Is it regrouping? Upon googling it seems regrouping is borrowing or carrying over collectively. What's the word for not borrowing and carrying over? It's supposedly to train mental computation.
Example 1: When evaluating $21-9$, one can perform borrowing or do the following:
$$21-9$$ $$=10+11-9$$ $$=10+2$$ $$=12$$
Apparently the students who do this first build up mastery of addition and then subtraction facts like 11-9 so the idea is to teach students that they can reduce 21-9 to 11-9 to train mental computation instead of borrowing 10 from 20.
Example 2: For $12+9=21$, one can perform carrying over or do the following:
$$12+9$$ $$=12+8+1$$ $$=20+1$$ $$=21$$
The idea is that students are supposed to have mastered different ways to make 10 (1+9,2+8,3+7, etc) and then to make 20 (11+9,12+8,13+7,etc) so students would want to extract from 9 that which they can use to make 20 with 12, which is 8 to train mental computation instead of carrying over 10 from 2+9=11.