 Amanda Wren, I teach kindergarten at Cleveland Elementary. And I'm one of the students and I teach at Jory Community School, I teach third grade. And want to jump in and help us at the last minute when our earlier teacher had to cancel out the one whose name you saw earlier. So let's get started. I have a couple of technical things to tell you. During this webinar, we really would like you to be able to participate as much as is possible so you can't participate by talking to us. We're going to ask you to participate by typing. And I'd love you all to look at the boxes you're seeing on your screen and you should have on the right hand side a box that says Q&A or it says questions, I think. And there are a few places where I'm going to ask you to type something into the question box. I actually, I'm going to go to the next slide where I'm going to ask you to do that. Our focus, as I said, is number sense. And I'd love you to type a word or phrase about what you think number sense means. And I apologize, it says here that to type it into the chat box, and it's actually called the question box. So please look for the question box and type a word or phrase that answers this question what is number sense. So it's a phrase, then phrase number sense we use often in education. And we're hoping that this is going to help us test the system and make sure that it's working. I'm sorry, I'm not seeing any responses. I've seen some things go by. Oh, okay. But I can't read them very fast. All right, well, I'm going to come over there. All right. So some people are asking that it's children's ability to know how numbers work and what they represent, deep understanding of quantity, place value, so there's a lot of things people are thinking about what it is that numbers mean. So to think a little bit more about that, we'll come back to that and we'll probably have some more answers a little bit later. So to think a little bit more about that, we have an activity that we want to do. I'm trying to get to the right slide. Here we go. All right, you saw one quickly already because I ran a little trouble with my controls. But here we go. So I'm going to flash some dots very briefly on the screen and I want you to type how many you see into that question box. So is everybody ready? How many did you see? If you can type that into the question box, that would be really helpful. There we go again. How many did people see? Usually, this is a pretty easy one. It's four dots. It's represented in a square like we see on it. It's represented in the square, so it's pretty easy to see the four. So we're going to try that again. And here we go. So I want to see some typing in the question box and I didn't see that before, but perhaps I'm not totally sure how that works, whether people are typing and I can't see them for some other reason. So we're going to try it. Oh, there I see them. I see. Now a lot of people are saying for. I see. All right. So we're going to try it again. We're going to look at our question box better. Let's get rid of that. And we're going to be here. How many did you see that time? Should be another quick one. Pretty easy to see what's going on. Pretty easy to see. It's just like it was four. The square was three in a different shape. Everybody said three because it was a triangle. It was really easy. All right. Here we go. It's going a little bit harder. You ready? Not really. Not really, probably. Some people find three in a line a little bit harder than three in a triangle, but I still see a whole lot of threes coming in. All right. Here we go with the next one. How many was that? Up. That's going really fast. That'd be four, four, four, four, four. Even though it wasn't a perfect square, people could see that four really, really easily. This is what we call perceptual subitizing. You perceive the three or four dots intuitively and simultaneously. You just know. We can do that with three and four. Our brains seem to be primed for that. We can all do it pretty easily as long as they're not overlapping each other. We can see them separately. We're going to try a couple more. They are, in fact, going to get a little bit more complicated. Here we go. How many did people see that time? I'm seeing a seven. I saw an eight. Seven, eight. We're getting a little bit of variety here. All right. Let's go back and look at it again. There's a triangle next to a square. Now I think now you've got a little more time to look at that. Most people are going to see that it is, in fact, seven, which was the majority of the answers, but it went pretty fast. It was a big number. Some people who put eight maybe quickly thought and thought of it as two squares or who put six quickly saw this and thought it was two triangles. Let's try again. How many did you see that time? That was a little harder. We're getting some eight. Seven, eight. Seven, eight. We had. We got a 10. All right. Eight. We're getting a little bit more variety. If we go back and look at it again, it's arranged in a way we often use if people use 10 frames in the classroom where you put five in one row and five in the other row. And if you've done that a lot, it's easy to see that it's eight because you see that there's two missing. Also, if you've played dice a lot, you might easily see that there's six that arrange what we used on dice and two on the bottom. But guessing the bigger numbers went by really fast. Nobody guessed a small number like three or four or five. Everybody guessed something bigger than that. It was quite obviously more than five and not more than 10. So there was a lot you were still seeing. All right. Another one. Is everybody ready? Here we go. How many were that time? How many were that time? It's a little bit more complicated. Does everybody guess? Okay. So you're getting 15. Yeah. I think we're getting a lot. Let's see. I think we saw some 15 with a question mark. I saw 13. I saw 12. Okay. All right. So if we look at it this time, I think we're getting all 15 now because you can see that there's three groups of five. We're seeing that there are, in fact, three groups of five. And you know that you can add those three groups of five together. Here we go. Last one. I didn't prep you. Here it goes. Last one. How many that time? Nine, 12, nine. A lot of nine, I think. 10. All right. So let's look back at it. It does seem like the majority was right there. There are nine. But it's much harder to see than the 15 was. 15 is a bigger number and harder to count. But because we had those three groups of five, like on dice, it was easier to see. I think, Lisa, I think some people see this as three triangles. You can kind of see a small triangle that's pretty classic up near the top. And then there's like a big spread out one and a squished one at the bottom. Sometimes people see a couple of groups of four with one left over. I've heard somebody say it was a triangle and a square or three and a four with two left over. So there's a whole bunch of different ways that people, when they don't have time to stop and count, tend to see this. But it's not nearly as easy as the 15 with the three fives neatly arranged like on dice. Right. So what we've just been doing with these last several slides is what we call conceptual subitizing. You perceive the parts and put together the whole. And all of this happens really quickly. It's often not conscious. It's still what we call subitizing. And this word subitizing, I mean, is just a word based on the Latin word for suddenly, because you suddenly know what you see. And it's a really basic part of our number sense. I wanted to give you a little experience of using your number sense, perhaps not in the way you had thought in the beginning of this webinar what number sense was. And the distinction between the subitizing and the conceptual subitizing is that the conceptual subitizing, you have to know a couple of number facts. I mean, if you're four years old and you do recognize the four and the three, but you don't know yet if it's four and three or seven, it's not obvious to you that that's seven. You might say it's