 And I would like to introduce our speaker as the last few folks come dribbling in Barbara Soneka PhD is going to speak to us today on how children how counting represents what children must learn and Dr. Soneka is the you're in the Cognitive Sciences Department at UCI so Barbara it's all yours I needed to get this microphone down so you could see my face Is this can you see my face? Can you hear my voice? So I'll start by telling you that there are three kinds of people Those who can count and those who can't That's my only joke That's my only joke now. I'm now. I'm a professor so get ready to be bored So what I'm gonna talk about today is the construction of number concepts And I'll start out By talking about why you might care about the construction of number concepts And then I'll give you some definitions of the terms. I'm gonna use Talk about the process of constructing number concepts, and I'm talking about in this case children children You all have number concepts. I assume but the really interesting number concept construction in terms of the numbers I'm talking about it happens between about two and four or five years Old so all the data that I'm gonna talk to you about our experiments from kids who are about two three four five years old Okay So I'll talk a little bit about the early and partial number concepts that kids Have and then the cardinality principle, which is a really big and important principle that kids have to Acquire in order to kind of understand what numbers are and then maybe one slide at the end on Future directions. All right, so first of all, why is this important? Well If you live in Society I'm gonna argue that you should care about Early childhood development. This is at this point. I'm just talking about early childhood development in general. So this is Why should you be interested in? Programs for example that would Help two-year-olds three-year-olds four-year-olds who Are poor who don't have a lot of resources who maybe are in families with just one parent? Why is this of concern to you? Because presumably you don't have two-year-olds at home anymore, right and maybe you have Grandchildren or you know your kids have kids. I guess that would be grandchildren who are two And three and four, but but you know, maybe they're fine. Maybe you're thinking they're all right. They've got Families they've got schools. They're doing fine. Why do I care about sort of why should the public care about investing in early childhood development? Well, it turns out that the cost-benefit ratio of early childhood development programs Just if you look at them as economic development programs is much better than most other economic development initiatives So for every dollar that's invested in early childhood development most economists Figure that we get about three to eight dollars back and here's how we get it back Kids who have good early childhood Development use less Special education services in school and they repeat grades less often all that stuff costs money to the school system the child welfare system is Used less by kids who had early childhood by families where there was early childhood intervention So there's just the the treatment and administrative costs of child welfare are lessened the justice system kids who have early childhood intervention so preschool are arrested less and serve less time in jail Okay Kids who have early childhood intervention are less likely to smoke less likely to have substance abuse problems and more likely to have private health insurance so Save a lot of money in the health system by intervening at early childhood, okay? And kids who have early childhood intervention are more likely to finish high school more likely to finish college And they make more money through their whole life until they're 65 or whenever we stop measuring it So they pay more taxes so if you really want to Improve things if you really if even if all you care about is the bottom line and you want to just Do some economic development in a community Really, you can't get a better deal than early childhood intervention So that's my little pitch for the importance of early childhood intervention So moving on to my research specifically why math This is my cartoon Because In a word early early math concepts early number concepts pre kindergarten math skills are the single best predictor of academic achievement through high school So if we if you measure kids when they're five years old just starting kindergarten And you look at their pre-reading skills their math skills What you know sort of number skills like can they do two plus three equals five that kind of thing Their attention skills how well they can sort of sit still and pay attention. Do they have behavior issues? Are they throwing spitballs and poking the kid next to them? Or are they you know unwilling to sit still in school all that all the stuff you can measure About a five-year-old the the thing that predicts how well the five-year-old will do all the way through high school Is their early math knowledge? Okay, even if they can't pay attention even if they're throwing spitballs Even if they don't know how to read if they have good number concepts when they go into kindergarten That's the best predictor of how well they'll do all the way through high school Which of course predicts that you know you do better in high school you do better in college and so forth, okay? Now a caveat is we don't know for sure that that Correlation isn't causation right so we don't know for sure that if you intervene and teach them better number concepts that They will do better through high school could be that the kids who are more academically inclined just pick up the number concepts When they're two three four years old, but it's worth a try So definitions When I say numbers, I'm particularly today talking about the natural numbers in other words positive integers One two three four five the regular the whole numbers starting with one going up not talking about rational numbers Fractions