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The World of Illusion Knitting

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PICKING UP THREADS

This was written in
2007
so is now very dated

The Mathghans project was still going on in schools across the country and one day a large parcel of navy and bright yellow yarn arrived at our door. It had been wrongly addressed and was destined for a school wanting to make an afghan in their school colours. When I contacted Mark to tell him what had happened he told me to keep it and that he would send another lot to the school. This was a thicker yarn than I would normally use so I decided to use it for a very simple project.

We only had one arrangement of half-and-half squares so this was to be another, more dramatic variation. The squares were arranged to look like dark spirals on a yellow background. It was called About Turn.

All of the afghans we had made so far had a place in our classroom, or workshop, teaching. Most of them had been for one of two reasons; either we had come across a mathematical idea that we could represent in an original way or we could make an attractive and eye-catching mathematical, or geometric, picture. The next afghan was different. It was born out of frustration.

In August I had been persuaded to take on a part-time Maths timetable, to fill a gap for a year, until the school was able to take on a full-time teacher. Amongst others, I was teaching a class of 14 and 15 year olds who were having great difficulty in identifying the different four-sided shapes. Theoretically they knew the definitions for square, rectangle, rhombus, trapezium, kite and parallelogram but were unable to differentiate between the drawings. To try to identify the problem we made a worksheet for them to colour in. It had a key where they had to show the colours they had used for each shape. This made it very easy to spot where the errors were occurring. It didn’t take long to find the source of the problem. They were seeing in 3D where they should have been applying the rules for flat shapes. Wherever there were shapes that looked as though they could be the side of a box they coloured in rectangles.

For example, a box drawn in 3D can have parallelograms on the side - like Cubism or Fibo-optic. When the pupils were asked to define a rectangle they would happily say it had to have four right angles and then continue to colour a shape which clearly did not have any right angles.

Metafourmosis started life as a worksheet for these pupils but cried out to be an afghan. The angles of the worksheet would have been difficult to knit so the design was adapted to fit on a grid so that it was easy to calculate the angles.

Our basic knitting rules, creating 45 degree angles were adapted, working on the principle that if a shape is to be twice the height it has to lose (or gain, depending on the direction) its stitches at half the rate it did before. For a shape to be a third of the original height it has to loose its stitches three times as fast. It was then an easy task to create shapes to cover a grid. We chose a different colour for each type of shape and used different shades for the various sizes and variations within each group. For example all squares were shades of brown, all trapeziums were shades of blue.

The worksheet was a success because it identified the problem. The afghan has been a greater success because, as with all our other afghans, it is large enough for a whole class to see  at once. It can be discussed with 30 pupils simultaneously as they can touch and point out the shapes they are looking at. It never ceases to amaze us that there is such variation in what people see.

Very recently I have taken Metafourmosis into another group of older pupils and was given another insight into someone’s thinking. The design is symmetrical and in the centre it has two squares, two kites and two arrowhead kites. These are surrounded by four identical rhombuses. Because the rhombuses have equal sides, their outer edges form a regular octagon. One particular boy pointed out this octagon but he could not see the shapes in any other way. It took a great deal of shouting, cajoling and argument before he began to realise what some of the others were seeing.

This may seem a very trivial matter but it has enormous implications for the use of drawings in text books, exam papers, instruction manuals and many other walks of life. We all tend to assume that everyone’s interpretation is the same.

Click here to see more about About Turn
Click here to see more about Metafourmosis

20b. THE WORLD WIDE WEB OF KNITTERS continued