I realized as I started this rant that I had way too many ideas, and, moreover, that many of them didn’t fit in the same rant together. So I’m dividing it up. This rant is on general ideas about fantasy education. There will be two separate ones on training in weapons and magic (since there are so many fighter and mage heroes), and one on informal ways of education that take place outside an academy or school environment. Thanks to the people who suggested this rant; it’s good material.

1) Teachers have to eat, too.

In some ways, the model of compulsory, free education has failed. That is, it has not apparently taught very many fantasy authors that such things can’t exist in the fantasy world without a basis.
Who pays the teachers to teach the children? If the school’s supported by taxes, it would require a government capable of collecting and channeling those taxes, and people rich enough to give them—not the usual starving peasants who, we are told, can’t pay the taxes for anything. If it’s a moneyed school, with each student’s parents or guardian handing over a fee, there are not going to be many poor students, if any. If teachers are supported by the whole village, they may not take payment in coins; on the other hand, there should be mentions of them receiving baked food from parents and so on. Perhaps the teachers are itinerant and wander about, coming to offer their skills for a month or a year at a time, for as long as the village can support them. Pratchett uses this in Wee Free Men. Of course, if such a thing occurs, the students are unlikely to get as structured or polished a schooling as they will when it’s more formal. They may know a great deal about the geography of the northern kingdoms, if they had a teacher who was proficient in that, and nothing at all about animal species of the southern wastes, if they never had a scholar who knew that.
The reason I’m making such a big deal of this is that I realized I don’t know how many of the teachers in fantasy make a living. They seem to teach for the joy of it and do something else during the night, or feast on food that comes from air. (Perhaps many of the secret, daring thieves of fantasy are actually starving teachers). Take some time to think about this. I know you’re anxious to get on with the good stuff, like your teenage protagonist standing up to his bully of a teacher and knowing more than she does, but if this school exists at all, it’s got to have a foundation.
The bullying teachers could use some work, too.

2) Teachers will often fall somewhere in between ‘perfect professor’ and ‘hateful bastard.’

This is the point where bias grabs me and holds me against the wall and forces me to admit that there’s some personal interest here. I’m a teacher as well as a student currently, so I’m used to what a classroom looks like from both sides. And the teacher’s side is often woefully lacking in any fantasy that takes place in a school. Teachers are heavenly or horrible at their jobs depending on whether or not they’re nice to the protagonist.
Not only is that rather simplistic characterization, I think it’s a bad standard to use. The teachers are there to perform a job, the same way that a soldier is. A soldier isn’t automatically a horrible fighter just because he’s on the opposite side from the protagonist; in fact, a lot of fantasies take the opposite route and emphasize that the Dark Lord has at least one highly skilled or trusted lieutenant he can trust to fight well and rally the troops. Fantasies that, instead, say a soldier is inferior in skills because he fights for the Dark Lord or because he doesn’t like the protagonist feel silly and thin to me. And so do fantasies set in schools where the teachers who like the protagonist are the ones skilled at imparting information, and the ones who don’t are bumbling and never should have been hired.
It’s much, much more normal for a teacher to be skilled at some portions of her job and not others. She might be more comfortable working with students one-on-one than teaching in front of a whole class. She might do very poorly at the practical side of things, but impart all that the students could ever want to know about theory, or vice versa. She might overemphasize her own particular specialty; on the other hand, students are going to learn a hell of a lot about that particular subject, and some of it might be interesting. She might be a laughing and congenial and fun friend, but a horrible teacher. (I’ve known far more like that than teachers who were horrible bastards through and through).
If your fantasy involves a lot of interaction between teachers, give as nuanced a portrait of them as possible. I’ve heard some people argue that it’s perfectly fine not to do this, because teenagers are famous for not noticing nuances. The trouble is, these are the same people who turn around and claim that their protagonists are perceptive, compassionate, and empathic far beyond their years. So take the heat or get out of the kitchen.

3) Some learned skills are time-dependent.

