How do children learn to count? Part 3

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.

How do children learn to count? Part 2

In my last post, I showed that children learn the meaning of number words in a peculiar but systematic fashion. Today, I'll continue trying to explain this odd behavior.

Important to this story is that children (and non-human primates) are born with several primitive but useful numerical systems that are quite different from the natural number system (1, 2, 3, ...). They can't use these systems to count, but they may be useful in learning to count. In this post, I'll try to give a quick summary of how they work.

One is a basic system that can track about 3-4 objects at a time. This isn't a number system per se, just an ability to pay attention to a limited and discrete number of things, and it may or may not be related to similar limits in visual short-term memory.

You can see this in action by playing the following game with a baby under the age of 2. Show the baby two small boxes. Put a single graham cracker into one of the boxes. Then put, one at a time, two graham crackers into the other box. Assuming your baby likes graham crackers, she'll crawl to the box with two graham crackers. Interestingly, this won't work if you put two graham crackers in one box and four in the other. Then, the baby chooses between the boxes randomly. This is understood to happen because the need to represent 6 different objects all in memory simultaneously overloads the poor baby's brain, and she just loses track. (If you want to experience something similar, try to find a "multiple object tracking" demo with 5 or more objects. I wasn't able to find one, but you can try this series of demos to get a similar experience.)

On the other hand, there is the analog magnitude system. Infants and non-human animals have an ability to tell when there are "more" objects. This isn't exact. They can't tell 11 objects from 12. But they can handle ratios like 1:2. (The exact ratio depends on the animal and also where it is in maturity. We can distinguish smaller ratios than infants can.)

You can see this by using something similar to the graham cracker experiment. Infants like novelty. If you show them 2 balls, then 2 balls again, then 2 balls again, they will get bored. Then show them 4 balls. They suddenly get more interested and look longer. However, this won't happen if you show them 4 balls over and over, then show them 5. That ratio is too similar. (I'm not sure if you get this effect in the graham cracker experiment. I suspect you do, but I couldn't find a reference off-hand. The graham cracker experiment is more challenging for infants, so it's possible the results might be somewhat different.)

You can also try this with adults. Show them a picture with 20 balls, and ask them how many there are. Don't give them time to count. The answer will average around 20, but with a good deal of variation. They may say 18, 19, 21, 22, etc. If you give the adult enough time to count, they will almost certainly say "20."

Those are the two important prelinguistic "number" systems. In my next post, I'll try to piece all this information together.

How do children learn to count? Part 1

How do children learn to count? You could imagine that numbers are words, and children learn them like any other word. (Actually, this wouldn't help much, since we still don't really understand how children learn words, but it would neatly deflect the question.) However, it turns out that children learn to count in a bizarre fashion quite unlike how they learn about other words.

If you have a baby and a few years to spend, you can try this experiment at home. Every day, show you baby a bowl of marbles and ask her to give you one. Wait until your baby can do this. This actually takes some time, during which you'll either get nothing or maybe a handful of marbles.

Then, one day, between 24 and 30 months of age, your toddler will hand you a single marble. But ask for 2 marbles or 3 marbles, etc., your toddler will give you a handful. The number of marbles won't be systematically larger if you ask for 10 than if you ask for 2. This is particularly odd, because because by this age the child typically can recite the count list ("one, two, three, four...").

Keep trying this, and within 6-9 months, the child will start giving you 2 marbles when asked for, but still give a random handful when asked for 3 or 4 or 5, etc. Wait a bit longer, and the child will manage to give you 1, 2 or 3 when asked, but still fail for numbers greater than 3.

This doesn't continue forever, though. At around 3 years old, children suddenly are able to succeed when asked for any set of numbers. They can truly count. (This is work done by Karen Wynn some years ago, who is now a professor of psychology at Yale University.)


Of course, this is just a description of what children do. What causes this strange pattern of behavior? We seem to be, as a field, homing in on the answer, and in my next post I'll describe some new research that sheds light onto the question.

SNPs and genes for language

Modern genetic analyses have told us a great deal about many aspects of the human body and mind. However, genetics has been relatively slow in breaking into the study of language. As I have mentioned before, a few years ago resarchers reported that a damaged version of the gene FOXP2 was responsible for the language impairments in the KE family. This sounds more helpful than it really was, since it turns out that even some reptiles have versions of the FOXP2 gene. In humans, FOXP2 isn't just expressed in the brain -- it's expressed in the gut as well. This means that there is a lot more going on than just having FOXP2 or not.

Over the weekend, researchers presented new data at the Boston University Conference on Language Development that hones in on what, just exactly, FOXP2 does.

It turns out that there is a certain amount of variation in genes. One type of variation is a Single Nucleotide Polymorphism (SNP), which is a single base pair in a string of DNA that varies from animal to animal within a species. Some SNPs may have little or no effect. Others can have disastrous effects. Others are intermediate. The Human Genome Project simply cataloged genes. Scientists are still working on cataloging these variations. (This is the extent of my knowledge. If any geneticists are reading this and want to add more, please do.)

The paper at BUCLD, written by J. Bruce Tomblin and Jonathan Bjork of the University of Iowa and Morten H. Christiansen of Cornell University, looked at SNPs in FOXP2. They selected 6 for study in a population of normally developing adolescents and a population of language-impaired adolescents.

Two of the six SNPs under study correlated well with a test of procedural memory (strictly speaking, one correlation was only marginally statistically significant). One of these SNPs predicted better procedural memory function and was more common in language-normal adolescents; the other predicted worse procedural memory function and was more common in language-impaired adolescents.

At a mechanistic level, the next step will be understanding how the proteins created by these different versions of FOXP2 do. From my perspective, I'm excited to have further confirmation of the theory that procedural memory is important in language. More importantly, though, I think this study heralds a new, exciting line of research in the study of human language.

(You can read the abstract of the study here.)

Finding guinea pigs

One problem that confronts nearly every cognitive science researcher is attracting participants. This is less true perhaps for vision researchers, who can sometimes get away with testing only themselves and their coauthors, but it is definitely a problem for people who conduct Web-based research, which often needs hundreds or even thousands of participants.

Many researchers when they start conducting experiments on the Internet are tempted to offer rewards for participation. It's too difficult to pay everybody, so this is often done in the context of a lottery (1 person will win $100). This seems like an intuitive strategy, since we usually attract participants to our labs by offering money or making it a requirement for passing an introductory psychology course.

If you've been reading the Scienceblog.com top stories lately, you might have noticed a recent study by University of Florida researchers, which suggested that people -- well, UF undergrads -- are less likely to give accurate information to websites which offered rewards.

Although these data are in largely in the context of marketing, this suggests that using lotteries to attract research participants on the Web may actually be backfiring.