What Is Counting-up Method?

The "counting up method" is a method of addition of relatively small numbers by counting. It is sometimes the preferred method where carries or borrows would otherwise be involved.

Example
58+3 is evaluated by counting 59, 60, 61. That is, you count 3 more than 58.
_____
In my opinion some of these odd ways of teaching addition and subtraction would disappear if we all learned to use an abacus or soroban. These tools use number representations that serve well and are not difficult to learn.