Mathematical Minds in Action: Implementing CCSS Math Practices and NCTM Process Standards

This edWeb podcast is sponsored by Sadlier.

The webinar recording can be accessed here.

This edWeb podcast helps educators intentionally develop students’ capacity to think and act like mathematicians, beyond mastering content alone. Being a mathematical thinker and doer requires complex proficiencies such as reasoning, problem solving, communicating, modeling, strategic decision making, and using precise mathematical language. These practices are essential for deep understanding and long-term success.

In this session, we:

• Unpack the connections between the Common Core State Standards (CCSS) for Mathematical Practice and the National Council of Teachers of Mathematics (NCTM) Process Standards
• Examine the research that supports teaching these proficiencies as central to effective math instruction
• Explore how Progress in Mathematics (Grades K–5) translates these practices into clear, actionable instructional strategies

Educators gain insights into how to shift math instruction from rote procedures to meaningful problem solving and discourse. The edWeb podcast also highlights classroom strategies and resources that support diverse learners in building strong mathematical habits of mind.

Listen to this session for practical takeaways you can apply immediately, a deeper understanding of mathematical practices, and tools to enhance engagement and achievement for all students. This edWeb podcast is of interest to K-5 teachers, school leaders, district leaders, education technology leaders, and curriculum and instruction directors and coordinators.

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Transcript

SPEAKER_01 0:00

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SPEAKER_02 0:18

Hello, everyone, and welcome to today's EdWebinar. My name is Ines Pezzo, and I'm the marketing director for William H. Sadler. With me today are Janine Ferreira. She's the senior vice president public Thank you. So today's webinar is brought to you by William H. Sadlier. Sadlier is a trusted name in education for generations. Some of our most notable programs include the legacy math program Progress in Mathematics for grades K through 8, From Phonics to Reading, which has been rated by All Green by Ed Reports, and Building Reading Success with Wiley Blevins, which has been an intervention foundational skills program created exclusively for Sadlier. We also offer Sadlier Math K-8 and the well-loved and successful Vocabulary Workshop series. for grades 1 through 12, and the most recently well-developed, and we're excited to announce this, Southerly Pre-K with White Label Evans, which is launching this fall. And Vocabulary Workshop Select, a digitally vocabulary work program aligned for today's current high-stakes assessments, both designed to give students a strong start and prepare them for academic success. And now, without further ado, it's my pleasure to introduce today's presenter, Dr. Kateri Thunder. Dr. Thunder, would you like to share a little bit about yourself?

