Pate’s grammar school in Cheltenham is in an odd position: a super-selective, super-successful school in an area where social conditions conspire against success.

It is on the wrong side of town, far from the Regency terraces, where two large housing estates are linked by an urban renewal area. Deprivation is high.

Pate’s A-level results put it among the top five state schools in the country. It gets 750 applications for its 120 places every year. But very few get in from the 13 local primary schools. Last year, none did.

“We felt isolated here,” says head Richard Kemp, who came to the school from Aylesbury grammar in 2000. To bring the school back in touch with its neighbourhood - and boost the chances of local children - Pate’s set up a curriculum enrichment programme for bright local primary children. With help from the Sutton Trust (its chairman, Peter Lampl, is a former pupil), the pound;40,000-a-year programme is now in full swing. Every week, 65 bright Year 5 children from local primaries come to the school in groups of up to 15.

Sharon Johnson, the smiling, energetic former deputy head who runs the project, collects the boys and girls in a minibus, takes the two-hour sessions and drives them back to their schools. The nine and 10-year-olds arrive and head straight for “their” classroom, a large, bright room refurbished for them. Last term, they produced a magazine on the subject of homelessness. This term, they are going to produce a video about what it is like to be a citizen of Cheltenham. They enjoy it all so much that it is heart-breaking to think that quite possibly none of the 10 bright children will get into the school.

That is one of the reasons why Sarah Whittick, head of St Thomas More primary, is one of three local heads who have so far declined to join the scheme.

“I believe all children are equal and I didn’t really want to say some are better than others,” she says. “And Pate’s is really, really selective - I thought parents could feel a lot of resentment when their children didn’t get in.”

But she is enthusiastic about another “good neighbour” scheme run by Pate’s, where a music teacher from the school comes to St Thomas More every two weeks.

Philip Tate, head of Whaddon primary, says the afternoons at Pate’s have worked well for his pupils. “It’s given them an opportunity to be stretched with other children of similar ability,” he said.

Local secondaries were also broadly in favour of the scheme.

“We don’t need to scout for talent,” says Mr Kemp. “We already have so many applications we can hardly cope. But the raison d’etre of a grammar school is to provide a particular type of education for able pupils, whatever their background.”