To me, the article isn't really about math... math is just used to illustrate how poorly we teach in America, and then talks about the reasons why, primarily that we do a really poor job of teaching teachers. Some key quotes:

- It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. …The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them.

- American institutions charged with training teachers in new approaches to math have proved largely unable to do it. At most education schools, the professors with the research budgets and deanships have little interest in the science of teaching.

- Without the right training, most teachers do not understand math well enough to teach it the way Lampert does. “Remember,” Lampert says, “American teachers are only a subset of Americans.” As graduates of American schools, they are no more likely to display numeracy than the rest of us.

- Left to their own devices, teachers are once again trying to incorporate new ideas into old scripts, often botching them in the process.

- In Japan, teachers had always depended on jugyokenkyu, which translates literally as “lesson study,” a set of practices that Japanese teachers use to hone their craft. A teacher first plans lessons, then teaches in front of an audience of students and other teachers along with at least one university observer. Then the observers talk with the teacher about what has just taken place. Each public lesson poses a hypothesis, a new idea about how to help children learn. And each discussion offers a chance to determine whether it worked. Without jugyokenkyu, it was no wonder the American teachers’ work fell short of the model set by their best thinkers.

And the most important two paragraphs:

The other shift Americans will have to make extends beyond just math. Across all school subjects, teachers receive a pale imitation of the preparation, support and tools they need. And across all subjects, the neglect shows in students’ work. In addition to misunderstanding math, American students also, on average, write weakly, read poorly, think unscientifically and grasp history only superficially. Examining nearly 3,000 teachers in six school districts, the Bill & Melinda Gates Foundation recently found that nearly two-thirds scored less than “proficient” in the areas of “intellectual challenge” and “classroom discourse.” Odds-defying individual teachers can be found in every state, but the overall picture is of a profession struggling to make the best of an impossible hand.

Most policies aimed at improving teaching conceive of the job not as a craft that needs to be taught but as a natural-born talent that teachers either decide to muster or don’t possess. Instead of acknowledging that changes like the new math are something teachers must learn over time, we mandate them as “standards” that teachers are expected to simply “adopt.” We shouldn’t be surprised, then, that their students don’t improve.

But this isn't news (even if the article is from 2014). I've linked to this report from 2007 many times before and I'll do so again:

http://mckinseyonsociety.com/how-the-worlds-best-performing-...

To find out why some schools succeed where others do not, McKinsey studied 25 of the world’s school systems, including 10 of the top performers. The experience of these top school systems suggest that three things matter most:

Getting the right people to become teachers; Developing them into effective instructors; and Ensuring the system is available to deliver the best possible instruction for every child.

In the US, we fall down on all three of these points.