SPEAKER_00 2:12

Absolutely. Thanks so much, Ines. It's so wonderful to see you all. I'm thrilled to hang out with you for the next hour to talk about math instruction and, bigger than that, the practices of math instruction. I've had the pleasure of teaching every grade band from three-year-olds through college math, and no matter what I'm doing, I'm guessing you're probably like me, you're asking questions of how can I do this better? How can I have a better impact on my learners? How can I do this more efficiently? Because there's so many decisions to make as an educator. So we're going to dive into thinking about all those decisions and how to be really impactful and efficient in those decisions through this lens of the math practices and the process standards. So how can we take things that we know and research and put them into action? So I I hope you will share lots of questions as we go. Please make sure that you're thinking and sharing what you really want to know about our work. Here's my goals for our time. And hopefully something about these goals is jumping out at you already. So we're learning ways to intentionally develop our learners' mathematical minds. Lifelong goal, right? And we'll know we've learned this by the end of our hour together because we will be able to unpack connections between the Common Core and math practices and the NCTM process standards. We'll be able to examine the research that supports how to teach these really effectively in our math instruction. And we're going to look carefully at what can this really look like? What are my actionable steps? And so we're going to use a textbook resource, Progress in Math from Grades K through 5, to just kind of see how would this actually translate into my clear, actionable instructional strategies. So overall, In over 60 years of research around what works in math education, we know that we want to grow proficient mathematicians, math thinkers, mathematical minds. And to do that, we need learners to develop expertise in complex skills, not just the content, but the complex skills of mathematics, the processes. And if we want learners to be able to do that, that means our teaching has to actually emphasize those complex skills and explicitly teach them. So overwhelmingly, 60 years of research has told us that. And over time, research has kind of defined what are these processes, what are these proficiencies more and more and more clearly. So in 1980, there was this landmark piece of work saying we need to think about problem solving and how can problem solving be central to our instruction. So we've known that for over 40 years. And then in 2001, the National Research Council said, hey, let's really like pull that apart and really look at what does it mean to be mathematically proficient? Well, there's actually five strands. There's strategic competence, conceptual understanding, adaptive reasoning, procedural fluency, and productive disposition. So you can still see that problem solving is part of this space, right? It's kind of the foundation of all of our thinking about what it means to be proficient, and how we're starting to tease apart these strands to see what else does it mean? Everything from disposition and attitude, right, through something that is really abstract, which is that reasoning, right, and that understanding, and then some things that can become more concrete with fluency and strategies, applying those strategies. And over time, we've tried to really, through our research, define these even more explicit And so you're probably familiar with NCTM's standards. They came up with content standards that say, here's the what of math that we should be learning. And then they also came up with process standards that say, here's the how we'll learn it. And they used problem solving, reasoning and proof, communication, representations, and connections to try to define what does it mean to engage with mathematical content and with our peers as they're working with the mathematical content. So all of this, we kind of put together and said, well, these are really great ideas across mathematics, education, research. We really are starting to understand what it means to be mathematically proficient, but how do we put that into practice? What does that mean in my classroom with my kids, right? That's what we all want to know. And so the math practice standards from Common Core tried to make the processes, the proficiencies even more actionable. So this chart is really trying to show you some of this alignment, that the math proficiency strands from the National Research Council, they connect to all of the math practice standards. And then you'll see there's five NCTM process standards, but you'll see them kind of connecting to multiple practice standards, right? All the way from the first math practice standards all the way to the eighth one, right? You kind of see them interweaving in there. So our work together today is we're going to really try to see what does research say is the best, most effective, most efficient and impactful way to engage kids in the math practice standards, which are linked and aligned to those process standards and linked and aligned to those proficiency strands. So you're going to see this chart, little snippets of it come up over and over again as we unpack each of those practice standards and process standards. I'm much research I'm pulling together for you so that you can take it and translate that into your classrooms. But I wanted to share with you some resources that are available in case you want to check them out. So there's the From Principles to Action and Catalyzing Change from NCTM, and they are really working to make these standards actionable. Then there's two resources from IES and What Works Clearinghouse about overall mathematics instruction. How can we support them? How can we support our learners? What do we know is most effective and efficient? And then overall in our instruction, math and beyond, what do we know? And how does that line up with what we know about teaching the proficiencies? And that also comes from IES and Visible Learning by John Hattie. So the links to some of those free resources will be in the chat in case you want to go and see the original research and check that out further. So here's our plan. Ready? We're going to jump in and for each practice, each process, I'm going to share a definition that comes from those resources, from that research over the last 60 years, particularly really emphasizing the last 10 years. What have we learned? What do we know about how to teach this and what those practices are? Then I'm going to share those research recommendations, right? How can we best teach those practices and make them actionable in our classroom? And then I'm going to share some examples. So if you are a note taker, this is your chance. You can grab your piece of paper, right? And you want to have kind of those four categories. What's the practice? What's the definition? What's the research recommendation? What's the example? And if you're not taking notes, then hopefully this will give you kind of like that mind's frame, right? Of, oh, each practice, this is what Kateri is going to be sharing. All right, you ready to dive in for the first one? Here we go. All right, so the first math practice standard is make sense of problems and persevere in solving them. And you're going to see links to problem solving, communication, and representation as we define that. So when you think of engaging your learners and making sense of problems and persevering and solving them, what do you think of? I tend to really think a lot about cognitive skills. And I think many, many of us do, right? And we kind of lean towards the cognitive skills of problem solving. Do learners comprehend the problem? Can they make some strategic decisions about tackling that problem and using what they already know, using the language of the problem, using their different strategies in that space. But there's an equally important part of making sense of problems and persevering and solving them, that persevering bit, right? And that idea of actually making sense of the problem. And those are the affective skills. So we know we also need to engage learners in self-regulation and some productive struggle. How can they try these problems and how can they say to themselves oh, not only will it make sense, I do believe it will, but I'm capable of doing this. I might not know the answer yet, but I have some things that I can use to help me tackle this and figure this out. So this practice means that we need to do instructional moves that engage both the cognitive part of our learners and build those skills, as well as the affective part of our learners and build their affective skills. Now we're gonna look, that's our definition of the