Core Knowledge and Cognitive Development
Lecture Intoduction
Lila Gleitman
Well hello everybody. I'm more than happy to welcome you all to the second annual Anne and Benjamin Pinkel Endowed Lecture, whose overarching topic is Mind and Brain Paradigms; this is the second year of this endowment for whom we have the Anne and Benjamin Pinkel Estate to thank for endowing this chair through their daughter Sheila Pinkel who unfortunately is in Laos and not here today, but we do thank them. It's especially fitting to have an endowed chair on this topic for Dr. Pinkel; although he was an engineer by training and got his BSE at the University of Pennsylvania, in fact his intellectual passion was the human mind and brain function. And so, that's what this series is about.
It's especially fitting also that the sponsor at the University of these lectures is the Institute for Research in Cognitive Science, which as many of you know is the only center for the study of cognition that's funded as a Science and Technology Center by the National Science Foundation, and the mission of this Institute is to study the nature of mind and the technologies that flow from an understanding of how the mind and other intelligent machines work.
Speaker Introduction
So everything is very fitting, but it's especially fitting that we have professor Elizabeth Spelke to talk to us today, because she is an internationally renowned scientist whose professional career has been devoted to understanding the human mind, especially by looking at infants. Professor Spelke was educated at Yale, and then at Cornell University, with Dr Ulrich Naisser, and she graced the halls of the University of Pennsylvania for many years as a professor in the psychology department, and then she fled for a while to Cornell University, and now she is a professor in the department of Brain and Cognitive Science at MIT. And I would tell you about her accomplishments and awards and honors except that they would occupy too much of the day. She is a member of the National Academy of Sciences, I should mention that, and then I'll simply mention the last award that she received, and that is she has just received the Distinguished Scientist Award of the American Psychological Association for the year 2000 to add to her very long list of accomplishments and recognition. So I just want to say a couple of words about Dr. Spelke's work in general, lest she talk about something else. Maybe it's easiest if you think back to the philosopher Emmanual Kant who famously argued that human beings come endowed with certain ways of organizing and categorizing the evidence that comes in from the world. Dr. Spelke has been one of the most important people, I believe, who's interpreted these kinds of ideas reframed them in modern psychological terms and set out to study them. Well you can see this is kind of hard if you want to find out how one natively does that sort of thing. You don't want to ask the infected population of people who have studied in school and been affected by the culture and so forth. It would be nice if you could ask untutored populations. Well it's hard enough to ask these questions, let's say, of rats who you've starved into submission, or to slave boys, but how would you ask questions like this of a baby? How could you say to a tiny infant who doesn't speak yet, "Oh baby, does this Escher drawing look weird to you?" or "Do you really believe two things could be in the same place at the same time?" And it's part of the great contribution of Professor Spelke's work that she's been able to do this, she's been able to tell us and to do herself, questioning of prelinguistic infants in this way, to find out how they conceive of the world. So I want to introduce her first as my dear friend, second as my esteemed colleague, finally as maybe the Dr Doolittle of the human species. Elizabeth Spelke.
Lecture :: Core Knowledge and Cognitive Development
Elizabeth Spelke
Thank you Lila.
I actually think we should have a change in plans, and I should continue sitting there, and Lila should continue giving this talk, since I think she can talk about my work more eloquently than I can. But since she's sitting down already maybe I'm going to have to keep going here. Let me just say first it's a great pleasure to be back at Penn where I spent nine very happy and very formative years. What I want to do today, though, perhaps a little in a spirit of contrariety, is talk about three very old ideas which throughout the time that I was at Penn and for many years after that I was quite convinced had to be wrong. These three ideas are, first, that human thinking differs utterly and profoundly from the thinking of other animals, that other animals may be endowed with all sorts of interesting instincts, including cognitive instincts of various sorts, but we humans are endowed with a faculty of reason which sets us apart qualitatively from all other animals. The second intuition, that human thinking changes qualitatively over development and in particular becomes more powerful, so that as children get older, they come to command a richer array of concepts and they come to be able to think thoughts that simply aren't available to the cognitive system of the younger child, and the third intuition, that somehow human thinking depends on language, that language is not only a vehicle for expressing our thoughts and communicating them to other people, but is somehow intimately involved with our capacity to have thoughts, especially all those thoughts that set us apart from other animals and let us single out abstract notions and so forth.
As I said, all three of these intuitions have been challenged very vigorously over the last 20 or 30 years or so. The first intuition, that somehow our thinking is qualitatively different from those of other animals, hasn't really in any sense been shown to be logically flawed, but it looks empirically very implausible, in light of a growing body of studies in comparative cognition which show close homologies between the perceptual capacities, action capacities, and many of the cognitive capacities of humans on the one hand and other primates and even more broadly other vertebrates on the other, and I'll be giving examples of this as we go along.
This doesn't preclude the possibility that there's some magical faculty that only we have, it does make it begin to seem less plausible that such a faculty should exist. Turning to the notion that human thinking becomes more powerful over the course of cognitive development, there have been stronger arguments that this simply can't take place. In particular, it's been argued that three claims all dear to the heart of the students of cognitive development, can't all be true. The first claim, the one captured by that second intuition, is that as children get older they come to have concepts that they didn't previously have. A prime example that gets pointed to that I'll be talking about in part today, is in the domain of number, where it seems that kids go from thinking of numbers as the kind of things that you count, natural numbers, to a notion of numbers that includes all of the rational, to a notion that includes the reals, and so forth. At each point, the notion of number becomes more general and more powerful than it had been previously. The second claim, that students of cognitive development are pretty fond of and that also seems, would seem pretty intuitive to all our grandmothers, is that these changes take place through learning, that if you want to take a child whose concept of number includes only the counting numbers and get them to understand the rational numbers, you would do well to send them to school and learn some mathematics. We don't come to change our concepts of number by growing bigger brains or getting hit on the head by a hammer or something; we come to change them through learning. But the third thing that both an overwhelming body of research and a whole bunch of theoretical arguments supports, is the notion that over the course of learning, what the child is able to learn is constrained by the concepts that she already has, the concepts that she brings to bear to interpret the learning experiences that she gets constrains the outcome of those learning experiences. So some of the most powerful empirical evidence for this view comes from the work of Piaget, who's shown in case after case that if he takes a child who doesn't understand what some aspect of number and tries to teach them that aspect of number, the child will systematically misinterpret the teaching that she's receiving. Actually, my favorite example of this comes not from Piaget but from work by Rochelle Gelman that was conducted here at Penn, where she undertook a study where she was attempting to teach children who understood counting about fractions, and specifically tried to teach them that there are numbers between the numbers that you count. So she very laboriously would sit down with such children and say, "look, you've got one and you've got two and you've got three, but between one and two, there's another number two-and-a-half" and in various ways drum this into the kids, there are these numbers, two-and-a-half. About the closest she got to teaching the kid anything, was that maybe a better way to count would be to go, "one, one-and-a-half, two, three". This is two now, this is one-and-a-half. But for a kid for whom number is defined by counting, all of this input about the other numbers was getting assimilated to this counting system. What should be obvious is that these notions can't all be true simultaneously; something has to give here. Jerry Fodor has argued that we shouldn't give up the idea that conceptual change results from learning, and we shouldn't give up the idea that the mechanisms, the primary mechanisms of cognitive development are learning, and we shouldn't give up the idea that learning is constrained by your existing concepts. In fact, he argued at length looking at a variety of different approaches to learning, that all of the in one way or another come down to processes of forming and testing hypothesis, but hypothesis require concepts in order to be formulated; if you want to test the hypothesis "one-and-a-half is a number" you need to already have the concept one-and-a-half in order to test that. If we keep those notions, we're going to have to give up on the notion that concepts change qualitatively over development and propose instead that the full range of conceptual capacities that we ever see expressed in us, in adults, are available to us prior to any learning and emerge just over growth and maturation. A similar argument by Fodor leads to the conclusion that learning a language can't change the way you think, just like learning other things can't change the way you think. Languages are of course learned and learning the words of a language means in one way or another forming and testing hypotheses about what those words mean, and that in turn requires that you have available the underlying concepts; you want to learn what the word "seven" means, you're going to have to test the hypothesis that the word "seven" means "seven" and for that you already need the concept "seven". So the argument is, that the venerable intuitions all have to be false, and somehow we should be looking for a deep continuities between thinking over evolution, at least recent human development, over cognitive development and independently of language.
