Uma visão da rede de teorias TAC-EOS sobre o papel das conexões matemáticas na compreensão da derivada
Resumo
A pergunta é respondida: Dada a expressão algébrica de , que conexões garantem ao aluno uma compreensão de que lhe permite esboçar o gráfico de e explicar sua relação com o de ? Para isso, foram aplicadas entrevistas e uma tarefa a um grupo de alunos, em que eles tiveram que esboçar o gráfico de e explicar sua relação com o da derivada, e sua atividade matemática foi analisada usando a articulação TAC-EOS como um ponto teórico referência. Os alunos que mostraram uma compreensão que lhes permitiu resolver a tarefa estabeleceram conexões de: diferentes representações, significado, parte-todo, implicação e característica. Por outro lado, a falha em estabelecer certas conexões é uma explicação plausível porque alguns alunos não resolvem a tarefa.
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Referências
ARNON, Llana et al. APOS theory: a framework for research and curriculum development in mathematics education. New York: Springer-Verlag, 2014.
BERRY, John.; NYMAN, Melvin. Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, v. 22, n. 4, p. 479–495, 2003.
BORJI, Vahid.; RADMEHR, Farzad.; FONT, Vicenç. The impact of procedural and conceptual teaching on students' mathematical performance over time, International Journal of Mathematical Education in Science and Technology, v. 52, n.3, p. 404-426, 2021.
BORJI, Vahid., et al. 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, v. 14, n. 6, p. 2301-2315, 2018.
BRAUN, Virginia.; CLARKE, Victoria. Using thematic analysis in psychology. Qualitative Research in Psychology, v. 3, n. 2, p. 77–101, 2006.
BUSINSKAS, Aldona Monika. Conversations about connections: How secondary mathematics teachers conceptualize and contend with mathematical connections. 2008. 183f. Disertación (PhD of Philosophy) - Faculty of Education, Simon Fraser University. Canada.
CAI, Jinfa.; DING, Meixia. On mathematical understanding: perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education, v. 20, n. 1, p. 5-29. 2015.
CHIGEZA, Philemon. Translating between and within representations: mathematics as lived experiences and interactions. In V. Steinle, L. Ball, & C. Bardini (Eds.), Mathematics Education: Yesterday, today and tomorrow. Proceedings of the 36th annual conference of the mathematics education research group of Australasia (pp. 178–185). 2013. Melbourne, VIV: MERGA.
COHEN, Louis.; MANION, Lawrence.; MORRISON, Keith. Research methods in education. London and New York: Routledge, 2018.
DOLORES-FLORES, Crisologo.; GARCÍA-GARCÍA, Javier. Conexiones Intramatemáticas y Extramatemáticas que se producen al Resolver Problemas de Cálculo en Contexto: un Estudio de Casos en el Nivel Superior. Bolema, Boletim de Educação Matemática, v. 31, n. 57, p. 158–180, 2017. DOI: http://dx.doi.org/10.1590/1980-4415v31n57a08
DOLORES-FLORES, Crisologo.; RIVERA-LÓPEZ, Martha Iris.; GARCÍA-GARCÍA, Javier. Exploring mathematical connections of pre-university students through tasks involving rates of change. International Journal of Mathematics Education in Science and Technology, v. 50, n. 3, p. 369–389, 2019. http://dx.doi.org/10.1080/0020739X.2018.1507050
LOBO, Rogério dos Santos. Utilizando a Derivada para inferir sobre a velocidade de contaminação do novo Coronavírus: uma possibilidade para as aulas de Cálculo. Revemop, v. 2, n. e202021, p. 1-18, 2020. DOI: https://doi.org/10.33532/revemop.e202021
ELI, Jennifer.; MOHR-SCHROEDER, Margaret. J.; LEE, Carl. Exploring mathematical connections of prospective middle-grades teachers through card-sorting tasks. Mathematics Education Research Journal, v. 23, n. 3, p. 297-319. 2011.
ELI, Jennifer.; MOHR-SCHROEDER, Margaret. J.; LEE, Carl. Mathematical connections and their relationship to mathematics knowledge for teaching geometry. School Science and Mathematics, v. 113, n. 3, p. 120-134. 2013.
EVITTS, Thomas. Investigating the mathematical connections that preservice teachers use and develop while solving problems from reform curricula. 2004 Disertación (PhD of Philosophy)). Pennsylvania State University-College of Education. EE. UU.
