Kids come to school with an intellectual starter kit," says Brian Butterworth, neuroscientist and dyscalculia expert. "In the starter kit for maths are language, memory, reasoning and so on. These are important for any school subject. But for working with numbers, you also need a special number tool. If you are missing that, maths will be difficult. If you don't have a screwdriver in a real toolbox, you can sometimes make do with pliers. But some things you can't do, and others take a long time, and nothing is efficient."
FACTS AND FIGURES
Children have had trouble with maths since education began. In the 1980s, the Cockroft report revealed a seven year difference between the most and least mathematically competent children in classes of 11-year-olds. Surveys across the world, including the supposedly more numerate Pacific Rim, show that at least one child - and up to three - in each class of 30 will be a long way behind.
Girls and boys are equally affected. Dyslexics are more likely to have difficulties with maths, and poor numeracy runs in families. However, it may not necessarily be biologically inherited: learned helplessness can be picked up at home.
Until recently, no label existed for trouble with maths. Now one does. But agreement on what the word dyscalculia means is hard to find. "I don't know what dyscalculia is," says Tandi Clausen-May, principal research officer at the National Foundation for Educational Research. "Different definitions seem to measure different things. If we are not careful, we will end up with a definition which says the children with dyscalculia are the ones who do badly in dyscalculia tests - which isn't a definition at all."
GENE OR MACHINE?
At the heart of this uncertainty about labels is a bigger uncertainty about what is actually happening inside children's heads when they do maths. Two scientific theories hypothesise two different processes. Simplified, the first theory says we all possess a "number gene". It determines that one special bit of our brains, probably the left parietal lobe, is dedicated to counting things - an ability we share with babies, apes and birds. Adults with damage to this part of the brain lose their ability to understand numbers. Children with dyscalculia, the theory goes, never have it functioning effectively at all.
This view is put strongly by Brian Butterworth of University College, London. "I'm a neuroscientist. I've seen patients where all of their brain is shot to pieces except this bit, and they can still do calculations. I've seen very bright young women, at top universities, who cannot do any maths.
It's not a memory thing. It's a special bit of the brain made for numbers, perhaps a very orderly bit, so we put very orderly things in there. When people can't do numbers, that bit of their brains isn't working properly."
It seems convincing. But, say other academics, most children who fail in maths have a whole range of difficulties, and hardly any of them, however bad they are, have no understanding of numbers at all.
"My research suggests there is no such thing as a single mathematical or arithmetical ability," says Ann Dowker, an experimental psychologist at Oxford university. "There's a very wide range of things that can go wrong with maths, and just as many explanations."
LOOKING AT INDIVIDUALS
Yet experts across these views agree on the starting point for helping these children make progress in maths: it is for teachers to look in detail at the child's mathematical make-up.
Screening tests and diagnostic materials (see resources, below) are important tools. Teachers need to work one-to-one with children and ask themselves: how do the children approach estimating? Can they see patterns? How effective is their long-term memory for addition and multiplication facts? How effective is their short-term memory: can they hold a sum in their head, for example, long enough to copy it down from the board? Can they cope with the words in word problems? Do they have spatial ability - recognising patterns of dots on a dice without counting them? Do they understand symbols: that the written number four means four things?
Some children, says Professor Butterworth, "don't have an intuitive sense of threeness, or fourness, or fiveness. If you show them four things, they have to count them to know that they are four."
Even if they know what four means, they may not be able to grasp that it comprises two twos, or half of eight, says Julie Kay, head of Mark college in Somerset, the only DfES Beacon school for teaching maths to children with dyslexia. "Dyscalculic children can't see how numbers relate to each other. They tend to see lots of disjointed facts, think, 'There's no way I'm going to be able to learn all that,' and give up".
"Tests are quick and easy, but they are no substitute for the teacher talking to the child and assessing what they can do and like to do," says Tandi Clausen-May. "I want to identify strengths, not weaknesses. You start with what the child can do and work from there."
If at all possible, experts agree, these children need one-to-one support because their difficulties are so variable and idiosyncratic. If they are in small groups they need work tailor-made for their level of understanding and approach. They also need an adult who can focus on ways of working, not on following a script or getting right answers.
They will inevitably work very slowly and need frequent repetition: topics apparently covered but then left for half a term will disappear from the child's mind without trace, says Julie Kay. Every new topic needs to be intermeshed with those that have gone before. "Children have to begin to see the relationships between numbers, between the fact that 10 and five are related to 50 and 100, and that halving and doubling are part of all of them."
