Visualização e Abstração no Ensino de Funções Contínuas
Resumo
Este artigo traz os resultados da análise de um relato sobre o uso da visualização no ensino da matemática como aliado para o entendimento do conceito de Continuidade. Focamos o momento em que alunos de Cálculo I, participantes em uma intervenção pedagógica, se atêm aos aspectos gráficos de funções reais, refletindo sobre o conceito em uma situação de ensino e aprendizagem. Adotando a perspectiva da abstração estrutural, buscamos esclarecer a estruturação das ideias concebidas pelos alunos ao representá-las visualmente como forma de construir/atribuir significados ao conceito de função contínua, ou ainda, como forma potencial de promover processos de abstração em matemática.
Referências
CABRAL; M.A.P. Curso de Cálculo de uma Variável. Rio de Janeiro: Editora IM- UFRJ. 2013.
DAVIDOV, V.V Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Soviet Studies in mathematics education, v.2, 1972. (Ed, J. Kilpatrick, Tradução J, Teller) Reston, VA: NCTM. 1990.
DUBINSKY, E. Reflective Abstraction in Advanced Mathematical Thinking. In D. Tall (Ed.) Advanced Mathematical Thinking, p. 95–123, Dordrecht, The Netherlands: Kluwer.1991.
FLORES, C.R.; WAGNER, D.R.; BURATTO, I.C.F. Pesquisa em visualização na educação matemática: conceitos, tendências e perspectivas. Educação Matemática Pesquisa: Revista do Programa de Estudos Pós-Graduados em Educação Matemática, v. 14, n. 1. 2012.
FREUDENTHAL. H. Mathematics as an Educational Task. Dordrecht, The Netherlands: Reidel, 1975.
HAZZAN, O. Reducing abstraction level when learning abstract algebra concepts. Educational studies in mathematics, v. 40, n.1, p. 71-90, 1999.
HAZZAN, I. e MITCHELMORE, M. The role of abstraction in learning about rates of change, Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia Conference, p. 278-285. South Australasia: MERGA. 2006.
HERSHKOWITZ, R., SCHWARZ, B.B., DREYFUS, T. Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, v. 32, p.195-222, 2001.
LIMA, E.L. Análise Real. Coleção matemática universitária, v. 1, p. 185-204, Outubro, 2008. Rio de Janeiro: IMPA.
MITCHELMORE, M.C., & WHITE, P. Abstraction in mathematics and mathematics learning. In: Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education, v.3, p.329-336. Bergen, Norway: PME. 2004.
MITCHELMORE, M.C., & WHITE, P. Abstraction in mathematics learning. Mathematics
Education Research Journal, 19(2), p. 1-9, 2007.
MOREIRA, P.C., CAMPOS, D.F. Descontinuidades na Passagem do Cálculo para a Análise: as noções de limite e de função contínua num ponto. Bolema, v. 37, p. 1017-1035, 2023.
MORETTIN, P.A.; HAZZAN, S. Cálculo: Funções de Uma E Várias Variáveis. Editora Saraiva, 2000.
NERI, C.; CABRAL, M.A.P. Curso de análise real. Rio de Janeiro: Editora IM-UFRJ. 2011.
NOSS, R., HOYLES, C. Windows on mathematical meanings: Learning cultures and computers. Dordrecht, The Netherlands: Kluwer, 1996.
PIAGET, J. Studies in reflecting abstraction. Philadelphia: Psychology Press, 2014.
PIAGET, J. (and collaborators) Studies in reflective abstraction (Recherches sur l abstraction réfléchissante). Philadelphia: Psychology Press, 1977.
PINTO, M. M. F. Students’ Understanding of Real Analysis. PhD Thesis, Warwick University. Ann Arbor, Michigan: ProQuest-CSA, LLC, 1998/2011
PINTO, M.M.F. & TALL, D.O. Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, v.22, p.2-10, 2002.
PINTO, M. M. F.; SCHEINER, T. Visualização e ensino de análise matemática. Visualization and the teaching of mathematical analysis. Educação Matemática Pesquisa: Revista do Programa de Estudos Pós-Graduados em Educação Matemática, v. 17, n. 3, p. 637-654, 2015.
PRESMEG, N. C. Research on visualization in learning and teaching mathematics. Handbook of research on the psychology of mathematics education, p. 205-235, Rotterdam: Sense. 2006.
RADFORD, L. The Epistemological Foundations of the Theory of Objectivation. Teaching and Learning Mathematics, p. 127-149, 2015.
SCHEINER, T. New lights on old horizon: Constructing mathematical concepts, underlying abstraction processes and sense making strategies. Educational Studies in Mathematics, v.91, n.2, p.165-183, 2016.
SCHEINER, T. Sense making in mathematics. Towards a dialogical framing. Proccedings of the 41st annual conference of the Mathematics Education Research Group of Australasia, p.669-676. South Australasia: MERGA. 2018a.
SCHEINER, T. Mathematics cognition reconsidered: On ascribing meaning. 2018. Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education, RUME, p.1234-1239.2018b.
SCHEINER,T.,& PINTO, M.M.F. Cognitive processes underlying mathematical concept construction: The missing process of structural abstraction. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (Eds.). Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, v. 5, p. 105-112. Vancouver, Canada: PME. 2013.
SCHEINER,T., & PINTO, M.M.F. The evolving framework of structural abstraction: a tool for and an object of research in knowing and learning mathematics. Manuscrito de circulação interna: notas de aula. 2014.
SCHEINER, T.; PINTO, M.M.F. Images of abstraction in mathematics education: contradictions, controversies, and convergences. In: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, p. 155-162. Szeged, Hungary: PME. 2016.
SCHEINER, T.; PINTO, M.M.F. Emerging Insights from the Evolving Framework of Structural Abstraction. 20th Annual Conference on Research in Undergraduate Mathematics Education. San Diego, CA, USA: RUME. 2017.
SCHEINER, T.; PINTO, M.M.F. Emerging perspectives in mathematical cognition: contextualizing, complementizing, and complexifying. Educational Studies in Mathematics, v.101, n.3, 2019.
SCHEINER, T., PINTO, M.M.F. Theoretical advances in mathematics cognition. Proceedings of the First PME Regional Conference: South America, p. 97-104. Rancagua, Chile: PME. 2018.
SKEMP, R. The psychology of learning mathematics. (second edition, first published in 1971) London: Penguin Group. 1986.
STEWART, J. Cálculo, volume I. 2001. São Paulo: Pioneira Thomson Learning.
TALL, D.O. How humans learn to think mathematically. Exploring the three worlds of mathematics. Cambridge, UK: Cambridge University Press, 2013.
TALL, D.O., & VINNER, S. Concept image and concept definition in mathematics with particular reference to limit and continuity. Educational Studies in Mathematics, v. 12, n. 2, p. 151-169, 1981.
TALL, D.O. (Ed.) Advanced Mathematical Thinking. Dordrecht: Kluwer:. 1991.
TREFFERS, A. Three dimensions: a model of goal and theory description in Mathematics Instruction-The Wiskobas Project. Dordrecht, The Netherlands: Reidel Publishing Co, 1987.
VAN OERS, B. From context to contextualizing. Learning and Instruction, v.8, n.6, p.473-488, 1998.
VAN OERS, B. Educational forms of initiation in mathematics culture. Educational Studies in Mathematics, v. 46, p. 59-85, 2001.
WILENSKY, U. Abstract meditations on the concrete and concrete implications for mathematics education. Epistemology and Learning Group, MIT Media Laboratory, 1991.
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