As for homeschooling being illegal, I know from the tutor who teaches my son English at the group for homeschooled children that he attends that there are parents from countries where homeschooling is not allowed who have gone on the run with their children rather than subject them to the misery of attending schools that they hate.
Some of them have given up good jobs and are living in poverty simply in order to protect their children from the horrors they've experienced in schools.
What if I'm done with school and get interested in politics but never learned anything about political systems and how they work because I was bored with that subject when I was younger?
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You can change public schools, you can even change to a private school the argument "schools are generally bad that's why I home school" doesn't work for me.
School is not only about education but also about learning social skills, making friends, learning to deal with stress, experience fruits of hard work, experience devastation. Well, in my school, Gymnasium, both philosophy and logic are mandatory subjects, as is ethic or religious education(pupils can chose between these two), mandatory subjects are also sociology and psychology and I personally find all of these subjects quite interesting, but they are not for everyone, it is not that others find these subjects boring but it looks like not everyone is capable of thinking in the way that is required in order to understand these subjects(logic especially).
Of course there is lots of stuff I learned in school that I don't remember today or have to look up again but I think it's important to give children a a wide range of options once they are done with school.
Interests change so much especially when you are young.
My husband is the same, he was interested in maths, science, and metalwork, and the knowledge he gained at school in those subjects has stayed with him, while the other things he left behind completely.