More than 400 schools were disrupted by late morning, as the government promised a bill to dock family benefits over damage.
A 15-year-old student had his hand blown off during a protest in northern France on Monday, authorities said, as demonstrations disrupted hundreds of schools on the eve of a nationwide day of action.
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A police union source said the student, in the town of Lens, appeared to have picked up a sting-ball grenade fired by riot police, hoping to throw it back, when the device exploded in his hand.
Students have been protesting over overcrowded classrooms, teacher shortages and crumbling facilities. Tuesday's day of action has been called by the Union syndicale lycéenne (USL), a high-school students' union.
Among the schools hit was the Vauquelin secondary school in Paris's 13th arrondissement, where "around ten hooded and masked young people came up to the entrance", according to headteacher Carole Zerbib. They "brought bins and inflammable items" to the doorway and then set them alight, she told French news channel BFMTV.
As a result, the school will remain closed for the day. "These are criminal acts that fall under the law and must be punished in the harshest possible way," said Jean-Didier Berger, France's junior interior minister.
Demonstrations and tensions were also reported outside the Voltaire, Sophie-Germain and Turgot high schools in Paris, as well as in the cities of Lyon and Lille. "Around forty higher-education sites" are also expected to be disrupted, Higher Education Minister Philippe Baptiste told the radio station Sud Radio.
By late morning, 413 schools had been or were still "subject to actions or disruptions since the start of the day", according to France's education ministry. That represents a 27% fall compared with the same time on Friday, although 40% of the schools had not previously been affected.
"Vandals pay" bill
In response to the violence seen in recent days, government spokesperson Maud Bregeon told news channel CNews that a bill known as "casseurs-payeurs", or "vandals pay", would be examined in the Senate on 27 October.
"It will allow us to dock family benefits… for damage caused by adults or minors" and "to introduce a form of collective responsibility", she said.
Mathilde Panot, who leads the hard-left France Unbowed (LFI) group in the National Assembly, told public broadcaster franceinfo that "there is no political response" to the high-school students' demands, only "a policing response".
Tuesday's day of action, dubbed "Act III", is backed by a broad alliance of education unions. The centre-left Socialist Party (PS) has also called on people to stand "alongside" high-school students.