In a recent Common Core debate, one of the advocates talked about how Common Core teaches children to "think mathematically". She illustrated her point by telling the audience that she was taught to "carry the one" when she added, but was never told why she was doing so. She was taught to blindly follow a memorized process with no explanation of the mathematics behind it.
That statement is simply not true.
Math books, and math teachers, have always explained the concept of place value.
Many of us probably can recall the little wooden blocks to represent ones, the sticks with nine lines across them to represent tens, and the larger wooden squares with lines running across and down their sides to represent hundreds. We used the smaller wooden pieces to build the bigger ones so we could see, for example, that ten little blocks made one stick.
The math books had pictures that used the blocks to show how place value worked. The place columns were labeled, with the mathematical place names and pictures of the appropriate block. The process started with ones and tens. The little block was above the ones column, and the stick was above the tens column.
A problem was given, such as 28 + 15. The text illustrated, and the teacher explained, that we…
always began with the ones place. So 8 + 5 was 13. And we counted out 13 little blocks. Then we put a stick down and put those little blocks on top of the stick.
If we filled the stick, we picked it up and "carried" the little blocks on top of it over to the tens column because that was where sticks belonged. If there were any little blocks left, they had to stay in the ones column. In the example problem, that meant that there were 3 little blocks left in the ones column, so we wrote down the number 3.
Then we moved to the left, labeled the tens column. In that column, we saw 2 + 1. So we counted out 3 sticks. But we had "carried" another stick into that column, so we had to include that as well. That made 4 sticks, or 4 groups of ten. So we wrote that down in the correct column, and we had our answer of 43.
The teacher moved us from needing to use sticks to using the numerical form itself to "carry" into the next place value when it was necessary. And if we think about it, we can probably recall that process as well, where we were told that no column could have more than one numeral in it. The text usually put lines between the columns to force us to write the numbers in the correct places.
The simplest and most direct process was taught so we would be most likely to both grasp the concept and accurately solve the problem, building both our basic math skills and our self-esteem.
For most of us, that process is now so automatic that we complete it without conscious analysis. But the fact that we do not think stop and think about why we "carry the one" every time we add does not mean that we do not know why. We do – which means that it was taught it to us.
To assert otherwise is simply ridiculous.