"Common core" math is an attempt to codify this style so more kids can get a deeper understanding of numbers instead of just blindly following steps. Like the people that created it noticed people like you and me (I do something similar but not quite the same) have an intuitive understanding of math that made us good at it that they want to replicate for everyone. But it seems like very few parents and teachers understand it themselves, resulting in a blind-leading-the-blind situation where it gets taught in a bad way that doesn't achieve the goal.

Also aside, in the method I was taught in school (and I assume you and GP from terminology), "carrying" is what you do with addition (an extra 1 can be carried to the next column), "borrowing" is for subtraction (take a 1 away from the next column if needed).