Figuring Out Fluency: Week 4

Hey math friends, welcome to week four of our figuring out fluency book study. We are going to be talking all about chapter 5, but this week is going to look just slightly different. So I am actually out of town, so I am pre-recording this. Um, but we are still going to be chatting all about chapter 5. I pre-recorded an interview with Rosala. Rosala is one of the co-authors to the companion books. So, we've got this main figuring out fluency book that we are reading together, but then there's also these companion books which are excellent and they dive a whole lot deeper into a specific domain and how the principles within the main figuring out fluency book apply to these different domains. So, she's one of those co-authors and I was able to get with her before the book study started and we chatted all about chapter 5 and talked about the different modifications or applications for the ideas in chapter 5 and how we can make that um what that looks like in an upper elementary math classroom. So, we aren't going to do Q&A this week because obviously I wasn't able to see your questions before I met with um Rose Alba and had this conversation, but this interview is a little bit longer and so that will be the bulk of our time together tonight. It is fantastic. I hope you enjoy it. Someone from team mix and math will be live with you in the comments to answer your questions and support you if you are here watching um live together with other teachers tonight and if not if you are um watching the replay. But I really do hope that you enjoy my conversation with Rose Alba and I look forward to being back with you live next week. Rosala, thank you so much for joining me to chat about chapter five of figuring out fluency. I'm so glad that you were able to join me for this. Yeah, I thank you so much for having me. I love talking about this stuff. So, I feel like it's going to be a great combo. Yes. And we're talking all about automaticity, which I think that there are a lot of like misconceptions or confusion about what automaticity is. And um this chapter did such a great job of making it very clear about what it is, what it isn't, and how we move students towards being automatic with those facts. So yeah, I thought I would kick us off with a quote that I had highlighted here. It's on page 108 and it says, "Umaticity is the ability to complete a task with little or no attention to process. Little thought, if any, is given to skills that are automatic. They appear intuitive or reflexive." I know that is what we want for our students, right? Like we want them to be automatic with those facts. Yeah, absolutely. I think what lies here is the misconception that automaticity is always kind of linked to speed and recall which we do want students to have but it's the matter of which we go about it right because there are automaticity um strategies and and things like that in place that I don't know if we're seeing enough in the classroom right because we're we're seeing that as like the outdated kind of version of automaticity as being fast. So we do things like give a lot of fact tests and speed fact tests in order to build that when there are a variety of ways to address automaticity. And at the end of the day, automaticity always kind of falls back to fluency. And I think even before kind of getting into automaticity, every teacher should have a real solid understanding of what fluency means and because even that has so many misconceptions around it, honestly. Um, and I I'm not sure if you've talked about this in from the other chapters, but fluency has three main components. You know, anybody reading the book will come across that in these chapters. um obviously flexibility, efficiency and accuracy. um where I think sometimes you know we kind of I don't want to say miss the mark but again it's just like our traditional way of teaching and where we have to shift um educationally but is the procedural fluency piece which we talk about a lot in this or John and Jenny did a great job talking about that procedural fluency piece in this chapter right because that was always linked also to speed and recall but procedural fluency is so much deeper than that, right? When we're thinking about speed, we're prioritizing like quick answers, right? Um, but what happens is there tends to be a lack of conceptual understanding behind that. So, it's not enough to say quick anymore. And I think immediately when we hear the words either procedure or fluency or even automaticity, we link it to really being fast. But we have to make sure there's an understanding behind that. Yeah. And you know on that same page on 108 there is this um I mean it's a graphic. What would you call that? Like a pyramid. What do they call that? Do you see? Oh, is it the the stack pyramid? The phases. Yes. There we go. That one. Yes. Okay. The phases. Yeah. Yes. And so I think a lot of times we jump to what we want the goal is that phase three is that mastery that automaticity and there is a section in here that talks about the fact that a lot of times we skip phase two which is the strategies which is the understanding. we go from this counting and the skip counting to all of a sudden, you know, trying to get students to be automatic with these facts and without the strategies really the only way we can get them to be automatic, though it's not really is memorization. And so we almost have to like backtrack and be like, "Oh, wait. Let's not skip phase two. Let's spend a lot of time in phase two because that is what um actually I I did pull this quote. said um page 110, automaticity is an outcome of learning concepts and strategies mixed with abundant exposure and practice. Absolutely. Practice is a big deal. I do want to talk about that especially with activities that are in this chapter, but something you said about memoriz memorization is key here. These phases are really important. And what I'm seeing and I think many of us do again because it's like that traditional model that we've always followed. We stay very long in that first phase of counting and we 100% tend to skip or maybe not even skip but breeze through phase two which is that derived section and we do want them straight into automaticity. But when we breeze through that phase 2 section um the derived section um we do heavily emphasize wrote memorization for the kids for our students. And when as we know when we're having students even if it's unintentionally wrote, you know, promote wrote memorization, we're really just asking them to mimic