 So welcome to Early Number Sense, Edition and Subtraction, the Landscape of Learning K2 to you with Kathy. We are so glad that you're able to join us in this webinar that was developed with ERLC and a grant from Alberta Ed. And I will now say hello and turn it over to Harry. Hi, Harry. Hi there and welcome to everyone who's participating in this webinar on behalf of the Edmonton Regional Learning Consortium. I told Kathy just a few minutes ago that I'm really grateful that she's willing to play with us in this way because this really means that we have the opportunity to follow up the conversation. And I think it's just wonderful, you know, when I see your names up in the list of participants, we don't have a huge group today, so we have this absolutely amazing opportunity to have this conversation with Kathy Fozno and I know that you're as excited about it as I am. I have to tell you that as I'm sitting here in my office and as Juan pointed out, we don't have to travel to participate in this session, none of us. But as I'm sitting here in my office, I look over at my bulletin board and I've got a quote up on my wall that says, Our work is driven by the desire to transform classrooms into communities of mathematicians, places where children explore interesting problems and like mathematicians engage in crafting solutions, justifications and proofs of their own making. And that, of course, was written by Kathy and it's something that inspires me on a daily basis as we plan our professional development. So again, I'm really grateful that Kathy is with us today that she's willing to participate and experiment with us in this way and please take advantage of this conversation. And certainly at the end of the session as well, please give us advice and feedback on the kinds of things that you would like to see the Edmonton Regional Learning Consortium doing to support and compliment the great work that you're doing in your classrooms as well. So welcome, Kathy, and I'll turn it over to you. Thanks, Harry. And hi, everyone. It's great to be here and this is really exciting for me. I haven't done many of these webinars, so we're really piloting this together. All right. Let's go to the first slide. And I just want to start by asking if there are any general questions that people have given where I left you when I was up there? Are there things you've tried that you're struggling with or excited about? Just a few general comments or questions first before we get underway. If you want to stick a smiling face up there or a hand, perhaps saying I want to say something? If not, we'll go on. All right. Well, feel free as we go on. You can stop me at any point. And I want you to really feel that you have the chance to get your own questions answered, too. So feel free at any point in our conversation tonight to stop and discuss what you feel you want to help with. All right. So this is the first slide I put together. And it's actually not math. It may look like it because there's numbers there. I told all of you that I was a painter in my free time. I never have much time to do it, but this is an old one I did a while ago. You may know Sudoku's. If you know Sudoku's, could you just put a green check up or a smiling face, something so I know you know what those are? Great. Michelle, what about you? Do you know what those are? Should I explain it? Okay. A Sudoku is a math puzzle that if you look across the column, sorry, across the row. Yeah, thanks, Wanda. You'll see that the digits 1 through 9 appear in somewhat of a random order. But there's 1 through 9 appears. And if you look down a column, 1 through 9 must also appear. And then if you break the whole array up into squares that are 3 by 3s. So yeah, that's great. Wanda sticking a nice dark line. And there you go, Wanda. That's up one more. Can you move it up one line? You got a 4 by 3, Wanda. Can you move it up? There we go. So up in that left upper square, there's also a 1 through 9. So with the Sudoku, you have to work your puzzle so that 1 through 9 appears in every row. 1 through 9 appears in every column. And 1 through 9 appears in every small 3 by 3 square. So anybody notice anything about the colors? Take a look for a minute at the colors of the numerals. Thoughts, questions, things you're noticing. OK, Michelle says I did know what 1 was, but I just hadn't done 1 before. Great. So all the 1s, yeah, Wanda says all the 1s are in the same color. They are. Notice the 2s are all the same color. The 3s are all the same color and so forth. So I had this idea that where the number had to be mathematically is also where it has to be aesthetically. In other words, if you do the mathematics first, the mathematics produces the art. Anyway, I thought I would just start off with giving you something to ponder about. Yeah, cool connection with math and art. You might want to try some other Sudoku, see if it happens again. OK, let's go on because that's not really our topic tonight. This is our topic tonight. I'm going to focus tonight on mini lessons for addition and subtraction. Hopefully you had a chance to try some mini lessons. Can you just put a green check up if you have tried some mini lessons since I saw you last? Yes. Yay, good for you guys. Super. Good. All right, so we're going to work pretty deeply with those mini lessons tonight. We're really going to explore the landscape of learning and look at kids. So here's a warm-up mental math string for you to begin with. This one's for you. If you would just work on the problems, obviously, if I were doing this in a classroom or you were doing them with kids, we would do one at a time and we might have discussions. But I'm saving a little bit of time here because I want to have you focus on something a little bit different. I want you to look at the numbers that I have put in my string and think about how are they related? What strategy does this string support? And what big ideas underlie this strategy? So, Wanda, maybe you could give us a little timer, maybe two-minute timer. Everybody can take up two minutes and actually solve the problems and think about how the numbers are related. And then I want to have conversation on what strategy does the string support and what big ideas underlie this strategy? Actually, if you feel like you've had enough time and you can stick a green check up to when you're ready for discussion, and then I'll know we won't have to go the full two minutes if we don't need it. Okay, thanks for that time, Rwanda. Lisa, I see that you checked it off and a bar head actually just did too. Gail's still thinking, Michelle's still thinking. So, Lisa, maybe we