Figuring Out Fluency: Week 2

Hey math friends, welcome to our week two book chat on figuring out fluency. If you are here watching live, even if you're not watching live, say hello in the comments. Let us know where you are watching from, how you're feeling about the book so far. I feel like I'm going to have to have a better like uh prompt for the chat because I think every week it's like how are you feeling about the book so far? Although I will say that reading your aha moments, I can tell that we are having um lots of light bulb moments. A lot of you are loving the book, which makes me so excited because it I just think is such a fantastic book. So, let me check my phone, make sure that we are live and that y'all can hear me and see me. Let me know in the comments if you can hear me. Okay. Um let's see. I can see lots of comments, so I know people are here. you know, last week I can see tons of comments and last last week I know we had like some volume balance. So, let us know if the volume sounds good for both of us and one of us is not way louder or softer than the other person. I cannot see any comments. I can't even see myself. Oh, there we go. Okay, got comments. So, welcome. Um, if you missed last week, also if you're here watching the replay, also say hello in the comments. We are so glad you're here. Um, I see we can hear you and see you. Wonderful. So, we have fixed the volume issues. Hopefully, that is no longer an issue. Thank you for those of you who let us know in the comments. Um, but for those of you who did are just jumping into the book study, welcome. If you were not here last week, the replay the replay for last week and the replay for all of these will live on the Mix and Math YouTube channel. So, if you can't come for any reason, you're welcome to watch the replay. But if you have not watched last week, my name is Britney Hegy. I am the original math brain behind Mix and Math. And then this is Christy. Christie is our lead math content specialist at Mix and Math. And we are actually, you're not going to hear too much from us tonight. So, we are here live with you now. We are going to chat here for just a couple of minutes. And then we are actually al together me Christy and then all of you watching in the comments we are going to watch an interview that I did with John Sanjiovani. He is one of the authors of the book figuring out fluency. I did a interview with him about this week's chapter a couple weeks ago and um so we are going to watch that all together. So Christy and I will be hanging out in the chat with you and then that interview is about 40 minutes and then after that her and I will hop back on live with you and we will do our Q&A. So we've pulled four questions that y'all asked on the Q&A post and so we will talk through those live, but of course while we're watching John's interview, we will be there to chat with y'all in the comments. Um, did I forget anything Christie? I don't think so. I'm I you I don't know if John needs an introduction. I mean, he's the author of the book, but he also is just such an amazing background in terms of a well- reggarded math educator. So, if you have not checked out any of his other books, he I know has several um in fact, we did one of his books last year, um productive math struggle. And so, I highly recommend all of his books if you besides this one if you need more to read. Yes. Although, don't jump into another book until you're done reading this one. But I will say um in our workshop that we do, I think I have a quote from him that um you know what, maybe I can go find it while we're doing the interview and pull it up because it's such a good quote, but he is a fantastic math educator. I think he is still um maybe like a you know what, I'm not even going to gonna say because I'm not 100% sure what his role is, but I think he still works in a district. And goodness, can you just imagine be being in his district and learning from him all the time? He is hilarious if you ever get to see him um present in person. He's wonderful. But we hope that you enjoy this interview with him this evening. So, we will see you in the comments and we will be back after this interview with John. When I play the video, if you could just let me know in the comments that you can hear and see it, that would be great. I'm like a little panicked with the technology after last week. I just want to start by saying thank you so much to you and Jenny for um writing this book. This book is fantastic. It is one that I think every single math teacher should read. It is definitely beautiful. Thank you. I appreciate that. And I know Jenny does as well. Uh we're proud of the work and we hope that it helps teachers, you know, continue to grow as as math teachers. Yeah. So this this week we're obviously talking about chapter 3. And I think that chapter 3 is so important because it really talks about that like foundation, the beginnings. It really sets the stage for building fluency. And I love that y'all just kind of like jump right in. On page 49, there is a section that I highlighted and it said, "Conceptual understanding of any topic is developed through the use of manipulatives and other concrete tools, visuals, drawings, and connections to meaningful situations." And so I love that you started in that place with this with these foundations. Obviously, this entire book is not focused on building conceptual understanding, but we cannot build fluency without conceptual understanding first. Yeah, that's that's a great takeaway. And you're you're right on the money. Um, conceptual understanding is is the foundation for everything. Do I understand an operation? Do I understand how to represent situations? And obviously that starts with a physical manipulative first. um and it proceeds to diagrams and then nodes um are connected back and forth before they're connected to equations and other representations. Um I think that's really important to know that we have to teach the operation and the properties within an oper of that operation and understand those concepts. But one of the things that can be overlooked is sometimes we spend too much time with a conceptual understanding and students then don't have that opportunity to grow out of it and become fluent in in the the computation or or whatever procedure that they might be working with. So we must start with conceptual understanding and make sure that's a really great place and then begin to move out of that when students show that they're ready. Yeah. And I love that you emphasize the importance of visuals and manipulatives. So, I was actually I've got a a first grader. He just finished first grade. He's moving into second grade. And so, it's so fun to get to do math with him at home. And, you know, he's really curious about multiplication. And so, we were, you know, I've kind of exposed him to it. he is, you know, working on some problems and he started explaining something to me and he's using mental math and he was he was actually doing the um the make a 10 strategy after he had you know done everything with the area model and he was like well if you have two nines you have like a nine here and a nine here and then you move one over and that makes 10 and eight. And I asked him I said what were you like when you were showing me these groups on a table there was nothing there. I was like what were you envisioning? and he was like, "Oh, cubes." But it was so neat to see him use that strategy without manipulatives physically there. And it was because he had that initial experience. And so I just loved that y'all jumped into like these are so important and it really builds like that mental picture that they use to basically apply these strategies later on. Yeah. So, I mean, it's funny you say that because every human, me and you included, use visual models and manipulations to um make sense of the world around us. And that could be through computation, it could be through spatial awareness with geometry and navigating maps and things like that. But um it's interesting to note that many of us actually think about numbers spatially in our heads and how close together they