2004 LECTURE SERIES

What Makes Human's Smart? Lessons from Children

Dr. Elizabeth S. Spelke
Harvard University
Tuesday, Feb. 3, 2004

Thanks very much, Jim. It's a great, great pleasure to be here and so wonderful to see so many people turning out for lectures on learning, memory, and the brain. It's really wonderful for someone like me to see this.

Now, of course, as we look across different animal species and all the different kinds of feats of learning and cognition that take place in the nature world, we see many, many examples of exquisitely complicated neuro and cognitive systems, giving rise to organized behavior. I'm thinking of things like the dance of the bees conveying the location of food or migratory birds navigating over long-scaled distances, finding their ways from one environment to another. But I think, at least my personal feeling is, that of all of the feats of cognition in the natural world, some of the most interesting ones are feats that the high school students in this room engaged in about ten years ago when they first started going to elementary school and leaning the basic cognitive skills that the--that they and the rest of us use every day of our lives, skills like reading and mathematics and systematic reasoning. So the question that I want to ask--and I'm sorry, Jim--I'm not going to be answering this question or the bigger question of what makes us smart, but I'm going to try to suggest ways we can think about it. The question I'd love to be able to answer is how is it that humans are able to develop and then use these kinds of complex cognitive skills?

Now, obviously, I am far from the only person to ask questions like this. Because these skills are so fundamental to our daily lives and because they're so vulnerable to impairment in people with neurological damage or in children with developmental disabilities, for these reasons, they've been intensively studied and continue to be intensively studied today.

Now, most studies ask how we go about reading or performing mathematics, either by studying skilled adults and asking how they engage in these activities, what processes go on in their brains when they engage in these activities, or by studying children who are in school and in the process of acquiring them. In that context, what I want to do tonight is suggest something that may be somewhat counterintuitive, that we may be able to get additional insight into complicated school-learned skills, by studying two further populations. One of them, as Jim said, human infants, who never--have not yet dreamed that such skills exist, and the other, non-human primates who will never get an inkling of what it is to read or perform an arithmetic calculation. And my reasoning is simple: When we, as adults, engage in complex cognitive skills and when we as children learn those skills in the first place, we do so by bringing to bear cognitive abilities that we already have, abilities that have a long developmental history, both in our ontogeny and in phylogeny.

More specifically, here's what I want to suggest tonight. That we as humans are endowed with a set of what I'm going to call core knowledge systems, specialized systems for representing important parts of the environment and solving important kinds of problems. And I'm thinking of things like a system for representing inanimate manipulable objects that lets us reason about how those objects behave, a system for representing other people, that lets us predict what they're going to do next and interact with them, systems for navigating through the environment, and systems for representing sets and numerical magnitudes.

Now, all of these systems, I'll suggest, emerge early in development and therefore are shared by human adults and human infants. And they also have a long evolutionary history so they're shared by humans and many non-human animals. Now, from these building blocks, I'll suggest, children and adults build new cognitive skills and they do so by bringing together, combining and orchestrating these different core knowledge systems. And in particular in many cases, including the case I'll be talking about tonight, I suggest that human language plays a role in allowing us to combine information from different systems and develop new concepts and new ways of understanding the world. So these are the general points that I want to be making. I want to make them by presenting you with a single case study in complex human skill development. I want to be looking at how we develop concepts like seven and how we come to be able to perform operations like adding 5 plus 4 and getting hopefully 9.
My talk's going to be in three parts. I want to tell you first about two core knowledge systems that have come to light through studies of human infants and non-human primates that I think are building blocks of these other abilities. One, a system for representing small, exact numbers, and the other, a system for representing large approximate numbers. Then I want to talk about a fascinating set of events that takes place between the ages of about 2-1/2 and 4, when kids first construct their natural number concepts by--when they learn verbal counting. Finally, I want to then turn back to us adults and ask how these core systems and also the language of verbal counting come together to allow us to think thoughts like three plus four is seven.

So let me start by introducing you to the first of these core systems, a system for representing small numbers of objects. Now, this system, I think, first came to light really clearly in some beautiful experiments conducted by the Developmental Psychologist Karen Wynn at Yale University. She's done studies with human infants where, essentially, she presents infants with puppet-stage kinds of events in a magic show. And the basic assumption behind these studies is that if you present babies with something that's impossible, that couldn't happen and they're sensitive to the fact that this event couldn't or shouldn't be happening then they, like us, will do a double take and will pay more attention to that event. So with that assumption, Wynn designed studies in which she presented a single object on a puppet stage then raised a screen so that the object was covered and then took a second object and dropped it behind the stage next to the first one and then asked infants, in effect, how many things are back there behind the stage by lowering the screen and revealing either one object or two. And what she found in this case was that infants looked longer at the single object, suggesting that that's not what they expected to see and that the expected number, or the more expected number, would have been two. Now, after that initial observation, Wynn varied the problems that she gave kids, she showed that they expected the one plus one event to lead to exactly two, not three or one. She also found that they could solve problems in which you began with two objects on the stage and then added a third and that they could solve problems where you began with two and removed one. They knew that two minus one should yield one object, not two.

Now, these studies, as I said, were all done using preferential looking methods, just measuring how long babies look at events where something goes as it's supposed to or in an unexpected way. But it turns out that this ability that Wynn discovered is very robust and it's been shown by my colleague at Harvard, Susan Carey, to manifest itself in other kinds of tasks that infants are given. And she's done a whole bunch of tests. I'm going to give you an example of just one.

