Boon Liang Chua

Generalisation and perceptual agility: how did teachers fare in a quadratic generalising problem?

One of the essential 21st century skills calls for individuals to think creatively and to solve problems using both conventional as well as innovative strategies (Learning and Skills Development Agency 2003). Particularly relevant to mathematics, this skill finds an academic home in the topic of pattern generalisation, which is about seeing the general in the particular. The idea of having to develop flexibility in making generalisation is not a new issue and for some time now, its importance has been emphasised by many researchers (Mason, Graham, and Johnston-Wilder 2005; Lee 1996). Lee (1996) calls the ability to see a pattern in multiple ways as perceptual agility. With even greater emphasis in today’s society, it seems reasonable to think that perceptual agility is not a luxury but a must-have for all students, rather than just for the more-able students. However, to nurture students and help them develop this ability, teachers must become versatile in engaging in different ways of seeing a pattern so as to be able to provide proper guidance and model how to do it. But are teachers perceptually agile themselves? This paper aims to investigate this issue through the asking of the following two questions: (1) Are prospective teachers capable of deriving multiple expressions for the same generalising problem? If so, to what extent? (2) What strategies did the prospective teachers employ to derive the expressions for the generalising problem? The article begins with a discussion of the methods, the classification scheme used for data analysis as well as results, and follows with a discussion of research findings.

Exploring the link between task features and generalisation

Research on students’ expression of generality in pattern-generalising problems has widely documented their difficulties in articulating and representing the functional rule in words or in algebraic notation (Hoyles and Küchemann 2001). But recent studies seem to attribute the difficulties mainly to the way the patterns are depicted in such problems (Stalo et al. 2006; Lannin, Barker and Townsend 2006). Apart from these findings, little else is known about what and how other features of the problems might affect students’ abilities to express generality. In this study, we speculate that students’ performance in pattern-generalising problems might be influenced by the five task features described below.

Topic study group no. 20: Visualization in the teaching and learning of mathematics

This paper was published in the Proceedings of the 13th International Congress on Mathematical Education held at Hamburg, Germany from 24 - 31 July 2016