Wisconsin Public Television University Place  Experimental Epistemology  Edward Hubbard 3/13/13 UP848  cc >> Good evening, and welcome to Wednesday Nite at the Laboratory. I am Dave Nelson from biochemistry here standing in for Tom Zinnen, who usually introduces these programs. Tom is from the Biotechnology Center at the University of Wisconsin. And I want, on behalf of the Biotech Center and of its partners in this program, Wisconsin Public Television, the Science Alliance--  I forgot the other one. [LAUGHTER] >> Alumni. >> UW Alumni, yeah. The Wisconsin Alumni Association, yes, to welcome you here and thank you for coming tonight. It's my pleasure to introduce Edward Hubbard of the Department of Educational Psychology and the Waisman Center at UW Madison, a new faculty member. Ed is director of the newly developed educational neurosciences laboratory, and his research interest is in the boundary between psychology and neurosciences, about the physical changes that take place in the brains of people, children I guess I most cases that he studies, as they learn. A separate and also interesting approach is the question of the effect of multiple inputs, sensory inputs, taste, touch, sound, and sight, in the process of learning. 

Some of you who have read a book called The Man Who Tasted Shapes know about the process of synesthesia. This is a research interest of Ed's that I guess we won't hear about tonight. The tools he uses are the tools of physics and biology, functional magnetic resonance imaging and electroencephalography, and we'll hear some of the reports of what he's able to do with those tonight. Ed joined the faculty in 2012. He did his undergraduate work at Berkeley in psychology. He took his graduate degree at UC San Diego, also in psychology. Then he spent a postdoctoral period in France where he explored the use of these techniques, fMRI and electroencephalography, to study learning. And then before he came to Madison he stopped for a bit in Vanderbilt where he continued to use these tools to look at the effects of learning math by small children on their brains. He's teaching a course this semester called Mind, Brain, and Education. It sounds interesting to me. 

And his intriguing title, as you see, is Experimental Epistemology: 'Twas the Night Before Pi Day. It's a pleasure to welcome Ed. [APPLAUSE]  >> Thanks a lot. I'm really excited to be here and to have the opportunity to share with you guys some of the work that I'm doing about math and number and to be able to provide a little bit of context for the links between education and neuroscience that we're exploring in the educational neuroscience lab. And I'm especially happy that I get to do this on, it would be even perhaps better tomorrow, 3/14, on Pi Day. That's also, it turns out, Einstein's birthday. So I will try and make sure that there are some references to both Pi Day and Einstein throughout the talk. And to begin with, I just have a little bit of humor for you guys. Some of you may already know this. For some people this may be new. 

But this is a sentence. And I can see some people who do indeed get my humor. For those of you that are struggling to get this, the square root of negative one is I, two cubed is eight, this is the sum symbol, and obviously pi. So, in honor of Pi Day tomorrow, I ate some pie. Now that I've explained it, some of you agree that it's very little humor. [LAUGHTER] But it actually leads to, I think, a very important question, a fundamental question, and this question was put very nicely by Warren McCulloch, who is one of the founding fathers of cybernetics, the idea that you can take our understanding of brains, our understanding of nervous systems, and apply that understanding to questions about robots, to questions about learning in computational systems. And he said one of the questions that drove him throughout his career was, what is a number that a man may know it and a man that he may know a number? And one of the things that he advocated for was this notion of experimental epistemology. And, of course, everybody wants to know, what is that thing? And so I'll tell you a little bit. So the first thing is that epistemology is the study of knowledge, how it's acquired, and the limits of that knowledge. It's a philosophical question that goes all the way back to the earliest philosophers. 

So, Plato and Aristotle argued and debated about these types of questions. How can we know anything about the world and how can we have certainty? But in the 20th century, especially in the middle of the 20th century, scientists started to realize that they could make empirical progress on these questions that had been tied up with philosophy for over 2,000 years. And nowadays, lots of different disciplines are contributing to this experimental study of these philosophical questions. Very few people typically frame what they're doing in that sort of term, but people that are interested in ethology and animal behavior are actually doing work that is relevant to this question of what are the foundations of human knowledge. Anthropology, of course, by comparing different ways of knowing the world. Developmental psychology, more my own field, neuroscience, education, lots of other fields are contributing to this exploration of experimental epistemology. We're really trying to get an understanding of what the basis of human knowledge is. And for me, I'll tell you guys a little bit today about what we know about how humans come to have an understanding of number. But before I move on to the empirical work, I just want to sort of preface the rest of what I'm going to say by including a nice quote from Albert Einstein about epistemology. He says the reciprocal relationship of epistemology and science is of a noteworthy kind. 

