The parochial system relies upon the Church. Charter schools rely upon the kind of localism that used to characterize public education in its earliest manifestations. Independent private education relies upon the American genius for association that so fascinated Tocqueville. This is so because school choice avoids the false choice of statism and libertarianism. Government can focus on what it can do effectively (sometimes too effectively), namely transfer payments, and leave administration and vision to those with a greater stake in the outcomes.
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