A pair of researchers at the University of Illinois conducted a study comparing math standardized test scores in private and public institutions. It seems like I've seen this study before. Except this time, the results are a little bit different. According to this study, it was the public school kids who did the best.
Talk about flying in the face of expectations.
The researchers looked at a variety of possible explanatory factors and found that the ones that seemed to make the difference were certified teachers and using "modern, reform-oriented" curricula. In other words, good teaching leads to good learning. Note that what made the difference was not what kind of governance the schools operated under. What matters in the classroom is what happens in the classroom, not the boardroom.
I think that has to be a key point here. Certainly, the guiding hand of governance is and will be a factor in what happens in the classroom. However, simply creating a private school (or charter school) seems unlikely to drastically improve learning, no matter what the rhetoric tells us.
The other interesting flip here is that the autonomy of private schools isn't necessarily working to their benefit. Take note all you, "the free market will force schools to improve" people. The researchers say that schools may be driven by parents' desire to follow a "back to basics" math curriculum rather than keep up to date with more effective modern ideas on math instruction. Yes, those schools may be following what the market wants, but the market is not asking for what research indicates is the best for student learning.
Frankly, this study is probably an outlier as everything else I read seems to be saying that public schools are at the bottom of whatever schooling hierarchy exists. But it does re-raise the crucial point that good teaching - more than any system of governance - leads to good learning. If we're serious about improving education, we need to look at what's actually working in the classroom, not who's calling the shots in the boardroom.