As in, children, teenagers, or adults learn them better or are more likely to have them. Either you put up with that, you come up with some plausible (very plausible) explanation, or you resign yourself to appearing to have a supergenius, almost completely implausible character.

Languages.

Children learn them better than adults or teenagers. Very soon after birth, a baby can distinguish the sound of her native language from the sounds of other languages, assuming that the adults around her have been speaking that language continuously. Babies absorb words at an astonishing rate, and young children only a little more slowly. The window of opportunity slams shut in the early pre-teen and teenage years. Teenagers and adults learning to speak second, third, or other languages are far more likely to have an accent and/or to use grammar or idioms that are too formal or not appropriate to the situation than people who learned to speak the language as children. This is also why ‘feral’ children left in the wild as infants and reared among animals until their teenage years are not going to be able to speak much, if at all; they’ve missed out on the most fertile learning period.

Sword-training.

I’ll say more about this in the fighter training rant, but allow me to say here that I will never understand why all those fantasies starring clumsy teenage boys start them on the sword during the period when they’re dropping objects, tripping over their own feet, and offering danger to other people. You want to give someone like that a sharp piece of steel? Wait until he gets comfortable with his body, for fuck’s sake. Even better, have him start training with wooden blades and so on as a child, so that the movements will be more natural to him than they will if he starts at sixteen or even later.

People-reading.

Children in dangerous situations may develop a keen sense of people’s reactions and triggers for particular emotions, especially anger. If the child in question had a happy home life, however, there’s no reason why she would “instinctively” know not to trust a stranger whom her parents trusted. That “instinctive” and “intuition” bit has to bear a lot of weight it wasn’t meant to bear, because authors are anxious that their protagonists not make perceptual mistakes. Stop it.. Knowledge of people in general, of other cultures, of atypical reactions in most situations, relies on personal experience, experience learned from others, cultural inheritance of stereotypes and truths, and experience gained from mistakes, not just on “intuition.” Instinctive perceptual geniuses make me tired. They also never, ever make mistakes, and perfect characters are boring.

4) In some places, education is the bull. In others, it’s the shit.

Don’t insist that your fantasy cultures adopt your own, or your country’s, or your culture’s, attitudes about education wholesale. Except in a handful of special situations, like alternate history, fantasy worlds are often disconnected from Earth and have no reason to imitate its attitudes about anything, most especially not this.
Is education valued? It may not be, and for any variety of perfectly good reasons, even among the nobility. Many nobles in our own Middle Ages didn’t learn to read and write, either because they were women or because there were trained scribes and monks to do that, and noblemen weren’t supposed to be scribes or monks. It may take too much time or money. It may be considered to impart undesirable attitudes; there go those damn university students, thinking they’re so much better than us, and rioting again, damn it! Perhaps education is very specialized. Someone who wants to become a linguist would find no value from a curriculum focused on teaching only mathematics and no foreign languages or grammar, unless he was the kind of person who appreciated knowledge for its own sake—and even then, could he afford the loss of money or time? There are all sorts of barriers to put in the way.
Conversely, there’s no reason to devalue education, either. The common stereotype is that a lot of peasants spit on education, but no reason is given why they should do so. Why? Come up with one, and make me believe it. Perhaps teachers are mostly wandering scholars and given to thievery, like gypsies. Perhaps the nobles are usually the educated ones, and the nobles do things like pass laws, arrest peasants for breaking them, and then point out smugly that the laws were written down—never mind that other law that says peasants aren’t allowed to read. Perhaps the village children who go off to the city and get educated always end up coming back, sighing, and saying that nobody understands them or their dreams, while acting snobbish and lording it over everybody. All of those are better than turning others ignorant just so that your protagonist will shine because she loves to learn.
Of course, that’s part of the point of this: setting should twine with and support characterization. Your characters did not grow up in a vacuum, and you’re trying to maintain the illusion that neither did they step out of our own world (most of the time) or your head. Rather than attempting to assert that your protagonist feels the way she does about learning because she’s just Special that way, it’s much better—and more interesting—to think about how her culture would have come to value or devalue education, and put part of the cultural attitude in her own.

