One day last May, Gillian duCharme, headmistress of Benenden School, the Pounds 14,000-a-year independent girls' school, drove out of her school's 250 acres of Kent parkland to spend a week supply teaching at a comprehensive in east London. The result is a gripping television documentary to be screened on BBC2 next week, painful in its exposure of the gulf between these two worlds, compelling in its scrutiny of classroom survival.
Viewers will acknowledge her bravery in putting herself on the line like this, but may come to wonder, as her week at Forest Gate Community School in the London borough of Newham progresses, whether it isn't something more akin to foolhardiness, or even innocence about the true extent of the divide she has stepped across.
"I'm curious," she says. "That's why I did it. And I like a challenge. I thought it would be a very, very good learning experience, and how often do you get to go inside a school for more than just a half-day visit? Especially a maintained school? Never. Also I don't mind making mistakes. Some people do, but I don't. After all, it's when you make the most mistakes that you're learning the fastest."
So what did she learn? "I don't think I came away with any elevated thoughts - I was far too exhausted."
In fact she learned that she would need to adapt her classroom style fast to survive at Forest Gate. She would need to be tougher on the dreaded mixed-ability 9T, and talk less and listen more if she was to meet all the needs in her streamed, but still very varied, French class. She learned that money in the Pounds 1,600-per-head-per-year school is "quite tight" and not nearly enough to provide good opportunities. "They go swimming twice a term, it nearly broke my heart. We have our swimming pool right there." And she learned that there were things that went on at the school that she questioned fundamentally.
One was the leadership style of head Andy Richardson, who keeps a tight, managerial grip on his school. "The place runs very well. You can feel the calm and order as soon as you walk in there. But I thought at the time, and I still think, that he should get out more and do some teaching."
Another is the role of the support teachers, there to help statemented pupils cope in mainstream classes. "But as far as I was concerned, most of them were sitting there doing nothing. Talk about an easy ride. I feel quite strongly about this. I just don't understand it. There must be support teachers who are very supportive, I may have just not struck lucky, but I thought: if you haven't got any money, why waste it on that? Why not take the students out of the classroom, if they can't cope, at least for some periods every week and give them extra help, and then put them back in the classroom?
All this business about you can't take them out of the classroom is rubbish. No one's going to persuade me that it's a good system in the way I saw it."
Which leads on to the thing she learned most from her week: to prize anew her own school's independence. "We wouldn't dream of having anything like that where I work, because we make our own decisions. We might try it, but we wouldn't carry on with it. And that's the thing I now value even more than I did, because if you've got a good idea and there's money in the budget, you can try it, and if it doesn't work, you can ditch it and try something else. "
The film is just one in a series about top people going back to their grassroots, but in this case the fit is tenuous. Gillian duCharme has never held any sort of permanent teaching job in England, let alone one in east London. A down-to-earth ex-grammar school girl from Kent, not at all the grande dame her title might suggest, she taught in the United States for 19 years before becoming head of Benenden, and retains both the instinctive brisk egalitarianism of America - of course all children are equal, and in an ideal world, they would get equal life chances - and the outsider's detatchment to what goes on here.
"You see, one of the great sadnesses of this country, to me, is the destructive envy. I can utterly understand that if you are in a situation where you have very little - and I have been in the situtation, I promise you, where I haven't even had the bus fare at one point in my adult life - you feel like that. But envy doesn't get you anywhere. The best thing to do is to see what you can do to improve matters."
On the programme, she ponders whether this could be through people paying more taxes for a better education system. In practice, it is more likely to boil down to seeing if some of the Forest Gate kids might be able to come and use Benenden's pool sometime: "not to be patronising, just because I think they'd have fun."
Out of term, she lives in Greenwich, in south London, which may explain why she found the conditions at Forest Gate unsurprising. "It wasn't a shocking experience in any way. I liked the kids, I really did, and I felt the work was generally good, although I didn't see much of the written work so I can't gauge it that well, and yes, I felt they could achieve their potential, although obviously the more opportunities you can give children, the better. One of the big challenges there is that some of them are not very good at English, and I know that literacy is one of Andy's main concerns."
Since the programme was filmed, Andy Richardson and one of his deputies have been to Bendenden to talk to the school's staff, and both sides continue to speak well, if carefully, of each other.
"Gillian is a nice person," says Andy Richardson. "I liked her, and she weathered the storms that came her way. Obviously it was all a bit of a contrived situation, having a film crew following a supply teacher around. "
But differences remain.
"What I couldn't get Andy and co to believe is that nobody shoves out money to independent schools. You're running a business, if you're in my job. I have to get students or we close down. We've got no endowment, not a penny. If you compare us to somewhere like Eton, we've got nothing. Yes, we might have more money coming in per capita, but we've also got to have fancier facilities, the parents demand it, and it's Pounds 50,000 every time you put a spade in the ground. We've spent Pounds 15 million over the past 12 years but we argue about every single twopence."
Andy Richardson says: "They've got lovely facilities. I said to her, when we were down there, 'do you realise my whole school would fit in your car park?'"
Back to the Floor: A Class Apart is on BBC2 on Tuesday, November 4 at 9.30 p.m.
* Television and radio, page 26