could start with you. What strategy do you think the string supports? All right. Okay, Lisa can't type in the chat box and I think she doesn't have a mic either, Wanda, correct? So, maybe you could give her a hand and how about I see that, Gail, you've written something in. You've got a mic too, Gail. Why don't you pick up your mic? I'll turn my mic off so we can hear you better. Why don't you tell us what you're thinking? I was thinking about the compensation where you're just moving one number, where you're going 39 to 40, so 92 plus 40, and then taking away your extra one at the end. Or the same with the others, changing 99 to 100, and then going from 100, adding the 32, and then taking away the one. Does anyone want to disagree with that? Anyone have a different idea than what Gail suggested? Bar head, what about you? Do you agree with her or disagree? Well, we noticed that you get the same answer in the first two, but the second one is easier to do. So, if we can just move around the ones numbers from the first number to the second one, we can make the problem as friendly as we want. That's interesting. So, Keith, what you're saying is that 99 plus 32 is a whole lot easier to solve than 92 plus 39. So, even though we could make 92 plus 39 into 91 plus 40, it might be a whole lot easier if we thought about it as 99 plus 32, and then we may get 100 plus 31 and use compensation there instead of the compensation with the first problem. That's really interesting. Yeah. And what happens then with problem three and problem four? What kind of switch has happened there? You want to put a smiling face up, or how about a hand? You want to put a hand up if you want to respond to that? Take a look at problem three and problem four, given what Keith said. We could, of course, solve 139 plus 94 is 140 plus 93. It's still a little messy, isn't it? But if we think about it as 134 plus 99, then it's a whole lot easier perhaps to think about it as 133 plus 99. Can you type in right here, Wanda, for us? Yeah, thanks. So, now using that strategy, what happens when we get to 199 plus 34? It's already pretty easy, right? We could think about that as 200 plus 33, 129 plus 97. Anything interesting to do there? Could we turn 129 plus 97, for example, into 127 plus 99? Can you type that in for me, Wanda? 127 plus 99. And we can turn that right into 126 plus 100. So, we're doing some interesting swapping here. What's being swapped? Anybody want to put a hand up to respond to that? What's being swapped? It's not just compensation. Take a look at the digits. We have barheads responding. Not only can we move the ones and tens around, we can also move the hundreds around to make even a friendlier number. Very cool. Yeah, yeah. So, where can we swap and where can't we swap? In other words, can we swap a unit digit into a tens place? It seems no, right? Why not? Why not? If anyone wants to respond verbally rather than typing, feel free to stick a hand up so I know. Remember to turn your mic on. Yeah, they have different values, Gail is saying. Yeah, if they have different values, then, of course, we can't. But if it's in the same value place and we switch it, we can make our problem really friendly. And turning 39 into 99 makes the problem nice and easy. Or turning 94 into 99 if you're trying to, in other words, swap your 9 and your 4 like in the third and fourth problem. This strategy actually was not on my landscape initially. This is a strategy that a second grader invented. We were in the middle of doing strings. In a school, I think it was in Missouri many years ago. And a girl made a mistake and wrote the problem down backwards. And then she got the right answer. In other words, instead of writing 92 plus 39, she wrote 99 plus 32 by mistake. And then she got the right answer. All the kids were trying to figure out why. And that lovely conversation ended up developing into this beautiful strategy, which now is on our landscape. And we call swapping. All right. So the big idea is that underlie this strategy. If you string it out, however, and think about, let's do 92 plus 39. And what if we were to write it as 90 plus 2 plus 30 plus 9? So there's 92 plus 39. So if we turn this problem into 99 plus 32, what's really happening? Is there a property involved? Because we're really turning it into 99. So what we're really saying is that this is equal to 99 plus 32. So to group it that way, what properties have I used? I didn't go now left to right. If you're having a conversation and you want some time to think about this, could you just put a green check up so I know? OK. I'll repeat the question. We talked about the strategy. We labeled it swapping. You actually swapped the digits. And you said it's because of their value. And I said yes, place value is a critical underlying big idea here. But there's something else at play. It is on the landscape. And so I asked you to think about properties. Gail's saying the commutative property. Gail, can you turn your mic on and tell us what's been commuted? Because I'm also seeing somebody else right with the yellow. Can you turn your mic on, Gail? I'm seeing the same thing as I'm seeing what I see in the yellow. That's what I was thinking about and how they're just being exchanged around. So when we exchange it around, so if we put the 31st and then the two, that would be the commutative property. But we're also associating. So we're deciding what pieces to associate to. We're actually turning the nine and the two. We're commuting them. But we're also associating the nine with the 90 instead of the two with the 90. So both the commutative property and the associative property are the underlying big ideas of the swapping strategy as well as place value. And so when we analyze the strings, if you think about how the string was constructed, notice that on purpose, the first two problems do give the same answer. I mean, a kid, of course, could use compensation here for the first one, but that's a strategy that he might already know if he thinks to use it. A string isn't designed to give people opportunities to use what they know as much as it's designed to support the development of a conversation around an interesting strategy you want to try to develop. So you need to craft your string in ways that are going to bring the relationships up to discussion. If I wanted to craft a string on compensation, I would have crafted it differently. This one was really crafted to bring up swapping and then a conversation on how if we take all of these add-ins, and I can't find my little hand seems to have disappeared, Wanda. Can you put a hand over by this 90? Yeah, thanks. Move