practice, and now we're gonna look to see what do we do? How do we do this? So there's four research recommendations, four ways that we can impactfully engage our learners in making sense of problems and persevering in solving them. I'm gonna share each one and show you some examples, okay? And as I'm sharing the examples, think about in your instruction, how do these mirror things that you're already doing, right? They might not be exactly the same, but how is something that you're already doing? Yes, that's the same. And maybe there's a little shift for you to make. Maybe there's a way that you could make a small change that'll have a big impact in implementing these research recommendations. The first one, explicitly teach learners to think about their thinking. That's that metacognition. That's that place where we're helping learners say, okay, I'm going to be in a problem and how do I tackle this? What's maybe a framework for thinking about problem solving that I can engage in that focuses not just on memorization, but on sense making? So this is an example from a The Saddler curriculum of the four-step problem-solving process. So this might be similar or the same as something that you do. There's not one right way to do it, but you'll notice that this problem-solving process really emphasizes that you're reading to make sense, to know, comprehend what's happening in this problem. And you're making a plan. So you're thinking through, huh, all right, I'm not sure what the answer is yet, but I have some ideas of what I might do first and second. and third to try to tackle this. And then write. So work your plan. Actually try this out. Actually draw some things and write some things and build some things and create some things and label things so that you can take this process that's in your mind and try to bring it down into paper or concrete materials. And then check. Don't just come to a space where you're like, wow, I think that's the answer, but actually check it and see, does that even make sense? If I bring what I have figured it out into the space of the context again. Would that make sense? If I tried this a different way, would I get the same answer? So this space, you can see on the right, an even more detailed version of this type of a framework to help learners learn how to think about their thinking, right? The example on the right has more details for learners who are readers and older, right? Fourth, fifth graders. So there's more details in it, but you'll see that same framework that you start as a young learner, read, plan, solve, check, and then you keep leaning into to that framework to help you think about your thinking. There are so many ways that you can engage kids in understanding, making sense of a problem and thinking about their thinking and having something that learners can lean into and recall. It's like a mnemonic, right? Is really powerful, really powerful, especially when you start to feel some of that stress because you're going to see some problems that maybe you don't know how to solve, right? Something that you haven't seen before and that can create some some affective response. The other thing that we want to do is we want to actually teach multiple problem solving strategies. We want to create a toolbox for our learners so that when they're in that space of thinking about their thinking, they can say, ooh, the strategic decision making. I need that task. I need that strategy to help me in this problem. So Polya has a really phenomenal list of strategies that can be helpful. And so that might be a place to lean, to look at what are some problem solving strategies and to be really explicit and intentional in teaching them. Here's how you might act out a problem. Here's how you might find a pattern and then use it to figure out a solution that makes sense. Here's how a graph might be able to help you. And you can see from this chart, right, that being really explicit and intentional in how over time we teach learners more and more problem-solving strategies that also teach them how do you use it in this situation? How do you make a decision, right, to make sense of the problem and persevere in solving it? Oh, that strategy didn't work. What's the one that I could use instead? Well, maybe I need to convert this to a simpler problem with simpler numbers and see if I could figure that out. The third and fourth recommendations are solving novel problems. That means something that when a learner looks at it, they're not sure what the answer is right away. They don't necessarily know exactly how they would tackle it immediately. Over time, kids will start to say, oh, this strategy works really well for me as a first step. But at first, we really want to make sure that we've got novel problems and that we explicitly model ways to represent those novel problems. We say like, hey, there's a way to model, a way to represent this problem with mathematical models and to see what's actually happening, to try out those strategies. So this is an example from a first grade lesson where we're learning how to use this strategy of act it out. So we're explicitly teaching a problem solving strategy. We're using that framework to think about our thinking of read, plan, write, and check. It's a novel problem. So many of the kids haven't seen this before, right? They haven't seen, oh, Pam sees two butterflies. Jake sees one more than Pam. How many butterflies does Jake see? And then in this space, engaging kids and actually understanding here's an example of how I might act this out, how I would model this problem, right? How am I going to represent it? And giving them a chance to try it as well. So those are the four recommendations for the first math practice. Explicitly teach learners to think about their thinking. Explicitly teach multiple problem-solving strategies. Select novel problems. And then explicitly model ways to represent those novel problems. Oh, I could use cubes for this? Great. I could pretend those cubes are butterflies or dragonflies or people. I can pretend there are lots of things, and that will help me use that manipulative to represent that problem. Here's a few other examples, just to help you connect to your current practice to see, am I doing these four recommendations? Have I already tried these out, right? Are they already part of my teaching practice? So the first one, talking it over with a partner, with a small group. Okay, I've got a fourth of a set of flowers red, the rest of flowers are blue. Are there more red or blue flowers? That's novel, right? I'm not quite sure how to figure that out. Well, I can lean on my mnemonic, my framework to try to make sense of this problem. I could think, okay, well, I'm going to read it. I'm going to plan. I'm going to write. I'm going to check. And maybe I want to help kids understand, well, I could act this out, right? That's one of my strategies. I can make this a little simpler, right? What if I said if a half of a set of flowers is red, right? How could that help me to problem solve here, right? Two other examples. Dan has a solid figure. It rolls. It doesn't stack. It doesn't slide. Which of these figures could be Dan's? Which is those three D shapes? Rolls, but doesn't stack and doesn't slide. Now I'm going to lean into some of my strategies that are in my toolbox and use it to help me make a plan. How could I solve this? And here, this bottom one, check this out. They're actually going to try to draw, right? Saying like, hey, you know what? Drawing can help you to resolve this problem. Drawing parts and holes can help you to figure this out. Maybe trying out an equation and seeing how an equation that represents the situation could help you solve. That's a strategy you could use. And I'm going to model for you how you might try it. And then on the next, another type of problem, you'll be able to use it. All right, so I'm guessing you have lots of ways that you're doing these four research recommendations. Hopefully there's something in there that is new for you. And we're going to jump into the second math practice. This one is reason abstractly and quantitatively. It connects to reasoning and proof and problem solving. So I love this one because this is one that I actually think many of us kind of shy away from, reason abstractly and quantitatively, especially in the elementary grades. It sounds a little intimidating, right? And if we're not feeling like competent math thinkers, math doers, it's hard to engage our learners in this work. So first, let's look at what does it mean? What does it mean to reason abstractly and quantitatively? What does it mean to think about that reasoning and proof and link it to our problem solving? Well, it It means that we're taking quantities and operations and we're contextualizing them, putting them in a context, a situation, and