I have a 10 year old who attended private school. I'm not sure if they used Common Core or not, but they definitely used an approach designed to instill an understanding of concepts vs. the rote arithmetic drilling I did as a kid.
I was skeptical at first, but mostly because the school did a poor job explaining the methods to parents, which left me to Google the methods so I could help my son when he had questions. That said, he does have a great understanding of numbers and is able to do a substantial amount of math in his head using this understanding. (e.g. 99+99+99 is just 300-3). It's essentially what I do in my own head, though with slightly different methods.