four and it's three. Right. So when we grown-ups do this and when we achieve it, we're using both the subitizing skill and our knowledge of how those numbers fit together. So it's both visual, what we see, and then internal, what we can think about. That's right, sort of skill. So one of the big ideas about number sense that we have developed here is that the quantity of small collection can be intuitively perceived without counting. Nobody was counting one, two, three, or one, two, three, four for those small chunks. And then we learned to put small chunks together again or not counting them. And probably most of you weren't even counting five, 10, 15 on that bigger one with 15. You saw three, five, and boom, there it was. It was 15. Conceptual multiplication. Right. Conceptual multiplication, indeed. So these big ideas are going to crop up again a few times in the presentation. And our big ideas are things that we've developed here at Erickson that are really ideas for teachers. They're not so much ideas that we want children to memorize, but they're ideas that we want teachers to give children opportunities to understand and explore. So they're like a guidepost for a teacher. What are the central ideas in number sense? This is one of them that a teacher needs to be aware of. Okay. So let's think about another big idea. Quantity is an attribute of a set of objects. So right now we'd like you to type in the question box a word or phrase, what do you think this means that quantity is an attribute of a set of objects? How can quantity be an attribute? What does that mean? That quantity is an attribute of a set of objects. Okay. I see someone said concept. We have tells you how many in a group. Defines how many are in a group. Quantity is an adjective. Ooh. Quantity is an adjective, not a noun. Very nice. Create a picture. Numbers are adjectives. Defines the amount. Objects can also have other attributes like shapes. Okay. Good point. All right. A lot of good thinking. So let's go on and think a little bit more about this big idea. This very important big idea, that quantity is an attribute of a set of objects. Three elephants might seem obviously bigger when compared to three mice. Three elephants could not fit in my living room and three mice could probably fit in one of my shoes. If you use the attribute of size, three elephants are bigger. But if we use the attribute of number or numerosity, they are identical. So that's one way to think about this big idea. Right. When you say they're identical, what we mean is the sets are identical in terms of number or numerosity. Okay. We're going to keep talking a little bit more about that and have another illustration here about the attribute of a set of objects that we're looking at. And this is something that got brought up in people's questions, that it's a descriptor or adjective and it's not the only thing. So if we think about this picture of roses, what attributes does this collection of roses have? Red color is an attribute that we could use to describe this collection of roses. Round shape is an attribute that we could use to describe this collection of roses. Sweet smell is yet another attribute that we could use to describe this collection of roses. Well, if they were real, sadly, my computer screen does not smell sweet and yours probably doesn't either. We need to scratch and sniff. Anyway. And finally, quantity is another attribute. One of many. I'm sure I could have come up with even more. Another attribute of this set of roses, there are three roses in this collection. So I think, I mean, this whole idea of quantity as an attribute is sort of so obvious. It's kind of dull. But the problem is that as an adult, you've gotten so used to this understanding and I think the point of us having a big idea that says quantity is an attribute of a set of objects is to help remind us as adults that this has learned knowledge, right? Kids come into the world subititizing, but they don't come into the world calling it three. Or calling it four. But they put that stuff together. They do have some ideas pretty early on about more and less and which match up. And if everybody gets one, so they have it. And it's actually pretty abstract pretty early. I'll move on to that. And numerosity is this abstract idea that exists apart from number words or written symbols. The words and symbols vary from language to language, but the numerosity does not. We can say that there are three elements or if we were speaking Spanish, we might say that there are three. But we're still talking about the threeness. And you might understand that without having any language to represent it. It's an abstract idea. And it also doesn't really exist in the things that exist in our grouping of the things unlike some other attributes. And humans seem to be biologically programmed to automatically perceive the numerosity of small sets. So I have another thing I'd like you to do in the question box. Well, no, never mind. Hold that thought. Be ready to type. So we've just talked about two of the big ideas. That quantity is an attribute of a set of objects and the quantity of a small collection can be intuitively perceived without counting. And they're part of this larger set of big ideas that we have developed. And this first slide are big ideas that relate to the things we think children are thinking about in preschool and kindergarten. Although they are still important things to be thinking about as children get older and even as adults thinking about what does it mean that quantity is an attribute and that it's abstract. It's something we still use in higher mathematics but it's something we've developed in the early years, hopefully, and continue to develop. And then as children get a bit older, they think about some other ideas about a number of cents. And I'm not going to read all of these and you will have them in your saved webinar. But children begin to think about big ideas related to place value of base 10, that we group by 10s and the positions of the digits matter. Children begin to think about big ideas related to fractions, to holes and units and equal parts and all of this. We start with this very foundational idea that quantity is an attribute and we use numbers to name specific quantities and then we build this huge, amazing, phenomenally useful system out of that. Another word for this sort of aspect of number is the cardinal value, right? The cardinal value of number, the 3, the 4, what it really means, how many is it? Right, and that also comes to play the carnality of a fraction, you know? We're getting a comment that says, will these slides be available to us after the presentation? The pace is quite fast. Okay, so that's helpful to us. They will be available and we're giving you a taste of some ideas and an experience of learning about math that is not, we know is faster than is ideal for learning, but we want to give you a taste that moves into understanding what's happened for teachers in the classroom. We think it's better that you experience some math even if it's quickly and at a surface level the way that we tend to teach it because then I think you'll have a better understanding when we get to describing what it looks like in a classroom. But you will get to review this presentation and I don't know the details about that but the folks from the National Working Group will let you know at the end of the presentation. So, here's my question I spoke about a little bit before. Do you think that babies under six months of age have number sense? Type yes or no in the chat box? Yes, yes, yes, no. Yes, no, yes, no, yes, no. Fair number yes is some knows. I think I prime people to think that the answer might be yes by the things that I said, which is good. I'm glad to know you're paying attention. Of course, some of you might have thought about this and read about this before. So, I'm going to go on and briefly. Now, when I like somebody put sort of, I'm going to give you... It probably depends on how you define number sense because if you think about it as the cultural piece