Nottinghamshire County Council, which opposes the Government's assessment system, has a policy of not applying for money to help pay for supply cover during national tests. As a result last year's key stage 1 tests took place in only a tenth of the county's 480 primary schools.
"There is a feeling in Nottinghamshire that [the national tests] are bad and that anything we can do to frustrate them will be beneficial to the children, " National Union of Teachers branch secretary Brian Helliwell said.
"If there is one child with special needs in the class, you need supply cover to carry out the tests. We are calling on our members not to do the tests whenever possible, and we have been very pleased with the response."
The education authority's own "value-added" system of measuring secondary schools' performance will soon be extended to primaries. The council hopes it will be adopted nationwide if Labour wins the general election.
Fred Riddell, education committee chairman, said: "By the next election we will have a methodology which we will be able to offer to David Blunkett with a view to replacing the panoply of tests and statistics and league tables. "
The LEA took legal advice on its decision not to apply for grants for supply cover, Mr Riddell said. "We were very careful not to be wrong-footed as a loony-Left council which was refusing to carry out the law. The Government invited authorities to bid for funds, and we declined. I am quite sure that if they could fault us legally, they would do so."