I'm not going to challenge every little piece of that, but what I do want to do today is look at again at those three intuitions and suggest that there may be something right about all three of them. My plan of attack today is first to talk about a line of research that seems to me to provide some empirical reasons to think, just some evidence, in favor of all three of these notions, people thinking qualitatively differently from other animals, and the qualitative changes happening over development, and depending on language. Then I want to step back from those data and suggest an approach to cognitive development that I hope will suggest a way around the problems posed by the field of cognitive development and more generally the challenges posed to each of the three intuitions that I just went through. I'm hoping to do both of those things relatively quickly, because I think at least some of you have heard me do that before, in a talk I gave here at Penn three, four years ago, something like. I want to go on after that, and spend most of today talking about new work, in which I attempt to apply this approach to cognitive development to the domain that has caused the most intense fighting and questioning about conceptual change, the domain of number, and see if we can make any sense of developmental changes in number processing and also in the ways in which we as adults conceive of numbers, by looking at development.
So let me start with space; as I said I'm going to try to go quite quickly here, but I do want to mention all the people who've contributed to this work, particularly Linda Hermer-Vasquez and Ran-chow Francis Wong are responsible for most of the studies I'll be talking about today, and Stephan Gutera and Gell Ross have been involved in some very recent ones. Now the work we've done looking at development of spatial memory and processes that use spatial memory, navigation processes, was all directly inspired by work that took place at Penn by Ken Cheng and Randy Gallistel some 15 years ago now. Research that suggest while animals of various sorts are very good at finding their way through the environment and getting from place to place, they do so by virtue of systems that are quite limited and encapsulated, by virtue of modular systems. Let me take you through a little of the evidence for modularity in other animals so you get a feel for the phenomena that we're going to be looking at. This just shows an overview of a couple of studies that Cheng and Gallistel performed, where they took rats who were hungry, showed them where food was in a test chamber, then took them out of the chamber, buried the food, and disoriented the animals, disoriented each rat, put the rat back in the environment, and now the question is, is the rat able to use information about the environment to reorient itself, and if so, what information does it use, and the measure of whether the rat's able to do that is whether they're able to go search for the food. Is that clear? What they found in these studies is that rats were able to reorient themselves and search non randomly for the food, but they oriented themselves by using only a subset of the information that was available in the environment. In these studies, walls might differ in brightness, corners might differ in patterning, and different odors came out of different corners, and all of those sources of information uniquely specified that the food was over here, not anywhere else. But what the rats tended to do, was search for the food in two locations: the corner where it actually was, and the geometrically equivalent opposite corner, suggesting that the only information that the rats were using came from the shape of the environment. In particular, the rats seem to have encoded that the food was either, say, in a corner where it's directly to the right of a long wall, since there are two such corners, rats searched in those two locations. In this first study, it looks like they searched the correct location more than the other, which maybe suggested there was some ability to use other kinds of information, but it turned out in followup research that Randy Gallistel conducted at UCLA, that was not the case. The reason they're a little better at this corner is they weren't completely disoriented. When you manage to totally disorient a rat, they seem to rely purely on the shape of the environment and not on any other information. An important point here was that they were also able to show that rats detected other sources of information and used those other sources of information for other tasks. For example, if you give a rat a task where it has to push through a white door as opposed to a black door, they can easily learn to do that. They can represent that a door or a wall is white, and they can represent that food is left of a long wall, but they don't seem to be able to represent that food is left of a white wall; they don't seem to be able to combine these sources of information together, and the argument is, this shows that the cognitive process that the rat is taking in about the environment are couched in separate modular systems. One more piece of evidence for the same thing before I get to our work. This comes from a different laboratory and uses a different kind of task, and it's a very brief and somewhat idealized summary of a set of experiments conducted by James Bigler and Richard Morris, again looking at rats, but now in a very different kind of paradigm. These rats were not disoriented, they knew where north was, they were moving around the environment, they were hungry, and the question was, how quickly and effectively could they learn where food was located? What I've graphed here are three different conditions. In the first condition, the food always occupied a constant position in the room, though the rat might enter the room from different directions, so that the egocentric direction of the food changed from trial to trial, but it was always in a constant position in the room, let's say the northeast corner. In this condition the rats quite rapidly learned to go directly to the right location to find the food. In the second condition, the food was always located directly at the base of a single distinctive landmark which moved around the room. Again, in this condition the rats very rapidly learned to head for the landmark to find the food. In the third condition, the food was located in a constant spatial relationship to the landmark, such as in this case, northeast of the landmark. In this condition, the rats didn't learn to locate the food. They learned something, because over trials what they started to do was search in a circle around the general vicinity of the food, but they never learned to go directly northeast of the landmark. It's as if the rat can learn, the food is in the northeast in the room, or the food is at the red cylinder, but they can't put these two sources of information together, to represent the food is northeast of the cylinder. Again, evidence for modularity in the spatial representations of other animals.
Linda Hermer and I looked at this evidence and felt here was a chance for people otherwise dreadfully bad at navigation, many of us myself included, here was a chance where maybe we'd be able to show ourselves capable of something that other animals were not readily doing. We developed a situation based very directly on the Cheng and Gallistel type experiments. I'll take you through the first study, because then I'll just go very rapidly through a whole bunch of findings. In the first study we looked at adults. We put them in a rectangular room. There are four corners in the room where objects could be hidden, we hid an object in one of those corners, let them see that, and then they closed their eyes and spin themselves around until they're disoriented, and then they open their eyes and we say, "go find the keys" or whatever it was that we'd hidden. When you test people in an all-white room, they perform very similarly to Cheng and Gallistel's rats. If you hide something in a corner with a long wall on the left, they'll search the two corners with the long wall on the left. The fact that they go to both of them only says that we succeeded in disoriented them; the fact that they don't go to the other corners shows they do use the geometric information to reorient themselves. So far they look like rats. On the other hand, when we add a single distinctive nongeometric cue to the environment, like a blue wall in this first study, now their performance looks very different from the rats'; they readily combine the geometric and nongeometric information and search at the uniquely correct corner. They do this over all, and when I present data I'll mostly show you first trials also, to show you they don't need to actually get coaxed into this over a period of trials; they do this very readily. So that's adults, but when we went on to look at children, who we tested through basically the same measure, except we're picking them up and spinning them around to disorient them, we get a very different pattern of results. In a nutshell, the children don't look like adults, they look like the rats. They use the shape of the room to reorient themselves, they don't use the blue wall to reorient themselves. This puzzled us, so we ran a bunch more studies, of which only a few [of them] are on here; we asked is it just blue walls kids don't like those, if we switch to more interesting objects, more kid-friendly objects like distinctive toys in different locations, will they break the symmetry of the room? Answer: no. We also asked, suppose we take objects as big as the child, a big plastic dummy of an object, and we place it on one side of the room and we hide something right next to it, will they go there as opposed to the side of the room that doesn't have the dummy on it? Answer: no. Then we thought, suppose we put the distinctive nongeometric information, coloring and patterning information, suppose we put that right in the corners where we're hiding something, will that affect children? Answer: if we keep the task the same, we're disorienting them, no it doesn't affect them, though you can easily get a child to learn to find an object that's been placed in a blue box if you change the task: you don't disorient themselves, you simply show them over trials in effect like what Biggler and Morris did with the rats, that the blue box is a moveable object and the toy stays with it, under those conditions they'll use the color of the box quite well to find the object, but in this situation where they're disoriented, they don't, they just use the shape of the room. Finally, we asked, what happens if we make the shape information less distinctive and the non-geometric information more distinctive. So Frances Wang first tested kids in a square room with a bright satin red fabric covering one wall, and then Stephan Gutterez come along, I don't have a picture of this, but tested kids further in a round room, a cylindrical room, no distinctive geometry at all and a very bright pattern on the wall. Well kids were very attentive to the bright pattern on the wall, and again if you change the task and give them a task where they're not disoriented and they have to respond to that pattern directly, they very readily learn to do that, but both in the square room and in the cylindrical room they failed to use the red fabric on the wall to reorient themselves and locate an object.
