I'm a father to a six year old daughter and a four year old son. Those who think that peace and harmony are humanity's natural state are wrong. Siblings naturally argue, fight, have tantrums, and have to be taught how to behave in a civil manner towards each other. It is a rare day in my home that a parent doesn't have to intervene with our children to keep things from spiraling out of control, and sometimes we intervene too late.
What children at this age need to learn surprisingly looks a lot more like the first year law school curriculum than it does like the topics on their plate at school. The first year law school curriculum is cannonical. Property, contracts, torts, criminal law, constitutional law, legal communication (aka writing and research), and civil procedure are national norms that socialize new law students in the legal culture.
Children need many of the same lessons.
Going from a two year old's version of property law, "What's mine is mine and what's your's is mine too," to an adult's conception of property, is a painstaking process whose corollaries must all be learned individually. Don't touch the neighbor's plants. Don't go on the neighbor's land. Don't take things at stores that you haven't paid for. Don't grab your sister's things. Don't break your sister's things. But, children learn a more expanded version of property law, because they also need to learn the moral, if not legal corrolaries of property law: share, even if you don't have to share.
Children quickly learn that it is useful to try to hold parents to their promises, and have a hard time honoring executory promises of their own, beyond a few hours. Countless moments are spent teaching children contract interpretation and exceptions. We'll go camping this week does not mean we'll go camping in the mountains this week. Impossibility can void a contract, for example, in the case of horrible rain storms or a funeral on the day we are supposed to go camping. Maybe does not mean I promise. Eating your dinner is a condition precedent to dad's promise that you get dessert. Contracts carry implied duties as well, like the implied duty not to whine about why you can't have more than what was promised.
Torts, which illustrated at great length in cartoons, also encompass many sublessions. The necessity of exercising reasonable care not to hurt others gets translated into look where you're going, no it is not O.K. to throw your brother off the porch steps, stepping onto the street without holding mom or dad's hand is a strict liability offense. Learning that lying is often bad is also a long and difficult process, as is learning the exceptions to the rule, such as that lying is O.K. when everyone knows it is a lie, and that fraud is generally only a serious issue when damages result.
Substantive criminal laws are routinely violated by young children who take time to internalize appropriate behavoir. Don't bite, don't pinch, don't push, don't grab, and don't steal are not lessons that can be learned in a single setting. The more advance elements of criminal law, like the exception to the general rules for self-defense, are best reserved for older children.
Due process at home is not as intricate of courtroom procedure, but those lessons have to be learned as well. Don't tattle unless you can meet the burden of proof. Don't offer irrelevant evidence. Unsubstantiated claims lead to pointless "he did, he didn't" arguments. Don't ask for judgment when you have unclean hands yourself. Appeals to the other parent are usually denied and are subject to a duty of full disclosure of the proceedings below.
The Constitution rarely comes into play in raising children, but issues of how free speech should be or can be are omnipresent, as are issues of equality, and the ever changing division of decision making power between parents and children with is its own model of federalism.
The nature of these lessons is no more apparent than when you actually arrive in law school. While, as I noted before, the notions of fair play and societal rules that govern us are not platonic ideals that we are born knowing, by the time you reach law school, the lessons you learn in your textbooks simply refine the moral intuition that you already have, which is one reason why law schools use the Socratic method. The students already, through a lifetime of experience, know the arguments on boths sides of almost every issue and now just need to learn to hone the presentation of those arguments. Law students are not like physics students learning new ideas for the first time. They took all these classes already in the course of growing up. Now, they simply need to learn how to attach case names and fancy labels to those ideas.