When they see the New York skyline appearing in the distance, one of them gasps: "So it really exists!" Our leader is Nigel, a warm Northern former headmaster – and he should be described as a Leader with a capital L. From the moment we arrive, the group looks to him for guidance on everything.
Robert Wilne, LAE's energetic headmaster, says the school should be judged not on its success at Oxbridge entry but on the route its students took to get there.
He says of his interest in politics: "When I was in year 2, my headmaster told me I should be in politics, as I could worm my way out of any situation, so it's always been something I felt confident I could do.
Physical abuse: One of the children informed us that the headmaster says that they should be beaten 20 strokes on their backs as a form of punishment when they do something wrong.
We were going in aged 16 and talking to headmasters about why they should be using it.
City of London School headmaster David Levin said: "What we should be saying is how can the good schools in the independent sector and in the state sector help those that are perhaps a little way down the track to get better.
The idea is that the process should be presided over by a "headmaster", a figure of similar stature to Martii Ahtisaari, the former Finnish president, who led the 2006 Kosovo talks.
The tragic suicide of a retired headmaster shortly after he was prescribed the antidepressant Seroxat should be seen against a background of the doubling of antidepressant prescribing over the past 20 years.
Christopher King, the chairman of the Headmasters' and Headmistresses' Conference (HMC) and headteacher of Leicester Grammar School, will argue that markers should be paid more so that exam boards can demand a higher standard from them.
"The headmaster wants everything his way and wants to run that school his way, but we should be asking: are the pupils there for the school or is the school there for the pupils?" he said.