decontextualizing them, taking the numbers out, taking the operations out, and just looking at those naked numbers. It also means that we're interpreting these meanings, the meanings of the values of the numbers, the meanings of the operations through different representations. And then we're thinking really flexibly about those relationships and how that can help us draw a mathematical inference? How can we infer, oh, that's the relationship that's happening there. Here's how I might describe it. Like that reading word in there, infer, we do it in math also. All right, so let's look at what does the research recommend? How do we effectively, impactfully engage kids in reasoning, abstractly, and quantitatively? There's three things that we do, okay? The first is that engaging and contextualizing and decontextualizing. So you probably do pieces of of this, putting the pieces together is going to be the important part. So here's a contextualized problem, right, where we're talking about a display of plants in trays of 10. And there's two different questions here. How many trays of plants do they buy in all? And how many more trays does one child buy over the other? So it's in a context. Then, after we've really made sense of that context, we want to help kids to De-contextualize, pull some numbers out and see how can I express what's happening in this context with numbers, with operations, using expressions and equations. So here are two, right? Here's where the first one is where they're adding or combining to show how many trays were bought in all. And the second expression is showing how many more one child bought than the other. And they're using subtraction for that. But then we want to contextualize again. We don't want to leave it out there in that decontextualized space, right? So the solution after problem solving and figuring this out using different representations, we want to put back into the context and say, oh, well, the answer, right, to that, the sum to that addition problem is three and four-fifths. But what does that mean, right? Three and four-fifths of what? So they buy three and And then, okay, so I've done this subtraction work. What does it mean, right? What does that number, that fraction that I've ended up with, what does it mean? And so I have contextualized again. So going back and forth is really important for our learners to make sense of this, to really start to reason abstractly and quantitatively, thinking about those numbers and relationships. Here are some other examples of ways you might do this. You might start with naked numbers. And have learners really think about, well, what would that mean in a context? Because those numbers and those values and those operations can mean different things in different contexts. Or you could start with a context, right? And maybe just provide like a little image support to say, well, how could I decontextualize this to make sense of it? What would it mean if I was just looking at numbers and operations? Or you could even engage learners in creating their own, right? That's a way of asking learners to contextualize and decontextualize, asking them to think of numbers, think of operations, and put them into words, and then to remove them from those words and see what those relationships are between the operations and between the quantities. So that's contextualizing and decontextualizing. And in that work of putting numbers into situations and then pulling those numbers and operations out, you're making connections among representations. That's the second recommendation. So here's an example. So we're looking to see how can I make connections between this contextualized story problem that there's three pairs of tennis socks in a package and three packages in each box and how many pairs of socks are in these two boxes. And we've got another representation of an image. So I can start to say, how do those words, how do those sentences match that image? How do I make sense of that, right? And then I have an equation. Well, where do I see this equation in this image and in this context? How do I see connections across these different representations? And then I also see, oh, we're trying to highlight like, oh, here's how I use parentheses. Here's how this associative property of multiplication works. So now I'm connecting it to this property, this relationship. And how does this relationship get represented by mathematical notation, by words in a context, and by images? So helping learners make those connections helps us to really think about that reasoning space. How can I see and talk about these quantities? And how can I kind to abstract those into some mathematical relationships where maybe I'm just seeing numbers and equations, just seeing the mathematical notation. The third recommendation is to teach related operations together. So this is something that you all probably do often, right? Thinking about inverse operations, thinking, well, I want to make sure that when we're talking about subtraction, I can also say, oh, a strategy for subtraction is think addition, or a strategy for division is think multiplication. There's also really important work around all of the language, all of those representations and how the contexts help show the relationship. between the operations. So in this context of 21 tennis balls with three tennis balls in a can, how many cans are there? We're working to help make sure that kids see, oh, there's groups, there's number of groups, there's a number in each group, and then there's a number in all. And those are my factors in my products. Well, that actually relates to division. They're just called different things. So this teaching of operations together can help me think real flexibly about operations and say, well, I'm going to be engaging with some, well, what maybe first appears to be multiplication, and then I can also use some division. Or maybe it is first appears to be division, and then I can use multiplication flexibly in that space. The other kind of relationship you might do is say, I'm going to pop back for a second, that maybe you want to show how addition and multiplication are related to each other, or subtraction and division. So all of the operations kind of work together. So are we helping kids as we engage in reasoning to see each of the operations as related to the others, right? Or are we teaching them separately? We want to teach them together. All right, so that's our second practice. We're moving right along. You ready for our third one? The third one is one of my favorites, constructing viable arguments and critiquing the reasoning of others, right? This is that space really of communication, but you're still going to lean on some of that reasoning, still going to lean on some of that representation. So the research tells us that this mathematical discourse, that third practice, is not just speaking, but it is also reading. It is also writing. It is also listening. listening. And because it's mathematical, it's also visually representing ideas and viewing them, seeing them, right? So I encourage you to think through this constructing a viable argument and critiquing the reasoning of others. Are you engaging learners in all of those different modalities? That's one of the things that the research is going to tell us. Recommendations are to engage in mathematical discussion or argumentation, right? To not always have a right answer, but to really debate it, to debate the process, to debate which answer makes sense or answers, to really establish this idea of math talk in a space where you're listening and speaking, you're reading and writing, you're viewing and representing, establish that as a regular routine and then use multiple modalities. So let's look and see what that looks like. Ready? Okay. So do you remember back at the very beginning, the first practice, we said that there's a framework that we can use for problem solving to help us really regularly think about our thinking? Well, that can help to create kind of guardrails, right? So that this mathematical discussion is something that can be safe, emotionally and cognitively safe for me. So I can engage in discussing, well, right now I'm reading, so I'm trying to really understand this, right? Or I can write, here's what I understood from reading this problem. So it helps to give me some safety net, right, for how I might engage in that discussion. And also, remember, we looked at that list of strategies. We want to make sure we want to teach kids explicitly some strategies for problem solving. Well, when we're in that space of discussion, that argumentation, we want to help them talk about those strategies. We want them to say names of strategies they're trying. Well, I drew a picture. Well, I used a graph to help me. Right. And so we're going to also support