There is no free market for schools. Why not?
Because the free market has utterly failed us in so many respects. Why should we let it screw up the schools, too? Would you prefer schools to be run like the auto industry? Or maybe the telecommunications industry. Health care? Banking?
1. Our schools get bigger, flashier. They get terrible gas mileage. They focus on one popular subject to the exclusion of all others, because it is what the parents want, and so it is what brings in the most money. A generation later, when the world changes, the schools are too big to change quickly, they go belly up, and our students can't compete with the Japanese.
2. The largest cities will have 3-4 different good school systems to choose from, but they will be so expensive that the poor and lower-middle class can't go. These people (often majorities in the cities) will be forced into lower-cost schools with shoddy equipment and terrible bandwidth, or will be forced to go to free schools in public places like the city center, where the schools will be so crowded almost nothing can be accomplished. Smaller towns and cities will generally have only one choice for schools, and the school corporations will rabidly fight any competing schools that try to establish, to the point of buying or burying any independent schools that get set up. Rural areas are just out of luck; if they want to go to school, they will be forced to do so by mailing handwritten letters to the nearest city (unless we privatize the mail, too).
3. Similar in distribution to the second example, but with coverage in rural areas. Individuals with learning disabilities will be ineligible to attend. To get into school, a complicated underwriting process will be necessary; any pre-existing knowledge identified by the examination will prevent the prospective student from taking classes covering those topics. This will be a broad definition; knowing the date the declaration of Independence was signed will be enough to make you ineligible for all history classes. If you know the Pythagorean theorem, you're not eligible for the entire 10th-12th grades. If the school suspects you lied about pre-existing knowledge, for example by doing well on an exam, you will be expelled from school with no chance for appeal. Costs will rise by 10% a year, yet somehow tuition will increase by 50%. Children under an employer-sponsored group plan (approx. 50% of the market) will have a significant advantage, as they can attend classes for which their pre-existing knowledge would otherwise exempt them so long as there are enough dumb children in the group to balance out the smarties.
Banking would be fun, but I need some breakfast. I think the point is made and over-made, anyway.