it over here by this 90 plus two that I've written. Yeah, that's fine up there, too. So if we really realize the underlying big ideas when the conversation starts in our classroom, it's important to examine why the swapping is working, not just that we can do it. And so by stringing out the add-ins as we've done here and then looking at what has happened that we actually have commuted the nine and the two, and then we've associated the nine with the 90 and the two with the 30 is a way to help children begin to examine both the commutative and the associated properties. Okay, let's go to the next slide. Stop me at any point, too. If anyone, if you want to say something or have a question about what we're discussing, please stop me. You know, put a thumb down or give me an emote so I know what you're thinking. Okay, now let's go to kids then. So we're going to go back to the real little ones. We're going to go to a mini lesson that could be done in K1. This is one of our billboards. I think I described some of these in the workshop when I was with you. There are three here, three shoe billboards. What I'd like you to think about in this one is how is this crafted? Here's one. One is pointing to the first one, the sandals. The second quick image that you would show after discussion on the first one is the middle one. And then the third quick image is that one. Now it is a string, meaning that these three problems are related. So what I want you to think about in this is what strategies would you expect to see and how would you represent them? So if this were a string that you're going to use with your K kids, what strategies would you see as the purpose? And then how much you represent kids thinking? And it will just show me with a green check when you've had enough time and want to start discussion. Yeah, feel free to use the whiteboard if it helps you. How are the three images related? That's a nice way to think about it. What strategies are children likely to say? You've shown the first one and you're asking how many shoes are there? What strategies do you expect them to use? Would you put your hand down? It doubles. Okay, who's responding with doubles? Can we tell who that is? Can we just stick a one of seven hands raised? Okay, that was Barhead. All right, Barhead, let's go to you. I'm going to turn my mic off so you can hear better. Would you tell us why you think doubles is involved? Who's going to speak? Looking at the first set, you have four on the top row and four on the bottom row, so four plus four. Whereas the next one you have four on the top row, four on the second row, or you're trying to do four plus five. So you use the doubles four plus four and then four plus five. And so the strategy that you're thinking is involved here is that if kids know what four and four is, then four plus five can be solved by doing four plus four plus one. And if we use that same idea and go to the third one, now we've got four plus three, but kids might also see that as six plus one, given the way they were arranged, or as three plus three plus one. They also remember at birth. We talked about this a little bit when I was there with you in the face-to-face, a little bit about children's ability to sabotage at birth. So groups of twos and threes, we know kids can see as a unit. So doubles, kids may not know four and four, but they may be able to think about it as two and two and two and two. And so the shoes in the first billboard, kids might do two, four, six, eight. They might say, I see four plus four. They might say, I see two plus two plus two plus two. And you could use circles and actually, whoops. Yeah, there we go. How do I get my circle around there? And you could draw the circle around. I can't seem to get that to move one. I was trying to draw a better circle there. That's better. Thanks. And so whatever kids say, the circle can be drawn around it as well. So yeah, this is about doubles and near doubles. And we're going to go to the landscape for a minute just so we can look at some of the things that we've been talking about. And we go back to the landscape. So this is the bottom part of the early childhood landscape. And let's see. Wanda, I've forgotten how to do it. What I'm going to ask you to do, Wanda, if you don't mind, is if somebody types in the highlighter tool, that's it. Right, thanks. OK, so there's a highlighter tool. You'll see it also on your screen. And if you would click on it, and if something has come up and you see it on the landscape, for example, one of the things that we just talked about in that last one was kids use of doubles. So there it is. Right? If anyone sees anything else on here that's sabotaging, we talked about sabotaging, counting, counting by ones. Kids could count, of course, if we go back and take a look at that earlier slide, kids, of course, could count by ones, couldn't they? We hope they're not. We hope that we've built the mini lesson to support them to go beyond counting by ones. But of course they could. Do you see any other ideas that might come up? What about this one? How about equivalents? It's seven. It can be thought of as four and three, or it can be thought of as three and three and one. It can be thought of as six and one. Anything else you see there? About cardinality. However, many kids say, knowing that it's a mount, anything else you see? Conservation. I'll color it in. If you are struggling with the tool, would you put your blue face up there? So I know I'm trying to find out if it's tool use or if you're thinking about the landscape. So if you're struggling with the tool, put the blue face up. If you're struggling with the landscape itself, put the blue face up. Okay, so Barhead's having thoughts about the landscape. Everyone's in deep thought. Michelle is still thinking about the landscape. I see somebody has circled skip counting. Highlighted it. Yeah, because kids could go two, four, six, eight. What I'd like to do, I'm going to come back to this landscape. I know you're all in deep thought right now. I want to go to a classroom of some children so you see what the kids are doing and then we'll come back to the landscape. Okay? I'm not going to leave the landscape. Don't worry. So we're going to go to a first grade classroom. We're going to take a look at clip 18. Wanda's going to load it up for us. And we're going to see a teacher doing a mini lesson that is about kids playing tag on the playground. And she tells the story that she was on the roof and she looks down and all she sees is heads. And she's trying to figure out how many kids are playing. And this is one of our crazy schools in New