are and how we're manipulating them. though we may not process that or be act, you know, aware that we're doing that, that that's something that we do. And so that's why it's important for for your student or your kiddo and and mine and and everyone else is to really have a strong visual um foundation, if you will, of the math that, you know, again, we're going to call that through the rest of our lives. Maybe not in every situation. Many situations we just, for lack of a better word, know it, know what 40 and 40 is. But there are moments in everyone's lives as mathematicians where um it's not instant recall. And so we have to manipulate the math in some way and we call on those visual um visual thoughts even even if we aren't actively aware that we're doing it. So such an important thing. And then when you're talking about elementary students, what a great way to talk to them about what did you see? How did you see it? Um, and not just for the efficient ways that they're doing it necessarily, but also potentially for less efficient ways that they may be going about um, computing or or just thinking about math in general and not necessarily correcting them, but infusing new ways that they might think about it. Um, because efficiency is an important part of this conversation. Yeah. And I think it's it's always um I was actually coaching a teacher in kindergarten and um you know she loved to use the situations where students like got it right or they were really efficient and um was overlooking the opportunities where they were like just maybe a little off in their answer or um maybe it wasn't the most efficient way. And I was like those are such fantastic opportunities for us to dive into and discuss. And so that's very similar to what you just mentioned. Um looking at the the things that they do great and the things that they're still like developing. Yeah. And you connect that to like even basic facts when you're first dipping your toes into that or or multi-digit computation for that matter. But we might have a situation where students doing 7 plus 9 on double 10 frames, right? And that they're moving three from nine over to seven because it's the first addend to make 10 and and and some more. And while that's fine, um it's typically more efficient to build off of the 10 that or the the number that's closer to the next 10 or something of that nature. So, um again, having that conversation with students, having it with just people, people don't want to do anything that takes longer than it needs to, right? But at the same time, they may have perceptions about which addin they have to start with or which number they prefer. And so, exposing students to the way that others think, especially their peers, um in those situations is so important. And and it's not something that can't it's not something that has to wait till fifth grade or fourth grade like kindergarten students can have these conversations if we are um you know ready to ask the questions. Yeah. And they they are willing and I think in some ways we maybe have some of these conversations in lower grades and then actually lose some of these conversations in upper grades. And I actually want to pull out one of the quotes that you had. It's on page 53 and it's talking about flexibility, but I I want to tie this to where I think we kind of go off in upper upper grades or upper elementary. It says as students start working with two and three-digit numbers, we need to make sure to retain the flexibility that is central to fluency. And I thought like I read that and I was like, you know what, whether we have these discussions or not, we do encourage a certain degree of flexibility with students when we're talking about like decomposing numbers and all of that. And then it's like once we get into two and two and three-digit numbers, we kind of lose we lose that flexibility. We lose that like creative thinking. And I don't know, I just I guess my wondering is why why do we encourage some of this in lower grades and then when we get into those bigger numbers and upper grades, we we don't see that as much. Yeah. And as you go through this upper grades, it's not just two and three-digit numbers, right? But we're also talking about decimals and fractions and and other number types. And I think and again I think is the statement here. I think much of that is the perception of what it means to do math and quote unquote the right way to do math and that lining up decimal points or you know carrying out uh computations in certain ways is the right way to do math. And so when there's a right way to do things we have to fall in line with what those right ways are. And so that means being less flexible. And I'm not speaking necessarily just to an algorithm, but often um I'll see situations where it's maybe something like 4 and 9/10 plus 2 and 7/10 and a student makes an error or something and and the redirect is well let's line it up. Let's put it on graph paper so we can keep our thinking straight and neat. Whereas you might be let's be more flexible about how we're thinking about it and how you might shift and make a five or a hole or something of that nature. I guess to answer your point or your question, um, often I think as we go through the grades, we start to think about there's a right way to do math and a specific way to do it. And when that happens, flexibility starts to get factored out. Yeah. How do we how do we challenge that? Is that one of those things? How do we challenge us as teachers like Well, actually, that's it. Yeah. And I mean, I think we as teachers have to see it and do something about it. And to be fair to teachers, I think many of us do think in flexible ways. um we just um sometimes mimic the math instruction that we receive. Um I know that as a fourth grade teacher, I thought about, you know, all the ways that I write about. I I thought about math in those ways, but I didn't do that with my students always because I didn't deem that to be the appropriate mathematics for for instruction. And so, um I think part of it is we have to be comfortable with flexibility. Um and in some cases we may have to improve our own flexibility and make sure that we understand the math more deeply so that we can um embrace those different approaches. Yeah. And actually it's really interesting. I'm jumping ahead a little bit, but in the back half of this chapter, there's a lot of talk about like um estimating and um well, yeah, estimating. And one of those strategies is um compatible numbers, which I think is specifically called out in Texas standards. I don't believe it's called out in common core standards. And that's actually one that's like really hard for me. Like the compatible numbers is just not how my brain works. And so as I'm going through and reading this, I was like, this is an area where I really have to focus on growing this strategy for myself because it's going to be really hard for me to support students and probably create learning opportunities for them about this strategy when I don't fully own that strategy for myself. Yeah, that's a great point. I mean I think growing strategies for ourselves and that's okay because all of us think in different ways. Um and so there are strategies that we may not go to or they may not be called to as often be they estimation strategies, computation strategies, comparison strategies, whatever it might be, right? And so being aware of the ones that we are not as strong with and finding time to practice them ourselves or making sure we call them out during instruction is really really important. Um but also recognizing ourselves that there are certain strategies we don't gravitate towards and knowing that there are other humans i.e. kids that may not go to ours and so that's all the more reason why we um have to to really lift up that flexibility um not just in our thinking but but recognizing that in others. Yeah. And you know I don't want to skip over we the there's a whole section about skip counting and I wanted to make Oh good I'm glad. Yeah. Let's talk about that one. So, let's talk about skip counting because we, you know, talk about strategies that we