By the time I get hold of students and try to get them to apply maths to science, usually all they remember is the algorithm - and I know that maths at our school is taught really, really well, so they would have gone through the stages. But the fact that the students rely so hard on their algorithms makes generalisations really difficult for me to teach. For example, I have a student who, at 17, still struggles with pre-algebraic concepts (he cannot, for example, reliably state a relationship between two things when given a visual, written or verbal scenario).

But that's a digression. I just wanted to say, yeah, that's how it's taught. It's just a shame that most kids (or teachers/schools) rely so heavily on the algorithm that they forget the other techniques and so can't back up a little to an earlier stage if they need to.