Looking at young kids, we see a pattern very similar to what we see in rats, very different from what we see in adults, and it raises the question, what's changing between, I don't think I even told you this, but between a year-and-a-half of age, year-and-a-half to two years which is what the age of the kids we were testing, and adults, and as a preliminary to asking what's changing, the interesting question, we needed to know when the change was taking place, so Linda Hermer for her PhD dissertation at Cornell, did a study on kids of a variety of different ages. Let me just jump to the bottom line: it seemed to be around age six that she first saw kids, five to six, she first saw kids starting to use the blue wall to break the symmetry of the room. Her further studies focussed on kids at that age in the range of 5 to 6 years of age, and she studied kids's performance in the reorientation task in relation to a bunch of other measures that she took in hopes of seeing what other kinds of cognitive factors would be correlated with success at the reorientation task. She looked at a bunch of different things, like how good their verbal memory was and how good their visual-spatial memory was, and various measures of language. I want to tell you about one measure of language comprehension and production. She looked at kids' comprehension of the terms above below front back left and right, by presenting them with a device where there's a single reference object in the middle, a red ball, and she gives them a green ball and asks them, "would you put this on top of the red ball" or "would you put this to the right of the red ball" and so forth, and just looks at where they put it when they're given a display where there's various different positions you could put the ball in. She also looked at production, through a slightly more complicated task involving the same general apparatus. She had two of these; one of them she and the child looked at together, and it had, say, the red ball in the middle and the green ball to its right; there was another one that the child's mother had in a separate area where the mother couldn't see this, and the mother's just had the red ball on it, and the instruction to the child was, "could you give your mom the green ball and tell here where to put it in the display. She should put it the same place we've got it in here." The question was just whether the child used spatial language to answer those questions. The findings were, at this age, there were basically kids; she was able to divide kids into two groups based on their performance in the reorientation task. There were kids who looked like the younger kids and only used the shape of the room, and others who looked more like the adults and used the blue wall. When she then looked to see what other of these measures correlated with that, the only thing that correlated was children's productive use of spatial expressions with the term left and right. Above and below didn't correlate, comprehension measures didn't correlate; that's a little unfair, because the kids were uniformly pretty good at the comprehension measures; we can't rule out the possibility if we'd made a harder comprehension task, we would have seen more of a correlation, but in any case, for whatever it's worth, this initial study suggested that what goes along with success in this task is successful productive use of relevant spatial language. She did one more study. This one was based on the Biggler and Morris paradigm that I told you about. In this study, children were not disoriented, they were, the game was, they were put in a room where there was a toy hidden under one of eight or nine I think it was identical looking containers, which were placed around the room in a circle, and a single landmark was also placed in the room, and on different trials the object would appear under different containers, but it was always immediately to the left or immediately to the right of the landmark, for a given child it would be consistently to the left of the landmark let's say. What she found, testing kids at about at the [garble] a little older in this experiment, was again there were two groups of kids. One group of kids, like Biggler and Morris' rats, learned to search a cup near the landmark; as the landmark moved around they would come over trials eventually to come search near the landmark, but they didn't specifically and correctly go to the left or to the right, and there were other kids who did go, did learn specifically to go to the right, the correct location, and again when she looked at a subset of these other measures, it was spatial language production, productive use of left and right which was correlated with children's success. At this point one could tell two very different stories. The story that would be in line with all of the arguments for why language acquisition requires a conceptual system adequate to the notions that the language to be acquired is going to express. By that notion, the story you should tell is, by some point children come to, through maturation let's say, come to have the conceptual resources to represent relationships of left and right. Once they have these, they use them in two different situations. They use them in reorienting themselves and locating hidden objects, and they also use them as a basis for learning these relevant spatial terms. The developmental pattern could go that way. But another possibility that tempted us was that the child developing language was actually doing some work in allowing them to capture the relationships that broke the symmetry of the room in the reorientation task and allowed them to find a hidden object. In an attempt to distinguish these two possibilities, and I see Lila's eyes narrowing at this heretical idea that I am sharing with you, in an attempt to distinguish these possibilities we went back to adults and then asked whether we would be able to turn an adult into a young child or a rat by occupying their language faculty with some other task. So what we did in this experiment was, they'd had three conditions; two conditions were the same as the first studies I'd told you about: objects hidden and the adults had to find it either in an all-white room or in a room with one blue wall. The third condition was the same except that prior to beginning the experiment, adults were given a little bit of training in shadowing, that is they listened to a tape recording of a prose passage being read, and they continuously repeated what they were hearing. This is a very attention-demanding task that makes you quite incapable of carrying on another conversation at the same time, either overtly or covertly. So adults are engaged in shadowing and while they're shadowing they're brought into the room and they're run through the study just as one would run it with a very young child, for whom you're not really using language, but we did point out different corners of the room and point to where objects were being hidden start the adults spinning around and then tap them when it's time for them to go try to find an object. As you can see, adults' performance now looks very similar to Cheng and Gallistel's rats. They still used the shape of the room to locate the object despite the fact that they were engaged in this very attention-demanding task. Their search was not random among the four corners; it was heavily concentrated in the two geometrically appropriate corners. But what they didn't anymore do was use the blue wall to break the room's symmetry and find the object uniquely. Let me just tell you one more study from this series. We did a whole bunch of them, but this one I think is all I'll tell you about today to end up this series. You can ask what is it about the shadowing task that impaired people's performance? Was it the fact that it used language, or was it just the fact that it's just a very demanding task that draws a lot of attention, has short-term memory demands and so forth. In an attempt to distinguish those possibilities we contrasted what happened with verbal shadowing with adults' performance in a different kind of shadowing task, where again they're listening to a tape recording and continuously reproducing what they hear, but this time the tape recording is playing a rhythmic, constantly changing rhythmic sequence, and they're reproducing it by clapping out the sequence. We did a few preliminary tests to convince ourselves that this was also attention-demanding—by some measures as demanding or more so than the verbal shadowing tasks. And you can see that it did have an effect on subjects' performance, if you compare during rhythm shadowing to non-rhythm shadowing, it's clear they're doing some more forgetting and making some more errors. Nevertheless the pattern of performance is the adult pattern, not the rat and child pattern; they're continuing to use the blue wall to reorient themselves.
The suggestion I ask you to consider, even you Lila, based on these findings, is that spatial language is actually doing some work for people in this task. It's doing work for children as they're coming to perform qualitatively differently from rats, and it continues to do work for us as adults. You can ask, how is this possible? How could the child learn relevant spatial language in the first place, if the child didn't already have, independently of language, the concepts that these spatial terms are going to pick out. How could a child learn an expression like left of a blue wall, if the child wasn't already able to represent that concept? Here's a sketch of an answer. The picture is that the child starts with a set of different modular systems of representation. Within these systems, all of the information that's going to come together in the expression left of the blue wall, is available. The geometric system contains information that distinguishes left from right. Remember the rats showed that they had this information by going to corners with a long wall on the left and avoiding corners with a long wall on the right. So within the purely geometric system there is information about left and right. And the proposal is, the child learns expressions containing the words left and right by relating those expressions to purely geometric representations. So if a parent or somebody says something like, "look at the thing on your left" and thing is an object with a distinctive geometry, that the geometric representation can pick out, the child has a representation that can now be mapped to the use of that word. The child also has systems of representations, systems for representing objects and surfaces and their properties like color, and by using those systems the child can learn words like blue. Now, what we need is that the child, having learned these words, has a language faculty that gives her a combinatorial syntax and compositional semantics, so having learned "left" in the context of expressions like "left of the mountain" or "left of the wall"; expressions with a unique geometic interpretation, and having learned "blue" in other contexts, the child doesn't have to learn what "left of the blue wall" means; the child gets that meaning for free by combining together the meanings of the individual terms. So the child's language faculty allows her to form the expression "left of the blue wall" and when she forms this it becomes a link between two otherwise separate systems of information, allowing the information in those different systems to be combined. That is the idea specifically for "left of the blue wall". It leads to this more general idea about one process that may go on over the course of cognitive development, and that I would argue does go on in the case of the development of spatial representations.
