"I won't speculate about what other people might do," he says unconvincingly. His syntax, however, keeps giving him away. He cannot help referring to his current job as education secretary in the past tense: "I tried", "I wanted", "I thought".
So while his mouth keeps saying "class sizes", you feel his mind has moved on to "crime figures". But this is more than just a job promotion. In the public imagination, Blunkett is about to ascend from the middle of the premier division to the champions league. Once the holder of a key cabinet position, he is about to be annointed as a possible successor to the prime minister himself.
Which makes it all the more frustrating that he won't talk about the future. "Just tell me," I say. "And I promise I won't tell anyone else." He laughs - head back, mouth open, not a camera in sight - and rolls down the car window. "I'll just whisper it out the window and then it'll be out in the open," he says, then laughs some more and taps me on the shoulder.
Blunkett is a rare phenomenon in New Labour, someone who manages to keep on message but still write his own script. With most cabinet ministers you could hold your breath and wait for "boom and bust","prosperity and opportunity" or "sustained economic growth", safe in the knowledge you will never be short of air.
With Blunkett the two phrases you are most likely to hear are "In Sheffield city council" - which he lead during the eighties - and "I know in my heart". Both are significant.
The first because it refers not only to his home town and constituency, but also his political apprenticeship. Unlike most of his peers, he cut his teeth in the municipal Labourism of the eighties. It was a bleak and challenging time for Labour councillors. They were faced with the choice of trying to make Tory cuts as painless as possible or risking rate capping or abolition.
"I go back to Sheffield every weekend which keeps me in my place. I'm not a cabinet minister there. I'm David."
The second is important because it implies an emotional intelligence that has been lacking in political leadership of all persuasions for some time. Blunkett does not have to go off the record to sound like a human being.
Recalling the first time he went to university he says: "I kept thinking 'What am I doing here?' You have to build up a very hard shell of confidence. But then you have to get rid of it or you get chippy." The hard shell may have gone but he has needed a thick skin in the job he is about to leave.
While he boasts of the extra cash he has extracted from Gordon Brown to modernise buildings and get class sizes down, teachers bemoan the culture he has bequeathed of performance related pay as well as state-set targets for children scarcely out of nappies. Has anything been lost in the drive for standards?
"I've found a similar problem to the one I had as leader of Sheffield city council," he says. "In trying to do things as quickly as possible you don't always take people with you. I regret that I haven't been able to raise teachers' morale."
Where public services are concerned his emphasis, he admits, has always been on the needs of the consumer rather than the interests of the producer. One wonders whether he will take on the police with the same tenacity as he has faced down the teaching profession.
Blunkett's arrival provokes complex reactions. On the one hand he is a minister; a man to impress and defer to. On the other he is blind; a man who needs assistance and, literally, guidance. The two are by no means contradictory. But for the many who have met neither a minister nor a blind person before, they can present a challenge. How do you behave in the presence of someone who is simultaneously powerful and vulnerable?
The standard response is to overreact. As his guide dog Lucy took him down some stairs at the Albany road school in Cardiff two teachers rush to hold him back. "It's alright," says Blunkett calmly. It is the patient tone of a man used to putting others at their ease before he can relax himself. In a country still awkward in its dealings with people with disabilities, it will take a while for some to realise Blunkett is the man to look out for not to look after.
As we drive away he talks about a little boy from the Congo, an asylum seeker, who has just started his first day at the school. "That must be very difficult," says Blunkett. "But it's better that he should start at this stage of the year, when they can give him more help," he says.
One only hopes he expresses as much care and compassion about the welfare of the little boy and his parents in his next job as he seems to do in this one.
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