that negative numbers decimals nothing like that nothing fancy just the whole exact numbers one two three four five and up When I say concept I mean mental representation that goes beyond purely sensory information So some information that you have in your mind about the world That's not just something your eyes or ears will tell you not just like a sound or a color But some kind of representation so concepts have content meaning the what what it is that you're representing what the information is They have a format. There's some Way that the concept that the information is stored in your brain This probably isn't that important for this talk so you can ignore it if you're bored And a conceptual role so concepts can enter into various kinds of calculations and that's part of how we define what the concept is so for example if I have Concept of five and I know it's exactly five and I can do addition and subtraction multiplication With five then that's that that defines what that's part of what defines what the concept is right? What I can do with it can I add it to other numbers? Can I subtract it that kind of thing and then when I say construction? like the construction of a concept I'm talking about a particular type of Process called conceptual role bootstrapping. It's also called quinian bootstrapping after the philosopher quine and I'll talk a little bit about how that works So here's a quick tutorial on conceptual role bootstrapping This is a kind of learning process that happens in lots of different domains where you start with a set of placeholder symbols so You've got so the green boxes are supposed to represent Sort of stuff that's in your mind already Knowledge that you have maybe you were born with maybe you have from earlier And then the white boxes that s1 s2 s3 s4. That's s is for symbols So those are symbols symbols symbols there in some kind of structure. They're not connected to anything else You know, so you've acquired some structure. Let's say you know that s3 is connected to s1 and s2 And it's connected to s4 but s4 is not connected to s1 and s2, right? So they have structure in terms of each other. They have connections Relative to each other, but they're not connected to anything else. You know So as a very Simple example. I was driving the other day Last week with my four-year-old and he had been watching a cartoon that had pyramids and mummies and ancient Egyptians and he First he explained to me that if someone dies and you put a lot of band-aids on them, they will get up and walk around again. That's what That's what he learned from the mummies And then and and then he asked what's an ancient Egyptian? So this is an example of this kind of conceptual role bootstrapping in a very very simple example, right? He knows enough English at age four to know that ancient is a modifier of Egyptian like Egyptian is some kind of noun and ancient is something about it But that's kind of all he knows so at this point ancient Egyptian is kind of like these white Boxes with s in them right there He knows that ancient and Egyptian go together and that ancient is some kind of Egyptian and the Egyptians can be ancient But he other than that he has no idea what it could be so I could explain to him Pretty easily Now you see my little cartoon shows the the boxes are filled in and they're connected to the other now It's all greens right so now all the knowledge is connected together I could say to my son Well, Egypt is a place and people who live there are called Egyptians and ancient means a long time ago So ancient Egyptian was things that are things were people that were in Egypt a long time ago So it's pretty easy for me to connect those placeholders to stuff He already knows he knows what a place is he knows what it means to say a long time ago And so now the resulting system that he has it includes ancient Egyptian But it's connected to stuff he used to know and so he's got a he's got more Representational power now now he can represent information about ancient Egyptians. He couldn't do that before right? If I had said to him a month earlier This coin was owned by an ancient Egyptian That didn't mean anything to him right But now I say this coin was owned by an ancient Egyptian and he can think oh it was owned by someone in Egypt a long time ago So this is kind of a simple Kind of illustration of how bootstrapping would work first you acquire the symbol It doesn't mean anything then it gets filled in with meaning Okay So in the case of number there's a whole bunch of innate Conceptual cognitive systems that are relevant to number that are in your brain already some examples Parallel individuation is the ability to pay attention to multiple things at once, but it's not really a number system The big one at the bottom here is the I guess I'll try the pointing. I hate pointers the approximate number system That is a number system I'll talk about in a minute, but it's it can't do integers it can't do natural number and I'll tell you why in a second Natural language quantification. There is some quant there are some quantifiable resources in language So for example when you speak you make a distinction between singular and plural You're marking the difference between one and more than one some languages have dual markers Some languages have trial markers that mark one two three plural In order to acquire language you have to have the ability to make those distinctions between one and two and three and more than that So that's sort of a number related system and then set-based quantification is another example we Babies can track up to about three things at once But they can't track for they just forget. So if you put sort of three balls in a box and Then you take two out the baby will stick its hand in and start looking for the other one Right, which is evidence that they remember that there were three but if you put four balls in a box and You take the baby takes one out they stop looking right which is evidence that once you get to four They just go