The idea is that first humans, like other animals, are endowed with a set of core systems of knowledge; I've talked about two: object representations that let you pick out things like blue walls, or trucks, and geometric representations that let you represent the shape of the surrounding layout. These systems have some critical limitations. They're domain-specific, so the geometric system applies to extended surfaces but it doesn't apply to movable object; that's why the kid can't use the toy truck to break the symmetry of the room; they're task-specific, so the geometric system is used for reorientation but it's not used for finding a movable object either by rats or by children; they're informationally encapsulated, so the geometric system takes in information about the shape of the surrounding surface layout but not information about the color of the same layout, it's sensitive to some but not all of the information that the child can detect; and they're representationally isolated (that's my cumbersome term but it just means that each of these systems gives rise to representations that interact with one another in ways that are privileged by the cognitive architecture but which don't interact promiscuously) so a child or a rat can't just readily combine information from the object representations with information from the geometric representations at will to form new combinations. But children, humans, are also endowed with a language faculty, and that language faculty includes a domain-general lexicon, so you can learn words for geometrically defined things and words for things defined in other ways, and a combinatorial syntax and compositional semantics and using these properties of language the child is able to construct new concepts whose components were already represented but whose combination could not be represented because the components were couched in distinct modular systems.
What I want to do for the rest of today is see whether I can take these ideas and use them to make sense of a much more tricky set of developmental changes, and also more tricky set of phenomena that we see in adults, all in the domain of number. I want to talk about work that's collaborative with a whole bunch of people, most notably with Fei Xu and Sinath Sifkin (????), and with my collaborators Dina Slastahan (???) in Paris and Susan Carey at NYU, but there's lots of people who've had a hand in this, and in particular recently Kirstin Condry and Kate Brody and Lisa Faginson have contributed in some major ways. I should say, despite my initial examples, I'm not going to be talking about very fancy number concepts, I'm not even going to touch concepts like one-and-a-half or the square root of two; I'll be happy if I can come up with an account of how a child develops the concept "exactly seven." I think even to get there one needs to go beyond what infants have and beyond what we see in any other animal.
As in the case of space, let me start with animals and the number representations that a century of research in experimental psychology and behavioral ecology reveals about them. There's a wealth of evidence that was I think most importantly assembled by Randy Gallistel and then very eloquently discussed recently by Stan Dehaene and his book The Number Sense, a wealth of evidence that animals represent approximate numerical magnitudes. Here's an example of a study that shows this; this is actually a study that was performed on laboratory rats that were highly trained. They were trained in a task where they were hungry; to get food they had to press one of two levers, but that lever would only deliver food if they had previously pressed another level a given number of times. In different runs through the experiment the number of times you had to press the other lever varied. What we see here is, when the critical number of times the other lever had to be pressed was four versus eight versus twelve versus sixteen, we see how many presses the rats, once they were trained, actually performed on that level. You see a couple things here. First of all, you see that the rats, the number of presses by the rat varies systematically and appropriately with the number of presses required: they always give a few more presses than are required because there was actually a big penalty if they stopped too early, and no penalty if they continued too long except that they were wasting their time, the time they could have spent eating food. So the first thing you see is that there's an appropriate relationship between the number required and the number given. Second thing you see is that there's variability, and the variability goes up proportionally as the set size increases. That's to say that these approximate representations behave the way other representations of continuous quantities behave in psychophysical experiments, to a first approximation they accord with Weber's Law and discriminability is proportional to set size.
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...in terms of number, but was number being represented at all by these animals? It's not too hard to think of alternatives. Maybe they learned to press for a given duration. Maybe they learned to press until they became tired to a given extent. One of these other sensory qualities that correlates with number could have been the basis for the response, if all we had was a single experiment like this. But of course, this is animal learning experimental psychology and we don't have just a single experiment, we have millions of experiments, in which all of these different alternatives are tested out. To make a very long and somewhat excruciating story short, you can show responses to number when every other sensory variable that you can think of is controlled. I think the most elegant experiments here are studies by Russell Church which show, in the auditory domain, that animals are sensitive to the number of tones that are presented and will reliably discriminated between tones of different numerocities even when you vary the lengths of individual tones, the total length of the sequence, and in fact they'll transfer that discrimination to numerical stimuli in other modalities, all suggesting that there is indeed a representation of number in animals like rats.
But the representation of number we see in animals like rats is really different from the representation of number that you see in the average four-year-old child that Rochelle Gelman and others have studied in such illuminating ways; Rochelle having done the studies here. Here are some of the most important differences. I'm talking about a child, a four-year-old child, who understands counting and who thinks of numbers as the kinds of things you count. One difference, one obvious difference has to do with precision. Counting gives you the precise number of individuals within an array, whereas the approximate numerical magnitude representations that animals shows, as I showed you, do not. So that's the first difference. Second difference, intervals work differently for the approximate numerical magnitude system versus for counting. For counting, the difference between two and one is like the difference between five and four; it's what you get by adding one or going up one in the counting sequence. Within the approximate numerical magnitude system the difference between one and two is like the difference between four and eight, because the system accords with Weber's Law. For the same reason, operations like averaging work differently in the two systems; there are some operations defined for one system that aren't defined for the other--adding one being the clearest case of an operation central to counting and that plays no role in the approximate numerical magnitude system. I want to spend a minute talking about what I think are the two most important differences between these two systems.
First of all, counting is an iterative system. When you assess the numerosity of a group of objects you focus on each object one at a time and you add one, you increment your counter as you go around for each of those objects. But, there's a variety of evidence, both in humans who, when we're prevented from counting we also represent approximate numerical magnitudes in the kinds of studies that have been done with rats, there's evidence both from studies with humans and also from studies with monkeys that the mechanism for assessing approximate numerical magnitudes is not iterative. I'll give you just one finding from very recent research by Liz Brannon and Herb Terrace. They trained a group of monkeys to respond, the monkeys were shown two arrays with different numbers of objects in them and the task was, they had to touch the two arrays in the order: smaller number larger number. They were able to get lovely, after the monkeys were trained, they were able to get lovely reaction time data on them. They'd put up an array of one object and array of two and the monkey would go one two, or they'd put up four and eight and monkey would go four eight. The task is clear, and they got reaction time data for all of these. The first thing is they got further evidence that the monkeys' representations accord with Weber's Law, by looking at the reaction time data. The time, when the monkey was presented with the problem, touch four then five, they were slower than when they were presented with the problem touch four then eight; that's what you'd predict for an analog magnitude system that accords with Weber's Law. On the other hand, if you compare their speed when they had to touch four then eight with their speed when they had to touch one then two, there was no difference. That suggests that however the monkeys are assessing the numerosity of the four and the eight, it's not through any iterative process of successively going around and counting them. If there were, you'd expect the four-eight to take longer than the one-two. So it suggests that there's some parallel, simultaneous process at work when you apprehend approximate numerosity than when you apprehend exact numerosity and there's similar evidence for the same thing in studies with humans. Final thing I want to signal about the approximate numerical magnitude is that, there's a sense in which the representation of individuals, individual elements, is not explicit in the approximate number systems or the approximate number systems you find in humans who are prevented from counting. The clearest evidence here comes from some recent research by Entrilagator and Cavanaugh where, they did studies that were not directly focused on number perception, they were focused on limits to our ability to attend to individual objects and follow individual objects through time. What a whole variety of studies in Patrick Cavanaugh's lab have shown, is that when adults are given a task that requires that they focus attention on one object, or move attention from one object to another, as those objects become too close together and too small within the visual field, we reach a point when we're no longer able to reliably select a single object for attention or move attention from one object to another. If an adult is given a task where there's like ten dots in front of them and their told, "focus your attention on the leftmost dot, and now every time you hear a tone, move you attention one dot to the right." This is a trivially easy task if the dots are big and well spaced and in central vision, but if you make the dots too small and too crowded together you lose the ability to do this. But once you're below the threshhold, below the point where you're able to selectively attend to single dots, you're still able to perceive a group of dots, and you're still able to make approximate numerical magnitude estimates. That says that in a sense, we can assess the approximate number of entities in a collection without being able to explicitly represent the individual entities that compose that collection. That's going to be crucial for what's coming.