York. You actually can go up on the roof and the teachers do do that. And so she uses an overhead projector and little circle discs and the kids are going to be telling her how many kids are playing and how they know. And then we'll have a conversation on her recording. Okay, Wanda, can you play the clip for us? You bet. I'm just going to jump in for a quick second. This should start automatically for you. If it doesn't, just check to see if there's a little play button. And if you don't mind, once the clip has finished, if you guys will just click on that green check mark because it'll be just a tiny different rate for everyone. That would be great. So thanks so much, and here it comes. Okay, so that was the end of the clip. I saw some blue faces, so let us know if you were able to hear the teacher and that little boy talking or not. Okay, if you were not able to hear it, that may be something to do with the players on the computer. Then let me know if for backup, if you have the disk that you got during the face-to-face day, I'm not sure if you're able to bring that with you today. But we could also watch it, everyone could watch it individually off of their disk as well if you have that in your computer. The page you see on the screen, it's in the folder, mini-lessons, and the page that you see on the screen is the page you want to find on your own CD. We've just played that first clip, and if you run your mouse over it, it will say clip 18, and that's the one we just looked at. I just got connectivity back, which mysteriously disappeared, and she does not have the CD. Okay, maybe you want to play the clip one more time for us, Wanda? Maybe Lisa would get it now? Uh-oh. Okay, Wanda, we're going to have a problem if nobody's able to see the clips except you and me. We have a bit of a problem. If you brought the CD with you, would you put a green check up so we know, because we can give everybody time to load it up if you've got it with you. Okay, Michelle has it with her, great. Go ahead and load it up. Gail doesn't have it. Lisa doesn't have it. Bar head, what about you? No. Wanda and Harry, what do you want to do? I mean, we can reschedule this, but without the video, it's going to be difficult to do what I've planned here. We've got several video clips to look at. I agree. Well, let's try this then. This is kind of around the boat wave, but how about I'll play it and I'll try to share my desktop while it's playing, and then you guys let me know. It might be a little bit herky-jerky, but we'll see if you're able to see it and hear it. Okay. I'm going to have to crank my volume. One second, I just have to do a tiny little fiddle. Okay, Wanda, I could see it fine. It looks like Harry you couldn't see it on the screen. What about the rest of you? Could you give us a signal so we know whether you were able to see it? No. No one's been able to see the clips. All right. Okay. So, I mean, we can do a few things together. I've planned some things without clips, and we can do those together. Unfortunately, a hunk of what I also wanted to do was with the video clips. Well, maybe you're right, Kathy. Maybe the non-clip stuff for now, maybe if it works at another time to get together and meet up and then do sort of the clip bits almost around here. Yeah, we'll do the clip bit. Okay. All right. So, let's go away from these clips, and I'm going to have to skip this next slide of the landscape because what I wanted to do was analyze with you the clips and what the big ideas were in the clips that the teachers were having discussion on and then also converse with you about where those ideas were on the landscape. We'll save that for next time. And I'm going to skip this one too, and I'm now going to go to a Grade 2 classroom. This one does not have a clip. So, I'm going to ask you to think about this string and think about how the problems are related, and I'm going to ask you to vote. Do you think this string has been crafted around compensation, splitting, jumps of 10s in adjusting, or moving to a landmark number? I guess I could put E's, none of the above. So, if you would want us to put the letters up for us. Yeah, great. Thanks, Wanda. Lisa, can you give us a thumbs down if you just need more time, or you're ready? Okay, forehead, what about you? What are your thoughts? All right, looks like everybody's saying C. So, somebody put a green check up there who'd like to try to convince us. And then turn your mic on. Who wants to convince us? You have lots of nice consensus, so it looks like everyone seems to agree. So, let's just hear one argument as to why. Let's see if it's the same thing everyone else was thinking. Gail, how about you? Can I call on you? You tell us what you're thinking. You voted to C. Great. So, tell us what you're thinking. That's okay. When I was looking at it, I was seeing the 43 plus 10, the 43 plus 20, and then just one number off on the 43 plus 19. So, that's why I thought it was easy to compensate for the next 10, and then go from there. Yep. So, the 10 plus 10 is the helper problem. So, that sets the stage. That's why it's the first problem in the string when I wrote it. So, as you write your own strings, you want to think out what kind of helper are kids going to need. If I put the easy problems up first, it might help them with the problems that come next. So, plus 10 is the helper. If that's too hard for most of the kids in your class, this is not the string for them. If they can handle plus 10, then plus 20 is a little bit of a challenge. Can you take another jump of 10? Another jump of 10. How easy is it for you to take jumps of 10s? And then we come to the problem where the third one, where we're hoping plus 19, might be thought of as plus 20 minus one. And then 68 plus 23. Notice I changed the 43. Here, I made it as 68. That was random. I changed it because now I want kids to begin to use what we've been talking about. So, will they think to do this problem as 68 plus 20? Or 68 plus 10 plus 10? And then add three. And again, it's not to make every kid use that strategy. That's not the goal. Except what children say. But your numbers are chosen very carefully to support children to start making use of taking jumps of 10s and then adjusting. 