Removing the Shackles of Euclid
A collection of articles reflecting on the maths curriculum
Key Stage suitability • Explanation
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Removing the Shackles of Euclid
A collection of articles by David Fielker indicating an approach to euclidean geometry free from the constraints of a conventional Euclidean development.
From the book's Introduction
Previous collections in this series have been of Mathematics Teaching articles by different authors on particular themes. This collection is different in that, certainly as the series progressed, the possibility of a separate publication became obvious, and the articles were written partly with that in mind. This means firstly that I am deprived of making the complimentary remarks characteristic of a compiler.
It should also mean that an introduction is unnecessary, since anything relevant to the series should already have been written as part of the articles. However, one or two things need to be said, or at least emphasised.
Whatever happened to...
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One thing I have found is that writing a coherent sequence of articles is much more taxing than putting together a single article. The lack of detailed forward planning left a welcome flexibility, but the time-scale meant a continual re-reading of what I had written to ensure continuity, an avoidance of repetition and a fulfilment of all promises.
The only broken promise was one to return, in the last article, to the Cockcroft Foundation List. Eventually it no longer seemed appropriate to take this up in that context, but perhaps it does in the context of some remarks now.
The relevant section in the Foundation List is called 'Spatial Concepts', and this for me is wider than geometry, since in spite of the derivation of the word I choose not to include ideas concerned with mensuration and trigonometry. Remove material ideas such as length, area, volume, scale drawing and bearings, and the Foundation List reduces to some fairly banal "recognise arid name" items, together with ideas whose depth depends very much on how you interpret the phrases "appreciate" and "understand and use". The third article, for instance, shows what could be implied by "understand and use...diagonal", Certainly there is no indication of where problem solving or investigational work is related.
Still, it is only a foundation list, and must be interpreted according to Cockcroft's introduction to it, though I hope I have said enough in these articles to indicate that I would have chosen the foundations quite differently. On the other hand, I have specifically criticised the examination boards for their restrictive syllabuses in geometry, and made it clear that conventional examinations give little scope for geometrical activity.
I should also emphasise that the geometry I am looking at is both euclidean and synthetic.
It is euclidean because I have not looked at possibilities for topology or for affine or projective geometry, although occasionally some of the properties of these other spaces have been used. I am not suggesting that other geometries should not be taught; indeed, they should be, and the same principles apply. 1 have merely chosen to restrict myself almost entirely to euclidean space. The series title indicates that what I am advocating is a freedom from Euclidean (note the capital) methods and from Euclidean content, which even in so called modern syllabuses still exert a restricting influence.
I have chosen to focus on a synthetic treatment of euclidean geometry, which means that what I have considered is a direct handling of the objects of euclidean space. It would take another, or perhaps several, series to discuss algebraic and transformational methods such as coordinate geometries (note the plural), vectors and matrices in the first category or motion geometry in the second. All these ideas are also important, but apart from considerations of time and space one of the reasons for choosing to deal with a synthetic approach is that I feel this is the one which is most accessible to pupils, particularly those whose attainment is low (to use current jargon). Algebraic methods are often hampered by algebraic techniques (or rather the lack of them); and transformations only become interesting when one combines them, and that is what pupils find difficult. I am oversimplifying here, but at least I have given examples of complex synthetic situations which are accessible to all children because the geometrical content necessary to an understanding of the problem is very simple.