Professor James Tooley, interviewed in this segment, studied how the very poor get education in India, Africa and China. The majority of those families, who earn just a few dollars a week, choose low-cost private schools over free government-run schools. He reports that their kids get better education from the less-qualified and less-paid private teachers than from their government counterparts, for about a dollar a week.
Throughout the show, Stossel pertinently questions the American public education system, on the basis of quality, price and principle, also pointing at the incentive problem. Other things being comparable (kids, neighborhood, etc.), charter schools perform better and cost less than government- and union-run schools.
Education is not different from other goods and services; there is no reason not to let the private sector provide it, on the contrary. As with any centrally planned service, the problem of economic calculation in the absence of private property re-surfaces resulting in political rather than rational allocation of resources. Were they private initiatives, money sinks like "No child left behind" and "Head start" would have been canceled already, making room for better programs.
The only rationale left for public school is universal coverage. But as we know from many examples, the free market better solves this problem by democratizing services to the masses and serving the remaining unfortunates with the help of charities. And it does so without incurring all the downsides of socialized education.
As a step towards complete separation of education and state, Stossel advocates a voucher system to regain some of the benefits of the private sector. Such vouchers have been shown to be successful, in randomized experiments, yet politicians still oppose them.