First off, I think you’ll have to concede that liberal democracies can be superior to monarchies. Its not like there was a free market and then everyone saw it was bad and decided to abolish it.
Moreover, the state can have the best anything in the world because they can simply raise the necessary funds via taxation. But this is cherry picking; it doesn’t reveal how wealthy society as a whole is.
[EDIT] On some level, facts won’t help you. If this guy is like I think he is, you will need a strong personality, backbone and intelligence to overcome him. Im not saying you don’t have these things. I hope you do. Another thing you can take advantage of is you have the internet right in front of you during class on your laptop (maybe). You can look up things as he’s saying them. Do a quick search of mises.org while he’s rebutting you.
There was also a thread about the cost of public schools in america… here
Overall just be bored with what he has to say. Liberal democracy theory is laughable. There’s no reason you have to have a state institution to have collective action. The voters who want education can just pay for it themselves and leave everyone else alone. Why are they allowed to attack American Citizens? They don’t attack Canadian Citizens? Is it because the American Voter owns other Americans? Would it really put a wrench in their plans so much not to be able to loot and steal?