By empowering our students to lead and serve lead for change is changing lives, transforming communities and improving our world. Learn more, check out the research and access free leadership curriculum now at leadforchange.org. On behalf of the American Federation of Teachers, I'd like to welcome everyone to today's webinar on success in mathematics teaching students with the quality of instructional practice in algebra. My name is Robin Vitucci with the FT, and I will be your moderator before we begin, I'd like to thank today's virtual conference sponsor lead for Change, which is celebrating its 10th year lead for change is a free leadership curriculum for grades six through 12 with the Community Service framework that is easily integrated into any class. Club or setting lead for changes. The nation's fastest growing, privately funded student leadership program with more than 15,000 educators and nearly two million students. Give your class, club or school a chance to win up to $10,000 in the lead for Change challenge. You can learn more about lead for change by clicking on their logo in the right side of your screen. We truly appreciate your support now. Let's watch a short video on how our webinars work. Hello everyone, welcome to our 2022 share my lesson virtual conference. My name is Kelly Booze, director of the American Federation of Teachers. Share my lesson before we begin. We'll go over a few housekeeping items. For those of you who have joined us many times before, you know that we make our webinars as engaging as we possibly can. So to get us started, please open up that group chat box and tell us where you are from. And why you are joining us today and what interests you about this particular topic? In addition to the group chat, if you're joining us live, you will be able to provide some different reactions throughout the webinar today, so let us know what you're thinking and throughout the webinar, whatever reaction you want to give, share it with us and share it with your fellow participants. At the end of this webinar, we will be facilitating a question and answer session. Use that Q&A widget to submit any questions that you want us to ask the presenter. If you have any technical issues, please also use a Q&A widget and one of our share. My lesson team members is there and ready to respond to you. If you would like a copy of the slide deck or any of the related materials, you can find those in the resource widget. For those of you who want professional development credit, you will be able to download a PDF certificate at the conclusion of this webinar verifying your participation today, you do need to answer the poll questions that you will see throughout the webinar. To access that certificate now, let's turn it back over to your moderator who will put up a sample poll question for you to try. The poll question is located directly in the slides. You can answer your question. And then hit submit. From all of us at share my lesson. Thank you for joining us today. Enjoy your webinar. Alright, and now here is your practice poll question. If you had the option to live in a city for a year, which do you choose, Paris London, Tokyo, Buenos Aires, Cincinnati or I'm staying home? So click click which one you want and then maybe you can tell us in the group chat why you chose that answer. I think I would pick Paris because I would like to learn a new language and I took French when I was in school and so I know a little bit of it. But then I could learn more and what's beautiful so. Will give another couple seconds so people have a chance to answer. Now someone suggested Italy. I also I think that would be a great choice too. London all right? Nokia because your your daughter lives there perfect, all right now. Let's see the results. Alright, a lot of Paris people, so we'll maybe I'll see you all there. All right now it's my pleasure. To introduce to you our presenter, Dr Erica Lidke, with the University of Delaware, you can read her bio on the right side of your screen. Thank you for joining us today and welcome. Thanks Robin, and thank you everybody for being here. I as Robin said I'm in mathematics education professor at the University of Delaware. My name is Erica Lucky and I I want to welcome you all. I know there are many many many things pushing on educators and teachers. Time in particular and energy right now. And so I'm really grateful you chose to share an hour of that time with me. I will introduce myself a bit more in a moment, but I wanted to say a word about the webinar today which is titled supporting mathematics learning with the quality of instructional practice in algebra framework. In this webinar I'm going to share a framework for thinking about aspects of teaching that supports students learning in algebra. And then I'm going to share some specific strategies for planning and teaching and some specific instructional routines that support these ideas. Given the time we have together today, this is really only a taste of this work. But it's my hope that it will provide some useful ideas and food for thought for use in the classroom with other teachers and with your students. And I also want to caution. I often say that while this work is specific to algebra and I'll talk about algebra a lot, it isn't unique to algebra. So it's my hope that you know some of these practices and ideas might resonate even if you're not teaching algebra specifically. C oh good. Alright so I I have this bridge image in here a fair bit. I saw a number of participants from Pennsylvania. This is the Ben Franklin Bridge in Philly, which is where I am right now and I think about one of the goals of this is to sort of build a bridge, a bridge from what we know about how students learn algebra and the kinds of instruction that supports students and learning algebra content to what that could look like in practice and how we can kind of translate that into into our instruction when we work with. Students. I think that I'm pausing here. Yeah, for a quick poll the bridge makes me think of getaways and I know we already had a question about getting away, but I'm curious your preferred getaway? Would that be on or near the beach in a vibrant city in the mountains by a lake somewhere else? So take a minute and answer that poll and you're welcome to share your reasoning in the chat. If you wish. Oh good. Some folks are answering the poll. Great, thank you. I think I am a beach person myself, I keep. Dreaming of time at the beach? It's very centering to me. Probably what I would pick. That will give folks another minute. Yes, as someone said, I find the water very