us, right? We're really just saying like here's here's how this works. You try it. you do it. And then we think that if they practice that enough that they do have that deeper understanding when that's not really the case and we see that when it comes to efficiency, which is the other part of fluency, one of the three main components of fluency, it's very difficult for students to be efficient when they don't have those derived fact strategies because they're memorizing. So it's very difficult for them to choose which of these strategies would be most efficient for me to solve this problem because they're not thinking of the variety of ways derived ways that they can approach that. So that comes along where we do want students to have basic facts in their long-term memory, right? where they can pull it out quick because the whole point of having them stored in their bank is because they have to get to problems that have a higher cognitive demand. And if they're stuck on 8 + 7 and it has nothing to do with the problem, it's only one basic component of the problem, they're going to be stuck there and can't move on to the things that demand a higher cognitive, you know, thinking, right? And we see that quite often. So, we tend to say, well, kids don't have number sense. They can't do this. You know, we we like to blame things on number sense a lot overall. Like, they don't have number sense. But I think it's really important to be specific about what about number sense in particular are students not lacking but haven't reached yet, right? Is it that, you know, they're having difficulty? That counting phase is really important. Are they stuck in that area? Are they stuck in some of these um procedural automaticity standards that we want students to have like doubling and things like that? Like we have to be very specific on what students need without just saying they they lack this, right? They can't solve this higher cognitive demand problem because they don't get the concept. Maybe they do get the concept, but they're stuck on that basic fact level, right? So we, you know, we want it stored in long term, but we got to make sure that they get the thing. Yeah. So I Okay, so I have a couple thoughts. One, I think we all have the same goal like we is we want is we want that um that quick recall. We want that like long-term they know the fact because especially with in upper elementary, that's what the more majority of teachers in the book study are in upper elementary education. And on page 112, it says automaticity frees up students to reason about more complex ideas. And we see that in our third, fourth, and fifth grade classrooms where they are working with more complex ideas and they don't have those basic facts and that gets in the way. It makes it really hard for them to, you know, work on grade level content. And so we see this problem here. We see, okay, this is where this would be so much easier if students had this, but they're not there yet. But I don't think like procedural fluency or just fluency in general is talked about enough. And so this book has given us such a great guide to say, "Okay, now we see this problem. Here's how we fix it. Here's what they're missing." Um, I think even just this chart on 108 where we can say, "Okay, they're not at phase three. Here are the things that we can start diving into. Like, have they mastered phase one? Like, can they skip count?" Like if we're even just talking about like I love the idea of bringing skip counting into upper elementary, skip counting by decimals, skip counting by fractions. Can they do that? Okay, yes or no. Can they do they have any strategies to be flexible with larger whole numbers and fractions and decimals? So I think that we see the problem and now going through this book we see we kind of have like a road map for okay these are the different things we can do to kind of like bridge to the solution if that makes sense. Absolutely. Absolutely. And I think what's important with this you nailed it with what you're saying is doing formative assessments along the way. Right. So, I know hearing the word assessments is scary, and I don't mean time test, and I don't mean any of that, but just really looking at your students, observing them, and seeing, okay, if we're on phase one right now, what can I do to move them forward? What can I do to support them? Right? And the derive facts section, which is phase two, which is really heavy and really important. Um, let's say they come into your grade level because you are upper elementary, but they are having difficulty. Maybe they did not have this in previous grade levels, you know, maybe they don't remember. Whatever it is, maybe it was like a short-term memory thing. They kind of learned it, forgot about it, moved on. If you're upper elementary, I think it's really important to make sure that you're revisiting these stages. Not that phase one is for our little guys, you know, or little kiddos, and then phase two is, you know, for this certain grade or age level. Absolutely not. I think we all have to address all three phases and make sure especially in upper elementary that the whole numbers are solid because if the whole numbers are not solid fractions and decimals will be a nightmare. It will be very difficult and we see that nowadays we see that decimal and fraction concepts are really difficult. I'm going to even say not for students but even for adults for you know we we see that in life fractions really it's scarce grown people so in order to have students kind of have that fluid fluidity they really need to make sure that the whole numbers are down pat right so even if it's upper elementary we have to concentrate on those three phases regardless yeah and you know what's interesting that you're talking about the the fact that we've got to make sure we're there with whole numbers. One of the things that I love about this book and what in this book and one of the reasons that I chose this book is because a lot of times, you know, I'll get messages from teachers and they're like, "But my students are at like a kindergarten level. My fifth grade students are at a kindergarten level." Which first off, I have lots of just thoughts about statements, blanket statements like that. Yeah. Anyway, but let's just like lean into that for a second. This book shows how like I think it's on page 113 where they're listing the um the eight automaticities we want students to have. These are things that are helpful in kindergarten all the way through 6th, 7th, 8th and beyond. And so when you're