082
• Pambudi, D. S., Budayasa, I. K., & Lukito, A. (2018). Mathematical connection profile of junior high school students in solving mathematical problems based on gender difference. International Journal of Scientific Research and Management, 6(08), 73-78. https://doi.org/10.18535/ijsrm/v6i8.m01
• Pino-Fan, L., Godino, J., & Font, V. (2011). Faceta epistémica del conocimiento didáctico-matemático sobre la derivada [Epistemic facet of the didactic-mathematics knowledge about the derivative]. Educação Matemática Pesquisa, 13(1), 141-178.
• Pino-Fan, L. (2013). Evaluación de la faceta epistémica del conocimiento didáctico-matemático de futuros profesores de bachillerato sobre la derivada [Evaluation of the epistemic facet of the didactic-mathematical knowledge of future high school teachers about the derivative] (Unpublished doctoral dissertation). Universidad de Granada.
• Pino-Fan, L., Godino, J. D., & Font, V. (2015). Una propuesta para el análisis de las prácticas matemáticas de futuros profesores sobre derivadas [A Proposal for the analysis of Prospective teachers’ mathematical practices on Derivatives]. Bolema - Mathematics Education Bulletin, 29(51), 60–89. https://doi.org/10.1590/1980-4415v29n51a04
• Pino-Fan, L., Godino, J, D., & Font, V. (2018). Assessing key epistemic features of didactic mathematical knowledge of prospective teachers: the case of the derivative. Journal of Mathematics Teacher Education, https://doi:10.1007/s10857-016-9349-8
• Pino-Fan, L., Guzmán, I., Font, V., & Duval, R. (2017). Analysis of the underlying cognitive activity in the resolution of a task on derivability of the absolute-value function: Two theoretical perspectives. PNA, 11(2), 97-124.
• Rees, P. K., Sparks, F. W., & Sparks, C. (1991). Álgebra. México: McGraw Hill.
• Rodríguez-Nieto, C. A. (2020). Explorando las conexiones entre sistemas de medidas usados en prácticas cotidianas en el municipio de Baranoa [Exploring the connections between measurement systems used in daily practices in the municipality of Baranoa]. IE Revista de Investigación Educativa de la REDIECH, 11, e-857. https://doi.org/10.33010/ie_rie_rediech.v11i0.857
• Rodríguez-Nieto, C., Rodríguez-Vásquez, F. M., & Font, V. (2020). A new view about connections. The mathematical connections established by a teacher when teaching the derivative. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2020.1799254
• Rodríguez-Nieto, C., Font, V., Borji, V., & Rodríguez-Vásquez, F. M. (2021). Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.1875071
• São Paulo [state]. (2012). Currículo do Estado de São Paulo: matemática e suas tecnologias. Secretaria da Educação, São Paulo, Brasil.
• Sari, P., Hadiyan, A., & Antari, D. (2018). Exploring derivatives by means of GeoGebra. International Journal on Emerging Mathematics Education, 2(1), 65-78. https://doi.org/10.12928/ijeme.v2i1.8670
• Sánchez-Matamoros, G., García, M., & Llinares, S. (2008). La comprensión de la derivada como objeto de investigación en didáctica de la matemática [The understanding of derivate as the object of investigation in mathematics education]. Revista latinoamericana de investigación en matemática educativa, 11(2), 267-296.
• Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2015). Developing pre-service teachers’ noticing of students’ understanding of the derivative concept. International journal of science and mathematics education, 13(6), 1305-1329. https://doi.org/10.1007/s10763-014-9544-y
• Steward, J. (1999). Cálculo. Conceptos y contextos [Calculus. Concepts and contexts]. Thomson, México.
• Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: stability and change in students’ conceptions. International Journal of Mathematical Education in Science and Technology, 38(5), 609–637. https://doi.org/10.1080/00207390701359313
• Yavuz-Mumcu, H. (2018). Matematiksel ilişkilendirme becerisinin kuramsal boyutta incelenmesi: türev kavramı örneği. Turkish Journal of Computer and Mathematics Education, 9(2), 211-248. https://doi.org/10.16949/turkbilmat.379891