They are dependent on each other. Epistemology, this sort of philosophical question, without contact with science becomes an empty scheme. Science without epistemology is, insofar as it is thinkable at all, primitive and muddled. So, Einstein recognized the importance of these sorts of links long before I was even born. But if we take that seriously, if that's our starting point, the links between science and knowledge, one question we might ask is, what do we know about the development of number? What do we know about even the cultural evolution of number? And so if we look, we can see that the earliest number representations, things like marks inscribed on a bone to keep track of, for example, they think this might be the number of days in a month, like a lunar cycle, because there are 28 and they repeat. We also see that here. Those sorts of systems are new, relatively speaking, relative to human brain evolution, relative to our existence as a species. So, math and number, in a formal sense, are these brand new things. And yet how then, if they are these new things, do they end up finding a location in the brain? How do they end up being something that all of us are able to do and acquire? Within just a couple of millennia from these earliest marks, we see the invention of very clear numerical systems, we see the development of geometry, and then things like even the modern proof of Fermat's last theorem. So, within just generations, we see this dramatic explosion of mathematical knowledge and mathematical understanding and how does that happen. 

And one idea that many of the scientists in the field have started to converge around is this notion that we humans come into the world equipped with a system that helps us to make sense of quantity. That system is typically referred to as the approximate number system. And this basic system really sort of works in a course way of just being able to roughly say there's more over there, there's less over there. And it's present even without education. So we can see the signature of this system in infants. We can see it in non-human animals. We can see it even in people who have the limited number vocabulary. So this tells us it doesn't depend on the acquisition of language but language probably augments those abilities in lots of interesting ways. And I'll show you guys just a couple of different ways we've been able to demonstrate the existence of this approximate number system. So the first one, this comes from Liz Brannon at Duke University, and she was kind enough to post this on Vimeo. 

And so here's the video. What you're going to see, you see this little boy here in this highchair, and at the center of the screen, first there's something to draw his attention, this sort of pinwheel. And then on the left you'll see displays of dots, alternating between eight dots and 16 dots. On the right, there's eight dots just repeating, but those dots change in size, they change in position, all these sorts of things. And then we can just see, what does the infant prefer to look at? So now you see his attention has been drawn. He's clearly attending. Here you've got 16, 8, 16, 8. Here you've got 8. He's looking back and forth. He's checking them out. 

Now, well that one's interesting because that one changes. The other one, that doesn't change so much. So you notice he's not looking at the other one at all anymore. He's just staring at the one that's interesting, the one that has the change of number. After a while, he double checks, looks to the other side, nope. And now he's staring at the one that's interesting, and you can see he's really actually even enjoying this. That's kind of astonishing. Look at that. It changes.  He double checks again. 

Nope. That one's not interesting. And back he goes. And so that's the end of one trial. And so you can see how researchers who are interested in these types of questions can drive really powerful evidence that infants have this sensitivity to various different quantities. Even in the absence of language, even in the absence of being able to ask an infant, well, what are you experiencing, what are do you know, we can just use their eye movements and we can use their behavior. Yeah? >> How do they know that it was quantity and not just variety? >> Yeah, so one of the things that they do across a lot of these studies is you change, for example, the size of the dots. So one concern is, let's say I've got eight dots and then I've got 16 dots. If I keep the dot size the same, of necessity, there's going to be twice as much surface area in 16 dots versus eight dots. And so, the fact that the infant can pick up on that change in number when we control for area helps to tell us its number. 

But if they couldn't discriminate that, if they couldn't recognize that there was that difference, then there wouldn't be a preference for that change. So, in both cases, both displays, there are things that are changing. The position of the dots changes, the size of the dots changes, but it's really only one that has a change of number. And so, being able to say, hey, it's that change in number that drives the attention. And I'll show you, actually, one other way that we can sort of be sure that it's number and also give you some sense of the limits of this sort of system. So there -- and her colleagues, she's a French researcher but this was when she was at Harvard for a while, they asked whether or not you could get this same sort of behavior across different sensory modalities. So here infants are presented with these displays of a particular number of shapes, and there's also, I didn't check for audio so I may just have to use my microphone, there's also voices that are producing a certain number of sounds. Let's see.  [VOCAL SOUND] Can you guys hear that? Okay. [VOCAL SOUND] So what you hear is that there are four sounds being produced and four numbers being shown. 

[VOCAL SOUND] And then they switch. So now there's 12 dots being shown and there's four sounds. And so the question is, will an infant look longer when it matches across vision and audition versus when they don't match across vision and audition? [VOCAL SOUND]  Let me just reattach the microphone here. So what we can see in this data which is really nice is infants will look at the congruent situation, so where the number  of voices and the number of shapes match, longer than at the incongruent situation, but that's only true for these very young infants when there's a three to one difference. So you see four versus 12, there's that three to one ratio. Or six versus 18, there's that three to one ratio. But here, when it's a four to eight difference, a two to one ratio, they don't notice that difference as much. They don't seem to be sensitive to that ratio, that congruence between the voices and the number of objects that are out there. Now, these are very young infants. These are actually tested in the maternity ward. 

So within about two days of being born, and we see this three to one ratio. By six months of age, you see that infants are sensitive to a two to one ratio. So six-month infants would be sensitive to this four versus eight. And by the time they get to eight months, they're even sensitive to, for example, six versus eight. So they become more and more sensitive. Adults typically can do about 10 versus 11 or nine versus 10. So we're really sensitive to these very subtle differences, but that's still an approximate system. Let me show you one more video. This comes from Jessica Cantlon and Liz Brannon, and what they did here is they had monkeys that they had trained to do basic arithmetic problems. So they showed them six dots, and then they showed them four dots going away, and the monkey just has to press the one that's correct. 

