As a student, I always understood why my professors wanted me to show work. It was so that, if I came up with the wrong answer but had gone about the problem the right way, I could still get partial credit. I was doing it right, I just switched a +/- sign somewhere or divided wrong or something.
As a teacher, that is still true. That's why I want my own students to show work and there's a lot of partial credit to go around.
But I have suddenly learned a whole new reason for it: if they get to the right answer by wrong reasoning I want to know about it so I can fix the problem in class before a test. The more I can get my students "talking" the more I can spot the flaws and improve their understanding, not to mention the better I understand them and the way they approach the question.
I've graded a number of classes and never noticed that before, probably because I never before took so much of the responsibility for others' learning outcomes on myself before.
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