I've told you about a bunch of difference between analog magnitude representation and the kind of number representations that we seem to have, at least from the time when we're about four years of age. But you can ask, would it be possible to train other animals to represent exact numerosity and to count just as well as four-year-old children do? There have been a bunch of studies that have done this, and I want to spend a little bit of time on one, because it suggests both a way in which the answer to that question is yes and another way in which it's no. This is a study that was done by Tatsuro Matsuzawa on a chimpanzee who he's continuing to study; there was about a week ago an article in Nature where he's continuing to study this chimpanzee's number abilities in various ways, but I'm showing you here some data from an early report some fifteen years ago where he had trained this chimpanzee, Eye, to respond with, he trained her to use the Arabic numeral symbols to designate sets of objects. In the initial stage of training, Eye would be given arrays that contain either, say, one red pencil or two red pencils, and given the Arabic numerals one and two, and she had to learn to press the numeral two when there were two pencils there and press the numeral one where there was one. When she got good at that, they added in three, when she got good at they added four, and so forth, and continued up, at the time they'd published this they'd gotten up to six; they've now gotten at least to nine. First thing to say, is Eye succeeds at this task. It is possible to train quite precise number discriminations at least in chimpanzees. But there's two things about Eye's performance that I think are really interesting. The first I'll show you. Look at what happens when Eye goes from discriminating one from two to discriminating one two and three. In this line over here you see performance on one, performance on two, and performance on three. What I want you to see is that at the end of training one versus two, she's reliably picking two, the symbol for two, when shown two pencils and the symbol for one when shown one pencil, but when three is introduced she continues to perform well on one, but she falls apart on two. I'm calling these symbols, "symbols for one and two", but Eye seems to have interpreted them as "symbols for one and not-one", so when three comes along the "two" symbol is what you use, it's just fine. Well, you could think, what happens as we move through training, what happens when she gets good at one versus two versus three and now we add four? Well, she stays good at one, now she stays good at two, but three falls to chance. Again, it seems "three" meant, when she was learning one from two from three, what she was really learning is "one" "two" and "not one or two". Well this pattern continues all the way up to six in these data and all the way up to nine in their more recent data. At no point does Eye seem to get the idea that each of these symbols designates a specific numerosity. Another way to see the same thing is to ask, is there any savings in learning as new symbols are introduced as the experiment goes on? Well, what you see here is performance divided into sessions, and you can see that there's essentially no savings. It takes as many sessions to learn "six" given that you've already learned "one" through "five" as it takes to learn "four" given that you've already "one" through "three". Both of these findings suggest, I think, that there's no learning set for number in this chimpanzee. She's able to learn the discrimination but she never gets the idea. There are these sets of discrete objects and each symbol applies to a set and each set you get by adding one to the previous set.
This is in contrast to little kids who resemble Eye in some ways, at the very early stages where they're learning the terms "one" "two" and "three". There's lovely research by Karen Wynn and others showing that when kids first start engaging in counting when they're about two or even two and a half years of age, they don't understand the number words any better than Eye does. So a very early counter understands that one means one, when Karen asks them "give me one doll" they'll give one, if she asks "give me any other number of dolls" they'll never give one, they know the other numbers don't mean one and one does mean one, but if she gives them a choice between two dolls and three dolls and asks which is the two dolls, they're at chance. This looks like Eye's performance here. Same thing happens at a little later point in development. Kids will get one and two right, but they're still treating four and the other numbers as if they apply indiscriminately to anything that's not one or two. Again, they look like Eye at this point. What you never see is anyone who looks like Eye at THIS point. Once kids work out the meaning of one and two and three, they seem to work out the whole count sequence. At that point, they know what all the number words in their count sequence mean, and in fact in further studies by Rochelle Gelman they even know that number words exist outside their count sequence, and that for any number, whether they themselves know the name for it or not, you can add one and you'll get another number. So they seem to get the whole system and Eye doesn't. We can ask, what's going on here.
One possibility investigated by Karen Win is that humans are endowed with a concept, "adding one", which other animals don't have. Research that seemed to support this idea or studies that, when conducted, using preferential looking methods with young babies, these were five-month-olds, the kind of methods Lila was talking about before, where you simply present babies with events in a puppet show where things either go normally or they go bizarrely, you look to see if the babies do a double take and look longer at the events we would say were bizarre, and she found that if you initially present a single object on a scene and then covered it up and then you added a second object to it, then removed the cover, kids looked quite happy and not terribly interested if there were two objects there; they looked a good bit longer if there was one object there or three objects there. And similarly, their looking time suggested that if you started with two and then removed one, they expected to see one, not two or other numbers. This suggested that kids are able to take account of simple additions and subtractions. Of course as in the case of the first study that I told you about with the bar pressing, you can think of a lot of other sensory variables that kids might be responding to other than number, and people who do studies of really cognitive development have no shortage of friends who are happy to point those other variables out to them. Wynn's study was followed by a large number of replications with controls for things like, maybe it's amount, continuous amount of "mouseness" that's being responded to here, or objects in particular spatial locations; these things have been controlled for in a lot of these studies. People have obtained the same results as Wynn and their evidence rules out the continuous variables of mouse coloring or spatial positions and so forth. You get these findings even when spatial positions change, when colors and shapes of objects change, behind the screen and so forth.
It looks like there is something interesting related to number going on in these studies. But we can ask, do these studies account for a distinctively human representations of number, the distinctive representations we see in the four year old child that we don't see in the chimpanzee or other animals, and I think at this point the answer has to be a resounding no. I'm running out of time so let me just give you two reasons to think that Wynn's study is not picking out what's special about human number representations. Reason one, other primates succeed at Wynn's task, but they don't succeed at, they don't get to the four year olds' insight about the natural numbers, but they do succeed at Wynn's task. We're not the only animal that does that.
Second, kids' performance in these addition experiments seems to be strongly limited with respect to the size of the sets to which it applies. So in further experiments by Wynn, she showed that while kids could add one mouse to one mouse to get two mice rather than one, they could not add five mice to five mice to get ten mice rather than five, even though the ratio differences are equally large. It looks in that study and in a whole bunch of others that have been done, as if the system that's at work in Wynn's task is able to track up to three objects but not more than that. It doesn't naturally provide a way of accounting for kids' insight at around age four that adding one can just go on forever.
These kinds of limits have led to an alternative interpretation of Wynn's addition and subtraction findings, according to which they have nothing to do with number at all. They only have to do with object representations. They show that babies can represent objects, both when they're visible and when they're occluded and they can keep distinct objects distinct in their representations, but they don't say anything about number representations. Stan Dehaene has argued that that can't be quite right for two reasons. One is, it doesn't at all account for other situations in which kids have been shown to be sensitive to number, when objects aren't involved at all, when they're given two or three tones or two or three syllables, there's a variety of different situations where babies have been shown to be sensitive to number. But second, it seems unlikely that it would be that babies would have no capacity to represent larger numerosities because other animals do. We've seen other animals can represent these approximate numerical magnitudes. That raises, at least, the possibility that babies should be able to do that as well. That's a possibility that Fei Xu and I recently tested through a preferential looking experiment, a simpler one than Wynn's. We simply relied on the standard finding that if you present the babies with the same thing again and again they get bored, and then if you present them with something new they get more interested, and we used this phenomenon to test whether babies could discriminate eight visual elements in a display from sixteen visual elements. The way we ran the study is, we had two groups of babies, one habituated to eight elements, one habituated to sixteen. We made the displays equal in size, and we made the total amount of black ink on the page equal as well, by making these elements twice as big as those. During habituation, the two arrays are differing in number, of course they're also differing in density, and they're differing in average element size. To make sure that babies, to set things up so that babies had to respond to number and not density or element size, we then tested babies with new arrays for which element size and density were controlled. One array had sixteen things, one array had eight things. Because we controlled density and element size for the test, we of course we not controlling overall area or amount of ink on the page during the test, but those had been controlled habituation. Basically we were following the logic here that Russ Church and others follow with animals, to test for number discrimination in a situation where the most obvious alternative candidate variables are controlled; half of them are controlled during habituation, the other half during the test, so none of them serves as a consistent cue to number.
