The predictions are aggregated from targets sent into the Department for Education and Skills by every headteacher in England. The Government wanted 85 per cent of pupils to reach level 4 in both tests by 2006 and for the results to be maintained to 2008. Instead, 79 per cent made the grade in English last year and heads believe it will remain the same this summer. In maths, they believe the proportion could increase from 76 to 80 per cent, although some past predictions have been over-optimistic.
Christine Gilbert, Ofsted's chief inspector, told authorities in November that too many primary schools had "unacceptably low" expectations.
John Gawthorpe, head of Mayhill junior, in Odiham, Hampshire, said:
"Probably heads think it is easier to improve maths scores than writing.
Writing is the ball and chain that is holding English scores back.
"I think the targets are more realistic now heads are looking at how their cohorts are doing. Before we were all in some ministerial wonderland where we would strive to reach their targets. I don't know any head who would lose a second's sleep if the government misses its target. It has no grounding in reality."
Predictions for 14-year-olds are also less than rosy. Although heads expect the proportion reaching level 5 in English to rise from 72 per cent to 74 per cent, they think maths scores will drop from 77 per cent in 2006 to 74 per cent while science scores will remain steady at 72 per cent.
But 16-year-olds should do slightly better. Predictions are for 60 per cent of pupils to gain five good GCSEs, up from 59 per cent. Mick Brookes, general secretary of the National Association of Head Teachers, said:
"There is no doubt that having aspirational expectations of children is where we need to be, but it needs to be in context. The people who know the children best are teachers and other people working with them. They can base targets on a child's experience and what can be done to encourage them to move on."
Trials of new shorter tests for 11 and 14-year-olds are due to begin in 10 local authorities in September. Pupils can take them as soon as they are ready.
In Portsmouth, heads are hoping for a rise on last year's maths test scores of 67 per cent. Margaret Griffin, head of Lyndhurst junior, has set her target three points higher. She said: "We are an optimistic band. Maths is a real priority in all schools this year, there has been a lot of training for teachers."
Results for children in care in primary and secondary schools are expected to rise slightly, but will remain well below the national average and government targets.
The Government set a target in 2002 that, by 2006, 60 per cent of 11-year-olds in care would reach level 4 in English and maths and 24 per cent five good GCSEs.
Information on last year's results is not available but it is believed the targets were not achieved, and they are predicted to be missed again this year. Local authorities predict 55 per cent of 11-year-olds in care will reach level 4 and 17 per cent will get five good GCSEs.