﻿ Counting Pane
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It is often difficult to know where inspiration comes from. Counting Pane was a sudden inspiration in our very early days of representing Maths in knitting. Perhaps it was suggested by the squares, to be found in many classrooms, shaded to show multiples of numbers.  Counting Pane shows all those shadings simultaneously. It remains one of our favourites and has probably had more attention over the years than any of the others. Does this make it the most important? Maybe.

It was the first time we used the technique of knitting in strips rather than working with diagonal squares and specific angles. The individual squares were separated by a stripe of the background colour and one column was separated from the next by a similar stripe.

Counting Pane has 100 squares in 10 rows. The first row represents the numbers 1 to 10, the second is 11 to 20, and so on. The colours across the top represent factors. Blue stands for 1, yellow stands for 2, red is 3, green is 4, raspberry is 5, brown is 6, baby pink is 7, baby blue is 8, orange is 9, and purple is 10.

If a square contains yellow it means the number divides by 2, if it has red then the number divides by 3. If a square divides by 2 and 3 it must divide by 6 so also contains brown.

No factors above 10 are included. All squares which remain plain blue are prime numbers. In the top row the squares with only blue and one other colour are also prime numbers.

There were two problems, at the designing stage. The first was that when we looked at the factors of all the numbers we realised there were more than we had thought about and we couldn’t possibly include them all. We could have colours to show that numbers will divide by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 but couldn’t go  higher. because there simply wouldn’t be enough room to show them all. Also and more importantly, the colours had to be easy to identify and anyone looking at the grid had to immediately know when they were looking at a particular colour, not left wondering if this shade of yellow was the same as that shade of yellow.

The second problem was working out what size the squares must be to fit the different numbers of stripes needed. We decided to use squares with twenty-four ridges of garter stitch as 24 will divide by 1, 2, 3, 4, and 6. We had to cheat a little where we needed 5 or 7 stripes as these numbers don’t divide evenly into 24.

The knitting process was easy as the afghan was knitted in ten columns with the navy blue, to separate the squares, knitted as extra stripes on the way. However it did need a great deal of concentration to ensure that the colour sequence was correct. One big advantage of knitting small pieces, like the 24 stitch wide columns in this case, is that there is very little to pull undone when a mistake is made. When the columns were complete all that remained was to join the columns with navy blue separating strips and to add navy borders all round, including a channel for the hanging pole.

I remember the finishing of this more vividly than anything else I have ever made. When it all went together it was a shock. I had previously assumed I knew most of what one could ever know about the numbers up to one hundred. Ben was there at the time. He was already proving to be a very able mathematician with far more talent and knowledge than I ever had – and he could knit and crochet. We laid Counting Pane on the floor and stared at it. There was far more there than we had anticipated. We sat on the floor picking out one amazing fact after another. They came thick and fast.

What struck me first was the large number of colours in the tens column (10, 20, 30, etc.) I must have known it as I was knitting the column but it only jumped out when it took its place with the other columns. I have often been known to talk about ‘nice numbers’. To me, nice numbers are those that have lots of factors because they are easy to juggle with. I had never classed multiples of ten as nice numbers. They were ‘just tens’. Everyone knows you add a nought if you want to multiply by ten or take one off if you want to divide by ten. That didn’t make them interesting enough for my nice numbers. Now they were the first thing to shout at me. That column was far more colourful than any of the others. There were more factors than anywhere else.

It is obvious, when you stop to think about it, that any number that will divide by 4 also divides by 2. Everywhere there was a green stripe representing 4, there would be a yellow stripe for 2. Similarly, every square containing grey, for 8, must also have green and yellow. A brown 6 always went along with a yellow 2 and a red 3. Oranges, representing multiples of 9, made an interesting sloping line. We could have expected this because adding 9 is the same as adding 10 and taking away 1 so the squares are 1 away from lining up. We hadn’t expected the lines of red that seemed to criss-cross in all directions. Bright pink only appeared in two columns but, as it represented 5, perhaps that was not so surprising.

The other big surprise, that came to change my perception of certain numbers, was the number of plain blue squares there were. Theoretically, a square was blue if its only factors were one and the number itself. This is the classic definition of prime numbers. It is sometimes stated as ‘a prime is a number that has only two factors’. In Counting Pane not all prime numbers are blue so it cannot be used as a reliable definition. 2, 3, 5, and 7 are all prime numbers but they have a coloured stripe. However, what was very apparent was that the blue squares were clustered in certain columns. It should have been obvious that there would be no primes in the even-numbered columns, because 2 divides into all the numbers in them. Whenever I think about prime numbers now I think about them in terms of which column they are in.

We have made three versions of Counting Pane. The first was bought by the Science Museum. It is slightly different from the others because we tried to resolve the problem of 1 looking like a prime number when technically it isn’t. For versions two and three we decided this was an anomaly we would have to live with and introduce as a discussion point when appropriate. Versions 2 and 3 have other slight cosmetic differences.

At the most basic level this afghan offers a very graphic way for children to understand the difference between odd and even and to realise that this is really the same as knowing whether a number divides by 2. Yellow only appears in half of the columns. These columns contain only even numbers.

Numbers which divide by 5 are also spotted quickly as raspberry is noticeably only in the 5’s and 10’s columns. Putting these two ideas together there is only one column with yellow and raspberry so this must be the column of 10’s. There are many questions that can be asked.

The school where I was teaching had several Support Teachers and, because I mostly taught groups where the lack of English was a serious problem, I had the same Support Teacher in the classroom with me for at least half the week. He actually had a degree in Classics and that was what he had taught in earlier years but he was brilliant at being able to communicate with these pupils. He was also intrigued by the items that appeared on the classroom wall and he was as likely as I was to spend long periods of time discussing them with groups of pupils. He was captivated by Counting Pane. He would lead groups, or individuals, through ways to express for themselves what these patterns might mean.  Sometimes, to the amazement of pupils, he would fly in through the ever-open door, study the pattern for a minute and disappear just as quickly muttering, “Just passing by … came in to check ….”

Counting Pane has formed the basis of many exhibitions and workshops. One memorable occasion was at the annual North-East Maths Fair held at Houghton-le-Spring, probably in 1997. We had six or seven afghans on display, some with suggestions for tasks to be attempted, others simply for talking about. Part way through the day a girl, aged 13, wandered in alone and was instantly drawn to Counting Pane - perhaps she was attracted only by the stunning colours as we later discovered she had been brought to the event by a teacher and didn’t really have much interest in being there.