gravitate towards and others don't. I think that we see a lot of students who use a lot of skip counting and don't have any strategies beyond that. So, there's actually two sides of this. I was like, how do I want to bring this up? Because I think I don't want to demonize skip counting because I actually think that we stop skip counting at at one point like maybe beyond once you get to like mastering your multiplication facts. How often do you see skip counting in classrooms anymore? But skip counting is so powerful. But then on the flip side though, we we we stop at skip counting and don't develop strategies beyond skip counting. So there's kind of like two sides to we need more skip counting in upper grades and we also need more strategies beyond the skip counting in lower grades. So counting is um a prime thing to go back to that every we all go back to. Counting is just a natural thing to do. What I would suggest or think about is that we don't count as well as we think we do in classrooms. And that's no shot. I mean at anybody or the work that they do. Um counting forward I think we do a great job with but I don't know that we count backwards as well as we need to. Skip counting is often by an interval tens, hundreds and what have you. And problem there is when you add and subtract you actually skip count by different groups and you're moving between those groups. And I don't know that I always did a good enough job with my second graders moving between skip counts. So we might start with 13 and count by tens. Um but then I wouldn't necessarily pivot and count by ones from there or something like that. So moving between skip counts is something that I think is I don't want to say taken for granted but just overlooked. Um skip counting backwards which lends itself to division. Um I don't think is is something that we do or spend enough time with. And again, when I say we don't do this, I'm not pointing out blame or or making blame. It's just something that I think we need to do a better job of. And and I think our standards sometimes are guilty of it because they don't actually point us in those directions or um they mask some of the bigger ideas. And sticking with counting for a second, let's talk about fractions and decimals. I know that elementary and middle school students aren't charge of counting by fractions and decimals, right? forwards, backwards, and in groups of, but boy, it's such a significantly important thing because it um you know, it's a lynch pin in in multiplication, division, and so forth. So, to your point, we need to do a much better job skip counting, right? Um and and and and changing our intervals and then doing it so much that we don't need to even think about it so that we can start to focus on other strategies. Like, it should be almost a fluency in its own, if you will. Yeah. And you know what's actually interesting too is with this skip counting like so much I mean this whole book is talking about flexibility like is having these different strategies. I even think in skip counting I'm going to go back to my son for a second when I was working with him on this problem. This originally was he was trying to figure out um well what would it be? Uh 9* or 3 * 6 and he was skip counting by threes which he isn't super fluent like skip counting by threes. And so he was like 369 and then he was like 11 and he was like that is 27 right? And so he saw because it was an area model. He was like oh he was like 369 and then he's like well I see nine and I see nine so let me just add 9 plus 9. And it was that like flexibility and strategies. And I think the same thing we're talking about skip counting being able to skip count. And it's not necessarily always just like let's say we're skip counting by you know fourths. Being able to say, "Okay, I'm gonna skip count by fourths and then, oh, I can go up a whole." I mean, just having that flexibility within the skip count. Um, yeah. Is so val. But the ability to maneuver between the ideas. So, three counts of six versus six counts of three, right? And he started with six counts of three and then saw something different and knew how to to move um or or change his his thinking. Um, that is grounded in conceptual understanding. And I think that conceptual understanding is often like do we understand the operation but it's not just that it's the properties of operations. And I know that many students can reverse addins and say or insinuate that they understand the commutive property right but that's different than moving between um add-ins or in this case factors and realizing that I can move them so that I can count differently or draw it differently or what have you. And um I don't know. I think properties are often memorized in certain ways and not really understood and therefore not used as well as they could be. Yeah. And I think you called that out in the book um specifically about how matching the property with like the actual um like problem. Like that is not a great activity to have students do. That kind of defeats the purpose. It's not that they can identify the property, but that they can use it. Yeah. I remember Jenny and I were talking about this and u we were talking about properties and um you know she's a huge advocate for understanding properties deeply but not just understanding them deeply being able to use them really well um and often the rhetoric around again the commutive property is you know don't call it the flip-flop property which by the way that's important but I think there's a bigger thing that we want to go after as math teachers and that is do my kids use it intuitively and do they understand um why it's helpful? Yeah, absolutely. And that's it. Understanding why it's helpful. And I I think it's a hard it's hard for them to understand why it's helpful when they haven't had the opportunity to see it and touch it and be it. That's right. And I think also part of this conversation is as adults, we take some of these things for granted. Therefore, we don't maybe emphasize or spend enough time with them because we just well, of course, it's good. You should of course you understand it and don't recognize that it's a major aha for kids and and again as adults we already know it right. Yeah, this makes me think we did um we did a book study or it was over mathematize it which is fantastic book. Um and we actually talked about the commutive property and um we were talking about it in the sense of multiplication and how it doesn't necessarily like depending on the context it doesn't necessarily make sense like 3 * 6 is not the same as 6* 3 when you have that context attached to it. And so I think, you know, hearing that and then also seeing the value of it when we're talking about like number sense and um fluency, I think it's really important that we have we're very clear with the goal like what are we trying to focus this learning experience on because um I think the I think the community property as a like tool for flexibility and like number sense um is fantastic and We don't want to just blindly use that if we're trying to actually like work through context like those two learning experiences are not the same. Sure. Well, and I think that what's important here is the difference between abstraction, right? And um representation and precision. So, let's talk about 3 * 6 and 6 * 3 for a hot second, right? Three groups or three baskets of six apples will yield 18 apples. But that representation is different than six baskets of three apples. Um, so that's part one of this, right? And I think the other one is something we talked about at the beginning and that is we feel like there has to be a right way to write something or there's a precise way to, you know, exact has to be like this. And when it comes to abstraction, 3 * 6 is 6* 3, that doesn't matter unless it's a um an explicit context as I just just spoke about. and and sometimes like, well, no, this number has to go first or that factor has to be second and and that's not necessarily the case. Um, and so I think we get caught in these haveto's and and it's only a have-to situation when there's an explicit context connected