So you see one dot, another dot coming in, and the monkey does two. Ten plus two, 10 dots, 12, they picked 12. Correct. One plus seven, one dot, seven dots, go in. Picks eight. One plus three. And four. And you can see the monkey is very good and is also very fast. So, one of the things, the monkey gets a little food reward, either a peanut or a juice reward, for each one that they get right. So obviously they're motivated to get it right. 

But what we can see is, as we make the problem harder, as we go from really having the correct answer be very different from the incorrect answer, both humans, so humans are not allowed to count in this situation, humans and monkeys both show a similar fall off in performance. As the problems get harder, monkeys start to do worse, humans start to do worse. And if we look at the amount of time that it takes humans and monkeys to do this, you can see that as the problems get harder, both humans and monkeys take longer when it gets harder. So they do worse and they take longer. So this suggests that even our ability to do some basic arithmetic depends on the same sorts of abilities that we see in monkeys. Yeah, jump in. >> How old are these monkeys? >> These are adults. These are actually Duke undergraduates. [LAUGHTER]  So, I'll leave aside any sorts of upcoming March Madness comments. But, given that monkeys can do these sorts of things, then it becomes an interesting question to ask about their brains and to ask, what's going on in the monkey brain when they're doing basic number things? So, Andreas Nieder and his group in Germany asked exactly that question, and in order to investigate this, they set up a simple delayed match to sample task. 

So a monkey sees a fixation point, then they see, say for example, four dots, then they have a delay and they have to remember how many dots did they see during this delay period. Then they're presented with one of two different displays. Half the time they're presented with a matching display, so four dots, and you can see the dots are different sizes, different position, they release the lever, they get a juice reward. If it doesn't match, then they have to hold the lever for that intervening second and then release it when the matching stimulus eventually appears. And then after the matched stimulus, when they release, then they get the juice reward. And what you can see here, this gray shaded area is during this memory period. So there's nothing actually being presented on the screen, and what you can see is that there are neurons in this part of the brain right in here that we call the intraparietal sulcus. So it's the fold that divides the parietal lobe into the inferior and superior portion. So within the intraparietal sulcus, especially down in the depths of the intraparietal sulcus, we see these neurons in the monkey brain that respond most strongly to a particular number. So, here's a neuron, for example, that responds really strongly to four objects. 

It responds a little bit less to five, also a little bit less to three, even less than that to one. And so what we can do from this data is we can show a sort of tuning curve where you get low responses to one, higher responses to two, three, four, and then this drop off for five. And one of the things that they do is, for example, I mentioned this already in response to your question, they control the area. So on some of the trials, the area of the dots is constant. Whether it's three dots or four dots, they're all the same size, and that means area is going to go up with the total number. On other trials, they make the dots smaller as you have more dots, and therefore the area is constant across all of the number of objects that are presented. And they show that there's no difference in how the neuron responds whether you've been presented with the standard or with the equal area condition. And then from this, they're actually able to show across a large number of neurons that there's kind of this bank of neurons that are tuned to individual numerosity. So there's some neurons that like five, there's other ones that like four, and one of the things that you can see is that those tuning whits increase as the number of objects go up. So this is this approximate ability to just understand, roughly, the number of things that are out there. 

But these neurons are not exact. And we can finally look at this by seeing groups of people. So this is a study that was done by Pierre Pica who is a linguist, Stan Dehaene who is a cognitive neuroscientist, he went down to the jungles of Brazil and actually talked with these people, the Munduruku, who have a fairly limited number vocabulary. So, they talk about one, and if you show them one object, they will typically use their word, pugma, for one. If you show them two, they're really good, xep xep, for two. But if you start to show them even three, it's more like the way we use the word "a few" in English. It's not precise. Beyond that, you go to four, you go to five, they seem to have words for a bunch and maybe a word for a whole bunch, but they don't have words for like 435,211. They don't have things like pi in their language. So, clearly, there are certain things that we're able to do with our number system that they're not able to do. 

But interestingly, on simple tasks of non-symbolic arithmetic, so very similar to what you saw the monkeys do, the Munduruku perform just like Harvard undergraduates. So, again, we see the ability to do some of these basic number things doesn't depend on language. Infants can do it, animals can do it, even these people with a very restricted number vocabulary can do it. That's all well and good, but eventually, if I'm really interested in the number... Yeah, go ahead. >> [INAUDIBLE]  >> Yeah. >> [INAUDIBLE] >> Well, one of the things is the humans, after a little bit of training, get pretty quick too. So that's part of it. If you just have one of us start out, we're always going to be a little slower, and then over a period of time you can show that the humans start to speed up. And you actually see that very nicely in this data. 

