The Most Important Thing to Learn
Math has a certain clarity. That clarity allows math problems to produce a very fine "aha" experience. I like math.
If I could teach students only one thing, I would want them to learn how to enjoy math. If I was teaching math, the first thing I would want my students to learn is that it can be enjoyable to solve math problems.
So, #1 on my curriculum, no matter how you measure it, is teaching students that math can be fun, and teaching them how to enjoy math.
How young can it start? A three-year-old shows you how high she can jump. How low can she jump?
Teaching this Skill
How do you teach this? I think, for very young children, you kind of have to teach them the experience. The problem is, they often aren't sure they got the answer correct. What you need is a hooray, smile, cheer, and a high-five. Um, actually, this depends on the situation, a "good job" goes a long way. The idea here isn't to "reward" the student. The idea is this. The student has a good feeling, because he/she thinks he/she solved the problem. But the student isn't sure, and doesn't quite know what to do with the feeling. So you signal that the feeling is valid and appropriate, and show the student what to do with the feeling.
For older students, say third-graders, you might want to embed the problem inside a game, so that the student is "winning" or defeating you. That tends to amplify the good feeling of suddenly getting how to solve the problem. (If competition is important, the "mature" problem-solver can imagine any problem as a contest, for example trying to beat nature.
For even older students, say sixth-graders, you are fighting their perception that math isn't fun. Of course, that perception probably is an accurate description of their experience so far. For these students, have them learn the aha experience in a nonmath context. Then when you get to math, they are in the habit of using their problem-solving skills to get an aha experience.
Of course, to teach this skill, you can't first tell students how to solve a problem and then give them the problem. You can't give them the problem and then tell them how to solve it. While they are trying to solve the problem, you should not be interferring with their thinking process. (Don't tell them what to pay attention to, don't give them hints, don't remind them of things they should consider, and so on.)
Why I Need Students to have this Skill
For my method of teaching, it is critical that they know and understand the aha experience. I teach math by solving problems. So they have to be creating this experience. Also, if students can't "get into" the problems, then they won't solve them. So I need them to expect that they might find enjoyment in the problems. For the traditional way of teaching math, this skill isn't needed. (However, there is little to no chance that the student will actually learn any math without this skill.) But it is a worthwhile skill for real life.
Introduction to Curriculum