Adding left to right
Add multi-digit numbers in your head from the left, keeping one running total, instead of using the right-to-left method built for paper.
Add two numbers in your head without the columns coming out backwards.
The trick
Start at the big end. Add the leading places, then the next, keeping one running total in your head as you go. There is no column to finish and nothing to write down, so the answer arrives in the order you would say it.
The running total gets revised upwards as you go, and that is not a mistake in the method — it is the carry arriving.
Why it works
Split both numbers into their places and addition rearranges freely, because it is associative and commutative:
Nothing in that says which bracket to do first. Right-to-left is not a fact about addition at all; it is a constraint of the written algorithm. On paper you commit one digit per column and never touch it again, so you have to work in the direction the carries travel — leftward — or you would be going back to erase digits you had already written.
A running total has nothing committed to erase. When the tens contribute 130 to a total of 600, you add 130 to the number you are holding. The carry is not a separate step; it is ordinary addition into a number that was never final.
It goes further
Because you build the answer from the top down, you can stop early and have an estimate. And the estimate has an exact guarantee: the partial total is always a little under, never over, because every place you have not reached yet contributes something positive.
The size of that gap is worth knowing. Two digits sum to at most 18, so the places below the one you stopped at contribute at most 18 + 1.8 + … , which is under 2 of the place you stopped at. Stop after the hundreds of a three-digit sum, and you are inside 200 — the hundreds count is right, or one too low, and never anything else.
It extends to a column of several numbers, which is where it beats the paper method outright: add all the hundreds, then all the tens, then all the units.
The bound loosens as the column grows — with m numbers the shortfall is under m of the place you stopped at, since each place now holds up to 9m rather than 18 — but it stays a shortfall, so a partial total is never an over-estimate.