So you can see here, this is the response time. So how quickly do you respond on individual trials. And the humans in red here, they're a little bit faster when it's easy and maybe a little bit slower when it's hard, but the basic finding that we both slow down, both humans and monkeys slow down when it gets harder is really striking. And the other thing is that you can see these two lines basically converge. So, for the slower trials, humans and monkeys are almost identical in terms of how long it takes them. And then you can also see these error bars. So, across different monkeys, some of them seem to have different tradeoffs. Some of them are more willing to get a few wrong to be able to get that juice reward when they get it right and be able to just do more trials. Other monkeys seem to be slightly more cautious, and they want to do it correctly and they're not so worried about if they get a bunch of trials done. They just want that juice reward every time. 

You might think that many of our students here are going to have that same sort of tradeoff. Do I want to get it done quickly or do I want to get it right? So, I don't think this is unique to the monkeys, but we can definitely see these sorts of individual differences. So, the question that I really want to get at, and this is a huge question, is how do we go from this, a sort of large primate brain that is able to discriminate approximate quantities to be able to take sets of dots like this and say that's more than that even without counting them, to a brain that is able to do things like integral calculus? And, strikingly, in this fMRI study, so here's the front of the human brain, the front of the monkey brain, here's these prefrontal region, and this intraparietal sulcus region where they found neurons that were sensitive to quantity. It's these same systems in prefrontal cortex and intraparietal cortex in the human even when you're doing calculus. So that already tells us something powerful, that the same brain systems are being recruited all the way from simple numerical things, non-symbolic quantity, on up to things like integral calculus. But, if we want to get from dots to calculus, we're going to have to break that problem down into smaller chunks. And so, one of the things that I've been exploring is how do we go from this approximate number system to an exact system? And this really requires a process of inculturation, a process of education that children are going through. And we see, for example, even in children from low SES backgrounds, they, at the beginning of kindergarten, often haven't acquired the ability to count and to use exact numbers. So we can see very clearly the impact of the environment on using these sorts of skills. But how does this happen at a brain level? I'm going to go ahead and skip that. 

But I can't record from individual neurons in a child's brain. The IRB won't let me. Their parents generally are a little uncomfortable with this idea. So, how do I do this? How do I get at neurons in a child's brain? And one of the ways that fMRI researchers have come up with is this adaptation method. So we've seen that there are these neurons that respond most to a particular quantity. So, for example, here you've got a neuron that responds most to 16. Interestingly, if I present the same stimulus over and over again in the same way that we did for the infant habituation studies. If I present the same stimulus over and over again, neurons in the brain start to respond less and less. You get this habituation or adaptation. And the neurons that respond most strongly are the ones that are going to adapt the most. 

So this neuron responds a lot to 16 dots, and so it adapts. It responds less strongly to the second presentation, to the third presentation, and to subsequent presentations of 16 dots. But a neuron that responded, for example, most strongly to 20, only responds a little bit when you show it 16. And so it adapts a little bit. And a neuron that responds most to 24, doesn't respond at all to 16, and so it doesn't adapt at all. So by repeatedly showing the same number of objects and by changing the position, changing the size, and then occasionally showing other numbers of objects, we can see how much does the brain respond to eight after it's seen a lot of sixteens. How much does the brain respond to 16 after it's seen a lot of sixteens. How much does the brain respond to, say, 24 after it's seen a lot of sixteens. And in this way, we can actually pull out a measure of what's going on in the brain, how are these neurons responding. And in a study that Manuela Piazza did in Paris in 2004, she showed, in fact, that it's the same intraparietal sulcus region. 

So you can see here, on the outside of the brain, here's the parietal cortex at the back of the brain, those brain regions are the ones that respond more to when a number changes. And critically, one of the things that she did to make sure, again, it's not just this preference for things that change, is on some trials she had the same number of dots but instead of having circles, she showed people triangles. And different brain regions, mostly down here in the temporal lobe that care about object shape and also a little bit more sort of up here in superior parietal cortex, showed responses to the shape change, but it was different regions from the brain regions that responded to the number change. So this really gives us some purchase on the number circuits in parietal cortex in the human brain. So we developed a child-friendly version of this. What we did is we presented, say, six dots. Six dots, six dots, six dots, six dots. The kids get bored. Their brains habituate. And then we presented them with either nine or five. 

And nine is relatively far from six. So we expected big brain response after you've seen a lot of sixes. Whereas, five is close to six so we'd expect a small brain response. And then in other runs, we told the children that they were helping Sally the Scientist and her team get back to Earth by finding planets. In other runs of the Sally the Scientist game, we presented them eight dots, eight dots, eight dots, eight dots, and then again we presented them five and nine. But now, after seeing a bunch of eights, we'd expect a large response to five and a small response to nine if their brains have built up these links between quantities and symbols. And these are all the children that we tested. You can see that they were between about six and nine. We also had a group of adults. These standardized scores, we typically expect them to be about a hundred with a standard deviation of 15, and you can see that across their math abilities, across math skills, and reading, these children were pretty typically developing if anything, especially in the reading you're starting to see, like these third graders, second graders, are performing slightly above average. 

So, these children are not learning impaired or have any other sort of difficulties. They're having great opportunities to learn, and so we can measure how their brain responds to that learning experience. And the first thing we found is just as we expected. We do get bigger responses to, for example, five as a digit after you've seen a lot of eights, and small responses to five as a digit after you've seen a lot of sixes. And you get big responses to nine after you've seen a lot of sixes, but small responses to nine after you've seen a lot of eights. So this is exactly what we've predicted. This crossover interaction. And this tells us, in fact, that their brains have already started to make the link between the number of dots out in the world and the number symbol. I saw your hand, yeah? Okay. If you want to interrupt, don't hesitate. 

