She got there by beating every other student in north-east Kenya.

At first, the young Somali can appear to be shy but that exterior belies an inner strength born of an intense competitive spirit. Asked to test a microphone by saying the first thing that comes into her head, she replies: "Number one."

In Kenya, access to secondary school depends on your mark out of 500 in an exam sat at age 13 or 14. A mark of 250 or more is considered good. Anything over 300 for a girl, in a system which still favours boys, is exceptional. Fatuma scored 364.

Grace Wachuka, an education specialist with the non-government organisation Care International, worked in the refugee camps at Dadaab for five years and has taken a special interest in Fatuma.

"In Kenya," she says, "for a girl to get over 300 marks means she is very bright. For a girl to do that in Dadaab is outrageous. Fatuma is one in a million."

When Fatuma talks of her life-changing exam results, she is a picture of frustration. "I was expecting to get 400-plus," she grumbles. "But the moderators cut some marks I think."

Midway through her second term at the Nairobi boarding school, Fatuma's presence here is still a surprise, even to senior members of staff who privately admit that they would prefer the handful of scholarships at Kenya's elite national schools to go to Kenyans.

Most of the other pupils in their regimented ranks of red and grey uniforms made it to this imposing school from the comparatively well catered-for suburbs of the capital or places like Central Province. The imposing institution, built under British rule from grey stone, is the alma mater for daughters of ministers, businessmen and judges.

But the refugee girl is not intimidated. "I don't care even if their father is President," she says without aggression. "I know where I came from. I know why I'm here. We sleep in the same beds, we eat the same food." It wasn't always so. Fatuma studied for her exams in a shack built from flattened, empty cooking oil cans provided by the UN's World Food Programme. There were at least 100 pupils to a teacher in her class and almost all the teachers were untrained volunteers.

Dadaab is a dust-blown trinity of overcrowded refugee camps, built to hold 45,000 refugees, on the arid plains that divide Kenya from its northern neighbour, Somalia. Today it shelters 270,000 people in conditions Oxfam describes as "conducive to a public health emergency".Some of the best stories have humble origins but few of them emerge from Dadaab. Understandably, Fatuma is a hero in the camps and the sometimes awkward teenager at Kenya High knows that thousands of refugee children are counting on her to blaze a trail for them.

When news of Fatuma's scholarship came through there was a rare party in Dadaab's Hagadera camp. The heroine of the hour remembers celebrating with fizzy drinks."School is not a priority at Dadaab – girls don't have an equal chance," says Ms Wachuka. "Fatuma has triumphed in very difficult circumstances." From the age of 12 she "had a dream" of going to a national school in her host country and wasn't going to be put off by naysayers who told her that refugee girls could not go. "It can be done," she says. "I've done it."

Her eventual aim is to study medicine and one day return to Dadaab as a doctor. "If there is peace in Somalia," she adds, she would like "to go and help people there where there are not enough qualified people." The teenager understands that she is a role model and has a simple message for other young Somalis. "You know education is the key to success. First go to school, work hard and choose a career. Work hard, aim higher and be nice to people."

This is almost exactly the advice Fatuma's mother gave her eldest daughter before putting her on a UN flight out of the refugee camp and into a world unknown to either of them. The culture shock must have been immense but has been managed with another maternal tip: "Don't take these things too seriously." The lawns and courtyards of Kenya High are eerily quiet for a school of nearly 850 pupils. The watchword here is discipline.

They are certainly a world away from Fatuma's first school in Kismayo. The Somali port is now the stronghold of the radical Islamic militia, al-Shabaab, where last year a 13-year-old girl was stoned to death in a sports stadium after reporting that she had been raped. Fatuma remembers the school she left at age eight as a place you "would hear gun shots and fighting ... You would see people killing each other."

After a lifetime of wearing the hijab in front of other people, the most difficult adjustment has been wearing the compulsory uniform of a skirt and a short-sleeved blouse. The awkwardness of the transition is doubtless compounded by being 15 and relatively tall. Fatuma carefully folds her gangly limbs into the smallest space possible but she is far from invisible.

She admits that her new life is not always easy. She misses her seven brothers and sisters and speaks to her mother by telephone only once a month. Her scholarship pays for boarding fees and uniforms but nothing more. There was no money to pay for the nine-hour bus ride to Dadaab during the Easter holiday, so she stayed in Nairobi. Faced with the brightest girls in Kenya Fatuma is no longer "number one". In her first term, she lagged behind in the two national languages, English and KiSwahili.

But there is plenty of reason to think she will catch up. Remarkably, she came near the top of her class in computer education, having never seen one before; and has taught herself to swim butterfly, having never been in a pool before reaching Nairobi. But it's not enough for her. "I don't feel good. In my school I used to be the best," she says. This is followed by a note of polite defiance that lands somewhere between a promise and a warning: "They are not brighter than me. They are just better at the moment."