Grammar schools used to be seen as the best way for bright children from poor backgrounds to improve themselves.
Education officials discourage 11-plus private coaching
It is a vision which has inspired teachers such as Stephen Nokes, headteacher, at John Hampden School, High Wycombe, Buckinghamshire.
"My vision for the school is that we address the issue that grammar schools used to do...of advancing social mobility," he explained.
"I'd be very disappointed if we were elitist and we strive not to send elitist signals out," added Mr Nokes.
But some academics believe these hopes are unrealistic in the face of the selection system for grammar schools.
Experts such as Professor Brendan Bunting, an educational psychologist from the University of Ulster, argue that there is a relationship between intensive coaching and success in 11-plus verbal reason tests.
Prof Bunting said two studies he carried out, showed that children who are coached for nine months or more improve their 11-plus scores by 30 to 40%.
The argument continues that it is the better off families who can afford such extra private tuition and this skews the social intake of grammar schools.
"I think it is very difficult to believe in a system which is so obviously open to the effects of coaching and there's a downside in terms of the selection bias that goes with coaching," he added.
Education officials and grammar schools across England discourage parents from seeking extra tuition for their children.
In Buckinghamshire, for example, parents are advised that the three practice tests taken in primary school prior to the test, are sufficient preparation for the exam.
Councillor Marion Clayton, Buckinghamshire County Council's cabinet member for achievement and learning, told the BBC, "Parents have a responsibility for their children. We can advise parents.
"We can advise schools and I know that schools give the same advice to parents that what we offer in school is the optimum preparation for the 11-plus," she told the BBC.
Councillor Clayton is sceptical about coaching making much difference to exam results.
"I'm not a professional. I take professional advice and I have absolutely no professional evidence to say that further practice over and above what is offered in schools will make a significant difference to outcomes," she said.
However Councillor Clayton's guidance made little impact on parents from Buckinghamshire interviewed by the BBC. One family said it had paid £1,800 on coaching fees.
And there is a suggestion that parents from other parts of England also ignore advice from education authorities and grammar schools to avoid coaching.
Five hundred and forty-four parents from 26 grammar schools completed a BBC questionnaire.
The questionnaire, although not representative of every parent from the 164 grammar schools in England, showed that 443 of the parents who responded, had either paid professional tutors to coach their children or had provided extra help themselves.
Two hundred and fifty-seven hired private tutors spending an average of £700 on fees.
Of the same sample, 294 said they earned more than £50,000 annually.
The answer to the problem of skewed selection according to Dr Rob Coe, professor of education at Durham University is for grammar schools to change their selection policies.
"If the brief was was to design a test that would bring the best GCSE results, I would go with verbal reasoning, but if the brief was to increase the number of children from socially less advantaged backgrounds, I wouldn't," he said.
His views are echoed by Stephen Nokes at John Hampden Grammar.
"I suspect that a lot of my colleagues would agree that we are here to bring social mobility.
"What we should be doing is identifying potential.....I would not pretend that any situation could be 100% fair."
And according to Prof Bunting finding a fair method of selection is a challenge schools and education experts have yet to meet.
"I would have great difficulty believing that on the basis of nearly 100 years that we have been trying to do that - I don't think we have been very successful."