Experts looking at testing in England's schools have said formal Sats tests might eventually be replaced by teachers' assessments of their pupils.
Ministers have agreed to the group's immediate findings, which include scrapping science Sats taken by 10 and 11-year-olds from next year.
Instead, teachers will assess pupils. But English and maths tests stay, Schools Secretary Ed Balls told MPs.
The tests will be moved back next year from May to June.
The experts endorse new, single level tests for English and maths - but say they need more piloting.
Key Stage 2 science tests discontinued and replaced with teacher assessment
proposed School Report Card should be developed urgently as league tables alternative
Key Stage 2 English and maths tests in June not May
extend trial of Single Level Tests as a potential alternative
might better teacher assessment permit a move away from externally marked national tests?
smooth primary to secondary transition with an extended project across both and primary "graduation" certificates
annual sampling of about 10,000 14-year-olds to monitor standards
They are seen as "a potential alternative to the current tests" - including being used in school league tables.
A statement from the group also said: "The government should continue to invest in, strengthen and monitor the reliability of teacher assessment to judge whether a move away from externally marked national tests might be viable at a future date".
At a news conference, members of the group of educationists and head teachers were asked repeatedly whether they stood by this statement and to clarify what it meant.
They said they did stand by it.
One, Jim Rose - author of last week's report on the primary curriculum - said: "If you had a situation where teacher assessment was so robust you were confident that the information it was delivering was as good as or better than national tests, my God wouldn't you go for it?"
Another, primary head teacher Gill Mills, said schools were using tests anyway so she did not think there would ever be an "either/or" situation.
But Ms Mills suggested the Key Stage 1 tests in reading, writing and maths taken by children aged six and seven provided a model.
Administered informally as part of lessons when teachers choose, and marked by teachers, the results are moderated externally but not published.
She said: "That's a good example of what's happening to move towards."
But the government insists this is not likely soon.
Schools Minister Jim Knight said Key stages 1 tests involved a high workload for teachers - and there was "a reliability issue".
"We don't think moderated teacher assessment is anywhere near good enough for us to use as an accountability measure."
The changes that have been made are unlikely to stop a planned ballot on a boycott of next year's Sats by two education unions.
The National Association of Head Teachers (NAHT) and the National Union of Teachers (NUT) oppose Sats because they say teachers are obliged to "teach to the test".
They argue that the focus on the Sats subjects is to the detriment of the wider curriculum and they oppose schools' results being published in league tables.
The NAHT's general secretary Mick Brookes said dropping the science test would make the curriculum more narrow.
"Clearly if you take out the bits being tested, you are going to narrow the curriculum even more."
The union would not be scrapping its plans for a ballot on Sats, he said.
The government has warned heads they have a statutory duty to administer the tests, and a boycott would be illegal.
The National Governors Association, which generally liked the report, said the proposal that Sats be conducted later in the summer term needed to be carefully examined, as it could have unintended consequences in schools with programmes of creative or out-of-classroom activities.
The expert group also backed the government's proposed "report cards" which would grade schools on behaviour and children's wellbeing, as well as test results and Ofsted reports.
Relying on the assessments teachers already make of their children's progress, as will now happen in science, is the pattern already established in Wales.
All the tests for 14-year-olds in England have been scrapped already following last year's marking fiasco.
But as a check on national standards, a random sample of 10,000 pupils will sit tests each year - as is done in Scotland.
Next year in a few hundred pilot schools, single level tests will replace Sats in maths, though the government is moving cautiously in introducing the new tests.
With the existing Sats, about to be taken by more than 580,000 primary school children in England, the questions cover a range of abilities so children can be scored at different levels.
In the new single level tests being piloted in English reading, English writing and mathematics, the questions are all pitched at a certain level of understanding.
Children are entered for the relevant level when their teachers think they are ready. So they take it with others of the same ability level rather than the same age - rather like a piano exam.
Liberal Democrat spokesman David Laws welcomed the proposed changes, though would go further and have more teacher assessment in English.
But shadow children's secretary Michael Gove said parents wanted clear, rigorous and transparent testing at the end of primary school.
He added: "By declining to stand up to outside pressure and retreating on the principle of external assessment I fear the secretary of state has failed the test of ensuring he defends what is best for our children."
The science community is pleased the science test is to end.
The chief executive of the Association for Science Education, Annette Smith, said it did not assess science as it happened in primary schools - as an active, practical activity which built on children's curiosity and observation skills.
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