Podcasts about cognitively guided instruction

Assessment in the Early Years Guest: Shelly Scheafer ROUNDING UP: SEASON 3 | EPISODE 13 Mike (00:09.127) Welcome to the podcast Shelley. Thank you so much for joining us today. Shelly (00:12.956) Thank you, Mike, for having me. Mike (00:16.078) So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond? Shelly (00:30.3) There's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students, we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words, our youngest learners are just starting to connect symbolic representations to mathematical ideas, let alone, you know, put letters together to make words. So When we think of these assessments, we need to take into consideration that primary students are in the early stages of development with respect to their language, their reading, and their writing skills. And this in itself makes it challenging for them to fully articulate, write, sketch any of their mathematical thinking. So we often find that with young children in reviews, you know, individual interviews can be really helpful. But even then, there's some drawbacks. Some children find it challenging, you know, to be put on the spot, to show in the moment, you know, on demand, you know, what they know. Others, you know, just aren't fully engaged or interested because you've called them over from something that they're busy doing. Or maybe, you know, they're not yet comfortable with the setting or even the person doing the interview. So when we work with young children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving. And this is why Shelly (02:47.42) they often or some may respond differently to the same proper question, especially if the setting or the context is changed. We may find that a kindergarten student who counts to 29 on Monday may count to 69 or even 100 later in the week, kind of depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent. informative and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? You know, what do we do? You know, we're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines. And it's these authentic situations that we can look at as assessments to help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends. Mike (04:11.832) So I wanna go back to something you said and even in particular the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words do. Talk a little bit about what you were trying to convey with that, Shelley. Shelly (04:27.548) So young children are doers. When they work on a math task, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting, if they group their objects, how they line them up, do they tag them, do they move them as they count them. They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding, finger counting, and even how they cluster or match objects. So I'm going to give you an example. So let's say that I'm watching some early first graders, and they're solving the expression 6 plus 7. And the first student picks up a number rack or a rec and rec. And if you're not familiar with a number rack, it's a tool with two rows of beads. And on the first row, there are five red beads and five white beads. And on the second row, there's five red beads and five white beads. And the student solving six plus seven begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the seven. They push over five red beads and two white beads. And they haven't said a word to me. I'm just watching their actions. And I'm already able to tell, hmm, that student could subitize a group of five, because I saw him push over all five beads in one push. And that they know that six is composed of five and one, and seven is composed of five and two. And they haven't said a word. I'm just watching what they're doing. And then I might watch the student, and they see it. I see him pause, know, nothing's being said, but I start to notice this slight little head nodding. Shelly (06:26.748) And then they say 13 and they give me the answer and they're really pleased. I didn't get a lot of language from them, but boy, did I get a lot from watching how they solve that problem. And I want to contrast that observation with a student who might be solving the same expression six plus seven and they might go six and then they start popping up one finger at a time while counting seven, eight. 9, 10, 11, 12, 13. And when they get seven fingers held up, they say 13 again. They've approached that problem quite differently. But again, I get that information that they understood the equation. They were able to count on starting with six. And they kept track of their count with their fingers. And they knew to stop when seven fingers were raised. And I might even have a different student that solves the problem by thinking, hmm, and they talk to themselves or they know I'm watching and they might start talking to me. And they say, well, 6 plus 6 is 12 and 7 is 1 more than 6. So the answer is 16 or 13. And if this were being done on a paper pencil as an assessment item or they were answering on some kind of a device, all I would know about my students is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer. What skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense? Mike (08:20.84) Absolutely. I think the, the way that you described this really attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening. It makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying. and sometimes they're written communication, and there was a lot that I could have also taken in if I was attending to those things in a little bit more depth. It also strikes me that this might feel a little bit overwhelming for an educator. How do you think about what an educator, let me back that up. How can an educator know what they're looking for? Shelly (09:17.5) to start, Mike, by honoring your feelings, because I do think it can feel overwhelming at first. But as teachers begin to make informal observations, really listening to you and watching students' actions as part of just their daily practice, something that they're doing, you know, just on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect them to do, what they might say next if they see a certain actions. And after several years, let's say teaching kindergarten, if you've been a kindergarten teacher for four, five, six, 20, you know, plus years, you start to notice these patterns of behavior, things that five and six year olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know, you know, what you're looking at. But before you say anything, I know that isn't especially helpful for teachers new to the profession or new to a grade level. And fortunately, we have several researchers that have been, let's say, kid watching for 40, I don't know, 50 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. listeners might, you know, know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So. When I was sharing the thinking and actions of three students solving six plus seven, listeners familiar with cognitively guided instruction, CGI, they might have recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in my first student, they didn't say anything but gave me an answer. Shelly (11:40.068) was using direct modeling. We saw them push over five and one beads for six and then five and two beads for seven and then kind of pause at their model. And I could tell, you know, with their head nodding that they were counting quietly in their head, counting all the beads to get the answer. And, you know, that's kind of one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student. coming next time, I might work on the second strategy that I conveyed with my second student where they were able to count on. They started with that six and then they counted seven more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding. was shown with my third student when they were able to use a derived fact strategy. The student said, well, I know that 6 plus 6 is 12. I knew my double fact. And then I used that relationship of knowing that 7 is 1 more than 6. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next. And I can also think about what I might do. And so it's this knowledge of development and progressions and how children learn number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students. And they can use those, you know, to guide their instruction for that student or, you know, look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with. Mike (13:39.17) You know, as a K-1 teacher, I remember that I spent a lot of my time tracking students with things like checklists. You know, so I'd note if students quote unquote had or didn't have a skill. And I think as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning, Shelly (14:09.604) well, I think it's important to know that if, you know, it takes us belief. We have to really think about assessment and children's learning is something that is ongoing and evolving. And if we do, it just kind of becomes part of what we can do every day. We can look for opportunities to observe students skills in authentic settings. Many in the moment. types of assessment opportunities happen when we pose a question to the class and then we kind of scan looking for a response. Maybe it's something that we're having them write down on their whiteboard or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs up or a thumbs down, know, kind of to check in on their understanding. We might not be checking on every student, but we're capturing the one, you know, a few. And we can take note because we're doing this on a daily basis of who we want to check in with. What do we want to see? We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say, for example, that our lesson today includes a dot talk or a number talk, something that we're going to write down. We're going to record student thinking. And so during the lesson, the teacher is going to be busy facilitating the discussion, recording the students thinking, you know, and making all of those notes. But if we write the child's name, kind of honor their thinking and give it that caption on that public record, at the end of the lesson, you know, we can capture a picture, just, you use our phone, use an iPad, quickly take a picture of that student's thinking, and then we can record that. you know, where we're keeping track of our students. So we have, OK, another moment in time. And it's this collection of evidence that we keep kind of growing. We can also, you by capturing these public records, note whose voice and thinking were elevating in the classroom. So it kind of gives us how are they thinking and who are we listening to and making sure that we're kind of spreading that out and hearing everyone. Shelly (16:31.728) I think, Mikey, you checklists that you used. Yeah, and even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking. So one of the things we did in Bridges 3rd edition is we designed additional tools for gathering and recording information during workplaces. Mike (16:35.501) I did. Shelly (16:56.208) That's a routine where students are playing games and or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this is something you're going to do with every student. You know, or even, you know, in one day because these are spanned out over a period of four to six weeks where they can go to these games. And we might even see the students go to these activities multiple times. And so let's say that kindergarten students are playing something like the game Beat You to 10, where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that, I might just want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And so those might, you know, of objects, you know, based on the number, they spin a four, can they count out four? And those might be kinds of skills that you might have had typically on a checklist, right, Mike, for kindergarten? But I could use this activity to kind of gap. gather that note and make any comments. So just for those kids I'm looking at or maybe first graders are playing a game like sort the sum where they're drawing two different dominoes and they're supposed to find how many they have in all. And so with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from the dot? And if one set of the dots on one side and then counting on the other. Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those. So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist-like. But I can target that time to do it on students I want that information by thinking ahead of time. Shelly (19:18.684) What can I get by watching, observing these students at these games? trust, I mean, as you know, young children love it. Older children love it. When the teacher goes over and wants to watch them play, or even better, wants to engage in the game play with them, but I can still use that as an assessment. Mike (19:39.32) think that's really helpful, Shelly, for a couple reasons. First, I think it helps me rethink, like you said, one, getting really a lot clearer on like, love the, I'm gonna back that up. I think one of the things that you said was really powerful is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're gonna have students engage with, be it a game or a, Shelly (19:39.356) and Mike (20:07.96) project or some kind of activity and really thinking like, what can I get from this as a person who's trying to make sense of students thinking? And I think my checklist suddenly feels really different when I've got a clear vision of like, what can I get from this task or this game that students are playing and looking for evidence of that versus feeling like I was pulling kids over one-on-one, which I think I would still do because there's some depth that I might want to capture. But it it changes the way that I think about what I might do and also what I might get out of a task So that that really resonates for me Shelly (20:47.066) Yeah, and I think absolutely, you know, I didn't want to make individual interviews or anything sound bad because we can't do them. just, you there's the downfall of, you know, kids comfort level with that and ask them to do something on demand. But we do want more depth and it's that depth that, you know, we know who we want more depth on because of these informal types of observations that we're gathering on a daily basis in our class. You know, might, says something and we take note I want to touch bases with that thinking or I think I'm going to go observe that child during that workplace or maybe we're seeing some things happening during a game and instead of you know like stopping the game and really doing some in-depth interview with the student at that moment because you need more information I can might I might want to call them over and do that more privately at a different time so you're absolutely on there's a place there's a place for you know both Mike (21:42.466) The other thing that you made me think about is the extent to which, like one of the things that I remember thinking is like, I need to make sure if a student has got it or not got it. And I think what you're making me think can really come out of this experience of observing students in the wild, so to speak, when they're working on a task or with a partner is that I can gather a lot more evidence about the application of that idea. I can see the extent to which students are. doing something like counting on in the context of a game or a task. And maybe that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to kind of see, is this way of thinking something that students are applying in different contexts, or did it just happen at that one particular moment in time when I was with them? So that really helps me think about, I think, how those two... maybe different ways of assessing students, be it one-on-one or observing them and seeing what's happening, kind of support one another. Shelly (22:46.268) think you also made me think, you know, really hit on this idea that students, like I said, you their learning is evolving over time. And it might change with the context so that they, you know, they show us that they know something in one context with these numbers or this, you know, scenario. But they don't necessarily always see that it applies across the board. I mean, they don't, you know, make these. generalizations. That's something that we really have to work with students to develop. they're also, they're young children. Think about how quickly a three-year-old and a four-year-old change, you know, the same five to six, six to seven. I mean, they're evolving all the time. And so we want to get this information for them on a regular basis. You know, a unit of instruction may be a month or more long. And a lot can happen in that time. So we want to make sure that we continue to check in with them and help them to develop if needed or that we advance them. know, we nudge them along. We challenge them with maybe a question. Will that apply to every number? So a student discovers, when we add one to every number, it's like saying the next number. So six and one more is seven and eight and one more is nine. And you can challenge them, ooh, does that always work? What if the number was 22? What if it was 132? Would it always work? you know, when you're checking in with kids, you have those opportunities to keep them