peaceful. I do too. It's a nice break from the chaos of working with students and teachers all the time. Something about the sound of the waves for me. Alright, so we'll share some results. Looks like we are a beach, a beach going crew which is great and you know, I hope that whatever place you you took and the sort of energy you know, the calm energy you find in your getaway that you you bring that with you throughout this webinar. So our time together today, I'm going to spend probably hopefully about 10 minutes, giving a little bit of background not only of myself, but about kind of what we know from research on algebra, learning and algebra, teaching about the kinds of things that support students to learn algebra. And then I'll share the instructional framework, which has five different components and we won't have time in this short webinar to dig into all five. So we'll focus on sort of three aspects of that, and I've listed them here, and hopefully there'll be some time for Q&A, and I'm trying also to keep an eye on the chat, so I encourage you. You can use the Q&A box if you have questions throughout. I'm happy to answer questions throughout. And you also can use the chat and I'll. It's just me. So I'll try to keep an eye on that and then at the end I'll share some resources and next steps. So one way I like to think about this as a way to start is to to share my why and So what does that mean? I the work I'm presenting on today comes from my Y and my why? I think about this doctor, Robert Berry, who's the former president of NCTM. Encourages us to not always start with what we have to do and how we're doing it, but to think about our why? Because until we're clear on our why the rest of the stuff doesn't. Doesn't always connect right? Knowing the why he says prompted me to ask myself a series of what and how questions. So in this picture is my why I started my career in education as a high school teacher in New York City and I see a lot of New Yorkers in the group today and I worked at a public high school for many years where I taught a lot of Algebra, 2 precalculus AP calculus class. This is one of my first calculus classes. It's a very serious group you can see. And my students were absolutely brilliant and demonstrated deep conceptual understanding of a lot of the calculus ideas. And but I was faced with this tension as a teacher, which was that sometimes my students struggled to get through problems. And often where that struggle point for them was was in the algebra and doing of the algebra. So they had like deep understanding of hard math concepts. But the sort of getting through and getting to the end was a challenge and a challenge for me as a teacher was. I would go look for help in teaching the content I was teaching to my students. I'd look for resources and I couldn't find them. There was a lot of support. For some or, you know, supporting elementary math students. But really, at the high school level and in the upper levels of high school, I struggled to find it, and so this led to some questions for me which was about how I could support these brilliant students to better learn the content I was teaching and how I could deepen my practice. So that I could better support students learning opportunities. And. I wanted resources to be able to do that and so my why was this? Search for resources and that's kind of what brought me to education research, so I wonder anyway, if you might take a minute and some of you already can be anticipated this question, but maybe take a moment in the chat to share your why. What brings you to this work? And you're welcome to comment on each other's wives as well. Someone said that's why I chose to teach algebra, right? I think it's yeah so I'm gonna make a case for algebra next. So as you're as you're sharing your why. So why algebra? One thing we know from a lot of education research and I don't know that we needed education research to tell us this, but that algebra serves as a gatekeeper for students, right? It's a course that a lot of students have to take to move on to the next level, but it's also the content in algebra serves as a gatekeeper for students, and some of that has. It has forced us as a field to think about improving access to students for students to algebra. We know success in algebra leads to the ability to take higher math classes, but we also know that. It's a predictor for positive outcomes for students. Yeah, we also know that students often really struggle in algebra, and this has actually led for recent calls for algebra to be abolished, right or completely reformed. Some folks claim it's an unnecessary gatekeeper, or that it's just simply not relevant to students and at the same time we're asking more students. To take algebra, and we're often asking them to do it earlier in their in their sort of school careers, right? So when we think about focusing on when it's offered to students, we also often don't think about the instruction and sort of what's happening instructionally when we offer algebra to students. And I would argue that this is all important, and these are important conversations to have. But I'd also argue that. Regardless, on your position of whether we replace algebra one with data science or things like that, we can kind of acknowledge that students encounter the ideas in algebra and need to develop their algebraic reasoning in their understanding. And when they do, we want them to be supported to learn sort of essential content deeply right? And in ways that are relevant to them, responsive to them and conceptually focused. So that's kind of where I'm coming from. I see a lot of folks weighing in on this that algebra is particularly relevant. Some folks are asking for strategies to help different groups who are learning algebra. So one question I have for you, and maybe you can share this in the chat as well, is why is algebra often so difficult for students? And I I'm trying desperately to follow the chat while I'm also talking, but it seems like there are some folks talking about confidence being an issue for some students gaps and misunderstanding, right? So students come maybe to an algebra class, but they're they have some some unfinished understanding from earlier courses. That's certainly one option. Interpreting words interpreting sort of some of the abstractions in algebra can be really hard for students. So as you're sharing what some of these ways in which algebra is difficult for students, you also might think about giving that. What do we think should be a key focus instructionally in algebra classes? In other words, how not? What I'm not