Can I chip in that I have taught in both sectors, and the thing that shocked me most about teaching in comprehensive schools was that I felt many children were underachieving for fairly simple and straightforward reasons, yet many colleagues seemed to regard this as normal and acceptable. In one case I even experienced out and out resentment for tackling it in my own classrooms.
This attitude was alien to me, as I felt they could have been much more exacting about things like: regular use of dictionaries, legible handwriting, correct spelling of the technical terms to do with the subject, appropriate layout of work (eg dating, page numbers and so on) so the thinking behind it was logical and good records of classwork were kept, individual use of modern text books rather than sharing scruffy old ones between three or even four pupils, appropriate physical layout and tidiness of the classroom to promote an ordered and scholarly environment, visually attractive, regularly updated and thoughtful displays of excellent work, annotated marking of pupils' work, and other things like that. All this should be the stock in trade of any school that has pride in the job, but disappointingly this was not always the case where I taught.
It seemed to me that some children were allowed to amble through the school day in a kind of underachievement haze, roaming from room to room carrying their worldly goods around with them like nomads, never properly settling, and never actually completing anything properly. I think this is an important part of tackling underachievement at source, recognising the need for stucture and order as a basis for a thorough education and more creative and holistic elements interwoven through this.
When parents are shelling out for independent education, I believe this is mainly what they are buying into, and this is one of the reasons I hope for more ebb and flow between the two sectors in the future. If more teachers worked in both sectors, and more parents used them, then it would soon become apparent that different practices were existing alongside each other, but that some were more effective than others. I honestly think funding doesn't always even enter into it - it's about an attitude of mind in many cases, and knowing what the possibilities are.
I am not criticising the maintained sector wholescale in this posting, by the way, as there are many excellent maintained schools that do all the things I have described and get good results accordingly. But I do think that instead of slating the middle classes for making an intelligent and informed choice about what they want for their children's schooling (very on message politically since the 1988 Act, so they are only doing what they are told they should be wanting to do), it would be more productive to free up the two sectors so that it was easier for children to move between both, and more children got the opportunity to experience both. That way we would be looking at standards and social inclusion at the same time, and we might make more progress, because the two are inextricably intertwined. To an extent this has happened in Early Years settings, so perhaps we need to roll this out even further. I think that has been part of the motivation behind the Academies programme, for example.