of the word, it's like no, but do they have some awareness of quantity? That's what we're going to talk about. Right, that's what we're going to talk about with this next slide. No four months old I know knows how to speak or sign a number reliably but this is really interesting study that I'm showing you an illustration from that was done in 1992 and these were four months old infants and this slide that you're seeing now and try to go through this slowly enough to give everyone a chance shows what happened, how they conducted the study. So, if you're looking at the top row of little pictures, the first thing that... and this is what the baby is seeing. There's a baby strapped in a little comfy chair watching things on a screen and they have those fancy psychological things that help them record how much the baby is sucking which shows interest in where the baby's eyes are pointing. And actually I think it's live. I think they're seeing this live not on the screen. Oh, that's a real object. But they're not where they... They can't touch it. No, they can't touch it. Jennifer's right. She corrected me and she's right. It's live but they can't play with these things. They're removed from it in that way but it happens they're watched actual objects. So they see a toy placed on this little puppet stage and then a screen goes up and then they see a second object moving at the hand behind the screen and the hand going back empty. Okay, so the screen goes up, right? And that object is still back there and then they see the hand coming back in with yet another object. Is that right, Lisa? That's right. And then they see the hand removing empty. Okay, the hand goes away. So then there's still the blue screen and no hands and the screen drops and the bottom shows two things. Some babies got to see one thing and some babies got to see the other. And on the left is the possible sensible outcome which is the screen draft and there's now two of the toys, two objects visible. And babies did not seem terribly extra interested in that. It seemed as if they expected it. They just kind of kept second and let it a little. But when the screen draft and they saw something that was impossible, whatever it was, it may possible only through the experimenter's trickery but something that they wouldn't expect. They seem to be puzzled. They looked harder and longer and sucked more. So it appeared that these formals really were developing a sense of quantity, not a sense of numbers in terms of a system or names, but a sense of quantity at this early age and understanding that if there's one and there's one more, it should not look like two and not look like one. Yeah, and I think that in this experiment kind of defines that sense of quantity as the ability to mentally represent two-ness, because when the baby is seeing the blank screen, if they know what it should look like, what they're able to remember is that there ought to be two of those things back there or something like it. So they're reacting to something that we think is probably quantitative. So that's the very beginning of number sense that most babies seem to have. The babies they've brought into the laboratory. And it makes sense that evolutionarily, that it would be important or that it could be useful. At least to know more and less. Absolutely. Where is there more shade? Where is there more progress? Then the question is, where does it go after that? Older infants often learn the signs or words for more and all gone before any other ideas. It seems to be something that they're still thinking about, having more, having none. And one-year-old can tell that a pile of five is more than a pile of two, even though they don't know any number names. They can't tell you how many more. But if they have the same type of object, they will pick the pile of five or six as more than the pile of two. And by the second half of the second year, most toddlers can understand and follow if you ask them to take one or give one or take two or give two. And they usually have words for one and two. They don't have in that first year of a lot of speaking words for three or bigger. I actually heard a wonderful story. Someone told me after I used this very same slide talking to a group of people about numbers since that her granddaughter went through a brief phrase where she would say one, two, and then two, two for three. And then more two if it was more than three. And it only lasted for a little while and more words than you have to figure. So she says, that's so cool. She's milking everything she got. And then preschoolers build on this, and they really are building a firm sense of the numerosity of three, four, and five. It doesn't mean that we think preschoolers only count, but we're not really going to talk about counting skills today. It's like a whole other kettle of fish related to number sense. Most preschoolers, by the end of preschool, have learned a lot of number words, and many of them learned them in order, but really understanding the numerosity and how they go together and break apart that three is three and that three is two and one, and that four is four and four is two and one, four is two and two, or four is three and one more. They really can get that for three, four, and five. And they can do things with other numbers like count objects and say if you put 10 in front of them and they've learned those words and they carefully count them that it's 10, but they usually can't fiddle around with the bigger numbers so well. And to have a lot of experiences with these little numbers really gives them a very strong number sense. And then with kindergarten, they solidify their numerosity and sense of number combinations going up to 10. And a lot of the kindergarten year is really building from fives on to 10. Isn't that right, Amanda? Very true. And first graders, you know, if they've got that solid sense, can build again to have a sense of what the numbers really mean up to 20. And what we're really trying to say here is there's a distinction about really understanding the numerosity of these numbers and how to use them, and just being able to say the number words and even count something out what that means changes. Lisa, we've got a question. What effect would the child's environment and experiences have on this? Do you want to take a stab at that? Well, I think there are young children who have, there is some of this natural sense of those small numbers, but I do think the flexibility of thinking about number combinations and really building on that in the preschool and kindergarten years, there's an effect of the environment if anybody ever asks about that. And sometimes in preschool and kindergarten classrooms, teachers focus a lot on counting out the right number or doing some basic addition and getting the right answer, and they may discourage children's interest in the playing around with the numbers. So there definitely is some influence on how people talk to children about numbers. I do think pretty much most three-year-olds are going to be able to see the difference between one and two and three and have those words for it where we go from there and can certainly have a lot to do with how grown-ups talk to them. I want to add an example. Do we have time for that, Lisa? Sure. So there's a study done by some people at the University of Chicago quite a few years ago, actually, that found a socioeconomic difference at kindergarten. So what happened was they would start with word problems. So they'd say, what's 3 plus 1? Or I had three apples and I got one more. How many do I have? So those are problems that are math problems presented through language, through words. And then they would present another problem which was the same math, the 3 plus 1, but they did it by putting out three objects, putting up a screen and adding one more. And then telling the child to make their math that was right in front of them look like they think the one behind the screen looks right now. And what they found was that kids from less-advantaged backgrounds were able to do the thing with the