Modelling arithmetic strategies

This thesis examines children's arithmetic strategies and their relation to the concepts of commutativity and associativity. Two complementary methods were used in this research: empirical studies and computational models.Empirical studies were carried out to identify the strategies children used for solving problems like 3 + 4, and 3 + 4 + 7, and the conceptual knowledge associated with them. Their understanding of subtraction problems where the minuend is less than the subtrahend (e.g. 6 - 8) was also considered. A study with 105 subjects revealed a variety of strategies and information about children's knowledge of commutativity and associativity. Four levels of performance of commutativity were also identified. A longitudinal study was carried out with 12 children in order to obtain details of children's changes in strategy, and to double check the results obtained in the main study. The strategies observed to be used by children over a 20 month period parallel those found in previous studies, which show a general transition to more efficient methods. However, the longitudinal study revealed that development of such arithmetic strategies is a slow process. Furthermore, the studies indicated that knowledge of commutativity is a prerequisite for associativity.Models of the observed strategies have been implemented in the form of production rules in a computer program called PALM. The process of implementation highlighted features of children's problem solving that had not been 'detected during the studies.In addition to models that describe the space of strategies, a model of learning has been implemented for the transition from procedural knowledge of commutativity to that of associativity. The model is capable of generalizing its inbuilt knowledge, for instance, its ability to solve 2-term arithmetic expressions, to allow it to solve more complex problems, such as 3-term arithmetic expressions. A further model has been constructed for learning arithmetic strategies that are more efficient than those already represented in the program. It learns specific rules by adding conditions for efficient problem solving to its previous general rules