okay. I can't remember right But so that's that's the parallel individuation system. You can remember things one at a time up to about three But set-based quantification lets you chunk things lets you move things into categories and remember more So if you put two cars in the box and two bunnies in the box The baby will can remember that right they can remember two cars and two bunnies even though It's four right It's so if there were just absolutely the limit on three then they couldn't do it But if you if you divide it into groups that are smaller than three then they can they it seems they can remember One level of hierarchy cars and bunnies and within each one two cars two bunnies like that So you can so set-based quantification is another kind of cognitive resource that you can use to Kind of process number information Okay, but none of these can do Numbers right none of these can do natural number none of these can represent like 27 if I wanted to communicate somehow that there were 27 people in this room if I asked You any person to go out in the hall and tell somebody make somebody understand exactly how many people are in this room like let's say there's 27 and You're not allowed to use the word 27 or write the number 27, you know What what would you do? I mean you can't you you just can't none of these none of these systems can do large Exact numbers they can do differences between maybe one and two two and three maybe up to six with the set-based stuff The approximate number system can do approximate numbers so you can I can look in the room and say Maybe there's about 40 people here, but the difference between 40 and 41. I can't see all right, so the Cognitive systems that you have in your mind before you learn the linguistic number system number words and counting are not powerful enough or Exact enough to do the positive integers. They can't represent that information accurately enough just like you can't see the molecular structure of that Bar stool by just looking at it We just our perceptual capacity is not that good So you need stuff from the outside you need a microscope to look at the bar stool and in numbers You need you need number words from your language and Accounting routine Okay So the placeholders in the case of number are the words themselves the list one two three four or five and They get filled in Gradually with meaning and that's how we get number concepts. That's what I'm going to talk about All right, so to make this point again that I just made the innate systems aren't enough The two big candidates that you might think could do number would be either the parallel individuation system this one or Or the other one so first I'll talk about this one parallel individuation is important It lets it lets us pay attention to things multiple things at the same time, but there's no There's no symbol for an integer in the system. There's no symbol for the number. So it's The little cartoon I've drawn here shows that you know if you're looking at a lemon and a strawberry and a strawberry You actually have to have a symbol in your mind for each thing. You need three symbols in your mind to look at three Objects to individuate three objects There's no symbol for three right. There's no three on there And what that means is that the more things you're looking at the more symbols you have to keep in your mind And so there are limits on working memory to how many things you can individuate so The way I think about it sometimes is if you go to If you take some kids to a park or playground and you have to watch them Let's say you take three kids and they're running around on a park with all the other kids You got to watch those three kids, right? You can't just like pick three kids and go home when it's time to go home, right? You can't just keep in your mind like three I brought three You have to actually Individuate you have to actually track each individual and make sure you bring home the exact individuals that you took To the park so that's parallel individuation and there's because there's no You know if you had a symbol for three then it wouldn't you could represent three as easily as you could represent 300 because it's just one symbol but But it doesn't work that way and so you can't take Five kids to the park six kids eight kids you can't watch them all right You like there's a limit to how many things you can watch at the same time That's why the parallel individuation system can't it isn't enough to support integer concepts because you need a symbol in your mind for everything you're looking at and You can only keep like three or four symbols in your mind, okay? So again, if I wanted to talk about 27 people being in this room, I couldn't use the parallel individuation system Not enough. I can't watch 27 people Okay, the approximate number system is the other number system. This is really the only kind of innate number system that humans and other animals have it's just an estimation system and Everything has it rats monkeys, you know pigeons every animal with a nervous system basically has an approximate number system but It's not it can't do integers because it's not exact enough So one of those pictures is 24 strawberries and one is 28 strawberries and you probably can't tell which one, right? Yeah, I don't even know I think I think the one on the right has 28 But The point is that you have in your mind. This is an analog magnitude system So you have one symbol in your mind for the number and the symbol gets bigger as the number gets bigger But it's not exact enough to tell the difference between something like 2010 and 2011 Right, there's no there's no symbol for an exact number in this system So again, this system is good, but it's just an approximate system What we're born with is an approximate number system, but it can't it can't support things like, you know Addition and subtraction exact addition and subtraction or multiplication and division with exact numbers It's not precise enough. You guys I'll see that. Yep. Okay, so