These are studies done now on somewhat older infants, old enough to crawl around and also old enough to adore graham crackers. And the way these studies work, she does--Wynn's--the cartoon here isn't a very good depiction of a graham cracker, sorry about that, but I'll describe the event right. She presents kids with two boxes and in one box, she takes a graham cracker, puts it in the box, then takes another graham cracker and puts that one in the box so one plus one. In the other event, just a single graham cracker goes into a box, the boxes are widely separated and the child is simply encouraged go for it, go for whatever you want. And the question is which box will the child go to. Here are some of the data that she and Lisa Figensen, her student, collected. If kids went randomly to the two boxes then they should be at 50 percent. But what you see is that when they're given a choice of one versus two, they do better than that. They tend to go to the box with the larger number of graham crackers. Same for when they're given two versus three.

So what both these studies and other studies suggest is that when infants are presented with objects that move out of sight, they conform representations of exactly how many objects are in the scene. But I think the most interesting findings to come out of these studies don't concern what infants can do, but concern the limits to what they're able to do. So let me turn to that next.

There are two interesting limits to infants' abilities to track things through time and figure out--maintain a sense of how many things are in a scene. The first of them is these studies work great if you present babies with solid, separate objects, things like dolls or balls or crackers, then you see infants succeeding. But when, instead of that, you do what Gavin Huntley Fenner, who used to be here at Irvine has done--and instead of presenting a solid object, you present infants with a non-solid substance like a pile of sand that's poured behind a screen, all of a sudden, the babies aren't so good anymore. They can't keep track of how many sand piles you've placed into a scene. Similarly, they fall apart if you present stacks of objects, instead of single objects. A pile of bricks and a second pile of bricks, they can't tell you how many piles of bricks there should be. And in a slightly different paradigm, if you ask infants how many parts of objects are there instead of how many separate objects are there, again, they fail.

I'll call all of these things a domain limit. This is an ability that infants show when they're presented things in the domain of inanimate, manipulable objects and they don't seem to show otherwise.
But there's also a set size limit. Babies don't do so well when the number of objects gets big. In fact, the number has to be very small in order for babies to show these abilities. Once you get up beyond three, they just don't show them. This has been shown in the original Wynn studies, where they succeed with one plus one equals two, but not four plus four equals eight. And it's shown very dramatically in the studies where the infants have to choose which box has the graham crackers where these are the data I gave you before, babies succeeding in a choice of one versus two or two versus three, but look what happens when you go to a choice of two versus four. As soon as you go up to four, putting four graham crackers in one box, babies' choices go random.

Now, at first, this might seem puzzling. How could a baby know that three is more than two, but not know that four is more than two? But I think the way to think about these findings is what babies are doing in these studies is keeping track of the number of objects in a scene and they have a limited ability to do that. They can do that with up to about three objects at any given time in any given place and you go beyond that and the system explodes, can't be used at all anymore and they choose at random. So there's this set size limit.

So that's what we've learned about babies. Those of you who were here three years ago and heard Mark Hauser talk about how his work will know Mark Hauser--I guess I don't know whether he talked about number studies here, but let me tell you a little about what he did. He's been involved for some time in a collaboration with Susan Carey to conduct the very same experiments that she and Wynn and others have conducted on infants, on human infants, to conduct them on adult free ranging rhesus monkeys. So Hauser studies these monkeys.

Here's an example of an experiment. This is like the graham cracker box search test where they get the attention of an adult monkey. They've got two boxes. They put different numbers of pieces of apple, the monkeys' choice foods, into the two different boxes and then they simply walk away and videotape the monkey and watch which of the two boxes the monkey freely approaches. And with that and other methods, you can ask about monkeys' abilities to keep track of objects over time.

When you ask that, you find first of all, monkeys are able to succeed at these tasks, when they're given the same small number of objects that infants are given. But monkeys also show the same limits as infants. They show the domain limit, failing if you present them with a non-solid substance or with attached parts of objects, and they show the set size limit, maybe about one bigger than the infants. They--monkeys might be able to go up to four, but they took, like human infants, are limited in the number of objects they can keep track of at any given time. The fact that we see these common limits across human infants and adult non-human primates suggests that a common system might be at work in the two populations, a system for keeping track of individuals.

Well, let me turn and ask does this system continue to exist in us as adults? Do we have an ability to look out and keep track of individual objects as they move around, move out of view, and are we limited in the same ways that infants are? Well, the suggestion that there's a special system that we have for keeping track of small numbers of objects has about at least a 100-year history. It's been proposed and debated many times over the last century whether such a process exists. Those who think it does call it a process of subatizing. And some of the evidence that seems to suggest we have this process comes from studies in which you present us with an array that has some number of objects in it and we have to say as quickly as we can how many objects are in that array.

When the numbers are as low as about three or four, we're very fast and almost equally fast to make that judgment. With higher numbers, the amount of time it takes us to say how many are there goes up just about linearly. That seems--that's been interpreted to reflect a process of verbal counting, but the idea is maybe you don't need that counting for the small numbers of objects. Maybe we can keep track of them immediately and in parallel. Well, can we and do we do so by the same system that infants and monkeys use to keep track of objects?

