Disclaimer: The following post is not a serious suggestion that we should revamp the way we teach students to learn addition. It's merely an interesting thought experiment.

It's true. You don't need all those addition facts. You spent hours and hours and hours memorizing 100 addition facts (0 through 9 plus 0 through 9). Okay, actually, you didn't learn 100 of them, because you learned that addition is commutative (5 + 7 is the same as 7 + 5, so you just need to memorize one of them).

But you don't need all those facts. You just need 25 of them (arguably, only 20*). Here's what you really need: 1. math facts for all whole numbers less than 5; and 2. and all the "fives" facts:

"Less Than Five" Table
0 + 0 = 0
0 + 1 = 1
0 + 2 = 2
0 + 3 = 3
0 + 4 = 4
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
2 + 2 = 4
2 + 3 = 5
2 + 4 = 6
3 + 3 = 6
3 + 4 = 7
4 + 4 = 8

"Fives" Table*
0 + 5 = 5
1 + 5 = 6
2 + 5 = 7
3 + 5 = 8
4 + 5 = 9
5 + 5 = 10
5 + 6 = 11
5 + 7 = 12
5 + 8 = 13
5 + 9 = 14

That's a total of 25 addition facts. And you can do all addition problems with just these, as long as you're willing to mentally split numbers that are bigger than 5.

Here's what I mean. Suppose you have to add 4 and 8, but you don't remember what 4 + 8 is. What do you do (besides counting on your fingers!)? You split the 8 into 5 and 3 (because, according to your "fives" table, 5 + 3 = 8). So now you have 4 + 8 = 4 + 5 + 3. Since two of those numbers are less than five, you know their sum: 4 + 3 is 7. Now use your "fives" table to get 7 + 5 = 12.

Let's try another. Add 7 and 9. Split one of those up using the "fives" table: 7 + 9 = (5 + 2) + 9. Since we have our fives memorized, we do 5 + 9 = 14. Now, since we know 4 + 2 = 6, 14 + 2 = 16.

We could have done the problem by splitting 9 instead of 7: 7 + 9 = 7 + (5 + 4). You know 7 + 5 =12, and since 2 + 4 = 6, 12 + 4 = 16.

The process is:

1. Split the larger number using your "fives" table.
2. If you now have two numbers less than five, use your "less than five" table, followed by your "fives" table
3. Otherwise, combine the number that's bigger than five with the five, using your "fives" table. This will result in a carry, and then you can use your "less than five" table.

* The "Fives" table is actually shorter than this. You only need to remember the pairings:

0 and 5
1 and 6
2 and 7
3 and 8
4 and 9

For example, consider the pairing 4 and 9. 5 + 4 = 9, and 5 + 9 = 4 (plus a carry).

This brings the total number of necessary facts to 20.

Or you could, you know, just bite the bullet and memorize all of them. :D