the placeholders are the number words and The counting routine you should probably go by the movie now So first kids learn the words Themselves and they first learn it as an unbroken chain typically one two three four five six seven They don't even realize that it's separate words at the beginning and then with a little more practice They learn that it's that these are different words that could be separable They learn to sort of start in the middle of the list and count up start in the middle of the list and count backwards stuff like that and they learn Rules of counting like you have to point to each thing once and only once you have to say the words in the same order You can't count anything twice. You can't skip anything So if they know all that you know if you take a Three-year-old or four-year-old and you show them Row of things and say how many you know, how many apples are on this table? Chances are pretty good that they'll go one two three four five and you think great the kid knows how to count So why am I calling them placeholders? I mean what do they not know that does making me say that these are placeholders, right? Okay, so here's an example if you could show the video Oh, can we pause it? Can you go back because I think they didn't get to hear the initial command so this is a researcher in my lab and she's Asking the the child to give a certain number of I think they're oranges little plastic oranges to the stuffed animal, okay Can we hear again so she asked for five and the child knows how to count They count they appear to count correctly So she said is that five and then she's So you all saw that she counted she counted out eight She counted correctly and then the experimenter said can you fix it? So there's five and she added more Right and whenever the child says yes, that's five. We say yay. We don't tell them. No, it's not Okay, okay, so this is an example of the kind of counting behavior Yeah This is the kind of counting behavior that you see routinely kids learn the words they know the list They know how they look like they know how to count right They're pointing to everything and they didn't skip anything and the girl went one two three four five six seven eight And then the experimenter said is that five and she went yes And she said, you know, I don't think it is can you fix it so it's five and what did the kid do? She added more right so it really looks like these are placeholders They have learned to say the words they have learned to point and say the words and count, but they don't understand what they're doing And it's understanding what you're doing that really involves some big Conceptual issues and those are what I studied. I'm gonna try to give you a sense today of kind of what things kids need to learn so What we find when we do this task this give-in task that I just showed you on the video is that kids performance falls into Discrete sort of stages so first kids are what we call one knowers Which means that they'll give you if you ask for one thing They'll give you one and if you ask for any other number they will give you just some random like a handful They just grab a handful and stick them out there, but they won't give you one So they they know that one means one and other than that they don't know Then they become two knowers So they'll give you two if you ask for two and they'll give you one if you ask for one But other than that, they don't know they'll just grab Then three knowers they learn what three means learn what four means So these are sort of discrete stages where they know one they know one and two one and two and three one and two And three and four and those one two three and four knower stages We call I might refer to as subset knowers because They they can count up to ten or higher, but they only know the meanings the actual cardinal meanings of a subset of those Words either one or one and two one and two and three one two and three and four So we call those subset knowers so after the subset knowers after the three knower level or the four knower level they do they figure out what's called the cardinality principle and One way to think of it is It's just the principle that when you count the last word you say tells the the number of things in the whole set But it's really deeper than that Because it's also basically when you figure out the cardinality principle what you're figuring out is that every number Plus one makes the next number so You've you understand that for if you're a far knower You know what four means and then the cardinality principle is the point is when you figure out the cardinality principle That's the point where you figure out that oh five is the next word in the list And so five must be the number of objects you get if you've got four and you add one So it's figuring out that the next word in the list means adding an item to the set of Objects right and once you've understood that This is huge, but then you understand how numbers work then you understand what 2010 means 2010 means the number of things you would have if you counted from one to 2010 and added one item with every Number word right. It's it gives you an Way of understanding what all the number words mean what their cardinal meanings are and it's a really important conceptual breakthrough so So it also sort of it happens at that point It's it's a kind of a freeing thing because up through four because of parallel individuation through Three or four you don't actually have to understand the cardinality principle in order to learn what one and two and three and four mean because you can track three things you can track Maybe up to four things by the time you're three or four years old so You don't need to understand how numbers work in order to learn that two means two You can represent two without that rule Okay, so the cardinality principle is the principle that you need in order to understand numbers above four Okay, so these knower levels we see one knowers two knowers three knowers four knowers right here I've