talking about whole numbers, fractions, and decimals, like the examples that they show about like using 25s, we can talk about using 25s with whole numbers, but that same thing, that same um automaticity or what is what are they calling it? The seven significant strategies. No, are you on 127? Can Can we You're on 113. Okay. I forget the name. There's like a chapter that's all about like there's like utilities and strategies and automaticities, but whatever they whatever the word is for what they have listed on 113. Those things are helpful with whole numbers, but we can apply that same thinking to decimals when we're talking about 500s. Absolutely. I would love to actually if you have a second go through I teachers are reading this you know reading this book along the way but I would love to actually talk about each one and if possible have the teachers just think about how that strategy the procedural automaticity strategy how it applies to their grade level but also what younger lower elementary teachers would also need because everything in math is interconnected right everything we know this so I would love if possible to kind of go through all eight not in depth but just I want teachers to reflect on that if possible you know think about where you're at what your students need and then what they needed before that you could possibly kind of fill in that gap for them yeah is that yes okay let's kick it off with breaking apart all numbers through 10 yeah exactly okay so let's talk about that so break apart is one of the most foundational strategies that you can possibly have because if you think about it we're using it in multiplication we're using it for deco de decomposition composition. So if you think about break apart all numbers through 10 yes you're thinking about basic numbers right whole numbers but how does that apply to decimals and how does that apply to fractions? That's super important as well right we know that composition and decomposition are really important for students to understand with those concepts in general. So, if we're doing it early on, right, with basic numbers, those break apart. Seven and three make 10. You know, 6 and four make 10. Well, what does that mean when we're talking about decimals in one hole? How many tths and how many t make that hole? Because once you throw a decimal in place, students start thinking these aren't real numbers now, right? That decimal in place or even fractions, right? When we want to make a hole with fractions, well, what does that look like? When I break one hole apart, you know what I mean? Obviously, a half and a half, students pick up as early as kindergarten, thinking about a half and a half, make this hole, but then when it gets a little bit more complex, as students get older, especially if they're working with mixed numbers and things like that, again, it's just a whole transition all the way through. So, I love for teachers to think about and even like um lower elementary teachers to think about this too. I know right now it's upper elementary, you know, reading this book and talking about it, but it's so important for lower elementary teachers to see where is this progression going because if I'm a lower elementary teacher and I'm not discussing half and half make a hole, you know, it's going to create a gap for our students that's going to hinder them along the road. So breaking apart is a big deal in our math world and we got to we got to do it. Yeah. And I think it's I think it's really good even in upper elementary. So one of the things that I like a hill I will die on is starting in working with students is starting with something that they know so that they have like they have that um that like a moment of confidence and then building onto that and like connecting back to that, right? Because all new learning is built upon previous understanding. So with that break apart strategy even in upper elementary thinking about like okay how did they use this in lower grades in a way that they're confident with let's start there and be like okay so may what maybe it is like breaking apart um 10 okay great well how does this same strategy work when we're breaking apart a thousand or when we're breaking apart one whole like what how can you approach these in similar ways and so I think that even having students be able to see oh I already know something about this. Now we're just applying it in a different way to a different set of numbers. So that breaking apart and then really the base 10 combinations. Yeah, I was going to say that that's just decomposition. So it's more of the same. One is within 10 which is a benchmark that we really need students to know and the other is just regular decomposition of all numbers. So that applies to again moving from whole numbers to fractions to everything. I want to touch on something that you had said that you the deal the hill you want to die on which I totally agree with you. I'm on the same page. Um but I just want to mention something that kind of is important with what you mentioned. When students have success, it's been shown that when you start students off with that little bit of success, something they already know, it actually helps them with productive struggle right after, with perseverance right after. So, if you just throw something at a kid and you say, "Okay, let's let's persevere through this. You got this." Um, there we're seeing that a lot of students give up quite easily. I mean, it's happened in every classroom. I've been a teacher forever. You're a teach, you know, like we've all been there where we give something to a student and they go, "Okay, I automatically don't want to see this. Please get this out of my face." They automatically shut down. They shut down. They They got to go to the bathroom, to the nurse. They got to they got to do all the things aside from the actual task. But it's been shown that if students are given that piece of success right before when the difficult thing comes up right after, they're more likely to sit through it, right? Because we're building it's really basically reducing the mass anxiety that naturally comes along when something is difficult, right? You can apply that to anything in our own lives. when we do something that we know is quite difficult but we meet some success before even attempting it, we feel like we can tackle it, right? Yeah. So, I just wanted to mention that to what you're saying, you know, I totally agree and there's there's research behind that that shows that. So, give them the thing that'll give