The timing is so perfect that it feels like a plot. Nearing the end of a long tour of a flagship city academy in Leeds, Ed Balls agrees to drop in on one last class and finds himself in the middle of a debate on whether MPs should be allowed to vote against their party.

The irony is probably lost on year 9, most of whom cannot know just how close the government came to melt-down over a threatened rebellion by backbenchers last week.

But for the secretary of state for children, schools and families, Gordon Brown’s closest ally, the seemingly innocent question is painfully close to the bone. It would be hard even for Balls to deny that the prime minister’s authority suffered a severe blow when he was forced to cave in to the demands of Labour MPs who were unhappy about the abolition of the 10p tax rate. Now more rebellions loom, which makes the subject of party discipline acutely sensitive.

So what’s the answer? Twenty pairs of eyes swivel towards the minister, awaiting his verdict.

“The deal is that all MPs vote for the party line on things that are in our manifesto. Otherwise we wouldn’t get anything done,” he says briskly. But he does admit that if the government doesn’t respect or listen to its MPs, “they get angry”.

Earlier, Balls has been treated to a performance by one of the school’s star pupils, a gifted pianist and singer. Appropriately, the song that 14-year-old Curtis Johns has chosen is called Give Me Strength. The sentiment must strike a chord with the education secretary, who has seen Labour plummeting in the polls, facing the real prospect of defeat in London’s mayoral elections this week and reeling from the first teachers’ strike in 21 years.

Recently he has also faced damaging accusations from his own colleagues that he has been cynically trying to position himself as a future leader. Blairite MPs and party activists who blame Brown and Balls for forcing the last prime minister out of office gleefully muse about how “beautiful” it would be if Balls ended up knifing his boss.

“I am very loyal to Gordon and always will be,” Balls retorts, dismissing the notion.

However, some at Westminster believe that Brown has privately told Balls that he does not want to serve another term, leaving the way open for his most trusted former Treasury adviser to take over. Balls won’t say if he’s interested in the job but he repeatedly fails to rule it out.

“Politicians who give the impression that the next job is more important to them than the current one aren’t politicians who win the trust of the public,” is all he’ll tell me for now.

How does he think Brown is doing? Has the prime minister exceeded his expectations? There is a long pause before he offers a diplomatic answer about times being tough, what with the economy and all. It’s been a “difficult, rocky period”, he admits.

The prime minister’s Labour critics have temporarily suspended hostilities pending the outcome of this week’s elections, but Balls is adamant that Brown must continue as leader, whatever the result. “It was not by accident that Gordon was elected unanimously and I don’t think anyone else is putting themselves forward,” he says.

If he is depressed by Labour’s tribulations, at least the visit to David Young community academy in Leeds seems to cheer him up. The spanking new establishment, which has facilities to rival most independent schools, is viewed as a model of Labour’s education ambitions. Created out of the amalgamation of two sink schools, it achieves soaring GCSE pass rates despite exceptionally high levels of deprivation.

Unlike some cabinet ministers, Balls does not always have an eye on his watch and lingers to talk at length to several of the pupils. If he wants to be prime minister he will have to learn to display more of this charm to backbenchers. His big problem is his close association with Brown. Unless the prime minister’s fortunes improve dramatically, Balls will be damned by association.

As he continues to walk round, chatting to pupils, it strikes me that Balls, who read politics, philosophy and economics at Oxford and won a scholarship to Harvard, must wish that his own children could attend such a school.

Instead, Ellie, 8, and Joe, 6, are stuck at a primary in Hackney, east London. which received such a disastrous Ofsted report that it is threatened with closure. The Jamaican-born head teacher recently resigned amid allegations that she was “driven out” by white middle-class parents.

Balls and his wife Yvette Cooper, also a cabinet minister, are supersensitive about the subject. Publicly he will only say that, like every parent, he wants the best education for his children.

Privately, however, the couple must be wringing their hands in despair at the position they find themselves in: if they intervene they will be accused of throwing their weight around; if they withdraw their children they will be accused of hypocrisy.