got what are called pre knowers. Those are kids who don't know even what don't even know what one means yet This is an example of some data. I'm just showing you the ages of the The kids who fall at the different levels and you can see there's a correlation with age kids Obviously start at the beginning and then they go through the stages in order So the on average the kids at the later stages are older But there's enough individual variation that for example if you look at three and a half year olds I've drawn a line at 42 months old, which is three and a half an individual three and a half year old child could really fall in any of those Stages so you you don't it we don't report our results just by age We don't say three three-year-olds know this and four-year-olds know that Because there's such big individual variation that it makes a lot more sense to say two knowers do this three knowers do that Cardinal principal knowers do this Okay So that's why I'm gonna tell you when I report findings I'll always be saying oh subset knowers failed cardinal principal knowers succeeded That's because just talking about their ages doesn't tell you very much Okay, yeah, I don't know what you mean one and two and going backwards Okay, all right, so I'll show you some I'll show you some studies. It looks like I'm going Much more slowly than I expected so I'll maybe skip some of the boring studies here So here's an example of a study that we did where we did the give-in task with kids And then we also tested their expressive vocabulary and their receptive vocabulary So expressive vocabulary is when we point to a picture and ask them what it's called Receptive is when we ask them to when we say Show me the Bicycle or the balloon or the and they have to point to it Okay, so speaking being able to produce the words versus being able to understand the words What we find is that children's number knower level is very highly correlated with their vocabulary So kids who are learning language faster who know more language also are no more Numbers So that's one thing that's sort of interesting to us because language knowledge and math knowledge isn't necessarily related for older kids But it looks like in preschool when they're first acquiring this system language is a really important medium of acquisition So the better the language skills are the better. They're there earlier. They're learning the number concepts. Okay Here's another here's a study that I did with Susan Gellman. That's a picture I should tell you these little pictures at the top are collaborators. That's my graduate student James Nagin This was my graduate advisor at the University of Michigan Susan Gellman. So Here's a study where we put we had a box And we put five things in the box and I would say to the child Okay, I'm putting five things in this box and then I would close the lid how many things the child would say five and Then I would say good now watch and I would either shake up the box or turn it around or take something out or put something in And then I would say okay now how many is it five or six? for example and We found that even one knowers would say that if I shook the box or rotated it or lined up the objects or stirred them around with A spoon then if it used to be five, it's still five. They knew the number word didn't change These are kids who don't know what five means right? They're one knowers But if I added an object or took an object away, then they thought that the number word changed So it looks like from very early on they have some sense that these numbers might be Labels for specific quantities, but they they're not sure how the label is matched up to the quantity Here's a study with Emily slusser who's a grad student here of mine who just finished her PhD last week And she's going to Wesleyan for a postdoc She wanted to know when children understand that number is a property of only discontinuous quantities So for example, there's some number of marbles there, but you can't really say there's some number of blue creamy stuff, right? So she did this study where we would show kids two Cups and these are representations of the different trials a cup with Some amount of liquid in it or powder and a cup with some discreet objects in it And we'd either say which cup has green or we'd say which cup has five And so for which cup has green the right answers are circled for which cup has five the right answers would be those Okay What we find is that? Which cup has green? All the kids do fine even the pre-knowers right here are above chance Chance is 50% because right you're choosing between you've got two cups in front of you And if you just choose at random you're gonna get it right 50% of the time So the origin of the graph is 50% you see that everybody does great when we ask them which cup has green But when we ask which cup has five They still surprisingly do really well the pre-knowers don't the pre-knowers are not different from chance So that that bar is the error bar shows that they are within the margin of error They might just be guessing but all the other groups even the one knowers Pick out the discreet objects when you ask them which cup has five So from very early on another thing that they've sort of figured out is that number is about discreet objects, right? Kind of interesting, but there's still a big Jump right between from one and two knowers up to three and four knowers is still a leap all right Then we have we wondered what would happen in languages that don't have such count mass distinctions as English because you think well Maybe the English speaking kids Understand that number is about discreet objects because we have count syntax in English when we talk about discreet objects We usually use count nouns, so we'll say like cups or apples or chairs or you know shoes And when we talk about substances we often use mass syntax like water a bunch of water a bunch of flour a bunch of sand Okay, but not all languages do that some languages don't have count miss and count