At the supermarket checkout recently, I recognised a mother from the school run. Instinctively I looked in my trolley. On the top sat a bag of revolting frozen chicken bits, bought for the dog, and a giant bag of steak cut chips, bought for the husband (to eat with a rump steak). All the lovely fruit and vegetables were buried below, out of sight. Fearful that I'd be cast as the kind of mother who serves my four children cheap chicken and chips, I began scrabbling around trying to hide them from view. It was pathetic. I knew the truth. Why should I care what she thought? But I did. If truth be told, I am ashamed to say I sometimes pass judgment on other's people's shopping trolleys myself.

As a result of Jamie Oliver sticking his head into the nation's most poorly stocked fridges, food is, today, still inextricably linked with both class and morality. Perhaps it has always been this way. As Sheila Dillon, presenter of Radio 4's The Food Programme recalls of her working class Methodist upbringing, "My grandmother would actually say, 'We are better than those people who go to the fish and chip shop.'" And so it is now. Feed your family junk and you're a bad parent as well as a bad cook. Feed them well, and you're parent of the year.

In 1901, B. Seebohm Rowntree published Poverty: A Study of Town Life. As part of the book, Rowntree studied the weekly food shop and expenditure of 18 working-class families in York and six servant-keeping families. In short, he found that the working-class families were underfed and experiencing a poverty equal to that in London, and not previously associated with provincial towns. An equally startling aspect of the study is that unlike Oliver, Rowntree did not go looking for what he sought to improve: "I did not set out upon my inquiry with the object of proving any preconceived theory," he writes, "but to ascertain actual facts, and that I was myself much surprised to obtain the above result." In 1914, there was further proof of the class disparity, once again made apparent without recourse to finger-pointing: ex-public school boys who joined the army were on average six inches taller than the rest of the male population. Diet was a crucial factor in this, as we know from the soldiers who found army food unexpectedly lavish.

Today, a time when the UN food price index has risen for six months in a row to the highest since records began in 1990, and when the juxtaposition of food and class recalls an easy cache of cliches – from council estate mothers shoving burgers and takeouts at their obese children to rich families browsing London's Borough Market blowing £50 on a tiny basket of organic produce – it seems pertinent to repeat the questions: What are we really eating? Is a family's approach to food shaped by economics and class or are other factors at work?

There's a square in Camberwell, south London, that is an ideal location to revisit Rowntree's study in miniature. Like many city postcodes, within a few yards it encapsulates a broad socio-economic mix: flat-fronted stucco houses versus housing association terraces, many sub-divided. With Tesco, Sainsbury's, Asda, Iceland and various fruit and veg shops dotted along the nearby Walworth Road, it is also far from a "food desert". The rules of my exercise were simple: four families each keeping food diaries and receipts for one week. No preconceived ideas allowed. While it is obvious that such an exercise is not scientific, it can at least provide an illuminating snapshot of how people actually eat.

I have been visiting the square on and off for 13 years to see my friends Ralf and Jessamy and their children, one of whom, Alabama, is now a 15-year-old vegetarian (seven years without meat) and the other, Jonah, a 10-year-old gourmand. Their house falls between the two extremes. Ralf, who is German and by his own definition working class, was my starting point, all the more interesting for not being English, having arrived in the square from Berlin 20 years ago.

He has always been a fabulous cook, taught from infancy by his mother and grandmother. He remains baffled at the way the British approach eating: "You have this middle-class culture of feeding the children crap at teatime and then sitting down to better food when they are in bed. That strikes me as very peculiar. In Germany, we don't have that. Good food is an occasion for the family." It is a point reiterated by the food historian Ivan Day who believes our class/food conflicts have clear historical origins, from us being a country with a clearly defined hierarchy, not least a monarchy: "I've seen Italian lorry drivers at a roadside caff eating fresh artichokes for lunch – working-class people all over Italy and France eating really finely but simply with it." Sheila Dillon blames the broken food system caused by industrialisation: "Creating a food culture is very difficult," she says. The term "foodie", she continues, is nothing but divisive, indicative of the haves and have-nots.

Ralf cooks a meal for us – pigeon breast with Jerusalem artichoke and parsnip mash to start, followed by goulash made with organic shin of beef, served with rice and broccoli. On the table is a whole pecorino, a gift from his mother during a recent stay.

Since I first met Ralf he has become a "dinner lady" for a local nursery. He brings the principles of his own kitchen – strong flavours, exciting dishes, seasonal, British produce, refusing to offer children an alternative (his daughter eats the same dishes without the meat) – to the table of up to 80 children, many of them underprivileged and unused to his ingredients.

