Ok, here's what I see from the trenches.
First, I think I see the big picture because I:
have a special ed student in pre algebra
have a daughter in regular 6th grade math
teach remedial math in college
tutor bright but struggling high school boys in math
have just finished 8 credits of graduate mathematics
I could write a whole long post, but I'd get bored in the middle and delete it. So here goes with the conclusion:
1. Certain school districts (mine) believe in good curricula and qualified teachers. (see the exception below) This combination makes all the difference in the world. The highly qualified teachers know what the students need to learn in math, and will find a way to do the direct instruction. (The exception in my district is, of course, elementary school.)
2. I have tutoring students in other districts who use good curricula but have poorly qualified teachers and students in districts with highly qualified teachers and poor curricula. The boys I am tutoring had a very solid foundation in basic math and are still struggling. The good news is a good tutor can fill in the holes quite easily.
3. All of my professors at the university that I attend use the direct instruction method. They go over every proof in the book in detail (even though the proof is in the book) and they give solutions to every problem they assign.