Like taxi drivers everywhere, the one who picks me up at Belfast airport has all the answers. "Take this trouble with the Orange Order marches," says the cabbie as he fires up his engine and pulls away from the line. "It's not the marchers who are to blame but the residents who object to them. That's been a deliberate republican strategy since 1986."
With the parades crisis thus explained, can the cabbie now throw any light on this power-sharing business? What about the Northern Ireland Assembly? More specifically, what does he reckon to the new minister for education?
"McGuinness?" He shouts the name as if it were an expletive. "In any other country, the government would take that man outside and..." But the details are drowned out by the noise of the cab hitting a section of road where the tarmac has vanished, melted down to the foundations by the heat of a burning barricade.
Is it usual for a taxi driver to have strong opinions about an education minister? Perhaps not. But then, however you look at it - whichever foot you kick with, as they say here - these are astonishing times. Day after day, in a vast Palladian palace on the edge of town, men and women who could never bear to breathe the same air have been sitting down together and going about the business of government.
It is like a verse from the Book of Revelations. Catholic sits down with Protestant, nationalist with unionist, republican with loyalist. In accordance with the Good Friday Agreement, the various functions of state are divided among the political parties according to a prescribed formula. And it is this which brings us to the most astonishing fact of all. For when it fell to Sinn Fein to nominate a minister for education, the man it chose was the party's chief negotiator, Martin McGuinness.
The decision immediately triggered a wave of protest among those in the unionist camp who opposed sharing power with Catholics. For Mr McGuinness carries more baggage than an airport carousel. Ask anyone here and they will tell you that this curly-haired former butcher's apprentice from Derry made his name as an IRA chief of staff and that he is the tough guy who faced down the British government.
Listen to the loyalists who assemble in public places to demand the removal of Mr McGuinness from his post and you will hear the argument in its most extreme form. The man, they say, is nothing less than an unrepentant terrorist who has presided over the murders of 1,773 citizens. Now here he is, controlling the future of their children.
And yet among those who have done business - education business - with the new minister, it seems that Mr McGuinness has already acquired another reputation. He is, says one Protestant teacher, "an extremely bright guy - obviously up to the job". Another talks of his enthusiasm and commitment, and few fail to mention his extraordinary personal charm.
One of the many trade union representatives who has already had an audience with him described a "very useful first meeting which established an ongoing relationship of openness and frankness". This man had sounded cagey beforehand, but emerged from the meeting saying that all present had been "greatly encouraged", and the conversation had even got round to the minister's hobby of fly-fishing.
To those prepared to put the past behind them, there are more pressing issues than who did what to whom and when. For if you scrape away the familiar surface of Northern Ireland's sectarian politics, it is the universal problems of social inequality, of deprivation and class division that are left staring you in the face.
"Graduation day," says my second cabbie of the day, pointing out the well-dressed melee outside Queen's university. Last week, his son had been there, picking up a degree in finance and business studies. And right opposite Queen's, its own facade echoing the gothic revivalism of the university, is the Methodist college - the Methody, as they call it, without a care for the confusion this abbreviation causes outsiders. For any outsider struggling to unravel the nomenclature of Northern Ireland's education system is already confused enough.
Publicly funded schools here are known as grant-aided schools, and they come in five main varieties. Controlled schools, predominantly Protestant, are provided and funded by the education and library boards, while maintained schools, predominantly Catholic, are provided by voluntary bodies and funded by the boards.
Around 4 per cent of children are now taught at integrated schools. While these may be controlled or grant-maintained, their aim, as the name implies, is to educate Catholic and Protestant together. Finally, there are the voluntary grammar schools, provided by voluntary bodies such as the main churches, and funded from a variety of public and voluntary sources, including limited fees.
Such a school is the Methody. Founded in 1868 by the Methodist church as "a public school for boarders and day pupils, irrespective of their denominations", it now takes 1,850 children aged 11 to 18 - around a quarter of them Catholics.
Today is the first day of the summer holidays, and a brief peace reigns in the main lobby with its gothic wood panelling, its stained glass that says "Perpetua Lux" and its poppy wreaths commemorating the dead of two world wars.