5) Common sense has a place in fantasy education, too.

So the protagonist is getting in fights at school every day. Or a teacher hates her so much and is so jealous of her talent that he flings fireballs at her every time he sees her. Or it’s absolutely vital that students have quiet to study, but the protagonist and her friends interrupt it with chatting, and the teachers smile at them, because they’re the best students, and do nothing.
I want to know why. Once again, situations like this are often the product of a) not thinking and b) wanting the protagonist to appear Special. They don’t make sense in the context of the school. The other teachers have no reason to support the fireball-flinging teacher, not even one revealed at the end of the book, yet they do. The other students complain—quite rightly, I would think—about the protagonist and her friends interrupting quiet time, but they’re represented as fussy, hidebound rule-respecters who matter much less because they’re not going to save the world. The protagonist gets in fights that injure her and/or others, and disrupt the flow of education, yet she’s completely ignored or coddled, depending on what the author wants the teachers to be like. (See point 2 again).
Once more, people tend to say, “Well, but the world always seems unfair when you’re a teenager!” That bangs up against the idea that the protagonist is supposed to be the most intelligent, just, and perceptive student in the whole school again. (It’s also the reason that I don’t read a lot of YA fantasy. I have no interest in reading about someone who feels the whole world is persecuting her and is right about it, not any more).
There are all kinds of more pragmatic, sometimes darker, undercurrents you could be using instead, which are much more interesting, and sensible, and even happen in real life. Perhaps the teachers won’t punish the other student the protagonist is fighting with because he’s the son of the headmaster. Perhaps the fireball-flinging teacher is using illusions, and this is supposed to be part of the protagonist’s training in saving the world. Perhaps the teacher enacts spells around the protagonist and her friends to keep their chatter from disturbing the other students, so the ones who complain are really only thinking that they hear something. There are all sorts of explanations, fair and unfair, that will make sense and fit far better than the teachers and other students being out to hate on the protagonist because she’s beautiful and intelligent and talented.

6) Students are often big fish in small ponds.

A student may feel she knows everything there is to know. Perhaps she really is the smartest student in her village school. And then she goes to university and is also the smartest student there—in fact, the smartestperson, well ahead of her teachers.
Huh?
This is where the idea of experience comes in. I don’t care how smart your five-year-old is; she’s not going to know more about his subject than a sixty-year-old archaeologist who’s been reading up on that field for forty years. She won’t have had the time, and especially she won’t have read the books and spoken with other scholars and traveled to the site of ancient ruins to look over them. Likewise, a thirteen-year-old may be very good indeed at writing down her thoughts, but that doesn’t make her an instinctive sestina composer, particularly if she’s never learned the rules for one.
Families, teachers who know their students personally, family friends, and so on may praise a student to the skies. It does not mean that she is the best at everything. It does not mean that she’s going to step into a bigger environment and wow everyone with her talent. (This is a lesson that, in our own world, high school students going to college for the first time often learn rather sharply). It doesn’t mean that she knows more or is better than someone who’s had experience, practice, and, above all, mistakes to guide them.
I’m banging this drum so hard because I find the “instinctive talent” bit overplayed even in fantasies where the school is not the focus of the book. The author just suddenly yanks it out of her ass the air when the character gets in a tough situation; she suddenly remembers some sword move she’s seen only once and executes it perfectly, or she composes a new kind of poem and recites it on the spot. Try actually having your characters think their way past the difficulty instead, or use the skills you’ve already established that they have. I find quality much more sympathy-provoking than quantity. Show me a heroine who knows how to shift soil, has trained in it most of her life, and uses that to foil the enemies pursuing them, rather than the one who suddenly discovers that she can move mountains because she once heard someone talking about the theory behind it.
That was fun.