In 1981 the author submitted that "many of the (then)
more recent school syllabuses remain disjointed and give
expression still to a school mathematics course as step-by-step
progression through a list of disparate topics".
The position has not changed. It is not yet generally
accepted that there can no longer be an accepted body of
mathematical knowledge that needs to be taught. The rapid
development of new technology and the introduction of the
microcomputer should enable the 'modern' mathematics of the
early 1960's to enhance the mathematical experiences of pupils
in a practical and comprehensible way and prompt a new
style of teaching and learning mathematics.
There is, however, a fundamental core of mathematics which
must inevitably find a place in the school mathematics
curriculum. In Part I of the thesis the emphasis is on a
method of presentation of certain key topics which illustrate
the basic pattern of a group structure. Former complications at school level of putting plane
geometry on a logical footing have to be avoided. The use
of complex numbers highlights significant and sometimes
rather difficult geometrical ideas. In Part 11 the author
attempts to show how some of these ideas may be presented
to extend the basic pattern to that of linear algebra.
The work culminates in Part III with the use of linear
complex algebra to present more vividly the symmetries of
the Platonic solids. The author anticipates the realistic
presentation of the aesthetic side of 3-dimensional geometry
and takes a look at its possible presentation through the
medium of the microcomputer.
At this early stage of the development of the ideas to be
discussed, there can be no formal testing of the results
by quantitative analysis. Evaluation of the viability of
the proposals will be qualitative and the comments of
'critical academic friends' will be included. The originality demanded of a piece of research goes beyond
the exposition. Here it will consist of new insights into
ideas appropriate to senior pupils in schools and a rewriting
of existing material often thought to be beyond their scope.
The work is supported by suggested lesson sequences, transcripts
of recorded presentations, and examples of students' work.
Subsequent development must face the question of assessment
and evaluation at sixth-form level of the proposed new style
of teaching mathematics. The author makes some suggestions
in the concluding chapter.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.