39. Notice I turned. Can you point to 39 plus 54 for me, Wanda? Thanks. Notice that here I put the 39 first. And yet, maybe kids might do 54 plus 40. That's minus one. So, the string is getting progressively harder as you go through it. And that's the beauty of the crafting of them. All right. Let's go to the next slide. Here's another grade two string. Same conversation. Think about what's the focus of the string. And then vote. All right, Barhead. Tell us why you like A and C. Turn your mic on and tell us why you're not sure which way to go and why you like both. Well, the first one, the helper, the one right underneath it is compensation. And then the next one is a double. And it seems to be double plus one. We didn't see that one as a compensation for 26 plus 25. But the 32 plus 28 could be. It could be 30 plus 30. Double plus one. Double. I think the big message is that look at the numbers before you decide on a strategy. So, at the very end of the landscape, of course, we want kids to look at all the numbers before they decide on a strategy, like Harry said. But, of course, we are also, as a teacher, crafting our strings to support the development of a repertoire of strategies. Because if we haven't done that first, we can't expect kids to just invent all of these strategies on their own so that they do have a repertoire. So, I think, Keith, what I'm hearing you say is that both are involved because to turn the double into a near double, often one does use compensation. I think that's your point. Can you just show me, if I'm understanding your point, can you just show with a face, whether you agree with my paraphrasing? Yeah, I said, yeah. And I think you're right. Why don't you let us know the rest of you how you're feeling about that. If you agree with Keith, you know, stick a face up there so we know whether you're agreeing or disagreeing. I think what Barhead's group, what the group is saying is that 49 plus 51 is a near double. But it requires compensation in order to make use of that double. And so the string is really supporting the development of both. Yeah, nice. Okay, let's go to the next slide. All right, here's one on subtraction. Kind of maybe a grade 2-3. You'll just show us with a vote. All right, so Lisa, I'm going to ask you, you're sort of the lone soul out there on this one, aren't you? So tell us what you're thinking. And maybe you can convince some of the folks that voted for Dee. I think you only have a chat feature. So Lisa's going to write her response in the chat room. All right, somebody want to respond to her? Lisa, is everybody seeing what Lisa's written? She says she was thinking of using compensation to preserve constant difference to create simpler equations. So I'm assuming you mean, Lisa, that you would take us look at the second problem. Are you suggesting, yeah, I'm going to write. I think what I heard you say right here, I think this is what you mean. That you would take three off of the 43 and turn this into 195 minus 40. Would you just, I'm going to put a question mark about around that. Is that what you mean? To just show us Lisa with a smiling face or a frown if that's what you mean? That is what she means. Okay, so I'd like everybody to think about that for a minute and actually solve both of those two problems. Do we get the same answer? And then if you would just show with a smiling face or let's use the hands. Put a hand up if you think Lisa's strategy works. Put a thumb down if you think it doesn't. We've really got a mix here. Bar head, let's hear from you. Can you turn your mic on and try to convince us? You disagreed, right? So could we hear from you as to why you disagreed? Well, when we're adding two things, we can compensate but when we're subtracting, we can't. I don't know any other way. There's no balancing when we're subtracting. Do you get the same answer for both of these problems? No. Lisa, what do you think? Has the bar head group convinced you? Lisa's in the chat room now and she says yes. Yeah. The bar head group, I agree with you. Compensation. If you look at the difference between, Wanda, can you put the hand on the number 192 in the second problem? Thank you. If we look at the difference between these two numbers, 192 and 43, think about what the difference is. And now think about what's the difference if you're at 195 and 40, the answer to those two problems isn't going to be the same. So we just made the difference in that second problem, 195 minus 40, the difference is larger now. So compensation doesn't work. But I think the other thing to take a look at here is if we look at, remember the first problem is always the helper in the string. And the numbers are chosen not to give kids opportunities to use a strategy they already know. If they know all these strategies, we don't need to be doing strings with kids. We could just give them bill sheets of unrelated problems and just ask them to solve them in as efficient ways as they can and then have them share their strategies. That's not guided reinvention. That's not supporting development. That's a test and then just having kids talk about what they did. If learning was so easy in mathematics that one could just tell a strategy that you used, why wouldn't the teacher do that? Why do we need kids to do it? The point is that math isn't learned just by being told. One really needs to engage in thinking about relationships with number to make sense of strategies. And so by picking the numbers in the strings really carefully and structuring the string with relations that are going to bring things to the fore for conversation, we really are supporting the development of children. So to go back to this first one where Wanda has written helper in pink, that's the easy problem. And that's the one that we hope kids already know the answer to. Because now when we do the second problem, that's a harder problem. And in fact, many kids will try to use an algorithm there that somebody may have shown them. Or they'll make all kinds of problems, they'll start trying to count backwards. It's hard. But if you already know what minus 40 is, then minus 43 is easy because it's only three more. You just have to take three more away. So the second problem is tightly connected to the first one. Now when you get to the third problem, the helper has not been provided. If kids are struggling with this third problem and you felt it was too big a leap and you wanted to give a helper, what would the helper be? Talk about that. What would the helper be for 378 minus 39? And when you think you know, I think I'm going to ask you to just use your faces when you think you know. If you put it on the whiteboard, then somebody is going to see it. Now I want people to have enough time to think about this. So just show us with a face when you're ready to respond, when you have a helper that you would stick in. All right. I think there's two groups still thinking. But Lisa, you responded pretty quickly to that one. Would you chat, would you put on the whiteboard the helper that you're thinking