When I decided to work part-time - to take two days off a week to look after my baby daughter - I expected that my Tuesdays and Wednesday would bring me face to face with dozens of other men doing the same thing. The larks we'd have! We'd scoot round the park together in groups, amassing armfuls of conkers. We'd gather round CBeebies and laugh at the VPL of Justin the resident mime artist. We'd stage dramatic readings of The Gruffalo. We'd pile up our charges on the squashy sofas of the local Café Rouge, and argue over the merits of pull-up nappies and last night's Doctor Who.
It never happened. For one thing, the baby-changing facilities in Café Rouge are only in the ladies. And for another, men pushing strollers on weekdays in my part of south-east London are as rare as really good hardware shops.
A survey published last week by the Equal Opportunities Commission (EOC) suggested that 79 per cent of men want to spend more time with their children. That's almost four out five dads declaring their desire to be left looking after the baby. The same organisation claims that the average father does one third of the childcare duties in his household.
So where are they all? Why do I not encounter other men in the supermarket, pushing the buggy with one hand and steadying a basketful of rice cakes with the other? Why am I always the only grown-up male belting around on my hands and knees across the padded vinyl surfaces of the toddler gym? Why am I the only man singing along at my local library's Baby Rhyme Time?
This last one isn't so isolating - the woman who leads the session each week has a voice like one of Homer Simpson's sisters-in-law, so I'm actually the only other member of the group capable of hitting all the notes in her version of "Wind the Bobbin Up". But in other circumstances - at clinics, particularly - my daughter and I have had to contend with the occasional raised eyebrow, the odd suspicious mutter. Beady eyes have watched us to see if this rogue dad is capable of dressing and undressing his daughter without breaking one of her legs.
It's not so bad as it once was. A male friend of mine who gave up his career 11 years ago to look after his son was obliged to deal with open hostility at the school gates. "I felt examined, stared at," he told me. "If they ever talked to me, they'd always ask me why I wasn't at work. They'd assume I'd only come to pick up my son because my wife was ill. And when they realised I was there every day to pick him up, they began to think that there must be something wrong with me instead."
In 2005, trotting around with a baby is one of the few opportunities available to white middle-class males to feel what it's like to be part of an unregarded minority. I scowl at the stick-person on the baby-changer instructions, who always wears a skirt as she scoops and bags stylised poo. I will not feed my daughter anything from the Heinz Mum's Own range. And I've grown to resent the gender stereotypes that are peddled in children's songs. My daughter's daddy does not "drive a big red car". He can't drive a car at all. And whereas "the mummies on the bus go chatter chatter chatter", the daddies seem to have missed the bus completely. Doubtless most of them are working late at the office - statistically, fathers spend more time at work each week than their childless counterparts.
There are millions of fathers for whom part-time work is an economic impossibility. While most men continue to earn more than most women, families will be under pressure to divide domestic responsibilities in just the same way that their parents and grandparents did - let mummy look after the baby, and daddy disappear each day to sell his labour.
And our employment laws do too little to encourage change. In Sweden and Norway, fathers get a month's paid paternity leave, and there is a financial penalty if they decide to sneak back to work early. Last week, the new Health secretary, Patricia Hewitt, rejected a proposal to introduce a similar scheme in Britain.
So once the statutory fortnight granted to British fathers is up, what happens? If a British dad works for a big company, then there's little his employers can do to prevent him from reducing his hours - apart from revising the rota with bad grace, and then denying him promotion for as long as he's with the firm. (It's a trap that many working mothers have been forced to make the best of.)
Having always been self-employed, I've never been forced to ponder this dilemma. But I was set right by a 34-year-old former insurance man who now spends two days a week with his daughters. "A lot of men don't do it because they think they'd be stalling their career for evermore," he says. "And they might be right. In a lot of workplaces it's considered a weakness or a flaw if you have to leave work to pick up a child because they're ill. And that's why most men don't work part-time. It's easy to say you want to do it, but it's difficult to do. And once you've done it there's no turning back - not while you're working for the same employer."
So what can be concluded from the EOC's new, cute, heart-warming Father's Day statistic? Might that 79 per cent simply be a measure of men's proficiency in the field of sentimental hypocrisy? On Friday nights, the pubs of Britain are full of blokes enthusing about their little soldiers and princesses, and ordering themselves another pint. Other research published by the EOC produced another statistic that provides a revealing context for the week's star number. Eighty per cent of the British population are satisfied with the length of hours that they work. How many of those surveyed, I wonder, were dads?
This inconsistency might offer a sly acknowledgement of the world of difference that exists between a day with baby and a day at work. Entertaining, feeding and comforting a small child from the 6.30 wake-up call to the last stunning bottle of milk is much more exhausting, much more demanding than any day I've ever spent at the office. In the office you can finish a cup of coffee. You can talk to people. You can eat lunch sitting down. Nobody, generally speaking, will flick spoonfuls of cheesy pasta on your shirt.
How many of the men surveyed by the EOC were being entirely honest when they said that they wanted to spend more time with their children? How much time, exactly, would they like? An extra two days a week? An extra two hours?
How many would have dared say that the pang they felt when they looked at a picture of their child on their desktop screensaver, frozen in a moment of idealised loveliness, was easier to cope with than the more rigorous and unrelenting demands of an ordinary day spent at home, banging xylophones, pointing at cats and dogs, wiping bottoms, making food, drying tears, kissing, tickling, giggling, cuddling and chasing?
How many would admit that compared with the easy gratification of scoring points off people in meetings, gossiping around the water cooler, surfing the net when nobody's looking, and reading a nice novel on the train home, the pleasures of a day with your child are rather more hard-earned?
Next time that someone comes round with a clipboard to ask them about fatherhood, they should do one of two things. Tell the truth. Or come and play.