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An attempt to represent geometrically the imaginary of algebra

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posted on 01.10.2012, 13:03 authored by Ruth K. Tobias
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.



  • Science


  • Mathematical Sciences


© R.K. Tobias

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A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.

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