Why Granny's knitting adds up to a complex mathematical equation

While the needles have stopped, she gives a cursory glance down, where a symmetrical nine-point star frames the centre of her handiwork: "Anyway, if we do trade the Mondeo in, I’ve told Bob a hatchback would be better this time."

You find yourself watching, fascinated, as she plonks the knitting more centrally in her lap and carries on talking. A smaller, cream, brown and blue nine-point star can now be seen in the centre of the outer one. They are different shapes, and the outer one has burgundy in it.

On closer examination, so does the inner one - it’s just that the colours are combined differently. The coloured yarns, housed in old coffee tins with holes through the lids, roll gently round her feet as the wool feeds through. There are no written instructions in evidence anywhere.

Ellen can’t understand why anybody should be remotely interested in what to her is a standard piece of knitting. A pattern, she reckons, is a lot less hassle than a plain job. "The pattern hides all the workings, so you don’t have to be so fussy. Anyway, with the blue and the burgundy, it matches two jackets instead of one," she explains.

Mathematician Mary Harris, attached to University College London, looks at it somewhat differently. "Ellen is actually producing an Archimedean spiral. She knows exactly when and where to decrease’ and the precise number of stitches she needs to lose at certain places, so the work takes on the exact shape needed for the purpose," she says.

"Experience has taught that it’s no good thumping a piece of lumpy knitting with an iron. As she says, she’s concealing the adaptations to the shape by incorporating the changes into a symmetrical pattern.

"She’s doing that by changing the colour of her yarn systematically as she goes along."

There’s nothing unique about Ellen - yet. But she’s a product of her time, and that time has gone. Subsequent generations, dismissing what women like Ellen do as "granny’s knitting", have obviously never tried it, says Ms Harris.

She explains: "In mathematical terms, Ellen has worked a band of translational symmetry with two alternating motifs on a reducing number of units, until another strip pattern takes up the journey to the centre of the spiral where she changes to another pattern - the star with nine-point symmetry. Working from the circumference in, she’s produced a geometrically perfect dome-shaped design."

It’s the equivalent of conducting a symphony without a score. As far as Ellen’s concerned, she’s knitted a beret that matches two jackets. If it fits, that’s fine. End of story.

Maths does not feature because she never did maths at school. "Maths classes were for the clever folk, not the likes of me," she laughs.

It’s the kind of remark that lights blue touch paper under Ms Harris, who was originally trained as a scientist, but who has spent almost a quarter of a century trying to bridge the gap between school maths and maths in the real world outside the classroom.

"The reason Ellen can do what she does is because she doesn’t associate it with maths," she says. "This is because people like her were pre-conditioned into believing that maths was not for them.

"If someone had told her all these years ago that she was mentally calculating and adjusting complex mathematical formulas she would have decided there and then that she couldn’t knit to save her life, and turned her talents to sewing."

Maths, says Ms Harris, is still looked on as a subject connected with high status technologies - an assumption that led her to front a recent "Maths in Work" project.

Carried out through the University of London’s Institute of Education, it was aimed at encouraging teachers to use crafts and textiles as a basis for learning mathematics.

The project involved Ms Harris and a team of colleagues travelling to different areas in the UK. Among the talks and lectures were public workshops to which the team invited lecturers, mathematicians, teachers, members of women’s institutes and people involved in trades, crafts and textiles.

Those attending were asked to try to work out the answers to some practical mathematical problems - the structure of a garment to fit someone’s head, a simple knitted jumper to fit themselves, using ten stitches to the inch, or fitting non-wrinkly lagging on a right-angled pipe.

Inevitably, says Ms Harris, the academics over-complicated everything. When asked to explain verbally what they were doing, they were typically completely stumped.

On one memorable occasion, a women’s institute stalwart stood up and silenced a teacher who was carrying out a long and laboured breakdown of the pipe-lagging problem.

"Oh for pity’s sake lad, all you’re doing is turning a heel on a bleddy sock."

On a different occasion, another woman was seen helping out somebody struggling to work out the simple sweater design.

"Your armholes won’t fit the top of your sleeves because you’ve got too many stitches, but if you just decrease by two at the end of every tenth row till you’ve got 80 stitches, they should match."

She denied she’d just done some pretty impressive on-the-spot mathematical calculating. She couldn’t possibly, she insisted, because she couldn’t do maths - her husband was always telling her so.

"Knitting patterns are a form of algebra," says Ms Harris, "and their real use is to set up the beginnings of the motifs within a particular number of stitches.

"Once that’s established, a knitter will usually abandon the pattern as their own thinking takes over."

Real knitters communicate in the same sort of algebra that a pattern uses, she says. It’s a skill that translates into a myriad of shorthand forms, according to the particular family or group of people passing them on.

It’s just not something that can be compartmentalised and presented in the formalised way that mathematicians seem to need - which is why they can’t deal with it. Meanwhile, the Ellens of this world invent and reinvent as they go along, according to whether the intended wearer has long arms and a short waist, is flat-chested or has a bum that will undoubtedly look big in it - if the ribbing at the bottom isn’t subtly increased.

They can be found all around the country, rattling off garments that the recipients will often take at face value and consign to the back of a drawer, without realising they probably own a unique and finely-crafted example of transformation geometry.

Treasure these gifts, says Ms Harris. Hand them down as heirlooms and give the mathematicians of tomorrow something to think about.