From patterns to generalization: how is creativity expressed by future teachers in Brazil and Portugal?
DOI:
https://doi.org/10.46551/emd.v9n17a08Keywords:
Figurative Patterns, Algebraic Thinking, Representations, Generalizations, CreativityAbstract
The study investigated how Brazilian and Portuguese prospective teachers in initial training solved problems of generalization of mathematical patterns, with emphasis on multiple resolutions and creativity. The research adopted a qualitative and exploratory paradigm, focusing on characterizing the participants' performance in solving tasks involving figurative patterns, as well as the dimensions of creativity that emerged in these resolutions and the variations between the two contexts. The study highlights the importance of integrating visualization and the exploration of multiple representations in the training of future teachers. It is possible to identify that the focus on visualization in Portugal promotes a more robust understanding of algebraic regularities and generalizations, and in Brazil, the use of symbolic notation favors the development of formal skills.
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ARCAVI, Abraham. The role of visual representations in the learning of Mathematics. Educational Studies in Mathematics, v. 52, n. 3, p. 215-241, 2003. https://doi.org/10.1023/A:1024312321077
BARBOSA, Ana; VALE, Isabel. As representações: escolhas eficazes na resolução de problemas. Educação e Matemática, n. 166, p. 19-24, 2022.
BARBOSA, Ana; VALE, Isabel. Visualization in pattern generalization: potential and challenges. Journal of the European Teacher Education Network, v. 6, n. 2, p. 23-35, 2015.
BLANTON, Maria; BRIZUELA, Bárbaras M.; STEPHENS, Ana; KNUTH, Eric; ISLER, Isil; GARDINER, Angela Murphy. Implementing a framework for Early Algebra. In: KIERAN, Carolyn (Ed.). Teaching and learning algebraic thinking with 5- to 12-year-olds: the global evolution of an emerging field of research and practice. Springer, 2018, p. 27-49. https://doi.org/10.1007/978-3-319-68351-5_2
BLANTON, Maria; KAPUT, James. Characterizing the nature of algebraic thinking in the early grades: the case of equivalence. In: ROMBERG, Thomas; CARPENTER, Thomas; DREMOCK, Fae (Ed.), Understanding Mathematics and Science Matters. London: Lawrence Erlbaum Associates, 2005, p. 57-70.
BRASIL. Ministério da Educação. Secretaria de Educação Básica. Base Nacional Comum Curricular: Educação Infantil e Ensino Fundamental. Brasília: MEC/SEB, 2017.
BRUNER, Jerome. Toward a theory of instruction. Cambridge: Harvard University Press, 1966.
CAMPOS, Mylena Simões; GUALANDI, Jorge Henrique. A generalização de padrões matemáticos na Educação Básica: uma pesquisa bibliográfica. Revista Brasileira de Ensino de Ciências e Matemática, v. 7, n.1, p. 226-254, 2024. https://doi.org/10.5335/rbecm.v7i1.15166
CAMPOS, Mylena Simões; GUALANDI, Jorge Henrique. Os reflexos de uma oficina na mudança das concepções de professores: um estudo no contexto dos materiais manipuláveis. Educação Matemática Debate, v. 4, n. 10, p. 1-22, 2020. https://doi.org/10.46551/emd.e202059
DREYFUS, Tommy. Advanced mathematical thinking processes. In: TALL, David. (Ed.). Advanced Mathematical Thinking. Dordrecht: Springer, 2002, p. 25-41.
ERICKSON, Fredrick. Qualitative methods in research on teaching. In: WITTROCK, Merlin. (Ed). Handbook of research on teaching. Nova York: MacMillan, 1986, p. 119-161.
GUALANDI, Jorge Henrique. Investigações matemáticas com grafos para o Ensino Médio. 2012. 117f. Dissertação (Mestrado em Ciências e Matemática). Pontifícia Universidade Católica de Minas Gerais. Belo Horizonte.
KAPUT, James. Teaching and learning a new Algebra with understanding. North Dartmouth: University of Massachusetts Dartmouth, 2000.
KIERAN, Carolyn. Learning and teaching algebra at the Middle School through college levels: building meaning for symbols and their manipulation. In: LESTER JR., Frank (Ed.). Second handbook of research on Mathematics teaching and learning. Charlotte: Information Age Publishing, 2007, p. 707-762.
LEIKIN, Roza. Exploring mathematical creativity using multiple solution tasks. In: LEIKIN, Roza; BERMAN, Abraham; KOICHU, Boris. (Ed.). Creativity in Mathematics and the Education of gifted students. Rotterdam: Sense Publishers, 2009, p. 129-145.
LEIKIN, Roza; GUBERMAN, Raisa. Creativity and challenge: task complexity as a function of insight and multiplicity of solutions. In: LEIKIN, Roza (Ed.). Mathematical challenges for all. Cham: Springer, 2023, p. 325-342.
MATTESON, Shirley Marie. The effects of multiple representations on students' understanding of rational number concepts. 2006. 325f. Dissertation (Master in Mathematics Education). University of North Carolina. North Carolina.
