The statement came as Nigeria prepares to make the four-year anniversary of the Chibok abductions on April 14, 2014.
"Since 2013, more than 1000 children have been abducted by Boko Haram in northeastern Nigeria, including 276 girls taken from their secondary school in the town of Chibok in 2014," the statement said.
The Chibok abduction sparked global outrage and reignited the fight against the ISIS-aligned terrorist group.
Some of the girls were finally freed three years later, following negotiation talks between the Nigerian government and Boko Haram.
But more than 100 of them remain in captivity.
'A grave violation'
The UN agency appealed for an end to attacks on schools and "all grave violations of children's rights," in the northeast of the country.
Children, particularly young girls, are vulnerable to attacks by the militant group and the agency said they have been "consistently targeted and exposed to brutal violence in their homes, schools and public places."
"These repeated attacks against children in schools are unconscionable. Children have the right to education and protection, and the classroom must be a place where they are safe from harm, UNICEF representative in Nigeria," Mohammed Malick Fall said in the statement.
Four years on more than 100 of the Chibok girls are still missing.
Yana Galang, whose daughter Rifkatu is still missing, says she is hopeful her daughter will come back.
Every year on the anniversary of their daughters' disappearance, the parents gather in Chibok to pray for their safe return.
And this year will be no different.
"I know in my heart that Rifkatu will come back. I have not given up hope," she says.
Last year, Boko Haram released a video where they showed veiled girls they claimed were Chibok girls saying they would never return.
One of the alleged Chibok students said: "We are the Chibok girls that you cry for us to return to you. By the grace of Allah, we will not return to you.
The Bring Back Our Girls group has long campaigned for the return of the Chibok girls through protest marches, rallies and sit-ins.
They have clashed repeatedly with the Jonathan and Buhari administrations as they put them over pressure over the return of the schoolgirls.
This year, this group is marking the anniversary on Saturday with a march in Lagos and a lecture in Nigeria's capital Abuja.
The Dapchi kidnapping
Schools have been long been a focus of attack for Boko Haram who recently kidnapped 110 schoolgirls from Dapchi, Yobe State, northeast Nigeria.
All of the girls have now been reunited with their families after they were freed by Boko Haram.
However, one of them Leah Sharibu remains in captivity because she allegedly refused to renounce her Christian faith.
Since the conflict started nearly nine years ago, around 2,295 teachers have been killed and more than 1400 schools destroyed, UNICEF said.
Osai Ojigho Director Amnesty International Nigeria joined in the voices calling for the Nigerian government to do more to keep schools safe in the country.
"The time is long overdue for the government to deliver meaningful action on behalf for all these victims of Boko Haram's crimes. This starts by doing more to secure the release of the hundreds still being held, including the remaining Chibok girls and Leah Sharibu, the only Dapchi girl still in captivity."
Amnesty also called for better record keeping of missing and displaced, especially in IDP camps.
"Far more support must also be provided for past victims," Ojigbo said.
"For the families of those still missing, the government should open a register for abducted people, ensuring that the tens of thousands of people living in displacement camps get the opportunity to register their loved ones."
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