them that success that win and then throw something that scares them right because I think too it it kind of puts them in a position of and this goes with all of these strategies. It puts them in a position of um you know if you just start with the with like the hard thing that they feel like they've never seen before they it's it's overwhelming. Yes. And then I also think it becomes a lot more challenging to think back to like what prior knowledge do I have that that can help me with this? Yes. Whereas when we start them with something that they know they get like that boost of confidence and excitement. It reduces, like you said, the math anxiety and it kind of is already putting them in that position to bring in knowledge that they have to and apply it to the next thing. And so I think this is why it's so important for us, I'm so glad you had us talk about this. I think this is why it's so important for us to as upper elementary educators think back to how this strategy was used previously so that we can start students there. That doesn't mean spend all day there. It means just give them that quick win and then move on to applying it to whatever the grade level expectation is um you know for whatever you're teaching. Yeah. And and you said something really key there. Don't spend all day there because that is we tend to do also. We want to you know reintroduce that prior knowledge and bring out that prior knowledge but then it becomes a whole lesson and we don't want to do that right. So, we want to make that's why those formative assessments are super important because then we already know what the students are coming in with and then we can kind of gauge from there. Um, but you're totally right. We don't want to we don't want to be there forever, right? Because then it becomes a whole whole other lesson and we don't want to do that. Exactly. Yes. Yeah. Absolutely. Going back to these um what are we calling these? Are these these are automaticity? Yeah. So they're basically yeah they're they're the procedural automaticities that students should have in place basically. So bas the foundational pieces as far as when it comes to uh procedural you know automaticity what we want them to kind of have down pack. Um, we did talk about breaking apart, you know, for 10 because 10 is a benchmark. Will is decomposition, but oh dear, you talked about 25, which is good, but we're missing 15 and 30, and that is in this book. And highly, it is very highlighted. There are a ton of activities. There are activities in this chapter. There are a lot of activities in the companion books as well. We do not gloss over 15 and 30s as benchmarks as well. You know, those are super important. And I think that I think we do a good job spending a, you know, decent amount of time on the benchmark of 25, especially when we're talking about quarters. Not physical quarters in the money sense, but, you know, as far as fractions and things like, you know, we introduce quarters in a variety of ways. Um, but 15s and 30s, I don't think in my opinion from what I've seen we just don't hit those as much as we should and they're really considered benchmarks as well because there's patterns in there that students should be picking up. Um, so this is so interesting to me because if I were to somebody were to be like, okay, what are the eight automaticities? I don't think I would have come up with 15 and 30 as one on like by myself if I were just in my own brain thinking about this. But then seeing it on paper and seeing the reasoning for it, I'm like, of course. Like I just And I think it's actually interesting. I'm kind of grouping 25s and 15s, 30s here. So 15s and 30s, super smart because of time. That makes so much sense. But even 25s, I think I have a wondering if like the benchmark of 25s is going to be there's it's going to be hard, not harder, but like there's going to have to be more of an emphasis on it in the classroom because students aren't naturally getting that experience with 25s in life outside the classroom as much. Like I don't think my child has ever seen us pay any we don't have cash ever. So I don't even think he's saying have that everyday experience with it which kind of means absolutely you know that we are going to have to there's going to be a little bit more of a lift in the classroom is my thinking um but 15s and 30s I was like yeah that makes so much sense when we're so much sense when we're talking about like you know we've got to be somewhere in 15 minutes and 30 minutes like that should be something that students are automatic with yeah it ties in perfect with time like you said and you don't really need to I mean we address analog clocks in schools But, you know, outside of school, students are still dealing with 15s and 30s, regardless if they're looking at digital or not. Um, but you're absolutely right, 25 is very difficult to bring in that real world experience because you can talk about coins and money all day, but really everything's digital. and you know so it's about kind of tying in then what can I apply here from the real world that I can kind of bring into this um and and geometry is a great way to do this as well believe it or not so it's not just tied into money or you know uh fractions but geometry is a big piece something that I love to do with students is um I mean you're familiar with number talks but photo number talks and That is just one of the best ways that you can possibly tie in um because then they're looking at at it in a visual format as opposed to an abstract thing where this coin represents 25, right? They can see it in like more of a visual format like you know if there's this amount of windows or this I love to bring in murals, right? and then have I mean I just take pictures randomly of everything everywhere and somehow make it a photo number talk but it's it's a great way to kind of tie in the 25s in that sense as well. So it's just about thinking a little bit outside of the box. Again there are a ton of activities in here where students can practice and play with the 25 but you're thinking about you know real world outside. You have to be a little bit creative in how you bring that in. For sure. Yeah. Well and so you were talking about um geometry and so I'll go the measurement route. That was the one of the other things that they said about that 15 and 30 is and it's not just 15 and 30, right? 