So we see this mostly for the digits, actually. We saw that less when we did the same things with dots, and we were sort of surprised at that. But the finding for the digits was really striking. And, critically, we see it in the intraparietal sulcus, so in that same brain region that we were already interested in for math and number. Brain imagers have this very unpleasant habit of putting the left side of their brain on the right side of the picture. So this is actually the left side of the brain, the left intraparietal sulcus. I could tell you guys all about why we do that to ourselves, but just know that we make things hard on ourselves. But then we look to see how does this develop? How does this change? And what we were sort of surprised to see was that the biggest effect was actually for the youngest children and that the effect got smaller in the first and second graders and then it was present again a little bit for the third graders. So there might be a sort of U-shaped developmental trajectory, that this might be something that becomes easier and then the children, as they become really proficient, start to build up specialized neural circuits to do this. So we started to explore directly what the developmental trajectory looked like. 

And we were interested in this question of precision. So, you guys have seen this sort of logic of adaptation here, where you get this sort of process of adapting the neurons that have been responsive most to the adapted quantity, and then you do the test numbers, and then you get out a curve like this. But, for children who now have a narrower tuning function, who are better able to discriminate the numbers, what we'd expect is that the adaptation curve would also be narrower. It would be more precise. And so, here you're seeing what the curve might look like if a child was relatively insensitive to those differences and what the curve might look like if a child was more sensitive. And we know from lots of developmental studies that as children get older they become more sensitive to these subtle differences in quantity. So we developed a specific model of how changes in sensitivity to that non-symbolic quantity would impact that results that we were seeing. And what we found, once we worked through all of this mathematically, is that in fact we would expect for children who are insensitive, so the blue curve, we would actually get a very large distance effect. So the response to five would be very small; whereas, the response to nine would be relatively large. For children who were more precise, we'd actually get a smaller distance effect. 

So the difference between the response to five and the response to nine would become smaller. And that sort of fits with what we already saw. So that allowed us then to make two additional predictions. One is that we would see that most of the action is in how the brain responds to the close deviants. So we'd expect a small response for children who were insensitive and a relatively large response for children who were sensitive. Whereas, the far deviants, we'd expect not much of a change. The other thing that we would predict is that on average we'd get a bigger response in children who were sensitive. You can see these two red lines. If we take the average of those, this would be the response. Whereas, if you look at the average of these two blue lines, you're going to get a smaller response. 

So we were able to make some very specific quantitative predictions about how that should respond. And that's in fact just what we saw. So, now we're looking at the brain here with a particular region and parietal cortex that was defined based on previous literature. We knew that we were interested in this. We also knew that we were interested in these visual areas down here in the temporal lobe and the fusiform gyrus because those are the brain regions that actually recognize the squiggly lines. They know that this particular squiggly line is six or that this particular squiggly line is five. So the fusiform is responsible for sort of reading. But then you still have to link that up with the meaning. What does that squiggly line mean. And so, we were interested in the process of recognizing and the process of understanding these number symbols. 

And what we can see is that, in fact, there's a nice response in these fusiform regions, these blue bars, but in the youngest children there isn't a response on average to the meaning of those numbers. So, on average, we see a small response, but we also get a big distance effect. Just what we had predicted based on this model. As children get older, interestingly, you see that this starts to change. So in first graders, we see that now they get a big response to five but not a big response to nine on average in these parietal regions that care about number magnitude. And one way to think about this is that children start learning to count with one, two, three, four, five. These children have been five. They know what five is. They don't yet really understand what nine is. They can count to nine. 

They might even be able to count to 30 or 40 or 100, but they don't really seem to understand what nine is in that way. And we can see that reflected in the pattern of brain activity here. Then you can see, for second graders we start to see more of a response, and then finally for third graders we're seeing robust responses in these parietal areas to the number symbols, whether it be a five or a nine. So that fits with exactly what we had predicted. The other thing that we saw is that distance effect, the difference between the close and the far actually changed with development in these intraparietal sulcus regions here. So you can see in this blue area. But when we look at this carefully, it actually turns out that a better predictor, a stronger predictor of the change in brain function is children's math abilities versus their chronological age. Now, those two things go along together. Children of course get better at math as they get older, but when we directly contrast these two, we can see that math abilities are better at explaining the change that we're seeing in the brain than just chronological age. So this tells us that what we're really observing here is the signature of educational experiences changing these parietal brain areas that care about math and number. 

