A conceptual approach to the early learning of Algebra using a computer

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Abstract

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.

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