60 and it's 90 and talking about um angles because you know we have 90 degree angle 360 degrees is around and my favorite activity from this chapter was on page 113 and it's that combinations um that combinations activity and when I wrote this down I wrote in the margin I was like love this game we need oh there we go love this game I was like we need to do this with fractions and decimals but I also think you can do this with like giving students angle wedges and they're drawing them and like maybe it's to make a 90 degree angle and so you've got different angles or different like benchmarks and they are those combinations um to make whether it's the 90 or 180 or 45. So um that's that activity is so versatile. Yeah. and and the combination of those angles because I love that activity, but even saying, you know, all right, I have two 30 degree angles and I want to combine them. What would that look like? Right? So, then they're really kind of working on the pattern of um you know, a 15, what did I say? 15 or 30, whatever the angle was that I combining it and then seeing what new angle that makes. Um so, yeah, I mean it's you can play so you can play with this so much, you know. Um there's a variety of ways. Um so those benchmarks are really important. 25 15 30 we can't lose sight of that. It's not just 10 even though we have base 10 system. Yeah. Okay. I want to add on to that real quick. So it's interesting talking about measurement um or not measurement conversions. I was just reviewing a video this morning about that angle measurement when you students get to um it's a fourth grade standard when they have the um missing angle. Yes. Yes. And they've got to figure out like what it is. Really, that is just a missing add-in problem. Like absolutely, that's all it is. And when students have this automaticity with 15s and 30s and really just number sense in general, like that becomes a much less challenging like that that's a that's a standard that usually has fourth grade teachers like pulling their hair out. Yeah. And when we see that it is that missing when they understand angles and they have this like number sense and recognize that it's a missing add-in problem that standard becomes a lot more like like approachable for students and teachers for sure and and it it goes back to not just the 15 and 30 benchmarks but decomposition right because it's really when you're finding a missing addin of any sort it's really just a decomposition if I have this number and I know that this missing part well what can the other missing part possibly be again exactly what we're talking about. We're addressing it in upper elementary, but it's getting hit on in in lower elementary. And that's why I feel like I love my lower elementary teachers because they are the foundation to all of this. So, any way that schools can support them makes the whole flow of curriculum of whatever you have in place so much more attainable for the students. We reduce a lot of math anxiety if students are seeing something consistently across grade levels. They're decomposing with kinder up second grade up to fifth. They're just decomposing different things. So, it's important to have, you know, we always kind of focus on our curriculum and kind of like our grade level. Sometimes we look ahead to see like, okay, what do the kids need later? Let me kind of prep them a little bit for that. But we got to look past as well. It really is that, you know, that alignment always focusing on vertical and horizontal alignment, right? Yeah. And you know, there is um such a great book. It's the math packed. Yes. Talked about that book before, but I even think with some of these granted, not these, we're talking about automaticities here, but what you just said brought me back to um I guess it would be like the seven um the seven strategies. Even just having alignment on like what we are calling these strategies as well from grade to grade like that is such an easy way to get to help we don't need to add complexity by every grade level calling these strategies something different and students are like wait this is a new strategy and it's like no it's the same thing you've been doing since kindergarten so just a thought there in the alignment of even just teachers seeing we are doing this across grades let's get on the same page with you know the this representation is a great representation to use. And if I just did a post the other day about um number bonds and how we use those in kindergarten all the way up through fifth grade and it's like just having having something that is the same commonality. Yeah, that commonality there. That's what I was thinking that commonality so that the like difficulty with numbers is not compounded with like difficulty with a new name and a new representation and like Yes. But what we like to do is we like to give it cutesy names too. So we have to be mindful of, you know, we want it to be math vocabulary, right? Where students kind of can be consistent with along the way. Um, we want to make sure we're not giving it cutesy little names every year a different one because it really just exists a student. So, I know some math vocab is harder to kind of touch on with the younger guys and I get that, but if there's a way to kind of, you know, bridge that in order for it not to be way off as they go through the grade level. So, you're totally right. They need that consistency in the name of these strategies as well. Um, absolutely. Yeah, absolutely. Okay, let's dive into the last four. So, we've got doubling and packing. What are your thoughts on those two? Oh my goodness, girl. This is one of my favorite favorite favorite strategies. I don't care where I am presenting wise. I could be talking about something completely different. I find a way to talk about doubling and halfing in it. I will always find a way. Luckily, this chapter already has it. So, I didn't have to like do that. Um, doubling and half. It's so underrated. It is so underrated. It is actually unfortunately every school has a different set of state standards, right? Or, you know, county, you know, district standards. Um, not everybody does a common core and I understand that. However, the commonality that I've seen in all the curriculums that I've studied and all the places that I've worked with, um, doubling is addressed very early on in standards, not really talked about much later on, and then we kind of lose it basically. We we don't touch upon it enough. I This is actually one of my favorite strategy. I know that decomposition and making I know that's foundation but that's also thrown in your face everywhere with every standard all over the place. Um but doubling and halfing is one of the most I think in my