screen and the dots. They were able to make four on their math just fine. But the word problems, they were nowhere near as good as the kids who had a lot more advantages. Or I'm not sure which is an advantage. I think that they had less-rich language skills or perhaps different perceptual skills. I know some studies have shown that the language development doesn't, some children who have limited language may actually be able to solve problems in a different way. They just can't tell you with words how they did it. Which in school we help people can tell about words. So we're going to move on now and I just put up the last statement. The second and third graders are using this number since they've developed to become much more flexible and efficient solving problems using operations and much bigger numbers. So we earlier showed you our big ideas for the basic foundation of number sense and I'm showing them again here and then mapping them to some key skills because the idea is the underlying concepts that the child must be understanding in order to do something and then the skills are what they can do. So understanding that quantity is an attribute of a set of objects and that small collections can be intuitively perceived without counting come out for children naming the quantity of sets and moving on to conceptual subitizing. And then another one that we haven't spoken about so much today is composing and decomposing the numbers and like as we have spoken about it I just didn't read out that big idea leads to fluency and composing and decomposing numbers and one of the things that we work on in our work of helping people understand foundational math is understanding the math and then also understanding how children think about it and then what do you see or hear what's happening and another tool for that I'm waiting for my slide to change another tool for that is something we call the landscape of learning it's a tool that we got from a group called the young mathematicians at work it's a group in New York City and you'll have to look at this more carefully when you can expand it but I wanted to give you the idea it's a tool for taking this learning that we've been doing and yes we've been doing it fairly quickly we've taken you through in 30 minutes something we might spend two hours on if we were doing a really in-depth professional development but we wanted to explain to you how our whole project works and it's another way to think about it and you see it looks really busy it's because there's a lot of ideas and models and strategies that children may be following along as they get to a complete under a solid number sense and a good understanding of addition and subtraction so this is a landscape for number sense addition and subtraction in particular there are other ones that cover other topics and it's essentially developmental in that the ideas, the strategies, models and concepts at the bottom of the page are those that children usually acquire first or use first but we like the idea of the landscape because it suggests that there are multiple routes to the same destination and our experience of the way children develop is that it's extremely variable and that it is very difficult to predict what is the next thing that someone should be learning based on what they know and even that what does it mean to know something I mean with very young children it's often something they can do and then they can't do the next minute and I notice there's a question about getting a printout on the screen and you'll get this when you get the recording I have the website where I got this from and you can see other ones there so when you get a chance to look at that more closely you can grab that website go to that website, yeah I'll tell you a little bit about what our group is early math collaborative and I advance to the next slide I think so Jennifer, tell us a little bit about the beginning of the early math collaborative okay, well we're here at Erickson Institute which is a graduate school and child development in Chicago so that means there are a lot of people coming here to get master's degrees to work with children and families between the eight children that is between the ages of zero and eight and their families and our project trains teachers so most of what we do is we train teachers who are currently in service working in school oh here we go, look at that so we do that really through three different mechanisms we do the professional development we also conduct research and most of our research is about teacher ed teacher development how do teachers learn what do teachers understand and what does their practice look like and finally we try to provide information so we have a pretty rich website that has a lot of video we have a book out and we're trying to write another one and trying to be somebody who can curate some information and really bring together advances in research and our understanding of teachers and classrooms and make practice better so we're trying to move to the next slide we'll see if we can make that happen or not but in the meantime I want to give you a brief overview of our work we started in 2007 originally we were working with pre-K and kindergarten teachers in the Chicago public schools since then we've expanded and we've done work with all the way up to fifth grade we've worked in a lot of different settings so for charter schools and child care centers and places in Kentucky and New Jersey children's museum and so forth and then we also have been working with other people who train so people who train teachers and people who teach at community colleges that's some of our more recent work trying to figure out how do we take what we're doing in terms of the professional development and help someone else to deliver that same kind of a learning experience for teachers so I want to tell you a little bit about one of the projects the one that was highlighted in green in the last screen and I'm having a little trouble with my controls as you can see sorry about that so it's called the Innovations Project because it's from a grant that we got called Invest in Innovations from the federal government $6 million funded by the U.S. Department of Education a generous grant from the CME Group Foundation and it's a four-year school-wide intervention in eight schools and the teachers that you heard from earlier and briefly recently are going to be who are going to be sharing with you are from these schools and I want to tell you very briefly a little bit about what we have been doing with these teachers they receive training that we've got we've got several different aspects to our training at the heart of it is our conceptual framework which is the whole teacher approach in early childhood we talk about teaching the whole child which means that you don't just teach knowledge you have to think about their aspect and you have to think about their physical self and so in the whole teacher approach we're trying to think about teachers knowledge but also think about their attitudes that are really important in mathematics and to make sure that we're really looking at their practices so keeping that in mind there are four different kinds of intervention that we use learning labs is when teachers come together at Erickson so we get them out of their school setting the point is to deepen mathematical understanding and get them excited we also try to model the instructional strategies that we hope that they're using in their classroom to give them a chance to participate in a mathematical learning community so they know what it's like once you've experienced it then you want to make one in your classroom and then we also have individualized coaching so we have math coaches who are visiting teachers at their schools supporting them to implement new strategies and to really reflect on their own practice and make their practice something that can be constantly improved and then two other parts of that of our PD are grade level meetings at the schools where the coaches who work sometimes sit down with a