FÉLIX SANDOVAL, G. C.; MONTEVERDE, A. G. Taxa de variação instantânea com magnitudes infinitamente pequenas: uma experiência no Ensino à Distância. Revemop, v. 3, p. e202114. DOI: https://doi.org/10.33532/revemop.e202114
FERRINI-MUNDY, Joan.; GRAHAM, Karen. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. MAA notes, p. 31-46. 1994.
FONT, Vicenç. Una perspectiva ontosemiótica sobre cuatro instrumentos de conocimiento que comparten un aire de familia: particular/general, representación, metáfora y contexto. Educación matemática, v.19, n. 2, p. 95-128. 2007.
FONT, Vicenç et al. Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA. Educational Studies in Mathematics, v. 91, n. 1, p. 107-122, oct. 2016.
FONT, Vicenç.; CONTRERAS, Ángel. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, v. 69, p. 33-52. 2008.
FONT, Vicenç et al. The emergence of objects from mathematical practices. Educational Studies in Mathematics, v. 82, p. 97–124. 2013.
FUENTEALBA, Claudio.; BADILLO, Edelmira.; SÁNCHEZ-MATAMOROS, Gloria. Puntos de no-derivabilidad de una función y su importancia en la comprensión del concepto de derivada. Educação e Pesquisa, v. 44, p. 1-20. 2018.
FUENTEALBA, Claudio et al. The understanding of the derivative concept in higher education. EURASIA Journal of Mathematics, Science and Technology Education, v. 15, n. 2, 1-15. 2018.
GARCÍA-GARCÍA, Javier. Escenarios de exploración de conexiones matemáticas. Números: Revista de didáctica de las matemáticas, v. 100, p. 129-133. 2019.
GARCÍA-GARCÍA, Javier.; DOLORES-FLORES, Crisologo. Intra-mathematical connections made by high school students in performing Calculus tasks. International Journal of Mathematical Education in Science and Technology, v. 49, n. 2, p. 227–252. 2018.
GARCÍA-GARCÍA, Javier.; DOLORES-FLORES, Crisologo. Pre-university students' mathematical connections when sketching the graph of derivative and antiderivative functions. Mathematics Education Research Journal, v. 33. p. 1-22. 2019. https://doi.org/10.1007/s13394-019-00286-x
GARCÍA-GARCÍA, Javier.; DOLORES-FLORES, Crisologo. Exploring pre-university students’ mathematical connections when solving Calculus application problems. International Journal of Mathematical Education in Science and Technology, 2020. DOI: 10.1080/0020739X.2020.1729429
GODINO, Juan.; BATANERO, Carmen. Significado institucional y personal de los objetos matemáticos. Recherches en didactique des Mathématiques, v. 4, n.3, p. 325-355, 1994.
GODINO, Juan et al. Una perspectiva ontosemiótica de los problemas y métodos de investigación en educación matemática. Revemop, v. 3, n. e202107, p. 1-30, 2021. DOI: https://doi.org/10.33532/revemop.e202107
GODINO, Juan et al. The onto-semiotic approach to research in mathematics education. ZDM –The International Journal on Mathematics Education, v. 39, n. 1–2, p. 127–135. 2007.
GODINO, Juan et al. The onto-semiotic approach: implications for the prescriptive character of didactics. For the Learning of Mathematics, v. 39, n. 1, p. 37- 42. 2019.
GODINO, Juan et al. Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. Educational Studies in Mathematics, v. 77, n. 2, p. 247–265. 2010.
GREENO, James. Indefinite goals in well-structured problems. Psychological Review, v.83, n.6, p. 479-491. 1976.
HIEBERT, James.; CARPENTER, Thomas. Learning and teaching with understanding. In: Grouws D. A. (Ed.). Handbook of research of mathematics teaching and learning. New York: Macmillan, 1992, p. 65–79.
HIEBERT, James.; LEFEVRE, Patricia. Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics, Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. 1986, p. 1–27.
INSTITUTO NACIONAL DE FORMACIÓN DOCENTE (INFD). Proyecto de mejora para la formación inicial de profesores para el nivel secundario. Área: Matemática. Ministerio de Educación, Instituto Nacional de Formación Docente y Secretaría de Políticas Universitarias. Buenos Aires, 2010. Disponible en: https://cedoc.infd.edu.ar/upload/Matematica.pdf. Acceso: 03 may. 2019.