interested. I mean I am interested, but I'm not talking about what content we should be teaching, but how should students learn algebra? So there's been another couple of other comments about understanding how algebra connects to real life, but also comments about language and number sense, right? And I'm going to share some of what we know about from research about what's why students struggle with algebra, and I'm curious to see the extent to which this resonates with some of the things you're sharing now, and some of your own experiences. So. So as I said, I'm an education researcher these days, and I have looked a little bit into the history of algebra in the US, and so you know, here's a picture of an algebra class of your if you will. And it's interesting to me to think about the ways in which algebra has and hasn't changed over the decades, right? So after we're used to be fun school algebra, I should say used to be thought of as generalized or rithmetic. The point of algebra was simple manipulation, equation solving, and so on. But the way we think about algebra has evolved since then to shift to focus on meaning making and reasoning functions and representations. There's been shifts to include cognitively demanding tasks for students, allow students to productively struggle with ideas, and I want to share a picture from probably the early 2000s algebra class, which honestly could have been my. This isn't my classroom, but could have been my classroom. The technology is about right. Of the wet, what we see right is that like while equation solving and stuff is still a part of algebra, what we wanna do has evolved, but it's really hard to do that. And you know my algebra classroom in the early 2000s looked a lot like this one here. I wanted a classroom to have students talking to each other, making meaning of algebraic ideas, but I wasn't always sure how to go about doing it and I wanted to think about how to support students to meet all of those challenges that you're sharing, right? Things like the standards for mathematical practice? Hard to do that when you're the one at the top of the room going through a procedure. I'm bored, right? Which is sort of who I was as an algebra teacher at the time. So I want to propose that I think that's sort of the root of some of these difficulties, but also maybe the answer to it relates to two different goals for teaching algebra and for math. More broadly, we often talk about algebra having twin goals, conceptual understanding and procedural fluency, and these two ideas sometimes get a little bit lost. And in particular, they're mutually supportive. And while we talk a lot about teaching, conceptually, we talk a little bit less about teaching procedural fluency. I think because we don't want to be people who teach students. You know who stand at the board and write on the overhead and like. Have students mimic what we're doing. But if we don't talk about that, then we can't talk about the ways in which they're combined and the way in which they inter intersect. So. I want to take a minute and defend the teaching of procedures, but I want to distinguish that from procedural teaching, right? So procedural teaching, which is sort of the stand and demonstrate type of instruction, is something that I think we in teachers we, as teachers, are often trying to do less at. We're trying to give students more agency. We're trying to make math more relevant to them. We're trying to have them take ownership. But that doesn't mean we're not teaching procedures, so I know that might sound a little semantic, but when I what I mean is that when I say procedures, I think about procedures as the steps or instructions for completing a mathematical algorithm or task right? And while algebra in many ways is the study of relationships, procedures are really prevalent. We teach students how to solve a system of equations using elimination. That's a procedure, and that's part of what we're doing. It should be the only thing we're doing. And so we also know that procedures are much more complex in algebra than in prior math, and that conceptual understanding and procedural fluency are intertwined and can mutually support each other. So it seems important to think about how we teach algebra, and particularly how we teach algebraic procedures. So I want to go back to the bridge for a minute. So what's going what I'm doing with my colleagues right now is we've been working to take research on what we know about how students learn algebra and the kinds of places that algebra can create difficulty for students. And the types of teaching strategies that support students algebra learning to develop a framework and a resource for teachers that highlight some of these key practices and can serve as a bridge from the need and the reality of the things that we need to teach in algebra and the skills we want students to have. So the types of instruction we want to see, right? So like what we want is this focus on understanding of key concepts and ideas and where we are. As some of you have mentioned in the child, is like that feels like. Sometimes there's a gap that we're having trouble getting there, and I would argue that through some of the ideas in this framework by focusing attention on these particular features of instruction, we can be bridging from the kinds of work we're already doing towards this more conceptual understanding. And with that we'll come all these other fun things that we really want students to be doing as well, right discussion, explanation, grappling, justification. Somebody put the standards for mathematical practice, right? Like these kinds of ideas. So that's the journey I'm hoping to take us on today and I'm certainly curious if any of this resonates with you. Also, the first thing I'm going to do is share the broader framework and then I'm going to drill down into specifics and give you just a taste of some of the resources that we're developing. And I'll say that we are working on a practice guide for teachers of algebra that's rooted in this framework, and so I'll share some of that work with you today. But we also have a resource for you that you can download that will allow you to take some of that with you. And I encourage I see that there are some conversation happening in the chat, so I encourage you all to keep keep talking to each other. OK, so this came out of both the research that others have done in algebra and my own work watching lots and lots of algebra lessons and identifying 5 instructional practices that teachers are engaging in that support students learning opportunities around algebra content