This is a question that the Psychologist Brian Scholl has been asking recently and he's been asking it with a different kind of task. And now I have to apologize to you because I wanted to let you all be subjects in this task. I put it on my computer and it crashed my computer every time I tried to do it so you're going to have to use your imaginations. I'm not going to actually be able take you through these studies. But the way the studies work is you're shown a screen that has a large number of indistinguishable-looking elements on it so they're all white circles or they're all green squares or something like that. And then you are--a subset of those elements are indicated to you. They'll suddenly start flashing. And your job is to pay attention to that subset, keep your attention on that subset, while all of the elements start to move, each element moving in a different direction. And the question is, can you do it? Can you keep your attention on these objects when everything is moving around randomly? Well, the answer is, yes, you can, but when you try to do this, you see the same limits that we see in the studies with infants and monkeys.

First of all, a set size limit. You can do this when you've got to keep track of one thing or two things or three or you can just about do it when you have to keep track of four. Actually, if I had had the demonstration there, I would be able to show you something that's quite depressing to me personally, which is that young people, all those high school students there, generally can keep track of at least one more element than people age can. This is not a happy finding; however, it's a small difference. Small. Basically, we're all, you know, we've got the same system here and it's got this limit of about three or four, depending on who you ask.

There's also, very interestingly, in Brian's recent work, a domain limit. This system only works when you're given separate objects to track. If instead, objects are connected together so that what you have to track are ends of objects, parts of objects instead of whole objects, or if instead of moving like an object, things pour from one location to another--he's tested that with adults--then you fall apart completely. You cannot do this task. So it looks like we're seeing the same limits here in adults that we see in infants and in monkeys, suggesting that a common system of representation is at work.
So to summarize what I've told you so far, infants seem to form representations of one, two, or three objects. These representations support a variety of actions like locomoting toward a box to choose crackers or looking at an event where something unexpected happens. The representations persist when objects are out of view and the representations show two signature limits, the domain and set size limits that I've been talking about. And monkeys and human adults show the same limit, suggesting that they've got the same system as well.

So let me move on to the second core system, a system for representing large approximate numerosities. Again, let me start with babies and with a study conducted by my student Faye Shu, which used an even simpler method to probe whether infants are able to represent about how many elements there are in a scene. Now, this method simply relies on the fact that if you show a baby the same kind of thing again and again, each time letting them look at the thing as long as they want and measuring how long they look at it, as long as it's the same kind of thing, their looking time will decline over successive trials. And then if you show them something new, like us, they'll perk up, they'll look at it longer. So with that general phenomenon in mind, what Faye did was to create displays of either 8 elements or 16 elements and present those displays to babies successively. So one group of babies saw a whole collection of different displays of eight elements. The other group saw a whole collection of different displays of 16 elements. And after they'd gotten thoroughly bored with this whole setup, she then presented them two new displays in alternation of 8 or 16 elements. And across trials, she varied the sizes of elements, their positions, the density of elements, the overall size of the area that they covered in such a way that if babies were to respond to any of those variables then they should look equally at the two test displays. But if they responded to number then boredom with 8 should lead to further boredom with 8 and more interest in 16 and vice versa.

Here are the data from that first condition. What you're seeing here is babies' looking time on the Y-axis and you're seeing that indeed, as you present the same thing, the same kind of thing, same number, again and again and again, looking time goes sharply down. I'm sorry the colors don't show up that well here, but I hope you can see this. After looking time has gone down, when you shift to the test and present a new number, babies look longer than when you present the same number you presented before, though in different displays. So this provides evidence that infants are discriminating between these arrays on the basis of numerosity.

Again though, what I think is most interesting are the limits of the performance that we can see with infants and let me turn to that. First of all, studies using this method have shown that infants' numerical discriminations are highly imprecise. So I just showed you a successful finding, babies succeeding where they had to discriminate 8 from 16. But when Faye repeated the experiment to test discrimination of 8 from 12, 6-month-old infants completely failed. Boredom with displays of 8 items led to boredom with displays of 12 items. No evidence that they could tell the difference between those numbers. So it's a very imprecise system.

Now, when she then started playing with the numbers involved, she discovered a critical limit on infants' discrimination and the limit comes from the ratio of the two set sizes to be discriminated. So at 6 months of age, babies discriminate 8 from 16 and they also discriminate 4 from 8 and 16 from 32. And they failed to discriminate 8 from 12, 4 from 6 or 16 from 24. If you wait three months, babies' discrimination gets sharper. By 9 months, they can discriminate 8 from 12. They still are imprecise though and if you make the ratio difference smaller, they will fail again. So this is a signature limit of this system. Discriminability depends on the ratios of the two numbers to be discriminated, a different kind of limit from the one we saw with the small number system.

Well, next question we might ask, this sensitivity to large numerosities, how general is it? Is it something that babies only show when you present them with visual arrays of dots? Is this some visual ability or is it a more abstract and general ability? And another student, Jen Lipton, has been asking this question by doing studies on infants where she presents not visual spatial arrays of dots, but auditory sequences of sounds. Now, the logic of the studies is the same as in the dot studies. Take one group of babies and bore them with one number of sounds, another group and bore them with a different number of sounds, and then test everybody with the two numbers and see whether babies are more interested in the novel number sounds. Of course, you can't measure looking time towards sounds so in order to conduct these studies, Jen borrowed a method from studies of speech perception that, for those of you who were here when Pat Kool talked a number of years ago, may sound familiar. This method capitalizes on the fact that when a sound is played from a speaker that babies can clearly hear to be to one or the other side of them, they will turn and orient toward the speaker. And after the sound ends, if it was really interesting, the babies will maintain their orientation toward that speaker for some period of time. So you can measure how long babies stay oriented towards the sounds as a measure of how interested they are in it.