thinking, to help them grow. Mike (24:23.426) I want to pick up on something that we haven't necessarily said aloud, but I'd like to explore it. You know, looking at young students work from an asset-based perspective, particularly with younger students, I think I often had points in time where there felt like so much that I needed to teach them. And sometimes I felt myself focusing on what they couldn't do. Looking back, I wish I had thought about my work as noticing the assets, the strategies, the ways of thinking. that they were accumulating. Are there practices you think support an asset-based approach to assessment with young learners? Shelly (25:06.278) think probably the biggest thing we can do is broaden our thinking about assessment. The National Council of Teachers of Mathematics wrote in Catalyzing Change in Early Childhood and Elementary Mathematics that the primary purpose of assessment is to gather evidence of children's thinking, understanding, and reasoning to inform both instructional decisions and student and teaching learning. If we consider assessments and observations as tools to inform our instruction, we need to pay attention to the details of the child's thinking. And when we're paying attention to the details, what the child is bringing to the table, what they can do, that's where our focus goes. So the question becomes, what is the student understanding? What assets do they bring to the task? It's no longer, can they do it or can they not do it? And when we know, when we're focusing on just what that student can do, and we have some understanding of learning progressions, how students learn, then we can place what they're doing kind of on that trajectory, in that progression, and that becomes knowledge. And with that knowledge, then we can help students move along the progression to develop more developed understanding. For example, again, if I go back to my six plus seven and we notice that a student is direct modeling, they're counting out each of the sets and counting all, we can start to nudge them toward counting on. We might cover, you know, they were using that number rec, we might cover the first row and say, you just really showed me a good physical representation of six plus seven. And I kind of noticed that you were counting the beads to see how many were there. I'm wondering if I cover this first row. How many beads am I covering? Hmm. I wonder, could you start your counting at six? You know, we can kind of work with what they know. And I can do that because of Shelly (27:31.928) I haven't, I've focused on where they are in that progression and where that development is going. And I kind of have a goal of where I want students to go, you know, to further their thinking. Not that being in one place is right or wrong, or yes they can do it, no they can't. It's my understanding of what assets they bring that I can build on. Is that kind of what you're after? Mike (27:58.51) It is, and I think you also addressed something that again has gone unsaid, but I think you, you, you unpacked it there, which is assessment is really designed to inform my instruction. And I think the example you offered us a really lovely one where, we have a student who's direct modeling and they're making sense of number in a certain way and their strategy reflects that. And that helps us think about the kinds of nudges we can offer. that might shift that thinking or press them to make sense of numbers in a different way. That really the assessment is, it is a moment in time, but it also informs the way that you think about what you're gonna do next to keep nudging that student's thinking. Shelly (28:44.348) Exactly, and we have to know that if we have 20 students, they all might be, you know, have 20 little plans that they're on, 20 little pathways of their learning. And so we need to think about everybody, you know. So we're going to ask questions that help them do them, and we're going to honor their thinking. And then we can, you know, like so again, I'm going go back to like doing that dot talk with those students. And so I'm honoring all these different ways that students are finding the total number of dots. And then I'm asking them to look for what's the same within their thinking so that other students also can serve to nudge kids, to have them let them try and explore a different idea or, ooh, can we try that Mike's way and see if we can do that? hmm, what do you notice about? how Mike solved the problem and how Shelly solved the problem. Where is their thinking the same? Where is it different? And so we're honoring everybody's place of where they're at, but they're still learning from each other. Mike (29:51.224) You know, you have made multiple mentions to this idea of progressions or trajectories, and I'm wondering if there are resources that have informed your thinking about assessment at the early ages. Is there anything you would invite listeners to engage with if they wanted to continue learning, Shelley? Shelly (30:13.008) I think Mike, had that question earlier, so just pause this for a second. Okay. I know you will. I just know it's right here. Mike (30:16.558) That's okay, no worries. We'll cut every single bit of this out and it will sound supernatural. Yeah, yeah. Mike (30:39.854) Brent's over here multitasking. Shelly (30:41.85) OK. OK, I'm just making sure that I'm not going to blow it. I think you're spot on. I think I thought we skipped something. No, it's up here. Mike (30:53.132) Okay, just pick up whenever you're ready. Shelly (30:55.108) Yeah, I just have too many notes here. Shelly (31:10.084) OK, I've got it. Do you want to ask the question again? Mike (31:12.258) Go for it. Absolutely. Yep. Are there resources you'd invite listeners to engage with if they wanted to keep