mass syntax so Emily did this study with Mandarin Chinese speakers in that case. We say which one has five Guh, which is a numeral classifier and They did fine when we had the classifier So saying five guh is like saying five of them five things, but when we took the classifier away Then in order to make it grammatical in Chinese we had to say I'm thinking of a word you tell me which cup goes with this word ready five Okay, you can't say just which cup has five in Chinese like you can in English when we did it that way They didn't do well So the Chinese speaking kids when they were pre knowers There's only three kids there, so we don't know but the subset knowers 33 of them in this sample were at chance And not until they figured out the cardinality principle Did they start to succeed on this task where they had to match the word five to discreet objects? So it's suggestive of an English Mandarin difference, but the methods weren't identical So we need to go back now and do it. We need to say in English. I'm thinking of a word Can you tell me which cup goes with this word ready five and we need to go back in Mandarin and do it exactly the way We did it in English To find out if there's really a language difference All right, and now I'll just talk a little bit about what kids know when they figure out the cardinality principle For a long time. I think people thought of this as a fairly simple rule They said you count and the last word you say when you count means how many things there are but I think I can show you some data here to Really show that it's a really big conceptual breakthrough and that there are a lot of sort of Big concepts that come along with understanding the cardinality principle Okay, so this is Susan Kerry my postdoc advisor at Harvard We did this work together One thing about Understanding the cardinality principle is understanding that the direction of mapping between the number word lists and set sizes is Forward more backward less so as you go forward in the number list that the sets get bigger as you go backward in the number list The sets get smaller. That's something you have to understand in order to understand the cardinality principle So we did this Sorry, and the second thing you have to understand is That the unit of mapping is exactly one word to one item So every time you go forward one word you got to add one item It's not like add one add two add three something like that. Okay So here we did this task Where we would show the child two plates of things and say for example, okay I'm putting six blocks on this plate and six blocks on this plate and now I'll move one Now there's a plate with five and a plate with seven and I'm gonna ask you about the plate with seven Are you ready which plate has seven and We found that basically the subset knowers pre knowers two knowers three knowers They were all a chance the four knowers and CP knowers were just a little bit above chance So just when they're either when they figured out the cardinal principle or when they're just getting ready to figure out the cardinal principle Do they grasp this concept of forward in the list means more stuff and backward in the list means less stuff all right then the task to Assess the their understanding of the units we would say I'm putting five erasers into the box How many erasers the child would say five good now watch and then we either add one eraser or two more erasers and say Okay, now how many is it six or seven? This is just to try to find out whether they understand that if you add one eraser at six if you add two erasers at seven, right? This one they all Failed except for the cardinality principle knowers so really This seems to be a big part of the cardinality principle understanding the direction of mapping and the unit of mapping is Really key to understanding how number words can mean set sizes Here's another experiment Susan Gellman and Charles Wright Ted Wright here at UCI were my collaborators In this experiment we'd say okay I'm gonna tell you a story about when frog and lion came to my house and I gave them some snacks and I tried to make sure their snacks were the same because they like their snacks to be exactly the same But sometimes I made a mistake and their snacks were not the same So I want you to look at their snacks and tell me if I did it right or if I made a mistake So here's frog snack, and here's lion snack And the child would the children were very good at saying That I made a mistake in this case in half the cases They were identical and half the cases they were different by one like you see here They'd say you made a mistake. Look. Oh look right there. You forgot a peach there, and I'd say oh I'm so silly. I can't believe I forgot one and then we'd say Okay, um frog has six peaches. Does lion have five or six? What do you think and? Here are the results This is the age to show you that the kids at higher knower levels were definitely older But here are the results by knower level pre knowers one knower two knower three knower four knower CP knower So up until the time that they understood the cardinality principle They didn't have any idea Whether these two sets that they're looking right at that they have just told me are not the same Might have the same number or not Okay Here's another representation of the same data C is for cardinality principle knower and these are the individual kids So the one knowers two knowers three knowers four knowers and pre-number knowers are down here and as you can see it's a really kind of Amazing finding there's just nothing in between all the subset knowers are down here all the CP knowers are up here and Age is on the x-axis So you can see there's some overlap in the age But whether they know the cardinality principle or not really predicts whether they succeed on this task So it looks like their understanding of one-to-one correspondence being the basis of numerical equality is Very tied to their understanding of the cardinality principle Okay, last study words for numbers we