with it, right? Yeah. And I think some you just said something about precision. I think also part of mathematics is understanding to what degree of precision you need. um just in just in life like because and I think there was an example that got brought up during that discussion. We were talking about a wedding and like if you if they were asking for like three bouquets with six flowers in in each but you brought six bouquets with three flowers in each like that's a that's a situation where like it it needs to be very precise too exactly how it's listed. But in other situations um it it doesn't need to be that precise. And I think about that that actually brings me back to like that whole discussion about estimation. Like so often in life like estimation is really a life skill because to some extent I think that their estimation is more valuable in my everyday life. Like not thinking about my career or anything like that but my everyday life estimation is often more valuable than a very precise number if I'm going through the grocery store. Yeah. Online shopping or something like that. Estimation guides so much. to decide. It guides your decision-m as you speak too. It guides your is my answer good enough decision-m it guides a lot and it's something that I've been working on and and really reading more about because estimation is something that I feel as though becomes a procedure in itself. It's a rounding thing. It has to have a right answer. But yet in the real world, estimates aren't that. There isn't always a right estimate. It's a close enough conversation. Um, and then we make decisions based on context about is it close enough or do I need an exact and how important is that in the world we live in today when you think about it, right? Um, we don't do a lot of paper pencil computation. We do it on our phones or with a calculator in some sort and then we estimate or determine if that answer is yeah, it's about that. Um, and so I would argue that it's more important today than it's ever been. And yet as teachers, we are less prepared to teach it than not necessarily we've ever been, but than we really need to be. I agree. And it's also a really challenging thing to teach because it takes time because it's not something that Yeah. Yeah. It it it's something that develops it just over time those those estimation skills and I think that makes it challenging. Oh, without a doubt. And it's challenging for a myriad of reasons, right? Developing number sense and developing computational skills and yada yada yada. However, it does take time. And if we take a an approach where we're estimating in some capacity almost every day with a focused discussion, if we're sure that we are not causing students to mimic the way we estimate, but instead exposing them to lots of different ways to think and estimate, and that means that we're comfortable with lots of different ways. Um, again, does it take time? Absolutely. Are we more gifted at estimating as adults than we think we are? Yes. Um, can we develop students with it? Absolutely. and and it doesn't necessarily take a full lesson every day. I mean, I used to remember just asking to my students, wherever you are, estimate how many hops it's going to be to line up. Um, estimate how many and just like that's quantity and then in class just little estimates of like what do you think the sun will be and we would write down four and then just find out. Um, so to your point, it does take time, but I think that maybe not the same type of time we're we're used to uh thinking about. Yeah. And two thing I have two thoughts to that. one, you know, I really appreciate in the book how you call out the need for explicit strategy instruction, like students have to develop um you know, just certain strategies. And so with that, you know, y'all talk about several um strategies for estimating, but then the other thing that I thought was really really helpful was that strate or that estimating is not the same as rounding. like rounding is one way to help us estimate. Um but that when we are rounding an answer versus um estimating to help us decide like what a reasonable solution is. Um that might we might estimate differently. Sorry, we might yeah we might estimate different in the sense that we might estimate to 25 rather than 220 or 30 because it's closer. So I did a terrible job of explaining what No, you didn't. Clearly, I'll go back to the explicit instruction. Oh, let's go back to the explicit instruction in a second. Let's talk about rounding for a second. Rounding is a procedure, right? Um that uh can help with estimation. Rounding is often the poster child for estimation because it's something that we can correct and say yes or no. And that's something that drives math instruction maybe too much. Um and it's something that can be tested. Right? I can test whether or not you can round. It's much more challenging to test whether or not you can estimate. Well, um, let's say it can't be done, but not necessarily in the the formats or assessment that we have. But let's talk about explicit instruction. Um, that's something that like I'm often misqued or mischaracterized in talking about. And so explicit like we do have to explicitly teach students how to do things in mathematics. There's never been an argument against that. I think is sometimes mischaracterized as well just let them discover it and they'll discover how to do it. And I don't think that is I know that's not really the message. the message is let them come to the table with their own ideas, right? And then be explicit on other ways to go about it or be explicit on how they can polish their understanding or maybe even abandon theirs and and acquire something else. And so to your point and to the point in the book, we have to teach students how strategies work, right? But we first want to give them an opportunity to kind of play with a situation to see if they already have understanding or to counter what like students might have misunderstandings that need to be addressed during explicit instruction. And if we're not aware of them or if they don't become aware of them themselves, we can't address those. Um so I'm glad you brought that up. We can't just hope that they'll learn how strategy works. We have we actually have to show them. Um but at the same time we have to be careful not to say this is the exact way the strategy has to always work or when it always works if that makes sense. Yeah. Well and I think what has come up so often in the book is that the explicit strategy instruction is often built upon something that they know or experienced. Is that correct to say? That is correct to say. Yeah. Yeah. Sometimes it's naming something that they did and and naming it and showing them how this works and then giving them very intentional I'm skipping ahead a little bit, but there was a section about um about number strings and how we want students to practice this specific strategy. And so we're very intentionally, you know, creating a learning experience for them to dive into that strategy. Um but it's explicit instruction is not okay, this is how we're going to do it. You are going to do it this way and you're always going to do it this way. that that that's it in a nutshell, right? So explicitly showing them how make 10 works but not explicitly saying this is how you use it. This is you will always use it this way. You always use it for these numbers specifically and you always start with this addin for for example. So going back to the 7 plus 9 or imagine it was 27 plus 39 always telling students that you give some to the first addin. Now you do it like I do it. Now practice it. Like that's fraudulent because 27 plus 39 is a situation where maybe I would give some 239 to make 40. Like you get the idea, right? And so um explicitly teaching them how to make a 10 is one thing. Telling them you always have to make that certain 10. That's not um necessarily a best practice because again there's different ways that students may think about those addins. So, um, yeah, I think we have some work to do there without and I think that as teachers sometimes we're not comfortable or