A display of silver trophies speaks of high achievement in every field, from hockey to choral singing, from boating to public speaking. On the wall is an architect's impression of a new performing arts centre, and the noticeboard is a polite advert for "Lazer-Twist Methodist College Pens - pound;2 each from reception".
When Martin McGuinness took up office, the Methody was the first grammar school he visited. By all accounts it was a low-key affair - an informal chat with governors and a brief tour of the buildings. But the visit "not only demonstrated the board of governors' determination to continue to work with the Department of Education at the highest level". It also showed a "courage to confront difficult, contemporary issues", according to the principal, Dr Wilfred Mulryne, writing in the college newsletter, Methody Matters.
And there can be little question that one of those issues concerns the future of Northern Ireland's 11-plus examination. In its present form, the exam meticulously grades children as A, B1, B and so on. And while some grammars can and do admit C or even D-grade pupils, top schools such as the Methody draw almost entirely on the highest achievers.
But there are many - and the new minister for education has been one of the more outspoken - who believe the 11-plus is a crude and divisive mechanism that has no place in a society struggling to promote a climate of inclusiveness and mutual respect.
Not surprisingly, the eventual fate of the system is a matter of concern to grammar schools of all complexions. As one senior teacher at the Methody puts it, the breakdown of voting in the Assembly is likely to be "more ideological-economic than ideological-sectarian". The minister, he says, "may well find he has got to move more cautiously than he would wish".
Somebody who might well prefer the minister to throw caution to the wind is Frank Kelly. The principal of Corpus Christi college, a Catholic maintained school off the Falls Road, he helped prepare a discussion document called A Fairer Deal For All Children, which sets out a case for scrapping the 11-plus at the earliest opportunity.
"Children who attend grammar schools have an unfair advantage over the majority of their peers in terms of social status and breadth of opportunities, including social networking and quality of environment," says the document. Positive discrimination in favour of grammars over past years amounts to "a human rights issue of justice".
Of the 11-plus, it says: "The test is nothing more than a competition. It is not a measure of educational attainment and yet its impact can affect the rest of a young person's life."
There is no display of silver in the entrance hall at Corpus Christi - just a trophy that says "NISFACoca-Cola five-a-side under 15 winners 1995", some group photos of the school's soccer teams and, high on the wall, a CCTV camera.
As far as Mr Kelly is aware, every one of the 900 boys at his school is nominally a Catholic, and it's a similar story in the staffroom. At an equivalent school off the Shankill Road, an identical monoculture will prevail, except there the culture will be Protestant.
It's a short walk from the Falls Road across to the Shankill, through poor streets where republican hunger strikers gaze down from murals and the park gates bear the initials of the Real IRA. Veer to the north and you pass through the extraordinary steel-mesh dividing wall called, somewhat inappropriately, the Peace Line. At once you are in a parallel world of equally poor streets where kerbstones are occasionally picked out in red, white and blue, but where the prevailing hue is orange.
Yet despite the strict colour-coding of everything around him (kids on the Falls wear green and white stripes while those on the Shankill wear orange), Mr Kelly is not a man to see things in black and white. His conversation is full of provisos and qualifiers, and he is keenly aware of the infinite subtleties and shades in the society in which he lives and works.
He wants Mr McGuinness to put more resources into special needs and social deprivation, he says, and to look at the way funds are distributed. And there's no reason on earth why such actions should leave the minister open to charges of favouring Catholic schools.
To demonstrate his point, he brings out a table showing the take-up of free meals in Belfast schools, long held to be a key indicator of social disadvantage. Sure there's a tendency for more Catholic maintained schools to be at the bottom and for more controlled grammars to be at the top. But third from bottom, claiming proportionately more free dinners than Corpus Christi, is Mount Gilbert community college. And Mount Gilbert is off the Shankill Road.
"So, it's Martin McGuinness you're going to see, is it?" says taxi driver number three as he cruises up the Prince of Wales Avenue and approaches the parliament building at Stormont. "By all accounts, he comes across as very likeable."
And so he does. Rather than remain behind his big desk, the minister brings himself in front of it, and sits down. "Nice to meet you," he says, apologises for the rush and modestly declines to mention that, this very afternoon, in the last session before the summer recess, his enemies will try to have him removed from the executive.
