In another post I wasn’t too kind to state involvement in education, saying it is a great crime against children, young adults and their families. Hopefully the following helps explain my thinking.
First, compulsory education for children. The state is forcing demand for a particular version of education and socializing on the poorest and most vulnerable. This is an assault on their status as free human beings and relatively reduces the role of their parents and the rest of the family in their life in favor of violent strangers. This can’t be a good environment for children to interact with each other and adults.
Second, property gets confiscated to fund state involvement in education. This taxation is itself robbery and also discourages production. Unfortunately, the state has a prominent role in financing education with other people’s money. It has far less incentive to perform well because of the nature of the organization and relations compared to the market, so the kids get less in marketable skills and the community gets an inflated cost of education and the money just keeps flowing. In higher education it also provides incentives for greater costs and debt and lower quality by encouraging existing students to over-consume and the admittance of the under-qualified.
Third, government control over teachers and content and other related privileges for the sector. Essentially, with the unionism, bureaucracy, legislated standards, barriers to entry and so on, it is restricting the marketplace even more which makes the poor quality and high cost even worse.
Also, when it comes to the state, even just the one in America, I think it is important to remember certain things, especially with the education of children and others involved: it threatens violence to dozens of millions of people if they don’t surrender their property to the tune of trillions annually; that it uses other people’s money to bail out connected failures; that it claims the power to tell people what they can’t consume and whether they can practice many trades or professions; that it has murdered hundreds of thousands of civilians in war; and more.
This is a rather dangerous institution to support forcing itself on education, especially of children and with it pressuring them to pledge allegiance. Indeed, it is possible that the very purpose of the modern system of state involvement in education is for it to have greater control over the people. 
Basically, I want the separation of education and state. Abolition brings an end to the government’s role in schools, which means four things: the end of compulsory attendance; the end of government control of content; the end of government control of who teaches; and the end of the government’s practice of taxing some people to pay for other people’s children to go to school.  But it doesn’t bring an end to education, if anything it will enable education to flourish after decades of the state.
I’m not a planner. I’m not sure who would teach kids, what kids would be taught, how much it would cost, how it would be structured and so on. That is for people to decide through their own plans, voluntary exchange and cooperation and it is what will reform education. In comparison, the state, with what amounts to an education-industrial complex, has been unjustly crowding out and criminalizing many plans in favor of their own.