He is the epitome of Jamie Oliver's dream of collapsing class boundaries by bringing good food to all and he was recently shortlisted for the Radio 4 Food and Drink awards. "He is fantastic," says Dillon, who sees him and his like as this country's only hope for closing the class gap by educating children to enjoy food. By contrast, Jessamy, who grew up in an upper-middle-class home in Dorset, cannot cook: "I really do not cook, literally never. My mum only cooked a very few dishes. I don't even cook the children's tea if he's not around. Often we'll go to the cafe round the corner. I'm not interested."

Ralf's shopping bill for the week totals £178, with organic produce bought at a farmers' market and an organic butcher (he will only buy from one, the others he considers not good enough). Two dishes – shin of beef on Saturday evening followed by pork belly for Sunday lunch – come to £65 (although leftovers will do one meal during the week).

With the figure of £65 in my head, I go around the corner to the small basement flat of Nichola and Harold and their five children, all under 11. Their budget for the week is £60 – around £70 at a push. Every item is laid out on the spreadsheet Harold compiles on his computer. The housing association flat has four main rooms, a kitchen just big enough for a table, where Nichola insists they eat together, a small sitting room area and two bedrooms. The five children share a room, the two youngest a bed. Nichola, who is a full-time mother, had her first child at 17 and grew up on a diet of oven meals, interspersed with Jamaican dishes, mostly because her mother worked. Harold is from Sierra Leone. They eat stews, omelettes, curries, soups and oily fish, all made from scratch with ingredients bought cheaply from Nichola's insistence of finding the best deals. They have themed weeks – vegetarian, Chinese, no carbohydrates – as a way of experimenting with different flavours and introducing the children to different foods.

Until recently, the family had an organic veg box delivered from Abel & Cole. Desperate not to repeat the menus of their own childhoods – for Harold, Sierra Leonean food dripping in oil and for Nichola meals such as fish fingers. Recipes are sourced on the internet and Harold uses Jamie Oliver's books. "I like the way he just throws it all together. He's like me."

None of the five children was fed by jars; sweets are a once-a-week treat; they use the eggs from the chickens in their neighbours' tiny allotment. As they sit telling me all this, I am struck by three thoughts. The first is that they are doing better than I am. I have a family roughly the same size, spend at least double, have more household help, infinitely more space and yet am nowhere near as experimental: Nichola's children eat capers in their potato salad and would happily accept a dish such as fennel risotto.

Second, they are Jamie Oliver's dream, proof that you can feed a family well once food becomes a shared pleasure. The third – and most satisfying – is what better family could one wish for to disprove the cliche that low income equals low nutrition? "A lot of our approach is to do with our children," explains Harold, who works for Lambeth Council. "I'm conscious of how important their food is to their learning and we wanted to change our diet. It is much easier to be healthy as a family rather than us having a separate shopping bill. When we look in our fridge now, we eat healthy, they eat healthy."

Education has always been at the centre of the debate about food and class. Kathryn Hughes, the historian and acclaimed biographer of Mrs Beeton, points out that while Beeton supposedly appealed to the well-appointed 19th-century home, her book was intended to educate the aspirational mistress, new to employing servants.

It is this idea of being "taught" that lies at the heart of Jamie Oliver's mission to get the masses to eat better. It is, after all, what Harold and Nichola's family has benefited from, albeit in an autodidactic way. However, when somebody else is doing the teaching, there is usually some sort of comment about class going on too: "My mother was a social worker, fresh from Cheltenham Ladies College, working in Hoxton," Hughes recalls. "She was responsible for checking the cupboards of wives of prisoners to ensure they weren't buying too much tinned ingredients. She'd never cooked herself, these women were over 40 and yet there was the implication that my mother knew better."

I am reminded of this when Spenta, who lives a stone's throw from Nichola, tells me that she relies heavily on takeaways – Indian, kebabs, fish and chips and KFC – to feed her two children, 11-year-old Yazard and 10-year-old Zoish. Two years ago, Spenta took the children on a 10-week social enterprise course for obese children. Mend – Mind, Exercise, Nutrition… Do it! – was established in 2004 and devised by Great Ormond Street Hospital for Children and the University College London Institute of Health with the aim of improving children's weight and health, effectively nipping obesity in the bud. Aware that weight in her family was an issue – "my brother is huge" – Spenta heard of the course through the children's primary school. Most of the other children there were from the neighbouring council estates, bar one from a public school in Dulwich. "They told me not to feed the children takeaways," she tells me, "but I told them I am a full-time carer of my mother, I do not have the time. So then they said, 'Well, buy healthy takeouts then.' They told me to look at the labels on the foods, but I haven't got the time to do that when I go shopping. Often, I've only got an hour."

