"One bullet plus one bullet equals? Two bullets!" This is how the children of the small Iraqi village of Jaraf were taught mathematics during two years of jihadist rule.

On Saturday, a neighbour opened the gate of their school for the first time since the Islamic State (IS) group was forced out by Iraqi forces last week.

The children took over the building and were soon playing football with soldiers in the main hall and jubilantly ripping up their IS textbooks.

"They brought new books... all of them Islamic," said Sanaa Ahmed, recounting the time in 2014 when IS took over her village south of Mosul.

"There used to be pictures in our books. They changed that, they said it was forbidden," said Sanaa, a lively 10-year-old wearing a pink woolly dress and a stack of white bracelets on both wrists.

"They brought us pictures of little girls completely covered, with the niqab (full veil) and even socks and gloves... I don't know how they wouldn't suffocate in there," she said.

According to the United Nations Children's Fund, 4.7 million children have been directly affected by the conflict in Iraq and 3.5 million are out of school.

In Jaraf, many parents decided not to send their children to school when IS took over.

"They were trying to control the children's minds, they would teach them things that encouraged them to kill," said Abu Salem, a 28-year-old father whose house faces the school.

- Songs praising Baghdadi -

"For example, in the mathematics class, they would learn what is a bullet plus a bullet, or a rocket plus a rocket," said Salem Abdel Mohsen, another father who did not send his children to school under IS.

"When the kids grow up, what kind of education will they have had? When they graduate, will they become a doctor? An engineer?," he asked.

"It won't be possible, of course they will become Daesh (IS)," said the young man, now a member of the local paramilitary force known as Hashed al-Ashaeri.

The cost of IS textbooks, a list for which is still pinned to a wall in Jaraf's Zeinab Bint Khadija school, also discouraged many parents.

A copy of the physical education book produced by IS -- complete with the group's logo on every page -- was lying on the floor in one of the classrooms.

Most of it consisted of instructions for normal exercises but the IS touch was evident in some lessons.

The first one in the book was "how to sit cross-legged properly" and another was called "the seven stones game", which apparently aimed to prepare for the ritual "stoning of the devil", part of the hajj pilgrimage.

Another class described an exercise to be performed whilst singing a song praising the IS caliph and to which the refrain goes: "All of you join and give allegiance to (Abu Bakr al-) Baghdadi."

"There was a teacher, she used to tell us that the Islamic State is good, that they give you food and money. Good money. Better than the army. 'Tell your parents to join us'," Sanaa recounted.

- 'We forgot everything' -

Jaraf is one of several hundred tiny and remote villages dotting the windy roads of the Tigris Valley in Nineveh province.

Before Iraqi forces started retaking swathes of land, the villages were in the heart of the "caliphate" IS proclaimed in June 2014, even more isolated from the rest of the world than they previously were.

The school in Jaraf would remain closed for days on end and even when IS sent a teacher down from Mosul for a few classes, many children stayed home.

Islamic State group teachers remonstrated the children every time they visited, telling the girls to wear the niqab and the boys to dress "Afghani style".

But the children said that was never enforced.

In Jaraf, there were no jihadist training camps for children such as those show in IS propaganda videos.

"The children didn't understand all this, they are just children, they want to play," said Abu Salem.

Most of the children there never ventured more than a few kilometres from their village and talk about jihadist rule like a benign inconvenience, a couple of school years with an unlucky crop of teachers.

"They tried to teach us their things but we were fed up with Daesh and we already forgot everything," said Sanaa.