On Nov 10, 2012, at 10:44 AM, Joe Niederberger <firstname.lastname@example.org> wrote:
> Instead, students should be introduced to formulas and > become very comfortable that "expressions with variables" are much like a language to express meaningful things, long before they are asked to "solve for x".
There is a natural progression of sophistication. Some teachers have a very keen sense of this, and some have no sense of this. Amazingly, even though there are many teachers with this sense and they produce very successful math students, this is not taught in teacher school. If any thing was to be taught in teacher school, I would have expected this to be that thing. It is not complicated. It requires no pretending or embellishment. It does not require us to believe absurdities like toddlers understanding vector algebra or ants understanding trigonometry.
The biggest problem, and Clyde's problem as well, is that the vast majority of teachers do not have longitudinal experience. They don't know how this process works because they have never seen the whole process work. They do not teach mathematics to a student or a class of students for six straight years, they are only responsible for one of those six years. This is something I would address first. If teachers had more longitudinal experience they would understand the progression. Can you imagine the effect this would have on a teacher's experience?