Students Constructing Problems
This has never been an open-ended request (though that might be useful). Instead, we are doing loox problems, for example, and I ask them to construct a loox problem for me.
Learning the Structure of the Problem
There are two reasons for doing this. First, to construct, say, a loox problem, the student has to understand the structure of the loox problem. If a student is having trouble with solving loox problems, it might be that the student doesn't understand the structure of the problem.
Thus, when the student constructs a problem of a given type, the student is exploring the structure of that type of problem. As a second benefit, you get to see if the student understands the structure of the problem.
Learning to Check Answers
To construct a problem, you essentially start with an answer and then produce the problem. That is the exact same path that you follow when you solve the problem, but you are taking the path in the opposite direction.
For example, construct a loox problem, you might decide that a loox has 3 ears. If you put 4 looxes in the box, you have 12 ears (3 x 4) in the box. If you add 2 dogs to the box, you have added 4 ears (2x2), so now there are 16 ears (12 + 4) in the box. Solving the problem is then just reversing these steps. You calculate that there are 4 dog ears in the box, you subtract 4 from 16 to get 12, the number of loox ears, then you divide by 4 to learn that each loox has 3 ears.
The path from answer to problem is (usually?) easier than the path from from problem to solution. That's one reason we give the problem in the format we do. But, the path from solution to problem is exactly the path that we expect students to take when we ask them to check their answers. For example, in the above problem, suppose the student concludes that each loox has 3 ears. The student, to check the answer, is supposed to take the same steps as would be taken in constructing the problem.
Therefore, asking students to construct a problem is very good practice for the skill of checking an answer.
Learning to Solve the ProblemWhen the process of constructing the problem is the same path as solving the problem, except in a different direction, the student can also learn to solve the problem. In a sense, when you construct a loox problem, you take information (the number of ears a loox has) and fold it into a different shape. To solve the problem, you have to unfold it. So learning to fold it can help with unfolding it.
Social FactorsAll of the social factors are working in the right direction for having the student construct a problem.
First, there is less emphasis on right or wrong -- the student is much less likely to perceive the possibility that the student can be wrong. And in some sense, this is not illusion -- in a very real sense, whatever the student does is right. And it is more like using a skill. So it fits the "flow" model of doing math.
Second, there is the chance to fool the instructor. They might not succeed, but if they do, students seem to like it. In any case, there is in some sense a superiority in being the person constructing the problem. The student can feel that superiority. They are also likely to learn, as modelled by you, that there is no inferiority in being the person solving the problem. I mean, I like doing math problems. In a real sense they are doing the work and I am having the fun.
And Maybe...One of my friends was trying to teach a student the concept of opposites. She gave examples, such as on and off, in and out, and tried to explain it. The student didn't get it. So she asked the student to give her a problem -- say a word and my friend would say the opposite. The student said "red".
There is a lot to think about there. How come some words have opposites and some don't?
So when you ask your students to construct a problem, usually nothing out of the ordinary will happen. But sometimes the student will construct a problem that leads you, the teacher, to a new level of awareness.
That's not all. Sometimes my students construct a problem that I like and then use. In essense, they are creative.
Once I was doing the exercise of what things in a category have names. I started I think with snow and rain. Then I asked for topics from the class. They suggested cheese, which turned out a little differently than I expected. So I learned something. They suggested shoes, which I had done before and was what I expected. Then a student suggested doing numbers. Well, that was what I most wanted to do. So I took his suggestion.
Using This Technique
The obvious situation for using this technique is when (1) a student is stuck, not understanding the problem, and (2) you are stuck, not understanding why the student isn't getting it and not knowing what to do next. It is an all-purpose technique and requires no thought from you.
It can also be a part of any exercise, if there is a structure to the problems.