confident in our own thinking and so we devolve to a just follow the way I'm doing it because I'm not sure if I can react to a different idea. Yeah. And I I also think something else that contributes to this and I am 1,000% guilty of this is when you I I think there's a lot of um power I guess in teaching one grade level and being in really understanding that content very deeply. But I also on the flip side of that, I think there is something that's missed by not teaching a grade level before or beyond and seeing how like this strategy may always operate like this at this in this grade or in this age. Um but there are applications with that strategy later on where that flexibility is is needed for students. And so, um, I think it for me it has been such a gift to kind of move into the role that I've been in the last however many years where I'm looking at a lot of grade levels and seeing in even in my own teaching of like, oh, I didn't consider how this is going to look different in each grade. That's great. That's a great point. I mean, I was fortunate enough in my career, um, early career to bounce around grades maybe because they were trying to find the right place for me. Who knows? Uh but the moral of the story is uh as you bounce around grades, you start to see the math across those grades and then you bounce into middle school or something of that nature and you see, oh this is how that came that's a third grade idea, right? And so that is super helpful. I think that that's where as a teacher you don't have the time to look back at two other grade standards and you do have resources that may help you print resources, but that's where um some coaches, excuse me, some teachers have access to an instructional coach or someone who can help facilitate those conversations, but but not everybody has that yet. And so to your point, it's really important to be able to see where the math may be going and to understand how it builds, where it's built from. um having access to another adult or some resources is really important. Um and just being mindful that your moment in time in math is not the only thing and that there's so much more to it, right? Yes. And the there is a chart on page 65 and you know I'm sure you haven't memorized the page numbers of I have. 65 begins with um Oh my gosh. It was the basic fact strategies and their experience. Oh, that's a great that's a Yeah. Yeah. Do you see that? And you're just like, I'm so proud of myself that we put this together. Like, this is such a strong chart. Yeah. I'm proud of of our work and really proud of the discussions we had as we plan these things and they make their way to print, but um you know, for every page, there's probably an hour of discussion. That's a lot, but you get the idea. Um and and so on. Go ahead. What were you going to ask? I apologize. Oh, no, you're fine. I was just going to say that that chart at the bottom of page 65 is such a powerful chart and I I want to sit on this chart, but I also want to bring it back to the basic math facts because that's the part of this chapter we've not touched yet. And I think that um this chart I kind of think has kind of been like sitting in my head for a while even before I read the book in the sense that a lot of the strategies like being a fifth grade teacher a lot of the strategies that I want students to have um they may be missing and they were strategies that really are developed when they are first learning their multiplication facts. And so I the reason I love this this chart on page 65 is because it's showing how when we are when we are doing this strategy instruction with these basic numbers, it is giving them the foundation that they need to be able to think flexibly with much larger numbers, fractions, decimals, multi-digit whole numbers. And so, um, if this is like the one chart I could print and put on like every, uh, in every school so we can just see how all of this like yes, we can go the route of just getting kids to know their multiplication facts, memorizing it, but at some point we are going to have to go back and develop these strategies, but at that point, we're working with much much more challenging numbers, right? And that's why basic fact instruction needs to start with strategies. The end goal is quick recall, not using a strategy. But the strategies then are the underpinning for so many other things. You you mentioned being a fifth grade teacher and you know working in fifth grade or other grades, there's always this idea of like I had to get them ready for middle school. I have to get them ready for next year. And the way to do that is just get them to that one procedure they're going to use next year. But what's misunderstood is that manipulation of numbers like you see with the basic fact chart on that page or in other situations that transcends grades. So the strategies I use with whole numbers work with fractions, work with decimals, work with integers. And there are times when middle school teachers need to reinvest in those strategies, right? But there's other times where as an elementary teacher, I just have to be I need to be aware that this is going to work. This is going to serve you your students in time and that knowing the strategy really well for fact recall or for multi-digit computation um is an asset that kids really need. Yeah. Well, and it there is um there's a section in the book that I highlighted and it was talking about let me see I need a page number so I can figure out where it is. I did write down all the page numbers here. Let's see. It was Oh goodness. That's okay. I can't remember where it was, but it was specifically talking about how we cannot keep students from like grade level content because they don't know their basic facts. And I wish I could remember exactly. Oh, here we go. Um, it is page 605, maybe 66. Okay. Oh, maybe I didn't write it down. I actually don't think that's it. But you know, you do know what section I'm talking about where Well, I do know that later in the book, we talk about barriers to fluency instruction and that basic fact recall can become a barrier because as teachers, we've been led to believe that if you don't know your facts, you can't work with multi-digit computation or fractions or things like that and that um the way facts are taught or that recall is still developing, those things become barriers. So maybe that's what you were referring to. But even so, there are barriers to fluency that we really have to be aware of because we can fall into those traps and and hold kiddos back with the best intentions really, not not thinking that this is a big issue and that he has to have this or he won't be successful. Yeah. And I think where I think where my brain was maybe going with this, you know, is that these strategies help them, you know, thinking about um what's one when we're thinking about like um the distributive property or break breaking apart like that is something that is super helpful for students in fifth grade, fourth grade, third grade, which are the majority in the book study is third through fifth grade teachers. And oftentimes they'll be like, "Oh, well, they don't know their multiplication facts, so we cannot they cannot work on multi-digit multiplication because they don't know the facts." And it's like, "What supports can we provide students so that they can continue to learn, you know, this strategy, you know, using the area model or just breaking apart, finding partial products. what supports can we provide so that they can still access this content while fluency with their multiplication facts is still being developed. Yeah. So that's a great question. I think one thing that I want to make sure that everyone's aware of is that basic fact recall improves the more you work with facts. And so if you're doing 23 time 14 or 23 time 89 for that matter, um you don't have to know all your facts to do that. like by doing the work you are going to revisit and and get another dose of practice with basic facts. So that's the first thing is that you need to work with facts in other contexts so that you can improve recall of them. So that's first. Second off, um