On top of a hill, surrounded by breathtaking views of verdant mountains chequered with fields of beans and maize, teams of teenage boys are playing football in a school playground. It’s a tranquil scene, but it isn’t all it seems. This is Rwanda, a country where even games have two meanings.

Ten years on from the genocide which led to the deaths of a million people, there is no PE here. Instead, there is something more profound: “physical and peace education”. The teacher picks the teams so children do not just play alongside their friends, and at half-time all the participants sit down for a chat. At Kanzenze school, as in almost every school across this central African country, there are pupils whose families have been killed.

Some of those children can glance across the classroom at a classmate whose father, brother or cousin was responsible for their loss.

Thirteen-year-old Murorakeye bears a deep machete scar on one side of her face. She lost her father in the massacres. Her name means “always happy”.

Rukundo looks younger than his 14 years and has a twitchiness which betrays some inner distress. He lives alone with his 16-year-old sister. When asked what happened to his parents and his four other brothers and sisters, he looks down, twisting his fingers together, and spits out a single word:

“genocide”. To play football together in such circumstances is an act of extraordinary trust and forgiveness.

Mere numbers cannot adequately convey the horror of the carnage which swept Rwanda in 1994. Orders were given by some in the interim government for the Tutsi minority to be wiped out, and most of them were; hacked to death with machetes, or bludgeoned with makeshift clubs or hoes. Husbands from the majority Hutu ethnic group were told to kill their Tutsi wives; Hutu mothers were told the children of their Tutsi husbands must die. And schools, now so symbolic of Rwanda’s attempts at reconciliation, played a very different role in those grim days. “There was a policy of dividing people,” says Emmanuel Mazimpaka, a local schools inspector. “For example, there was a teacher who asked his class: ‘When you have 100 Tutsis and then you kill one Tutsi, what is the number which remains?’ The pupils who studied like this were taught that a group of people were there to be killed.”

Rwanda’s schools were cradles of hate long before 1994. A century ago, the Belgians who colonised the country saw the Tutsis as superior, and imposed quotas which gave them privileged access to education. After independence in 1962, a Hutu government retaliated by reversing the rules to favour children from its own group.

Racial hatred - never a problem in Rwanda in pre-colonial times - grew in schools as it did elsewhere. And so the education system, along with the government, the media and some of the churches, was at the political centre of Rwanda’s genocide. When the killing began, schools were at its physical centre, too. Tens of thousands of Tutsis took refuge in schools and churches, and in those places of sanctuary many were killed. But while the physical result, the appalling loss of human life, is obvious, the psychological damage is much harder to quantify. No one really knows how the children who lived through that time will function as they become adults. A Unicef survey found nine out of 10 children had seen dead bodies or body parts, eight out of 10 had experienced a death in the family, and a third had seen children killing other children.

Kanzenze’s headteacher, Jean-Berchmans Barihuta, lifts the lid of a desk in one of his school’s bare classrooms to display a strip of wood nailed roughly to its underside. This is a muffler to prevent the lid from falling with a bang, he explains. There are children here so traumatised a loud noise can cause them to pass out with terror.

“We have children who might fall to the floor just because someone drops a pen on to a piece of paper too loudly,” says Mr Barihuta. “It can take them two or three days to come back to school if something like that happens. We have to ensure their friends remain with them. When they return, we try to make the environment feel friendly for them, and we tell them not to be afraid.”

In his tiny, windowless office, Mr Barihuta produces a list of the children in each of his classes. There are several columns to this list: numbers of boys, numbers of girls, numbers of orphans. More than a quarter of the pupils here are orphans, he says, enumerating the reasons on his fingers: genocide, war, Aids. A war in 1997 and 1998, perpetrated by the Hutu forces ousted after the genocide, took as many lives in this school as the events of 1994.

Gisenyi province must have some of the unluckiest schools on earth. In 2000, a volcano erupted, scattering the local population and destroying a school. It is hardly surprising that apocalyptic religious sects are gaining a foothold in many local villages. Or that Mr Barihuta’s school has such a high drop-out rate, that he has 10 classes in the lowest primary year group, but only one in the highest.

