Steve Leinwand and Steve Fleischmann said:

In mathematics instruction, a chasm exists between research and practice. For evidence of this gap, look no further than the mismatch between what research says about developing students’ conceptual mathematics understanding and what we actually do. An example is the way we teach math content in elementary and middle schools. A growing body of promising research shows that if initial instruction focuses exclusively on procedural skills, then students may have difficulty developing an understanding of math concepts.

AND

On post-tests, the students who received only meaningful, or relational, instruction performed better in applying the procedure and solving the equations. In contrast,

the students who first received procedural instruction on how to solve an equation tended to resist new ideas and appeared to apply procedures without understanding.[emphasis added]

AND

the form of instruction humorously but accurately characterized as

yours is not to reason why, just invert and multiplymay not enhance the performance of many students. Alternatively, instruction that places a premium from the start on meaning and conceptual understanding may improve classroom productivity.

And yet, despite Common Core and other efforts, our procedural emphases still persist in many, many math classrooms. And parents clamor for them.

FYI, this is from twelve (12!) years ago…

**Image credit:** maths, Shaun Wood

Of course, there has been little movement towards more conceptual understanding in the Common Core era: the concomitant high-stakes testing guaranteed that would be the case. But even if the assessment piece of the Common Core initiative were as progressive as the Standards for Mathematical Practice, you can’t legislate a sea change in actual teaching in a nation such as ours. Procedural instruction reflects more than a century of “how it’s always been done,” and we lack a sufficiently large base of constructivist math teachers to turn the tide.

You’re completely correct that it’s the testing driving things away from real progress. Higher ups saying that they want you to teach “Higher Order Skills” then evaluating based standardized multiple choice tests doesn’t fool anyone. There’s also the paperwork issue… people have no idea how much the level of documentation of process required of teachers has increased, and it’s a whole lot easier to write up a procedural lesson than an inquiry/exploratory activity. I am thankfully in a position where I can do what’s best for my students and not worry about having to defend it, but the new teachers in my district are certainly not.

many factors causing the gap:

parental personal narratives as you mention, but also those of the teachers, who many themselves have only experienced procedural mathematics, unaware themselves of the inherent structure and sense making of mathematics. This though – we can mend should we be ardent and honest in our conversations with professionals. If we call forth a clear signal and speak in simple terms about what mathematics IS.