KASTBERG, Signe. Understanding mathematical concepts: The case of the logarithmic function. 2002. 2009f. Disertación (PhD of Philosophy) - Dean of the Graduate School, University of Georgia. Georgia.
KENEDI, Ary Kiswanto; et al. Mathematical connection of elementary school students to solve mathematical problems. Journal on Mathematics Education, v. 10, n. 1, p. 69-80. 2019.
KULA-ÜNVER, Semiha. How do pre-service mathematics teachers respond to students’ unexpected questions related to the second derivative?. Journal of Pedagogical Research, v.4, n.3, p. 359-374. 2020.
LUGO-ARMENTA, J. G.; PINO-FAN, L. R.; RUIZ HERNANDEZ, B. R. Significados de Referencia del Estadístico Chi-cuadrada: una Mirada Histórico-epistemológica. Revemop, v. 3, p. e202108, 21 jun. 2021. DOI: https://doi.org/10.33532/revemop.e202108
MEEL, David. Modelos y teorías de la comprensión matemática: Comparación de los modelos de Pirie y Kieren sobre el crecimiento de la comprensión matemática y la Teoría APOE. Revista Latinoamericana de Investigación en Matemática Educativa, RELIME, v.6, n.3, p. 221-278. 2003.
MHLOLO, Michael. 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, v. 16, n. 2, p. 176–191. 2012.
MHLOLO, Michael.; VENKAT, Hamsa.; SCHÄFER, Marc. The nature and quality of the mathematical connections teachers make. Pythagoras, v. 33, n. 1, p. 1-9. 2012.
MICHENER, Edwina. Understanding understanding Mathematics. Cognitive Science, v. 2, p. 361-383, 1978.
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS (NCTM). Principles and standards for school mathematics. Reston, VA: NCTM, 2000.
NATIONAL RESEARCH COUNCIL (NRC). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. 2001.
NEMIROVSKY, Ricardo.; RUBIN, Andee. Students’ tendency to assume resemblances between a function and its derivatives. TERC Working 2-92. Cambridge MA: TERC.1992.
NICKERSON, Raymond. S. Understanding Understanding. American Journal of Education, v.93, n.2, p. 201-239, 1985.
OEHRTMAN, Michael.; CARLSON, Marilyn.; THOMPSON, Patrick. Foundational reasoning abilities that promote coherence in students’ function understanding. In Marilyn Carlson & Chris Rasmussen (Eds.). Making the connection: Research and practice in undergraduate mathematics, MAA Notes Volume, 73, p. 27–41. Washington, DC: Mathematical Association of America. 2009.
ORTIZ, Juan; FONT, Vicenç. Significados personales de los futuros profesores de educación primaria sobre la media aritmética. Educación Matemática, v. 23, no 2, p. 91-109. 2011.
PAMBUDI, Didik et al. Mathematical connection profile of junior high school students in solving mathematical problems based on gender difference. International Journal of Scientific Research and Management, v.6, n. 08, p. 73-78. 2018.
PINO-FAN, Luis.; GODINO, Juan.; FONT, Vicenç. (2011). Faceta epistémica del conocimiento didáctico-matemático sobre la derivada. Educação Matemática Pesquisa, v.13, n.1, p. 141-178. 2011.
PINO-FAN, Luis.; GODINO, Juan.; FONT, Vicenç. Una propuesta para el análisis de las prácticas matemáticas de futuros profesores sobre derivadas. Bolema, Boletim de Educação Matemática, v. 29, n. 51, p. 60-89. 2015.
PINO-FAN, Luis.; GODINO, Juan.; FONT, Vicenç. Assessing key epistemic features of didactic mathematical knowledge of prospective teachers: the case of the derivative. Journal of Mathematics Teacher Education, v. 21, n. 1, p. 63-94. 2018.
PINO-FAN, Luis.; GUZMÁN, Ismenia.; FONT, Vicenç.; DUVAL, Raymond. Analysis of the underlying cognitive activity in the resolution of a task on derivability of the absolute-value function: Two theoretical perspectives. PNA, v. 11, n. 2, p. 97-124. 2017.