It is hardly surprising, either, that schools find it impossible to teach history as a separate subject. Rwanda’s history is taught instead as part of “civics”, in which children are asked to reflect on the racial harmony believed to have existed in pre-colonial times. If it was possible then, pupils are told, it is possible now.

And yet there is something irrepressible about Rwanda’s youth. At the Nyarutovo catch-up school, a special facility supported by Unicef for pupils who have missed out on education, Marie-Claire Mukeshimana smiles as she leads her fellow pupils in an English language lesson. “A sock,” they all chorus carefully as she points to items of clothing on a chart. “A necktie. A watch. A trousers.”

As she leans against the wall outside in the sunshine, Marie-Claire might be any other teenage pupil. Yet her very presence bears testimony to an astonishing determination to be educated, which is far from unique here.

“I live about 12 kilometres away,” she says, indicating somewhere on the other side of the mountains. “My friends and I start walking to school at five o’clock each morning, and usually we arrive at eight. We are a little afraid of being attacked on the road because it’s dark, but the authorities have instructed their officers to keep watch for us.”

Marie-Claire is 14, and this year she hopes to complete Primary Two. “I was in Primary Two when my father died,” she explains matter-of-factly. “There were some problems of war and we fled, so I abandoned school. When we returned I helped my mother to cultivate our land.” Now she is back in school, starting again. Even at Nyuratovo, where two years are covered in one, she will be 17 before she finishes primary school. And yet she hopes to complete secondary school and then go on to further study. “I want to learn, to know,” she says when asked to explain why she goes to such lengths to attend school. “I want to be able to speak French and English. I want to get a vocational training so I can get a good job.”

Marie-Claire is not alone. All over this country, there are outbreaks of almost impossible optimism. There is something in the spirit of the place which cannot be destroyed by genocide, war, disease or poverty.

To reach the School of the Assumption at Birambo you have to turn off the main road and then, on a good day when the mountainous dirt track has not been made treacherous by rain, to drive for an hour and a half. Or, if you are a pupil at the school, to walk with all your worldly goods, your term’s supply of soap and your few items of clothing, for several hours. It takes some of the girls two days to get from home to school.

Approaching Birambo, two landmarks are immediately visible outside the school. The first is a mass grave, one of three in the village, its headstone a tribute to the thousands who died here. The second, more unusual for Rwanda, is a huge satellite dish.

The headteacher, Sister Marthe Ntuyumve, laughs a cheeky, almost flirtatious laugh when asked how the dish came here. “It was given to us by the president,” she explains. “He came here after the school reopened in 2000 and we told him we needed this technology. He said, ‘OK, you will have it’. So when he got back to Kigali he asked the company to give us a phone line and a satellite dish.

“Electricity is no problem here. We have a very good supply. Except when it rains hard, and that brings down the wires.” In Rwanda, it rains hard almost every day for several months of the year.

The school still has to pay the internet connection charge, a massive pound;100-plus each month, equivalent to the salaries of five teachers.

But Sister Marthe, who gained a passion for computers and computer games during a posting in the Philippines, thinks it worthwhile. “If the children didn’t have anything to eat I would give up everything,” she says. “But we grow much of our food. We have potatoes, beans, rice and maize flour.”

Sister Marthe believes the future is bright. “One thing I am sure of is that Rwanda in 10 years will be different from what it is now,” she says.

“As we are on the move towards this big globalisation, it is important for the poor also to access the world.” Across the corridor in a small computer suite, a huddle of pupils is doing just that. Ariane Rugumaho, aged 16, is emailing a teacher from England who spent a year at Birambo. As she does so she’s listening to an internet radio station, the sound of the British Eighties pop group Bow Wow Wow echoing incongrously through the room.

James Smith, director of an English charity called the Aegis Trust which is building genocide memorials in Rwanda, is here to see if the school would make a good site for a live webcast on April 7, the 10th anniversary of the genocide. What, he asks, could Ariane say to pupils across the world who wanted to know about the events of 1994?

“I could say that in another 10 years our country will be among the developed countries, because we are going to work very hard,” she says. “I could say that in Rwanda things are different now, it’s changed. I think I could say it will be ‘never again’.”