group of teachers all the third grade teachers or all the kindergarten teachers or all the first grade teachers about once a month to help build a community of practice there and this is really how we're hoping that our intervention will continue after there's no more money to pay for us to do it that the teachers at the school will be able to take on themselves the jobs of talking to each other about math and the final part that doesn't directly involve the teachers is what we call a leadership academy where we ask principals and also some assistant principals to come here to Erickson and learn a bit more about what good math looks like and what math is so that they develop their math eyes and they have a network of administrators so they can talk with each other and learn more. We want to talk a little bit after we've given you a taste of what we do for teachers helping them think about math and helping them think about children and we've told you a little bit about all the places we've done that and now we're going to invite two of our teachers from this innovations project to talk to us a little bit about what they've learned and how their practice has changed. First up is Amanda Wren who is a kindergarten teacher and so really the first thing I think we want to hear about Amanda is to what degree your understanding of the math has changed and in what way? Well as I was thinking about how I thought about number since before I became involved with Erickson it was mentioned before that kindergarten is focused on the numbers 1 through 10 a lot and being flexible with that and I think my idea of number since was that I could tell if the students knew the number, if they could name the number and if they could join two numbers together and come up with the right answer. So you mean like name the numeral? Yes, the numeral and know, recognize it. And then what was the other piece? Put numbers together? Be able to take two numbers that they could name and then tell me what they were if they were joined together like in addition. So two and three together is five. Yeah, okay. Being able to learn from the PDs and from my coaches I developed a deeper sense of all the different things that come before and add to and make up a child's number since and so it allowed me to deepen my own understanding of what we can do with numbers and what numbers mean to a child and to myself as well. So as I implement that in the classroom I realized that I am able to know more what to look for as the child is explaining their thinking about a number and the different clues I can look for as you guys saw the landscape of learning it's very it's not linear and it's not like a rigid sequence where I can teach the kids okay now they have this and I can move on to this but it's going to look different for each child and more aware of what I can be looking for in order to move them up to that landscape of learning. And when you started the project you were a second year teacher is that right? Yes, it was actually perfect timing for me because I was still developing my teaching philosophy and my practices and so as I was learning with Eric's then it was really perfect because I could implement the things that we were learning in the classroom right away and to figure out what it meant for the kids in my class the things that research was saying. I think you have an example to share with us and hopefully my pictures are going to work. Yes, well in our classroom we start each morning you'll see hopefully a picture soon but we call it our Wreck and Wreck attendance chart and it's the Wreck and Wreck is a tool we use in math and it has 5 white beads and 5 red beads on the top row and then there's the same set of 10 beads on the bottom row so it makes up 20 total. But we've also taken that concept and turned it into a chart where children can put their names on a stick and then put it in one of the slots each morning just to record their attendance and to let them know. So how many kids are in your class? This year we have 24. Okay and so your Wreck and Wreck chart on the wall has like, I saw it, it has 3 rows of 10, right? Yes, that's 5 red and 5 white on each row. Yes, so we modified it so it could be up to 30. Okay and little pockets, right? Yes, little pockets they can slip their name in so that's the first step in the morning. And then as we gather as a group and start talking about just daily routines we move the Wreck and Wreck attendance and start over and the kids have a chance to show me and the rest of the class how they can figure out how many kids are with us today and the beauty of the Wreck and Wreck is that because kindergarten it's really important to be flexible with those 5s and 10s and be able to show how they can see it in different ways the kids have a chance to not only explain their thinking but also build upon each other's strategies of oh I see that the first 2 rows are also then with name sticks so I know that if it's 10 and 10 that's 20 but then another child might see oh well I see there are 5 here and 2 on the red so that would be 7 and so just showing the different ways that they are familiar with the numbers. So the structure of the red and the white pockets kind of helps them group the number of kids into these 5s and 10 groupings and so the conversation ends up being about that. Yes and one thing I also love about it is it's accessible to the kids no matter where they are and their understanding of number sense because you can still be working with cardinality and count each number up to the last number which was 23 on the day that the picture goes and then so they can use that so what's that little girl on the left saying in the picture she was just demonstrating her excitement over the realization that if all 10 sticks are filled on a row then it's easy to know it's 10 and so that's an efficient strategy that she can use to show the class how she figured out the total. I'm going to move back now that I figured out a little tool to try this this is just showing that the children they come in and they just put them wherever they want to and so when it gets to the point that all 23 children are there it might not be all organized like this at first. Right, so if it's not organized at first then what do you do? Sometimes a child will volunteer oh let's make it easier to count other kids who are more interested in showing their creativity and counting or something that they've been thinking about and then show the different ways they grouped them in their mind like we were talking about the dot display and some people see it as four and three and so the kids might see it in different ways. Great, so then you make sure that there are multiple ways and you always try to ask more than one child how they figured out the number? Yes, that's become a big deal as someone being able to offer a new strategy or build upon another student strategy and they love to show the different ways and you told the story before about the picture on the right-hand side the two children both putting their hands up on there and what was happening there. So the first child on the left had come up and demonstrated to the class how she was going to count by fives because she knows five reds and five voids and then the second child while the first student was still up there raised his hand and said I like her strategy but I also see how we could use it in a different way and so he chose to do it by the fives that were the same color and then we talked about how they ended up with the same answer but got there a different way. That's awesome. Thank you so much Amanda for sharing about what you're doing in kindergarten and now we're going to turn to Juan Aracendi from Jordan Elementary School who teaches third grade and has also been participating in our project for the last three years. So Juan maybe you could start by as Amanda did talking a little bit about how your own personal sense of what number sense is has shifted a little bit. Sure. I was born in a different country, I was born in Mexico and I was sharing with previously about how I