These were six-month old infants, what do we find? In our first study, where the discrimination was between eight and sixteen, the babies succeeded. Those who were bored with eight looked longer at sixteen, those who were bored with sixteen looked longer at eight, overall longer looking at the new number than at the old one. When we then repeated the study with eight versus twelve, babies failed. So, for young infants, as for other animals, these approximate numerical estimates are not at all precise, they break down when the difference between the two numbers gets smaller. The fact that babies failed to discriminate eight from twelve is consistent with some much older evidence from Prentice Starkey, that babies also failed to discriminate four from six. So, we have evidence for a system for representing approximate numerical magnitudes in babies. But this can't be the only numerical system that babies have, because if it were, babies would fail to discriminate two from three, but they succeed at that. Also, if the approximate magnitude system were all that babies have, they wouldn't be able to do addition, because remember they can't do five plus five to get ten rather than five, yet we know they can do one plus one to get two rather than one. So, how to account for all these findings?
Here's a proposal. The proposal is that babies have two systems of representation that are relevant to the construction of a concept of number. One is a system for representing and tracking objects. It's limited to approximately three to four objects, so they can't use it to keep track of higher numbers of things. It captures the number of objects in an array exactly, so they can discriminate arrays with two objects from arrays with three. It captures the effects of adding or subtracting one object from an array when the number of objects is small, as in the Wynn experiments, but it includes no explicit representation of objects as forming a set. In effect you can think of this as a representation that incorporates the information "There's an object in this scene, and there's an object in the scene which is distinct from the first object." But no explicit representation that these two objects form a set with cardinality two. There's also a second system, the one that's been studied in animals and that Fei Xu and I found in kids, a system that has no upper bound on set size, but that has a bound on discriminability, set by Weber's Law. This system does not allow for addition and subtraction. In particular, the operation "adding one" is not an operation in this system, but it does include an explicit representation of sets that allows "more or less" comparisons. But note, as I told you earlier from the Entrilagator and Cavanaugh work, in those sets the individuals that compose the set are not explicitly represented in a way that would allow separate attention to and manipulation of the individuals. So, they've got a system for representing sets, and a system for representing individuals, each quite limited, and these two systems appear to be unrelated to each other early in infancy. Neither system by itself could account for our human number representation, both because neither system has the power of the four-year old's counting system, and also because there's evidence for both of these systems in other animals who, when confronted with a learning task like the one Matsuzawa gave Eye, never come to the induction, "Oh, you just keep adding one and you get all the natural numbers."
So how do children come to that notion? Here's a suggestion. The suggestion is that they come to this notion by combining the representations from those two initial systems together. In particular, from the system of object representations they get the notion of number as picking out a set with an exact discrete numerosity to which you can apply the operations of addition and subtraction, and from the system of approximate numerical magnitude representation, they get a representation of sets that have no upper bound, that can continue indefinitely, and that have a cardinality such that they can be compared to each other, one set can be seen as having more than the other. They develop these by combining these two representations together at the time they learn verbal counting.
At this point we can ask Fodor's question: how could a kid ever learn verbal counting if they didn't already have this system that combines the two kinds, the various components of these number concepts together. Here's a stab at an answer based largely on data that Karen Wynn collected, some of which briefly reviewed for you already on how children learn the meanings of the number words, and extended by some recent research that Kirsten Kandry and I have been doing, probing a little further kids' understanding of number words as they're beginning to learn to count.
The idea is that the first step that the child takes, right at the beginning of learning to count at about age two or two and a half, is that the child relates, the child learns the singular-plural distinction, if they're learning a language that has it like English, they learn that distinction, they learn that the word "one" goes with singular terms, and that the other number words go with plural terms, and they relate singular forms to arrays containing an object, they relate these two arrays given by representations given by the object tracking system, and they related plural terms and the other number words to arrays, representations given by the approximate numerical magnitude system. At this point, when a child hears an expression like "one dog", that connects to a representation of a single object, that happens to be a dog, when they hear expressions like "some dogs", "two dogs", "ten dogs", or "look at the dogs", that connects to a representation of a set, an approximate numerical magnitude. At this point, though, these two representations are completely separate from each other. Now according to Karen Wynn's data, children are in this state for about nine months before they learn the meaning of the word "two" on average. It's not easy to learn the meaning of the word two, but it eventually happens, and what I propose is going on when kids learn that word is that they learn that expressions like "two dogs" are special because they pick out a unique object representation, a representation of an object and of another object, while also picking out a unique approximate numerical magnitude, namely a representation of a small set. So the idea is that "two" is the first word that links these two systems together for the child. The next step in Wynn's data, about three months later on average, is that kids learn the meaning of the word "three", I would assume through the same kind of process. And now, the child is in a position to notice what happens in the counting routine when they go from two to three. In particular, within the object tracking system, the progression from two to three corresponds to the operation of adding an object; within the approximate numerical magnitude system it corresponds to the operation of increasing the size of the set. Then the final step is for the child to realize that every one of these number words behaves like two and three; it picks out a set of discrete individuals, that is to say an entity defined by the combination of a system for tracking individuals and a system for apprehending sets and their numerical relationships to each other, and further that the progression from one number to another in the count routine is always like the progression from two to three, it involves adding an individual to increase the size of the overall set. That's the story. Is there any reason whatsoever to think that this story could be right?
I want to end today by I hope addressing that question a bit obliquely, by telling you about some research that returns us to studies of adults, and tests out a prediction of this picture of number development. The prediction is the following: if our concept of the natural numbers, exact numerosities, is constructed by using language to bring together representations of individuals with representations of sets, then these number concepts should continue to depend on language for us as adults. On the other hand, when we as adults are given tasks that only require that we represent approximate numerical magnitudes, those tasks should not depend on language, even if those tasks are set up to be superficially extremely similar to the exact number tasks. I want to talk about lines of research, one conducted using behaviorial methods and the other conducted using a couple neuroimaging methods, that were both undertaken to test those predictions. First let me talk about the behavioral studies; these were all conducted in collaboration with Sonnet Sifkin on Russian-English bilingual subjects, and the idea behind the studies was the following: we took people who were fluent both in Russian and in English and we attempted to simulated in these adults the process of learning elementary arithmetic facts in school, by teaching them new facts about numbers, facts that they were already able to compute on their own but that they didn't know by rote. We taught them three different kinds of facts. We taught them the results of a set of two digit additions, for example, and they had to know the exact answers there, so for example eight-seven plus fifty-four is exactly 141, not 131. We also taught them new exact number facts involving smaller numbers but addition in novel bases, base six or base eight, and finally we taught them some new facts about approximate numerosities, such as that the cube root of 1300 is approximately 11, where the alternative that was given was distant from the correct answer. All of these facts were chosen so that people would not have the answers off the top of their heads, and so that they would benefit from training on the facts. But what we did is we presented some facts in Russian and trained people in Russian, and presented other facts in English and trained them in English, taking care on half the problems to make the answers the same in both of the two languages, so that if we were to see training effects it wouldn't simply be due to subjects getting more practiced at saying the words in one language or another, it would really be specific to those number facts.
Here are the findings of that first experiment. What I've graphed here, it's kind of complicated, but the grey bars show the time to give an answer, the white bars show the number of errors that are made, this is training data now and it shows what happened over the course of the first day of training and the second, there were two days of training, in each of the two language and for all three types of facts, and the simple bottom line is, subjects improve; training benefits subjects on both languages and on all of these tasks. They get better with training. Now the question is, what will happen if we test them on these facts, both in the language that we trained them in and in their other language. If exact number requires for its representation participation of the language faculty, and in particular the words that you use to link the different components of exact number together, then training in one language should yield better performance when subjects are tested on that fact in that language than when they're tested in the other language, and if approximate number is not dependent on language in this way, then you would expect training not to have a language-specific effect.