learning, Shelly? Shelly (31:28.368) You phrased that a little bit different. What I answered was, what are some of the resources that helped you build an understanding of children's developmental progressions? Do you have that question? Or I can jump on from what you asked, too. Mike (31:35.5) Okay. Yeah, let let. No, no, no, let me let me ask the question that way. Shelly (31:42.202) Okay. Mike (31:45.774) Okay, how did we have it in the thing? Can you say it one more time and I'll say it back in the question? Shelly (31:50.768) What are some of the resources that help to build an understanding of children's developmental progressions? Mike (31:56.504) Perfect. What are some of the resources that helped you build an understanding of children's developmental progression, Shelley? Shelly (32:05.34) Honestly, I can say that I learned a lot from the students I taught in my classroom. My roots run deep in early childhood. And I can also proudly say that I have a career-long relationship with the Math Learning Center and Bridges Curriculum, which has always been developmentally appropriate curriculum for young learners. And with that said, I think I stand on the back of giants. practitioner researchers for early childhood who have spent decades observing children and recording their thinking. I briefly mentioned Cognitively Guided Instruction, which features the research of Thomas Carpenter and his team. And their book, Children's Mathematics, is a great guide for K-5 teachers. I love it because I mean, the recent edition has QR codes where you can watch teachers and students in action. You can see some interviews. You can see some classroom lessons. And they also wrote young children's mathematics on cognitive-guided instruction in early childhood education. So I mean, they're just a great resource. Another teacher researcher. is Kathy Richardson, and some listeners may know her from her books, the developing number concept series or number talks in the primary classroom. And she also wrote a book called How Children Learn Number Concepts, Guide to the Critical Learning Phases, which targets pre-kindergarten through grade four. And I love that Kathy writes. in her acknowledgments that this work is the culmination of more than 40 years working with children and teachers observing, wondering, discussing, reading, and thinking. Shelly (34:11.692) It is. So spot on to the observations and the things that I noticed in my own teaching, but it's also still one of the most referenced resources that I use. And if podcasts had a video, I would be able to hold up and show you my dog eared book with sticky notes coming out the all the sides because it is just something that. just resonates with me again. And then I think also maybe less familiar. Mike (36:38.958) I think you mentioned giants and those are some gigantic folks in the world of mathematics education. The other piece that I think really resonates for me is I had a really similar experience with both CGI and Kathy Richardson in that a lot of what they're describing are the things that I was seeing in classrooms. What it really helped me do is understand how to place that behavior and what the meaning of it was in terms of students understanding of mathematics. And it also helped me think about that as an asset that then I could build on. Shelley, I think this is probably a great place to stop, but I wanna thank you so much for joining us. It has really been a pleasure talking with you. Mike (37:28.95) Say thank you again, but definitively. Mike (37:35.48) Brent, how do you feel about that? Mike (37:41.966) do you want to jump in? Yeah, feel free. Mike (38:04.194) is Is there a question I could ask that would set you up? Mike (38:14.574) can work that in to a new ending. Mike (38:42.872) Do you, there something that you want to add though, Shelley? Cause we can, we can edit it, edit content in, and we can sequence content in too. So if there's something that mattered to you, we can absolutely add it. Mike (39:14.542) Let's do a question like that then. Mike (39:35.02) What if we see... Mike (39:40.578) Why don't you, why don't, Yeah, why don't you say it? Go ahead and say it the way that you you it was going to flow out and then we'll we can edit this in definitely. Mike (39:59.191) Okay. Okay, go for it. Yeah, yeah. Mike (40:06.676) can I tell you this is one of the smoothest podcast recordings we have had? There's nothing to be sorry about. Mike (40:18.562) There, okay, I was, can you ask the question again, Mike? So that it's clean. Mike (40:29.752) Are there resources you would invite our listeners to engage with if they want to continue learning? Mike (42:26.328) I think that's a great place to stop. Shelly Schaefer, thank you so much for joining us. Mike (42:38.602) That was perfect. Yeah, fantastic. I'm gonna cut roll. No, there's nothing to be sorry about © 2025 The Math Learning Center | www.mathlearningcenter.org

I first learned about Problem Types with Cognitively Guided Instruction in the book Children's Mathematics. The group of researchers studied children and found that when given open ended questions children move through a progression of understanding… when given no explicit instruction.