I might have students use tools to make sure they're accurate with those facts. Um I might want to stress and reinforce strategies of thinking versus counting. Counting is a practice at the beginning for certain facts. Um, for multiplication, two, fives, and tens are counting facts. But when you get into multiplying by four, that's when you're going to start using a thinking strategy, per se. And I bring that up because I don't know that we always do a good job revisiting those thinking strategies. Um, and so kids go right back to counting because that's the one that's most comfortable. Yeah. And in those cases, if I have kiddos that are counting and that's their preferred recall approach, then I need to practice the heck out of some skip counting, right? So my opening routine every day might be skip counting by fours. um or skip counting by sixes or something like that. I might do that over and over and over again. Um and just make sure that if you're going to keep counting, that's not efficient in this case. We're going to get really good at it. Um there's obviously opportunities to give kids basic fact charts and tools like that. I think that's a good resource to have, but the one thing we want to keep in mind is that they may divert to that too quickly or prematurely. Um, and so I might have to take into um account some some other tools or modify them. Meaning if you know you're multiplying by if you know twos and fives facts, you might get a multiplication chart that does not have twos and fives on it, right? Or those might be blacked out so that you're only using it for facts that you're not um you're you're not strong with or you don't have quick recall of yet. Yeah. Okay. So, I'm gonna ask you a question. We had a teacher in the mix and math community who was asking about hanging multiplication charts on the walls in classrooms. She said she's a coach and the majority of her math classrooms have like their like a multiplication chart hanging on the wall. And she was like, I'm not sure how I feel about that because in on one hand um you know it's a great support for students who need it, but on the other hand that's their go-to is like let me just look at the wall for this fact rather than actually thinking. Do you have any thoughts on that? I I mean I think out of context is a hard question to answer. Um because every kid, every group, every class is a little bit different. I think I would I think I would say it's it's a good idea for most cases, right? I think that I would be mindful that if I see that Britney's going through it over and over and over and over again instead of trying to use some mental tool, right, or some mental strategy, then maybe I'm going to start to withdraw or pull it back a little bit from from that individual per se. Um again though I can manipulate the chart so that it is not necessary for certain situations i.e. multiplying by two like I might take that entire row right off the top. Um but one other thing I keep in mind is this every time you use that chart that's one more dose and finding the product of four times seven right and every time you every dose you have is one more step towards not needing that chart. Um, right. And I think one thing that we need to do a better job of, and this is not directed towards that coach, this is really, this is really myself, and that is just having better conversations with kids about, but do you need it? Um, and having them be metacognitively aware of like, yeah, the chart's there, but I don't need it for this one. Um, and I don't know if I always trusted my elementary students to think that way. You know, sometimes you treat them as little kids and that you have to give them directions on everything. Um, and we really want them to have agency about stuff like this. That is probably I think one of my favorite things about this book and one of the actually one of the quotes that I wanted to bring up, but one of the favorite things about this book is just how much um encouragement there is for us as educators to talk to our students about their thinking, ask them what they're thinking, why they're thinking that. And um and and just what you said, it's like I think it's really easy to not trust our students and be like, "Oh, you're you're a second grader. You're a fourth grader. You don't know anything." Yeah. Like you're not thinking at that level. And they are so they they are so aware of their thinking because they may not be aware of their thinking if we don't ask them the questions, but they have such rich thinking. Yeah. That it's so insightful when we do take the time to ask. I think we overlook the fact that they're people. I don't mean it like the way it sounded. That sounded horrible, but um they have ideas and thoughts and feelings and stuff too. And sometimes they are treated just like little kids with the stereotypical way that kids are treated. And that doesn't mean that they're not developing and growing and still learning stuff, but kind of we all are, right? Um and they have ways of looking at the world and thinking about the world that don't always get valued because I don't know. I mean, I think we're getting into a different conversation alto together, but schooling has been an act of compliance for many students, and that's really not what we want education to be. Um, and so math, I think, is probably the biggest uh example of that where it's be like me or else. Um, that's not really how mathematicians think and do. And engineers are creative people. They aren't always rule followers. So, I mean, I think we have some work to do there in our profession and and it starts with us. Yeah, I heard someone say that you're either there you're either like creative or you're like good at math. And I was like, "Oh my goodness." Like the you to be great at math, there is an element of creativity like creative thinking, flexible thinking. Like it just I couldn't believe that those two things were contrasted. Yeah. No, that's that's a great that's a really good point. Um and it's something I'm really passionate about now. uh and and I'm starting to work on something new about this opportunity for agency and what can we do to help our students become um that their their own agents for math. Yeah, I love that. I look forward to reading that. And I'll close us out with this with this quote because it said that traditional basic fact instructional practices of memorizing facts and using times tests have stripped students of their mathematical agency. Agency is a behavior where you feel like you can figure something out. And when we say memorize, we are essentially saying don't figure this out, just remember it. And I know that that quote was about basic like math facts. But I think that is just like math in general. When we tell students just memorize this, just remember this procedure, follow these rules, we're saying you don't need to figure this out. And we strip students of that agency. We do. The one thing we sometimes can be well one thing that can be lost from time to time is that there are students who do think through procedures and do acquire through memorization. And so we do have to be mindful that there isn't like just like we don't want to say every kid should memorize, we should also not be saying like don't ever memorize either, right? And I think that's true for basic facts when it comes to 397 plus 413 like that's not going to be a memorized situation. Right. Um, so, so like that that's not possible at all. But I do think that the importance of understanding and recognizing and and and respecting that humans think and learn in lots of different ways means that while we don't want every kid to memorize, it's unfair to say that nobody can either. And so finding a balance and more importantly individualizing for students is is this conversation. And I don't think individualizing is always mis is always understood well, right? It's recognizing that we learn different ways in different situations um and being responsive to that, right? Yeah, absolutely. And really, I mean, that's what we want as educators is to be like responsive to our students, responsive to their