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.

The probability archive : from essence to uncertainty in the growth of knowledge

From a ‘cultural science’ perspective, this paper traces one aspect of a more general shift, from the realist representational regime of modernity to the productive DIY systems of the internet era. It argues that collecting and archiving is transformed by this change. Modern museums – and also broadcast television – were based on determinist or ‘essence’ theory; while internet archives like YouTube (and the internet as an archive) are based on ‘probability’ theory. The paper goes through the differences between modernist ‘essence’ and postmodern ‘probability’; starting from the obvious difference that in a museum each object is selected by experts for its intrinsic properties, while on the internet you don’t know what you will find. The status of individual objects is uncertain, although the productivity of the overall archive is unlimited. The paper links these differences with changes in contemporary culture – from a Newtonian to a quantum universe, progress to risk, institutional structure to evolutionary change, objectivity to uncertainty, identity to performance. Borrowing some of its methodology from science fiction, the paper uses examples from museums and online archives, ranging from the oldest stone tool in the world to the latest tribute vid on the net.

Predicting the consumption of foods low in saturated fats among people diagnosed with Type 2 diabetes and cardiovascular disease. The role of planning in the theory of planned behaviour

The present study tested the utility of an extended version of the theory of planned behaviour that included a measure of planning, in the prediction of eating foods low in saturated fats among adults diagnosed with Type 2 diabetes and/or cardiovascular disease. Participants (N = 184) completed questionnaires assessing standard theory of planned behaviour measures (attitude, subjective norm, and perceived behavioural control) and the additional volitional variable of planning in relation to eating foods low in saturated fats. Self-report consumption of foods low insaturated fats was assessed 1 month later. In partial support of the theory of planned behaviour, results indicated that attitude and subjective norm predicted intentions to eat foods low in saturated fats and intentions and perceived behavioural control predicted the consumption of foods low in saturated fats. As an additional variable, planning predicted the consumption of foods low in saturated fats directly and also mediated the intention–behaviour and perceived behavioural control–behaviour relationships, suggesting an important role for planning as a post-intentional construct determining healthy eating choices. Suggestions are offered for interventions designed to improve adherence to healthy eating recommendations for people diagnosed with these chronic conditions with a specific emphasis on the steps and activities that are required to promote a healthier lifestyle.

Tendering theory revisited

This paper discusses the content, origin and development of Tendering Theory as a theory of price determination. It demonstrates how tendering theory determines market prices and how it is different from game and decision theories, and that in the tendering process, with non-cooperative, simultaneous, single sealed bids with individual private valuations, extensive public information, a large number of bidders and a long sequence of tendering occasions, there develops a competitive equilibrium. The development of a competitive equilibrium means that the concept of the tender as the sum of a valuation and a strategy, which is at the core of tendering theory, cannot be supported and that there are serious empirical, theoretical and methodological inconsistencies in the theory.