of? Okay. You did it in the chat room. 378 minus 40. Well, because if 378 minus 40 is easy, then 378 minus 39 is just taking away one less. Now, of course, kids are going to have, they might feel a bit of a challenge here. They might think, do I take another one away or do I add one back in? But that's what you want them thinking about. Because as Hallie said, on the horizon at the end of all of this work, maybe by grade three, end of grade three, we're hoping we've developed grade one and two teachers. We're hoping we've really developed a whole nice repertoire strategies for addition and subtraction. And kids have to be able to mentally, you know, in grade three, we'd want them mentally be able to just say, oh, 378 minus 39, that's easy. I could do 378 minus 40, you know, and then adjust. Okay. Let's go. Now, there's a clip here. And I think what I'm going to do with the clips is the following. You've all got this CD-ROM that Wanda and I took these clips off of. And Wanda, you can leave the PowerPoint up. Correct? Yeah. And so what I would like you to do, all of you, is the clip numbers for you to find. And the picture is on the PowerPoint here. So you can see if you can use the hand, Wanda, and point to the clip of, clip 76. Yeah. Go down to the icons and keep going forward right there. Those are the icons for starting the video. And this is the page. It's one of the mini lesson pages in the folder of the CD-ROM. I gave you marked mini lessons. And if you take a look at 76, you'll see a boy there that does solve the second problem in that string. And then you'll see a girl, and he decomposes to solve it. And you'll see a girl who uses constant difference. But then there's another clip. And this is a discussion on the third problem in that string. And to remind you what the string was, I'm just going to go back here for a minute. The third problem is 378 minus 39. At least it told us a really nice helper would be 378 minus 40. The boy in this clip, clip 79, struggles with this problem something fiercely. He starts decomposing it all. He starts with 70 minus 30, eight minus nine. He gets one. And then other kids say, no, it's not one. It's minus one. And then he doesn't know whether to subtract 39 from 300 or add 39 to 300. You'll hear the discussion on this clip where he really is terribly confused. And many of the children in the classroom become confused with this third problem, which is a signal to you when you're in the middle of doing strings. I just took a jump too fast and too far. And I'm not scaffolding enough. So you've got to think on your feet. You can't just pick up one of my books and say, but I'm doing the string on page 35 in Kathy's book. Wouldn't teach him to be so easy if we could all do that, right? Instead, you have to think on your feet. And you have to think, OK, the string isn't working right now. What am I going to insert? So Lisa's suggestion is one. And what you can take a look at in the next slide, which I'll put up for a minute. You see, would you continue with the fourth problem if you see that kind of struggle happening with problem two, but then particularly with problem three? And I would hopefully answer to that would be no. You don't have to think out, but how are you going to finish the string? So what I want you to do right now, even though you haven't seen the video clip, you can look at it later. And what I'm telling you about the video clip is that the children struggled through problem two and really struggled at problem three. With problem two, they eventually got it and began to see how easy it was to just take three more away. But then the third problem, they were totally lost again. So what I want you to do right now is craft a string, just two or three more problems, thinking on your feet. What would the next two or three problems be that you would do instead of problem four and five? OK, if my question is clear, could you just show me with a smiling face or a blue face if my question is clear of what I want you to work on? Great. What about you? You know what I want you to work on? You just show us with a face. OK, good. OK, so everybody be working on that. And Wanda, maybe you can give us a whiteboard. And when people think they have just two or three more problems. In other words, imagine problem three didn't work. And so now you're going to put up a new problem four, a new problem five, and a new problem six. What are your three problems going to be? And then type them, when you finish, type them on the whiteboard. You might want to use your, this time your A in the text bar so that your string is within a rectangle. And I don't know, Wanda, do you want to use a timer here so people have a sense of why don't we stick a timer up? Let's give people two minutes again. When you put your string up, could you also type in your name so I know who wrote it? Just underneath it. Just type your name. OK, this is hard, isn't it? I know. I mean, doing strings is hard enough. But then when you really have to think on your feet and think out, what would I give if it's not working? And of course, often what we come in with as our plan doesn't work. And we do have to think on our feet. So I see bar heads because you've got your name on it. The 192 minus 41, the first one up there. Who's this that? Could you just put a green check up so I know 192 minus 41? Could you signal me with a green check? OK, so the first one up there is yours, Lisa. OK, and the one, I see that Wanda is typing one in. And the one on the bottom, 252 minus 30. Who's this that? Can you put a Lisa? OK. No, Lisa is the 192. For the one above. OK. All right, so let's discuss what we've got up there so far. Let's take Lisa's first. Only because you put hers up first, Lisa. So Lisa, you went back to the 192. So 192 minus 41. Let's all think for a minute. What might children do with that problem? They've already solved 192 minus 40. So most likely they're going to take one more away, right? I would think so. Although it's interesting, they might not only because there's no regrouping or anything needed, it's just going to be 2 minus 1 is easy. Its kids know the algorithm. And sometimes kids come in knowing the algorithm even though we've not taught it initially up front. We're trying to do mental math, but a sibling shows it. And then kids get habituated to using it without deeply understanding why it works. So Lisa, your string is going to encourage kids to go from the help of minus 40 to one more, one more, and one more. And it's certainly going to help them solve 192 minus 43. But what I'd like you to think about is you've guided the kids to get the answer to 192 minus 