The controversial tests for seven and 11 year olds face a shake-up, under plans announced by the Education Secretary, Charles Clarke.
There will be a more light-touch approach to testing
The impact of school tests for seven year olds in England is to be reduced, amid concerns from parents and teachers that they cause too much stress.
Mr Clarke, speaking in London, promised a more creative primary school curriculum, in which the significance of formal testing will be downgraded.
From next year, greater emphasis will be given to teachers' assessments, with tests in English and maths becoming part of a broader measure of pupil performance.
The national targets for test results for 11 year olds will be postponed for two years - with schools allowed to set their own targets.
Mr Clarke, whose predecessor Estelle Morris resigned when schools missed these targets, denied the changes signal a government climbdown after protests by teachers and parents.
And he sent a strong message that the principles of testing, targets and league tables would remain.
"I refuse to return to a school system that fails its children through a lack of public accountability and proper monitoring. That is anathema to progress," said Mr Clarke.
Testing was an important way of finding at early stage where improvements needed to be made, he said. Among pupils who fail their tests at the age of 11, only 12% later achieve the benchmark of five good GCSEs.
The more relaxed testing regime will see a greater emphasis on assessments by teachers - which run alongside the externally-marked tests. The two measurements will be integrated, with the teacher assessment "the key overall judgement," said Mr Clarke.
The tests at seven have provoked some of the fiercest criticism, with claims that they put an undue amount of stress on young children.
Teachers, who have threatened to boycott tests, argue that they add little information to their own professional assessments - and that too much school time is given to preparing for the tests.
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The government has always defended its system of tests and targets as being central to its efforts to raise standards.
And in evidence, it has pointed to this year's results of an international league table of literacy skills, which shows English pupils among the best in world, with only Sweden and the Netherlands achieving a higher standard.
In the tests for seven year olds, children are examined in English and maths - in areas such as spelling and handwriting - and 11 year olds also take tests in science.
The Conservative education spokesperson, Damian Green, said that the government should be consistent and abolish other national test targets.
"I have called for all these arbitrary national targets to be scrapped. If Charles Clarke took that advice he would make a significant difference to our schools," said Mr Green.
And he attacked the changes to targets as a "cynical exercise in disguising failure".
The Liberal Democrat education spokesperson, Phil Willis, said that the changes represented a "partial u-turn", but they failed to sufficiently reduce the burden of the testing and targets system.
John Bangs, head of education at the National Union of Teachers, said the changes did not go far enough. "[The government] has recognised the logic of the argument against the tests but lacks the courage to abandon them.
"Any suggestion that the restructured tests are a valid assessment of pupils
or teachers is a nonsense."