opinion important things to have students have down for number sense in particular right it is not enough to just know 6 plus 6. It's just not right. We have to be able to incorporate incorporate that doubling and halfing because it helps with multiplication and division and not basic multiplication and division. I'm talking about extended multiplication and division. You know what I mean? There are some things, you know, something that drives me crazy is when I see a multiplication problem that is something, let's say 13 times two, and students are taking out a pencil and paper for that. You know, I I'm not saying pencil and paper is not good, but there are some things that we can just think about cognitively, mentally, you know, do some mental math work on that. It's just doubling a thing, you know. Um, you can use a bunch of different strategies to get that answer, of course, but at what point can we just possibly do that mentally? It's the same thing when it comes to division. We should be able to see numbers and half that, you know, and then it it deals with other concepts that I don't think we realize. Doubling and halfing is very important to things like even and odd. You know, it's very easy to do an even and odd lesson with students and have them concentrate on what those two things are. But are we applying that when it comes to computational strategies? I don't think we do that enough. I think we do even and odd lessons. This is what even and odd numbers look like. But knowing if a number is even or odd helps you think about what strategies do I want to use to compute because this is probably going to give me a remainder and then maybe I don't want to do that. So again, like the doubling and halfing ties into all these other things that's really important and I do think it needs to transition to more of a mental math skill within reason because some numbers I mean they're going to drive us crazy trying to double or half them. So obviously within reason. Um, but it's something that I think that I push teachers the most in this area, you know, as adults for themselves because I don't think we double in half enough. So then we kind of I I don't think it's purposefully, but tend to avoid it in our classroom, right? So really try to build that within adults first and then have them running through life doubling and halfing as much as they can to bring it into their rooms later. I think that this was a strategy that like I or like just a skill that I owned until my adult life. And I think one of the reasons I love doubling and love having is because and actually I've talked about this like every single week in the book study. I love this the com the conversation that comes up around student actually using like multiple strategies or using like an automaticity and then changing their strategy to like carry out the problem. And I think doubling and having oftentimes gets us to a point where the problem we are working with doubling and having can actually make it a little bit easier for us to do. And so I'll give you an example. So my um my son, he just finished first grade and he's like really curious about multiplication. Um yes, I know that like standards it's in third grade, whatever, but he's just curious about it. And so we were looking at this um it was an array and it was three * six. So it was um you know six rows of three. and he is not like super fluent with like skip counting by threes yet. And so he was like 3 6 9 and then he didn't know what the next one was. He saw nine here and nine here and he was like, "Oh, I'll just essentially what he did was like double in half, right?" Because yes, he was like, "Oh, well, I can just do like 9 plus 9 and that's easier." Because then he used make a 10 strategy to figure out what 9 plus 9 was. And so I think that it's like it's situations like that where it's like we having that skill of doubling and halfing allows us to make problems maybe just like more manageable for us to to compute mentally. Yeah. And and it falls back to one of the components of fluency which is efficiency, right? Efficiency is really all about selection. That's all that is, right? Are they selecting? We can solve any problem with any strategy. mostly most every problems with every strategy, right? Um but this is where efficiency falls into place. When there is a problem that you can solve any which way, but the most efficient way would be a doubling and halfing by students not having that ability or lacking that strategy. It's really hindering them, right? Because we're focusing on getting students to be more efficient and letting them select something that makes the most sense for that, you know, problem. That's where doubling and halfing can come in and actually help students and again they build confidence in that, right? Because they're building their mental math skills as well with that. It's interesting when I'm in classrooms and students are first kind of approaching doubling and halfing, they really they're they are gassed up. They love this. So they'll come up and they'll be like, "Miss on a guess what 11 and 11 I can double you know, so it's like they get so excited. They throw out these random numbers and I'm like, "All right, what is it? Go for it." You know, especially when they halfing is a lot harder than doubling. Um, it's almost like we see subtraction is a little bit more difficult for students than addition. Halfing is a little bit more difficult than doubling. They really get pumped up when they can do it, right? they just think that they're these stu superstars and they are because it's a skill that you know it's kind of being picked up on more recently than in the past like with our traditional textbooks and things like that. So yeah, and something that you said made me think about um there's a quote on page let me find it. Um actually I think I wrote it down. It is oh on page 110. I can't remember if I've already said this but something that you uh said made me think of this. It said automaticity improves accuracy because the need for computing is gone taking with it the potential for computational errors. Yes. So while we could while students could use um like an algorithm or something to solve it, they actually have maybe less potential for errors by using a strategy like doubling or halfing um to make it a like friendlier problem to work with. So, I thought that was just really interesting because in the example going back with my son, like if he would have continued skip counting because the next thing he said was like 11 and then he