From The Rebirth of Education:

Whereas formerly only the elite may have gone to private schools, there has been a massive proliferation of private schools, especially in Asia and Africa.  These budget-level private schools are producing better learning outcomes, often substantially better, than publicly controlled schools – even for the same students – and often at much lower costs.

But:

This isn’t to say that across the board, private schooling is better than that available in government-run schools; in general, the evidence that private schools outperform government schools in well-functioning systems is weak.  In the United States, where there has been the opportunity to do the most rigorous experimental studies, most researchers agree that the private sector edge in learning is nothing like a full effect size [1 standard deviation], almost certainly not even a tenth of an effect size, and some legitimately dispute whether the private sector causal impact is even positive.

Yet the fact remains:

Broader than just the success of specific interventions inside government schools is the observation that even in low-performing government systems one finds excellent schools, but also, even nearby and even operating under apparently exactly the same conditions, terrible schools… The problem is not that government schools cannot succeed, for in nearly all developing countries some of the very best schools are government schools.  The problem is, as the LEAPS study authors emphasize, “when government schools fail, they fail completely”…

Case in point: In Mexico, “essentially all of the weakest-performing schools – those more than 100 points [2 standard deviations] below the average – are government schools.”

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My main objection: I strongly suspect that private schools have a big cost advantage over public schools even when they don’t have much of a learning advantage.  This effect is easy to miss in the First World because there is relatively little demand for cheap adequate private education when there’s free adequate government education.  But religious schools strongly suggest that private education for the masses can be provided at WalMart prices.  And if parents were paying their own money, WalMart pricing would probably dominate.