So using this method, Lipton found that babies will orient towards sound sequences in the beginning, that when you play the same number of sounds again and again, their orientation time goes down. And when you change to twice as many sounds or half as many sounds, their orientation time goes up. So this shows that babies can discriminate numerosities in sound sequences, as well as in visual spatial arrays.

What are the limits on this performance? Well, we were very interested to see that babies showed the same limits here as they show with dots. So at 6 months of age, they succeed with a 2 to 1 ratio and fail with a 3 to 2 and 9 months of age, they succeed with the 3 to 2, fail with a 5 to 4. The very same limits. You can predict how babies will do with these sound sequences from knowing how they've done with dots, beginning to suggest that there may be a more general and abstract ability to represent large numerosities, something that isn't specific to the visual system or the auditory system, but that's more central.

Okay. Let me get back to Mark Hauser and the other species of monkey that he loves to study. This is a cotton top tamarin, a particularly adorable juvenile cotton top tamarin. There are many nice things about these tamarins, besides the fact that they're so cute. One of the things that's nice about these tamarins is that Mark has discovered that you can take essentially the same auditory head-turn method that's used to study speech perception in infants and you can use it with them and they'll show the same pattern. They'll get bored and stop attending if you play the same thing again and again and then if you play something new, their attention will rise. So using this method, he and I devised studies of numerosity discrimination, spontaneous, untrained number discrimination in these monkeys.

And the findings look a lot like our findings with 9-month-old infants. Monkeys succeed at discriminating numerosities. They show a ratio limit. They succeed with a ratio of 3 to 2, failing with a narrower ratio, just like 9-month-old infants do, suggesting, I think, that a common mechanism is at work here in human infants and in non-human primates, once again.

Well, what about human adults? Do we also have this ability to discriminate numerosities when we control for things like for how big the items they are and how spread out they are in a scene and so forth, do we have this ability and do we show the same ratio limit? Well, this time, you can be a subject. I am going to show you two arrays of dots in rapid succession. I'll show each one long enough for you to get a look at it, not long enough for you to count anything so don't even try. Take a look at the first array and the second array and then tell me which has more dots, array one or two.

Group: Two.

Male: One.

Spelke: One? Come on. That was supposed to be an easy one. Now, be honest.

Group: Two.

Spelke: Two. Thank you. You all saw, I'm quite sure, that this had more than that, even those elements were smaller. You did see that. Don't tell me you didn't see that. Yes, you did. Okay. You all should've succeeded at that task. Here's another one. This one you're allowed to get wrong. Okay. Which array has more, one or two?

Group: [Various answers.]

Spelke: They were not the same. They were not the same.

Group: [Inaudible.]

Spelke: How many people vote for one? How many vote for two? Most of you were wrong. The one--the first array actually had more. This array actually has a little bit more. There are 26 dots in this array and 25 in the other. That's too small a difference for adults to reliably discriminate, okay? But that's the phenomena. Here are the data that Hillary Barth obtained when she conducted this study again and again, testing at different ratios. What she found is that adults can easily perform, unlike some of you, adults can easily perform this task at the largest--with the largest ratio difference. As the ratio narrows, performance goes down, showing the same signature that we see in the infants and the monkeys.

Well, the nice thing about adults, especially Harvard college sophomores, is that you can give them boring tasks again and again and learn all kinds of additional things that no self-respecting infant or monkey would sit still for. So Barth went on to ask, what if we make this harder? And instead of asking which is more, this array of dots or that array of dots, give people an array of dots and then a sequence of sounds and ask which is more numerous, the array of dots or the sequences, are there more dots are or there more sounds, okay? And you can do this with the same numbers you use to ask about two arrays of dots, okay?

When she did that, we got a surprise. First of all, what she did is she took a set of numerical comparisons at which people were performing just about halfway between perfect and chance, okay? So she took comparisons where, if it's purely visual, comparing dots to dots, you're going to be not perfect, but you'll be somewhere in the middle there. And then the question was if you take those same numbers, how much worse will you be if you have to compare dots to tones. And the answer is almost not at all. This surprised to all the subjects in the study. They all thought it was going to be way harder to compare dots to tones, but in fact, there's no reliable difference in their performance in those two cases.

Well, moving right along, we've still got the compliant Harvard undergraduates so Barth moved along and asked, suppose we give people a non-symbolic arithmetic task, like an addition task. We present one array of dots, we present a second array of dots, then we'll ask people to add those two arrays together in their minds, think about the sum, and compare the sum to a third array, decide whether the sum is bigger or smaller than the third array. We can ask whether people can do this. Now, the students are saying absolutely not. There's no way that I could do this. That's one task we can ask them to do. And when you do and compare their performance at comparing the sum for the comparison array with their performance when they just got that sum physically present on the screen at all once, you see they're almost as good at the addition task is they are at the simple comparison task.

