What is the purpose of public education?Or to put it another way, why is it compulsory for children to spend thirteen years of their lives learning in school; what benefit do they get from it? Assuming that the children benefit, what do their parents get out of it? And what of society at large? Do we benefit from educating other peoples' children?
In the first instance, that of the child as an individual - we live in a big and complicated world. Centuries ago you could learn a trade from adults around you without any formal instruction. Almost everyone you knew would be close enough to walk over and say hello. Any contracts and tenancies into which you entered would be on traditional terms. Unless you were rich you would handle very little money in your life, and mostly make deals by paying in goods, or remembering what you were owed. And your rights in your local political system were likely to amount to "rise in rebellion if it all gets too unbearable, otherwise put up and shut up".
So the need for education was minimal.
Now, on the other hand, just to live quietly in a small town, work at a nondescript job, and stay out of trouble, you need to:
Read and write, quite well - you will sign contracts to secure your dwelling place, your employment, your communication with friends and family. More informally you will get written instructions to assemble your furniture; you will receive birthday cards; you will want to write to your MP or Councillor about your concerns.
Be able to handle arithmetic, compound interest and other mathematics - you may take on debt to buy a home, get an education, or just to get something you really want. If you don't understand how interest works, you are likely to pay much more than you need to, and may cause problems for other aspects of your life.
Have a knowledge of complex systems - to know your legal rights; to know how to get involved in politics; to keep up with what's going on in the world; simply to find the right bloody form to fill in at the post office, requires not just personal knowledge of how society works, but also knowledge of who to approach with your questions, how to judge the trustworthiness of sources etc.
If you don't have an adequate education you will not be able to find a job, or at least not one that's interesting and has any chance of going anywhere; so if you want to do interesting things at work, and have enough money to do interesting things with your free time, some kind of education is essential.
Also, you need a certain baseline of education in order to be able to get more - many information sources assume a certain level of competence. For example, almost all companies assume that you can read - and that you can read in English. More and more, it is assumed that you can access the Internet, and that you will be able to make use of information you find there. Supermarkets assume that you can't add up very well, so that they can make more profit, and it is very much in your interests to prove them wrong.
Enough about the individual - most people realise at some point that education is worthwhile for them personally, albeit it takes some of them years after leaving school (as it happens this is why I am in favour of offering free courses at Further Education colleges to all, up to A-level.
It is in a parent's interest to educate their child; a parent should want the best for their child, and the best cannot be had without an education. If you want your children to decide what they want to do in life, and to achieve their potential, then spending some time teaching them what life has to offer, and developing their skills and knowledge, is fairly vital.
The tricky thing, obviously, is that all parents have limited expertise, and also limited time; so if home-schooling were universal, many budding scientists would get a great education in the humanities, but not learn the first thing about science because their parents have no expertise in it; children who could have been great novelists will end up as lab technicians because their parents didn't have the literary chops to light the touchpaper; and children who could have done anything will get hardly any education at all, because their parents don't know enough, don't have enough time, or simply can't be bothered.
Like so many aspects of life, by pooling the responsibilities and costs of a major requirement we can raise the floor - make the worst experience better than it might have been - and make opportunities more equal.
With all the flaws and failings that our current education system undoubtedly has (a scary proportion of children reaching age 11 still functionally illiterate, for starters), it is still possible for a child whose parents have little education and less interest in education to make it in life; to learn what they need in school to go out and make the world their oyster.
Educated people are more productive; educated people are better citizens; better education leads to less inequality in a society; an educated workforce is the biggest draw for companies to base themselves near you, both on the micro-scale (look at the number of tech start-ups around Cambridge) and on the macro-scale of countries in the global economy.
If you have any belief in equality of opportunity, a good, free, compulsory school system is the foundation stone of equality - and if you don't believe that equality of opportunity is a good thing, please justify yourself below, because even our darling Coalition pay lip-service to the idea.
A story from my father - when he was a fairly young adult, credit cards first became available. Some of his friends went "Ooh, free money", and spent up to their credit limit because they could, be it on suits or cars or whatever - and were paying it off - and regretting it - for years.
e.g. if you do not wish to receive marketing telephone calls, you can register with the Telephone Preference Service. This is free. There are several companies who will charge you money and then register on your behalf. They are deceitful bastards. Finding this out takes either bitter experience or some research.
Translation into American - essentially free education up to finishing High School, no matter the age of the student.
 At least in theory; a range of practical hurdles do exist between baby Sam and Sam who has achieved their heart's desire.