the language that they're using. We're going to support their their ability to name a strategy and use a strategy to think about and regulate that problem solving and then to make a plan and check it. The other thing we can do to make math talk this regular routine is to have times throughout our day where we're engaging or throughout our math time where we're engaging kids regularly in different ways of communicating about math. So here are some examples. You might have a talk it over or a let's talk, right? And you're asking more of an open question where kids might write and talk. They might think, pair, share, right? They might draw and talk. might draw and show and then write back to each other. You might have a write about it where they write first, right? And then they're sharing it with a peer or showing it to a peer. And each of these different ways of formatting are saying things like, explain how you know. We're saying things like, well, is it true? Is the statement true or is it false? And if it's false, could you explain why? I could even add, if it's true, can you explain why? Right? So something where there's a bit of predictability I know we're going to be doing a think-pair-share. I know we're going to be doing a let's talk. I know that we're going to be able to be engaging in writing and then exchanging our writing and giving each other feedback. Now, the thing that I'm focused on is the math talk. So I feel safe because I know we're going to engage in math talk regularly. And then what is it that I'm going to argue? What is it that I'm going to be thinking about and discussing and critiquing the reasoning of others about. The third recommendation is about using multiple modalities. So making sure that we're not just talking out loud, right? Or that we're not just writing, but that we are actually engaging in speaking and listening, reading and writing, viewing and visually representing. So that might be a space where the strategies that you're explicitly teaching help you to look at them and say, am I making sure that kids are really engaged in all of these modalities? Or do I tend to lean on one rather than the other? How can I make sure that as we're engaged in our discussion, there's a chance to write, there's a chance to read, there's a chance to listen, there's a chance to represent? It might, these are two other examples of routines. It might actually just be spread across your time where you say, we're going to ask, what do you notice? What do you wonder? You might be familiar with that routine, right? You might ask when you're doing that contextualizing and decontextualizing to explain something about the context or the number without the context. And so anytime that there's a chance to explain, matching that with multiple modality these can be really powerful. All right, guys, we are rocking. We are rolling. So I hope that you're adding questions into the chat. I hope that you're seeing some places where you are doing really great work and finding some little spaces to make some shifts. This is math practice four, model with mathematics. So you'll see it really kind of encompasses many of the process standards because you're representing and making connections, you're using the modeling to problem solve, and you have to do some reasoning, right, when you are modeling. So let's look and see what the research really says about this modeling with mathematics. Well, it sounds similar to that reasoning space, right, where you're decontextualizing this context, the situation, and abstracting it into a math representation. And then once you've taken this and you've put it in a math representation, you're describing what are the relationships I see? What other representations or tools can I use to really understand this, to really describe all the relationships? And then analyzing those relationships to make a prediction and bring it back into the context. So you'll, you hear some similarities, right? Some overlap. And that's because mathematical modeling is the heart of all work that we do as mathematicians, right? We're taking all the content that we know, we're taking all the strategies and representations that we know, and we're using it to model, using that mathematics, and then to help us make those analyses and predictions. So how do we do this with kids, right? How do we do this with kindergarten through fifth grade? Well, two recommendations that we're gonna unpack. The first is to really make sure you're using those multiple representations. There's five types of representations that you wanna look through and really make sure that you're hitting each of them. You wanna make sure that you have physical representations. So that is manipulatives, acting things out, right? It could be real world objects, right? Like if you're thinking counting bears, base 10 blocks, 10 frames with counters on them. You also want to have visual representations. So you want to look and see, well, have I shown kids how to draw something that they've been using as their manipulative, right? So in this example, you can see they're really practicing. Well, they built with the 10 frame. Now what does it look like to draw it? They built with the base 10 blocks. What does it look like to draw it? And how can I do that efficiently? You want to also have visual representations that are things that are mathematical tools like tables and graphs, number lines, right? What are all the ways that we organize and show information mathematically? We've got even like graph, dot paper. What are ways that we can show using a bar model? All of those are visual, and they're all ways that we can take things that are happening in the situation and model them. Then verbal, right? We want to make sure that we're engaging in language, not just mathematically precise language, but also real world language, because that's where mathematical modeling intersects, right? It's a real world situation. And then we abstract that into some mathematical representations. We also want that context. And we want some symbolic representations. We want to make sure that we're showing the mathematical notation, the different ways that you can orient an equation. the different ways that you can represent a number or a value or an operation, right? Like the division sign, you can show that so many different ways. So all of those five types of representations across your instruction should be happening so that kids really build up some of those mathematical modeling tools. And then the other piece of research is to implement the CRA method. So the CRA method, it stands for a concrete representational and average Sometimes people refer to representational as pictorial. So they'll say CPA method or they'll say symbolic. So concrete, symbolic, abstract, CSA method. But really what it means is you want to have kids, all learners, all ages, engaging with something concrete and parallel modeling. What would that look like if I created a visual, a picture, and then parallel modeling? Parallel modeling, what does that look like when it's something abstract, right? When it's that symbolic representation and how do I talk across all of those models to really make sense of them? How do I look and see how are these multiple levels of representation similar and how are they different? That parallel modeling through the CRA method is something that actually research has found is effective with all groups, all populations of learners, all ages of learners So sometimes I like to point out, if you feel like you did not care for algebra or calculus, it might be because you didn't get to use some concrete materials and see how those connected to representational or pictorial representations. And then how they also connected to the symbolic. You might've had to just jump into symbolic and that wasn't really priming your brain to make sense of it, right? So all ages, all learners engaging in these three parallel, modeling, concrete, representational, which is that picture, that image, and then abstract. Here's that example of base 10 blocks, right? Can you see all three levels being parallel modeled here? I can see this is representational base 10 blocks. I can see the symbolic, right? Or the abstract with the equations and the place value table adds a little visual in there, right? Then I'm asking kids to draw. So I've got representational again. I've So there's one level that's missing that I have to make that decision as a teacher to pull out the actual base 10 blocks and let them see those connections. Let them see how are each of these visuals and each of these abstract representations connected to the manipulatives. Here's another one. We could do that without numbers. We can do it with geometry, with shapes. Are there ways that we can help kids see the manipulatives and then also connect that parallel