Addition, for me, is intimately connected up with my concept of a number. When I think of numbers in my head, I often think of the number in connection with its constituent parts, and when I divide these parts up into equal pieces I “get” multiplication. However, on top of this bare bones thinking, I also conceptualize numbers strongly by their size, thinking about the number first as the most significant digit in the number and proceeding down to the less significant digits. Which makes me wonder, do we teach addition backwards?
The standard grade school algorithm we are taught for adding numbers starts with the least significant digit and then proceeds to higher significance digits. Thus, for example, to add 123 and 237 we first add 3 and 7, obtaining a 0 and a carry of 1. Then we add 2 and 3 and that carry of 1 to get a 6 with no carry. And then we add 1 and 2 to obtain 3. So our total is 360. But when I was learning to add numbers I didn’t work this way. I worked the other direction. I started with 1 and 2 and add them to get 3. Then I would take 2 and 3 and add it to get 5. Finally I would add 3 and 7 to get a 0, but then I needed to add that carry of 1 back to the 5 and obtain 6. Of course my answer was the same, 360, but I did the addition starting with the numbers that matter the most in the magnitude of the number (1 and 2).
Which makes me wonder if teaching addition starting with the least significant digits first and then moving to the most significant digits doesn’t cause kids to never really grasp how a number represents magnitude or size? By performing the largest magnitude terms first, it seems that you’d get a better idea about the size of the resulting sum, which you then increasingly refine as you go to lower significant digits.
Of course I think teaching both ways of performing addition (as well as any other variation) is probably better than teaching just one way of performing addition. But it would be interesting to see if kids who are taught to add “backwards” have a better grasp about orders of magnitude than those who are taught to add “forward.”