Here are the results of that first study. These, the first two sets of bars here show you what happened after subjects were trained in Russian on new two-digit arithmetic facts. You can see that they're faster when they're tested in Russian than when they're tested in English. (Now, Russian was their first language, so you could have thought, well, they're just going to be faster in general in Russian), but in fact we see that when we trained them in English, we get the reverse pattern; they're now faster in English than they are in Russian. In both cases they're faster when they're tested in the language that they were trained in. The situation was slightly more complicated for the novel bases, but the same general conclusion applies. For the novel bases, subjects were better overall when they were trained in Russian than when they were trained in English. But within each of the two languages, they're faster when they're tested in the language of training than when they're tested in the other language, again showing an effect of language-of-training on these representations of exact number facts. But when we get to the approximate number facts we see no effect of language-of-training at all. It doesn't matter in what language you come to apprehend that the cube root of 1300 is about eleven; having learned that in either of your two languages, that information is equally available to you in both of them.
We've gone on to do a couple more experiments; I'm pretty much over time so I'll go through them very very quickly. We asked, could we make a more focused comparison by taking a single operation, two-digit addition, and contrasting performance when you're trained to give the exact, versus trained to give an approximate answer? Here's an exact problem, here's the same problem but the two candidate answers you have to choose between, neither is exactly right, one is close and one is far away. Again we took Russian-English bilinguals and compared training and transfer effects for learning exact answers versus approximate answers to the same problems. (For any given subject, they either got the exact version or the approximate version, and across subjects we can look at the effects there.) Training, again, you see a training effect from day one to day two, for both languages and for both types of problems, and for testing you see the same pattern that I showed you from the first experiment. When subjects learn exact answers to two-digit addition problems they do better in the language they were trained in than in their other language, both for training in Russian and for training in their other language, in English. But when they were given the same problems but only had to estimate the answer, we see no effect of the language of training. Finally, in the last study, we asked "is this all specific to arithmetic, or are these effects that we'll see whenever people have to represent number, whenever they have to learn something about number, whether they're doing an arithmetic problem or not?" So in the last study, instead of trying to take adults back to their elementary arithmetic classes, we tried to take them back to their elementary social studies classes by inventing two new countries and making up little essays in which we told people facts about these countries. So, they read one essay in Russian and one essay in English and in each of the two languages some of the facts that they learned were numerical facts. "The country of Cap Nopa became a democracy 74 years ago." And some of the facts that they learned were non-numerical facts: "The first president of the country was a lawyer." In both cases they were subsequently tested through the same kind of two-choice multiple-choice test: "The first president was a lawyer versus a doctor" "The country became a democracy 74 versus 72 years ago" etc. They're learning these two stories and then they're tested on facts, both numerical facts and non-numerical facts. What I showed you initially was the training data; they get better at both the stories. When we get to the test data what we see again is an effect of language-of-training for the numerical information, but it's not an effect that you see across the board for everything you memorize, because it doesn't show up for the non-numerical information. In all of these situations it looks like for us as adults, language is continuing to play a role in our representation of exact numerosity. Since I'm out of time I'm not going to show you any pictures of brains, but you can ask me about them if you want in the question period; if I have to I'll show you pictures of brains where we look at patterns of activity in different regions of the cortex as people are solving exact versus approximate arithmetic problems. This is all research conducted primarily by Stan Dehaene, and it supports the same general conclusion: exact representations recruit areas involved in language and verbal memory, approximate representations do not.
Let me conclude, or at least summarize. I've suggested that human cognition begins with a set of core systems of knowledge, that are largely constant over development, don't undergo radical changes, and are largely shared by other animals. They're not, in general, unique to us. I've also suggested that each of these systems has some critical limitations, limitations that appear both in animal studies like the ones that Randy Gallistel and Richard Morris have run, and that also appear in studies of young children. These systems are each domain-specific; they encompass some of the things that children are able to perceive and think about, but not other things; they are task-specific; they are used for certain purposes but not others; they're informationally encapsulated; they take in some but not all of the sensory information that the child detects; and they're representationally isolated; they give rise to distinct and separate representations of the world. I've also suggested that new representational capacities emerge when children learn language, because language provides a medium within which information in these distinct encapsulated systems can be combined, and I gave two possible examples of this, one in the domain of spatial memory and the other in the domain of number. Because these new representations depend on language, we would expect them to be unique to humans, we would expect them to allow children to develop new concepts, concepts with old components but that involve new combinations, and we would expect the thoughts that involve these representations to depend on language in adults. These are all the things that I want to suggest were right about those initial, venerable intuitions that I started out with. Lest Lila give up on me altogether, notice that I'm not becoming...the fact that I'm arguing that our thinking differs qualitatively from other animals and changes qualitatively over development and depends on language, does not involve arguing that somehow our thinking is a totally mysterious, unanalyzable process, that we can't relate to its foundations either in development or in evolution. On the contrary, what's special about our thinking seems to depend not on some wholly new different faculty that only humans have, other than the language faculty, what it depends primarily on are the same core systems of representation that we find present throughout our lives and also present in other animals. This gives me the hope that we can continue to investigate these systems through a combination of research in comparative cognition and cognitive development. Thank you.
Questions & Answers
John Trueswell
I have a question about Eye. Your argument that you're making, that these connections require a language faculty, seem to have their critical test in comparison to primates, right, the humans versus the primates, because the primates, if we can argue that they don't have the language faculty, but have other cognitive capacities which are close to humans', then the critical ones are like the case of Eye, and it struck me that one of the big differences between that Eye was trained about numbers and the way that we are trained about numbers is that Eye was trained with each number coming one at a time, whereas the child is trained with all of them.
Elizabeth Spelke
That's absolutely right. The child gets the whole counting routine and then has to figure out....
Trueswell
Has anyone tried to train chimps that way?
Spelke
No, if there were a single study that I could talk anyone into doing on training, a single chimp training study, it would be that. Another things would be though; my strong bet is that if you were to train a child following the same sort of procedure - it would be probably an impossible experiment to run because you'd have to raise the child in some controlled environment where they never heard anyone counting, but my bet is if you were to train a child on this sequence, or if you were to train Eye on a different kind of dimension where the relevant learning set isn't for number, it's for some other property, I'd be astonished if it turned out that getting the sixth symbol took as long as getting the fifth and the fourth and the third; I think we'd see her showing a learning set, so my hunch is that the difference is, that's real that you point out between the way Eye learned these symbols and the way little kids do. My hunch is, that's not going to be what accounts for the difference, but you're quite right, it's an important experiment to do.
Trueswell
...train the primate like the humans. And then, in relation to that, has anyone tried to do, run primates in a maze study, that the rooms...
Spelke
Yes, I'm embarrassed by these findings. Mark Howser and Stephan Guetterre have been running these studies on cotton top tamarinds, and they perform totally differently from the rats or the children or the adults. They don't use the shape of the room at all, which everyone we've ever tested does, and they very readily learn to go left or right of a landmark. One thing that's true - now wait a minute, I'm not going to give up right away - for a year before they were run in that study they were run in experiments where, to get food, they had to climb to, food would be found in particular trees and not other trees, and the particular trees changed from day to day, and what cued them was a non-geometric landmark right at the base of the tree. They were trained on this for a year, and I want to wait and see what happens in the absence of that kind of training, where you could easily imagine, not that there would be a direct representation "left of the landmark", but that there's be some complicated response strategy which over the course of the year the animal could learn, like "look for the landmark, now turn left", right, that would be a strategy it could use, two modular separate representations without directly combining the information, just chain from one to the other. The studies need to be done on untrained monkeys, and we don't know yet how that will turn out.
random questioner
You said that verbal counting seems to be the linking mechanism. Has there been any investigation, does it have to be verbal, what about deaf children?