So then Barth did a final twist on the method. Suppose you have to add an array of dots to a sequence of sounds and then compare that sum to another array. At this point, the subjects all said, I quit, I'm going home, there's no way I can do this. And she said, stay and guess and I'll give you extra cookies and come on. And the restful astonished all of us. People were as good as cross-modal addition as they are at visual addition and not noticeably worse at cross-modal addition than in any of the other tasks I gave you.

Now, of course, these are college students, they've learned symbolic arithmetic. So the next question to ask is, is this an ability that you get after years of memorizing your addition tables and performing calculations in your head or is this an ability that comes for free by having this system of quantity representation? Well, that's not an easy question to answer with adults, but we can answer it by turning to 5-year-old children.

So Barth and Kristen Lamont and their collaborators designed a cartoon version of the numerical comparison tasks that I just presented to you to use with 5-year-old children, 4- to 5-year-old children, children who had not yet started elementary school and therefore, had received no instruction in symbolic arithmetic. Here's how the task worked. First she shows an array of blue dots. She says, look, we've got some--oh, let me take you through these in sequence. Let's start with a comparison test. Here's some blue dots, oops, now they're all covered up, and here come some red dots. Tell me, are there more blue dots or more red dots? Simple visual comparison. And here are the results. You see two things. Chance is 50 percent. You see that kids can do this task. They're not perfect, but they're well above chance. And you see that they show the ratio signature. The closer the two arrays are numerically, the less well they do. So they can compare on the basis of numerosity and they show the ratio signature.

Then Barth went on to ask about a cross-modal comparison. She shows an array of blue dots, she covers it up and says, now, listen, and plays a sequence of sounds. And then she asks, are there more dots or are there more tones? And she can do this for the same numerosities that she used in the visual comparison. What you see here once again in green is the purely visual case and in blue is the cross-modal case. And what you see is there's precious little difference for the 5-year-olds, just like the adults, between their abilities to do this with a cross-modal comparison versus a visual comparison. You don't need years of experience with symbolic numbers to have these abilities. So they can compare the dots. They're almost as good.

So now let's turn to addition. You start with a single--this is the comparison task that I've already shown you. Here's her addition task. Start with a single array of blue dots, cover it up, and then say, hey, here's some more blue dots. They're going back there too. Now all the blue dots are there. Now, here are some red dots. Which is more? Are there more blue dots or more red dots? And as in the case of the adult studies, you can see how they do on this task when the blue dots appear in these sets and have to be mentally added with how they do up here where the blue dots appear all at once. Um, here are the findings again for the third time for the visual comparison versus the visual addition and you can see that for kids, as for adults, they can do this task almost as well as they can do the comparison.

So in summary, it looks like human infants are able to represent large numerosities when continuous variables are controlled and they show a ratio limit on discrimination. Looks like these representations are abstract, but there's common representations formed for visual spatial arrays on the one hand and sequences of sounds on the other. They show the same ratio limits in infants. Human adults and non-human primates also have these abilities. They also show a ratio limit, suggesting that we're looking again at a common system of representation over development. And in children and adults--we don't know about infants yet--these large number representations can be used to perform
non-symbolic addition.

So, so far in this talk, I've told you about two systems of number representation: a system for representing small exact numbers of objects and a system for representing large, approximate numerical magnitudes, each with a different set of limits and a different set of processes. But let's turn now and consider for a moment the child who is starting school and learning arithmetic in school.
Now, for this child, teachers will assume that children are dealing with one system of representation, not two, that there's a single system for representing numbers. What's more, this system has no set size limit, children don't stop enumerating things when they get to three, they keep going beyond that, and it has no ratio limit, it isn't imprecise, although we have trouble when I flash dots at you discriminating 26 from 27. When you can engage in verbal counting, you know that 26 and 27 are not the same number and you can order them appropriately. So the system that children need to be working with when they get to school has to represent exact whole number values with no upper bound and be usable to enumerate anything, not just objects or arrays of dots or the kinds of things we've been studying so far.

So the question is where does this new system come from? Well, the evidence suggests that this system is constructed by children between the ages of about 2-1/2 and 4, as they learn the language of verbal counting. There's been beautiful work looking at children's learning of counting and it's been surprising to many of us, including me, I was taken up very short when I had a 2-1/2-year-old child who Karen Wynn ran in her studies and I saw what little he knew about numbers. It takes kids about a year and a half to figure out what the words in a counting routine mean.

They've got all the words. Their grandparents are very happy to see this. Looks great. What Karen showed, to my horror when she showed it on my own child, is that at this point in development, children know almost nothing about what these words mean. They know what the word one means. They know that one means one. And Karen showed this by simple task where she gives kids a whole bunch of fish and says, can you put one fish in the pond? And the child does it. But they have almost no clue what the other number words mean. When she asked them to put two fish in the pond or four fish in the pond or eight fish in the pond, using words that the child uses herself when she engages in this counting routine, what the children do is grab a handful. No relation between the number the grab and the number Karen asked for, okay? They simply grab a handful. So it looks like at this point in time, one means something like one, but all the other number words just mean something like more than one, collection, a bunch, okay?

