I think for someone having difficulty understanding the subject, it is simpler to obey certain rules. You may know intuitively to consider 2 as 8/4ths, but how does one teach that?
I laid out a simple method that should always get the novice the answer. Your shortcuts may be valid, but not for the novice.
SD
Understanding very basic mathematical concepts (WRT learning algebra) should be a given. Perhaps this is the problem with the students ... they don't know how or why 2 = 8/4.
I have a simple philosophy concerning math: work smart not hard. This reduces the probability for error. Of course, if the fundamental concepts are not understood, the students are going nowhere fast, until they can grasp some basics.
I employ this philosophy daily, as I am my children's algebra teacher. :-) If the shortest distance between two points is a straight line, then there is no use traveling in circles. I explain other valid methods for solving a problem once they realize that the concepts just aren't as difficult as initially portrayed in the book.
Nothing worse [for motivation] than a student feeling overwhelmed at the outset by all the official rules and steps. Mine simply know better (from experience) to assume that the problem is with them. Thus, they approach the subject with confidence rather than with a sense of dread. And if they are the ones who are a bit dense at times, it's no big deal. They just keep working at it until the light goes on. If my explanation A doesn't do the trick, there is always B, C, D, and so on.
At the start of the last school year, my son began Algebra I (Saxon). Immediately it was apparent that negative numbers were frying his brain. He was obviously frustrated. So, we stopped right there. I spent several days strictly working with him on subduing the evil forces of negative numbers. There was no way I was going to expect him to continue without mastering these basics. The strict focus solved the problem, and we moved on. He has become very strong in the subject, and this year he has moved easily through Algebra II. He clearly has a deep understanding of the concepts. I believe that the most important equation to solve is the one with the fear factor.
/RANT :-D