Rounding Up  Season 1 | Episode 7 – Cognitively Guided Instruction: Turning Big Ideas into Practice  Guest: Dr. Kendra Lomax  Mike Wallus: Have you ever had an experience during your teaching career that fundamentally changed how you thought about your students and the role that you play as an educator? For me, that shift occurred during a sweltering week in July of 2007, when I attended a course on cognitively guided instruction. Cognitively guided instruction, or CGI, is a body of research that has had a massive impact on elementary mathematics over the past 20 years. Today on the podcast, we're talking with Kendra Lomax, from the University of Washington, about CGI and the promise it holds for elementary educators and students. Well, Kendra, welcome to the podcast. It's so great to have you on.  Kendra Lomax: Well, thanks for having me.  Mike: Absolutely. I'm wondering if we can start today with a little bit of background; part history lesson, part primer to help listeners understand what CGI is. So, can you just offer a brief summary of what CGI is and the questions that it's attempted to shed some light on?  Kendra: Sure, I'll give it my best try. So, CGI is short for cognitively guided instruction, and it's a body of research that began some 30 years ago with Tom Carpenter and Elizabeth Fennema. And there's lots of other scholars that since then have kind of built upon that body of research. They really tried to think about and understand how children develop mathematical ideas over time. So, they interviewed and studied and watched really carefully what young children did as they solve whole-number problems. So, you may have heard about the book ‘Children's Mathematics,' and that's where you can read a lot about cognitively guided instruction and [it] summarizes some of that research. And they really started with whole-number computation and then have kind of expanded into areas like fractions and decimals, learning about how kids develop ideas about algebraic thinking, as well as early ideas around counting and quantity.  Mike: Uh-hm.  Kendra: So, there's a couple of books that are kind of in the CGI family. ‘Young Children's Mathematics' includes those original authors, as well as Nick Johnson and Megan Franke, Angela Turrou, and Anita Wager. That fractions and decimals work was really led by Susan Empson and Linda Levi. And then, like I mentioned, ‘Thinking Mathematically' is the text by the original authors that kind of talks about algebra. So, in all of those texts that summarize this research, basically, we're trying to understand how do children develop ideas over time? And Tom and Liz really set an example for all of us to follow in how they thought about sharing this research. They had a deep respect for the wisdom of teachers and the work that they do with young children. So, you won't find any sort of prescription in the CGI research about how to teach, exactly, or a curriculum. Because their approach was to share with teachers the research that they had done when they interviewed and listened to all of these many children solving problems, and then learn from the teachers themselves. What is it that makes sense to do in response to what we now know about how children develop mathematical ideas?  Mike: I mean, it's kind of a foundational shift in some ways, right? It reframes how to even think about instruction, at least compared to the traditional paradigm, right? Kendra: Yeah, it's less a study of how best to teach children and really a study and a curiosity about how children bring the ideas that they already have to their work in the math classroom, and how they build on those ideas over time.  Mike: Definitely. It's funny, because when I think about my first exposure I think that was the big aha, is that my job was to listen rather than to impose or tell or perfectly describe how to do something. And it's just such a sea change when you rethink the work of education.  Kendra: Definitely. And it feels really joyful, too, right? You get to be a student of your students and learn about their own thinking and be really responsive to them in the moment, which certainly provides lots of challenges for teachers. But also, I think, just a sense of genuine relationship with children and curiosity and a little bit of joy.  Mike: Definitely. So, I'm wondering if we could dig into a little bit of the whole-number work, because I think there's a bit that we were talking about with CGI, which is really the way in which you approach students, right? And the way that you listen to students for cues on what they're thinking is. But the research did reveal some ways to construct a framework for some of the things you see when children are thinking.  Kendra: So, if you read the book ‘Children's Mathematics,' you might notice or recognize some of those ideas, because CGI is one of the research bases for the Common Core state math standards. So, when you're looking through your grade-level standards and you see that they're suggesting particular problem types, number sizes, or strategies that children might use, much of that is based on the work of cognitively guided instruction, as well as other bodies of research. So, it might sound familiar when you read through the book yourself. And what CGI helps reveal is that there's a somewhat predictable sequence: That young children develop strategies for whole-number operation for working with whole-number computational problems.  Mike: Yeah. Can you talk about that, Kendra?  Kendra: Yeah. So, young children are going to start out with what we call direct modeling, where they are going to directly model the context of the problem. So, if we give them a story problem, they'll act out or model or show or gesture, to show the action of the problem. So, if it describes eating something ( makes eating sounds ), you can imagine, right, the action that goes along with eating? And we're all very familiar with it. So, they're going to show maybe, the cookies, and then cross out the ones that get eaten …  Mike: Uh-hm.  