needs, responsive to the conversations that are happening in the classroom. It is such a like math is very much like a science, but it's so much of an art and being in the situation with your kids. So, it can be when we allow it to be, right? And that means that we have to be comfortable with it and and admire it ourselves and and let go of some of the stuff that we were led to believe to be true. Uh such as this is a right way and there is a right way and I know the right way. Yeah. Well, I just appreciate you and the work that you did in this book because it is very much um giving us the permission to challenge maybe like what we thought fluency was, you know, at some point just because of how we learned and giving us the strategies and the tools and the activities and games um and routines to do it differently. So, thank you so much. Thanks. Um, and thanks to Jenny for being a great partner in this work and um, yeah, we're proud of it and we think it's the beginning of a conversation rather than the end. Yeah, absolutely. Well, we still have four more weeks in the book study, so it is going to be great. But thank you so much for chatting with me about chapter 3. Awesome. Thank you for having me and thanks for all you that are part of the book study and investing in yourself over the summer. I always admire that. All right, let me know in the comments what you thought of that interview with John. He is just wonderful. Such a wealth of knowledge and I thought it was um he just did a really great job bringing some clarity to some different things in chapter 3. So now we are going to jump in and do a few Q&A or yeah, we're going to do the Q&A with a few questions that y'all asked. Um, okay. Let's see the very first one here. Christy, I'll let you kind of like if you want to jump in, summarize this. Um, so we can kind of Sure. I This is a question that I noticed. Um, some variation of it came up quite a quite a bit with like what do we do about students who lack number sense or haven't been taught these strategies before? Um, like how do we or they come to us and they're two to three grade levels behind. like how do we plug the hole, right? How do we make up for lost time? Um, and the I think there were so many different variations of this general question. And I think the first thing that I would say about this is that you are not necessarily going to make up for years and years of lost time in one year. But that even just doing these, you know, using these strategies for one year and having the kids be exposed to it will absolutely help them go back and plug those holes for lack of a better term with those basic fact strategies, right? And I think that that chart that we um that you referred to in your chat with John on page 65 is such a good example of that where we may be teaching grade level content of adding multi-digit numbers or adding decimals but we are using a basic strategy like um make a 10 or doubling or add a group or whatever it is. And I think even just um having kids exposed to that strategy at grade level content allows them to then have that mindset of oh now this this way of thinking is in their brain. And so when they do see a basic fact like you might use omega 10 for you know 7 plus 5. I think I used that last week. it can kind of um you know light bulb can go off in their brain to think like oh this way of thinking that I've been using for adding fractions is something that can also be used here. Um so that's one thought that I have and I want you to add on to that. Yeah. So, I think that and you know, I kind of feel like I said the same thing last week because I totally jumped ahead and I was like praising the chart on page 65, but I think that it's to some extent if our students are like have these holes or they're lacking these strategies, um naming the strategy and showing them how it applies in a bunch of different ways is really helpful. So, when we're talking about like even just um I think I can't remember if it was this chapter or last week, I think it was this week. We were talking about like decomposing numbers and they talked about decomposing like nine into being five and four, but then we can decompose three4s into two fourths and 1/4. And I think explicitly showing students that connection can help them both like build that that strategy that they're that they're missing and apply it to grade level content content. And I think the other thing to remember too is um I have some feelings when we say like oh kids are three grade levels behind because like what exactly does that mean? because math is so um like broad, right? Like students thrive and are maybe feeling um have like unfinished learning in different domains. And so, but I'm going to say this, I had to preface it with I don't love, you know, this, but let's say we're thinking, okay, a kid is three grade levels behind and we think that they need a complete year of instruction on this stuff, but they likely have picked up some type of understanding. their brain is more developed. And so where a five or six year old needs a lot of time with the make a 10 strategy within 10, our fifth graders or our fourth graders or our third graders don't need as much time. And so giving them that exposure to the make a 10 strategy with a 10 and then connecting it to um like a decimal is actually like a much quicker process. Um, but they may just need that that call out with those simple numbers first so that they can see that connection. So, hope hopefully that made sense. But, um, I think it that chart on page 65 just to me I'm like start with numbers that they feel comfortable with and connect to grade level content to show them how it helps them with their current work. Start with where what where they are, right? Starting with this um asset mentality versus a deficit mentality. thinking about what they do have and what they do know and building on that as opposed to oh my gosh here's everything they don't know um is going to be so powerful just in your own brain and then exactly like you said I think starting with what they do know and starting with those those simpler numbers even for students who maybe are fluent in those um basic facts it's still going to benefit them to see it play out in this simple way before it gets extended to this more sophisticated or complicated um type of number. Yeah, absolutely. There is we could probably spend a whole 30 minutes just talking about this question because I know that this is something that we are very passionate about. We we work on creating progression guides or concept guides, learning road maps. They've had a bunch of different names over the years. Um but I want to be mindful of time. So, let's kind of go through our other questions here. So, this was one actually that we promised we would bring back from last week. And I think last week it was talking specifically about parents. This week it's about like how do we get the school on board? But really what this question is is like how do we get everybody other teachers the school administrators and parents on board in teaching math in this way and focusing on fluency in this way. So I obviously have some thoughts. Christy, do you want to jump in or do you want me to take this first? Um you go ahead. Okay. So, actually, you had really good thoughts on this. I mean, I've got thoughts, too. Okay. Yeah. You you felt when when you said, "Let's do this question." I have a lot of thoughts. So, I know you've got thoughts and then I'll I'll supplement. I'll add in. I definitely wanted to talk about this question. But so my another thing that they that was an aha moment for me that I really loved in this chapter is they reframed that phrase they don't know their facts or a student doesn't know their facts as they don't have a strategy. It's not necessarily well it is that they don't necessarily know their facts but they don't know their facts because they don't have a strategy to attain that fact or to um work through that fact. And so I think just reframing this whole idea of they don't know their facts as they don't have a strategy is actually extremely helpful if you're trying to get other