No, memorizing steps is not the same as understanding. Memorizing steps is MEMORIZATION. What you say makes no sense at all. Someone can understand the equation it does not mean they are going to memorize 10 steps. You can tell someone to read a page and than recite it back, if they can not recite the entire page, it does not mean they CANT READ. It means they can not memorize the entire page. Some people have fantastic memory, some people can memorize it if they do it over and over(Which works better with people who dont have amazing recall).
SHOW YOUR WORK MEANS DO ALL THE STEPS, That is memorization, that has nothing to do with auctual learning. You simply dont understand the difference. I sure hope your not a teacher since you would label a student incompetent for not having a great memory. I dont really understand how anyone can confuse the two.
Why do you think students have to memorize do times tables many times? TO MEMORIZE THEM! You can get the question manually by actually doing the math in your head. Whats 9 x 9 = or you can simply memorize 9X9=, 9x8= so on.
Just because you memorize material does not mean you understand it. I can tell someone to memorize a page and read it back, however it does not mean they understand what they actually read. It means they memorized it. You should try learning the difference between understanding materiel vs memorizing it. They are obviously not the same thing.
People with long division often can not remember all 10 steps since they get lost after 3-4 steps, so they often use the short cuts. The short cuts get the same answer, and these particular student might do fine on long division with shorter steps but when you get them too many steps they just become lost. However the short cuts are still a correct way of getting the answer but not everyone can watch the teacher and do all 10 steps. Its kind of like memorizing all of the states, it does not mean you are stupid for not knowing them, but you never took the time memorizing them. Just because you remember something does not mean you understand what you read. I have memorized enough material to pass a test, I really did not understand what I was reading but I knew the answers.