would say this picture has four turtles Show me another picture with four turtles and we would have the alternative picture here controlled So that it had the same area as the target or the same contour length as the target Sometimes it had the same color something like that. So in this case, that's the right answer over there. That's the wrong answer Sometimes we would say I asked them to match the pictures on the color of the items or the mood of the items So we'd say pick another picture with orange turtles or another picture with happy turtles So this one is the correct answer is this one where they're orange But you can see that the wrong answer matches the target in mood. They're happy So if you matched happy with happy then you'd get the wrong answer. Okay, so we'd say that mood is a distractor in this trial All right, so for the color and mood trials the pre-number knowers did not do so well But the subset knowers and the CP knowers all did great So if we asked them to show me orange turtles show me happy turtles, they were fine But when we look at the number trials They didn't do well at all the CP knowers could match on number But the subset knowers and pre knowers pre number knowers could not in fact, they were significantly below chance Indicating that they continued to match on the distractor So if they just had no idea and they were guessing they'd be at 50% But they were below 50% and I think what happened here is that we said Show me some orange turtles great. I know how to do that show me some happy turtles Yes, I know how to do that show me another picture with five turtles. I have no idea But I'll just match the happy ones again, right Okay So if we make number the only possible match so that here's happy orange turtles and we've got sad green turtles on both sides But here's four sad green turtles and here's eight sad green turtles This is matched for area area or contour length and this is matched for number Then they're just basically at 50% So if they've got no color or mood to match on then they just they're just guessing they have no idea again until they figure Out the cardinality principle of counting. They don't even know what dimension of experience. These number words are referring to Okay, so I'm sure I've shown you all these kinds of studies about what kids Learn when they learn the cardinality principle all these really big concepts Basically for the Chinese kids the idea that numbers are a property of discrete discontinuous quantities The idea of the mapping going the direction of the mapping and the unit of the mapping between number words and set sizes the idea that one-to-one correspondence is the basis for the numerical equality of sets and Just just identifying the dimension of experience that number words are talking about Knowing what it is for something to be five versus ten that you're talking about how many there are Okay So hopefully I've convinced you that there's a lot of big conceptual Development going on in this domain at this age and so the future directions What I'm about to start now is a head start study with the Orange County head start because we actually find Really big differences between high-income kids and low-income kids in the age at which they're doing this I actually when I started out doing this research. I didn't think it was relevant to education because the kids that I Deal with are typically pretty high-income kids whose parents really have it together and are bringing them into university laboratories to be experiment participants and experiments and I Can tell you as the parent of young children that if you can actually get yourself organized to make an appointment with a University lab and take your kid in there for an experiment. It means that things are going pretty well in your life And we actually find okay. Can I have one minute to finish on my last slide? We actually find that kids from low-income families Might not have these concepts in place by the time they start kindergarten or even first grades We see kids in kindergarten in first grade who don't understand the cardinality principle now I've just shown you what a huge thing the cardinality principle is My kids I can tell you and probably your kids and grandkids Understood this when they were two and a half or three years old I know because I'm a number researcher and I was doing with this stuff doing that the Experiments with them in the bathtub at home because I was curious about whether they understood it The both of my sons understood this by the time they were two and a half or two years and ten months or something And in our lab we typically see it by three and a half Across a broad range of high-income kids Low-income kids might not understand this when they're five when they're six because nobody's counting with them and That causes huge problems for them in school. Okay, so we're doing four groups of children The the focus is on low-income bilingual kids whose first language is Spanish We're doing an intervention in Spanish and English and pre-testing and post-testing them in both Spanish and English to find out what their conceptual changes and In addition to the language learning And then we've got three control groups middle-income Spanish English speakers low-income English monolinguals and middle-income English monolinguals Okay, and the intervention we're doing is just a simple linear board game It's kind of like playing shoots and ladders Except that it just goes from one to ten you spin a spinner and you move either one space or two spaces and you count you say, okay Three, you know, I'm on space three. I spun the thing. I got a two so I go four five and whoever gets to ten first is the winner and Believe it or not some research has shown that this kind of intervention if you play this game maybe for 15 minutes Once a week for four weeks Massively improves their number understanding. All right. Thank you. This is my lab my collaborators These are all my students who did the all the research that I was showing you