Required Skills: a mental model of quantity, fluent rote counting up to at least 10, a mental model of addition.
In my last post, I discussed the fact that “counting on” is not in fact obvious to most kids. So how does one teach a kid to count on without interfering with their natural number sense but also without waiting for them to discover it themselves?
The key idea is to use a kid’s natural mental models of quantity and addition. What are these models? Most young kids figure out quantities by counting. Through their real-world experience, they also tend to have a robust understanding of what it means to “put numbers together.” That means that kids are, in fact, able to answer questions like “What do we get if we put together 2 cookies and 1 cookie?” They will probably answer it by actually showing the numbers using counters or fingers (or even actual cookies!), but they are already able to solve the problem. As you’ll see, that’s a crucial feature of working with mental models: when a child already has the requisite mental model, they will already be able to solve the question you’re working on. The issue will be that they will not yet be using an efficient method.
All right. Say that we have a child who can figure out a question like “What do 8 and 1 make when put together?” by counting, and say that we want them to do so without the tedious counting of 1, 2, 3, 4, 5, 6, 7, 8, 9 every single time. In other words, we want them to count on. Where do we start?
Here’s our crucial observation: instead of teaching them to count on like it’s separate and different from what they were doing before, we work in the context of the child’s original mental model. We take what they were doing already (which is counting from 1), we work with their perfectly valid process, and we point out that there’s a faster way to do it! Here’s how it works:
Parent: Hey, can you figure out what 5 and 2 make when put together? You can use these counters.
Child: *gets out counters, makes a group of 5 and a group of 2, counts 1, 2, 3, 4, 5, 6, 7.* It makes 7!
Parent: Great job! Now I’m going to teach you a faster way to do this. Let’s arrange your counters in the two groups of 5 and 2 again. Remember, we want to see how many this makes, so we’re going to count all of them together. Let’s start counting with the group of 5: 1, 2, 3, 4, 5…
Hey, wait a minute! We got up to 5. That’s not surprising, is it? That’s a group of 5, so we’d expect to get to 5 if we counted this pile. So really, we could count these quickly, without even looking at them. Let’s try it. Can you count these quickly, without even pointing to them?
Child: 1, 2, 3, 4, 5.
Parent: Very good! Now remember, we’re trying to count all of them. And we’ve already counted the first group! Keep going…
Child: …6, 7.
Parent: Nice work! Look at that… you added the numbers without having to actually to point at any of the counters in the first group. Let’s try it again! Let’s add 7 and 1. *makes a group of 7 and a group of 1.* Now, let’s count how many we have all together. Can you start by counting this group of 7? Try to do it quickly, without actually pointing to each counter.
Child: I can’t!
Parent: Remember, there are 7 counters in this group. We’re putting together 7 and 1. Think about what happens when we count 7 things.
Child: Ohh, right! 1, 2, 3, 4, 5, 6, 7.
Parent: And we need to count all of them, so keep going.
Child: …8. So there are 8 in total! 7 and 1 makes 8.
Parent: Very good! So remember, when we’re putting together two groups, you can actually count the first group quickly, without having to point at each item. That’s a good trick to make putting together numbers faster.
At this point, the child has reached the first stage of “counting on”: they can figure out the total number of items in two groups put together by counting the first group quickly, without actually having to point at every single counter in the group. This will make it faster for them to add.
Note that even though the child now knows this shortcut, they may very well forget it. The good news is that you can remind them within the context of their own mental model and their own methods. “Are you putting together 6 and 2? Yes, you can figure this out by counting all of them if you like. But remember, this group has 6 items. What numbers are you going to say if you count them first?” The child will not feel like they were doing something wrong by counting from 1: they will merely be reminded that there’s a faster way to do it, and that perhaps they should take advantage of it.
This example illustrates a recurring theme of teaching via mental models: a new method will not seek to replace the child’s original way of figuring things out. Rather, it will build on top of it. If a child forgets a new shortcut, they will still be able to figure out the question, and it will be possible to remind them without invalidating the old method. This is because both methods are built on the same mental model. We will see many examples of this in the coming months.
Note that if a child is not yet fluent with rote counting, this may not work, as they may get stuck after saying “1, 2, 3, 4, 5” and not be able to continue the sequence, instead starting at 1 and saying “1, 2.” If that happens, practice rote counting and adding by counting from 1 for a while and come back to counting on when the sequence is fully internalized.
In my next post, I’ll describe how to work on transitioning from the “fast counting” version of counting on to the traditional version.
This is the second in a series of posts about counting on. Here are the first, third, and fourth entries in the series. These lay out precisely how I teach counting on, as well as my motivation for teaching it like this.