Does one-to-one really add up?
The assumption that individual tuition is the perfect solution for primary pupils who struggle with maths is being challenged. Gordon Brown has pledged to spend pound;35 million a year on it, with other funding from schools' budgets. By 2011, at least 30,000 seven-year-olds should be receiving the extra tuition through a government intiative titled Every Child Counts.
But the details of how it will work are not yet finalised, and some academics question whether one-to-one tuition is the right approach for all pupils.
One local authority which is ahead of the game is Hackney in East London, where a scheme known as Numeracy Recovery originated. The borough has a team of nine specialist teachers assigned to its primary schools providing individual sessions.
It might seem unlikely that a child could reach the age of six without their teacher realising they cannot count. But Judy McGrail, numeracy recovery teacher at Southwold Primary in Hackney, said children become adept at covering up their misunderstandings.
She said: "A class teacher can't possibly know exactly where every child is. It's a very difficult task. We have a team of nine numeracy recovery teachers in Hackney and we have all found similar problems.
"Counting is one such problem. In class it will look as if a child can count because they are successfully chanting with all the other pupils, but they may have problems."
Tina, a neatly-turned out child with impeccable manners, had just such difficulties counting. The six-year-old used to mix up 13 and 30. Now she whizzes through her maths activities with Mrs McGrail.
In a session on reading and writing three-digit numbers, Tina is shown several methods. First she reads from a board: "100, 200, thirty hundred, forty hundred ..."
Judy corrects her and they carry on. They then work together placing plastic blocks on to a grid marked with different columns for hundreds, tens and units.
Mrs McGrail said: "This is what she'll be doing in class next term - she has caught up with the rest of the class."
Over the last three years, 412 children in Hackney have been singled out because they were not predicted to receive a level 2 by the end of key stage 1. After intervention, 83 per cent of them went on to reach the grade.
Kathy Secular, Hackney's lead tutor in numeracy recovery, said: "One-to-one is a practical decision on our part. We thought long and hard about individual teaching because of the economics - schools are keen to 'hit' four children instead of one. But I think it might look like children are making very similar errors, but the misconceptions behind those errors can be quite different."
However, not everyone agrees that individual tuition is the answer. Margaret Brown, professor of mathematics at King's College, London, said: "We did some research in the 1980s and found teachers got better results if they worked with small groups or three or four children than with a single child. With a single child you can target their problems more accurately, but the debate in small group helps pupils to make progress."
A small study by Michelle Parker, then at the University of Newcastle, looked into how teachers could get the best results with children working in pairs. Pupils played a computer game in which Zilt the pirate had to find buried treasure by walking a given number of feet along an empty number line which had only the end point labelled. Some children counted from the beginning. Others counted from the end point. But two pairs hit on the strategy of calculating which number would be in the middle and counting from there. The teacher got the children together to share ideas and afterwards only one pair - the least able - carried on using the longer methods.
The question of how the tuition should be provided is important because local authorities may have several systems to choose from. This contrasts significantly with the way literacy catch-up sessions were introduced.
Every Child a Reader is mainly the introduction nationally of the existing Reading Recovery scheme, but For Every Child Counts the Government commissioned a review by Sir Peter Williams, chairman of the Advisory Committee on Mathematics Education, which is due to present its findings in June. Rather than endorse a single programme he may take the same route as the review into phonics carried out by Sir Jim Rose, a former primary chief inspector. He recommended no single programme, but simply said that teachers choosing a phonics programme should ensure they meet ten core criteria. There are 21 phonics programmes on the standards website.
Maths programmes which are used as well as Numeracy Recovery include Maths Recovery, which was designed in Australia and introduced to the UK through Liverpool University, and Catch Up Numeracy, developed in Norfolk.
The Catch Up programme is based on the research of Professor Ann Dowker at Oxford University. In her study What Works for Children with Mathematical Difficulties, for the then Department for Education and Skills she found the four most common areas of difficulty were: memory for arithmetical facts, word problem solving, representation of place value and the ability to solve multi-step arithmetic problems. She also pointed out that children's grasp of maths varies very widely.
Dr Graham Sigley, head of Catch Up Numeracy, said: "Our programme is very individualised. Each child has a profile of learning needs created by the teaching assistant using standardised tests.
"It is one-to-one but we know a couple of schools where if they have children who have issues with the same component, such as word problems, at the same level, they have those children working together."
Professor Brown suggests teachers bear in mind their pupils' different strengths and weaknesses, and do not pin too great a set of expectations on a single scheme.
"Low attaining children can be moved forward by working in small groups or individually," she said. "Some will need just a bit of extra tuition, but sometimes it just takes a long time for people to take things on board. We keep looking for magic bullets, and we can improve undoubtedly, but we are never going to get miracles."
Maths for rats and monkeys
How numbers are learnt is a hot topic. Research has discovered that animals including rats, monkeys and lions, like human babies, all have some approximate idea of numbers.
Chimps can tell that two is larger than one, but animals can mostly only distinguish individual amounts of one, two and three. As numbers get larger, they find it harder to discriminate. A rat or pigeon could tell 45 is smaller than 50, but not the difference between 49 and 50.
While it seems there is evidence for this in-built sense of estimating magnitude, there is no agreement that a sense of number is innate.
Children can count fluently to 10 by the age of three, but one researcher spent several years teaching a chimp to count from 0 to 9. The difference between the two brains is the ability to learn language.
Last month, the Primary Review published a study of research into the science behind learning by Usha Goswami, of Cambridge University and Peter Bryant, of Oxford University.
They stated that learning to count is, at first, a routine much like a nursery rhyme. But at 3, children start to learn the idea that certain words refer to exact numbers. This knowledge, along with other number facts such as two and two making four, is stored in the language areas of the brain, making language development a prerequisite for learning not just a,b,c but 1,2,3.