They become very resistant and frustrated when the teacher wants them to show their work and deducts points for not doing so, it makes no sense to them, why should they have to show their work as long as they got the right answer? What work was there to show since...it wasn't work?
Yes, this is true. I think that when teachers say "Show your work," they should define what they mean. I suspect that most teachers mean, "Show it the way I wrote it out in the examples on the board." Yet their statements could easily be interpreted as "Show YOUR work, as in, what you had to write down." Sometimes, some kids just have to write the answer.
At the same time, I see huge value in learning how to write out a stepwise solution to a math problem. It's great to be able to solve problems in your head, but how can you explain (and justify) the solution if you can't also write out the logical steps it takes to get there? IMO, this is a critical skill for anyone who will ever be in a position that requires explaining something to someone. These situation would include writing something (a scientific paper, a book, a proposal, etc.), giving a talk, explaining to your boss and colleagues why your cool new method is so much better than the old one, convincing people that they should invest in your ideas, and so on.