About six to nine months later in Karen's studies, children work out the meaning of the word two. And at this point, if you ask for two, you get two. If you ask for three, you get a handful, more than two. But not an exact number. Another three to six months later, kids figure out the meaning of the word three. Next step after that, maybe six months later, they get the general principle. Something magic happens and they figure out the general principle that each of these words in their counting routine picks out a distinct exact number and that you can use counting--if somebody asks you for six things, you can use counting to give them exactly six. It's a very long time to get to this point. And then after that, we've recently shown once kids get to that point, they really get control of the logic of this system and they can use it even with numbers that they can't count to. So you--if you take a 5-year-old, as Jen Lipton has done, assess how high that 5-year-old can count--say the 5-year-old can count to 100, you then--she then brings out a jar and says, look, I've got 129 marbles in this jar. Now, watch what I'm going to do. And she takes one marble out. Are there 129? The child will say, no, okay? If she puts another marble back in, they'll say, yes, there is. They've worked out the logic of the system, but it takes them a very long time to do that.

So how are they doing it? Well, stepping back and considering the representations of number that we see in infants suggests an account of children's learning the meanings of the number words and the counting routine. In that first step where children only know the meaning of one, they seem to treat one as roughly synonymous with "a" and to apply it, whenever their first system picks out an object in the scene. So the word one applies, just in case you've got one object, you know, an object in that scene. All the other number words seem to apply when the second system of representation says there's a bunch of things out there. You apply that, the other number words to a bunch of things represented by the second system.

Nine months later, when kids figure out the meaning of the word two, what they may be doing is seeing that this word applies when their first system is representing an object and another object and the second system is representing that there's a set there, a small set. Learning the word two has to involve bringing these two systems together so you have both the individuals and the cardinal numerical value. Similarly, learning the meaning of three involves linking that word to a representation from the first system that I describe of an object, another object, and another object and using the second system to represent you've got a bigger set of objects here.

Now, once children have learned the meanings of two and three, Wynn and others have never found a stage where children separately have to learn the meaning of four. Once they get up to three, the next step seems to be to understand the whole system. Now, it makes sense, I think, that you wouldn't find kids understanding four if this object system maxes out at about three. But how could they understand it? Well, they could understand that the progression from two to three involves two changes. The first is you add another object to the scene and the first system for representing objects can convey that. And the second is, you increase the size of the set, okay? And with that induction, all that's left to do is to apply those principles to all the other number words in the count sequence and you can arrive at the conclusion that each word in the count list picks out a set of individuals one more individual than the previous word, a larger set than the set picked out by the previous word.

These suggestions are not new to me. There's an elaborate argument that Paul Bloom and Karen Wynn have made, but this is just the point that children end up with. I think the work on infants and monkeys suggests why they get there. Because of the tools they have to work with, in trying to solve this problem figuring out what people are talking about when they're using these words.

Okay. Now, once they get there, they have taken a step beyond any capacity that we see in any other non-human animal. No other animal has ever come to this induction of the natural number counting sequence that children spontaneously arrive at, at about the age of 4. And the most dramatic cases that I think show that other animals don't make that induction are just cases where animals have been trained. In one case of a chimpanzee, trained over a period of 20 years to use symbols for numbers and by brute force, this chimpanzee and also a parrot that some of you may know or know of, by brute force, they can be trained to relate specific symbols to specific numerosities, but this general productive principle that lets you represent number with no upper bound, that kids seem to grab onto it about age 4, they never get.

So what this suggests then is that this uniquely human system of knowledge of number that comes in sometime around 4 years of age comes from putting together three different systems: the core system of object representation that I started this talk with, the core system for representing approximate numerosities that I moved to next, and the language of number words and verbal counting that serves to link these two systems together.

Well, what I want to do in the last minutes of this talk is to turn away from preschool children and straight back to adults and ask what, if anything, does this say about how we process number. When we think the thought seven, what is going on in our minds and does it have anything to do with what's going on in the minds of children who, when they were initially acquiring the meaning of that word and coming to figure out the workings of the counting routine. Is this kind of system at work in us?

Now, notice that it doesn't have to be. It could be that we first apprehend natural number concepts by putting together these three systems, but once we've got them put together, those concepts could take on a life of their own. Our representation of the number seven could be purely abstract, divorced from any of these three systems of representation. Now, that's a possible state of affairs. But I think a growing body of evidence is suggesting that that's not what actually goes on, that when we represent large exact numerosities, we use the same three systems that children used in building those concepts in the first place.

And there's two general kinds of evidence that suggest this. The first, which I'm just going to mention, not really talk about, but I'd be happy to say more about it in the question period, if people are interested, is studies of neurological patients with impairments in numerical abilities. Now, if normal number processing in adults depends on these three systems then you might expect that there isn't just one way to become impaired at numerical processing. There might be three different ways, each leading to different patterns of impairment. And in fact, studies of individual patients are consistent with that. There's the best evidence for the roles of these two systems. Some patients who are just fine at the dot comparison tests, they're great at representing approximate numbers, but their language is impaired, they're impaired at talking about number and they're also impaired at exact numerical calculations. You have another set of patients whose language is just fine, but they have impairments to their basic sense of numerical magnitudes. They do very badly on dot tasks. And they can rattle off some verbal facts, show a different pattern of impairment. So patients are one way to get evidence for this kind of organization in our mature representations of number.

