Prospective Middle-School Mathematics Teachers’ Knowledge of Equations and Inequalities

Abstract

Over the last three years, in our EarlyAlgebra 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 equiva- lence 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 models and representations and to build students’ abilities to switch be- tween them (in line with the theories of Dreyfus 1991, and Duval 1999). The results have shown the negative effect of closure on generalisation in symbolic representa- tions, the predominance of single variance generalisation over covariant generalisa- tion in tabular representations, and the reduced ability to readily identify common- alities and relationships in enactive and iconic representations. This chapter uses the results to explore the interrelation between generalisation, and verbal and visual comprehension of context. The studies evidence the importance of understanding and communicating aspects of representational forms which allowed commonali- ties to be seen across or between representations. Finally the chapter explores the implications of the results for a theory that describes a growth in integration ofmod- els and representations that leads to generalization.