The 11-plus? It's going to be a big debate - the biggest debate, he says. A research team from Queen's university has been gathering evidence on the subject, and the results, together with a comparative study looking at the situation in other parts of the world, will be published in September.
The ensuing debate will last until spring, he says. And only then will his department decide what action to take. "We are very determined to deal with this issue as speedily as possible," he says. But he is reluctant to pre-empt the debate.
"My party's opposed to the 11-plus, and I'm also opposed to it," he says. "But, obviously, there are many opinions out there, and I have a responsibility to listen."
But he does speak of the thousands of children who go away "with the label of failure around them" - speaks with feeling, too, as he himself famously failed the 11-plus, going on to be taught at a Christian Brothers' technical college, which he left at 15. "We obviously have to deal with this problem," he says, "and we have to ultimately create a situtation where, as I said on the day I became minister for education, the cornerstones of our approach to education would be the need for equality, excellence, accessibility and choice."
He laughs wryly as he recalls subsequent meetings with senior officials who reminded him that there should have been another cornerstone - affordability. "But it's a matter of the right strategy," he says. "You could have all the money in the world for any aspect of government, but if you haven't got a proper strategy, it's hopeless and a waste of money.
"And we cannot afford to waste money. We want to use whatever budget we have to bring about a fair system of education which will be of benefit to children - a system of education of which our society can be proud. And the difficulty at the moment is that I meet many people on a daily or weekly basis who tell me that they are not proud of this 11-plus examination."
It's surprising, he says, how many prominent people have whispered in his ear in recent months that they, too, failed the dreaded exam. "That was obviously a big setback to them at that age, but they recovered from it. How many people don't recover? How many people give up?" Before long, the man who is at this very moment being described by loyalist demonstrators outside the building as an "unreconstructed terrorist" is invoking such sensible and familiar concepts as "joined-up thinking".
Roughly 40 per cent of the population in the north, he says, is under 25, "and roughly 37 per cent of those young people are living in poverty. Now that's very stark. That hits you in the stomach like a donkey kicking you. It certainly hurts me to hear it and it clearly shows that, in terms of how education rises to that challenge, there must be a responsibility on everybody within government to approach this in a joined-up way.
"Clearly, no matter how good your education system is, if we have children who are living in poverty and in areas of severe social deprivation, our ability to educate them will be curtailed by the living conditions they endure."
Will his enemies not be looking for every opportunity to accuse him of sectarian bias? They have already accused him falsely of intending to force the Irish language on all pupils. Is he not too obvious a target?
"I come from a community that has for a long time been discriminated against and treated unjustly by successive unionist administrations and the British government," he says, "and I think I've learned too much in relation to how conflict breaks out and perpetuates iself to make similar mistakes in an administration which I'm a part of."
He will, he says, be "scrupulous and very sensible" about not giving preferential treatment to the Catholic community. "For me, that would be a disaster. It would go totally contrary to everything in which I believe. And I certainly think that to set about putting in place a strategy which possibly five or 10 years up the road will allow people to raise these issues as causes of conflict or as reasons for instability within our society would be a serious mistake on our part."
He talks frequently about a society "emerging from conflict", about the need for stability and his desire to create "a society capable of evolving into something better than we are now". When a number of Protestant children walked out of schools in protest at his appointment in December, he wrote to them, inviting them to discuss their worries with him in person. Some took him up on his offer. "We sat down for an hour and we talked about it. It helped me, and I think it helped them, and very soon after that the pupil protests ceased."
And while the future may lie with integrated schooling - one of his first decisions was to agree to the setting up of two more integrated schools - he accepts equally that, given the hoped-for improvements in the political situation, it could be "that people are actually capable of having their children taught in controlled and maintained situations - that at the end of the day, we can effectively teach people that we have to respect one another, respect the fact that we are diverse communities, that we have different allegiances, that we have different aspirations, but that it is possible for us live together and at peace with one another."
He has to go now, he says. "Sorry it was so short. I've got a busy day today." Outside, in front of the grand entrance and for the benefit of two television crews, a member of a loyalist group, accompanied by a clergyman, is reading out a petition to the Assembly and a letter to Tony Blair, demanding once more the removal of Northern Ireland's minister for education.