Teacher's reflections on their knowledge and practice of teaching high school functions
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We report a characterization of the reflections made by high school mathematics teachers when discussing their resolutions of three numerical tasks related to the concept of function understood as a mathematical procept. Thematic analysis was used to code and categorize the reflections. Six teachers with at least 5 years of professional experience and an average age of 30 years participated. The results show that the teachers reflected on their actions in the tasks; their content knowledge; their teaching practice; and their professional training. As a result of these reflections, they recognized the need to improve their knowledge and practice of teaching functions in a more proceptual sense. The results contribute to deepen in how to make teachers reflect and incorporate a much more structural vision of functions in their teaching practices and, at the same time, incorporate the importance and role of their professional training in their reflections.
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Adu-Gyamfi, K., y Bossé, M.J. (2014). Processes and reasoning in representations of linear functions. International Journal of Science and Mathematics Education, 12, 167–192. https://doi.org/10.1007/s10763-013-9416-x
Google Scholar CrossrefArcavi, A. (2016). Promoviendo conversaciones entre docentes acerca de clases filmadas de Matemáticas. Cuadernos de Investigación y Formación en Educación Matemática, 11(15), 385 – 396. https://revistas.ucr.ac.cr/index.php/cifem/article/view/23839
Google Scholar CrossrefArtigue, M. (1995). La enseñanza de los principios del cálculo: problemas epistemológicos, cognitivos y didácticos. En M. Artigue, R. Douady, L. Moreno y P. Gómez (Eds.). Ingeniería didáctica en la educación Matemática. “Una empresa docente.” Grupo Editorial Iberoamérica.
Google Scholar CrossrefArtigue, M. (2000). Teaching and Learning Calculus. What Can be Learned from Education Research and Curricular Changes in France. In E. Dubinsky (Ed.), Research in Collegiate Mathematics Education, IV. American Mathematical Society.
Google Scholar CrossrefBall, D. L., Thames, M. H., y Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389 – 407. https://doi.org/10.1177/0022487108324554
Google Scholar CrossrefBardini, C., Pierce, R., Vincent, J., y King, D. (2014). Undergraduate mathematics students’ understanding of the concept of function. Journal on Mathematics Education, 5(2), 85-107. https://doi.org/10.22342/jme.5.2.1495.85-107
Google Scholar CrossrefBoyer, C.B. (1959). The History of the Calculus and its Conceptual Development. Dover Publications, Inc.
Google Scholar CrossrefBraun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3, 77-101. https://doi.org/10.1191/1478088706qp063oa
Google Scholar CrossrefBrodie, K., y Shalem, Y. (2011). Accountability conversations: mathematics teachers’ learning through challenge and solidarity. Journal of Mathematics Teacher Education, 14, 419 – 439. https://doi.org/10.1007/s10857-011-9178-8
Google Scholar CrossrefChamoso, J. M., Cáceres, M. J., y Azcárate, P. (2012). Reflection on the teaching-learning process in the initial training of teachers. Characterization of the issues on which pre-service mathematics teachers reflect. Teaching and Teacher Education, 28(2), 154 – 164. https://doi.org/10.1016/j.tate.2011.08.003
Google Scholar CrossrefCharalambous, C., y Pitta-Pantazi, D. (2016). Perspectives on priority mathematics education: Unpacking and understanding a complex relationship linking teacher knowledge, teaching, and learning. In L. English y D. Kirshner (eds.), Handbook of international research in mathematics education (3a ed., pp. 19-59). Routledge.
Google Scholar CrossrefChesler, J. (2012). Pre-service Secondary Mathematics Teachers Making Sense of Definitions of Functions. Mathematics Teacher Education and Development, 14(1), 27-40. https://mted.merga.net.au/index.php/mted/article/view/17
Google Scholar CrossrefDewey, J. (1993). How We Think. University of Wisconsin Press.
