Visualization in pattern generalization: Potential and Challenges
Ana Barbosa, Isabel Vale
This study tries to understand how pre-service teachers, for basic education, solve problems involving the generalization of visual patterns, identifying: the strategies used; the difficulties presented; the role of visualization in their reasoning; the factors that influence their generalizations. We followed a qualitative methodology. The participants were 80 pre-service teachers. During the classes of a Didactics of Mathematics unit course, they solved a sequence of tasks involving growing visual patterns. Results showed that students were able to use different strategies, but also that some dimensions of the tasks can have impact in their reasoning, provoking, sometimes, a shift on the strategies used and the emergence of difficulties of different kind.
Visualization, Patterns, Generalization, Functional reasoning
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.
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. (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.
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., Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage (Paul Chapman).
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.
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. (1989). Finding and Using Patterns in Linear Generalising Problems. Educational Studies in Mathematics 20(2), 147-164.
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.
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.
Vale, I. & Pimentel, T. (2013). Raciocinar com padrões figurativos. In A. Domingos, I. Vale, M. J. Saraiva, M. Rodrigues, M. C. Costa & R. A. T. Ferreira (Eds.), Investigação em Educação Matemática 2013 – Raciocínio Matemático, pp. 205-222, Penhas da Saúde: SPIEM.
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.