>Again, I don't understand how one can separate concepts from algorithms.
There is, I feel, an enormous difference between concepts and algorithms.
To understand the concept of division is to understand what it means to group things, compare things (as in ratios), etc. A child in first grade shows some basic understanding of the concept of division when the child shares something fairly with a friend. And this child can do it without knowing anything about a long division algorithm.
An algorithm is simply a step by step procedure that "gets a job done". But there are many many algorithms that accomplish finding quotients. Yet, we find it necessary to spend so much time teaching the one long division algorithm to nearly all our students.
Ask a student who has learned the long division algorithm to find 2 divided by 8. I guarantee that in nearly any class of 5th or 6th grade students, some students will answer "4", indicating that they may know an algorithm, but don't understand the concept. On the other hand, give a 5th or 6th grade student two dollars and ask that student to divide the money fairly among eight people, and chances are that student will find a way to do it, indicating that they understand the concept of division, regardless of whether they know the long division algorithm.
Sorry, but right now, I just feel that our students are capable and deserve to spend their time on more important mathematics than that long division algorithm. With all the mathematics that exists in the world, the long division algorithm hardly deserves the attention it is given.