Google Scholar CrossrefDreyfus, T. (2002). Advanced Mathematical Thinking Processes. In D. Tall, (Ed.), Advanced Mathematical Thinking. Springer. https://doi.org/10.1007/0-306-47203-1_2
Google Scholar CrossrefDreyfus, T., Kouropatov, A. & Ron, G. (2021). Research as a resource in a high-school calculus curriculum. ZDM Mathematics Education 53, 679–693. https://doi.org/10.1007/s11858-021-01236-3
Google Scholar CrossrefDubinsky, E. (2013). High school students’ understanding of the function concept. Journal of Mathematical Behavior, 32(1), 83-101. https://doi.org/10.1016/j.jmathb.2012.12.001
Google Scholar CrossrefDubinsky, E., y Wilson, R. (2013). High school students’ understanding of the function concept. The Journal of Mathematical Behavior, 32(1), 83-101 https://doi.org/10.1016/j.jmathb.2012.12.001
Google Scholar CrossrefEdwards, C.H. (1979). The Historical Development of the Calculus. Springer. https://doi.org/10.1007/978-1-4612-6230-5
Google Scholar CrossrefEisenberg, T. (2002). Functions and associated learning difficulties. In D. Tall (Ed.), Advanced mathematical thinking (pp. 140-152). Springer. https://doi.org/10.1007/0-306-47203-1_9
Google Scholar CrossrefEspinoza-Vásquez, G., Zakaryan, D., y Carrillo, Y.J. (2018). El conocimiento especializado del profesor de matemáticas en el uso de la analogía en la enseñanza del concepto de función. Revista Latinoamericana de Investigación en Matemática Educativa, RELIME, 21(3), 301-318. https://doi.org/10.12802/relime.18.2133
Google Scholar CrossrefGrattan-Guinnes, I. (1980). From the Calculus to Set Theory 1630-1910: An Introductory History. Gerarld Duckworth & Co. Ltd.
Google Scholar CrossrefHatisaru, V. y Erbas, A.K. (2017). Mathematical Knowledge for Teaching the Function Concept and Student Learning Outcomes. International Journal of Science and Mathematics Education, 15, 703–722. https://doi.org/10.1007/s10763-015-9707-5
Google Scholar CrossrefHitt, F. (1994). Teachers’ difficulties wiht the construction of continuous and discontinuous functions. Focus on Learning Problems in Mathematics, 16 (4), 10-20.
Google Scholar CrossrefHitt, F. (2003). Dificultades en el aprendizaje del cálculo. Décimo Primer Encuentro de Profesores de Matemáticas del Nivel Medio Superior. Universidad Michoacana de San Nicolás de Hidalgo. Morelia, México.
Google Scholar CrossrefHitt, F., & González-Martín, A. S. (2016). Generalization, covariation, functions, and calculus. In Á. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The Second Handbook of Research on the Psychology of Mathematics Education (pp. 3–38). Sense Publishers.
Google Scholar CrossrefHitt, F. (2017). El Aprendizaje del cálculo y nuevas tendencias en su enseñanza en el aula de matemáticas. Encuentro Internacional en Educación Matematica. La Educación Matemática como Herramienta en el Desempeño Profesional Docente. Cúcuta, Colombia. 6 – 15.
Google Scholar CrossrefNCTM. (2000). Principles and standards for school mathematics. NCTM.
Google Scholar CrossrefOehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics, MAA Notes (Vol. 73, pp. 27–42). Mathematical Association of America.
Google Scholar CrossrefPino-Fan, L., Parra-Urrea, Y., & Castro, W. F. (2019). Significados de la función pretendidos por el currículo de matemáticas chileno. Magis, Revista Internacional de Investigación en Educación, 11(23), 201-220. https://doi.org/10.11144/Javeriana.m11-23.sfpc
Google Scholar CrossrefPonte, J.P. (1992). The history of the concept of Function and some educational implications. The Mathematics Educator, 3(2). https://openjournals.libs.uga.edu/tme/article/view/1764/1672
Google Scholar CrossrefPonte, J. P., y Chapman, O. (2016). Prospective Mathematics Teachers’ Learning and Knowledge for Teaching. In L. English y D. Kirshner (eds.), Handbook of International Research in Mathematics Education (3rd ed., pp. 275 – 296). Routldge.