that I think could work to bridge this conceptual understanding and procedural fluency idea. And two of them focus specifically on teaching procedures. And I'll talk about them individually in a minute, while the others the other three. Focus on making and building connections. So the first idea is that one thing that teachers do to support students learning in algebra is they support students to make sense of procedures right to connect procedures to underlying concepts, and they do that in a few concrete ways, which I'll share in a moment. The other thing teachers do is support students in developing procedural flexibility. Oh no, some folks, media players cut out. We have a tech person, sorry I'm just noticing in the chat Susan. Is this something that you can bring? Somebody is on that wonderful we have. Sorry so we want to support students in developing procedural flexibility when they're learning procedures and I'll talk a little bit more about what that means. The other thing, and some of this might look a little bit familiar, right? Is that in order to support students to learn algebra deeply and to both make connections between procedural fluency and. Conceptual understanding is we want student teachers can support students to make connections between algebraic representations, tables, graphs, equations, things like that. The other thing teachers were doing is situating the mathematics of the lesson. Uhm? In the context of other topics in algebra and the broader domain in math and I'll talk about that more later, but we often teach algebra as like a discrete set of topics like today as linear equations and tomorrow with quadratics, without supporting students to see how they fit in a larger storyline, both in algebra, but in math more broadly. And the final instructional feature relates to the fact that algebra is much more abstract than math kids have encountered prior, and that creates difficulties for students. And so one way that teachers can support student learning in algebra is to connect these sort of abstract algebraic ideas to concrete or numerical underpinnings. And again, I'm going to talk about each of these, but I wanted to give you the big the big picture. So. Here we go. So given our time together today, I'm really going to focus on three. Although I'll talk about the last two briefly, and for each of these three, I'm going to describe the instructional practice and give you some examples of some concrete ways that teachers can do things. You know. For example, may help students make sense of procedures. The other thing our team has done is thought about what this looks like in in practice in a classroom. So we've developed a set of questions that can support students in focusing in these areas. We've thought through some deliberate planning. If you're interested in focusing in these areas, what might it look like? And then we've developing instructional routines and I'll talk more about what those are that really focus kids on these particular practice practices and instructional features. So for each of these three, I'm going to share sort of all four. Elements and then in the short time, we'll probably have left, I'll at least talk about the last two. But again, as I said, it's a bit of a taste. So when we talk about making sense of procedures, what we mean is supporting students to attend to the mathematical meaning or concepts underlying the procedures, right there steps in the solutions they generate. It allows students to develop an understanding of why a procedure works and what the different quantities involved in a procedure represent and can support some of the things that you all identified in the chat right? Having students make connections to the real world, for example. So there's different ways that teachers can do this. One is to focus really clearly on making meaning of individual steps in procedure. Another is to focus on the goal of the procedure and make meaning of the solution generated by the procedure. I keep talking about systems of equations, that's what I was working with. Some pre service teachers on so it's in the forefront of my mind. But you know, we are frequently frequently teach the procedure for, let's say, solving a system of equations using. I don't know substitution, you know we solve for X, we solve for Y. We plug it back in and we check and we get these two answers. But we also could focus on what that that answer means, right? What it means in the context of the system, what it means in the context of the graph. A third thing that can support this is attending to why a procedure works mathematically, and highlighting the mathematical properties underlying a procedure. So one example, or I'm sorry I'm going to actually focus on this first part for a minute, but as we're thinking about what it means to support students to make sense of procedures, one way to do this is to is through the questions that we ask students, right? And to design questions as we're teaching our lessons, or as we're going through instruction that call students attention to these ideas quite explicitly. So there's a lot of questions here in general about making meaning of procedures, but these first two can help focus on making meaning of the individual individual steps in a procedure. So, for example, asking students why does this particular step work mathematically? What allows me to do this? What were the steps involved in solving the problem and why did they work and why did they work in this order right? These are the kinds of questions that can really focus students attention on, giving meaning to the procedures that they're doing. And then there's a number of other questions here that address some of the other components, but you can see how asking these kinds of questions might focus students attention on really developing meaning around the procedures that they're working on. In addition to providing questions, another way to focus on meaning making is through deliberately planning instruction designed to do this right, and our team has been thinking of planning in in three sort of levels and the analogy we've been using is one of the ocean and I should pause and say like I am in no way a marine biologist, so I'm going to say something probably biologically inaccurate here, but the way I think about it is like the surface of the ocean has some cool sea life that when you go snorkeling you can see, right, you get. Bishop different colors and things like that. But if you dive a little bit deeper into sort of that middle layer, there's all sorts of different ocean creatures and plants, and maybe you get to see things