But I want to end today by talking about some studies of adults, partly because I can get you to work again, I can do these--some of these on you. Studies looking at adults engaged in simple arithmetic tasks and asking when adults perform these tasks, if they have to operate on exact numbers, do they activate all of these systems and in particular, do they activate language? And when they have to operate on approximate numbers, they don't need to get exact answers, then do you see the language system dropping out and this core approximate number system taking over? So that'll be the question.
Let me start with studies that we did on adults who are proficient in two languages. The reason we did these studies is there's a very commonly-reported phenomenon among bilinguals. You can be utterly at home in your second language, you can dream in that language, your whole family can only speak that language. You can even feel that it's been so long since you spoke your first language that you're not even that facile with your first language anymore. But if you are such a bilingual and you learned arithmetic in your first language, chances are that when you go to a restaurant and you have to verify the addition on the check, you'll go back to your first language to do it. This is a very commonly-reported phenomenon.

So the question is, why does this happen? And the work I've told you about so far suggests an answer. You want to know whether the addition on the check is exactly correct and representing exact numbers may depend on language. If that were right, then if we gave you a task where you only had to get--only had to decide whether a sum, the answer was in the right ballpark then you might not see this effect of language. And that's what we set out to test. So Sonet Sifkin and I did a series of studies on Russian-English bilinguals. These are all people who were native speakers of Russian, came to the United States when they were mostly in their teens, were now college students or graduate students very proficient in English so they were proficient in two languages. And to try to get a handle on the role of language in different kinds of number representations, we took--followed the logic of doing a training study teaching them some new facts.

So what we did is give them 2-digit addition facts, this like 49 plus 63. They had to decide what the right answer was. And, of course, if you give someone a fact like this, they'll take a little time and they'll go through the calculations in their heads with the carrying and everything and eventually come up with the right answer. But if you give them the same problem again and again, day after day, eventually it becomes like four plus six is ten. You don't have to calculate it anymore. You just immediately know the answer. So we did that. We trained people on some exact facts. We also trained people on some approximate facts, in both case giving them practice until they became very fast and very accurate at retrieving those facts. Now, the way the study was run is for any given subject, half of the facts that we taught were taught in Russian, the other half were taught in English. For any given fact, it was only taught in one of the two languages. And after everybody was really good at retrieving the facts, we then tested them on all the facts in both languages.

First look at what happened with exact addition. For facts that were trained in Russian, when they were tested in Russian--I should say up here we have error rates, which were very low, and here we have the time it took to come up with the correct answer on the trials when the subjects were correct. So here you see what happens when you were trained on an exact fact in Russian and tested in Russian. You're faster than when you were trained in Russian, but tested in English. But this isn't an effective native language because when you were trained on a fact, these same Russian-English bilinguals, native speakers of Russian, when they're trained on a fact in English, they're faster in English than they are in Russian. This goes along with all the anecdotes that says when you learn new facts about exact numbers, what you learn is specific to the language in which you learn it.
But look what happens for the approximate numbers. No hint of an affect of language of training on performance, okay? Learning a new approximate number fact in Russian makes you just as fast at coming up with the fact when you're asked in English as when you're asked in Russian. So what we think is going on is that when you need to--when you only need to approximate, your learning can be supported by this system of core knowledge, which is completely independent of language. It's in species like monkeys that don't have language and creates like human infants who haven't learned language yet. Language plays no role. But when you have to come up with exact facts, it does. So that's the first phenomenon.

What about the rest of us? What about monolinguals? When we have to represent exact numbers, do we use a language to do this? Well, this is a question that's been studied recently by the Belgian Psychologist Cathy Lumare, and she used the following kind of task. Actually, I'm not going to do this as a demonstration with all of you. I won't ask you to tell me as quickly as possible which is the correct answer here. I'll just demonstrate for you the kinds of problems that she does. She presents a problem in Arabic notation and in one condition, people have to choose the exactly correct answer from a near-miss, okay? So this is a problem that you can't solve just by getting to the right ballpark. You've got to get exactly the right answer. That's one kind of problem that she does. Another kind of problem that she does, same problem, but now, neither answer is exactly correct. One is almost right and the other is way off and you have to choose the one that's close to the correct answer.

Now, the question is, when you're doing either of these tasks, can you do them by simply operating on the Arabic symbols and going straight to an abstract representation of number or do you have to go through English? Do you have to go through--or in her case, Belgian French--a natural language in order to solve the problems? To get at this question, Lumare exploited the fact that in Belgian French, some number words are longer than others. So words like 16 and 15 are each basically one-syllable words. It's very quick to say 16 plus 15 in French. Seventeen and fourteen, on the other hand, are each two-syllable words. It takes longer to say those words in French. So if, when you're solving these problems you're activating and using a verbal code of the numbers, you should be slower on a 17 plus 14 problem than on a 16 plus 15 problem. So Lumare looked at that comparison both for the exact task and with the exact same problems for the approximate task. And here's her results. I've just given the times. Performance was near perfect, but here's how long it took. For the approximate problems, no difference between the speed with the problems that are short--are quick to say and the problems that take longer to say. But for the exact problems, you see there's a substantial difference. If it takes longer to say it, it takes longer for subjects to push the button indicating the correct answer. So what this suggests is that for monolinguals, as well as bilinguals, exact addition is activating a linguistic representation and approximate addition is not.