Google Scholar CrossrefPreciado-Babb, A., Metz, M., y Marcotte, C. (2015). Awareness as an Enactivist Framework for the Mathematical Learning of Teachers, Mentors and Institutions. ZDM: The International Journal on Mathematics Education, 47(2), 257 – 268. https://doi.org/10.1007/s11858-014-0657-0
Google Scholar CrossrefRasmussen, K. (2016). Lesson study in prospective mathematics teacher education: didactic and paradidactic technology in the post-lesson reflection. Journal of Mathematics Teacher Education, 19(4), 301 – 324. https://doi.org/10.1007/s10857-015-9299-6
Google Scholar CrossrefSaylor, L. L., y Johnson, C. C. (2014). The Role of Reflection in Elementary Mathematics and Science Teachers' Training and Development: A Meta‐Synthesis. School Science and Mathematics, 114(1), 30 – 39. https://doi.org/10.1111/ssm.12049
Google Scholar CrossrefSchön, D. (1983). The reflective practitioner: How professionals think in action. Basic Books.
Google Scholar CrossrefSfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22(1), 1- 36. https://www.jstor.org/stable/3482237
Google Scholar CrossrefSmith, C. R. (2015). Continuous professional learning community of mathematics teachers in the western cape: developing a professional learning community through a school-university partnership. Unpublished doctoral dissertation, University of the Western Cape, Bellville, South Africa.
Google Scholar CrossrefStar, J. R., y Smith, J. (2006). An image of calculus reform: Students’ experiences of Harvard calculus. Research in Collegiate Mathematics Education, VI, 1–25. http://dx.doi.org/10.1090/cbmath/013/01
Google Scholar CrossrefSteele, M.D., Hillen, A.F., y Smith, M. (2013). Developing mathematical knowledge for teaching in a methods course: the case of function. Journal Mathematics Teacher Education, 16(6), 451–482. https://link.springer.com/article/10.1007/s10857-013-9243-6
Google Scholar CrossrefTall, D. (1997). Functions and calculus. In A. Bishop, M. A. K. Clements, C. Keitel-Kreidt, J. Kilpatrick, & C. Laborde (Eds.), International Handbook of Mathematics Education. (pp. 289–325). Kluwer.
Google Scholar CrossrefThompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. En J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). NCTM.
Google Scholar CrossrefTrevisan, A.L., Ribeiro, A.J., y Ponte, J.P. (2020). Professional Learning Opportunities Regarding the Concept of Function in a Practice-based Teacher Education Program. International Electronic Journal of Mathematics Education, 15(2), 1-14. https://doi.org/10.29333/iejme/6256
Google Scholar CrossrefWatson, A. & Harel, G. (2013). The Role of Teachers' Knowledge of Functions in Their Teaching: A Conceptual Approach with Illustrations from Two Cases. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 154-168. https://www.learntechlib.org/p/167701/
Google Scholar CrossrefYouschkevitch, A. P. (1976). The concept of function up to the middle of the 19th century. Archive for History of Exact Sciences, 16, 37-85. https://link.springer.com/article/10.1007/BF00348305
Google Scholar CrossrefZandieh, M., Ellis, J., & Rasmussen, C. (2017). A characterization of a unified notion of mathematical function: the case of high school function and linear transformation. Educational Studies in Mathematics, 95(1), 21-38. https://doi.org/10.1007/s10649-016-9737-0
Google Scholar CrossrefZuccheri, L., & Zudini, V. (2014). History of Teaching Calculus. En: Karp, A., Schubring, G. (eds) Handbook on the History of Mathematics Education. Springer. https://doi.org/10.1007/978-1-4614-9155-2_24
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