Okay, we’ve seen Hogwarts. We’ve heard about Beauxbatons and Durmstrang. Presumably, many or most countries have schools of magic. Now, according to Rowling, Hogwarts has about 1000 students (which doesn’t really gibe with what we’ve seen in the books, but ah well.) We’ve seen no signs that there are other schools of magic in Britain - Draco Malfoy mentions in Goblet of Fire that his parents had considered sending him to Durmstrang for a proper Death Eater-style education. And Hogwarts is really the center of Britain’s magical community, at least as much as the Ministry in London is. So presumably it’s the most highly-regarded school in the country - if the headmaster of a school is also a major political figure, the school must be fairly important. Yet it’s not at all exclusive, and apparently it doesn’t cost anything to attend (Harry’s never paid, and Dumbledore makes mention of a fund to buy books and supplies in a memory - no mention that the school itself costs money.) So it probably is the only magical school in Britain (who would go anywhere besides the most reputable school in England since they all have the opportunity?), and even if substantial numbers of wizards educate their own children, this suggests that the wizarding community is quite small. Which is actually pretty consistent with the fact that most wizards seem to know who their fellows are, at least enough to name them.
There’s no mention at all, though, about primary education, and there doesn’t seem to be much in the way of post-secondary education. One would think there would have to be some elementary schools - at least in London and Hogsmeade - to teach readin’, writin’, and 'rithmetic (especially since such mundane subjects get very little attention at Hogwarts.) Clearly the little children of wizarding families are not attending Muggle schools, because most wizards seem to be completely ignorant of even the basics of Muggle society. Even Bob Ogden, a Ministry member, went around in what I think was meant to be a women’s bathing suit. (Though Harry, at least, wears jeans and t-shirts and sweaters under his robes. In fact, sweaters appear to be a normal article of wizard clothing. Apparently, when they’re not in robes, they dress in a fashion pretty similar to Muggles. They forget all of it when traveling into the Muggle world, however. But I digress . . . )
Anyway, there’s no mention of how little wizard children are educated, though they seem to at least possess the rudiments of literacy and mathematics when they appear at Hogwarts. Are their parents expected to teach them (or obtain tutors) all on their own? What gives?
And the only post-secondary education mentioned is the three years of Auror training. Presumably some other fields require it, but it sounds as if professional education, rather than a liberal arts approach, is the norm for those who are educated beyond high school. Even teachers at Hogwarts don’t need certification or any such post-secondary education; while Dippet doesn’t let 18-year-old Baby Voldemort become a teacher, clearly if there were a certification process requiring further years of education, Voldemort would have already known it. Hagrid doesn’t seem to have ever left Hogwarts, and he was also permitted to teach.
So high school students are being taught by those with no more than a high school education. And what are they being taught? Virtually all of their education is practical rather than in any way academic. Astronomy is the only natural science they seem to learn; there’s no evidence from what we’ve seen that they learn even the rudiments of chemistry in Potions, or of biology in Herbology and Care of Magical Creatures. While the former is perhaps understandable (wizards probably have little need to figure out chemistry - though it seems funny to imagine that despite their advanced abilities, they know less than Muggles about the fundaments of reality.) Herbology and Care of Magical Creatures are entirely practical - no discussion of common descent, or of ecology, and certainly nothing of biochemistry. Already their education sounds a good deal easier than my high school education.
We’ve seen no signs of anyone taking a basic literature course, nor a composition course (though they apparently write plenty of essays.) The closest to a foreign language is Hermione’s Ancient Runes class (perhaps Dumbledore and Crouch picked up the foreign languages they know through practice.) Arithmancy is presumably some form of math, though the “-mancy” ending suggests that it involves something akin to divination - their only math class is apparently an elective. There are no classes in Muggle History (though presumably Muggle Studies touches upon it) which makes sense until you consider that the magical community must be tiny. My high school had 1,100 students, serving a town and surrounding area of probably fewer than 10,000 people altogether. Hogwarts covers more ages, so even if we assume that significant numbers are educated at home or by tutors, that probably means the wizarding community in Britain has fewer than 10,000 members. It seems hard to imagine that Muggle History wouldn’t tie in pretty thoroughly into their own lives.
Basically nothing of an ordinary high school education seems to be covered at Hogwarts, and even if many professions have their own schools, it doesn’t seem likely that Auror school, for instance, requires any sort of liberal arts education. Teachers are apparently not particularly well-trained. Percy got a job with the Ministry with no higher education (whereas most government jobs, even pretty bureaucratic ones, seem to require a bachelor’s in the United States.) We haven’t seen much about other careers; presumably Healers are trained somewhere but all the evidence suggests that the wizarding populace is pretty uneducated.
Clearly Rowling didn’t flesh this out a lot when writing the books (there must be some sort of provision for teaching basic reading and math to young wizarding kids, at least.) Am I missing something? Is there any evidence that wizards have some sort of more complete educational system?