Melvyn Catton holds up a large white cylinder with numbers printed on it. He points to pairs of them. “What’s going to happen between 14 and 25, or 25 and 36?”

A hand shoots up. “Is it add 11?” asks the pupil.

Mr Catton beams. “Yes, add 11. It looks as if there’s a pattern here. Rosie?”

“It’s 11.”

“No matter what way we go diagonally to the right, it seems to be 11. How about if we go to the left?”

“Nine.”

“Is it nine? Don’t eat those Smarties, I need them for the next lesson.”

“What about 3 to 12?”

“Nine again.”

“It is nine. How about 63 to 72? Sam! Stephen?”

“It’s nine.”

“Well, it looks as if we’ve started a pattern here as well, a pattern on a vertical spiral. Spiral as we go round, where do we go to?”

Lots of hands. “Up, up.”

“Remember,” says Mr Catton to Year 5 at Radwinter primary school, “what I told you about number and mathematics. If you’re having fun with numbers then you understand mathematics. Sadly, if we are finding sums hard, then that means we haven’t really understood maths. Maths is fun.”

A hand waves in the air.

“Yes, Sam?”

“There’s nines!”

“Sam’s quite right, there’s a vertical row of nines.” Mr Catton hands out small versions of the white cylinder. “I want you to play with the numbers drum. Find out what it can do.”

Excitedly, Duncan points out a spiral of numbers to his neighbour. They start plotting their results as Melvyn Catton reminds the class: “I’m not very good at tables. I need to learn my tables because I can’t play with numbers if I don’t learn my tables. Right?” Some pairs of children work with superimposing tracing paper over the “Numdrum” and marking three numbers which add up to 10 or 100. Meanwhile, Duncan and his friend are crestfallen. Their spiral pattern has hit a snag. But Melvyn Catton explains that patterns do not always work out:

“Sometimes, finding out why is just as important.”

He explains: “I only recently got the Numdrum and I still haven’t found out all the things it can do. One obvious thing is the big drift across ability: you can do simple up and across relationships, but you can also do open-ended investigations. It is giving children who don’t see number patterns strategies for getting there. ‘There’s nine,’ says Sam. OK, he may not be able to tell me why, but he’s found it.”

The class is bursting with discovery. Lauren has been using a tracing paper cursor over the drum to find two numbers which added together make a third. She has found a rule for th cursor: “14 and 33 make 47, not 44 and 33, because you go up two when you move round one for any number.”

So, how can a simple plastic tube with a number grid wrapped round it add up to a prime tool for mathematical investigation? John Harrison, inventor of the Numdrum, is a Cambridge mechanical sciences graduate who had a sucessful career in engineering. In retirement, he has been coaching children in maths and physics up to GCSE. “I was teaching this nine-year-old boy who was very behind and found it miserable. I made a number square, then screwed it up in frustration. Then I had a thought. I unrolled it and stuck it round a cardboard tube. I could point out number relations: 28 to 43 was five along and up one level (a decade). A big smile appeared on my pupil’s face. He was about to add up on his fingers, but he grabbed the drum and counted round. Then he stopped, traced a spiral with his fingers and said, ‘Oh, look, there’s the table of 11.’ He took one home.”

Mr Harrison gave two number drums to his grandchildren, who said: “Why don’t you show our headmaster?”, who is Melvyn Catton. He immediately fastened on the idea of patterns, and, he says, “we started to see more and more of them.”

What John Harrison at first saw as a way of simplifying two-figure addition and subtraction, a kind of rolling number square where children could easily cross decade boundaries, turned out to be a revolutionary new tool for exploring number.

“Children need something more visual than a piece of paper, but without too many distractions, a visual representation of a number system,” says Mr Catton.

That was in 1999 and Mr Harrison started production on the kitchen table. Tours of MathFests during Maths Year 2000 raised enthusiastic responses among children. “They would grab it and shout ‘Look, look, what it does!’,” says Mr Harrison.

Teachers keep finding more uses for it, and John Harrison, teaching more and more demonstration lessons, keeps developing more cursors. “I made the first when an old friend said, “Interesting, but why hasn’t it got a cursor? You could get lost.”

The latest development is an “Ages” cursor, based on a family of six, from grandpa to small boy. Obviously, as in the real world people age at the same rate, so the ages will increase together, which gives simple exercises, such as moving the cursor up one level to show everyone’s ages after 10 years.

More complicated sums arise with questions such as, “How old will Grandpa be when Mum and Dad together make 101?” or, “Make a cursor for your own family out of blank tracing paper. How old will your mum be when you are 16?”