There are few more cheering ways to start the day than the discovery that a couple of Tory Cabinet ministers are fighting like cats in a sack. It is especially piquant when one of the protagonists is Michael Gove, probably the most widely loathed Education Secretary in years. For much of last week, he was trading blows with the Home Secretary, Theresa May, about which of them is tougher on Islamic extremism. These two rivals for David Cameron's job have kissed and made up, in public at least, but the important thing to remember is this: they are both wrong.

The row started when someone, later revealed to be Gove, spoke to The Times and appeared to suggest that the Home Office wasn't taking a strong enough line on combating Islamism. The Home Office then took the extraordinary step of releasing a letter written by May, in which she demanded to know whether Gove's department had been warned in 2010 about a supposed "Trojan horse" plot by Islamists to infiltrate state schools in Birmingham. Tomorrow, the education watchdog Ofsted is due to publish reports prompted by the allegations, which surfaced in an anonymous letter; the letter's provenance is unknown and it has been dismissed in some quarters as a hoax. But reports on two of the schools in question have already been leaked. One suggests that students are not being protected from the "risks associated with extremist views"; the other that governors involve themselves inappropriately in the running of the school and staff feel intimidated by the school's leadership. Five schools are expected to receive Ofsted's lowest rating.

In essence, the spat between Gove and May comes down to a question about how individuals become extremists, and whether there is a linear progression from religious teaching in schools to radicalisation as young adults. But there is a bigger problem which neither minister acknowledges because they are complicit in creating it, along with virtually all their coalition colleagues. Building on the last Labour's government's mistakes, they have created a dog's breakfast of a state education system where different categories of schools operate under entirely different rules. We now have half a dozen different types of schools, including community schools, foundation and trust schools, "faith" academies and free schools, all receiving public money. Some of them have to follow the national curriculum; others merely have to teach a "broad and balanced curriculum". Some have governors appointed on secular lines, while different types of "faith" schools are allowed to appoint between a quarter and a majority of governors for religious reasons. In the case of "faith" academies and free schools, a religious test can be used in all teaching appointments, while teaching staff do not even have to be formally qualified.

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Such variance strikes at the heart of a unified state system in which all children receive a broadly similar education, equipping them for the secular society which the UK has become. Instead, there is not just a mish-mash of different regimes but an open invitation to any group which wants to promote its own ideology at state expense. For the moment, the focus is on the role of Islam, but other religions are just as keen to get involved in state education. I am not a great fan of the shadow Education Secretary Tristram Hunt, but he is right on the money when he talks about "a worrying pattern of religious interference by governors, attempted hi-jacking of appointments, syllabus restrictions and cultural conformity".

Whatever Ofsted has to say this week, the bone-headed determination of the last Labour government and the present coalition to hand state education over to poorly qualified people, solely on the grounds of religion, has already produced one publicly funded disaster. At the Al-Madinah Muslim free school in Derby, part of the school is due to close shortly after being judged inadequate and chaotic by Ofsted. Women teachers complained about being forced to wear the hijab, while Ofsted found that "the basic systems and processes a school needs to operate well" were not in place. Outside the state sector, Ofsted has expressed concern about an Orthodox Jewish school in north London where some pupils at the boys' primary school reported that they had been slapped by teachers, while inspectors found that the children had a "very limited" understanding of other cultures. Then there is the independent Olive Tree Muslim primary school in Luton, which has been threatened with closure after Ofsted accused it of promoting Salafi extremist beliefs.

There is a paradox here. In my lifetime, religious observance has declined dramatically in this country. Yet the number of "faith" schools has gone in the opposite direction, encouraged by mainstream politicians who appear not to understand the importance of keeping religion out of the public sphere. "Faith" is only one aspect of personal identity, and an increasingly irrelevant one in modern Britain, so why give it such a privileged role in state education? We don't have Liberal Democrat or Arsenal supporters' schools and I can see no justification for pouring public money into religious ones.

This is the conversation David Cameron needs to have with his warring ministers, whose dispute is currently under investigation by the Cabinet secretary, Sir Jeremy Heywood. But that would require the Prime Minister to acknowledge an incoherence at the heart of government policy. Ministers want to discourage religiously inspired extremism; they also aspire to give religion an ever-bigger role in public life. They can't do both; these problems will not go away as long as we have a fractured state education system which gives a ludicrously inflated role to religion.

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