More than 4,700 schools are to pilot less stressful tests for seven-year-olds this summer.
The project offers schools greater leeway in administering tests to youngsters, but its main feature is that only teacher assessments will be reported to local education authorities.
Schools have welcomed the pilot, although some say they would prefer to have the option of using just teacher assessment without the tests. They fear that the pilot will not provide a true picture of teacher assessment as staff will look at test results before making judgments.
Research from Manchester university has shown that giving teachers the chance to see test results before making their assessments has a "considerable effect" on diminishing the extent of disagreement between the two.
Local authorities taking part in the project must devise their own systems to moderate teachers' assessments within national guidelines. A team at Leeds university will assess the methods used.
David Barlett, assessment co-ordinator with Birmingham council, where 284 of its 295 eligible schools are taking part, will start moderation visits this term.
He said: "We will observe, for example, a guided reading session and use that as a focus for talking to the teachers. Schools do not need to provide any evidence over and above what arises from their normal teaching. We will focus on three children in each school.
"The broad thrust of the pilot has been well received, but as an LEA we wanted to go further and make the tests and tasks optional, and have something more like the foundation-stage profile."
In Cornwall, all 240 of the authority's schools are taking part, and moderation visits will go ahead in June.
Julia Bond, the county's primary adviser, said: "We will make half-day or full-day visits to 25 per cent of schools to meet teachers and ask for samples of five children's work.
"Teachers feel positive about it because there is autonomy, a recognition that they are able to make these judgments."
Two LEAs, Northumberland and Blackburn with Darwen, have pulled out of the project, saying not enough of their schools wanted to take part. Peter Carpenter of Darras Hall first school in Northumberland said he was disappointed.
"I and my staff were keen to take part in the pilot as we don't feel the tests are particularly helpful," he said.
The National Union of Teachers and the National Association of Head Teachers said the trials should not have required that the tests and tasks be used.
The unions have told members that the pilot should mean less work. They also expect teachers' judgments to stand if there is any disagreement with the local authority.
John Bangs, head of education at the NUT, said: "We want to guard against LEAs imposing excessive and unrealistic assessment procedures on teachers."
More on testing next week in Wonder Years, a TES supplement on the primary strategy