Well, let me end with one final line of evidence, trying to get at the processes involved, in exact and approximate number representation. This comes from studies that were done in collaboration with Stanus Las Dahawn, a wonderful cognitive neuroscientist. And they used the exciting technique of functional brain imaging, functional magnetic residence imaging in this case, a technique that allows you to give perfectly normal adult subjects a set of problems to perform and then to ask what parts of the brain are most active while we're performing those problems by scanning someone in a magnetic residence imager for changes in blood flow, which is an indirect measure of neural activity in one or another region.

So the way these studies were done is Dahawn devised a series of exact addition problems and a series of otherwise very similar approximate addition problems. Subjects had one period of time when they were doing exact addition problems again and again, and another period of time where they were doing approximate addition problems again and again. And the patterns of activation of blood changes to the brain in the approximate condition were compared to those in the exact condition to ask, are there regions in the brain that are more active when you're representing exact number and doing exact addition tasks than when you're representing approximate number? So here is the one and only picture of the human brain that I will show you. You're looking at a left hemisphere. Here's the front, here's the back. And what you're seeing here in these pretty colors are the areas that were more active during the exact calculation task than during the approximate calculation task.

And there's two interesting ones. One is an area that's often activated when people need to translate from a sensory or qualitative code to a symbolic code, like a language code. The other is an area right next to a primary language area in the brain, an area that's often activated in tasks of verbal memory where you have to retain verbal material or generate verbal material. Two areas, you see them only on the left side of the brain. If I had shown you the other side of the brain, you would not see areas, you would not see differences appearing in those areas. This looks like a sign of a role of language and verbal memory in exact calculation tasks, above and beyond what we see in the approximate tasks.
So what I think these three lines have researched, the bilinguals, the people doing short versus longer mental arithmetic problems, and the neuro-imaging studies are suggesting is an interesting link between the process at work in us when we do numerical tasks and the processes that are at work in infants and young children. And in particular, two links. Human infants and non-human primates represent large approximate numerosities without language and in the tests I just told you about, it looks like when adults are called on to represent large approximate numerosities, we don't use language either.

On the other hand, human children begin to represent large exact numerosities when they learn number words and verbal counting, important parts of their acquisition of their native language. And adults, in line with this, continue to use language when they represent large exact numerosities and perform exact arithmetic.

So it looks like when we think thoughts like seven or operations like seven plus five, we're using three different systems of representation - the two core systems that I've been talking about tonight and the language of number words and counting that orchestrates them. Two of these systems, of course, are common to use and other animals. The third is unique to us and becomes uniquely available to us when we learn natural language.

So let me just close up here. I started the title of the talk with the question, "What makes people smart?" Sorry, Jim, I didn't give you an answer to that question, but did give you a set of suggestions. I suggest that there are two ingredients to the complicated cognitive tasks that educated humans excel at. One is a set of core knowledge systems that we share with other animals and with much younger humans and the other is a set of combinatorial capacities that are unique to us that emerge in childhood and that in the examples I've been giving you, either are, are or closely linked to, natural language.

Now, with this perspective, we can consider some more general questions. One question is how many core knowledge systems are we likely to have? Are there a million different ones in our brains, guiding our behavior in many different situations? I actually think that there are rather few core knowledge systems. Looking across all of the work that's been done on human infants and the work that's been done on adult non-human primates, I only see good evidence for about four systems: the two that I talked about tonight; another system for representing con-specifics, other people or other monkeys as agents with goals; and a system for representing space for purposes of navigation. I think what may allow for the richness and diversity of human cognition is not the richness and diversity of our core knowledge systems, but our combinatorial capacity that can take a small set of elements and generate a very large set of products from it.

Well, how many combinatorial systems are we likely to have? I've only given you evidence for one natural language, but, of course, humans are superbly good at manipulating symbols of many different kinds, like maps, scale models, I talked a little about Arabic notation tonight. And these systems also may be important ways of combining information together. I think the work that I told you about though on Arabic notation suggests that when we use these other symbol systems, we may actually be building on a capacity that's itself rooted in natural language. Natural language may be the most fundamental system we use for combining information from diverse sources and creating new concepts.

Finally, let me turn to the most important question, the one that one needs to answer in order to claim they've given an answer to the question what makes people smart, the systems I've been talking about, how do they work and how do they get orchestrated together to create beings who can read and calculate and do all these complicated things? Well, these are hard questions, but I'm optimistic that in the lifetime of the high school students, if not in the lifetime of the rest of us, we're going to be making very serious progress towards answering these questions. And I'm optimistic about this for two reasons.

First of all, because the core knowledge systems are not unique to educated humans, they're shared by other animals, we can study these other systems in those other animals with a whole wide array of methods that we get from comparative psychology, from ethology and behavioral biology, and especially from neuroscience. I think this is going to give us great power in triangulating in on each of these core systems and discovering the workings of the building blocks of our complex scales.
The second reason I'm optimistic is--has to do with the fact that these combinatorial processes seem to depend on language. Language is something that we've learned a lot about and that we are in a position to learn still more about through the wide-ranging set of methods and linguistics, in neurolinguistics and in those who study language acquisition.

So my hope is that by putting all of these disciplines together, in places like this wonderful center that Jim has created here, we will be able to make headway on these problems and begin to answer what I think may be the most interesting puzzle about human cognition, which is how we're able to start with a set of biologically-given capacities and arrive at a set of systems of knowledge that go far beyond what our evolutionary history could have possibly prepared us for.
Thank you.