that are that are a little bit more vibrant and a little richer. And if you want to go down even deeper, you see all the like really wild animals, right? Like they're really interesting fish and whales and sharks, and what and what not. But you need kind of scuba gear and some deliberate planning to get there. And so the deliberate part comes out because there might be really good reasons to be at the surface. Like sometimes snorkeling is fine, but you might have a reason to dig deeper and so we try to provide a planning framework that takes that into account. There we go. So for example, if you want to think about supporting students to make meaning of individual steps in a procedure, you might start at those sort of ocean surface all the way on the left here, right? And start with some questions for yourself in planning. What are steps in the procedure? What concepts underlie the procedure? How does each step work? How can I show or explain these connections to students as we work through a procedure? If you wanted to be maybe more at the middle ocean level, you might ask yourself, OK? Well, how could I plan my lessons such that students identify the concepts underlying each step in the procedure? How can I support students to explain why the steps work right? Having the students sort of take some of the ownership of that? Let's say you really want to go scuba diving, and you want it to go way deep, right? And see some I don't know. Hammer heads or something. You might ask questions like, well, how can I design instruction then that encourages students to do this? Through exploration. How can I support students to make these connections? What problems or tasks could I pose? And I would. I want to focus on this last bit because it feels the hardest and I want to propose this notion of instructional routines as a way to kind of design instruction that focuses in this way, so I don't know how familiar folks are with instructional routines, but. And instructional routine, well routines in general, right? Or an essential part of math class. I had a routine for how students collected calculators at the beginning of my class. Routines are predictable structures. Students don't have to think that hard. They sort of know what's expected of them. And instructional routines are similar. They provide students with a predictable structure. They always look the same. The content changes, but their routine is a predictable structure. They allow students to focus on the mathematical ideas and sensemaking rather than on what they're supposed to be doing. And they support students to engage in some of the mathematical practices. They're also somewhat flexible in that they allow for curricular integration, and I'll leave this Robert Berry quote up here a little bit, but he talks about instructional routines as an essential part of math class because they give structure to time and interaction, letting students know what to expect in terms of participation. They support classroom management and organization and promote productive relationships for learning. So these are, you know, predictable, structured learning experiences. But again, the content changes, but the structure remains the same and I want to acknowledge here if you not. Seen this book, but Grace Kelly Monica and Amy Lasanta and colleagues have an excellent book called Routines for Reasoning that focuses on how instructional routines work, why they support students, why they're particularly good for multilingual learners and for students with disabilities, and how they connect to some of the standards for mathematical practices. So I wanted to put a pitch in for this book. So before I share the instruction routine that we created, I wondered how much experience folks had with instructional routines. So here's your second pole. Do you use them all the time? You've done a few. You've dabbled, but not much never. But I wanna hear more. Oh no, not interested, no thanks. So take a moment and. Answer This poll. I am going to share as as this is as you're answering, I am going to share three specific instructional routines. Tie to these are the ideas that I'm I'm sharing with you today and I also at the end of the webinar have a resource for you which gives longer descriptions, examples, and even a template for planning your own. So if this feels like it's going by very quickly, you'll have the resource to dig into a little bit deeper on your own. I'm just gonna give folks another minute to reply to the poll. This was new for me. I did not teach with instructional routines when I taught high school, but I do a lot of work with pre service teachers who are learning to teach. We're learning to be teachers and professional development work around instructional routines and I think they can be really powerful. I'm just gonna let a few more folks weigh in. It was told there's a bit of a lag on the pole, so feel free to weigh in in the chat if you wish. Wait for one or two more replies and hoping the tech issues got worked out. Alright. Or about told to wait for 75% almost there. One more person. Ah, there we are, let's see. Ah, alright, so I've you a few of us have used them all the time. A few of us have done a few. Oh I'm, I'm glad grateful that nobody chose E 'cause then then it. I guess it would be time to leave the web and are. But so this is wonderful. My hope is that there's some new information for everybody regardless, kind of where you fall on this spectrum. So what would an instructional routine that focuses on making meaning of individual steps in a procedure? For example, look like? So we've developed a routine that we're calling, annotate and explain, and this instructional routine is intended to focus students. On making meaning of individual steps in a procedure. As a way to both support students to learn algebraic procedures but also connect to the concepts underlying those procedures, right? And this routine leverages something called a worked example, which comes from a lot of research on things that support students procedural flexibility actually, and the what works Clearinghouse has a really nice synthesis of research showing how powerful worked examples can be. When you use a worked example, rather than asking students to solve a problem, you provide them with a fully worked out example and focus their attention on analyzing that example and what this does is it changes the cognitive activity for students and can focus their attention on key ideas, right? So the argument here is not. You never have students solve problems, but that sometimes it's useful to change the nature of the activity that