MIELICKI, Marta; FITZSIMMONS, Charlies; WOODBURY, Lauren; MARSHAL, Hannah; ZHANG, Dake; RIVERA, Ferdinand; THOMPSON, Clarissa. Effects of figural and numerical presentation formats on growing pattern performance. Journal of Numerical Cognition, v. 7, n. 2, p. 125-145, 2021.
NCTM — National Council of Teachers of Mathematics. Principles to actions: ensuring mathematical success for all. Reston: NCTM, 2014.
OECD — Organization for Economic Co-operation and Development. The future of Education and skills. Education 2030. Paris: OECD Publishing, 2018. https://doi.org/10.1787/54ac7020-en
PITTA-PANTAZI, Demetra; SOPHOCLEOUS, Paraskevi; CHRISTOU, Constantinos. Spatial visualizers, object visualizers and verbalizers: Their mathematical creative abilities. ZDM Mathematics Education, v. 45, n. 2, p. 199-213, 2013. https://dx.doi.org/10.1007/s11858-012-0475-1
PORTUGAL. Ministério da Educação. Direção-Geral da Educação. Aprendizagens essenciais de Matemática para o Ensino Básico. Lisboa: ME/DGE, 2021.
PRESMEG, Norma. Visualization in High School Mathematics. For the Learning of Mathematics, n. 6, v. 3, p. 42-46, 1986.
RIVERA, Ferdinand. Teaching and learning patterns in school Mathematics: psychological and pedagogical considerations. Dordrecht: Springer, 2014. https://doi.org/10.1007/978-94-007-2712-0
SILVER, Edward. Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, v. 29, n. 3, p. 75-80, 1997. http://dx.doi.org/10.1007/s11858-997-0003-x
STACEY, Kaye. Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, v. 20, n. 2, p. 147-164, 1989. https://doi.org/10.1007/BF00579460
STYLIANIDES, Gabriel; SILVER, Edward. Reasoning-and-proving in school Mathematics: the case of pattern identification. In: STYLIANOU, Despina; BLANTON, Maria; KNUTH, Eric. (Ed.). Teaching and learning proof across the grades: a K–16 perspective. Reston: NCTM, 2009, p. 235-249.
TRIPATHI, Preety N. Multiple representations: an important tool for understanding mathematical concepts. The Mathematics Educator, v. 13, n. 8, p. 438-445, 2008. https://doi.org/10.5951/MTMS.13.8.0438
USISKIN, Zalman. Goals of the algebra curriculum: a reaction to Martin’s paper. In: HOUSE, Peggy; COXFORD, Arthur. (Ed.). Connecting mathematics across the curriculum: 1994 yearbook. Reston: NCTM, 1994, p. 162-168.
VALE, Isabel. Das tarefas com padrões visuais à generalização. Atas do XX Seminário de Investigação em Educação Matemática. Viana do Castelo, 2009, p. 35-63.
VALE, Isabel; BARBOSA, Ana. Exploring the creative potential of mathematical tasks in teacher education. International Electronic Journal of Mathematics Education, v. 19, n. 4, p. 1-12, 2024. https://doi.org/10.29333/iejme/15075
VALE, Isabel; BARBOSA, Ana. Pensamento algébrico: contributo da visualização na construção da generalização. Educação Matemática Pesquisa, v. 21, n. 3, p. 398-418, 2019. https://doi.org/10.23925/1983-3156.2019vol21i3p398-418
VALE, Isabel; BARBOSA, Ana. Visualization: a pathway to mathematical challenging tasks. In: LEIKIN, Roza. (Ed.). Mathematical challenges for all. Cham: Springer, 2023, p. 283-306.
VALE, Isabel; PIMENTEL, Teresa. Raciocinar com padrões figurativos. In: DOMINGOS, António; VALE, Isabel; SARAIVA, Maria; RODRIGUES, Margarida; COSTA, Maria Conceição; FERREIRA, Renata. (Ed.). Investigação em Educação Matemática: raciocínio matemático. Lisboa: SPIEM, 2013, p. 205-222.
VALE, Isabel; PIMENTEL, Teresa; BARBOSA, Ana. The power of seeing in problem solving and creativity: an issue under discussion. In: AMADO, Nélia; CARREIRA, Susana; JONES, Keith. (Ed.). Broadening the scope of research on mathematical problem solving: a focus on technology, creativity and affect. Cham: Springer, 2018, p. 243-272.
VALE, Isabel; PIMENTEL, Tesera (Coord.). Padrões no ensino e aprendizagem da Matemática: propostas curriculares para o Ensino Básico. ESEVC, 2009.
Yin, Robert Kuo-zuir. Case study research: design and methods. 4th ed. Thousand Oaks: Sage, 2009.
ZIMMERMANN, Walter; CUNNINGHAM, Steve (Ed.). Visualization in teaching and learning mathematics. Washington: Mathematical Association of America, 1991.
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