Primary schoolchildren are getting full marks in maths at school despite not knowing how to multiply, add or subtract small numbers in their heads, experts say.
The national curriculum expects pupils to know how to do basic addition and subtraction using numbers up to 20 by the end of their third year in school.
However, academics from the Institute of Education tested 259 children in schools judged outstanding by Ofsted and found none of them knew all their basic number facts even though their maths was 'above average'.
The findings come as a separate report today warns that A-levels are failing to teach pupils enough maths to study physics and engineering at university.
It raises fresh concerns about the way students are being taught at A-level, with fears that they simply learn how to pass the exams rather than given an adequate grounding in the subject ahead of university study.
Richard Cowan, of the Economic and Social Research Council-funded project at the IOE, who led the primary school maths test research, said: 'Ignorance of number facts is not the barrier to success in mathematics it is often believed to be.'
During the tests Professor Cowan asked children to perform a series of sums within three seconds. Only 61 per cent knew more than half the sums and only 1 per cent knew all of them.
The children were also asked what strategy they used to solve each problem. Only 10 per cent relied on their knowledge of sums on most problems, and none did on all of them.
Professor Cowan said: 'The current national curriculum suggests children develop their knowledge of facts initially through counting, then by using principles until the facts are well established.
'It is a compromise between a traditional emphasis on knowledge of facts as a basis for success in maths and a progressive emphasis on understanding principles.
'Many people agree with the traditional view and think children should spend more time learning facts to become competent in arithmetic and progress in mathematics. This study does not support the traditional view.
'We are not saying that fact knowledge is irrelevant, just that it develops more slowly than the current national curriculum allows and that this does not jeopardize children's mathematics progress.'
He said: 'Facts help children grasp principles, and applying principles helps children learn facts.'
'If parents want to help their children, then they should encourage them to use their knowledge of principles to solve problems, not get them to memorise the answers to problems like little parrots.
'We agree with the current national curriculum that there is no point in teaching that 3+5=8 if the child doesn't know what that means and can't solve the problem by counting', he added.
The report into A-levels maths today says that students are left struggling with their degree courses because they lack a good understanding of maths, according to the study commissioned by the Institute of Physics (IOP).
It raises fresh concerns that students are simply being taught to pass exams at A-level, with pupils taught how to use equations 'like recipes', rather than understanding what they actually mean.
The study is based on a survey of 393 physics, engineering and computer science students and 40 UK academics, as well as a small number of interviews.
The findings show that more than half of the academics questioned said students were not well prepared to deal with the maths in their degree course.
The vast majority of academics said this lack of maths 'fluency' would be an obstacle to students achieving their potential in the future.
The study also found that almost half of the students questioned thought that the maths in their course was more difficult than they expected.
But nearly four fifths thought they were able to deal quite or very well with the mathematical content of their degree.
Some of the students questioned said that they were unaware that physics contained so much maths.
Overall, 91 per cent of academics said that students joining their degree course lacked maths knowledge.
The report concludes that many academics are concerned about teaching to the test at A-level.
A number of academics felt that mathematics was taught too heavily to exams at A-level - in so far as students are taught to learn various equations and techniques by rote in order to pass exams, rather than being taught how and why the equations have been developed.
The report found that many academics and youngsters believe students should be able to practise maths more, in the same way that a musician practises musical scales.
The study also raises concerns that there is a big gap between physics and maths A-levels, warning that they are treated as two separate subjects, when they should be closely linked.