And we see this also when we look just in the youngest children, in the kindergarten and first graders, and those are the ages where we'd expect the most intense change between not knowing your number symbols and knowing your number symbols. And we see that actually this relationship is even stronger in that small group. Even though there's fewer children in that group of course, because it's just our kindergartens and first graders, the relationship is stronger. Yeah? Jump in. >> Have you or anybody looked at this type of work in relation to children who have been in Montessori schools? >> We haven't done this with Montessori children in particular. These children were all children that were in private schools in the Nashville area, and part of the rationale for that is that the school district would allow us to come in and also observe what was happening in the classrooms. So we actually have data on what the teachers were doing in the classroom, and we're still working on analyzing that, linking that up. But we also asked the parents of these children how much preschool experience do they have, how much daycare experience did they have, how much did they read to them at home. And we're still also making sense of all of those relationships between preschool experience, the in-class experiences, and the changes that we're seeing in their brains. But these are exactly the questions that we think are critical to really explore. 

If we think that we're seeing the signature of education on the brain, then we need to think seriously about what those educational experiences are, and we're still working through that even with this data set. That's also part of why I'm here doing exactly the work that I'm doing now.  Just looking at the time, I'm not going to be able to get through all of this. So, let me jump to one more concern. So, so far we've got a system now that can allow us to approximate quantities, and we add number symbols to it. But all we've gotten, if we think about tomorrow, we think about Pi Day, all we've really gotten is three. We don't even have 3.1 yet, much less 3.14159 and so on. All right? So we still have a long way to go if we really want to understand the origins of the full complement of human numerical abilities. And I'm currently starting a project with a number faculty member in Ed Psych, Percival Matthews. We're interested in questions about fractions. 

How is it that children begin to understand fractions? And we know that children typically begin instruction in this in third and fourth grade. We know that lots of children struggle to learn fractions. Why do they struggle so mightily to learn fractions? One idea is that the human brain actually is poorly adapted to understand fractions. But another idea is that teaching methods and the particular notation that we use might itself not be very well adapted for trying to understand ratios and teach fractions. So, one piece of evidence that makes us think that the second story, that it's something about the way we're teaching fractions, might be more correct is that it turns out that monkeys can do certain basic proportion and ratio judgments. So, here you can see Andreas Nieder's group again, this is another study from his group in Germany, they had monkeys do another version of this delayed match to sample task, but here they presented them with two sets of lines that were in specific rations. So, for example, one-fourth, so the bottom line is one-fourth as long as the top line. Or the same length, fourth-fourths or three-fourths. And then here they were able, they introduced a number of controls where, for example, the reference is either the same length or the test line is the same length, and what they were able to show, first, is that monkeys were indeed able to do it. And, in fact, if you have humans do it, you have monkeys do it, their performance is almost identical across these ratios. 

So, human abilities and monkeys' abilities again show this sort of similarity. And then they recorded from neurons in, again, the intraparietal sulcus in these same brain areas in monkeys, and they were able to find, again, that there are certain neurons in parietal cortex that care about these proportions. So, for example, some of the neurons respond most strongly when you're presented with a ratio of one to four and less strongly when you go to two to four, three to four, and four to four. Other ones respond most strongly to, say, two to four, so one-half. And still other ones respond most strongly to, say, three to four, so three-fourths. So this already tells us that human fraction understanding builds on something that we share with our non-human primate relatives. And then they asked, what about in adults? What if I show you, for example, using that same adaptation paradigm but now I'm showing you roughly one-sixth but in different forms of the fractions so the actual numbers are different. Five over 30, two over 12, four over 24. I adapt you to one-sixth. Or I show you, for example, in this case, what is this, two-thirds, right? Or, sorry, yeah two-thirds, one-half, and then you're presented with the German words, because they're in Germany, you're presented with the German words for one-half or one-third. 

I won't go into all of the details, but the key finding is that indeed when you do this sort of thing, you get these intraparietal sulcus regions that are going to respond more strongly to a difference in the numerical fractions. So, adults have built up these brain circuits that allow them to make the link between, say, five-thirtieths and two-twelves, four-twenty-fourths, but how do they get there? We know monkeys can do proportions, we know humans can make these symbolic fractions, but there's this huge gap, and nobody has done a single study investigating that gap at a neuroscience level. So this is a project that we're just now starting to try and see how do we go from monkey abilities to do proportions to adult human abilities to deal with symbolic fractions. And we're hoping that that helps us to get from three to 3.1 to maybe even 3.14. I'm still not sure if it gets us to 3.14159, but at least it gets us to some of the brain bases of these concepts. I'm going to, again, go quickly through one idea, which is this notion of the mental number line. So people also learn to link numerical magnitudes with spatial locations. And, in western cultures, we tend to associate the left side of the line with small numbers and the right side of the line with large numbers. And we think that this depends on those same frontal parietal networks, those same regions that are involved in number are also, it turns out, involved in space. And we can see these links between numbers and space in just one example here, this behavioral task. 

So I just ask people to tell me if a number is odd or even. And in one run, I asked them to respond to the odd numbers with their left hand and the even numbers with their right hand. The next time I say now I want you to instead respond to the even numbers with your left hand and the odd numbers with your right hand. What they don't know is I actually don't care about whether it's odd or even, I care about how big the number is. I care about whether the number is one or two or three. And what you see here is that people are faster to respond to a small number with a response on the left side of space. So here we're just showing the difference in the reaction time, the amount of time it takes you to respond with your right hand or your left hand. And if you're slower with your right hand, say it takes you 550 milliseconds, and with your left hand it takes you 520 milliseconds, then this difference will be positive. If, on the other hand, you're faster with your right hand, then this difference will be negative. And we can see this relationship. 

