I started teaching my older girl when she was 4.5. At this point, she could count, subitize, and show numbers on her fingers and that was about all. We started slowly and gently by introducing addition. And as soon as we got started, I got my first inkling of the fact that obvious things were not in fact so obvious. Since my goal was to make sure that my kids didn’t use any shortcuts that they didn’t thoroughly understand, this was a dilemma.
Let me illustrate. Do you know what 9＋1 is? If you’re reading a math blog, you almost certainly know that it’s 10. Furthermore, you know that to add 1 to any number, you can simply take the next number in the counting sequence: 65＋1 is 66, 118＋1 is 119, and so on. Similarly, to add 3 to a number, we can count 1, 2, 3 starting from the number following our number. In fact, this works in general: to add something to a number, we can start at the number following our original number and count up to that something. This method of addition is called “counting on.”
Why does this work?
This is the kind of question that tends to stump people. For most of us, that fact is either completely intuitively obvious or memorized by rote. There is no “why.” It just is.
Except that if you’ve tried to teach this to young kids, you’ve probably noticed that they do not feel this way. If you ask little kids to put together 9 and 1, kids tend to start counting at 1. Assuming they don’t lose track, they’ll say the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 before they arrive at the conclusion that the answer is 10. They will not use the shortcut. It is in no way obvious to them.
That brings us to the next question. Since it’s not obvious to a little kid that 9 and 1 together make 10, what do we do? Most answers to this question fall into the following two categories:
(a) Teach a kid directly that this works. Instruct a kid that a number plus 1 is the next number and then practice, practice, practice.
(b) Let a kid discover this fact through experience. A typically developing child will eventually figure out that if you put a number and 1 together, you get the next number. After all, kids often have to figure out what happens if you take one more of a thing…
Out of those two options, I prefer (b). That’s the option that doesn’t rob kids of their innate numerical sense and instead replace it with “it’s true because the teacher told me so.” Unfortunately, option (b) isn’t very efficient, and it also tends to break down when a child is faced with concepts that aren’t often encountered in the wild. When that happens, even teachers who started out preferring option (b) tend to default to option (a). First, we’ll teach it, then we’ll cross our fingers that it’ll eventually make sense!
My preferred option is neither (a) or (b): it involves working with a child’s mental model of the objects and operations to derive each new step. Like option (a), it is directed and efficient. Like option (b), it makes deep sense to the child and leaves them with their mathematical confidence intact. It is, in fact, the best of both worlds.
This is the first in a series of posts about counting on. Here are the second, third, and fourth entries in the series. These lay out precisely how I teach counting on.