I don’t think that it is effective idea because it doesn’t quite test the same skills as a normal exam.
Making up a good question is a hard thing to do. You have to be able to predict the deepness/difficulty of solving the problem before you actually solve the problem. It is surprisingly easy to create a problem, while attempting to create a deep problem, that ends up being too deep. It is also hard to write question that is phrased nicely and well-written. This skill is not necessarily related to the ability of a person to solve the problems that are given to them (most real world questions are caused by a known problem, not generated from thin air).
Another thing is that people will only ask themselves questions that they know the answer to. The intuitive thing to do is grade the exam based the range of content covered. The examiner could provide a list of topics that the students should ask themselves questions about. Even this is not effective.
Say that one potential topic is “traditions of different religions”, to name the most random topic I can think of. If a student chooses to ask himself to describe the practices associated with Hinduism, it is safe to assume that he knows equally as much about all other religions? With a normal exam, a student does not know which religion he will be tested on, but in this case he can choose the religion.
If a tell a person to describe a random religion (like Sikhism) and he describes it well, then it is relatively safe to assume that he knows a lot about other religions. However, if you give him the choice of religion and he chooses Christianity, is it safe to assume that he knows a lot about all religions, or that he picked that certain religion because it is the only one he knows anything about?
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Another problem is that it is impossible to give a objective score. In a normal exam, a teacher can create a solution guide (prior to the exam) that gives all of the required details for each answer and how many points each required detail is worth. No matter who the teacher is, the final score should be relatively independent of who scored it. With an exam that the students write, it is hard to do this.