Pensamento algébrico: contributo da visualização na construção da generalização<br>Algebraic Thinking: contribution of visualization in the construction of generalization

Authors

DOI:

https://doi.org/10.23925/1983-3156.2019vol21i3p398-418

Keywords:

Generalização, Visualização, Pensamento Algébrico.

Abstract

Para que os alunos sejam competentes em álgebra, devem compreender os conceitos e relações muito para além da manipulação simbólica, o que implica que o seu estudo se inicie nos primeiros anos com o desenvolvimento do pensamento algébrico. Naturalmente, a formação dos futuros professores deve acompanhar esta tendência. Assim, apresenta-se parte de um estudo de natureza qualitativa, no qual se procura caracterizar o pensamento algébrico de futuros professores do ensino básico (3-12 anos) na resolução de tarefas envolvendo padrões figurativos. Os resultados mostraram que os participantes usaram estratégias visuais e analíticas, tendo prevalecido as primeiras, e que as maiores dificuldades surgiram nas questões de generalização distante, frequentemente influenciadas pelo tipo de estratégias usadas.

For students to be competent in algebra, they must understand concepts and relationships far beyond mere symbolic manipulation, which implies that its study begins in the early years with the development of algebraic thinking, where the search for patterns and generalization in figurative contexts play a crucial role. Thus, we present part of a qualitative study, which seeks to analyze the algebraic thinking of future primary school teachers (3-12 years) in solving tasks involving figurative patterns. The results showed that the participants used visual and analytical strategies, having prevailed the former, and that the biggest difficulties.

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References

ARCAVI, A. The Role of Visual Representations in the Learning of Mathematics. Educational Studies in Mathematics, v. 52, p. 215–241, 2003.

BALL, D. L. Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, v. 21, n. 2, p.132-144, 1990.

BARBOSA, A. O contributo da visualização no desenvolvimento do raciocínio funcional. In A. DOMINGOS, I. VALE, M. J. SARAIVA, M. RODRIGUES, M.C. COSTA & R. FERREIRA, (Eds.), Actas do Encontro de Investigação em Educação Matemática. Penhas da Saúde, Portugal: SPIEM, 2013, p. 51–80.

BARBOSA, A.; VALE, I. Visualization in pattern generalization. Journal of the European Teacher Education Network, v. 10, p. 57-70, 2015.

BIEDA, K. N.; NATHAN, M. J. (2009). Representational disfluency in algebra: Evidence from student gestures and speech. ZDM - The International Journal on Mathematics Education, v. 41, n. 5, p. 637- 650, 2009.

BLANTON, M. Algebra and the elementary classroom: Transforming thinking, Transforming Practice. Portsmouth, NH: Heinemann, 2008.

BLANTON, M.; KAPUT, J. Functional thinking as a route into algebra in the elementary grades. In J. CAI, & E. KNUTH (Eds.), Early algebraization. A global dialogue from multiple perspectives. Berlin: Springer, 2011, p. 5-23.

BLANTON, M.; KAPUT, J. Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, v.36, n. 5, p. 412-446, 2005

BORROMEO FERRI, R. Mathematical Thinking styles and their influence on teaching and learning mathematics. Paper presented at the 12th International Congress on Mathematical Education, Seul, Korea. Retrieved in march, 5, 2017 from http://www.icme12.org/upload/submission/1905_F.pdf. 2015.

DUVAL, R. Commentary: Linking epistemology and semio-cognitive modeling in visualization. ZDM - The International Journal on Mathematics Education, v. 46, n. 1, p. 159-170, 2014.

DUVAL, R. Geometry from a cognitive Point of View. In C. MAMMANA & V. VILLANI (Eds.), Perspectives on the Teaching of Geometry for the 21st Century. Dordrecht, Netherlands: Kluwer Academic Publishers, 1998, p. 37-52.

GARDNER, H. Frames of mind: the theory of multiple intelligences. New York: Basic Books, 1993.

GIANQUITO, M. Visual Thinking in Mathematics. Oxford: Oxford University Press, 2007.

GOLDIN-MEADOW, S.; KIM, S.; SINGER, M. What the teacher’s hands tell the student’s mind about Math. Journal of Educational Psychology, v. 91, n. 4, p. 720-730, 1999.

KAPUT, J. What is algebra? What is algebraic reasoning? In J. KAPUT, D. CARRAHER, & M. BLANTON (Eds.), Algebra in the Early Grades. New York: Lawrence Erlbaum Associates, 2008, p. 5-17.

KAPUT, J. Teaching and Learning a New Algebra with Understanding. Dartmouth, Massachusetts: National Center, 1998.

KIERAN, C. Overall commentary on early algebraization: Perspectives for research and teaching. In J. CAI & E. KNUTH (Eds.), Early algebraization. A global dialogue from multiple perspectives. Berlín, Alemania: Springer-Verlag, 2011, p. 557-577.

