Two posts ago, I presented some rather odd data about the developmental trajectory of counting. It turns out children learn the meanings of number words in a rather odd fashion. In my last post, I described the "number" systems that are in place in animals and in infants before they learn to count. Today, I'll try to piece all this together to explain how children come to learn to be able to count.

Children first learn to map number words onto a more basic numerical system. They learn that "one" maps on to keeping track of a single object. After a while, they learn "two" maps onto keeping track of one object plus another object. Then they learn that "three" maps onto keeping track of one object plus another object plus another object. All this follows from the Wynn experiments I discussed two posts ago.

Up to this point, they've been learning the meanings of these words independently, but around this time they notice a pattern. They know a list of words ("one, two, three, four") and that this list always goes in the same order. They also notice that "two" means one more object than "one," and that "three" means one more object than "two." They put two and two together and figure out that "four" must mean one more object than "three," even though their memory systems at that age don't necessarily allow them to pay attention to four objects simultaneously. Having made this connection, figuring out "five," "six," etc., comes naturally.

So what is that more basic number system? One possibility is that children to learn to map the early number words onto the analog number system I also described in the last post (the system adults use to estimate number when we don't have time to count)?

Something like this claim has been made by a number of well-known researchers (Dehaene, Gallistel, Gelman and Wynn, to name a few). There are a number of a priori reasons Susan Carey of Harvard thinks this won't work, but even more important is the data.

As I described two posts ago, very young children can hand you one marble when asked, but hand you random numbers of marbles if asked for "two," "three" or any larger number. They always give you more than one, but they can't distinguish between the other numbers. Following Wynn, these are called "one-knowers." Slightly older children are "two-knowers," who can give you one or two marbles, but give you random amounts greater than 2 if asked for another other number. At the next stage, the child becomes a "three-knower." Usually, the next stage is being able to succeed on any number. I'll call those "counters."

Recently, LeCorre and Carey replicated this trajectory using cards with circles on them. They presented the children a card with some number of circles (1 to 8) and asked the kid, "How many?" One-knowers tended to reply "one" to a card with one circle, and then guessed incorrectly for just about everything else. Two-knowers could count one or two circles, but guessed incorrectly for all the other cards. Three-knowers could count up to three, but just guessed beyond that. Counters answered correctly on essentially all cards.

So far this doesn't tell us whether children learn to count by bootstrapping off of analog magnitudes or some other system. Carey and Mathieu LeCorre published a paper this year that seems to settle the question. The setup was exactly the same as in the last paper (now with cards with anywhere from 1 to 10 circles), except that this time the children were only briefly shown the card. They didn't have enough time to actually count "one, two, three..." The data for one-, two- and three-knowers didn't change, which isn't surprising. Both the "3-object" and the analog magnitude systems are very fast and shouldn't require explicit counting.

However, counters fell into two groups. One group, about 4.5 years old on average, answered just as adults. When they saw six circles, their answers averaged around "six." When they saw ten circles, their answers averaged around "ten." This is what you'd expect if they have mapped number words onto the analog magnitude system.

However, the other group, which was slightly younger (average age of 4 years, 1 month), guessed randomly for cards with 5 or more circles, just as if they didn't know how to count. However, these kids can count. If given time to look at the cards, they would have said the right number. So despite the fact that they can count, they do not seem to have their analog magnitude system mapped onto number words.

This means that the analog magnitude system isn't fundamental in learning how to count, and it actually takes some time for children to learn that mapping even after they've learned to count. Carey takes this as meaning that the analog magnitude system doesn't play a fundamental role in learning to count, either, and there are other reasons as well that this is probably the case.

One remaining possibility is that children use the "3-object system" to understanding the meanings of 1, 2 and 3. This seems to work nicely, given that the limits of the system (3 objects in children, 4 in adults) seem to explain why children can learn "one," "two," and "three" without really learning to count. Carey actually has a somewhat more nuanced explanation where children learn the meanings of "one," "two," and "three" the same may that quantifiers (like "a" in English) are learned. However, to the best of my knowledge, she doesn't have an account of how such quantifiers are learned, and if she had an account, I suspect it would itself hinge off of the 3-object system, anyway.