"Never underestimate, even in secondary school, how far back in maths you have to go," says Steve Chinn, maths consultant and former head of Mark college. Children who cannot grasp the algebraic principle that 3x + 6x = 9x may in fact never have grasped the concept of nine times anything.
These children can and do find ways around their difficulties. They may be unorthodox and slow, says Steve Chinn, but they are better than nothing.
"If you don't know the answer to 8+7, you may be able to understand that eight is bigger than seven and count on from there. Or start with two sevens. Or two eights. I've just assessed a little boy who hadn't been told you can count on to do subtraction. His counting back is hopeless, but his counting on is superb. So, for 15-8, once he knew he could count on from eight to 15 he was fine."
Chinn thinks children need to become problem solvers. "You tell them, 'If you can't do it this way, do it another way. For example to add nine, lots of people just can't add 10 and take one off. So you may have to count on nine, and you may have to do it on your fingers, and with luck you may eventually learn that if you add nine to something ending in six the next number will end in five. Or you may not. But you have to keep trying to teach them to link things, so ultimately they get a sense of what numbers are about. You aren't just teaching them how to get round a particular problem: you're teaching them a whole philosophy."
PILING UP PROBLEMS
Some difficulties dyscalculic children face may not be anything to do with numbers. If they have dyslexia or other language disabilities they may need extra help with mathematical language. "We teach maths like a foreign language in some ways," says Julie Kay. "What's a trapezium? It's not a word you find in any other context. Take away? For our students that means Indian or Chinese."
If children's short-term memory is poor, they need simple instructions and prepared sheets rather than work to copy from a book or a board. If their long-term memory for number facts is poor - and, as Brian Butterworth points out, we're all bad at remembering things we don't understand - then they need to work from what they can do, for example using fives and twos to replace the seven times table. They also need calculators and number grids so they can succeed in the branches of maths where number memory and even some number understanding is not essential.
Experts agree that many dyscalculic children are humiliated in maths lessons, which adds a hugely damaging overlay to their problems. But they should be told that even some mathematicians are poor at calculation, says Brian Butterworth. They manage by finding ways round their difficulties, which is a message all children should receive in the classroom. "The way the curriculum is structured, if you don't do well on the arithmetical side then you get discouraged, and your teachers don't encourage you because they think, 'If you can't even do arithmetic, what chance have you got with calculus, or algebra, or geometry, or topology?' But in fact you can off-load the calculating in these topics onto a computer.
"These kids suffer every day, because they have a maths lesson every day.
They suffer much more than dyslexics, because their condition isn't recognised and they think, and their parents think, and their teachers think, that the reason they can't do arithmetic is because they are stupid."
They stay thinking that way, too. Dr Dowker quotes a study of 37-year-olds in the 1990s by the adult Basic Skills Unit, which found a quarter of them were so weak or afraid of maths it handicapped their daily lives.
Clare Trott is maths support tutor at Loughborough university and founder of the Dyslexia and Dyscalculia Interest Group. She sees many people who struggle with maths when they come into her centre: psychology or sociology students unable to work with percentages or graphs, or trainee nurses, unable to calculate reduced drug doses for children. "My feeling is that people get through to 16, breath a huge sigh of relief, and then they find they do need maths and all the original difficulties surface again."
If anything is to change, says Steve Chinn, teachers need to protect children from maths shame as well as develop their understanding. "The 10-minute mental maths warm-up turns off a significant number of children.
So you say to a child: 'I'm going to ask you what 83+97 is, but first I'm going to ask you 80+90, so remember that because I'm going to come back to you.' You let them write it down. You don't put them under pressure. It's quite easy to change your objectives, so you don't humiliate the child and you still stimulate their brain."
* Dyscalculia Guidance and Dyscalculia Screener, Nfer-Nelson, www.nfer-nelson.co.uk
* Diagnostic Interviews in Number Sense, H Denvir and T Bibby, www.beam.co.uk
* The Mathematical Brain, Brian Butterworth, Macmillan
* The Trouble with Maths, Steve Chinn, Routledge Falmer
* Dyslexia and dyscalculia interest group: http:ddig.lboro.ac.uk
* Department for Education and Skills advice: www.standards.dfes.gov.ukprimaryfaqsinclusion56233