The impact of visualization on functional reasoning: the ability to generalize
Palavras-chave:
Functional Reasoning, Patterns, Generalization, Visualization, LearningResumo
With this study we try to understand how pre-service teachers, from basic education (for 3 to 12-year-old students), solve problems involving the generalization of patterns in visual contexts, identifying: the strategies they use; the difficulties they present; the role of visualization in their reasoning; and the factors that influence their generalizations. Considering the aims of this study, we followed a qualitative methodology. The participants were 80 pre-service teachers. During the classes of a Mathematics unit course, these students solved a sequence of tasks involving pattern generalization in figurative contexts. Our results showed that students were able to use different types of generalization strategies, but also that some dimensions of the tasks (e.g. type of pattern, nature of the figures) can have impact in students’ reasoning, causing, in some cases, a shift on the strategies used and the emergence of difficulties of different kind.
Downloads
Referências
Barbosa, A. (2010). A resolução de problemas que envolvem a generalização de padrões em contextos visuais: um estudo longitudinal com alunos do 2.º ciclo do ensino básico. Tese de Doutoramento em Estudos da Criança: Universidade do Minho.
Barbosa, A. (2011). Patterning problems: sixth graders' ability to generalize. In M. Pytlak, T. Rowland & E. Swoboda, Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education, pp. 420-428. Rzeszow: ERME.
Barbosa, A., Vale, I., Palhares, P. (2012). Pattern tasks: thinking processes used by 6th grade students. Relime, 15(3), 273-293.
Becker, J. & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 4, 121-128.
Blanton, M. & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446.
Davidov, V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Reston: National Council of Teachers of Mathematics.
Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the Teaching of Geometry for the 21st Century (pp.37-52). Dordrecht: Kluwer Academic Publishers.
Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Erlbaum.
Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades. Mahwah, NJ: Lawrence Erlbaum/Taylor & Fran- cis Group & National Council of Teachers of Mathematics.
Lannin, J. K. (2005). Generalization and Justification: The challenge of Introducing Algebraic Reasoning Through Patterning Activities. Mathematical Thinking and Learning, 7(3), 231-258.
Lannin, J., Barker, D. & Townsend, B. (2006). Algebraic generalization strategies: factors influencing student strategy selection. Mathematics Education Research Journal, 18(3), 3-28.
Lannin, J., Ellis, A. B., & Elliot, R. (2011). Developing essential understanding of mathematical reasoning: Pre-K-Grade 8. Reston, VA: NCTM.
Lee, L., & Freiman, V. (2006). Developing Algebraic Thinking through Pattern Exploration. Mathematics Teaching in the Middle School, 11, 428−33.
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 65 – 86). Dordrecht: Kluwer Academic Publishers.
Mason, J. (2008). Making use of children’s powers to produce algebraic thinking. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades. Mahwah, NJ: Lawrence Erl- baum/Taylor & Francis Group & National Council of Teachers of Mathematics.
Mason, J., Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage (Paul Chapman).
ME-DGIDC (2007). Programa de Matemática do Ensino Básico. Lisboa: Ministério da Educação, Departamento de Educação Básica.
NCTM (2000). Principles and Standards for School Mathematics. USA: NCTM.
Noss, R., Healy, L. & Hoyles, C. (1997). The Construction of Mathematical Meanings: Connecting the Visual with the Symbolic. Educational Studies in Mathematics, 33(2), 203-233.
Orton, A. & Orton , J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics (pp 104-120). London: Cassell.
Pólya, G. (1981). Mathematical discovery: On understanding, learning and teaching problem solving (Combined ed.). New York: Wiley.
Radford, L. (2006). Algebraic thinking and the generalization of patterns: a semiotic perspective. In S. Alatorre, J. Cortina, M. Sáiz & A. Méndez (Eds), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, 1-21.
Rivera, F. (2007). Visualizing as a Mathematical Way of Knowing: Understanding Figural Generalization. Mathematics Teacher, 101(1), 69-75.
Rivera, F. D. & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40, 65-82.
Sasman, M., Olivier, A., Linchevski, L. (1999). Factors influencing students’ generalization thinking processes. In O. Zaslavsky (Ed.), Proceedings of the 23th International Conference for Psychology of Mathematics Education, 4, pp 161-168, Haifa, Israel: PME.
Stacey, K. & MacGregor, M. (1995). The influence of problem representation on algebraic equation writing and solution strategies. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th International Conference for Psychology of Mathematics Education, 2, 90-97.
Stacey, K. (1989). Finding and Using Patterns in Linear Generalising Problems. Educational Studies in Mathematics 20(2), 147-164.
Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.
Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 267-283). Hillsdale, NJ: Erlbaum.
Vale, I., Pimentel, T., Cabrita, I., Barbosa, A. & Fonseca, L. (2012). Pattern problem solving tasks as a mean to foster creativity in mathematics. In Tso, T. Y. (Ed.), Proceedings of the 36 th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 171-178. Taipei, Taiwan: PME.
Warren, E. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67, 171-185.
Yin, R. (2012). Application of case study research. Thousand Oaks, USA: Sage.
Zazkis, R., Liljedahl, P. & Chernoff, E. (2008). The role of examples in forming and refuting generalizations. ZDM: International Journal in Mathematics Education, 40, 131-141.
Publicado
Como Citar
Edição
Seção
![Creative Commons License](http://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png)
Este trabalho está licensiado sob uma licença Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.