Just before he was assassinated, John F Kennedy was reading a book called The Other America: Poverty in the United States, by the distinguished social commentator Michael Harrington. It was about the poor and socially isolated whose lives many Americans knew nothing about. It had impressed the President so much that he was about to pass new social legislation when he was murdered.

In this country, a parallel phenomenon – what a modern-day Harrington might call "The Other Britain" – has been exposed by a series of shocking events that culminated last week in guilty verdicts in the trial of Karen Matthews and Michael Donovan. They were warned by the judge to expect long prison sentences for kidnapping Matthews's nine-year-old daughter, Shannon, in Dewsbury, West Yorkshire, last year.

Matthews's plan to make money out of Shannon's abduction has been presented as the work of a criminal mastermind, even though the sequence of events suggests something much more ad hoc and chaotic. Inevitably, the case has been compared with that of Baby P, who died after months of abuse in north London. "Pure evil" is how the detective in charge of the inquiry described Matthews, and his grandstanding for the media struck a chord.

We have been bombarded with unflattering photographs that show her as slumped, overweight and looking more like a woman in her 50s than her actual age of 33. She has never been out to work, and is the mother of seven children by at least five fathers, existing in such a state of ignorance that she described the two children with (probably) the same father as twins, even though they were born in different years.

Not surprisingly, no one has had a good word for Matthews since her conviction. "Karen just goes from one bloke to the next, uses them to have a kid, grabs all the child benefit and moves on," said John Bretton, father of her eldest child. The managing editor of The Sun, Graham Dudman, made the same allegation on Friday morning's Today programme, accusing Matthews of using her children "as a meal ticket" so she could get a bigger TV or council house. This is a favourite claim of the right, which likes to imagine that the working class – contemptuously lumped together as "these people" – are drunk, lazy and uneducated, but also capable of making actuarial decisions based on self-interest. The Daily Mail went further, describing Matthews as "lazy, sex mad and living on benefits, a pathetic symbol of broken Britain".

In fact, it is more accurate to suggest that Matthews provides an insight into an aspect of this country of which most of us are still barely conscious. The problem isn't one of entire communities blighted by feral behaviour; on the contrary, it was Matthews's neighbours in Dewsbury who campaigned and made the little girl's disappearance a national story, despite having neither the education nor the contacts of Madeleine McCann's parents. It was also local people who realised before anyone else that Matthews's version of events didn't add up.

The Chief Constable of West Yorkshire, Sir Norman Bettison, got nearer to the truth when he talked about "hidden, secret parts of our community" two days ago. We simply do not know what goes on behind closed doors, and even when someone guesses that something is badly wrong, social workers are reluctant to act. In a depressingly familiar pattern, a worried neighbour reported Matthews to officials at Kirklees Council three times in six years, yet Shannon was removed from the child protection register. The Dewsbury MP, Shahid Malik, has led calls for a public inquiry, but such demands miss the point; what's needed is a much greater willingness to acknowledge the abuse that takes place within individual families, and the stratagems adults adopt to conceal it.

Not every hard-up family in Dewsbury abuses its children, even if a cycle of deprivation – teenage pregnancy, incomplete education, low expectations – is repeating itself. What the authorities aren't good at, in West Yorkshire, or elsewhere, is spotting families where abuse is occurring in myriad forms, from beatings and incest to rape and deliberate isolation. Most of us don't live in a "broken society"; we love our families and friends, raise money for charity and campaign for causes we care about.

But we need to acknowledge the existence of apparently respectable households where a father can rape his daughters and make them pregnant on 19 occasions; where girls of 16 or 17 can be packed off to the Indian subcontinent to be married to uncles or cousins; where toddlers suffer a catalogue of unexplained injuries, including a broken back; and where teenagers can grow up in strict religious families without being allowed to make friends from different backgrounds, leaving them vulnerable to bullying and sexual abuse.

This is The Other Britain. It is real, and its victims need help as urgently as the socially excluded whose plight moved President Kennedy half a century ago. But it has little to do with the fantasy world of the popular press, where feckless women who live in council houses get pregnant just to get a widescreen TV.