Years 2 to 6 students' ability to generalise : models, representations and theory for teaching and learning

Over the last three 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 models and representations and to build students’ abilities to switch between 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 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 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 commonalities to be seen across or between representations. Finally the chapter explores the implications of the studies for a theory that describes a growth in integration of models and representations that leads to generalisation.

Teaching basic finance with Excel, and without pain

The five quantities of interest in elementary finance problems are present value, future value, amount of periodic payment, number of periods and the rate of compound interest per period. A recursive approach to computing each of these five quantities in a modern version of Excel, for the case of ordinary annuities, is described. The aim is to increase student understanding and build confidence in the answer obtained, and this may be achieved with only linear relationships and in cases where student knowledge of algebra is essentially zero. Annuity problems may be solved without use of logarithms and black-box intrinsic functions; these being used only as check mechanisms. The author has had success with the method at Bond University and surrounding high schools in Queensland, Australia.