I was never taught this but I read a book in the fourth grade and it's a huge help.

Imagine adding $124 + 589 + 276 + 985 + 231$.

If I see the $4$ and the $6$ that's $10$ and can forget it. I can see the $9$ and the $1$ and that's $10$ and I have $20$ so for. All that's left is the $5$ for $25$. I note the $5$ and carry the $2$.

Notice that is a *lot* easier then doing $4 + 9$ is $13$ and $13+6$ is $19$ and $19+5$ is $24$ and $24+1$ is $25$ so I not the $5$ and carry the $2$.

Then when we do the next column, we have the $2$ we carries and $2,8,7,8,3$ and the $2+8$ is $10$ and the $7+3$ is $10$. That leaves $2$ and $8$ and that's $10$. So that's $30$. We write $0$ and carry $3$.

Note: that is easier than $2+2$ is $4$. $4+8 =12$. $12+7=19$. $19+8=27$. $27+3=30$.

ANd the we have $3,1,5,2,9,2$ so $9+1=10$ and we have $3,5,2,2$. Okay, a bit of creativity. $2+2 = 4$ and $4+5 = 9 =10-1$. We can "borrow" from the $3$ and $4+5 +1 = 10$ and that leaves us with $2$. So that's $22$

The answer is $2205$.

Thing is... it doesn't make things worse. And what ever gives us more flexibility will *sometimes* make things better and never make things worse.

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Addendum:

On the other hand we have what David Stork wrote:

As an aside: While mental arithmetic may be an amusing diversion, you will almost never use it in real life... and in the future even fewer people will use it even more rarely. It is like using an abacus.

From my perspective, there are so many better things to focus on than mental arithmetic... such as memorizing basic integrals... that you'd do better to devote your time to those.

I can't argue with that. If tricks help, use them. But if the just confuse and give you things to frustrate you.

To my mind this is similar to the the tricks of multiplication:

$379\times 513 = $

$(400 - 21)(500 + 13) =$

$400*500 - 21*500 + 13*400 - 21*13 =$

$(4*5)10,000 - (20*500 + 1*500) + (10*400 + 3*400) - (20+1)(15-2)=$

$200,000 - (2*5*1000 + 500) + (4000 + 1200) - [(20*15 +15-2*20 -2)]=$

$200,000 - 10,000 -500 +5000 + 200 - [30*10 + 15 - 40 - 2]=$

$200,000 -5,000 -300 - [300- 25 -2]=$

$200,000 -5,000 - 600+27=$

$195,000 - 600 + 57= 194,427$

So *should* you do it this way. Well, only if intuitively it makes sense and is *easy* for you do it that way. It certainly isn't anything I'd feel comfortable *teaching* someone to do.

You could also do $379\times 513 = (400 - 25 + 4)(500 + 25-12)= ((\frac 12*10-1)*100 -\frac 14*100 + (\frac 12*10-1))(\frac 12*1000 + \frac 14*100 - (\frac 2*10-2)(\frac 12*10-1))$

and ... in theory that should be really easy. It's just place holding. But should you do it this way... Well.... probably not....I wouldn't.

But I take great comfort in knowing that I can *device* these ways of doing things. And that I understand why they work.