Three quarters of parents want schools to be allowed to set their own curriculum, and almost all reject league tables as an accurate way of judging their performance, research shows.
But 64 per cent think teachers should be paid according to their performance.
The findings come in a poll commissioned by the Policy Exchange think tank which this week called for "crude" league tables and routine Ofsted inspections to be replaced with new report cards.
Its proposal is based on practice in Alberta and Ontario, Canada, and New York and would see schools graded A-F according to a range of indicators including pupil progress and teacher turnover.
The Policy Exchange report recommends a "new deal for teachers" that would mean higher starting salaries, higher status and more autonomy. But there would also be reduced pensions, with schools allowed to develop their own local pay structures and sack incompetent teachers with greater ease.
The YouGov poll of 2,161 adults found that 87 per cent believed schools should be judged against a range of factors, rather than just results.
Helping Schools Succeed: a framework for English education: www.policyexchange.org.ukimageslibimages352.pdf.