to a visual? And then what might be the abstract? Well, there's lots of really abstract things, honestly, about shapes. We talk about them, their angles, talking about the number of sides, all of the mathematical words I'm going to put onto these images and these manipulatives. Those are abstract. calling something a pentagon, calling something a hexagon, calling something a trapezoid. That's all abstract. So I want to do parallel modeling. All right. You got your checklist of what you're doing, right? Things that are going well, things that you're trying and something that you need to try again. Okay, so here we go. Math practice five, use appropriate tools strategically. So you'll see the connection to problem solving and representation. Well, it really is. You've got these problem solving strategies and you combine it with what you know about representations and the use appropriate tool strategically means that you're going to strategically select and appropriately apply tools that include manipulatives, different types of graphic representations, using a calculator, all of these different tools that exist mathematically. How can I use those representations? How can I use those strategies really strategically? When's the right moment to do it? So there's three recommendations and research around using appropriate tools strategically. The first is to make sure that you're using estimation to monitor your problem solving strategies and solutions. So you're not just teaching estimation, which I have totally done, right? I've taught rounding as a separate thing. And then we get into adding and subtracting, multiplying, dividing. I never come back to it. Well, the point of estimation is to help us engage in that monitoring of problem solving strategies and to say, oh, I figured out the solution. Yeah. Okay. That fits what I estimated or no, it's totally off. Where did I go wrong? Okay. And you can see that in this tool where we're explicitly teaching the metacognitive process to say, did the strategy I choose, did the manipulative, did the representation really work, right? My estimate can help me do that. My metacognition, right? The questions I'm asking myself can help me evaluate that. So I might make a plan Think about which strategy I want to use, which tool. Estimate before I compute. But then when I get to a solution, pause and really think through, okay, was this reasonable? How does this compare to my estimate? Was my estimation strategy really off or was my strategy not the appropriate one? Was my tool not the appropriate one? The other thing that we want to do is make sure that we practice this strategic tool selection. We want to deliberately practice. So that means we're actually saying, hey guys, we've got a goal right now. Our goal is to think about using estimation to help us figure out what would be a reasonable answer, to help us identify the representation that would be most efficient here, right? Whatever that goal might be, we want to say it to kids so that then as they try try their practice. They may not be novel problems anymore. They may be more of tasks that they're familiar with, but now they're really thinking about it. They're engaging in that metacognitive thinking to figure out, is this the right strategy? Is this the right tool for this situation? That's the deliberate practice of a strategic tool. Here are some other examples of that deliberate practice, right? So I might say, hey, we're going to estimate using compatible numbers. Here are some problems to try. What might be the way that you're using compatible numbers? Did the way that you use it work? So I'm also engaging in that metacognitive thinking to evaluate. I might ask like, hey, okay, you need to decide. Do you think you can figure this out mentally or do you think you need to create something with your pencil and paper Is there something you need to draw? Is there something that you need to write? Which works really well for you? That's that critical thinking. Where it might be, I talk it over to say, well, which would be easier? Why would it be easier to use mental math in this situation and to draw or write in this situation? So we're deliberately practicing saying, we're going to think about our decision making right now. Which tool should we use? Which strategy? Which representation? Are you starting to see a lot of intersection? Because I feel like when I look at the math practices, the deeper I go, the more I think, oh, well, that sounded like math talk. That sounded like sharing my reasoning and critiquing someone else's. Yeah, because I'm also strategically choosing a tool. And then I'm talking with a peer about it or with my class about it and defending it and critiquing each other. So you end up doing multiple practices together. All right. Math practice six. We're rocking. We're rolling. This one is attend to precision. Attend to precision. Sometimes people think this means get the right answer. That's not what it is, okay? Attend to precision is putting together some of that reasoning with the communication, with the modeling to say, I need to use precise definitions in order to reason about this accurately. I need to use precise notation to communicate and model about this really accurately. I need to think about what are the pieces and the parts of different representations So that I can use those representations accurately. And what's the vocabulary? I need to be really precise with the words that I'm using so that my reasoning makes sense to someone when I communicate it to them. And when I describe my model using different modalities, that my vocabulary is accurate and precise. So it's not just... Did I get a right answer? It's really thinking, I want to make sure that I'm putting these things together precisely, right? Where someone else will be able to understand it and it is mathematically true. There's two research recommendations around attending to precision. The first is analyzing worked examples. So a worked example could be something that is correctly worked all the way through and analyzing it. It could be something that's partially correct, right? But there's part of it that's not done yet. And it could be that it's incorrect. So together, you'd want to look through and say, well, what can I notice in this example about the precision of the notation around the vocabulary, around the definition that's used? So take a look and see these two examples here at the bottom. And this conversation that we could have around the parentheses, around how do I know, oh, I'm using a division symbol or am I using an addition symbol? Where do they each go? I could have an analysis of this worked example and make sense of what is the notation? What are the language, the definitions? Am I using the order of operations correctly? What does order of operations even mean? Okay. Here are some other ways that you might be analyzing worked examples. Sometimes people will say, hey, what's the mistake, right? My favorite mistake, what's the error? That's a way that you're analyzing an incorrect worked example, right? So you're looking to see, well, I think they're missing something. There's something that I need to attend to precisely about this representation and the language that I'm applying to it. And this one about Kara and Andy, right? They're having to look at this book and measure and see why might their answers be different from each other. That is a way that you're analyzing partially correct or an incorrect and a correct example to see what do I need to attend to precisely about a measurement tool? What do I need to understand about that and make sense of it? Here's another one to talk it over. Which figure does not show three-thirds and explain why? So again, analyzing those incorrect and correct worked examples to really attend to the precision of creating equal parts. The second research recommendation for attending to precision is to explicitly teach mathematical language and notation and to really emphasize that meaning making. So again, returning to those multiple modalities, right? But to make sure that we're not thinking kids will just, you know, be sponges and learn the words and know really deeply what we mean about them, but to also have some explicit time when we're teaching what the words mean. So it might be that you're showing examples your modeling examples. Maybe you have a word wall. Maybe you have an image wall, right, that has words with it. Maybe you're defining it in multiple ways or with multiple examples and non-examples. And these are great examples of solid figures, right, where they're showing here's a cone that can roll, a cube that can slide, a cylinder that can stack. So we're really trying to help kids understand those words, roll, slide, and stack with sentences, with images, to explicitly learn those words. We