While Spenta's lifestyle did not change – except to insist the children took more exercise – Yazard now pesters her for healthy food and refuses to eat Kentucky. "Yazard loves vegetables, wants the semi-skimmed milk and brown bread," Spenta tells me. "He complains about takeout, but my mother puts him straight saying: 'Your mother is not a cooking machine! Today she just cannot do it!'"

Spenta's weekly bills, about £139, are the second highest spend. Their diet, on the other hand, relies the least on fresh produce. Is this class-related? Spenta does not think so. If anything, because she is Zoroastrian, she sees herself as middle class. She knows they should not be eating takeouts as much but she is breathtakingly honest about the restrictions of her life. Her mother, Bapsy, had done all the cooking until she fell ill. Spenta found herself at 40 with no skills of her own having to fend for the whole family. Recently her own health has begun to suffer because of the physical and emotional strain. She is clear, too, about the psychological function of the food she buys: "Food is a reward. Other people go out to restaurants, or dancing, and this, I would say, is my reward. Whatever money we have got we spend on food. As long as it is tasty and hot, I am not bothered what it is."

Over on the other side of the square live Regina and Andrew, both actors, and their adopted two daughters, Megan, 11, and Lily, 12.

"When I first met Reggie," Andrew said, "she would go out and have a McDonald's and these big tubs of Coke and I couldn't believe it… this classy girl!" Reggie laughs and confirms it's true. By anybody's standards, she grew up in a privileged home. Until her father lost all his money on the stock exchange – he was a millionaire – her parents travelled by Concorde and dined in the best restaurants, often taking their children along with them. Food was appreciated in a restaurant environment, but Reggie's cooking confidence was squashed at school. Meat loafs would end up in the sink – "'Poor Regina,' my home economics teacher would say. 'She tries so hard.' I am aware that lots of our middle-class friends haven't even tried a McDonald's. I do enjoy it but I suppose I do know when to stop."

Andrew grew up as one of six with a mother who cooked everything from fresh. He is responsible for all the cooking and shopping now, spending around £100 a week, mostly in Iceland, where he is constantly looking for the best deals. He makes his own bread every Friday and likes to use fresh ingredients. More than anything, though, the family's approach to food is shaped by the two girls, Lily, 12, and Megan, 11, they adopted five years ago. Lacking any structure in their previous lives, the girls needed routine and ritual. Now, they all eat the same dish on the same day of the week, week in, week out, to ensure the children feel secure. Both had to be weaned off their former diet of chicken nuggets and chips.

Pasta now features twice a week where once there were chips. Roast chicken is served every Sunday, meatballs are given with rice. There's ratatouille in the winter on one evening, replaced by salad nicoise in the summer. There is no deviation from the ritual. It's up on the wall on the girls' Charlie and Lola week planner.

For Reggie, the girls' new diet is directly linked to their improved health: "Megan had a permanently runny nose when she arrived, permanently. That has now stopped. Their colour wasn't great either." This brings to mind a line from Rowntree's Poverty: "It is to be regretted that so many children are brought up in the Workhouse, as they can hardly fail to become affected by the pauper taint."

Food for children – young children at least – is generally uncorrupted by notions of class. They do not look at a friend's lunchbox and see that the chocolate bar and salty crisps could mean more than the food itself. They just think their friend is lucky to be given such treats. Food to a child is simple and if you're lucky, it turns into a simple – and healthy – pleasure. When I think about the four families, there is one trait which is the same in each. In every family, there was at least one child who wanted to be allowed to cook regularly. For Ralf and Jessamy, it was 10-year-old Jonah. For Nichola, it was Lenise, 11. For Spenta, her son Yazard, 11, and for Reggie and Andrew, 11-year-old Megan. In every case, the parents were willing in spirit, but all expressed the sensation of a sinking heart at the idea of a chaotic kitchen that needed to be cleared up afterwards. Can it be a coincidence that all these children are around the same age? Could it be true, as Sheila Dillon told me, that our national wellbeing ultimately lies in educating and encouraging the children, rather than berating their parents? Hold on to that thought as your child spreads flour over your kitchen floor. I know I am going to.