Pythagoras might not have known how to 'knit one, purl one', but a couple of Lancashire teachers have found that knitting is an excellent way of explaining his theorem. Elaine Williams looks at a pioneering pattern for maths teachers.
Pat Ashforth and Steve Plummer are passionate about maths and knitting. Over the years they have knitted a series of geometrically patterned rugs, some of which are now hanging in the Science Museum in London. But these are not just attractive, intriguing exhibits; they're teaching tools as well.
One of Pat and Steve's creations, "The Pythagoras Tree", demonstrates that crucial little equation "the square of the hypotenuse is equal to the sum of the square of the other two sides" - saving hours of sweated explanation or chalking on a blackboard. By looking at the ingenious "tree" pattern and counting the stitches, even the most maths-phobic individual can achieve enlightenment, they claim.
When they're not teaching - he is head of maths at Walton high school in Nelson, Lancashire; she is a part-time supply teacher - Pat Ashforth and Steve Plummer are writing knitting books, visiting craft fairs to demonstrate their maths work and attending conferences to show what can be achieved with a ball of yarn and a pair of knitting needles.
The eminent mathematician, Sir Roger Penrose, was so impressed with their work that he donated a piece of his own geometry, a non-repeating pattern known as Penrose Tiling, to be made up into one of their rugs. Now their work and pattern design has come to the notice of the British Handknitting Confederation (BHKC), which is offering free yarn and knitting needles to any school that wants its students to pursue maths by knitting rugs in an Ashforth and Plummer design.
Pat Ashforth took up knitting seriously in 1986, following a kidney transplant that sapped her of energy and limited her leisure pursuits. But she wasn't interested in the kind of knitting her mother used to do, involving flowery and intricate patterns. A mathematician with a fascination for the beauty of simple geometric shapes, she decided to explore what she could make out of knitted squares and limited colours. She was helped by an inheritance of countless balls of wool in various shades of blue."My father had been a policeman and my mother would knit him a jumper every year," she says. "There was always wool left over, but, as she never seemed to use the same blue twice, I was left with all this wool in different hues."
By splitting her squares diagonally, knitting them half one shade of blue and half another and then joining them up into a simple sweater, she became aware that she had a design with endless potential for lessons on rotation, reflection, symmetry and tessellation. At the time she was teaching maths at Denbigh high school for 11 to 16-year-olds in Luton, where 95 per cent of pupils were Asian. "When I wore my geometric sweaters the kids were always coming up and touching me, tracing the shapes with their fingers," she says. "Many of them struggled with English, but they loved the shapes and patterns they could make."
Pat Ashforth's pupils were not the only ones to show an interest in her knitting. Steve Plummer, a mathematician with an arts training, had arrived at Denbigh as assistant head of maths and soon caught on to the teaching potential in her creations. She taught him to knit and together they spent hours creating mathematical designs. "I had always been interested in the design side of maths, and suddenly I could see so much that we could do and teach with this," says Steve.
They began to visit craft fairs and maths conferences, picking up and sharing ideas whenever they could, and from the mid-Nineties were trawling websites to make contact with knitters and knitting companies everywhere.
Five years ago, their work came to the attention of a yarn company in the US, Brown Sheep, which published some of their patterns in a book called Woolly Thoughts. It also asked them to design an afghan - the small knitted blanket or throw rug to be found in many US homes - using the mathematical methods they had pioneered. Many American families now have their design books. Once they started working on a larger scale the couple realised the afghan provided the perfect medium for their patterns.
They have built up a collection of more than 40 such afghans, all with a mathematical message, which they exhibit in fairs and conferences around the UK.
Romance as well as maths blossomed over the knitting needles - they are now partners in both senses of the word, and many of their afghans hang in the home they share in Colne, Lancashire. They bought the large, Victorian terraced house because it was ideal for their intention of running maths and crafts courses.