KRUTETSKII, V. A. The psychology of mathematical abilities in school children. Chicago: University of Chicago Press, 1976.

LANNIN, J., BARKER, D.; TOWNSEND, B. Algebraic generalization strategies: factors influencing student strategy selection. Mathematics Education Research Journal, v. 18, n. 3, p. 3-28, 2006.

MA, L. Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. Hillsdale, NJ: Erlbaum, 1999.

MASON, J. Expressing Generality and Roots of Algebra. In N. BEDNARZ, C. KIERAN & L. LEE (Eds.), Approaches to Algebra. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1996, p. 65-86.

MOLINA, M. Integración del pensamiento algebraico en la educación básica. un experimento de enseñanza con alumnos de 8-9 años. In M. H. MARTINHO, R. A. T. FERREIRA, I. VALE, J. P. PONTE, (Eds), Actas do Encontro de Investigação em Educação Matemática-EIEM. Póvoa de Varzim, Portugal: SPIEM, 2011, p. 27–51.

NEILL, S. Classroom nonverbal communication. London: Routledge, 1991.

NCTM. Principles to actions: ensuring mathematical success for all. Reston: NCTM, 2014.

NCTM. Principles and Standards for School Mathematics. Reston: NCTM, 2000.

PIMENTEL, T. (2011). Um programa de formação contínua e o desenvolvimento do pensamento algébrico de professores do 1.º ciclo do ensino básico. In M. H. MARTINHO, R. A. T. FERREIRA, I. VALE, J. P. PONTE, (Eds), Actas do Encontro de Investigação em Educação Matemática-EIEM. Póvoa de Varzim, Portugal: SPIEM, 2011, p. 3–26.

PRESMEG, N. Creative advantages of visual solutions to some non-routine mathematical problems. In S. CARREIRA, N. AMADO, K. JONES & H. JACINTO, (Eds.), Proceedings of the Problem@Web International Conference: Technology, Creativity and Affect. Faro, Portugal: Universidade do Algarve, 2014, p. 156-167.

RADFORD, L. The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, v. 26, p. 257–277, 2014.

RADFORD, L. Iconicity and Contraction: A Semiotic Investigation of Forms of Algebraic Generalizations of Patterns in Different Context. ZDM - The International Journal on Mathematics Education, v. 40, n. 1, p. 83-96, 2008.

RIVERA, F. (2011). Toward a Visually-Oriented School Mathematics Curriculum: Research, Theory, Practice, and Issues. Dordrecht, Netherlands: Springer.

RIVERA, F.; BECKER, J. Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM - The International Journal on Mathematics Education, v. 40, p. 65-82, 2008.

SWAFFORD, J.; LANGRALL, C. Grade 6 students’ pre-instructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, v. 31, n. 1, p. 89–112, 2000.

VALE, I. Das tarefas com padrões visuais à generalização. In J. FERNANDES, H. MARTINHO & F. VISEU (Eds.). Actas XX SIEM - Seminário de Investigação em Educação Matemática. VC: APM, 2009, p. 35-63.

VALE, I.; PIMENTEL, T.; BARBOSA, A. The power of seeing in problem solving and creativity: an issue under discussion. In N. AMADO, S. CARREIRA, & K. JONES (Eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect. Cham, CH: Springer, 2018, p. 243-272.

VALE, I.; PIMENTEL, T.; BARBOSA, A. Ensinar matemática com resolução de problemas. Quadrante, v. 24, n. 2, p. 39-60, 2015.

VALE, I.; PIMENTEL, T.; BARBOSA, A.; BORRALHO, A.; BARBOSA, E.; CABRITA, I.; FONSECA, L. Padrões em Matemática – Uma proposta didática no âmbito do novo programa para o Ensino Básico. Lisboa: Texto Editores, 2011.

WAKEFIELD; CONGDON; NOVAK; GOLDIN-MEADOW; JAMES, K. H. Learning math by hand: The neural effects of gesture-based instruction in 8-year-old children. Attention, Perception, & Psychophysics, v. 81, n. 7, p. 1-11, 2019.

WARREN, E. Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, v. 67, p. 171-185, 2008.

Published

2019-12-20

How to Cite

VALE, I.; BARBOSA, A. Pensamento algébrico: contributo da visualização na construção da generalização&lt;br&gt;Algebraic Thinking: contribution of visualization in the construction of generalization. Educação Matemática Pesquisa, São Paulo, v. 21, n. 3, 2019. DOI: 10.23925/1983-3156.2019vol21i3p398-418. Disponível em: https://revistas.pucsp.br/index.php/emp/article/view/44297. Acesso em: 22 nov. 2024.

Issue

Section

Finalizada - Educação Algébrica