The effect of different representations on Years 3 to 5 students' ability to generalise

Over the past 3 years, in our Early Algebra Thinking project, we have been studying Years 3 to 5 students' ability to generalise in a variety of situations, namely, compensation principles in computation, the balance principle in equivalence and equations, change and inverse change rules with function machines, and pattern rules with growing patterns. In these studies, we have attempted to involve a variety of representations and to build students' abilities to switch between them (in line with the theories of Dreyfus in Advanced mathematical thinking. Kluwer, Dordtrecht, pp. 25-41, 1991, and Duval in Proceedings of the 21st conference of the North American chapter of the international group for the psychology of mathematics education, vol. 1, pp. 3-26, 1999). The studies have shown the negative effect of closure on generalisation in symbolic representations, the predominance of single variance generalisation over covariant generalisation in tabular representations, and the reduced ability to readily identify commonalities and relationships in enactive and iconic representations. This presentation will use a variety of studies to explore the interrelation between verbal and visual comprehension of context and generalisation. The studies showed in a variety of contexts the importance of understanding and communicating aspects of representational forms which allowed commonalities to be seen across or between representations.