Developing Thinking in Geometry
A ‘Desert Island’ book
I do not know what the current rules for Desert Island Books are but this qualifies as one. In the spirit of the modern age the rules are probably negotiable, so that one can ask for a computer with Java-enabled software which will run the disk which accompanies the book, a dynamic geometry package and a certain amount of material starting with A4 sheets of paper and a printer. And perhaps some other young people who may or may not be called a maths class. And so back to real life!
Developing Thinking in Geometry is a wonderful collaboration of six authors, which sets out to provide a means of developing one’s own geometrical thinking with the very specific intention of helping others to do the same.
The book is made up of four blocks each of which deals with a theme worked out in three chapters dealing with Language, Reasoning and Visualising and Representing. Each of these chapters sets a variety of tasks, offers comments and opportunities for looking back and closes with a section on Pedagogical Perspectives.
The tasks include some old favourites especially perhaps to those who have attended ATM meetings or conferences but most have a very fresh feel to them and are very carefully chosen to contribute to the purpose of the book. There is a lot to work on here as some of the tasks take time. The text includes some hints which enable the development to be followed, but the comments make very clear that the tasks are simply starting points in the process of observing one’s own thinking. This second level of work requires time and patience as some of the awareness does not come readily. The resulting awareness is used to indicate what other learners, chiefly in classrooms, face and go through.
Invariance is a major theme and the use of dynamic geometry provides opportunity for many new insights. The disk provided my first encounter with dynamic geometry in which one has control over certain properties in a diagram and can observe ‘what changes and what stays the same’ in a new way. At next opportunity I will sign up for a complete package and continue the investigations.
For anyone wishing to reunite the insights and development of geometry using the traditional Euclidian metric approach and the ‘modern’ transformation approach there is plenty to work on here, and it is especially useful to have the two side by side.
Always calling the reader on and prodding towards further insight, the concrete visual objects in 2-D and 3-D are replaced with the challenge to visualise in a series of Imaginings where words prompt mental images which can now be conjured up and manipulated with increased facility.
The place of questioning, posing of problems, is given much attention and the significance of different phrasing will help to make us more aware of what difficulties, limitations and opportunities we offer those in our classrooms.
Geometry took its place in human culture as a set of practical procedures which in turn gave rise to a form of reasoning in which once certain starting points were agreed a whole structure of certainty could be built up. The processes of reasoning are made explicit and discussed at all stages of the book and the place for holistic and non-linear thinking is made plain. The stages of problem solving from engagement, through conjecture, the search for the means towards the certainty, presenting the proof and then looking back are all built into the text and its demands on the reader (or rather the worker!).
There are many key phrases which are clearly the distillation of the authors’ wide and deep experiences which will provide reference points for many practising teachers. A sample is:
- “A task worth doing affords opportunities for multiple outcomes”
- “Say what you see - watch what you do”
- “Convince yourself that...”
- “ convince a friend...”
- “ convince a sceptic”
The last words of the book remind us that:
“I cannot change others; I can however work at changing myself.”
“It is amazing what an influence this can have on others”.
At the lowest level there are enough ideas to set up lessons on many syllabus objectives to make it worthwhile. But if that were its only use it would be grossly under-utilised. The book will help anyone who wants to carry out some of that working on oneself. It needs a lot of time. It would be a wonderful book to work through in a group over a period of time and the earlier in one’s career one starts the process, the better.
Richard Knottenbelt • Victoria High School, Masvingo, Zimbabwe
Developing Thinking in Geometry
Edited by Sue Johnston-Wilder and John Mason
The Open University in association with Paul Chapman Publishing
2005
ISBN 1-4129-1168-0
ISBN 1-4129-1169-9 (pbk)