One really sort of nice thing, we can show that it's space that counts and not your hands just by having you cross your hands. Now you're faster to respond to small numbers with your right hand on the left side of space, and you're faster to respond to large numbers with your left hand on the right side of space. I can also have you do it, for example, by pressing foot pedals. I can have you do it by moving your eyes in space. So this tells us that this link between numbers and space doesn't depend necessarily on things like typing experience. It really seems to be something about the abstract understanding of space and number. And when we start to look at brain circuits that are involved in number, we see that these brain circuits here are also intermingled with brain circuits that are involved in moving your eyes around in the world and grasping. And so when we think about the ways that we teach children number, one way we teach children number is to use their fingers, to count on their fingers, another is this number line. And so part of what we think education is doing is it's taking that system that we inherit from our primate ancestry, that sort of basic approximate number system, and it's linking it up with all of these other representations. It's linking it up with fingers. 

It's linking it up with space. It's linking it up with language so that we can do so much more with numbers than what we would have been able to do if we just had this approximate number system. And so when we look at this, I'm, again, going to jump through this, when we look at brain activity patterns here, we can actually see. So, here in red, I've plotted from a study that we did in Paris. These brain activities in red are what we see when you're doing calculation, when you're actually doing basic subtraction. The blue is when you're moving your eyes. And critically, we can see, both in the front of the brain, calculation and eye movement overlap, and in the back of the brain. So in frontal cortex and in parietal cortex we see this overlap between eye movement and calculation. And so this suggests to us that some of the same brain systems that are being used for moving your eyes in space, for paying attention to the world around us, are also being used when we're making sense of numbers. And I'll show you guys just one example of this, one test that we did. 

So here we had people do this basic non-symbolic arithmetic task. So on some trials we told them that they were going to add. So they were told to add by seeing the letter A, and they were shown this array of dots for three-tenths of a second, so not long enough to count up the number of dots there. Then they were told again, don't forget you're adding three-tenths of a second, then they're again reminded they're going to add, and here they're presented with one of the responses, and they have to pick the set of dots that mostly closely corresponds to the sum of those dots. Now remember, we saw that monkeys could do something like this. We saw that college students could do something like this, but this particular version is pretty hard. So performance was not great, but it was not as if they were completely randomly guessing either. And then critically, we also recorded brain activity when people were moving their eyes. And we looked, in particular, in these areas in the back of the brain. And we trained up one of these statistical learning algorithms. 

So we took a program that was fed in a bunch of brain activity data, and that brain activity data was then used to classify eye movements as either did the person move their eyes to the left or did they move their eyes to the right. And the algorithm was able to do this. This is not a surprise. You can see it does it pretty well. Anything above zero is good. Anything above one is really good. So about half the subjects it's really good, for the other half of the subjects it's only good. But then we asked, once you've got that same statistical learning machine, can you then feed it in brain activity when people are doing arithmetic. And if we think about this mental number line idea, if you're adding, it should be like moving your eyes to the left, and if you're subtracting, it should be like moving your eyes to the right. And we were able to show that people weren't moving their eyes, but the brain activity was similar to as if they had been moving their eyes. 

So we see that you get successful classification of these arithmetic operations where leftward saccades and subtraction kind of go together, rightward saccades and addition kind of go together. Now, you can see that the effect is weaker. So it's clear that it's not like you're really moving your eyes, but it's kind of implicitly, internally like you're moving your attention around in the world. So this suggests that the same brain circuits for space are also being used, in part, when we're doing basic calculation. Let me skip to one really nice finding here. This is not our work. This is work that was done by Tang and their colleagues in China and in the United States. They had people do some basic arithmetic. They had them just look at symbols that were not numbers, then look at number symbols, do addition or do comparisons. So just tell me which of these two numbers is larger. 

So, they're presented with, for example, seven and five. They press seven is the larger number. They have the same thing where they had native Chinese speakers do it. And what's interesting, so I've been telling you about brain circuits like the intraparietal sulcus that are shared between monkeys and humans, but there also these different brain regions that are recruited for different humans whether you're a native English speaker or a native Chinese speaker. So for the English speakers, when they were asked to do this, we see language areas, Broca's area and Wernicke's area in the left hemisphere of the brain are more active for the English speakers than for the Chinese speakers. And as the task gets harder and harder, you see the difference between the English speakers in the white bar and the Chinese speakers in the purple bar becomes more and more pronounced. Conversely, we see greater activity in a pre-motor area, an area that's involved in moving your fingers, in the native Chinese speakers. And this might be related to, for example, abacus experience. And, again, just like what we saw before, as the task gets more difficult, the difference between the Chinese speakers and the English speakers becomes larger. So, one of the things that we can see from these types of studies is how different educational experiences start to impact the brain. 