Kendra: … right? So, they're really going to directly model the action or relationship described in the problem. And they're going to also represent all the quantities in the problem, which is different. What they learn over time is to count on or count back. So, some of the counting strategies where they learn, ‘Gosh, I don't want to make all the quantities in this problem.' It becomes too difficult, too cumbersome. And they learn that they could count on from one of the quantities or count back. So, in that cookie example, maybe there are seven cookies on a plate, and I have two of them for dessert, right? ( makes eating sounds) They go away. So, in direct modeling, they're going to show the seven cookies. They're going to remove those two cookies that get eaten, and then count how many are left. Where in counting on—so they have had lots of experiences of direct modeling—they can say, ‘Gosh, I don't really want to draw that seven. I'm going to imagine the seven … ‘  Mike: Uh-hm.  Kendra: ‘ … And I can maybe count backwards from there.'  Mike: So, like, 7, 6, 5.  Kendra: Yeah. Right. So, I don't have to make the seven. I can just imagine it. And I keep track of those two that I'm counting back.  Mike: That totally makes sense. And as a former kindergarten and first-grade teacher, it's an amazing thing to actually see that shift happen.  Kendra: Right? And it's really specialized knowledge that teachers develop to pay attention to that shift. It's easy for everybody else to kind of miss it. But for teachers, it's a really important shift to pay attention to.  Mike: I used to say to parents, when I would try to describe this, it's something that we almost aren't conscious of being able to do. But it's a gigantic step to go from imagining a quantity as a set of ones to imagining a quantity that is a number that you can count back from or count forward from. It's a gigantic leap. Even though to us, we've forgotten what big of a leap that was because it's been so long since we took it.  Kendra: Yeah. That's one thing I love about studying children's mathematics, is, like, you get to experience that wonderment all over again …  Mike: Uh-hm.  Kendra: … in the things that we kind of, as adults, take for granted in how we think about the world.  Mike: Yeah. I think you really clearly articulated the shift that kids make when they move from direct modeling, the action and the quantities, to that kind of shift in their thinking and also their efficiency of being able to count on or count back. Is there more to, kind of, the trajectory that kids are on from there?  Kendra: There is, yeah. So, after children have had lots of experiences to direct model, and then learn to become more efficient with that, and counting on or counting back, then they might start inventing. We call them invented algorithms, which is a fancy way to say that they think about the relationship between quantities and start putting them together and taking them apart in more efficient ways. So, they might use their understanding of groups of 10, right? So, in that example, with the cookies—seven cookies and eating two of them—I might know something about the relationship with fives …  Mike: Uh-hm.  Kendra: … Five and two make a seven. So, they start to develop some sense of how numbers go together, and how the operations really behave. So, in addition, I can kind of add them in any order that I want to, right? So, we see these called in the Common Core standards, Strategies Based on Place Value, Properties of Operation, and the relationship between addition, subtraction, or multiplication, division.  Mike: That's super helpful to actually connect that language in Common Core to what you might see, and how that translates into, kind of, what one might read about in some of the CGI research.  Kendra: Right. It'd be lovely if we all had the exact same ( laughs ) names, wouldn't it?  Mike: Definitely. One of the questions that I suspect people who might be new to this conversation are asking is, what are the conditions that I can put in place? Or what are the things that I might, as a teacher, be able to influence that would help kids move and make some of these shifts. Knowing that the answer isn't direct instruction. I could get a kid to mimic counting on, but if they're still really thinking about numbers in the sense of a direct modeler, they haven't really shifted, right? So, my wondering is, how would you describe some of the ways that teachers can help nudge children, or kind of set up situations that are there to help kids make the shift without telling, or …  Kendra: ( chuckles)  Mike: … like, giving away the game?  Kendra: Totally. Yeah. That's one takeaway that I'm always on the lookout for when people hear about CGI and this trajectory that's somewhat predictable.  Mike: Uh-hm.  Kendra: Let's just teach them the next strategy then, right?  Mike: Right.  Kendra: And what's important to remember is that these are called invented algorithms for a reason.  Mike: Uh-hm.  Kendra: Because children are actually inventing mathematics. It's amazing. Kindergartners are inventing mathematics. And so, our role is really to create the right opportunities for them to do that important work. And like you're saying, when they're ready for the next ideas that they're building on their existing knowledge, rather than us kind of coming in and trying to create that artificially.  Mike: Uh-hm.  Kendra: So, again, like, Liz and Tom really kind of taught us to be students of our students as well as students of teachers.  Mike: Uh-hm.  Kendra: So, what we've learned over time … some of the things that teachers have found really productive for supporting students to kind of move through this trajectory, to create increasingly efficient strategies, is really about thinking about carefully choosing the problems that we've put in front of students.  Mike: Uh-hm.  Kendra: So, paying attention to the context. Is it familiar to them? Is it reasonable for the real world? Are we helping kids see that mathematics is all around them.  Mike: Uh-hm.  