Images of balance and order
It all depends how you look at them. At first, I saw stars on the page, surrounded by wormy things in S shapes, the stars being the centre of attraction. Later, the stars were just the background. Each worm had its own route, resembling a river, road and bridges to me. There was a lot of everything - never ending and never beginning."
What was Suzanne talking about? Notions of the infinite maybe? She, and other members of her Year 8 class, had been asked to look at an Islamic pattern for homework as part of an assessment on tessellations. This was designed to help the teacher to find out what they had remembered and understood from earlier work, particularly about transformations of shapes. (Figure 1) When we taught together many years ago, we used Islamic patterns in the classroom, and in a maths club, in various ways. Our interest was reawakened when we visited Seville for an international congress on mathematical education in 1996 and saw the wonderful geometrical designs in the Alcazar palace and other places. The Islamic religion forbids representation of the human or animal form in religious art. As a result, creative talent turned to calligraphy, scrollwork and floral designs, and the exploration of geometric forms based on the subdivisions of the circle on which the patterns shown are based. The circle is the symbol of unity and represents the great arch of heaven. While Christian art is diverse, Islamic art, dating from the 7th century, is linked by a unifying concept of composition based on balance and order.
Islamic patterns can provide a rich context for classroom discussion, enabling teachers to assess their pupils' knowledge of geometrical ideas and vocabulary. Pupils can analyse a pattern, naming shapes, describing symmetries and transformations, finding angles, explaining their thinking and justifying their answers.
For example, there are triangles in the pattern for Task 1. Are the triangles equilateral? How can you be sure? What assumptions do you need to make?
At a seminar for teachers last year, we asked them whether the dodecagon shown with dots in the pattern of overlapping circles was regular or not, and to prove their answers. What do you think?(Figure 2) Even colouring a pattern systematically can be a mathematically worthwhile activity and encourage different ways of seeing the overall design. This is colouring with a mathematical purpose and not just a way of producing pretty pictures.
How do you see this pattern? (Figure 3) When it was coloured in two different ways two different perceptions appeared.
One (Figure 4) has a 3-D feel while the other (Figure 5) seems very flat; it has the sense of patterns of squares growing out from the original shape. How many squares will there be in the next layer?
Producing a copy of a design, either from scratch or using a square or triangular grid, is not always a easy as it looks, but you can soon tell when you have gone wrong! This work can encourage pupils to consider the pattern from different points of view: is it made up of shapes fitting together - tessellations - or of lines going across at different angles? Sometimes the work involves following instructions carefully, as Task 2. Catherine, one of Suzanne's classmates, wrote for her homework: "It was a very complicated pattern. When I began to colour it in, I realised that it was more the lines, and not the stars, that were the pattern. I also realised - especially when I was drawing the pattern myself - that there were not actually three different lines, it was the same line just twisted round."
It is good for pupils to be encouraged to talk, or write, about maths and their impressions of what they see and are doing in their own words. Such discussion can also be extended to so that pupils can develop their use of more formal, mathematical language; Catherine and Suzanne and others in their class were able to use more technical language when asked. They were able to show what they knew about rotations, for example, and different types of symmetry.
Some Year 6 pupils were asked to look at this pattern (Figure 6) and, when answering the questions, to show off as much mathematical knowledge and language as they could. Zainab said: "I can see stars that have 12 sides, and I can see a few different hexagons that are regular and irregular. They have six sides. There are a few equilateral triangles. They have angles of 60 LESS THAN . The stars are dodecagons. There are lots of lines of symmetry."
The teacher was able to build on these initial comments to extract more information about what Zainab and her friends knew.
It is, of course, much harder to extract information when you are given a complex pattern to look at than when just given a single shape. But it is also much more interesting to think about the symmetries of a large pattern.
Who says that mathematics is not beautiful? Exploring Islamic patterns can help to give pupils a sense of wonder; they remind us of our rich cultural heritage and of the many links that the subject has with other aspects of the curriculum.
Kath Cross is an HMI working in the Office for Standards in Education's school improvement division. Anne Haworth is a research associate at Manchester University
They both write in a personal capacity
WHERE TO FIND OUT MORE
ISLAMIC PATTERNS. By J Bourgoin. (Dover Publications pound;4.95). Contains photocopiable patterns Available from David amp; Charles, tel: 01626 323200
ARCHITECTURE OF THE ISLAMIC WORLD. Edited by George Mitchell. (Thames amp; Hudson pound;19.95)
ISLAMIC PATTERNS. By Keith Critchlow. (Thames and Hudson pound;16.95)
TEACHING MATHS THROUGH ISLAMIC ART. Victoria amp; Albert Museum. Free. Tel: 020 7942 2184
GEOMETRIC CONCEPTS IN ISLAMIC ART. By Issam El-Said and A Parman
World of Islam Festival Publishing, 1976. Out of print, available in libraries