students are engaging in. So here's an example that would look at solving single variable equations, right? We'll solving for X. And so I'll walk through the routine really specifically. But you can see on the left a worked example. And this is a textbook problem. This came right out of a textbook. Negative 3 ( X + 2 = 16 - X and in this routine students explore why each step in the procedure works. So you'll see that this is already worked out to a solution. We're not asking students to solve the problem, but you also see that there's some stuff missing, so we want students to annotate by filling in what's missing. In other words, how are we getting from the first line to the second right? Students might draw in the distribution. For example, how we getting from the second line to the third, and so on? But the key is, they're not just annotating then. They're also explaining right. So for each step in each annotation, why does this work mathematically? What's happening? And so again, the work students are doing then in this routine is they're analyzing rather than solving, and you can envision taking out this example and popping in another right, but the structure remains the same, which is the routine part? Of the instructional routine. One nice thing about this routine is you can envision in it as a warmup, for example, or you might adapt it to use problems from your curriculum or your textbook. You could extend it to be a longer in class investigation, but I think it has some flexibility in that regard. So how does it work? So before teaching right you would choose the procedure you were focusing on that you were interested in, and probably for yourself identify what are the key concepts underlying each step of that procedure and then select what I like to call an exemplar right. It's an example that is serving as an exemplar of the ideas that you want students to surface and workout that solution completely for yourself. Then you would present the worked example without the work that shows how to get from one step to a next and ask students, either individually or together, to annotate that example and show that quote UN quote missing work, and explain why each step works in their own words. Another opportunity is to give students some questions to prompt their thinking and discussions. So you might imagine it students working in small groups or even the whole class. You might put out some prompts to have students think about. As they work, so the next slide shows the same example with some questions right? Why did we do blah blah blah in this step? Why does it make sense to do that? How did we solve for X? Well, this strategy always work questions that can help students as they're talking about and answering. That are explanation. So here's an example of. Students work. And I did type out the students explanations just to make it easier to read. But you can see that the student drew those group of students actually drew in the distribution and then they wrote the negative 3 on the outside of the parentheses means you multiply. It also means take negative 3 copies of X + 2, which as a side note I thought was so delightful and what a great springboard for discussion. What does it mean to take negative 3 copies of X + 2? And how does that relate to the next line? Right? How does that relate to negative three X - 6 and then the students wrote the distributive property allows me to multiply the three with each term inside the parentheses. So here the students work is giving us the foundation to talk about why you're distributing and why this step works. In the next step, the student says we can add 3X to both sides of the equation, since adding the same thing to both sides maintains the equivalence of the expressions and as a teacher you're like jumping for joy that your students said that right, and there's a nice example of connecting the procedure to its underlying concept. We know that a lot of times what students do when they solve equations is they sort of indiscriminately add, like you know, they might add the three X to the three X and then the three X to the 6th or something. But this focuses students on kind of understanding why we're adding the same thing to both. Sides what's the mathematical idea under that? And you can kind of see through the other student explanations, similar ideas there. So before we move on to the next instructional feature and some more instructional routines, I invite you to sort of share some thoughts and reactions, either in the chat or questions in the Q&A. Do you see promise in this kind of routine for your work with your students, or do you have any questions you can share those in both the Q&A and then in the chat but, and I'll keep an eye on these, but I I am going to move us forward. OK, so the next practice is supporting procedural flexibility. So developing procedural flexibility within and across procedures means noticing structure to help choose which procedure to use under what conditions and in what ways, right it allows for the comparison of procedures and pathways to choose an efficient solution strategy and this is important because it allows students the opportunity to see and understand structure and deepen their understanding. But it's also a key aspect of fluency and I'll show you an example of that in a minute. So there's lots of different ways to support procedural flexibility. You might support students in nosing, noting multiple pathways through a single procedure, right? You could solve by distributing first, or by dividing. Identifying how to know what to do at each step in the procedure. Identifying conditions for choosing this procedure. What about this? The way this is set up makes me want to solve it using elimination rather than substitution, for example. Comparing multiple procedures for their affordances and limitations, and this is a big one, and this is the one I'm going to talk about and share an instructional routine with you about. So again, one thing we could do is start with questions that allow students to focus their attention on procedural flexibility, and you can see examples of these questions here, but you might think about questions like this last one. Why did I choose the strategy to solve the problem? What about the problem made me think it was a good strategy? Could I use a different strategy? Which one do I prefer? Which strategy made the problem easier for me to solve and buy? Similarly. Uhm? You might see you might think about planning in the same kind of way, right? Sometimes we want to be at the surface, so there are ways to think about showing students multiple