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```Date: 05/20/2003 at 13:26:12
From: Tonya

When I was in school, we were taught to add numbers from right to
left. Now my daughter's teacher is teaching her to add left to right.
How can I help her to understand both ways? The type of learning is
different from when I was a kid.

25             25
15             15
--             --
40 (my way)   30
10
---
40  (new way)
```

```
Date: 05/20/2003 at 23:45:27
From: Doctor Peterson

Hi, Tonya.

I'm not sure I agree with teaching this as a method of addition, but
I do see a reason for doing it. It shows on one hand that there are
different ways to do arithmetic - it isn't just a magic trick that you
have to do the way you are taught or it won't work; and on the other
hand, that it is possible to think about what you are doing and why,
rather than just blindly following rules. The danger is that some
teachers might just teach it as another set of rules, without showing
the meaning behind it. Let's look behind the scenes at both methods.

The traditional way (which was developed over centuries as the best
method for paper and pencil, since it requires the least writing or
erasing) starts at the right. We add the "ones" column, and get
5+5=10. We can't write this as the ones digit in the answer, because
it is more than one digit; so we break it up as 1 ten and 0 ones, and
write down the 0. The 1 ten is added to the tens column in the
problem, giving us 1+2+1 = 4, which we write as the number of tens in

The basic idea behind this doesn't really depend on the order of the
columns: we add tens to tens and ones to ones, and if the "ones" go
beyond 9, we move the tens into the tens column. The "new way" does
exactly this; the only real difference is that it "carries" after
adding the original numbers, rather than as part of the tens column
when it is first added. That is, you add the tens to get 20+10 = 30;
and you add the ones to get 5+5 = 10; and then you add these together
to get 30+10 = 40.

We could do the "new way" from right to left as well; the order
doesn't matter. Let's compare that with the traditional way:

new:      old:

1  <-- tens from 5+5
25        25
+ 15      + 15
----      ----
5+5 -->  10         0 <-- ones from 5+5
2+1 -->  3         4  <-- tens from 2+1, plus 1
----
40

I did two things differently from the "new" method you showed: First,
I switched the order of the 10 and the 30, just to make it feel more
familiar; second, I didn't write the 0 in the 30, since that results
from adding tens only, and there will never be any ones in it. As you
can see, the 10 you get in the old method, which is written in two
separate places, is written all in one line this way, but still in the
same columns. The 2+1 from the tens, plus the "carry" of 1, are
written separately now, but still added in the tens column. So all the
same work is being done, but it is perhaps a little clearer why we do

I think it's clear that this method is less efficient with pencil and
paper, because there is more writing. And working from left to right
in the traditional method would require erasing, because you would
write 1+2 = 3, but then have to add 1 and change it to a 4 after you
added 5+5. That was done in earlier days, because erasing is no
trouble on a sand table or chalk board or abacus; it was only with the
introduction of arithmetic on paper that the right-to-left method was
needed.

But the new method works well when I add in my head, and I suppose it
could be argued that the new generation will have more reason to add
in their heads than on paper. It's easier to work left to right in
your head, because you remember numbers from left to right. I would
say something like, 20 + 10 is 30, 5 + 5 is 10, 30 + 10 is 40. There
think "25 + 5 is 30, plus 10 more is 40." With my own children, I like
to have them solve a problem and then tell me how they did it, to
emphasize that there are lots of ways to do it, and each problem might
have its own special trick. If children get used to thinking, rather
than just following rules, it can only help their understanding of
numbers. My only concern is to make sure that they aren't just being
taught more rules, and getting confused as to which they should
follow.

understand both methods, but to understand why they are the same, and
what that teaches about addition. I hope the teacher is doing the
same.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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