What has actually happened is that there is such a need for the scheme among native English speakers that the child receiving the extra help these days is just as likely to be white, working class and male as a first-generation immigrant.
The scheme is doing a great job, but when teachers tell you about the backgrounds of the children needing this help, you wonder how far schools should have to step in to compensate for the inadequacies of parenting.
Many primaries are finding that before they can expect their charges to get much out of the curriculum, they have to teach them basic speaking and listening skills they should have learned at home. In increasing numbers no one, it seems, has the time, or interest, to do so.
Take Tom, an eight-year-old at a primary in a mixed, but by no means deprived, area of north London. Tom is brought to school by his dad, who is self-employed, and uses the time between parking the car and getting to the school gate to set up the day's business on his mobile.
Tom is collected from school by his mum, but she is too busy texting and talking to friends to ask Tom about his day. At home, television or the computer entertain Tom while his mum makes tea. Tom eats what he's given, so not much chat there, especially as he has it by the PC or telly and his parents like a takeaway in front of a good DVD (on their own telly in the front room "for peace and quiet"). At bedtime, he is told to undress, have a shower, and go to bed.
It's not an unhappy household, just one where there is no conversation.
Tom's teacher says that while he is clearly capable, he cannot construct a sentence - orally, let alone on paper. He has no idea how to form a subsidiary clause - even a simple one using "but" or "because", and as for picking up inference from a text, forget it.
"Nobody around him uses language for anything beyond giving instructions,"
she says. Like many of his classmates, Tom simply hasn't had enough exposure to language to see the point of "Wow!" words used to enrich language, or to decode inference in text.
A passage which referred, for example, to winter "stealing up on the animals in the wood" floored Tom's class, although one of them offered the thought that "the badger's nicking stuff". That, says the teacher, was something. The default position for most of her children is to sit and let things happen. The other day, the same teacher recounts, she was cheered to see a child chatting as she walked to school with her mother. Then she saw that the mother had her iPod on and was in her own little world. Reece, six, doesn't say much. His parents, who have four other children, have invested heavily in technology to keep their offspring off their - and each other's - backs; portable DVD players in the car, Gameboys and Xboxes. Ask Reece if he talks to his brother and sisters and he replies "What for?" Ask him if he likes to have stories told to him, and he says unhesitatingly:
"Nah, DVD's gooder - it's got pictures."
Nor is this uniquely a class-related problem. In a leafy Hampshire school, where 95 per cent of pupils are native English speakers from affluent families, one teacher speaks of children who have been practically brought up by a succession of foreign au pairs and who, consequently, have poor speech patterns and have had minimal conversation since birth.
One of them, Alice, is a lively six-year-old. She interrupts and has difficulty following instructions. When she wants to answer a question, she uses haphazard grammar, onomatopoeia and extravagant gestures. Both her parents work and she is on to her fourth au pair. This one is Serbian. By the time Alice's parents notice that she isn't academically doing as well as she should, it will, naturally, be the school's fault, her teacher says wearily.
As for the EAL children, Talking Partners' original target group, the teachers I spoke to said that although they may need some language help, they are generally pretty good at communicating. Why? Because at home their parents value the traditional aspects of family life, such as conversation and making time for each other.
Fiona Leney is a journalist