En route from patterns to algebra: comments and reflections

Abstract

This article is a commentary on the papers in this issue of ZDM on Understanding Generalization in K-12 Algebra. It tries to interpret and understand at least some of the reported phenomena within a constructivist framework. The latter locates meaning not in the external representations but in the individual's activity on and with them. This activity on the other hand is strongly regulated (but not determined) by social contracts and belief systems. From that, and considerations of the more general mathematical context, various suggestions for further and extended research can be drawn. One special aspect is that generalization processes will have to be complemented by some kind of instruction on the conventional algebraic symbolism and its usage.

Using patterns generically to see structure

“…1, 2008) of the research journal ZDM -The International Journal on Mathematics Education suggests that the promotion (intended or not) of number-pattern-spotting is as strong as ever in the mathematics education community. The issue contains a short editorial, nine research papers, and a thoughtful (but too polite) commentary by Willi Dörfler (2008).…”

Towards a Methodology for the Characterization of Teachers’ Didactic-Mathematical Knowledge

“…A literature review shows papers dealing with inservice teachers and algebra (Doerr, 2001;Dörfler, 2008;Wilkie, 2014) but none specifically on inservice teachers' knowledge and patterns. This research proposal builds on the teacher knowledge about patterns in relation to some components of the DMK knowledge, on specific areas, which is not taken into account in the above researcher papers.…”

Flexibilidad en la resolución de problemas de identificación de patrones lineales en estudiantes de educación secundaria

“…lineal, cuadrática, inversa etc.) se suelen utilizar para hacer la transición de la aritmética al álgebra, introduciendo, así, la idea ISSN 1980ISSN -4415 http://dx.doi.org/10.1590ISSN /1980 de generalización de un patrón para un número cualquiera n (DÖRFLER, 2008;LIN;YANG;CHEN, 2004;MASON, 1996). Este tipo de problemas tiene varios apartados de creciente complejidad.…”

“…La dificultad mostrada por estos estudiantes para resolver el problema en el caso general muestra la necesidad de ayudarles a resolver problemas identificando regularidades e invariantes (DÖRFLER, 2008), para lo cual es necesario organizar datos para poder identificar patrones o reglas de formación, por ejemplo mediante tablas o escribiendo una sucesión numérica.…”