That's it for how children learn to count, unless I get enough comments asking for more details on any point. For those who want to read more, there are many papers on this subject at Carey's web page.

Children first learn to map number words onto a more basic numerical system. They learn that "one" maps on to keeping track of a single object. After a while, they learn "two" maps onto keeping track of one object plus another object. Then they learn that "three" maps onto keeping track of one object plus another object plus another object. All this follows from the Wynn experiments I discussed two posts ago.

Up to this point, they've been learning the meanings of these words independently, but around this time they notice a pattern. They know a list of words ("one, two, three, four") and that this list always goes in the same order. They also notice that "two" means one more object than "one," and that "three" means one more object than "two." They put two and two together and figure out that "four" must mean one more object than "three," even though their memory systems at that age don't necessarily allow them to pay attention to four objects simultaneously. Having made this connection, figuring out "five," "six," etc., comes naturally.

So what is that more basic number system? One possibility is that children to learn to map the early number words onto the analog number system I also described in the last post (the system adults use to estimate number when we don't have time to count)?

Something like this claim has been made by a number of well-known researchers (Dehaene, Gallistel, Gelman and Wynn, to name a few). There are a number of a priori reasons Susan Carey of Harvard thinks this won't work, but even more important is the data.

As I described two posts ago, very young children can hand you one marble when asked, but hand you random numbers of marbles if asked for "two," "three" or any larger number. They always give you more than one, but they can't distinguish between the other numbers. Following Wynn, these are called "one-knowers." Slightly older children are "two-knowers," who can give you one or two marbles, but give you random amounts greater than 2 if asked for another other number. At the next stage, the child becomes a "three-knower." Usually, the next stage is being able to succeed on any number. I'll call those "counters."

Recently, LeCorre and Carey replicated this trajectory using cards with circles on them. They presented the children a card with some number of circles (1 to 8) and asked the kid, "How many?" One-knowers tended to reply "one" to a card with one circle, and then guessed incorrectly for just about everything else. Two-knowers could count one or two circles, but guessed incorrectly for all the other cards. Three-knowers could count up to three, but just guessed beyond that. Counters answered correctly on essentially all cards.

So far this doesn't tell us whether children learn to count by bootstrapping off of analog magnitudes or some other system. Carey and Mathieu LeCorre published a paper this year that seems to settle the question. The setup was exactly the same as in the last paper (now with cards with anywhere from 1 to 10 circles), except that this time the children were only briefly shown the card. They didn't have enough time to actually count "one, two, three..." The data for one-, two- and three-knowers didn't change, which isn't surprising. Both the "3-object" and the analog magnitude systems are very fast and shouldn't require explicit counting.

However, counters fell into two groups. One group, about 4.5 years old on average, answered just as adults. When they saw six circles, their answers averaged around "six." When they saw ten circles, their answers averaged around "ten." This is what you'd expect if they have mapped number words onto the analog magnitude system.

However, the other group, which was slightly younger (average age of 4 years, 1 month), guessed randomly for cards with 5 or more circles, just as if they didn't know how to count. However, these kids can count. If given time to look at the cards, they would have said the right number. So despite the fact that they can count, they do not seem to have their analog magnitude system mapped onto number words.

This means that the analog magnitude system isn't fundamental in learning how to count, and it actually takes some time for children to learn that mapping even after they've learned to count. Carey takes this as meaning that the analog magnitude system doesn't play a fundamental role in learning to count, either, and there are other reasons as well that this is probably the case.

One remaining possibility is that children use the "3-object system" to understanding the meanings of 1, 2 and 3. This seems to work nicely, given that the limits of the system (3 objects in children, 4 in adults) seem to explain why children can learn "one," "two," and "three" without really learning to count. Carey actually has a somewhat more nuanced explanation where children learn the meanings of "one," "two," and "three" the same may that quantifiers (like "a" in English) are learned. However, to the best of my knowledge, she doesn't have an account of how such quantifiers are learned, and if she had an account, I suspect it would itself hinge off of the 3-object system, anyway.

That's it for how children learn to count, unless I get enough comments asking for more details on any point. For those who want to read more, there are many papers on this subject at Carey's web page.