The use of autonomy in the development of mathematical concepts in primary and middle school children

Abstract

The research reported in this thesis is divided up into three main sections. The first section begins with a brief review of the literature on methods of conducting psychological research in the classroom. It is argued that an interest in the psychology of education must be followed by a methodology which takes into account the fact that schools are social institutions within our society, and that they have functions other than the teaching of cognitive skills. Learning which takes place in schools may therefore be seen as being embedded within a social milieu. The classroom research therefore begins with an extended period of observations and interviews. Initially, the investigator operates as a non-participant, and gradually, as his presence becomes an unnoticed part of the school's routine, certain limited interactions are carried out with the children. Conversations and informal interviews are also held with the head and members of the teaching staff. From all of these encounters, a simple model of mathematical learning in the classroom is put forward. This is supported by two scenarios taken from actual classroom situations. A hypothesis on mathematical learning in the classroom is then developed which states that children must experience a certain amount of autonomous activity if they are to formulate 'higher order' strategies for dealing with mathematical representations, structures, and problems. Section two of the thesis contains a series of experiments which are designed to test aspects of this hypothesis in a controlled setting. This is accomplished by using three specially designed portable 'toys' which each contain rules or patterns relating to mathematical systems. Positive evidence is found for the claim that an autonomous condition is more effective than a yoked, control or combined experience (hybrid) in promoting mathematical conceptualisation as scored by written and verbal measures. In the third section of the thesis, the toys are brought into the classrooms in an attempt to integrate them with more formal instruction. Results of these efforts are assessed and documented via interviews, photographs, and samples of classroom work. Educational implications and suggestions for further research are then briefly discussed