would have skip counted from actually I was working with a group of fifth graders who were doing this. they were skip counting things and all of their if you mess up one skip count and then you do all of the rest correct everything's off and um and so we talked about like what are some other ways that we can find these they working on division so they were trying to get their their multiples and I was like what are some other ways how can we use you know like what you know about numbers and reasoning all of that to get these rather than skip counting so that just goes back to again this if we have these automaticities actually makes the likelihood of error maybe a little bit less because we're not there's just less room for you know whether it's computational or skip counting or whatever those mistakes are. Yeah, absolutely. I mean it's the it's it's making sure that they have the foundational pieces in place so they can move on to things that have like that high like more cognitive demand basically, right? they're not stuck on an algorithm where they could have errors and then actually see losses over wins and successes which you know what I mean then students feel a particular way. So you're absolutely right. I absolutely right. Yeah. So okay we've got two more. So fraction equivalents with fraction families. Yes. I'm actually I think we could even combine this because it's fractionally equivalents with fraction families. There's also the conversions between fractions, decimals, percentage, right? It really to me this section is every all of this is foundational things that students need and it touch you know it touched upon something that we've already discussed. This is there's a fluidity here. This has to be seamless. So while all of these foundational pieces of procedural automaticities is important for when the students are starting off and they're younger, it really transitions to these harder math concepts that seem completely scary, but it's just more of the same. If we think of it and we have students think of it as this is just more of the same. I decompose this with regular num with whole numbers. I can decompose this with fractions. if they have a solid foundations of fractions, then they can convert things to decimals because they're not seeing this as different entities, right? And I think that's what tends to happen is they feel like, okay, they understand the decimal unit, um, but they don't understand the fraction unit. Well, isn't just like whole numbers transitioning to decimals and decimal I mean, they're called decimal fractions technically, right? Because they're one and the same. we just isolate the names. Um so so the fact that they understand need to understand that equivalence is really important. So that's another foundational piece to this procedural fluency part is that they need to understand the equivalence of these things and then they are able to do those conversions between these things. But students need to stop seeing every math concept as it's a separate entity. And I do have to say that tends to happen because of the way curriculums or scope and sequences are set up. We tend to do things unit by unit. I have thoughts on that. We'll leave that for another day. When we do things unit by unit, which I understand it's part of it's part of the game. It's part of, you know, what is thrown at us. I understand that spiral review needs to be hit heavy when it comes to that because it's not enough to say, "Well, we're going to deal with our decimal unit right now, do something else, then we're going to hit our fraction unit and expect kids to just somehow merge these things together. They really need to see that interconnectedness and they really need that spiral review and not a spiral review in the sense of a worksheet which you can do but spiral reviews in the sense of games routines you know this book the companion books do a really good job with that where it gives you centers and all game all of that basically all of those resources I know you have a ton of resources as well oh my goodness you have an abund abundance of resources that teachers can use but those given to students are important because it is that they can't just isolate I'm going to learn decimals now remember whatever I need I got to remember memorize whatever I need to have to memorize in order to get through this test and the veno test and then now we're moving on to something else and I completely removed what I learned previously because I have to focus on this thing you know so we have to kind of remove that mindset for students and we do that by really interconnecting these as much as we can. Focusing on those equivalencies, those conversions and talking about numbers, talking about numbers as you know these are not different things. This is again like I said more of the same you know wasn't breaking apart 10 more of the same. We do with whole numbers, we do it with decimals, we do it with fractions and this is what it looks like back to back al together. And I think that so I think also with fractions and decimals too like this was one that I thought was really interesting and I and I had this like sense of with in upper elementary this might be the one that has like maybe the most um I don't know if push back's the right word but like because it's not technically like in the standards until sixth grade when they start actually having to convert between because like fourth grade is and first are introduced to decimals but it's only um tths and hundreds and then even in fifth grade we're working with decimals but there's no converting between fractions and decimals so I think that that is sixth grade correct don't quote me on that but so I so I sense if you know with us being like a three five group it's like oh but is that one that they really need to get now like or do we wait until middle school and I think that with this it's this has to be based on understanding. It is. We're not talking about like either memorizing these like me. We're not talking about memorizing that 25 hundreds is the same as 1/4, but it's more so that reasoning and they have to they have to be able to see and experience and visualize like, okay, this is 1/4, but let's also think about like if we were to take a um a dollar and partition it and I I know I just talked about like kids are exposed to money. If we were to take a dollar or something else and partition it into four equal parts, what would be in those parts? It's 2500. So students are doing work that would allow for these connections. It just