strategies or multiple methods, but pointing out that there are others or showing one method in pointing out that there are others. We might also design instruction in which you're explicitly comparing Christie multiple procedures and having students see the connections between them. Or you might encourage students themselves to explore and compare across multiple procedures. And design instruction that foreground students doing that work. So I'm going to share another instructional routine that also leverages worked examples. That's a way to think about designing instruction that has students really thinking about comparing procedures. So we call this one side by side worked examples. And in this root instructional routine, you're using two or more fully worked out problems side by side, and the focus of the routine is on the comparison right? And again, these could be problems that come from your curriculum. These could be problems you design, it could be a warm up, it could be a longer longer part of your lesson. So here's an example with systems of equations, and this is the same system solved in three different ways. So Nico is using. I think it's substitution. Savvy is using elimination. Elliott is using what I've heard called the equal values method or setting them equal and they all get the right answer and these are all mathematically correct and valid approaches to the problem, but they lend themselves to nice questions of comparison right? And by giving students so this is what I would share with my students. Giving students work already completed allows them to then analyze and compare across and try to figure out what's happening so they have to do the work of figuring out that, oh, Nico manipulated the second equation and made it X equals, and then that's what allowed him to do substitution, right? Oh, I guess savvy added the two equations 'cause the wise went away. Those are the kinds of analysis that we would want. And again here to reflection quote. Sorry I missed something. So before teaching you would select the problems fully, write them out and then share them with students side by side. And again the things we want to focus students on our similarities and differences between the methods, efficiency or ease of use. And as I was starting to say before reflection questions can really help with that. So here is an example of some of those questions. Right, how is each student reasoning through the problem? Well, these strategies always work. Is there another reasonable strategy you could solve this by graphing? Does that always work? What about this particular problem makes one strategy more efficient or easier than the other? Oops, sorry I think I forgot something. So yeah, so. This instructional routine is a nice way to focus students attention on comparing procedures, and again I would invite you to share your reactions in the chat. Share any questions you have in the Q&A. And then we can. I'll share one more instructional routine with you in just a moment, but I wanna be mindful of time. I'm going to give folks a minute to share thoughts in the chat as I set up the next one. OK, so. As we talked about, there's lots of different types of connections that support students learning in algebra, and the one I think we often use most commonly is this idea of connecting between representations. Right and thinking about graphs, equations, contexts and tables. The second is situating the mathematics in a given lesson to the broader story line of math, and this third, connecting between numeric and abstract algebraic ideas, is connecting these algebraic abstractions to their underlying numeric or concrete ideas, and I'll talk a little bit about those at the end, but I'm going to focus right now on connecting between representations. So connecting we often use different representations, right? So here is a a graph, a table and equation representing a linear relationship. And I think we know it's important to use and present these, but it's the connections between them that really support student learning and these connections get support students in seeing how ideas are developed and represented across representations and become a little bit more flexible problem solvers. And so we could again support students in connecting representations within problems or across problems by the types of questions we ask, right? And here are a few. So what information do I learn from various representations? What are the advantages and disadvantages? And then you might have questions that focus on the connections. Can I connect what I see in one representation to something else? How do I describe the connections? And then I love this one. When I change something in one representation, what happens to the other representation? Why do you think that happens, right? If I change the leading coefficient in a linear equation, what happens to the graph? And that really forces students to look at and see connections? And again, the sort of same planning ocean analogy is here, where we might think about at the surface level showing some of these connections to students at a slightly deeper level, supporting students to articulate, explain or annotate, or at a deeper level, designing activities that allow students to do that work on their own. And so one such activity is another instructional routine, and this is the third one I'm going to share, and this one comes from Grace Kelly, Monica and Amy listen to exceptional work around instructional routines. Called connecting representations. This is a bit of an abbreviated version because of the time, but here students explore the connections between two representations by noting key features of 1 representation. And the other to deepen their understanding of the idea. So for example, here you see three linear graphs. And we start out by asking students how are these graphs similar and how are they different, right? So this is 3 examples within the same representation. They represent three different linear equations. The next thing we asked students after they sort of discussed the similarities and differences is we give them two equations now notice. Two of these equations match two of the graphs, but the third is missing. And so we asked students which representations match and how do they know, and importantly, we asked students to make connections visually. How do you see one? How do you know? How do you see the 1/2 in the graph? For example, for each matched pair? And then finally we ask students to create an equation for the graph without a match, right? Translating that understanding into a new representation where