The designs on their afghans, mostly knitted in garter stitch with washable yarns, range from simple counting games, such as snakes and ladders, through to mazes, patterns based on the series of numbers discovered by Fibonacci, the medieval Italian mathematician, and more complex mathematical ideas such as space-filling curves and fractals.
Pat Ashforth remembers vividly how one afghan, which she had crocheted to demonstrate a mathematical curve called a Peano, had caused heated debate among academic mathematicians at an Oxford conference. The curve is formed out of a line that touches the corner of every crocheted square. The couple called the afghan their "Peano Beano", but delegates got themselves into a stew over whether it was a Peano curve or a Hilbert curve. When she returned home from the conference, Pat Ashforth found a book that told her she had created a Hilbert open Peano curve.
Steve Plummer says: "We can present academic mathematicians with mathematical problems in our designs that they can have difficulty sorting out. They are looking for algebraic solutions while we are dealing with the properties of shapes. Ours is an elegant approach. I trained in art as well as maths, but, because of the shortage of maths teachers, the art side had to take a back seat. Knitting has enabled me to combine my artistic side with my maths."
They both take the afghans into school as teaching aids. Pat Ashforth says:
"They feel nice and look good, they can be touched and counted. There is just so much maths we can get out of them."
Their pride and joy is an afghan called "Counting Pane", which is 10 squares down, 10 squares across. Along the top row, each square is given a colour - blue for the first, yellow for the second, orange for the third and so on. Each square from one to 100 that can be divided by two is given a yellow strip; each square that can be divided by three is given an orange strip. So some squares end up with many coloured strips, some only one or two. "It is wonderful for learning the times tables as well as the patterns of numbers," says Pat Ashforth.
If there are lessons to be learned by looking at the afghans, there's even more to be gained in their manufacture. Pat Ashforth and Steve Plummer have made up patterns for five afghans to go with the BHKC promotion they are recommending as a first project for teachers to make with pupils. These afghans have been selected for ease of construction - simple garter stitch with no necessity to get an even tension - and the maths work that can be generated from the planning, the making, and the finished item.
Mark Barnes, BHKC chairman and director of the King Cole yarn company, says: "Knitting can teach children the practical side of maths, and it's good for their motor skills. It's also therapeutic; it can be very relaxing. It was popular in the Eighties, when handknits were in fashion, and it's making a comeback. If we can get children knitting, that has to be a boon."
Julie Gibbon, a maths teacher in Northumberland, has taken the de-stressing message on board. Taking advantage of the offer of free yarn and needles, she has got 70 Year 6 pupils at South Tynedale middle school, Haltwhistle, knitting an afghan as a way of relaxing after their Sats. "It's been a steep learning curve as many of them couldn't knit," she says. "But they've loved it. It's certainly been a way of relieving post-Sats stress. They've learned the concepts of increase and decrease, and have had a lot of pleasure working out the shapes and patterns they want to make."
Julie Gibbon is familiar with the Ashforth Plummer afghans from attending maths conferences and has her own copy of the "Pythagoras Tree". She says:
"I once had a student who really couldn't grasp the concept of Pythagoras's theorem no matter how I tried to explain it. So I got out my knitting and the penny dropped almost straight away."
To contact Pat Ashforth and Steve Plummer,tel: 01282 864273. For details of the BHKC offer, write to: King Cole, Marrie Mills, Old Souls Way, Bingley, West Yorkshire BD16 2A8.
* THE PYTHAGORUS TREE
To demonstrate Pythagorus's theorem - that the square of the hypotenuse is equal to the sum of the square of the other two sides - Pat Ashforth casts on with one stitch, then adds a stitch for each new line until she forms the diagonal. Then she decreases by a stitch a line to form the square. To work out how many stitches she needs to pick up to form the next square, she divides the diagonal by 1.4, which is approximately the square root of 2. "That's the only calculation you have to do for many of our patterns," she says. "It's beautifully simple."