thought I knew how to do math and how I thought it was good to do math but I didn't really have number sense. I learned the steps to solving problems, I learned the steps of how you solve it in an efficient way but it didn't really make sense in my head like I just knew the procedure to get to an answer and when I started teaching, this is my eighth year teaching sixth year teaching sixth, a third grade I in the beginning that's how I was teaching my students like this is how you do it step by step but I realized that my students were learning the procedure of doing things but then I started understanding what they were doing. So now because I have a better number sense I can explore with my students how to teach them to be flexible with numbers, how to teach them to decompose decompose numbers, how to teach I'm teaching them now how to represent it in different ways and teaching different strategies that they can actually implement in different problems as I teach everyday. Something interesting in what you said is that I want to ask a little bit more but it sounds like you're saying that your knowledge about number makes it possible for you to teach in a different way that still you have that kind of understanding yourself you really couldn't have opened it up to other strategies for example is that correct? And one thing that you had shared with us that I put on the slide is this understanding about numbers not just being a unitary thing that you could express the example I gave that you could express 20 as 80 less than 100 or five more than 15 or two tens or more and more ways. And before like I said I knew the steps to solving problems but I didn't really understand what it meant. And now coming to Everson for all this workshops and having my coaches I had the luxury of having three different coaches throughout the year in which it was really good because I learned something new from each one of them just understanding myself what these ones are explaining I'm more open to sitting there and modeling their strategies and I was very left in the beginning of well yeah that sounds great but I really don't understand what you're saying so now having a better number sense myself really allows me to model their thinking and it's not just what they're telling me but when I see my other students getting confused with it I can stop and I can explain it to them with my own words that I know it might be easier for them to understand instead of just oh yeah I made five circles what the five circles represent things like that. So it seems to me like you feel that you have changed your teaching about number sense because you've used your own new thinking about number sense. Right, absolutely and five years ago six years ago when I started teaching math or in third grade I just thought this is how you do it and this is how it works. Don't ask questions, just do it this way because this is a way that you're going to get to the right answer. Now I don't even tell them how to get to the right answer I start my routines everyday with having a number talk I just put something up on the board and then we solve it with different strategies and my students share their strategies and they understand that it's more than one way to get to the right answer and they say I know you're not looking for the right answer they're only looking for the strategies how we can get there. And yes I can model two or three two years ago it became really overwhelming having so many strategies and then knowing which one was better for them so through one of your PD's that came here over the summer or they were telling us yes you can have 20 strategies but you need to think about which one works for them narrow it down to two or three that are going to be more efficient just get skin that be very creative with numbers and start going in different directions but then you lose them just have to concentrate on the strategy that you know are more effective for all of them. So we have a couple of examples of these sorts of number talks and problems that Juana was talking about that you can see this is things that you want to you wrote while the children were speaking to you. Right so I started my math block it's 75 minutes and I started the first 10 or 15 with a number talk and I just wrote eight times six and half of my class already knows that it's 48 but they were not allowed to say that so they had to prove that that was the right answer so someone started with well you can make eight circles and if you notice I made a mistake and I was putting eight little lines on it and then one of my students said there's a send-in you can make eight circles and put six in each one and that will show how many I did eight and they were like I think you made a mistake I'm like what do you mean and I didn't even recognize my mistake I didn't even notice it and they were like well you have eight circles you cannot have eight that's in it because you have eight times six I was like oops I'm sorry you said it led to an interesting conversation that one child said you could have made six circles with eight in it Right so we were talking about how to place numbers for number models and one of my really smart girls said well if you have plain numbers you can switch them around because there's no story that goes with it Do you have what kind of numbers? Plain or naked numbers as we call them Ordinary numbers, naked numbers numbers that we're just talking about the number and we're not talking about what it means in the world Right so she pointed out she said you can switch them around because we know that we call them short cuts for multiplication and I'm trying to develop a better vocabulary with them so we're changing a few of the words that we use in class but for now they knew it was called the turn around short cuts so we proved it and you can see on the tape diagram underneath we counted by six, twelve, eighteen twenty-four and then someone said well you can also make six boxes and come by eight so I modeled and all of the things that you guys are seeing on the screen I think some of my students are telling me to do writer you tell me what to do I will do it because this is why you go ahead and cross it out because they had said make eight circles and put six in each and then you were fixing it because they said wait you now if a child had said and maybe sometimes this happens make eight circles and put eight in each then that would have been recording what they did and hopefully in the conversation with the children someone would have said oh that's too many so I modeled their strategies and then are we going to see the next picture so this is a number talk where you're really trying to get children to think about this number fact that eight times six equals forty-eight or six times eight equals forty-eight and then by the way I don't write the answer until we prove it two or three different ways you write down that eight times six is just eight times eight leave it blank and then they have to prove it and then you all agree on the forty-eight right and the second slide that you guys are seeing right now my students were having a difficult time with writing number models for division problems so I gave them the word problem on the top and I'd like to start my lesson I'm just going to read it if it's hard for you to say Mary had 28 tickets and she wanted to share them with seven friends and how many tickets does each friend get right so they were able to answer the question they were able to show it with a number with a picture proof picture or picture proof as some people call it or in a way they were able to do all of that but the number model didn't really match what these were doing or what the problem was saying so what I did today I decided to use the same problem and I'm using the same numbers underneath it says Mary had 28 tickets and she gave four tickets to each friend how many friends got tickets right and those two problems we took like 35 minutes just moving those problems and it seems like a long time but we have very good conversations about why does