43. But then we still have to think about, but have we really done a string yet? In other words, we've got four very tightly, very tightly crafted problems that lead kids to the ability to get the right answer for 192 minus 43. But they may not really be thinking about a strategy here or a big idea only coming backwards. They didn't struggle as much on 192 minus 43 as they did on the 378 minus 39. So what our goal really needs to be is to help kids begin to think out, OK, if that's where the struggle really was, what are we going to come in with to help kids begin to think about problems like that? I see that, Barhead, you also started with the 192 minus 41. And then you played around with that a little bit by doing 39 and then 42. That's a bit more of a challenge. And that's nice because you're really making kids have to think now about the adjusting. Let's take a look at what Gail has done. Give me a signal if you think you know what Gail is doing with her. I think this is very nice, Gail, what you did. If you think you know what Gail's playing with and why she went to those numbers, could you just put up in a moat face? In other words, if you're confused and you have no idea why Gail went to such big numbers, put up the blue face. And if you're thinking, oh, I never thought about that, but I see what she's doing and I like it, put up a yellow. Okay, Wanda wants some thinking on this, Lisa. Not Lisa, I'm sorry, Gail. Gail, you have a mic, right? All right, so Gail, could you turn on your mic and tell us what you were thinking? As I was going through, I was trying to go back to something similar to what they were comfortable with with taking away the whole group to 10 first. So I went to taking away the whole group to 10, but I wanted a fresh number so that they were thinking about what they were doing again. And then I went to just changing it by one. And then I wanted to see, on my next one, were they getting it again this time, now that they've seen it a couple more times and have been kind of helped through it. And there, I think you've really, you've got a really nice handle on the crafting of strings because what you've really done is you've provided another cluster. You aren't just providing problems to get kids to your answer. You're providing a cluster to get them to think about it again. Bari, have you played around with that a little bit? I see that you're trying to jump around there to get kids to really think out, do I add one? Do I take one away? How do I adjust? And that's nice too. But what I particularly like here about what you did, Gail, is you gave another cluster to get them to think about it. You provided a helper. Remember, Lisa, you said earlier, what the helper would be for 378 minus 39 would be 378 minus 40. So that's sometimes a helpful thing to remember as well. I'm going to go to the next slide for a minute. This is the teacher's revision. So, you know, the kid, Daniel, boy, in that clip, struggles on 378 minus 39. What Michael does when Daniel finally finishes is he provides Lisa's helper. So that Daniel and other kids can then go, oh, my gosh, that would have been easy. I could have used that. If they don't do that, the teacher can then say, if this problem had been in there, would that have been helpful? Could we have used it? How would we use it? And then the 19 was chosen by Michael in the last problem. You could move the hand all the way down, wander right to that one. The 19 is chosen because it's a small number. And again, to encourage kids to think about, well, what could we do here to help us? And again, if kids struggle, and if you think that that's too big a leap right away, I would do 371 minus 20. And then 371 minus 19. It's often nice to end your string without a helper so the kids have to make it. The purpose of the string is not to always craft it so tightly with all the helpers that the only thing we've done is sort of held the kid's hand to get a strategy up that we wanted him to use, but really there's no empowerment. There's no thinking that's really happened. It's more that we've led them to our answer. So there are some more clips here that one can look at. And again, on the PowerPoint, which Wanda will have up for you, it tells you the clip number. And this one is clip 83. You'll find it. The PowerPoint also shows what it looks like so you know what page you're looking at. Wanda, can you move the hand right up to the top of where it says mini lessons up on the top right there? That's how you know what folder you're in. And what I'm thinking might be a good thing to do is I don't know whether we really need to convene and talk about the clips. I think what would be helpful after a session like this is to go back and take a look at the clips that are in the PowerPoint and get familiar with that CD-ROM that I gave you, and then revisit the landscape. I'm going to go back to the landscape here now for a minute. This is the upper end. We looked earlier in the PowerPoint in the beginning of our session at the bottom of the addition and subtraction landscape. Here we're looking at the upper end of the landscape. And with addition, keeping one ad in hold and taking leaps of tens, we looked at some strings that are going to develop that. We looked at some strings about taking leaps of tens back and adjusting. The clip just prior to this that I can't show you is Michael doing a string with his kids where he's varying, adding on, versus removing. In the very beginning of our session, I had you do a string to get you started. It was a string on swapping. Getting yourself to a landmark number, we didn't look at that, but that is also something that is on the landscape. Using constant difference, we explored how that becomes, how compensation doesn't work for subtraction, but instead we need to keep the difference constant. We've got a little bit of time left. I'm going to return to the landscape I left you with earlier in the beginning of this. Fast forward through some of these. Because I don't want to short change those of you that work with real little kids and I left you kind of hanging when we were looking at K1 stuff because you said you were struggling with the landscape. If you take a look at the clips that, for example, the clips that are here, which you can do after the session, and if you would make a note, I'm going to type some things in. Wanda, can you give me a... Actually, no, I'll do it this way. We'll go to the landscape. What I want you to take a look, when you look at those clips, these are going to highlight some of the things. And I think, Wanda, can you save the changes? Can you give me a nod to that? Yes, good. I'm