Proof in Elementary Geometry
Uses powerful images to provide convincing reasons for the truth of theorems in geometry
Key Stage suitability • Explanation
|Proof in Elementary Geometry
Item Ref: ACT018
|Proof in Elementary Geometry - PDF
Item Ref: DNL028
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Proof in Elementary Geometry
This book will interest all those who are concerned with the current state of geometry in school. The concept of proof is a vital part of what mathematics is all about.
In this book, the author claims that ‘seeing is believing’ and using powerful images to provide convincing reasons for the truth of many theorems in geometry. Students are more concerned with memorising proof rather than being convinced by them. Using a ‘tracing’ on top of a ‘diagram’ we can often show clearly the truth of assertion. In other words: we can prove it.
There is a profound ambivalence about the issue of proof in mathematics in general, and about proof in geometry in particular. Many of us remember having first met proofs with an injunction to ‘learn this off by heart’ and whether or not we understood and appreciated them seemed far less important in the eyes of the teacher than that we knew them.
In this booklet, persuasive words and pictures are used to put the case, rather than strict logical arguments. The aim is to convince, rather than to prove.
The PDF download version of this book contains all the pages in colour and again in black and white for those that wish to use it the monochrome version.
There is also a page at the back of the book which is suitable for copying on to a transparency for use with the activities.
ISBN: 1 898611 17 3
Geoff Giles lived and breathed mathematics education throughout his working life. When he joined the University of Stirling as a lecturer in Mathematics Education in the nineteen sixties he was already certain that all children could understand and enjoy mathematics if they could access it in the right way. This led him to begin work on his own series of booklets and, in particular, on the use of concrete materials. A short time later he started DIME Projects – Development of Ideas in Mathematical Education – to allow the development and sale of his ideas in the form of experimental literature and teaching aids.
DIME grew fast, and within a few years was bringing in some £12,000 from the Scottish Education Department alone for school materials supplied to three major curriculum development schemes. At the same time the publication of twenty-four booklets, originally written by Geoff for the Fife Mathematics Project, began. When a major London curriculum development involving over 100 secondary schools, the SMILE Project, started using the DIME booklets, it became clear that a new stage had been reached. Since then DIME materials have sold all over the world.
The first items that Geoff developed in the range of concrete materials were the DIME Solids to use with the Build Up booklets, Rotagrams which are used to compare angles, the Probability Kit which offers a wide range of practical experiments in probability, and TakTiles which support practical discovery work in shape fitting and symmetry. From these beginnings the DIME range of teaching aids continued to expand. Geoff's book Algebra through Geometry, which uses TakTiles to give meaning to algebraic simplification, had a significant influence through the Key Stage 3 National Strategy when it was featured in a training video.
With geometry so close to Geoff's heart it was natural that he sought to make its study not just easier, but more meaningful. In particular, he was long unhappy with children's ideas of proof. This concern led him, in 2001, to write Proof in Elementary Geometry, in the hope that it would help to clear up some of the misunderstandings that bedevil the treatment of proof in geometry in the secondary school. His death in 2005 sadly brought an end to his life-long work, but his ideas and materials will continue to inspire mathematics teachers everywhere for many years to come.
In this booklet, I will use persuasive words and pictures to put my case, rather than strict logical arguments. The aim is to convince, rather than to prove.
I am happy if students reach the stage of seriously thinking through the issues involved, and making up their own minds about the truth of the matter. Whether they are right or wrong does not concern me too much, I am confident that personal involvement in the issues will lead them inevitably to a better grasp of the mathematics involved.
But this gives the wrong impression. Of course I wish to provide proofs, but these should not be dead proofs. They should be reached as a result of the student’s own personal considered thought, not foisted on him as the conclusions of bygone experts whose words are to be treasured and memorised. In other words, they should be living proofs.
As a first example the question is raised of’covering a chess board (with two opposite corners missing) with dominoes'.This is explored and a common sense proof that it is impossible is put forward. The same methods are then used on other shapes and sizes of board. Finally the more difficult question is tackled of’covering these boards with straight trominoes'.
In working through each of these it becomes apparent that there is a point at which the person involved accepts its truth, giving him a sense of personal ownership, or conviction.
In Part 2 we are concerned with proving early theorems in geometry in a similar personal and dynamic way. By using diagrams and tracings in this way, we show the value of such an approach.While such a hands-on practical and experimental approach may seem hardly satisfactory as a proof, on reflection it must be admitted that it is both generally applicable and, more importantly, leads to the feeling of personal ownership. On the grounds that they lead to this personal ownership and understanding, such presentations must be acceptable as proofs.
The most significant section, Part 3, deals with Motion Geometry proofs of many of the angles in a circle theorems.
If a tracing of a pair of lines moves over a clock face, angles are all related to angles between the hands of a clock, thus making the work far more attractive and palatable to younger students. This is helped further by the dynamic nature of the proofs.
The work continues in Part 4 by showing how easily the theorem about the Mid-points of the Sides of a Triangle can be seen and proved, by using two transformations. And this is followed up by discovering another theorem that just has to be true, and giving the proof.The final example concerns a nice theorem that I actually discovered by thinking about transformations in this way.
The purpose of this book is to help you understand the thinking that lies behind this new approach to geometry, and to let you consider the value it could have for your own students.
As the over-riding goal is the understanding of the mathematics, your active involvement and participation in the work is most desirable.
With this in mind, all figures that involve tracings are repeated on translucent paper so that you can actually cut them out and use them with the appropriate Diagrams.
Having successfully done this, you will wish to see how these ideas work with students using an overhead projector. You will probably want to enlarge the tracings and Diagrams on your photocopier before copying them onto your acetate sheets. Then everything is ready for you to Introduce a group of students to your own choice of topic, or even the whole class.
First I suggest you attach a blank sheet of acetate by one edge to the O.H.P. This will enable you to show an acetate Diagram under it.Then a tracing placed over it can be moved easily without disturbing the Diagram.
Suppose you choose Part 3.
Produce all the Part 3 diagrams and tracings on acetate. Have all these in two envelopes and you are ready to start.
Put Fig. 31 on the OHP. Get them to visualise what will happen if the circle is folded as shown. Do they all agree that if C falls on A, then D must fall on B? This could lead to much discussion.
Let it all come from them. Don’t suggest how they might look at it. Remember the sole objective is they themselves should think out what is happening.
Now move on to having Fig. 32 under the acetate flap. Place the tracing of Fig. 33 on top of the flap, and over Fig. 32 (which gives Fig.34). Ask them what will happen if the Tracing is given a 5 minute turn clockwise.
They should be able to tell you where points A', B', and C will move to. Ask them how much more the Tracing must be turned before A B is parallel to AD. Lead them on so that they discover that in fact LADC=LABC because both are equal to 2-A B’C.
In Part 1 there is really no need for OHPs, but in Part 2 they do help substantially in understanding the theory.
In Part 4 you may even find them necessary to make sense of the ideas being developed!
Covering a Chessboard with Dominoes
Understanding through the Use of Transformations
The Easy Way to’Angles in a Circle'
Discovering New Theorems for Yourself