One thousand children go into a school one at a time, and walk past 1,000 lockers. The first child opens every locker, the second child closes every other locker, the third child opens or shuts every third locker, the fourth child opens or shuts every fourth locker and so on through to 1,000. At the end of all that clattering, which lockers will still be open?
It took the pupils in Mary Jo Weisenburger's gifted class at Maumee high school, Ohio, a whole day to discover the answer to this problem (given at the end of this article), using charts, cardboard, spreadsheets and formulae. But then most of them are 10 years old. And it wasn't just the answer that she wanted them to find out.
"When children first come into my class, if I give them a problem they sit and look at it for 10 minutes, then say 'I can't do this'. Because they are gifted they can absorb knowledge, they can remember anything, but they haven't learned how to think. I want them to extend their thinking, to see patterns, to solve problems, to ask deeper questions."
Serena Troyan, who teaches science to gifted eighth-graders (Year 9) at Maumee high - and is currently Ohio's Gifted Teacher of the Year - is similarly enthusiastic about the benefits of stretching pupils. "I used to be such a pure scientist," she says. "The only way I would have kids present their material was in a lab report. Now I have them doing plays and poems and videos, using all their skills to learn science."
This kind of teaching could soon be on offer to England's brightest children, following Labour's pre-election promise to introduce special provision for gifted, talented or exceptionally able students. Ironically, President Bush's 2002 budget cuts will probably put an end to proposals for a modest, nationwide programme for gifted pupils, but locally funded schemes (the US education system is highly devolved) will be unaffected.
In Maumee, a small district serving 3,100 schoolchildren in the neat pastel clapboarded suburbs of Midwestern Ohio, the mix of working, middle and upper-class families have had a gifted programme running in their schools - four elementary, one middle and one high - for 26 years.
A child can enter Maumee's programme from the first day of school. Classroom teachers are expected to pick out children who, under Ohio's definition, "acquire skills... at a faster rate, with more depth and greater complexity than other children", and consult with Maumee's full-time district co-ordinator for the gifted and talented, Ellie Slotterbeck, on how to meet their needs. Teachers of gifted children receive 20 hours of extra in-service training.
But it is at second grade (Year 3) that the gifted education system really kicks in. Under Ohio law, every child in a state school has the right to be tested for "giftedness" twice a year. In Maumee this starts with an annual IQ test for all children, taken in class; children scoring more than 130 are identified as gifted. Children with borderline scores, children whose teachers believe they underperform in tests, children who show evidence of unusual creativity, and children whose parents appeal against their initial results then go on to a second test, individually administered by a psychologist. This is a serious business - each psychologist's test costs the district $250 (pound;178). Last year 60 children sat second-stage tests.
For Judd Morrow, 18 and leaving Maumee high school with a handful of scholarships for a business degree, entry to the gifted programme in second grade, 10 years ago, was uneventful. "All I remember is that we got snacks," he says.
But for others, admits Jim Lowry, co-ordinator of the school's gifted programme, the identification process is more traumatic. "There are parents who don't agree with our assessment regarding their children," he says. "There are also parents who say we are doing special things for these kids that should be done for their kids. Sometimes parents' and children's feelings are hurt."
But for those pupils who do make the programme - 15 to 20 per cent of the 200 to 250-strong year group - specialist classes will gradually increase and, unless they cannot keep up with the work, continue throughout their school careers.
From third to fifth grade (Years 4-6), gifted pupils spend one day a week away from their usual classroom doing special projects - thinking skills, electronics, ancient Egypt, local ecology - with Mary Jo Weisenburger. Sixth-graders moving to middle school see a substantial jump in provision - they spend every morning, when the whole-year group is timetabled for English and maths, doing those subjects in a gifted class.
In seventh and eighth grade, special classes are added in science and social studies, so those students identified as gifted in all subjects - between a quarter and a third of the total gifted group - move into the gifted group for more than half the day. Others attend a mix of gifted and regular classes.
Meanwhile, annual testing continues, so the gifted class has to avoid getting too far ahead on the standard curriculum, in order to accommodate newcomers. At times, says Joe Lowry, this has meant the gifted class easing up halfway through the week. "We used to have something called 'the fifth day' and people used to joke that on the fifth day the gifted class rested."
When the system works well, the gifted class becomes a haven of flexible, imaginative teaching within a heavily textbook and skills-oriented school system. Pupils set their own essay titles and assessment targets, they make videos, write poetry and plays and follow their own interests.