has to be intentional and it is I just want to make sure that I communicate that it's not this like thing that they have to memorize. It's not something that we have to avoid because it's not until sixth grade that they convert between them. Exactly. It's more so just connecting getting being intentional to connect different visuals, representations, things like that so that they can make that connection and know that 2500s is equivalent to 1/4. Absolutely. Absolutely. And you nailed it. I mean, we aren't addressing that until later, but the fifth grade foundation that we do is helpful for when students move into sixth grade and actually have to do that work. And the visualization piece that you mentioned is key and it's key to defining what fractions and decimals are. When students physically see a tenth in decimal form and physically see a tenth, you know, on a 10 x 10 grid, 100 grid, whatever you're deciding to use as your math manipulative and tool, when they physically see a tenth in either decimal form or um fraction form, they're able to see again, wow, this is the same. this is the same. It just has a different format. It has a different It doesn't really have a different name because it's the same name, but they can see how the concepts are interrelated. So, they're definitely not doing those conversions that they do in sixth grade, but we lay the groundwork for those sixth grade teachers. So, that's super important. Yeah. Yes. So, those kind of really the the fluency pieces that I'm hoping, you know, as everyone's reading that this chapter, you take away, especially doubling and halfing. Actually, all of them. The fifth 13. They're all They're all equally important. The underdog. I can tell. Yeah. Yeah. The underdog. Yeah. Exactly. I do, if it's okay, Britney, I'd like to kind of mention some tips if it's okay with you. Yeah. That I really want my teachers to take away with that are in this chapter, but I hope that it's not glossed over. So, I I kind of I don't know what page it's on, girl. I'm gonna be honest with you, but I wrote it down. Okay. So, there are just some tips here that I just I I hope they're takeaways for people. Um, one is, and this is just a list. I'm just going to go through it. We don't even have to go through each one, but just keep in mind because fluency is so important. The three components of fluency are so important, and these procedural automaticities are so important. But the way we we are changing the way we teach, right? So, you have to kind of have some things that keep you motivated along the way because it can tend to be very frustrating when we're doing something new and we're not seeing success as educators right away. So, something to keep in mind is it takes time for this to happen, right? I'm throwing out that doubling and halfing strategy. I'm throwing this out here left and right. It takes time for us to build our strength in that. Let you know, let alone the kids. Like we really have to keep that in mind. It takes time. We really want to also spotlight put the spotlight on when kids are using these strategies. So when you're doing things like number sense routines, let's say in the beginning of class, highlight when students are doing this on their own, when they're working with the derived facts, when they're transitioning from phase one to two, when they're transitioning from the counting to a derive, that's super important. So we want to spotlight them. We want to practice so much and I don't mean bombard them with I call it clerical work because it's just a sack of worksheets. You know what I mean? No one wants to do that. We don't even want to do that. I don't want a sack of work coming home with me, right? But getting them to practice in ways that gets them moving, that gets them talking, communicating. We know that those are keys to student learning, right? Moving, communicating, all of that. And we have practice all next week. Like that is all we're talking about next week is practice. It is so important. It's so important. And resources like this and resources like you have are great resources for teachers to take on. But the key is practice, practice, practice. And one last thing I'll say is do not gatekeep strategies. Please. One thing I do is I'm very uh passionate about working with multilingual students. Yeah. and we tend to see teaching lessons being very segregated when it comes to specific groups. Um I am very much an advocate of you know you can't gatekeep some strategies gatekeep strategies for some students and not others. Everybody should have access to all of this all of these strategies regardless of where you feel that they're at because sometimes unfortunately there is deficit mindset that comes into play. It's just a societal thing that has been around since forever. Um there's a great book to work on deficit thinking and thinking of things as a strength uh based model. I forget who wrote that, but there's a book on that. Is it strength? Um Karen Karp. Hold on. I just It's um I just shared a a post on it the other day. Of course in downstairs so I can't even It's the one with the superhero. I don't know if that helps. It's blue. That's all I I think. Yes. It's by Beth Coette and caring for Yeah. Strength based teaching and learning in mathematics. Yes. Exactly. So that helps with kind of shifting our deficit mindset to that strength-based model. We really need to see students in that strength-based model. And we do not want to gatekeep strategies. to not say this student cannot do doubling and halfing. They're just not there. They can't do it. Like everyone there. So tips to take with you because you know it can get frustrating along the way and we can feel defeated along the way when things aren't going smoothly right away. So thank you so much for sharing those tips. That is a great way to like close out our reflection on chapter 5. I thoroughly enjoyed our conversation. Thank you so so much for joining me um and just chatting about chapter 5. Thank you for the work that you did on the companion books and this was just a great discussion around like what is automaticity truly like what does it look like? How do we get there? So thank you so much. Yeah. Oh, thank you for having me. I had a great time. I appreciate it and I look forward to more discussions along the way. I hope you enjoyed my conversation with Rosalba. I told you she was wonderful. So, I look forward to being back with you live next week. We are going to be reading chapter 6. And we are almost done. We are only a few chapters away from closing