ACT and SATWould you recommend to your students to actually solve the mathematical problem or use the PITA (plug in the answer) method? Especially for simplifying rational expressions.
I also agree with 4,5, and 6. One note however; you cite simplifying radicals as an example type of problem. Often an answer is given that is not fully simplified, e.g. sqrt(72) might have distractor answers of 3sqrt(8) or 2sqrt(18). A calculator will confirm that these are equivalent to 6sqrt(2) so PITA won't work.
Another concern with PITA is incomplete solutions. 3 is a solution to x^2-x-6=0, so checking this answer confirms that it is correct. However the full solution of 3,-2 will be listed also.
A last concern with PITA is having an answer that works algebraically but is not a solution to the problem -- not in the domain, negative lengths, etc...
This is an interesting question because it does encourage us to think about what our end goal is. What is most important? Is it that students get the right answer through their own methods or that they use an ascribed way of reaching the answer that they are looking for? Surely our job as teachers should be to show them a range of different techniques or strategies and encourage them to come up with the answer using whatever method they choose, be it PITA or otherwise.
During the actual exam it is just the time taken that matters. If the PITA method is able to save time I would recommend students using it in the exam. Though it is important for students to realize that they should know how to actually work out problems as a list of answers is provided in only a few instances. And even arriving at the right result using PITA requires a considerable amount of skill. A student cannot use it with no idea about the question.
I wouldn't encourage students to do this (although many of them probably will), because while getting a good score on the test is important for college admissions, that approach won't help them at all when they get to college.