Could you add some glossary or quick summary of what PCK  or CGI stand for? It would be nice if this post had value without having to read the full cited text or having to look at the original links for context. I'd be up for a longer explanation and reformulation in your own words of why exactly people think they're bad at math and alignment and ESPECIALLY, what's the next step to fix that? (Currently I feel a little click baited by the title, in the sense that the content doesn't seem to justify the article without delving deep into it. Side not, I'm probably not the intended audience of this post as I feel I'm pretty good at math and enjoyed it my whole life. I think the intended audience would have even less patience than I did)

Thank you for excerpting this, even if it is rough.

This is fascinating.  I did a few semesters of teaching undergrad math while working towards my math MS, so this was completely downstream of the K-12 education system and I saw none of it.  But I did notice a lot of different attitudes towards math that fit with the observations above.  In the end, it was hard to teach to students who didn't care, and I much preferred tutoring since I could ask questions and try and really get into the gears of their misunderstanding and replace it with functional gears. Preferring this as often as possible to memorization or drilling a routine!   In classes there was so much misunderstanding because of people who had only memorized, and imperfectly, so when the "rule" didn't work, they were lost.  (e.g., "FOIL" for multiplying two binomial terms instead of understanding distribution of multiplication) 

Anecdotally, I was one of those who decided not to teach because I saw how low paid and how draining it was on teachers.   :(

Anecdotally, I was not very good at math in school (enough to pass, but not much better), mainly as a result of a lack of motivation. 
I got interested in math (and subsequently better) when I learned set theory (not in school), as I felt like I understood more about why arithmetic works the way it does.
It was hard to motivate myself to learn math when it felt like pointlessly executing algorithms that computers are so much better at anyway.
This is obviously just an anecdote, but I feel some sort of exposition to set theory in schools, though maybe not at the beginning of math education may be beneficial.
I am now better at math than I used to think, and actually enjoy it, and I think that you are right that many more people could be also.
 

The only benefit of public schools anymore, from what I can tell, is that very wise and patient parents can use it to support their children in mastering Defense Against the Dark Arts.

Well, that and getting to play with other kids. Which is still pretty cool.

This may be, perhaps, an under-appreciated function of (public) school schooling!