Tinkering: Improving on Your Answer
In the usual "insight" problem, your brain (with luck) eventually produces an answer to the problem. This answer can be immediately recognized as correct, because that's the kind of problem it is. So you get an "aha" experience.
In "tinkering" problems, the answer is not recognizable as correct. This answer also is usually wrong. To be more precise, that answer can be and usually is a step in the right direction, but it is not an optimal answer.
Therefore, in these tinkering problems, the answer has to be checked. Depending on how good of answer you want, the answer has to be improved upon. The process of checking usually suggests how the answer can be improved upon.
Activities with Tinkering
Science is a giant tinkering problem. Yes, there can be an aha experience when you think of a good theory or notice a relationship between variables. But you have to check your answer (a.k.a., test your hypothesis in a scientific experiment). And in fact, the process of building a theory is never finished. And in fact, any relationship between variables can be endlessly qualified (it holds in this situation, it doesn't hold in that situation).
Gendlin describes a process called focusing, which is a way of learning what your feelings are. Focusing is essentially a giant process of tinkering. You start out with the first answer that gushes from your brain. Then you use your brain to check if that answer has validity. If it does, you use your brain to figure out how that answer is wrong and how to improve on it.
Trying to define a word is also a tinkering problem. People in our culture think it should be easy to define a simple word, because they "know" the meaning. But it is very difficult to define a word (or describe the features used to perceive something). Producing a correct answer can (and should be) a giant tinkering process. (The more general task is explitizing expert knowledge.)
Tinkering in Math
Math, as a general rule, does not have many problems that require tinkering. One exception is the task of defining any mathematical concept. So, as a general rule, math is about "aha" experiences, not tinkering. However, tinkering captures one aspect of mathematical thought -- the desire to be rigorously correct.
However, the task of rigorously defining a word is a very useful teaching experience. Take rectangle. Just to produce the first thought can be useful. But if a student knows how to check, or if a teacher helps with the checking process, the student can be induced to provide a rigorous definition of rectangle. That, it turns out, is a very good way of building a good mental model of the rectangle.
Usefulness of the Tinkering Skill
Tinkering is also a very important skill for other activities. Sometimes a poem, story, essay, or letter comes out almost perfect the first time and any attempts to change it make it worse. But usually, writing is a matter of tinkering. You can tinker with a painting, a musical composition, or a choreographed dance. You can tinker with your golf swing. You can tinker with your hair or how your house looks. You can tinker with your approach to teaching, an advertising campaign, a recipe, or how you portray yourself to other people. You can tinker with a motor to try to get it to work.
Goals in Teaching
Ideally, you want your students to appreciate the process of tinkering. This includes that the want to get a good answer, and that they know how to work on improving their answer.
Of course, to appreciate the process of tinkering, students first have to have the concept of tinkering.
The first two exercises below are exercises in tinkering. The third teaches the concept of tinkering.
Exercise: Confirmation Bias
Exercise: The Concept