>The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence. As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.
Teaching for understanding means that teachers are responsible for ensuring that students are understanding. If a student is mistaken about understanding something, but the teacher doesn't probe their understanding to expose their misconceptions, that's not "teaching for understanding".
Common core encourages repetition through its focus on multiple representations. One might study linear growth as repeated adding, as a table, as a graph, and in applications to various real-life phenomena. Common core places emphasis on the student being fluent (as the author states, common core has fluency as one of its three major focal points) with all of these representations, and also in seeing the connections between them. This repeated exposure brings out misconceptions, builds understanding, and (over time) results in fluency.
I really don't see why the author has a bone to pick with common core since the sort of practice she describes would fit perfectly into a common core curriculum:
>I memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side?
Common core (and contemporary education movements) are against "rote" or "procedural" learning. They would be against making up a song to memorize f=ma, and merely using that song to plug-and-chug through a small collection of problem types.
One recent example I saw (a colleague works on coaching teachers in common core) was a class of elementary students who could correctly multiply 4/7 * 5/9, but couldn't shade in 1/4 of a square. They memorized and rehearsed the procedure for multiplication, but never built understanding of what they were doing.
The unfortunate thing is that they are able to demonstrate fluency in this skill - and they will likely score well on standardized tests as a consequence of this fluency. This skill, however, is shallow - and will be easily forgotten without continued practice. Furthermore, when the time comes to learn proportional reasoning, or rates of growth, or any other thing that has to do with fractions, they will have nothing to build their understanding on.
I have to make a concession to the author, however. It is easy to get this impression of common core from the sidelines. Most teachers, departments, and schools were dumped into the core (which is merely a set of standards) without much support or training. Implementing the core requires a major shift in how one approaches teaching, and whether it is due to a lack of understanding, a lack of will, or most likely - a lack of resources, many classrooms are merely cargo-culting the sorts of things that common core demands.
My favorite introductory book to the subject is https://amzn.com/0325052875 happy to chat!