I believe that children are rational beings from the moment they are born. This means that they explore the world around them. They simultaneously try to understand it and achieve goals they have. These goals may be very simple, like getting rid of this feeling of hunger, or quite complex, like wanting to become a professional actor. The solutions they employ may also be quite simple or complex, depending on the problem. For example a hungry baby may decide to cry out with a certain kind of way and then mother will come to give me some milk.
Of course the example I gave is simplified and in some ways erroneous. A small baby does not know language yet, so he won't formulate the causation in quite those words. But I believe that it still catches the process that goes through a childs mind when he decides to cry in just that one special way, when he is hungry, or a little bit different way, when he has wet himself, or wants company.
Living with a child is not easy. There are many, many things a small child does not know - things that may jeopardize childs health or life, or someone elses. It is up to parents to help child get what he wants and considers important and simultaneously help him avoid and learn about those dangerous things in his environment.
A small child that runs to the road in front of car, probably doesn't wish to die. It is much more likely that he saw something interesting on the other side of the road, or that he considered that big flat road to be a great place for playing. If in such an acute situation the parent grabs the child and then shows him that he was about to get run by a car, he'll probably be greatful for you saving him. And then it's time to continue helping him with his goal.
Human being is capable of forming and attaining goals. This means that human being forms goals, decides the methods to use in order to attain one's goals. These goals and methods have results that may be wanted, or unwanted. These three things for a triangle of goal oriented action.
For example libertarins typically divide into following main categories. Utilitarists, Natural Law proponents and as a rather new category Popperians. Utilitarian considers the outcomes as the most important facet - the goals and methods are only valued as parts of getting to this goal that is considered good. Natural law proponents see the methods of action as the one thing above all else and goals and results are unessential. Popperian libertarians consider all three important. It is important to have good and right goals, good methods and one of the reasons this is so is that thus the results will be good, but more importantly because it is right to act so.
Another important thing to understand is theory of fallibilism by Karl Popper. He has presented a forceful philosophy, which states that we cannot be certain that we have objective truth about something. We can merely create theories and test them against real world. If the theory fails any test we can device, it has been falsified and thus is not right theory. But on the other hand, a theory that has passed all the tests anyone ever thought about, is not necessarily true. That means that we can never be certain that we have objectively true theory, but we can be certain that certain theory is objectively wrong.
A big part of human life is positive problem solving. Every solved problem leads us to better situation and gives us new problems to solve.
Due to fallibilism, the idea of authority is insane. Expertise on the other hand is valuable, as long as the expert can explain his theories to others, or create solutions to problems others have.
How does this relate to children and family life?
Family life is full of problems to solve. Children wish to understand the world around them, just learning to speak is a lot of rewarding work. The family needs to decide how to spend their vacations, whether children attend school, and if they do, which school, etc.
Since we are all fallible, it is irrational to expect a certain person to automatically be right and to force others to follow him. Still this is how every child rearing book describes family life - parents must be the authority, always right, no matter how wrong they are. It is a lot better to actively search for good solutions, i.e. solutions that all parties prefer. These are called common preferences.
The purpose of parenthood is not to raise children into a certain kind of mold. It is about helping children understand the world around them, about making family a good place to live for all family members. This can only happen, when everybodys problems are taken seriously and they are being solved. Having problems that go long unsolved is good for no one. Learning that problems are something we cannot solve and that we have to live with them, is even worse, for parents and children alike.
It is also important to notice that children, even though they are rational, have little knowledge, or understanding about the world when they are born. They are not born all knowing about matters of morality, or speech, or any other sphere of knowledge. It is very important for parents to help children learn new things about the world as they become relevant, especially in the matters of morality.
Children, as all humans, learn things by creating theories, conjenctures about the thing they are interested in and then testing them against reality. So when one explains a child anything, it is important to give good explanations, and not just tell that this is how it is. A good explanation, that is not necessarily verbal, helps the child understand the matter with more depth - assuming that the parent was right in his explanation.
Some more notes regarding libertarianism. Popperian view to politics and morality results in some preliminaries and conlusions. I'm mainly speaking about private morality, and will separately indicate, when I talk about politics, or law making.
Human is goal oriented being and capable of shaping his goals and methods in order to lead a fullfilled life. This happens using conjecture refutation process.
It is evident that the goal-method structure should be such that one is capable of attaining the goals. Thus conflicting goals and methods are a bad idea.
Libertarianism is a theory of law i.e. it is a theory about what is legitimate law and what is not (see Wikipedia on Libertarianism and Anarcho-Capitalism). It uses the principle of non-conflicting theorems and achieves a consistent and good theory of law. Still the proponents of natural law and utilitarism often cling to only one part of goal oriented human action and forget that all three are important.
One cannot say that means are more important that goals, or vice versa. In order to fullfill one's life, in order to have good solutions, both means and goals need to be right. That is the only way of getting toward ever better outcomes.
Another thing many libertarians forget is that markets are only small part of human interaction. If people have very bad ideas about private morality, libertarianism cannot work. If most people want to kill, or imprison drug users, they will do it, with or without the help of government. We should help people better understand morality and solutions to problems in society in addition to trying to reduce the detrimental control that government has on our lives.