Restructuring the Learning Experience
Introduction
When I teach math, I try to make the experience for the students like my experience when I go boogieboarding or skiing. (Boogieboarding is like surfing except that the board is a hard syrofoam and you lie on the board an try to ride the ways.) You can think of skiing, or surfing, or many different games, sports, or crafts.
I conceptualize skiing as a problemsolving exercise. The idea is that humans have an unconscious structure that can learn. When it learns, you develop a skill. You also enjoy the learning process. I have a certain skill in boogieboarding. I enjoy using the skill so I boogieboard. When I do, I run into problems. When I solve the problems, I improve and enjoy myself.
Et tu, Math
Of course, I think of math as a problemsolving exercise. Math can be fun for exactly that same way that skiiing can be fun  it is enjoyable when you solve a problem.
Differences
Now let's think of the differences between skiing and math. In math, you are given a problem to solve. You either succeed or fail. In skiing, no one gives you a problem, or tells you that you have a problem. Instead, it just sort of appears. Then you deal with it. Or fall. But there is no one there to tell you that you failed.
Instead, in skiing, your conscious intent is to enjoy yourself by exercising your skill.
Second, in skiing, you can choose your level of difficulty. You might not do that wisely. But you know roughly your ability level. You will not choose a slope that is too easy, because there are no problems for you, you are not using your skill at a high level, and as a consequence, the easy slope is boring. You will not choose a difficult slope because you will just fall and that won't be any fun.
You can also choose to be adventuresome  choose a difficult slope for your given skill. Or you can choose to be conservative, and choose an easy slope that just moderately challenges you. Which you choose depends on your personality and your mood at that moment.
In math, the problem is chosen for you. Someone may or may not do a good job at that. The key thing, probably, is that the experience in skiing is that you are choosing your level of challenge, and in math you are not.
Restructuring the Math Experience
If students have a mature perspective towards math, you probably do not have to work so hard to make math like skiing. That might help a little, but it is not so important. The worry is students who do not have a mature perspective towards math. They are likely to see math problems as a test of their skill; they are more likely to see the potential for failure rather than the potential for enjoyment.
For these students, it is useful to restructure the math experience so that it is more like skiing.
One simple trick is this. Suppose you are giving problems. You can have a lot of problem sets available, at differing ability levels. You rate the difficulty of the problem sets. Then you let the students choose which difficulty level they want to work at.
For example, I was going to assess students' ability and then give them problems suited to their ability. But then I realized I could let them just choose what level of problem they wanted. A student might pick problems that were too easy. But so what? What did I care? Actually, the student might be right. And if the student was wrong, the student would probably learn that problems which are too easy are also boring. And if the student was so afraid of failure that the student would rather do easy boring problems? The easy problems are probably best for that student.
Some students picked problems that were too difficult. Again, I couldn't find any drastic consequences. They just moved to easier problems. And they probably learned something about choosing too difficult of problems. Some students pretended that they had solved the difficult problems. Again, what did I care?
So, as a general principle, try to reconceptualize the student's task as using a skill. Ideally, they perceive no problem, they just perceive progress.
Avoiding the RightWrong Milieu
The standard perspective is that the teacher asks a question, or gives a problem, and there is a right answer. The student tries to get the right answer. Then the student learns, from the teacher, whether the answer is right or wrong.
This is not a good milieu for math. Usually, it can't be avoided. But avoid it when you can. Sometimes, the question can be rephrased as one of opinion. Or the task can be changed, maybe to a process of discovery.
For example, I was going to do a class discussion where the students would try to list all of the numbers. That's a problem. They were going to have successes, but they were also going to have failures. I might point out some of those failure.
I happened to think of the interesting philosophical question of whether numbers were discovered or invented. At the last minute, I changed the whole perspective for my exercise. First, we discussed the difference between inventing and discovering. Then I assumed the numbers did not exist and let them either discover or invent the numbers.
Think of the difference in perspective. Discovering/Inventing the numbers was an activity. They succeeded, of course. If they never discovered any fractions, that wasn't a failure, because no one said they had to find all of the numbers.
And given the freedom to explore, they do just boring things. They of course first discovered the counting numbers. But they discovered other numbers when they could, and they recognized when some numbers were interesting, and they explored the different ways of describing/constructing numbers.
I didn't correct any answers. This reduced the sense of failure. And I can't find any serious consequences. For this particular situation, I believe that "number" does not have a technical definition in mathematics, so there really is no right or wrong answer. (But if someone asked, I did give them my opinion.)
As an aside, there was an unexpected benefit. It turns out that one of the basic exercises in fundamental number theory is constructing numbers. And the class came up with all three methods of constructing numbers. That was very cool (though they didn't realize it and I didn't have a chance to explitize it.) Also, one type of problem I like to give is my frodo problems, which involves constructing sets of numbers according to rules. My guess is that my students would be better at the frodo problems, because they now had practice constructing numbers out of other numbers (which is what happens in the frodo problems.)
So, as a general principle, try to reconceptualize the student's task so it is less like a problem with a right and wrong answer, and more like a journey, experience, or matter of opinion and judgment.