it work on the top when we do it this way when there's 28 divided by 7 and why I cannot write the same number model for the second problem and we were looking at how are they the same, how are they different using math in real life like how is this helping you understand math what do you know, I teach bilingual so a lot of my students their English is very limited so taking step by step the problem and seeing what do you see what do you visualize what or how are you going to solve the problem I don't start with a strategy that I want to teach them I say okay here's the problem how can we solve it and then they share what they know and then a day after you've done this you might have thought about what they know and you use that to make up your problems based on their homework because they notice they get the right answers but their number models didn't really match what the problem was saying that's what drove my lesson for today they knew a number of facts like 28 divided by 7 or 18 divided by 3 but the picture they drew didn't seem to fit with the way the word problem was the picture was fine it was just the number model underneath it wasn't you're saying there's just a lot of complicated parts to that and so your understanding of those complications and the need to model and write and let kids build by writing different problems is all arisen out of and I'm pretty sure in the past before taking this class I said Erickson and all of this I'm pretty sure that I was okay with just getting the right answer but now it's not about just getting the right answer it's about really understanding what is going on and the problem and how you're going to model it how are you going to prove that's the right answer and the students being able to identify when it's a set of things that you're putting together when it's a set of things that you're separating where it's a set of things what numbers really mean so what is that at the bottom of the page there I gave them what we call them again make it numbers and I used exactly the same three numbers that I used on the top in my students now are able to come up with their own division problems multiplication problems I said okay tell me a story one of them is Carlos has seven boxes of books and each box has four books and how many books does he have in all and another one is that there's seven boxes and 28 books in those seven boxes and how many books are in each box and so they came up with a multiplication problem and a division problem and they were able to tell me because of that so we've gotten an interesting picture of number sense being played out in third grade thanks Juana and in kindergarten thanks Amanda and if we want to think back way to the beginning before kindergarten this is being built with some very small numbers and for instance in preschool we might start to build this by asking children to show us three on their fingers and they might hold up the three fingers their thumb and their index finger and their middle finger and show that it's three and some of the ways we can have children be more flexible is sometimes hold up different fingers or we begin this by asking children can you show me three on two hands and that kind of question that's really one of my favorite preschool and early kindergarten number sense activities is asking children to show numbers on their fingers more than one way and using one hand or two hands and that kind of thing is what eventually leads to children being able to model multiplication when they're in third grade well one of the things that we haven't talked about so far in our discussion on number sense is the common core of course that's not being adopted in every state in the nation and I know that they're controversial for some folks but I would say that overall the common core practice standards are really focused on kids thinking and the process of doing the math and they emphasize flexibility and a fluent kind of understanding the ability to really generalize the knowledge going deep as opposed to surface and I think that we chose number sense because it's so core to all the math experiences that children will have there are certainly other important content areas in mathematics but I think number sense gives you a nice way to think about the pre-K to third grade span and I was just speaking about fingers but those dot cards that we were using before those again are a great tool that can be used in preschool, in kindergarten, in first grade, in second grade, in third grade and I've actually seen people do amazing dot card math talks with middle schoolers too but with the youngest children you simply might start by building on that sense of numerosity and having a matching game that isn't about having two identical cards but having two cards that have the same numerosity so three that are in a line and three that are in a triangle four that are in a square and four that are in a line maybe two different colors of each or there's a lot of different ways you could make dot arrangement cards of those numbers just from one to five and ask people to build it and then as they get older you might be using those dots to build that conceptual number sense with an array that let them quickly see how many and multiply it out or with different dice arrangements and they're knowing their addition facts so that's the kind of thing that really builds because number sense is really about thinking and making sense about our number system and number situations and what we can do so as I said a few moments ago we've given you the whirlwind tour of the early math collaborative both what we do and who we are and how it has been working out in the Chicago Public Schools and as you can tell from what we've done we are about the early math, what is math early on and foundationally and we're also very much about collaboration that we work together in a variety of ways to build the number sense of teachers so that they are building the number sense of children and again in other areas of mathematics so we'd like to take a few moments to find out if you have questions that you would like me or Jennifer or Amanda or Juana to answer about what we've been talking about and I know we've talked about a lot of things but we really wanted to even so I feel like we've only given you very much the tip of the iceberg but we wanted to frame the whole picture for you. So this is the time if you have a question to go ahead and type it in the question box and we'll keep an eye on that and see if something comes up because we'd really be glad to have that kind of interaction going on but in the meantime I wanted to ask Amanda maybe would you mind if I ask you to tell us just a little bit about what parts of the work that we did together you think might have been most important for you because I bet it's different for different teachers but for you to learn what you need to learn and change your own teaching practice what part of the professional development was most helpful? I think well I learned a lot each time we met at Erickson and probably like a lot of people are feeling now it's a lot to take in and I think the most important part for me to be able to apply that to my classroom instruction was to be meeting with my coach and also grade level meetings with the two other concurrent teachers at my school because it gave us a chance to sit down together look at student work figure out what was really going on in their minds and then plan our instruction accordingly to that because usually it's really hard for us to find time to sit down together and see what's happening in other classrooms and find areas that we need to support the kids so that was really a good discipline to start meeting with each other and really taking a close look at the work thank you how about you Launa what would you say was important and I remember you saying you were reluctant yes at first I think you said something like why should I? why should I change the way I'm teaching for me I think we know that you really did change and you have a sense of what you did but what do you