Sixth-grade (Year 7) pupils at Maumee investigate clowning; they create their own clown costumes with $5 (pound;3.50) a head to spend at a thrift store, and write and perform clown sketches for an old people's home. Seventh-graders write journals and argue ethics.
Serena Troyan's eighth-graders carry out research projects. Megan Mutchler, 14, who plans to be a biochemical engineer, worked with chemical analysis companies to find out why her local creek never froze in winter (diesel pollution, Megan discovered).
In ninth grade and high school the gifted programme expands again to include modern languages. Students judged gifted in all subjects now spend five out of seven daily periods in separate classes. The emphasis here is on acceleration rather than extension - some students, finishing the high school curriculum several years early, will be taught at college level or even go to local universities part-time. Meanwhile, the rest of the year group continues to be taught in mixed-ability classes, in some cases up to the age of 18.
It is this mixed-ability teaching from beginning to end of the US school system that some argue necessitates special provision.
"Students at the upper end deserve the right to go further," says Jim Lowry. "They need to reach their full potential and they often can't do that in a regular classroom, where the teaching is dumbed down to the middle."
But with most students picking and choosing courses from 14, moving in and out of classes, one effect of the gifted programme is to create a tightly knit group of 12 to 18 students in each year who stay in the same classes, some from the age of eight.
For the students this has pros and cons. In ninth grade, Judd Morrow opted out of the gifted maths and science programme while remaining in it for social studies, English and languages. "I wanted to be with friends who weren't in the honours (gifted) class," he says. "I sacrificed a learning opportunity for a social opportunity, and I'm glad I did, because I got to be with two different groups of people."
In fact some students opt out of the gifted classes anyway, particularly in seventh grade (Year 8), according to Ellie Slotterbeck. The workload is heavier than it would be in regular classes; at the same time peer pressure tends to make boys drop out. By high school, gifted classes which in elementary school had equal numbers of boys and girls are likely to contain at least two-thirds girls.
For those who stay, the closeness of the group is a major attraction, says Karah Kinkaid, 18 and about to start veterinary training. "We know each other so well we're not afraid to give our opinions and disagree with each other," she says.
In any case, she adds, the alternative is unattractive. "I opted out of honours English in eighth grade because I wanted to be with friends, but I was so bored I went back."
Gifted students know that the extra choices open to them are resented. "The regular students think we don't get homework," says Sean Hafermann, 13. "They talk about how easy bridge (gifted) students have it, how we're always doing fun stuff. I think it's maybe that they struggle with grades and they'd like to be able to understand stuff right away like they think we do."
The resentment does not always stop at the staffroom door. "There's a section of staff who think gifted teaching is a challenge and look forward to it," says Jim Lowry. "But among the faculty there's also a perception that teaching a gifted class can be a negative experience.
"I think sometimes these negative elements on the staff feel that the students aren't only smart, they're arrogant. Or that they're not just bright, they're lazy. I think they can be intimidated."
The division is particularly hard on children who just miss the IQ cut-off, agrees Ellie Slotterbeck; there is talk of introducing more differentiation to regular classes.
Intake to Maumee schools is rising, and the head of the high school, Jim Cellio, is frank about the gifted programme's appeal to middle-class parents. "There's no question about it - it's a strong selling point, and in our public relations pitch we identify our strong gifted and talented programme."
In the elementary schools, says Ellie Slotterbeck, individual teachers regularly push for less well-off kids to be admitted to the programme. "I have teachers saying, 'But, but, but... this child is in his or her second family, this child has food stamps'." These teachers will sometimes help parents appeal for a second-stage test. By high school, though, Jim Cellio - uncomfortably aware of the correlation between IQ tests and social background - reckons all children in the gifted programme are middle-class.
And you can see the appeal. The peer group is keen. The teaching is exciting. The classroom bubbles with ideas - every boy and girl in sixth grade gifted class claims to read for pleasure, and not only Harry Potter but biography, history, Edgar Allan Poe.
Maumee hopes to draw in more children who score high in IQ tests but perform less well in school subjects. "We have to put together a more individualised programme, to say, 'Can we work with this child, what are the interests of this child?'" says Ellie Slotterbeck. Giving that kind of attention to gifted children releases regular class teacher time for children in the middle, she argues.
Children at the top of the ladder have special needs, just like those at the bottom. And gifted children are a national resource, requiring nurture and investment if the country is to see the benefits of their brains. "I had a superintendent say to me once,'The cream will always rise to the top'," says Ellie Slotterbeck. "I said, 'Yes, but do you want it sour or sweet?'" The lockers with square numbers (1, 4, 9, 25, 36 etc) will remain open