So going back to your question  about, for example, Montessori, we might actually be able to identify, depending on different types of instruction, different ways that people are using these brain circuits to accomplish the same goal. And I just want to finish with one of the potential implications of this work. What about children that have real difficulties learning math? So, everybody's heard of dyslexia. There's a sort of math version of dyslexia called dyscalculia. Think of calculation. So this difficulty with math affects children even, say for example, when they're 10 years old. And one idea that's out there, and this has been tested by a number of different groups now, is that that approximate number system, that foundation that I was telling you about before, might be impaired in these children. We know, for example, that children whose mothers drank during pregnancy often, fetal alcohol syndrome, have problems. Your children that are born with low birth weight have problems with math and number, and we can see anatomical abnormalities in these intraparietal sulcus regions. But some children, perfectly normal gestation, perfectly normal experiences, also struggle mightily with math in the same way that some children struggle to learn to read. 

And so the suggestion is that in those children maybe it's this approximate number system, maybe there's differences in the intraparietal sulcus. And if that's true, then we would expect when we test children on a measure of their sensitivity to non-symbolic quantity, what we would see is that children who are struggling to learn math, dyscalculics, in this case 10-year-old children, are going to be worse than your typically developing children. So, in this case, closer to the left is better performance. So you can see the distribution of adults that they tested, distribution of 10-year-olds, distribution of 5-year-olds, these 10-year-old dyscalculic kids will look just like typically developing 5-year-old children. So we can see that there is a difficulty in just understanding non-symbolic quantities in these children. And then, in an fMRI study, this was done by Gavin Price and Daniel Ansari up at the University of Western Ontario in Canada, they were able to show that when you asked children to compare these dot arrays, there's a small region in the intraparietal sulcus in this same core brain region where the dyscalculic children aren't recruiting this brain area, whereas the typically developing children are. So we can see this at a behavioral level, and we can also see this as a neural level. So then what do we do? And one idea that Anna Wilson, who's in New Zealand, and Stan Dehaene developed was this idea of maybe using games. Maybe games are going to be a great way to do this sort of training that needs to be done to improve this non-symbolic quantity. So, they created a game that changes in difficulty, and it focuses on three key number skills. 

One is the ability just to compare quantities and pick the larger one. So your goal is to always try and pick more so that you can win the race. You're competing, in this case, against a crab, and you want to get there as quickly as you can, so you want to pick the one with more. And the second thing that they do is build these links between the symbol itself and the quantity. So just the sorts of things that I was looking at with the fMRI study I told you about. And then finally, they link that up with that spatially organized mental number line. So just the sorts of things that I've been telling you about from a neuroscience perspective, they were inspired to build into a game to help children. And they do find that there are benefits for the children after playing this game. There are still questions about how long lasting those benefits are and whether or not those benefits transfer. But how about at the other end? Can we also see differences in the brains of people who are, for example, professional mathematicians? And here we now see increased gray matter density, so increased cortical thickness in some of these parietal areas in professional mathematicians. 

And in particular, we can even see that that difference between professional mathematicians and non-professional mathematicians, people that aren't, is actually related to the amount of time that they've spent as mathematicians in the field. So, one interpretation of this is that being a mathematician forces you to use your brain in these specific ways and that leads to changes in the structure. The other possibility, though, is that people who already had these structural differences, maybe they acquire math better, maybe they're better able to use math, and so that's part of why they then go on to have successful academic careers as mathematicians. So, how about one of the most famous of mathematicians and scientists? Does he also show differences? And Dean Falk and her colleagues down at Florida State University recently uncovered pictures of Einstein's brain that were taken shortly after he passed away. His brain was preserved, and by careful inspection of all the folds and bumps in Einstein's brain, they were able to identify several interesting differences, and they highlight them here in red. So these are in some of the parietal and occipital parietal areas. There are differences in the structure and the shape of Einstein's brain in these parietal regions that we're interested in for math and number. They also note, right here, this sort of shaded region, it's very unusual that that region typically, so you can see in the left hemisphere how there's these clear divisions between this superior and inferior parietal and the occipital lobe, here there's a region that seems to be part of the parietal lobe but should be, based on exactly where it's at, part of the occipital lobe. So there are also these sorts of differences there in Einstein's brain. And maybe, it's tough to know now, but maybe that's part of where Einstein's genius came from. 

So, just to sort of wrap up here, I'm mindful of the time, what I want to argue is that this human ability to understand number builds on these basic abilities, builds on this approximate number system that's present even in infants, even in animals, even in people that don't have all the rich number vocabulary that we do. But an exact concept of number depends on building on that approximate number system so that we're able to understand number symbols and link those with quantities. And it's not only language but it's also space, higher order reasoning I didn't even really get to touch on for example, coming up with the years and years and work that went into Andrew Wiles' proof of Fermat's last theorem. That clearly involves a lot of reasoning, a lot of staying on task, keeping all sorts of information in his mind over the years that he was working on this. But, importantly, we also see that the same cognitive processes can be linked to different brain systems depending on the types of experience you have, and, conversely, we see that differences in brain structure and differences in brain function also relate to differences in math ability. So, with that, I just want to thank all of my collaborators in Nashville and in Paris, and if you're interested, you can just search Ed Neuro Lab Madison or you can email me. Thank you all, guys. [APPLAUSE]