The reason the thread took this turn was my example of an elementary school state test. There was no work to show, it asked to list three fractions in order, smallest to largest. Then they wanted the process to be explained. I did it one way and the test example did it another. Both were sound methods but when I asked if my answer would be marked wrong because I did it differently the answer was,"I would hope not." If the answer isn't yes or no then the test is flawed.
Did all of the fractions have a common denominator? I am guessing not. So, to ensure that the students did not just guess at the correct order, they asked for the method. Since I again guess that the method answer was not just a multiple answer, but the student had to write their work, a machine could not grade the answer. The teacher probably couldn't answer you directly since she wouldn't be grading the test, but if it were solid math then it would likely have been graded correct (say, not caring if the student found the lowest common denominator but realizing that the denominator was required to match and that whatever you multiplied the denominator by that you also multiplied the numerator).
Let me ask a question. You are taking a math test, and they do exist on job apps, they ask you to figure the total for 6 tickets at $17.50 per ticket without a calculator. Many would just multiply it out, I prefer to make it an easy number to work with. I add 17.5 to itself to get 35 and triple it, $105.00. That is done in seconds in my head but I think it be wrong a state test.
Again, doing math in a different way doesn't make it wrong. You in some way made it a more complex formula, even if it makes it easier to do in your head. Instead of 6*17.50 you did 3(17.50 + 17.50) which was (6/2)(2*17.50). Of course all yeild the same answer, this is the beauty of math. However, without showing your work, it would be difficult for someone to help you, should you have made an error along the way. It is simple math for you because you understand the algebraic concepts.
Like I stated, maybe a poor analogy by using music which is another mathmatecal language, showing work is essential practice to
build a
foundation on. If you had not had a grasp of manipulating numbers in multiple ways, you would not have discovered that you could find a shorter method for solving a problem. It is difficult to understand the need for showing work when it you have an aptitude for numbers (I did). I ended up losing quite a
bit of credit for not showing my work (it was easier to do in my head). Even though I did my own work, the teacher was quick to
point out that half of the answers were generally in the back of the
book and how were they to know if I did the work or copied the answers?
Math is logical. Since very few schools teach logic anymore (which may be why we have so few good public speakers), it is one of the few outlets that our brain has the opportunity to use that skillset. Why would you want to deprive students of that? We should embrace it.