This is, perhaps, the first real explanation I’ve seen of what Common Core math is trying to teach. It’s a laudable goal, but it just doesn’t work for the vast majority of people. It certainly did nothing for either of my kids, one of which is relatively gifted in math and undoubtedly use different tricks than they taught in Common Core the same way I did when I was in school, where I was taught the algorithms we all know and love.
Personally, I don’t know if you can teach “number sense” at least the way it is being taught today. It seems to me you should start with the simpler, wrote methods that are less error-prone and then add “number sense” to show other, more memorable ways to solve the problem. Or maybe you encourage students to come up with their own methods and allow them to teach it to their peers.
Of course, there are some people who just don’t have any number sense whatsoever and won’t no matter how you teach it or who teaches it.