When I was an elementary student, we were taught the traditional algorithms for mathematical computations. Which, for addition involved “carrying the one”. To solve 127+89=___ we would write:
and then compute the sum using columns and “carried ones”. Like this:
This method worked well for me. I understood that 7+9=16 ones, and 6 would be written into the ones place (now called units) for my total sum. I understood that the 1 from the 16 needed to go with the other tens, and thus 1+2+8=11 tens. I put the 0 into the tens place of my sum, and “carried” the 1 into the hundreds column. To finish the computation I had 1+1=2 hundreds and my total sum of 216. Yay for me, I found traditional addition methods easy to work with. I was using number decomposition in my method, without knowing it. So, how will I help my students who find the standard algorithm a little difficult? The ones who believe they are “adding one” to the tens column rather than adding ten. The answer is, make the sum visual.
There are many ways to make what is happening during the addition process visual! What they didn’t teach me in school is that my thought process can be written out using the partial sum algorithm. Actually writing out the decomposed numbers creates more understanding of number and place value relationships. I was, at first, skeptical of the value of new math teaching strategies, but now that I have used them I can’t get enough.
Solving my original problem with the partial sum (or instructional) algorithm would look like:
Each number can be decomposed so that the sum reads:
to make the addition easier, we add the similar numbers first, and then find the total sum:
I love the simplicity of decomposing numbers for problem solving. It gives students a clear, and easy to understand, way of thinking about large numbers. It seems to come naturally to kids, as with my son and his flowers. He picked them and sorted them by size all of his own design. He spent time looking for the ones that fit between. He just sensed a logical progression. To take this thinking further pictures and manipulates can be used. There are many resources available to teachers for helping students visualize unit measurement. One source that I like is: Ed Helper. They offer printables like the one below. You need a membership to use their pages, and there is a small fee.