PIRIE, S., & KIEREN, T. A recursive theory of mathematical understanding. For the learning of mathematics, 9(3), p. 7-11, 1989.
PIRIE, Susan.; KIEREN, Thomas. Growth in mathematical understading: how can we characterise it and how can we represent it? Educational Studies in Mathematics, v. 26, p. 165-190. 1994.
POLYA, George. Mathematical Discovery, John Wiley & Sons. 1962.
POLYA, George. Cómo plantear y resolver problemas. México: Editorial Trillas. 1989.
PURCELL, Edwin.; VARBERG, Dale.; RIGDON, Steven. Cálculo diferencial e integral. Pearson. 2007.
RODRÍGUEZ-NIETO, Camilo Andrés. Explorando las conexiones entre sistemas de medidas usados en prácticas cotidianas en el municipio de Baranoa. IE Revista de Investigación Educativa de la REDIECH, v. 11, n. e-857, p. 1-30. 2020.
RODRÍGUEZ-NIETO, Camilo Andrés. Conexiones etnomatemáticas entre conceptos geométricos en la elaboración de las tortillas de Chilpancingo, México. Revista de investigación desarrollo e innovación, v. 11, n. 2, p. 273-296. 2021.
RODRÍGUEZ-NIETO, Camilo Andrés; et al. Mathematical connections from a networking theory between Extended Theory of Mathematical connections and Onto-semiotic Approach. International Journal of Mathematical Education in Science and Technology. 2021. https://doi.org/10.1080/0020739X.2021.1875071
RODRÍGUEZ-NIETO, Camilo Andrés.; RODRÍGUEZ-VÁSQUEZ, Flor Monserrat.; FONT, Vicenç. 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. 2020. https://doi.org/10.1080/0020739X.2020.1799254
RODRÍGUEZ-NIETO, Camilo Andrés.; RODRÍGUEZ-VÁSQUEZ, Flor Monserrat.; GARCÍA-GARCÍA, Javier. Pre-service mathematics teachers’ mathematical connections in the context of problem-solving about the derivative. Turkish Journal of Computer and Mathematics Education, v. 12, n. 1, p. 202-220. 2021.
RODRÍGUEZ-VÁSQUEZ, Flor Monserrat.; ARENAS-PEÑALOZA, Jhonatan Categories to assess the understanding of university students about a mathematical concept. Acta Scientiae, v. 23, n. 1, p. 102-134. 2021.
SÁNCHEZ-MATAMOROS, Gloria; GARCÍA, Mercedes.; LLINARES, Salvador. La comprensión de la derivada como objeto de investigación en didáctica de la matemática. Revista latinoamericana de investigación en matemática educativa, v. 11, n. 2, p. 267-296. 2008.
SIERPINSKA, Anna. Some Remarks on Understanding in Mathematics. For the Learning of Mathematics, v.10, n. 3, p. 24-36. 1990.
SKEMP, Richard. Relational understanding and instrumental understanding. Mathematics teaching, v. 77, n. 1, p. 20-26. 1976.
SKEMP, Richard. Relational understanding and instrumental understanding’, Arithmetic Teacher, v. 26, n. 3, p. 9–15. 1978.
SKEMP, Richard. (1980). Psicología del aprendizaje de las Matemáticas. Madrid: Ediciones Morata S.A
UBUZ, Behiye. First year engineering students’ learning of point of tangency, numerical calculation of gradients, and the approximate value of a function at a point through computers. Journal of Computers in Mathematics and Science Teaching, v. 20, n. 1, p. 113–137. 2001.
VERÓN, Manuel Alejandro; GIACOMONE, Belén. Análisis de los significados del concepto de diferencial desde una perspectiva ontosemiótica. Revemop, v. 3, n. e202109, p. 1-27, 2021. DOI: https://doi.org/10.33532/revemop.e202109
VOSKOGLOU, Michael. Application of grey numbers to assessment of the understanding the graphical representation of the derivative. American Journal of Educational Research, v. 5, n. 11, p. 1167-1171. 2017. doi: 10.12691/education-5-11-9.
YAVUZ-MUMCU, Hayal. Matematiksel ilişkilendirme becerisinin kuramsal boyutta incelenmesi: türev kavramı örneği. Turkish Journal of Computer and Mathematics Education, v. 9, n. 2, p. 211-248. 2018. DOI: 10.16949/turkbilmat.379891.
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