might want to act them out to get into those multiple modes. This is another way you might explicitly teach some math language and notation. Here at the bottom, we're showing two different types of graphs, a tally table and a bar graph. And we could compare to see what are the pieces and the parts of each one. Why is it that there's some spaces where there's titles and labels and tally marks? And why is it that in a bar graph, we've got numbers that are labeled, right? What do I need to attend to to figure out how to create my own bar graph or my own table chart? All right, that was math practice six. We're going to move into math practice seven and eight, and those will be our last two. You ready? So math practice seven is look for and make use of structure. So this is one where you're really leaning into the idea of pursuing, finding, applying patterns, equivalences, and relationships. Mathematics is highly structured. Sometimes people will talk about it as being black and white. Well, math is actually also very gray or we wouldn't be able to critique each other, right? Communicate, create discussion and arguments. There couldn't be multiple correct answers or multiple representations or multiple strategies that are possible. If it was just black and white. So there's definitely gray, but within that gray, there's really cool structure. There's patterns, there's things that are equal to each other. There's relationships that we can lean on as we move along vertically in mathematics to build on what we already know. There are two recommendations for this math practice. We want to emphasize patterns and equivalencies and connections and not just isolated ideas or rules that expire. We want to make sure we engage even our youngest learners in this algebraic functional thinking. Okay, so here are the patterns. Here's a whole bunch of examples, right? Making sure that you're saying like, well, how can I explain in my math journal how to use a model to represent something? Which model is being represented by this equation or by this representation by this table? If I'm counting, if I'm skip counting, can I find any patterns that could help me? Can I find any patterns in how I'm creating equal parts that can help me understand the relationship between denominators and between numerators? So all sorts of spaces where we're asking kids to look at representations, the numbers, the the tables, the information that we're gathering together and saying, what patterns do you notice? There's patterns in number words, right? That can help us to understand how we say number words as numbers get larger and as they get smaller. We have to ask the questions. We have to engage them in thinking about that. The second recommendation is around engaging the youngest learners in this algebraic and functional thinking. So we don't want to wait. We want to engage three, four, five, six, seven-year-olds in thinking about algebraic thoughts, function thoughts, right? What are some of the patterns when we're counting? How can I predict what will happen next in a numerical pattern? How can I take something like start with four, double it, add two, add one, add six? How can I represent that different ways? How can I, instead of putting a number, use a picture or a letter to represent a quantity? And then what happens if that quantity varies? That's a variable. And we can do that kind of thinking with our youngest learners. We can really ask them to think about, well, what might happen if in the context of a game where I have a score, right? And someone has a score that's 10 points more than another person. How can I try to translate these relationships into some kind of a representation or mathematical model to make sense of it. That is algebraic thinking, right? Taking two graphs that are picture graphs, like they are down here, right? And saying, how do I compare those two picture graphs? That's some functional thinking, right? How are two graphs the same? But I'm using a picture graph. So I'm doing it in a really developmentally appropriate, accessible way for our youngest learners. And here comes our eighth and final practice. We're going to look for and express regularity and repeated reasoning. So this is that space. Sometimes people will ask, well, where is fluency part of this? Well, fluency falls in right here, right? So not only do we want kids to have accurate, appropriate, flexible strategies, but we want it to be efficient. And we want that to happen because they've seen some sort of repeated reasoning. They've got some regularity that they can express to make it more efficient when they're solving problems. especially with basic facts, right? And then larger and larger facts. The other piece of this practice is that we want them to be able to express generalities. We want them to actually reason to create an algorithm, right? To be able to say, oh, this is the procedure. This is how I typically can add numbers to make sure that I'm really being efficient. I do the same thing many, many times. That's an algorithm, right? Oh, I can create an equation from this. That is repeated reasoning, right? To say, oh, this is how I could represent this idea with variables or with numbers, with operations. So let's look at the recommendations. There's two. We have to make sure that we're asking questions that lead to generalizations and mathematical inferences. We want to point out those places where there is repeated reasoning, right? Regularity, something that we can pull out and say, oh, here's the pattern. And we also want to explicitly teach some strategies that are generalizable, that are efficient, that you could use anywhere, which wraps us all the way back to that first math practice, doesn't it? All right. So here are some examples of questions that you could pose to help kids get to a generalization. How does shading and marking off help you find the product of two fractions? What have you observed about a rule for multiplying fractions? How would you express that? How does making a bar graph make it easy to compare information? When does a zero have to be placed in the quotient? What does that zero in the quotient indicate? How do you order fractions mentally without coloring fraction bars? I love this one. Can the product of two fractions less than one be greater or less than either of the original ones? See how those are big open questions and they're asking kids to say, oh, I see some relationships that can help me, not just in this space, but then transfer it to a new novel problem, right? I can come, I can generate some rules. I can generalize some inferences and say, this is typically what happens. This is the structure I've found. This is the regularity and reasoning. And then those generalizable strategies, really thinking through what are the strategies that you're going to lean on that you're going to teach learners, right? They might be those polio strategies. They might be strategies that have been identified by Jennifer Bay Williams and John San Giovanni and the research that they've done. So you can see, like, this is a list of all the addition strategies that a second grader might need to learn, right? Being able to count on, being able to make 10, being able to do doubles. thinking about the place value. Those are all things that I can use no matter how large numbers are or fractions or decimals. Similarly with subtraction, I might think about, okay, I want to make sure that kids can count up and back, that they can make 10 and that they can think addition. Those are the strategies that I want to explicitly teach so kids can name them and use them again in a different scenario. So those are the two recommendations for the eighth math practice. Y'all, it's been a whirlwind. We've really done it, right? Okay, so now the beauty of this is it has been recorded. So if you're thinking, well, that was a good first look at those math practices and really understanding what they are. And now I really want to think about the recommendations again and the examples. You are welcome to listen to the recording again, right? And watch it and really look for places where you are already doing this work. You could just make a small shift to move forward. So thinking about how do we grow those proficient mathematicians, right? Are there ways that you're already developing that expertise in those skills? And are there ways that by developing those, it's because you're emphasizing and explicitly teaching them? What are the strategies that you already use? See if you can name them and identify them and then use the research to help you hone in on instructional strategies that will be even more powerful. Thank you for hanging out. I hope that you feel like, yeah, I've got some ideas about how I can intentionally develop my learner's mathematical minds through these practices.