To rub salt into the wound, the logistical challenges of sending their youngsters to Grazebrook school are immense. Every Sunday night the entire family decamp from their home in Castleford in West Yorkshire, where Balls and Cooper have neigh-bouring constituencies, and travel to London for the week. Taxpayers pick up the tab for the grown-ups’ first-class tickets (an open return from London to Wakefield is just under £300), although the couple pay for the children’s fares.

Earlier in the day I joined the family on their trip north. Cooper sits with Maddy, aged three, and her sister Ellie, who has been off school with a sore throat that day.

Maddy soon clambers onto her mother’s knee and the three spend most of the journey doing a jigsaw puzzle. Balls, meanwhile, does not flinch when one of the girls tries to put a pink cap on his head and happily goes off to buy everyone bacon sandwiches.

I note that Cooper has what looks like a pile of briefing notes on her lap, but she does not seem tempted to read them; time with the family is precious.

Interestingly, there are no big suit-cases. This family have two of everything so they don’t need to haul clothes and toiletries up and down the country. The practicalities of ferrying three young children between two properties, while holding down two cabinet jobs, are mindboggling but Balls and Cooper somehow pull it off.

“Until something goes wrong,” Balls says wryly. “Then it’s really bad.” They have a nanny, who travels with the children when they cannot, and there are grandparents who bail them out.

With a joint salary of about £280,000 they can easily afford the nanny, but Balls sympathises with those who struggle to meet the huge costs of childcare. However, he has stern words about the widespread practice of paying nannies and home helps cash in hand.

“As someone who comes from years in the Treasury, married to a Treasury minister, I can’t condone that at all. I don’t think the black market in childcare is the right way to do things,” he says.

Now 41, he had what he calls an “excellent” private education at Nottingham high, an all-boys’ school. “I think, probably, if you can make it work, coeducation is better. That would be a personal point of view, not a government view,” he says.

Does he feel that he missed out by going to a single-sex school? “Yeah. The girls’ school was over the road. In my day there wasn’t a great deal of contact.

“There used to be a square piece of ground with a fence round it on the girls’ side, called the MTA – the mixed talking area – and boys and girls in the third, fourth and fifth forms were allowed to go in and do supervised talking between 1pm and 2pm.” He laughs.

Despite sending his own children to a local school, he says that he isn’t against private education: “My mission is to make sure that every school in the country can be a good school and give pupils the high expectations and sport and cultural opportunities that I had at my school.”

There are still 638 schools where less than 30% of pupils achieve five GCSEs and Balls admits that there is a long way to go.

Critics within the party, such as Frank Field, the respected former welfare minister, despair at the pace of progress and have argued for children to be forced to stay on at school until they are proficient enough in English and maths to get a “leaving certificate”.

Balls dismisses the idea (“think of the knock-on effect on class sizes”) but believes that state schools can learn from the best private establishments. He talks enthusiastically about plans for Wellington college, the prestigious public school, to sponsor an academy in Wiltshire aimed at the children of military personnel.

He also sympathises with parents who cannot afford private education and go to desperate lengths to secure places at faith schools, many of which, he acknowledges, are “brilliant”. Like David Cameron, the Tory leader, he refuses to condemn the many parents who privately admit to attending church services simply to be able to send their children to a faith school.

“There’s always a sense of grappling with your faith. The idea that you can sort of start making judgments on people’s faith is a very hard thing to do. I think you have to leave that up to individuals and their local priest or vicar.”

As minister for children, Balls has vowed to make Britain the best country in the world to grow up in. Like every parent, he says that he worries that youngsters are maturing too fast and he is particularly worried about their exposure to nasty material on the internet.

“Did you know that YouTube is only supposed to be for over-13s?” he says. “I’ve only just found that out.”

He worries about what his own children might watch on the popular video-clip site and believes that public service broadcasters have a duty to protect very young viewers from distressing news and footage.

Our time is up and he is soon due to deliver another speech. He turns to his aides and asks what it’s supposed to be about. ”I prefer not to think about it too much beforehand,” he confides. “I do it better that way.”

Some at Westminster will see this as a sign of arrogance. Others will say it is the type of supreme self-confi-dence that gets a politician to the top.

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