going to highlight some of the things I want you to take a look at when you look at those clips. I want you to take a look at this idea of equivalence, how strings are being crafted to bring up the idea of conservation, even when the teacher was on the roof looking down and kids are giving different answers. Also, if a kid is doing 3 plus 4, and he solves it as 3 plus 3 plus 1, think about hierarchical inclusion. It's not just doubles and using doubles for near doubles. It's not just using compensation. There's a kid on one of the clips who solves 3 plus... No, sorry. He solves 3 plus 4 plus 3 as 5 plus 5. In other words, he turns it all around. He thinks of the 6 as 3 plus 3, and he's got 4, and he moves the counters, and in his mind, he turns it into 5 plus 5. So what I want you to think out, sometimes it's easy for us to spot the strategies in the string. We say, oh, he's using compensation. Oh, he's using doubles and near doubles. But it's really important for us as teachers to also understand the big ideas that underlie all of these strategies because we're really developing commutativity and associativity. Not at all that kids need to know these names. They don't. But we need to help them understand why the doubles and near doubles works. And near doubles requires an understanding of hierarchical inclusion. It requires an understanding of equivalence. It requires an understanding of conservation. And so we've really got with our representation to find ways to circle what kids are saying and write out the add-ins and then look at the add-ins and how kids are moving them around and regrouping so that we're really developing in them this whole foundation for early addition and subtraction in the K-1 classrooms because that's going to be the foundation for when kids come to the last part of the landscape. And we're talking about the upper end in grade 2 and of course even though this is a K-2 session we know addition and subtraction does carry into grade 3. Teachers are still usually working on it. And all of these strategies that we worked on in grade 2 that are going to come back around again in grade 3, the commutativity, the associativity, the big ideas that subtraction can be thought of as removal or as difference. It can be used to find missing add-ins we can regroup, we can swap. All of this is about commutativity and associativity as well. So I hope you'll enjoy now going back and doing more mini-lessons with your kids and also taking a look at that folder. In this session I only use clips from the third folder which is called mini-lessons. So it would be really helpful if you could go back and take a look at the clips that are in that folder and really examine the children's thinking with the landscape in mind. And Wanda will leave the PowerPoint up for you. Yes, okay. I didn't see that Wanda thanks for noting it. Somebody asked what's cardinality? I could answer it but I think I'd like to see if somebody in the group feels like they could answer it first. Does anybody think they could respond with an explanation of what cardinality is? I'm thinking of a bishop. Not that kind of cardinal Wanda. Or have you a big group there, right? You've got four or five people sitting around a table. Anybody in that group want to take a stab at it? What's cardinality? We're still talking to each other. Yes. So actually I don't know who asked the question but I'm really glad you did because it looks like a lot of people are pondering it. Anybody want to give a shot? What is cardinality? Lisa, great. You don't have a mic but if you want to put something in the chat room the last number counted says the number in the set, my goodness. That absolutely comes right out of a glossary I think. Lisa, that's perfect. I wish I'd written it. So if a kid is counting, one, two, three, four, five and then you say, so how many do we have? A kid who understands cardinality knows that five is the amount because it's the last number that he said. You'll often hear very young kindergarten kids. Young, I mean, developmentally, young on the landscape. They'll go one, two, three, four, five and you say, oh, you count so nicely. So how many do we have? And they say one, two, three, four, five. Yeah, so how many is that? One, two, three, four, five. In other words, they almost think that you're asking them to count. They don't really deeply understand the purpose of counting. They don't really deeply understand that the number that you end on is not the name of the cube. It's really the amount of the whole set that you've got five things. Yeah, changing counting into the context of what you have. I actually never say to kids, Wanda, would you count these for me? The reason is then I'm telling them the strategy. Instead I put a bunch of things out and I say, how many do we have? How could we find out? Some kids will say we could count. Other kids will sabotage. Some kids might count on. Some kids might move counters around and make smaller groups using compensation and make it into something they know. So by saying how many do we have and how can we find out, opens it up into a more open-ended, for an open-ended response instead of you tell kids to count. They're using your strategy and you have no idea. You've told them what to do. You might even be keeping them at a very low level. What are the questions before we sign off? Anyone else have a question that's come up for you either in this session or one you thought of? All right. Well, I see that we're at our 7.30 time. So, you know, I really enjoyed chatting with all of you and it's so terrific. Wanda, you did a great job. Well, I don't know if you all could see behind the scenes, but Wanda was helping me a great deal with this. Yay, Wanda. Thank you. She was really all these signals. I was overcome with all the technology and remembering my PowerPoint and all the things I wanted to say. And forgot to use half the tools that Wanda taught me to use. And Wanda, behind the scenes, was moving them all and writing things on the whiteboard for me and so forth. I really appreciate it, Wanda. Thanks so much. I'm happy to be here. I just thank you as well for your time today. It's always interesting when the technology doesn't quite work in the way that you had anticipated and we're always being back admiring how you dealt with that and continued with the very interesting sessions. So, thanks very much for your time as well. You're welcome, Harry. Thank you. All right. Thanks, everybody. See you again. Kathy and Wanda. Have fun with many lessons. Yes. Kathy and Wanda, perhaps we could just stay on briefly to just chat after everyone else signs off.