stakeholders on board because I think most people can agree my you know my child doesn't know their facts or the the you know third graders don't know their facts or the fifth graders don't know their facts or the students at this school don't know their facts. Well, if you say to them, "Well, guess what? When you're saying they don't know your their facts, what you're actually saying is they don't have strategies." And so, what's the best way to remedy that? Let's teach them strategies. And that's the the power of of actually teaching through strategies as opposed to teaching through memorization is now if they don't remember a fact, they actually have a tool in their toolbox or they have something that they can fall back on to recall that fact or to work through how to find that fact as opposed to um you know they memorized it and it's gone. It's gone. they don't have a mental like filing cabinet or somewhere in their brain where they can recall that. So, um really to me reframing it as it's not it reframing they don't know their facts as they don't have strategies can be really beneficial in in your case if you're trying to convince someone to kind of shift their their perspective on it. Yeah. And I think for me, like kind of where I came at this was a lot of times people just need to understand why. They need an answer. And so I'll give you like a little analogy, real life analogy before I got on the call and then we'll come back to this is, you know, we are considering moving. And so we I were working with a new realer and she's a friend and I told her, I said, I ask a lot of questions because I need to understand. And so she sent me this house. we thought it was overpriced and so I need to understand how they got to this price like what is the square footage and the assessment and like I was asking all of these questions because I just needed to understand why and I think to some extent when we're talking about families or other educators of the school and I think someone last week said like how do we get them on board without just saying oh well research says help them see why and I I don't want to keep like I promise I've read pages in the book outside of page 65 But again, the chart on page 65 to me is such a beautiful reason for why we teach facts, teach basic fact fluency in this way because we can see that these strategies are strategies that they need to know for upper for upper elementary into middle school into high school. I've had high school teachers reach out and be like, I'm so glad, now granted, this is a model, the area model, but the area model is built on the distributive property or the break apart strategy. And so I've had high school teachers reach out and say, "Oh my gosh, my students who understand the distributive property um from their elementary years are so much more um like prepared for high school level math. And so if we can help other teachers, if we can help parents see like I understand this is different than how you how you learned it, but this is actually going to help them succeed or better understand concepts. Like this is the foundation. This is going to help them when we get to upper grades. And yes, we could teach them like just to memorize now and fine, maybe they succeed, maybe they don't, but at some point they are going to hit a wall because they don't have this fluency, because they don't have these strategies. And so to some extent, I think like we we can we can we can talk to parents like that. We can help them understand. And when they have an answer to their question and they understand, I think that we get um a lot less resistance. And also sometimes them asking questions is not even resistance. It's just they just need to understand like I just need to understand why this house is overpriced or priced the way that it is. So yeah, that was kind of my thoughts on it. 100%. Yeah. Um okay, let's see. Um okay, this we need to come back to this one. This is a really big that was a really big question. I was like, we've only got like two minutes. Actually, we don't have two minutes, but we're going to make two minutes. Um, okay. This one was, are word problems more valuable than standalone problems? And then really, I think we pulled this one because of that second question. Like, how do we bridge the gap for students who struggle or haven't built that conceptual understanding? Like, the focus of this book is not conceptual understanding. It is fluency. But the authors made it very clear that fluency is built upon conceptual understanding. I think there's a quote in there from I don't even remember who it is, but it's I mean NCTM it's the authors of adding it up. Um I mean fluent it specifically says that procedural fluency is built upon conceptual understanding. So how do we support students who don't have that conceptual understanding? And I feel like Christy, that is like what we do all day every day. Like that is the work that we are doing. Um I know we talked about it last week. And actually um I mean we have so many resources to support with that. We've got our our workshop that talks about the different representations that we use that helps students um develop conceptual understanding. Mix and math 360. Our program is all videos and support resources for teachers and students that develop conceptual understanding. So, there are some great resources out there. Um, and a lot of times, like in my opinion, it starts with us kind of making sure that we deeply understand the math that we're teaching so that we can create and facilitate learning experiences that develop students understanding in this way. So, Chrissy, I don't know if you have anything to add to that. No, no. I I mean my my first reaction is always um think about CRA, right? Concrete representational abstract, the CRA model, the CRA sequence. You just released a blog post on that. I released two blog posts today. One on so if you are not familiar with CRA, I highly recommend that you check out um our blog on um CRA. If you just go to vixmath.com and you go to our blogs, it's um one of the the newest ones. And then we also have one on uh multiple representations and teaching through or teaching with representations and CRA and multiple representations are are closely linked but those are really the most powerful ways to build conceptual understanding is connecting different representations and um connecting the the CRA or in this book they they sometimes call the CSA concrete semiconcrete abstract. It's also called concrete pictorial abstract. They all mean the same thing. Basically, this this um sequence of starting with hands-on, moving to drawn and visual models and then um moving into abstract or symbolic representations of the same concept and how powerful and it's so been proven so many so many times how powerful it is at helping students um develop conceptual understanding. Yep. Absolutely. So the answer to this question is if you hang out in the mix and math world for any length of time, you will you will learn ideas and strategies and best practices for um really developing that conceptual understanding in students. So all right, we are at time. We are so glad that you're here. Thank you so much. Whether you're watching live or watching the replay, I hope you enjoyed this evening. We've got four more weeks of the book study. Next week we are reading chapter 4. Yes, I cannot remember who is on for next week as far as the book chat, but it will be on Thursday again. Um, and we hope to see you there live. Do we have any announcements or anything that I'm forgetting about? No. And next week you will not see my beautiful face. You will see me, another team mix member. So, you get to meet you'll get to meet her. It's her debut for book chat. So, um, definitely give her some love in the comments when you see her next week. And I will be there in the comments. I'll I'll join you all, but I won't be on stage. Yes. Okay. Yes. I forgot it was Meg next week, but it will be a great chat and we are just thankful that you are here. So, um, I hope you all have a wonderful rest of your day and we cannot wait to talk math with you again very soon. Bye everyone.