Spelke
That's a wonderful question. My bet would be, it certainly doesn't have to be oral. My bet would be that a deaf child using sign language or some other system with the properties, a child with a language faculty, who develops some other kind of way of expressing faculty, my bet would be they would be just as well off as an ordinary child learning the words "one" "two" and "three", but again, I don't know of any relevant research.
same questioner
And then what would be your speculation about the lowland gorillas that have learned American Sign Language?
Spelke
Do I have to? I don't think other animals can, are truly capable learning natural human languages, so my bet would be that, if we looked in detail at what lowland gorillas were doing we'd see performance like Eye's: we'd see all kinds of interesting learning capacities, but it wouldn't be, they wouldn't have the character of the capacities that our combinatorial language faculty immediately gives us. That's just my...[end of tape]
another questioner
...for...what it seems...there's been studies done with...the other thing is, the sign language done with chimps, it has been shown that they can actually approximate and make up their own language, they have done that. So they do have some capabilities.
Spelke
Do you want to answer that one Lila? I'm not an expert on language, but we have some here.
Lila Gleitman
There's that minimum controversy.
Spelke
I think I don't want to get any further into the animal language controversies.
another questioner
I have a question. What is....the language facilitating the representation. What would happen if, have there been any studies about multiple languages, that kids who speak two different languages...
Spelke
Where I'm really eager to do them. The question was, what about bilingual kids, and how do they learn things. I'm really eager to study kids learning arithmetic in bilingual classrooms. The work that we've done on adults, of course, it's really important to know does this generalize to kids. If it does, then I think we need to look very hard at bilingual education and face the possibility that for certain kinds of material, information taken in in one language may not immediately and readily transfer to other language, and a child learning in a classroom that shifts from one language to another could actually have more to learn than a child learning in a monolingual classroom. But this needs to be investigated. At this point we don't know what's going on. I have anecdotal, I can give you anecdotes from my own children who are bilingual, and who at one point were trying to learn arithmetic both in English and in French and finally we decided no, we're going to pick one of these languages and stick with that and ignore everything going on in the other language because they were getting a lot of interference, but this needs to be looked at systematically.
another questioner
I want to give you the opportunity to show us brains. But first, the Russian and English were all the training and testing with full words, or were they done with Arabic numerals?
Spelke
Words
same questioner again
So I guess, I know English, but what would you expect to be the result if all the training and responses were done with Arabic numerals, which presumably would transcend the lexicon and the specifics...
Spelke
I would expect that subjects would translate those Arabic numerals into whichever language they prefer to use, for doing arithmetic. So, these subjects who learned all their elementary arithmetic in Russian, I would expect to translate those numerals into Russian. The reason that we wrote everything out in words was precisely to prevent people from doing that. But behind that specific question is a more general question. Let me ask it for you, I think this is the real question: How crucial is language here, or is language just one symbolic system? Is what distinguishes us that we have a language faculty, or is what distinguishes us that we have a symbolic faculty, which can express itself through language but can also express itself through other symbolic systems like Arabic numeral notation? That, I think, is a really interesting open question. There's some evidence from imaging studies and other kinds of studies that Stan Dehaene and others have done, that the representation of Arabic numerals is in some ways distinct from the representation of number words. On the other hand, it does seem as if most people confronted with numerals translate them into one or another, into words in one or another language. I think possibility is that, what's fundamental to the construction of number is a language, but once you have that, you can now express, you can now develop other symbols that you can manipulate for the same representations. But really that's just a long-winded way of saying, "really good question". I don't know the answer. You asked for these findings, I'm not going to wait for a groundswell of interest here, I'll give them to you really quickly. The imaging studies used much simpler tasks, and we didn't try to train people on anything. We simply gave them simple, single-digit addition problems. In one block of trials, they had to pick the exactly correct answer, where the alternative was a near miss, in the other block of trials neither answer was exactly correct, and they had to pick the one that was closer. It's the same set of problems in one case with an approximate, requiring approximate addition and in the other case exact, and the subjects were told in the case of the approximate addition, don't try to calculate the exact answer, you'll do best if you just try to get in the right ballpark. They were told that in advance. The first study used fMRI. These are Stan Dehaene's studies, really, and he's separately publishing a whole bunch of further analyses, looking at performance in each of these tasks, but the stuff that we published together just looks at what happens when you subtract activation on one of these tasks from the other. You see in yellow the areas that are more active during approximate addition than during exact addition, and what you see is a lot of bilateral activation of the inferior parietal lobes. These are area where it's been previously shown that if people suffer a lot brain damage they're apt to have impaired number sense. Impaired sense of relative relationships among different numerical magnitudes. But for my purposes the more interesting data are what you get in blue, which is what you get when you subtract the approximate from the exact, these are the areas that show greater activation during the exact tasks, and in particular you see an area that's lateralized on the left, in the inferior part of the pre-frontal area on the left. This is an area which in other studies has been shown to be active in a variety of verbal memory tasks. Tasks like, "I'll give you a noun, you think of an appropriate verb: knife-cut", those kinds of tasks show activation. So, this is not a primary language area but it's an area involved in language and verbal memory that's more active in that task. The final study that we did used the same two tasks but it asks, "are we seeing these differences" (fMRI temporal resolution is terrible, so we don't know when these differences are taking place) do they have to do with stuff that's happening at the decision stage, or do they have to do with the actual initial encoding of these numbers? A better method for asking that is to shift from fMRI to event related potentials, where you get not nearly as good spatial resolution but you get very good temporal resolution. What Stan's data show here are the patterns of activation that you see between two and three hundred milliseconds after the onset of a problem. The general spatial patterns is kind of a noisy version of what we saw in fMRI, the left lateralized greater activation in blue is for the exact task and the bilateral activation [in the] parietal areas is for the approximate task, but the fact that we're seeing this less than three hundred milliseconds after problem onset, means we're seeing it well before the answers ever appear, so well before the subject is trying to decide which of these two answers is the right one or which one is closest to the right answer. This suggests that very early on in initially encoding these things, we can look at these Arabic numerals we can either activate a language specific representation of exact numerosity, or a language independent activation of approximate numerical magnitudes. So I think those go together. It's very preliminary work, obviously there's a great deal more to be done if we want to understand how the brain does arithmetic, but I think it's an initial piece of evidence that goes along with the behavioral data from adults and the developmental data as well. Thanks for asking me about it.
questioner
I wonder why you talk about approximate numerosity. Most of the experiments, all of them except for the last one for the Russian, seems just about approximate size. If I compare a cup of sugar with a tablespoon of sugar I'm not in any way approximating the numerosity of the grains of sugar.
Spelke
Size is controlled. There's a sense in which approximate numerosity does behave other spatial dimensions do, but these specific experiments that are tapping representation of numerosity control for size. So, for example in the stuff that Fei and I did, the arrays with the eight things in them were twice as big as the arrays with sixteen things in them during the initial presentation, so it's not a cup of sugar versus a tablespoon of sugar. We can talk about it more later.
Aravind Joshi
Referring to language, the only thing you used was composition, but composition is required for other kinds of activities, spatial tasks and so on. In what sense...
Spelke
I used two properties of the language faculty. One is that the syntax-semantics are compositional. The second is that the lexicon is domain general, and I need both of them. There may be other domains, like purely within the spatial domain, in fact I'm sure there are, where combinatorial processes take place, but I would argue that in those cases they're taking place within relatively modular systems of representation. Other animals, I would expect to be able to do those, I would expect them to look good at computing different spatial relationships. What I think is distinctive about us is that we can combine anything with anything else. I was just talking about number concepts today, but if you look at human reasoning. We can combine thoughts about numbers with thoughts about grandmothers. I can say, "my grandmother's 104 years old" or "she loves the number thirteen"; we can combine anything with anything else and that requires not only compositionality, it requires a domain general system of representation and I think that the lexicon of the language provides us with something very close to that. For sure there are constraints on what can be lexicalized, but they aren't constraints that obey the limits on any of the other cognitive domains like the spatial domain. We can say "left of the long thing" but we also say "left of the blue thing" or "left of the prime number" or "left of" anything that we can develop a word or an expression for.
