Next week Kathryn Blair will be among the new pupils at the all-girls Sacred Heart High School in Hammersmith, west London. The Prime Minister and his wife, Cherie, chose the school for her, secure in the knowledge that it lives up to the reputation of girls' schools for high academic standards. At GCSE it is one of the highest scoring state schools in the capital. Last year 69 per cent of its pupils achieved five or more top grades. The previous year the figure was 76 per cent, when it was one of the country's most improved state schools. The Blairs also considered another all-girls school for Kathryn, Lady Margaret in Parsons Green, Fulham. It does even better than the Sacred Heart in the league tables.
The two schools typify the academic success of all-girls schools, emphasised by examination league tables. Each year girls schools top the tables, particularly at GCSE. But if there appear to be sound academic reasons for choosing single-sex education for a child of the millennium, in other ways the leader of New Labour is being thoroughly old-fashioned in his choice.
For single-sex education is in decline. Most of the schools created after the 1944 Education Act were single-sex. Few questioned the need for segregation since the sexes were being prepared for different roles. Boys were being educated for work and girls to be wives and mothers and work in the home.
The comprehensive revolution in the Sixties brought dramatic changes. Many supporters of comprehensives believed that segregation by race, religion, ability and sex were all equally wrong, and research began to stress the benefits of coeducation. By 1996, there were only 227 single- sex girls schools and 197 single-sex boys schools in the state sector.
A growing number of independent schools also became coeducational during the Seventies and Eighties. Even more took girls into the sixth form. There are now 208 members of the Girls' Schools Association of fee-paying secondary schools compared with 254 a decade ago. Parents, too, have been converted to the belief that the mixing of the sexes during adolescence is "natural", although those with sons are more enthusiastic about co- education than those with daughters. Two years ago, a survey of parents found that only a third of those educated at single-sex schools wanted the same for their children.
The decline in all-girls schools may not yet be over. Ministers' decision to allow parents to ballot on the future of grammar schools may pose a threat. No one yet knows how many parents will vote for the end of grammar schools in their area. Many of the most successful single-sex state schools are grammars. Among independents, boys' schools are likely to continue to expand or maintain their market share by admitting girls.
Girls' schools have used their league table superiority to defend themselves fiercely. They say that girls are more likely to take maths and science in single-sex schools, and that their confidence soars.
But are they right? One of the largest and most influential studies of the decade carried out by Prof Alan Smithers and Dr Pamela Robinson, then at Brunel University, for co-educational private schools concluded that parents should not be swayed by single-sex schools' claims that they produce the best exam results. They topped the league tables, Prof Smithers said, because they tended to attract the brightest pupils, not because of segregation.
Girls' schools tend to be single-sex for historical reasons. They do well because, like the best football teams, they can attract talent from a wider area than their less well-known coeducational counterparts. Earlier this month, research commissioned by the Association of Maintained Girls' Schools and published by London University's Institute of Education suggested that all the studies in the last 20 years pointed in the same direction: what mattered was the intake of girls' schools, not the fact that they were single-sex.
Prof Smithers says: "There are lots of excellent single-sex schools but their excellence derives from getting together a lot of bright girls who then spark off each other and who are a joy to teach. If you put boys and girls who were equally bright together, then you would get the same kind of performance."
The position of girls' schools in the league tables is bolstered even further, he adds, by the fact that overall girls do better than boys at GCSE. At A-level, boys are beginning to catch up: although more girls pass, boys tend to get more A grades.
Girls' school heads, however, are adamant that, academically, they do make a difference. Jackie Lang, head of Walthamstow Hall in Sevenoaks, Kent, says: "You do not get stereotyping in girls' schools. Statistics show that girls are more likely to go for science and maths if they are in mixed classes.
"If a 13-year-old girl is in classes where boys are saying 'what do girls know about physics?', then sometimes she will believe them. In a girls' school you are more likely to feel comfortable with things that have traditionally been a male domain."
The counter-argument, says Prof Smithers, is that girls who are segregated from boys may choose science subjects because they are able and not because they are in a single-sex school.
But if Kathryn Blair is no more likely to become a nuclear physicist by attending the Sacred Heart than by going to the mixed comprehensive down the road, surely there are other advantages? Mrs Lang has no doubts. "The advantages in all-girls schools are far greater than the academic ones. It is more a question of confidence. For almost the whole of human history men have been dominant: it has been a man's world. A girls' school gives you a microcosm of a woman's world where you can get the kind of confidence that men have had for all that time. It is a crash course in being top dog."
An Australian research project found that girls in mixed schools were much more likely than boys to rank themselves in the bottom half of the class. In all-girls schools they were as likely as boys to put themselves among the high-fliers.
Prof Smithers, however, is sceptical about arguments that boys dominate mixed classes and undermine girls' confidence. "If that is the case, why are girls doing so much better in exams than boys?" he asks.
His study of life after school suggests that pupils were divided about the effect on their self-esteem of being educated together. While some girls from mixed schools said that boys dominated lessons, others argued that the boys "didn't get a look-in". One girl complained that all-girls schools were "a bit bitchy" with hysterical competition to do best in even the smallest spelling test.
If the champions of girls-only schools emphasise confidence, those in the co-education corner argue that educating boys and girls together is "natural" and that pupils from mixed schools find it easier to adjust to life at university than those who have led a supposedly sheltered existence in single-sex schools.
That, too, turns out to be a myth. Prof Smithers also found the ability to cope with the first year of university was determined by personality rather than by type of school.
Mrs Lang says: "The atmosphere in a coeducational school isn't particularly natural. Until they become couples the boys tend to keep themselves to themselves, and so do the girls. Gone are the days when girls' schools were ivory-towered convents."
Indeed, all the research evidence suggests that the Blairs' decision to pick single-sex education will make little difference. But that is not a reason for saying they should cease to exist. Prof Smithers believes that they should continue for a reason that the Prime Minister would approve of: parental choice. "Parents should follow their instincts. If they have worries about social adjustment, they may opt for co-education. Some parents with shy daughters may want the gentle and less threatening environment offered by the less academic girls' schools. Others will want single-sex education for religious reasons. It is important that parents who want this type of education should be able to find it."