they're creating something on their own. And again, how do they know? And showing those connections? Uhm? I'm actually skip slide. There we go, so this is what that routine looks like all together and you might imagine again providing questions for students to discuss as they're doing this work. So that is a short explanation of the Connecting representations routine because I want to make sure that I'm closing out. With the last couple of ideas. So again, I invite you to share questions in the Q&A or thoughts and reactions in the chat, but I did want to end. I'm going to do it, I was just given 5 minutes. I did want to end quickly explaining these last two ideas because I think they can also be really powerful. So the first one I'm going to talk about is situating the mathematics. One way students develop algebraic understanding is by connecting the math of a lesson to prior or future content or to the broader domain of mathematics, right? Math concepts and ideas are interconnected and sometimes in school math we teach them as if they're not related to each other, so building connections between and across mathematical topics allow students to see a through line in what they're learning. They can connect algebra topics to each other to related topics or to the broader domain of math, and it allows them to see how they fit together. So one example of a teacher who I was working with who was doing this and we're working on creating some instructional routines for these as well, is her the goal of her lesson was in teaching students to graph linear inequalities, but she started with a warm up in which students found the slope of a line between two points. In this particular case. And then she asked us the students to graph this linear equation, which they did. And you can see this equation had the same slope from the warm up well. Then she reminded students that they had learned they'd spent a whole time learning systems of equations and using graphing to solve this linear system. And she had students do that, and they did. Right there it is and then she used that work that prior knowledge to get into what she was really focusing on that day. Oh, I'm sorry and they identified the point of intersection, but what she was really focusing on that day was solving linear inequalities and in teaching them how to do that she was able to make connections to the prior knowledge, the process for graphing linear inequalities is very similar to the procedure for graphing a system of linear equations, right? But it yields a different set, a different solution space and she was able to have. Interesting conversations with her class about how the solution to the system so this system of inequalities was different than the solution to this system of equations but point out key connections and make a through line. And the last thing I'm going to talk about, I'm gonna do it is connecting between numeric and abstract algebraic ideas. So one thing we know is that one of the reasons students struggle with in algebra is that it is more abstract than prior school mathematics. And so connecting abstract ideas to their numerical foundations can support students in kind of getting over this hurdle, right? So, one way to do this is by using numbers to develop generalized rules and properties. Thinking about how the pattern of raising 2 to the 4th 2 to the third 2 to the 2nd, 2 to the 1st, 2 to the Zero, informs what we understand about two to the negative 1/2 to the negative two, and using that to abstract to the rule for negative exponents as an example. Using pictures and manipulatives to illustrate abstract ideas or leveraging concrete examples or experiences. And I want to just share one quick anecdote, which I thought was so delightful. I worked with a teacher who was teaching systems of equations using substitution and he said to his students if I have four quarters and you at a dollar, would you trade with me and the students are like of course and he said why and they said, well, because they mean they say they're worth the same amount and he said, OK, so I want you to think about the Y as a dollar. And the two X - 3 is 4 quarters, right? They represent the same amount. So would you trade the why here? Would you trade the dollar for the four quarters and use that to sort of support students in understanding this abstract notion of substitution in ways that were very concrete for them and that really supported students to develop their understanding. So I'm going to end there here. All the pieces of the framer put together and share that I have a handout for you that. Mirrors what what is on this slide that brings together all 5 components of the framework. And as I started out by saying we're in the process of developing a resource for teachers that defines each of these instructional features. Each of these five categories talks about each of these individual components in depth, and develops instructional routines that if this was something you wanted to focus on that you would have an instructional opportunity to do so. And we're also sharing with you a resource that includes the three routines I talked about today. With examples as well as a graphic organizer, should you want to sort of create your own so I wanna end there with thank you. I know we have about 30 seconds for questions at this point, but I there is my email and my Twitter and I invite you to reach out to me. Should you have any questions and I just want to thank you again for sending the time for folks who are are tuning in after the fact. Thank you for spending the time and please reach out. Thank you very much. Great, thank you so much Doctor Litke. And yeah, for any participants we are out of time for the Q&A, but you will be able to email. Let me go back to that. Go back an email or Kelly Boos rejoining you again. Sorry, we can email the presenter if you have any questions that we can answer later, and I want to thank the audience for joining us and we have a quick reminder video before we close out. You can download your certificates and enjoy the rest of your day. Hi everyone, Kelly booze rejoining you again. I hope you enjoyed today's webinar as much as I did. I want to go over a couple reminders and I have one big favor to ask of you. First, you should now be able to download that PDF certificate for your participation. Today you can access that PDF certificate using one of the widgets, the one with the checkbox. From here you should be able to open up that PDF certificate and download it. The certificate will be saved to your name for up to a year. 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