think was important about the way that we did the work together that helped you I think it's very similar to what Amanda just said having a meeting with other teachers and talking about the work but I think what I enjoy the most was it's actually right now meeting with my coach having a one-on-one with her having this conversation very rich conversation about what multiplication really means and how I can actually teach it in my classroom it's not about necessarily getting the strategy to how I teach it but get my own understanding and what it means and how I can apply it and how I can deliver that to my students because I think we all learn how to do certain things tonight but it doesn't really mean that we understand them which is more about understanding the concepts myself and the content and then being able to explain it to my students that's interesting so it sounds like what you're getting from your coaching at least right now is the depth of the content these math coaches let me tell you it is hard to find people who bring together the kind of classroom experience that you need to be a coach to be able to help a lot of different teachers and have that math content so I feel really good about the people that we've got on our team and I know you've got a terrific coach I wonder though if there are times when the coaching really is more about strategies or more about ways to organize things and if that's useful too or maybe really the math content is key I'd just love to know what you both think I would say it has both components in it it's not just the content but it's also the delivery of the strategies that I can model with my students so she's been teaching us how to represent numbers for this student so they can actually have a better sense of what number sense is but she does it in such a way that it's more for me for my students isn't that like this is how you do it Wanda this is how you teach it but look this is what it means this is how you can show it and then I take that and make it my own and show it to my students we've had a couple of questions coming in that I want to get to I know it took people a little while to get typing one question that really relates in some ways to what we were just talking about is again a question about pre-k teachers and the pre-k setting and I mentioned the use of fingers and dot cards as great ways to build number sense and build on the subatizing skill also any board games with dice can really give chance for children to build on their skills and I love for pre-k to make my own dice two and three twice so they keep seeing one, two and three a lot and if they're rolling two of them they have small numbers to combine and all of those can give you a sense of what children are understanding and that leads to another question someone asked about math assessments a couple of people asked about what math assessments are useful in the early grades and I'm not going to go out there for any particular commercial publisher but one of the things that I think is really useful for a teacher to understand their children's math or to do the kind of math talk activities, number talk activities which in younger grades might involve what Amanda was talking about using something like a rec and rec attendance chart and you could use a similar thing with preschool and you might use it in a different way instead of having the whole class be hearing about it it might be that children go put their thing in and the teacher with one child figures out how many kids are there that day and that cycles through all the children and eventually you would get a good sense after a couple of weeks about how each of your children is putting these numbers together So you're suggesting you would do it one on one because each one of those little short interviews would be something of an assessment Exactly, and the kind of number talk that Juana talked about which again there are ways to do those number talks with preschoolers and kindergarteners you can actually do dot things where you flash it and ask children how many and they show on their fingers and then you hold it up and ask them how they knew especially with the four and five they might be talking about number combinations and you can describe what they say and those can be good records for you to understand how children's number sense is developing and I think that's a really good formative assessment I really don't want to get behind exactly which commercial assessment is best but I would think assessments that ask children to solve problems and they give you some evidence of how they solve the problem are better than something where you just see how many right answers they got and you would need to look at the tools yourself and then another question somebody had was do we have any resources to recommend about number sense and the two that I can definitely recommend are the young mathematicians at work series that we mentioned if you look at that landscape of learning slide it takes you to a website and they have a lot of different resources starting with younger children and moving up through I think middle school math in thinking about how children develop their number sense and they have books and they have videos you can go and check that out the young mathematicians at work and I think that the website is called context of learning but if you Google young mathematicians at work you'll get there and then our own website which is earlymath.erickson.edu I believe that's correct but anyway look up Erickson Early Math or the Early Math Collaborative we have a website and there's a lot of videos of children there's also other activity ideas at various grade levels from pre-K through third grade there's some book examples on how to get math out of certain books and we also our collaborative has written a book called the big ideas of early mathematics what teachers of young children need to know and you can get more information about that at the website and that goes much more deeply into the big ideas that we just touched on very briefly here and the website is earlymath.erickson.edu but anyway if you put Early Math and Erickson into your Google you will definitely get there I've tried it or our name the Early Math Collaborative but we have a variety of ideas there and we can lead you to the book we really tried to make it a book that is approachable for all sorts of people giving examples of how young children do math the examples in the book are from preschool and kindergarten but we know that many first and second grade and third grade teachers have also read it and found it useful for their understanding the foundation of where things go with their children so thank you very much for joining us and asking questions and I'm going to turn it back to the National Working Group folks to give you some concluding information about how it is you will get your recording of this webinar and any more information they have about upcoming webinars. Fantastic thank you so much to all of you on the Erickson side that was a thorough and interactive and fun understanding of all of your work so we appreciate that so we want to as always thank the Bill and Melinda Gates Foundation for their support and helping to make the technology behind all of this happen although I have to say all of our presenters do this out of the goodness of their hearts to get good information out to the field so thank you to the Early Math Collaborative and to Erickson Institute. At this point we do not have our next series of webinars planned however if those of you on the phone have suggestions of what you'd like to hear more of whether it be related to a specific initiative you've already heard from or other content areas that you have not heard about yet please go to our website and you can find our email address there send us your suggestions because we do plan to put together a next series of webinars that will be coming up this summer or late fall so again thank you to everyone who participated thanks to our presenters and we hope you have a great rest of the afternoon.