Learning by trying something, doing it wrong, and then trying again, is important and is to be encouraged. At the same time, though, aim to help your child do a thing correctly the first time around. For example, your grade schooler is gluing together a plastic model. Since it will be difficult to take apart once put together, it is fairly important she learn to do it right the first time. She will develop increasing skill through practice, i.e. through trial and error.
Your children learn to "consider the possibilities." Putting puzzles together is a good example. Exploring the possibilities comes up in numerous situations.
Any time your child wants you to decide, to tell him how, to choose, or to figure out the right answer, consider the possibility of suggesting he consider the possibilities. Yes, at times you simply show him how, give him the answer, or solve the problem. Much of the time, though, encourage trial and error and explore various possibilities. This approach to learning has the desirable side effect of helping your child become socially and emotionally independent.
A second concept involved in trial and error learning might be thought of as successive approximation. your first approach to a problem does not work or turns out incorrectly. You then look at why it did not work, and your next effort takes this into consideration; you come a little closer. For example, your adolescent hurriedly paints a fence. She steps back, observes a few missed spots, and touches them up. Her hurried effort approximated a good job and her touch-ups really got the job done. Writing a theme for school similarly goes through successive approximations.
How do you help your children learn to use successive approximations? You encourage them to look critically at what they have done and to try to improve on it next time, whether it is their behavior while visiting a friend's house or school work. You encourage an orientation to successive approximation when you refrain from always giving the correct solution, or telling how to do something.
Learning through mental rehearsal may be the highest learning skill. What is mental rehearsal? Learning through mental rehearsal is more of a thinking process than a doing process. You imagine in your mind's eye or ear what something would look or sound like. You mentally try out each of the possibilities for solving a problem or completing a task, as if you were actually doing it.
How do you help your child use her capacities for mental rehearsal? If your child is presented with a problematic situation and asks what should she do, you say, "Let's think about it for a little while before you do anything. Let's see if you can imagine all the ways of dealing with it, and then think through what will happen if you follow each of the possibilities." You are encouraging your child to use mental rehearsal as well as demonstrating the technique. If your adolescent is experiencing a lot of anxiety about a debate, you can say, "Imagine yourself in the room where the contest is going to be held. Now imagine the debate is about to begin and think through everything you expect to happen (in sequence) before your turn. Now it is your turn. Think about getting up, walking to the microphone, looking at the audience, and giving your speech. Run this through a few times in your mind to see what the problems might be, what other people are going to do and say, and what you are going to say. Make mental notes about what you are not sure of, and what you need more information about." After the debate, your adolescent says, "It was not as bad as I thought it was going to be. It was almost like I had been there before." It is really true; she had been there before, in her mind.
With your preschooler, encourage her to slow down and think things through a little before starting an arts and crafts project, or participating in a special ritual at church, or making her first solo excursion to the grocery store. With your grade schooler, encourage her to think through an arithmetic problem before starting to work it, or how she is going to hold the bat before her turn at plate, or how she is going to get to school the first time she tries it on her own. There are innumerable opportunities for mental rehearsal, and your children should be made consciously aware of the technique and encouraged to use it.