SPEAKER_02 59:58

Thank you so much, Dr. Thunder. I think that was an amazing presentation. We had so many amazing comments. People were very excited and it was an incredibly valuable information because we could see what we're doing that works, like you said, right? And what other tips and tools and strategies that can help us teach math in a more strategic and effective and efficient way. So that was amazing. We are going to look at some questions and also we're going to have a poll up with two questions. If you'd like to participate, that would be great. One question is, how do I engage my English learners in math talk?

SPEAKER_00 1:00:40

It's a great question, right? Because I think often we kind of step aside and think, well, if they are not speaking in English, I can't engage them in math talk. But we can, because remember those multiple modalities, right? So we can ask learners to be speaking and listening and reading and writing in their heritage language. That's profoundly important. Then we can ask them, and I'll often do this, I'll partner learners with learners who speak the same heritage language and then pose for them after they've in that space to try it in English with another partner. That space of viewing and visually representing is another space, right? Another way that we can engage those English learners by saying like, is there a way that you could draw this? What parts of it could you label, right? And then showing that to a peer and talking through the parts that you can, but not feeling like you have to have a full long conversation in English, right?

SPEAKER_02 1:01:42

Yeah, that's perfect. You know, math is a very universal language, so you can use it anywhere and you don't have to worry about translation too much. So let's look at another question. Why are novel problems important for problem solving? And when are problems that are similar to each other helpful?

SPEAKER_00 1:02:06

That's great. So novel problems are ones where kids look at it and they're not sure immediately how to solve it. it. So that's really important. That's, you know, one, you're not always going to be with your learners. So you're not going to be by their side. You can't present every single problem to them. So you want to empower them to be in a situation where they don't know how to solve the problem and they can lean on those metacognitive strategies to say, what could I try? What do I understand about this? How could I tackle this? That helps them to have a space to practice that transfer and that application. So novel problems are really important. Problems Problems that look just like each other, sometimes called exercises, those are great for that deliberate practice to say, hey, we're deliberately practicing our estimation. We're deliberately practicing using compensation as our strategy so that we can make that a fluent strategy that's in my toolbox. So that then when I hit a novel problem, when I don't know how to solve, I can say, hmm, would compensation work here? I know how to do that really well. And you have some tools to help you figure that out. So a balance of both types of problems is really important.

SPEAKER_02 1:03:13

That's great. Yeah. I think, I think this is going to be one of those webinars that people are going to come back to because it's just, there's so many things also to think about and, you know, digest and see how we can apply them when we're teaching, you know, kids. Let's go to, and people are still, it looks like people will stop filling in the poll. I'm going to close it off now and let's keep going. So I, again, I wanted to thank you again. I'm so, you know, I feel like I've learned so much. I'm not a math teacher, but it's making me want to teach because it's so very interesting. And I wanted to also go and also thank Sadlier School for sponsoring this webinar. So Sadlier Progress in Mathematics K-5 is more than just a math problem. It's a pathway to long-term success in school and beyond. We invite you to learn more by visiting SadlierSchool.com or you can scan the QR code on your screen. And you can learn more about PIM and it's a brand new program. So it's actually very, very exciting. And we also wanted to share with you that we're going to have two more Sadlier EdWeb webinars, one with Dr. Wiley Blevins on prevention and intervention in reading when, why, and how, October 6th. And beyond definition, Learning Words Inside Out with Dr. Doug Fisher on November 18th. So we cannot wait for you to join us then for those webinars. And you don't want to miss these sessions. So again, we were very excited to have you here join us with Dr. Thunder. And we're glad you joined us today. And we hope to see you at another one of our ed webinar very soon. Have a wonderful day.

SPEAKER_01 1:05:10

We hope you enjoyed this EdWeb podcast. If you'd like to receive a CE certificate, you must watch the video recording. Recordings and quizzes can be found in the EdWebinar archives. Please visit home.edweb.net podcast for more information.