Learning to Count to Two

The thing about toddler learning is that it’s incredibly weird. The things they say are totally bizarre, just normal enough to make them sound and act like visitors from another planet.

You can understand why someone like Jean Piaget would decide that the way to study early learning is to basically treat them like tiny psychiatric patients. Susan Jean Mayer puts it like this in “The Early Evolution of Jean Piaget’s Clinical Method”:

The diagnostician stands against his or her very conception of reality in attempting to grasp and characterize another’s dysfunctional stance in relation to that reality. Psychoanalytic diagnosis, then, implies the need to close a profound psychological distance between doctor and patient in regard to their understandings of the world; in Piaget’s work, this psychological distance lay between adult and child.

A fair criticism of Piaget is that he treated toddler’s minds as essentially dysfunctional compared to logical, rational, adults. Still, he’s capturing something real with this stance towards children, and it’s that the way young kids think is just weird. And nowhere is that weirdness more clear than in early number learning.

But peer just a little bit closer at that weirdness, and the wonderful thing is that it can actually make sense. That’s simultaneously a vindication of Piaget as well as a criticism. There is a kind of semi-coherent, bizarre but sensible toddler rationality at play, if you can listen closely enough.

Here’s an example from home. My son is two and a half, and counting is now part of his life. He knows the number sequence but in this very destructive toddler way. I noticed this on Monday because while I was ignoring him and drinking coffee he found a pad of sticky notes. I guess a part of me heard him counting: one, two, three,…nine, ten, nine, ten, nine, ten, eleven, thirteen!

Friends, he had taken a whole pad of stickies and stuck them all over the microwave.

Clearly, then, the boy doesn’t have what early learning people call “one-to-one correspondence.” He knows the number sequence and he knows that when you have a bunch of the same object that the number sequence is relevant but… there doesn’t seem to be any understanding of what those numbers mean.

And yet.

Something else I’ve noticed is that he can completely reliably identify two objects. I noticed this while reading. If I ask him, “are there one or two people in this picture?” he will answer correctly 100% of the time.

“two!'“

“one!”

So, what does that mean? It means that the word “two” actually does mean something to him. He understands it. He knows it. So does the number “one.” He has some sort of number sense.

Emboldened by this, I started asking him a slightly different question while reading. For pages just like the ones above I started asking him, “how many people are there on this page?”

His answer: one, two, three, four, fix, six, seven! Or: one, two, two, two, three! Or even: eight, nine, ten!

So, that’s very interesting.

My thinking about this is, OK, he knows the number sequence, sort of. But that’s totally unrelated to his understanding of magnitudes. He doesn’t have a “three” concept yet, but he does have a “one” and “two” concept for sizes of sets. He doesn’t yet know that you can measure the size of a set with the number sequence.

But! I bet he could learn that. And then, I bet, we could move on to the “three” concept. And that, I bet, is how he could learn that the sequence can be used to measure sets, which is the big-kid one-to-one correspondence idea.

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There are a few things that are worth taking away from this.

First, this whole episode exemplifies a perspective on children’s thinking, one that is important for anyone who cares about learning. There are often little steps that are invisible without close observation. There are things that seem completely irrational, but they’re actually sensible within a limited domain. The manifestations can be bizarre—they’ll say or do wacky things, those things may or may not involve a microwave—but there is almost always a kind of logic at play.

Second, over the past few days, I actually have been able to model counting “one, two” to identify two objects. This amounts to a sort of principled explanation for something that he already knows. He knows that there are two people in the picture; I am implicitly giving a reason why. This feels so close, at a deep level, to how learning works in my classroom, that I feel confident that it’s the same mechanism at play.

The recipe is: (1) get good at a little thing; (2) give a big explanation for why the little thing works; (3) use the explanation to extend what you know to the big thing. This is one way we learn from examples.

Finally, there very clearly is research about early number learning, most of which I’m ignorant of. My friends Kristin Gray and Jenna Laib pointed me towards Doug Clements’ learning trajectories site, and I can sort of see where my son falls in his sequences. Under “Subitizing,” I see him along the path towards becoming a “Small Collection Namer”:

Names groups of 1, 2, and 3 with increasing accuracy. Most children of about 34-39 months of age can accurately name groups of 1, 2, and 3. Many children learn to recognize and name groups of 4 about 6 months later. The child is able to recognize small groups without relying on a model or matching strategy. 

In the “Counting” trajectory, he’s certainly not a “Corresponder” but he might be a “Reciter (10)”:

Verbally counts correctly to ten. Shows some correspondence with objects, but often exhibits performance errors in correspondence (e.g. skipping, double-counting). 

What I don’t see is how Clements would relate these two developmental sequences—what I described above is how I think they might be related, and I assume that others think that too, but I don’t know.

My final thought is that I’m amazed that this is still new to me. This is our third kid, and I guess I’ve been down this road twice already. But it moves so fast! They grow up so quickly! In a few months he’ll be a Chanter and a Reciter and a Corresponder and… ohmygod am I crying? Augghghgghhblurblurb, sniff, OK, I’m alright. I’m going to go take a long walk and scroll through baby pictures on my phone until I feel better.