The stakes for schools are high. Once established, the 11-year-old tests in English, maths and science will be used to measure primary schools' performance in exactly the same way that GCSE results measure secondary schools. Already there are complaints from some schools that others have cheated, that children have been primed in advance after teachers looked at the questions and that some have been given help or extra time.
But teachers are also raising fundamental questions about the tests themselves. They say that the standard varies from subject to subject, that marking is inconsistent even within subjects, and that there is too big a range of marks within a single grade (level is the technical term). Some schools complain that marking for English has been too tough.
Fee-paying schools are as concerned as state schools and say they will be making representations to government advisers at the School Curriculum and Assessment Authority. They believe maths, described by one school inspector as "like a Common Entrance paper [for public schools] 30 years ago" was much harder than English or science.
Robin Peverett, director of education for the Incorporated Association of Prep Schools, said prep schools welcomed the fact that the maths had been challenging but said: "A number of questions used difficult language. As a general rule it might have seemed that it was more a test of English than of maths."
Mike Russell, head of Malmesbury school, a state primary in Tower Hamlets, London, is critical of the way the test has undermined children's confidence and their feelings about the subject. "Most of the children, including the clever ones, walked out of that test feeling that they were no good at maths. They felt the test was an exposure of what they can't do rather than a test of what they can."
Even the prep schools, whose general complaint is that the tests are too easy, admit that the extension paper sat by only the ablest candidates was very tough. Only a small proportion are expected to reach the required level, and those who do are supposed to be at the same standard as an average 14-year-old. But are they?
Teachers are highly sceptical that standards are the same at the ages of seven, 11 and 14 when national curriculum tests are taken. At each stage each child is placed on a level from 1 to 8. For 11-year-olds level 6 is the highest for which papers are set. Most children are expected to be a level 4 with the less able at 3 and the more able at 5.
So what will the levels mean? Is a child who gets a level 5 or 6 at the age of 11 performing at the same level as a 13-year-old? Dr Peverett says: "It would be nice to be able to say to parents that a child who does very well at 11 is well on the way to GCSE. But it isn't true." John Kenwood, head of Bourne School, Eastbourne, said: "If you say to a parent a child is level 4, they will want to relate it to how the child did in the national tests at seven. And it won't relate." That is because, teachers say, children achieving a level 5 at 14 will have covered many more topics than those getting level 5 at 11. And their answers will reflect their maturity.
Schools also point to the wide range of marks within each level. Brian Bissell, chair of the Incorporated Association of Prep Schools' education committee and head of the Blue Coat School in Birmingham, pointed out that the range of marks for a level 4 is very wide. In English at his school the range was between 47 and 68. "We think that needs subdividing." Since each level is meant to cover two years' work, that would mean overhauling the entire system.
All schools are worried by the reports of cheating, so far unproven, which are being investigated by the School Curriculum and Assessment Authority. Mr Bissell believes the tests "have the potential to provide a very good service for children, teachers and parents. But they need further adjustment to their administration and marking."
Quite simply, he says, if they are to be seen to be fair, the papers must not be opened until the day of the test, as with GCSE examinations. This year, schools were allowed to open them a week in advance to check they were in good order.
On marking, he believes all elements of one subject should be marked by the same examiner to give a fairer picture of what children can do. As it is, spelling and handwriting were marked by one person and writing by another. "Children have lost marks in the handwriting tests because they have reverted to a stilted form of handwriting. The writing in their essays has much more character."
The School Curriculum and Assessment Authority is keeping the tests under review. Officials say the complaints that maths was harder than science and English have been taken into account in the marks, which have been adjusted.
They contest the view that a level 5 at 11 is not the same as at level 5 at 14 and point to a series of studies which they have done in maths and science. In English, because 14-year-olds do a Shakespeare paper and 11-year-olds do not, it is more difficult to establish comparability.
A spokeswoman said: "We are fairly happy with the structure of the tests. We recognise that schools are still on a learning curve. Tests like these have not been part of the primary school culture."