By Elizabeth Piper

LONDON (Reuters) - Setting her sights on making Britain "the world's great meritocracy", Prime Minister Theresa May unveiled reforms to widen opportunity in schools on Friday in an appeal to "frustrated" Britons who voted to leave the EU.

In her first major policy speech since becoming prime minister in July, May moved to tackle education, which has long divided Britain and often crushed politicians' attempts at reform.

She addressed those Britons whose vote to leave the European Union in June, she said, was a rejection of establishment politics that have failed to improve their lives and was a clear message: "They will not be ignored anymore."

"They want change. And this government is going to deliver it," May said in her speech in central London.

She said her government would set out other social and economic reforms, "but there is no more important place to start than education", promising all children a place in a good school to answer working class fears that "their children will not enjoy the same opportunities they have had in life".

Increasing capacity and encouraging successful institutions, including fee-paying private schools, to work to boost standards at other schools would help overcome the complaint of a shortage of good establishments especially in poorer areas, she said.

But it was her bid to allow more single faith schools and selective "grammar" schools that caused uproar in Britain, where governments have tried to phase out academic selection of children to end what is described as a 1950s two-tier system.

Jeremy Corbyn, leader of the opposition Labour Party, said expanding grammar schools would further divide the country.

"I want a good education for every child," he said.

TRADITIONAL CONSERVATISM

May rejected the criticism, saying she wanted to increase schooling options in a policy that taps into a rich vein of traditional conservatism in the suburbs and rural communities in so-called "middle England", a key electorate for the new leader.

Found across England but located mostly in the southeast outside major cities, grammar schools select children by using an examination taken around the age of 11.

Critics, including some in her own ruling Conservative Party, say the test comes too early and instead of inspiring all children, condemns those who fail to second-rate schools.

Education in Britain is a notoriously thorny issue. By making it her first domestic policy, May, who herself was educated at a grammar school, has signaled her intent to break with policies pursued by predecessor David Cameron and to engage with a traditional conservative electorate who could be vital for her political future at an election due in 2020.

Many of them voted to leave the EU and are keen for the new prime minister to press on with Brexit. May has said she will not trigger the formal divorce procedure this year while her government prepares a negotiating stance.

It could also underline the influence of one of her long-time advisers, Nick Timothy, who last year called for the creation of new selective grammar schools to give parents more choice and has frequently praised his grammar-school education for offering him chances his working-class parents did not have.

"Politicians - many of whom benefited from the very kind of education they now seek to deny to others - have for years put their own dogma and ideology before the interests and concerns of ordinary people," May said.

(Reporting by Elizabeth Piper; Editing by Alison Williams)