Mathematics Magazine for Grades 1-12
Addition of two numbers:
Let's add these two numbers in our heads (i.e., without paper). Can you do that? Our first attempt is to do it like most of us do on paper: 6+7=3 carry the 1 (13), 1+6+2=9, 1+3=4, 2+9=11.
The answer is . . . now what were those numbers? The problem here is memory, not mental arithmetic. That's why people use paper (or an abacus, or their fingers), to help out their memories.
Memory is why speed arithmetic experts (I call them "arithmetickers") usually add numbers like these from left to right. It is a little more complicated that way (you have to back track). But, you end up saying the answer from left to right, just as it is normally said. It is easier to remember a number from left to right.
Let's try again: 2+9=11, 1+3=4, 6+2=8, 6+7=3 and that previous 8 should have been a 9 (because of the carry). I actually remembered the answer, 11493, that time. It's still a test of my memory, but not bad. It may take you a little practice to be able to do that.
Addition of columns:
How about this addition problem? Add these up in your head. It's not too tough to add up the right column, remember the carry, and add up the left column, just as you would do with a pencil.
In the right column, a speed arithmetic person might group the 9 and the 1 (10), the two 5's (10), and then the 7+2+4 (13) to get 33.
Something similar we use for the left column.
A few people group elevens instead of tens. Instead, what I do is add 29+37=66, then 66+15=81, then 81+21=102, 102+32=134, 134+85=219, and 219+44=263. Isn't that slower? Maybe. But I have little to remember, just the sum so far. It becomes very fast, if you practice doing it that way.
That is how a person with an abacus (or a person counting with fingers) would do it. And an abacus is just a way of remembering the latest sum. That's something you can easily do without an abacus.
Older speed arithmetic books dwelt almost exclusively on tricks. Here are some of those tricks (which you can deduce on your own, instead of memorizing this table):