Adding Left to Right

Date: 05/20/2003 at 13:26:12
From: Tonya
Subject: Addition in Math

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
Subject: Re: Addition in Math

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 answer.

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 
it: we add tens, we add ones, and we add the results together.

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 
are lots of other ways to add in your head; I might be more likely to 
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.

I hope this helps. Definitely do help your daughter not only to 
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/