A conceptual approach to the early learning of algebra using a computer
This thesis describes an investigation into the conceptual understanding of algebra by early learners (age 11-13 years) and how a computer-based approach may be used to improve such, without any consequent loss of manipulative skills. The psychological framework for the investigation centred on the importance of the individual child's construction of a cognitive framework of knowledge and the relevance of the current state of this to the facilitating of concept acquisition. As such it incorporates elements of the developmental psychology of Piaget, Ausubel and Skemp. Furthermore, in order to assist in the synthesis of a sufficiently broad psychological theory of education it was necessary to postulate the formulation of a new integrated bi-modal model of learning. This is described, along with details of its application and significance to a theory of cognitive integration which is designed to promote versatile learning (after Brumby, 1982) in mathematics through a relational linking of global/holistic and serialist/analytic schemas. The research comprised two initial investigations followed by the main experiment. The results of the initial investigations with early learners of algebra showed that the dynamic algebra module written for the research produced a significant improvement in the children's conceptual understanding of algebra. The main experiment sought to further clarify this improvement and to compare and contrast it with that produced by a traditional skill-based algebra module. In order to facilitate this comparison, the performance of 57 matched pairs of pupils from two groups of three parallel forms of the first year of a 12+ entry co-educational secondary school was analysed. The results of the investigation confirmed the value of the dynamic algebra module as a generic organiser (in the sense of Tall, 1986) for the understanding of algebraic concepts, producing a significant difference in conceptual understanding, without any detrimental effect on manipulative skills. Furthermore, the beneficial effects of the programme were such that its results showed that it had provided a better base than the skill-based approach for the extension of algebraic understanding past the initial stages and into more involved areas such as linear equations and inequalities. The findings of this research show that the use of a module based on a computer environment, with its many advantages for conceptual learning, prior to the more formal introduction of algebraic techniques, is of great cognitive value. They also provide evidence for the theoretical model of learning proposed in the thesis, and suggest that for the production of a versatile learner in mathematics, more attention should be paid to the integration of the global/holistic abilities of the individual with his/her serialist/analytic abilities. The implications for the future are that such abilities, and hence mathematical competence may well be improved in other areas of the curriculum by the use of the computer within a similar theoretical framework.