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London, England (CNN) -- A crowd of students marched through central London on Wednesday, disrupting traffic and shouting slogans in a protest against an increase in university tuition.
Students also protested in other cities across Britain, including Leeds, Cambridge and Birmingham. Police said there had been 15 arrests in London and one each in Oxford and Glasgow, Scotland.
Three of the arrests happened after students filled Whitehall, the street that runs from Trafalgar Square past the prime minister's residence toward Parliament. Police penned the protesters in at either end to try to limit criminal damage, and a line of police in fluorescent yellow vests kept the students from reaching Parliament Square.
"We're kettled, we're freezing, everyone needs the toilet, but we're having fun," protester Clare Solomon told CNN, using a local term for being penned in.
Solomon, the president of the University of London student union, said police had been containing them for five hours as of 6 p.m. (1 p.m. ET).
"Obviously, people are starting to get angry now. Everyone is really hungry," she said.
But she thought the demonstration was a success. She estimated 20,000 to 25,000 people had turned out.
Police said they could not give an estimate of the size of the London crowd.
"Students are starting to self-organize," Solomon said. "If we can get 25,000 on the streets with no major organization, it's a sign of things to come."
British universities should not follow the American model, where students have to pay large tuition fees and seek loans or grants themselves if they can't pay, she argued. That excludes poor people from higher education, she said.
Students held signs saying "Stop Education Cuts," "Education not Segregation" and "Unite and Fight."
They managed to break into a police van inside the containment area in Whitehall after smashing the windshield, spray-painting the sides and trying to topple it. Other students stood in their way and tried to stop them.
Some students set off firecrackers in the crowd.
Otherwise, unrest did not appear to reach the level seen during similar protests two weeks ago, when protesters invaded the building holding the headquarters of the Conservative Party.
The students are angry about government plans to allow universities to charge as much as 9,000 pounds (about $14,200) a year in tuition fees, a substantial rise from the current cap of 3,000 pounds (about $4,700).
The government wants to raise the fees and scrap some subsidies for university students as it tries to cut a massive budget deficit.
Defenders of the plan say universities will not necessarily charge the maximum they're allowed to, but education experts have disputed that.
In the protests two weeks ago, students spray-painted anarchy symbols, broke windows and set off flares at the Conservative headquarters before being forced out.
Some set fires and shot off firecrackers outside the building.
That violence marred what was otherwise a largely peaceful protest. The National Union of Students said 50,000 demonstrators were on the streets in that protest.
Two officers were injured in the London activities, London's Metropolitan Police said. A female officer broke her hand and a male officer suffered leg injuries, police said.
CNN's Richard Allen Greene and Jo Shelley contributed to this report.