Kendra: Paying attention to the quantities that we select. So, if we want them to start thinking about those relationships with five and 10, or as they get older with hundreds and thousands, that we're intentional about the quantities that we choose for those problems.  Mike: Right.  Kendra: And then, of course we know that students learn a lot from not just us, but their relationships and their discussions with their classmates. So, really orchestrating classroom discussions, thinking about choosing students to work together so that they can both learn from one another, and really just finding ways to help students connect their current thinking with the new ideas that we know are on the horizon for them.  Mike: I would love for you to say a little bit more about number choice. That is such a powerful strategy that I think is underutilized. So, I'm wondering if you could just talk about being strategic around the number choices that you offer to kids. Can you say more about that?  Kendra: Sure! It's going to depend on grade level, of course, right?  Mike: Uh-hm.  Kendra: Because they're going to be working with very different quantities early in elementary and then later on … One thing I would say, across all of the grade levels, is to not limit students whenever possible. So, sometimes we want to give problems that kids are really comfortable with, and we know they're going to be successful. But if I'm thinking of how they develop more efficient strategies, sometimes the growth comes in making it a little tricky. So, giving quantities that are just a little bit beyond where they're counting as young children, so they develop the need to learn that counting sequence. Or, as we're working with older students, if we know that particular multiplication facts are less familiar to students. Giving them that nudge by creating story context, where they can really make sense of the action of the relationship that's happening in it, but maybe choosing that times seven that we know has been tricky for kids, right?  Mike: Yep.  Kendra: So, I would just encourage people to not shy away from problems that we know pose some challenge to students. That's actually where a lot of the meat and the rigor happens. And, but then we also want to provide support inside of those, right? So, working with a partner.  Mike: Definitely. Kendra: Or making sure they have access to those counting charts. That's one thing I would say across grade levels.  Mike: Yeah. So, you made me think of something else. It's fascinating to have this conversation, Kendra, 'cause it reminds me of all the things that I had to learn over time. And I think one of the things that I'm wondering if you could talk a little bit more about is, the types of problems and how the problem that you choose for a given group of students might influence whether they're direct modeling or they're counting on or whether they're using invented algorithms. Because I think, for me, one of the things that it took a while to make sense, is that the progression isn't necessarily linear, right? Like, if I'm counting on in a certain context, that doesn't mean I'm counting on in all contexts or direct modeling or what have you. So, I'm curious if you could talk a little bit about problem types and now how those influence what things students sometimes show us.  Kendra: Yeah. I'm glad you brought that up. When we describe, kind of, that trajectory of strategies, it sounds really nice and tidy and organized and like it is predictable in some ways. But like you're saying, it also depends on the kind of problem and the number size that we're putting in front of children. So that trajectory kind of iterates again and again throughout elementary school. So, as we pose more complex problem types … so, for example, the cookies problem where I have seven cookies, I eat two of them and the result is what's at the end of the story, right? The cookies left over.  Mike: Uh-hm.  Kendra: If I now make that problem, I have some cookies on a plate. I ate two of them, and I have five left over. All the kindergarten teachers, actually all the elementary school teachers …  Mike: ( laughs ) Yes.  Kendra: … can automatically recognize that's going to be a more tricky problem, right?  Mike: Uh-hm.  Kendra: Where do I start!? Especially if I'm direct modeling, right? We know they start and follow the exact action of the story.  Mike: Absolutely ( chuckles ).  Kendra: So, as we pose more complex problem types, you're right. You're going to see that they might use less efficient strategies because they're really making sense. They're like, ‘Wait, what's the relationship that's happening in this story? Where do I begin? Where are the cookies at the beginning, middle, and end of this story?' So, we see that happen throughout elementary school. So, it's not that direct modeling is for kindergartners. And that invented algorithms are for fifth grade. It's that as new ideas get introduced, as we make problems more complex, maybe increasing the number size or now we're working with fractions and decimals …  Mike: Uh-hm. Kendra: We see this happen all over again. Kids begin with direct modeling to make sense of the situation. Then they build on that and get a little bit more efficient with some counting kinds of strategies. And then over time with lots of practice with that new problem type, those new numbers, um, they develop those invented algorithms again.  Mike: So, this makes me think of something else, Kendra. How would you describe the role of representation in this process? That could mean manipulatives that students choose to use. It could mean things that they choose to draw, visual models. How does representation play in the process?  Kendra: Yeah. So, oftentimes I hear people say, ‘This student used cubes. That was their strategy.' Or ‘This student used a drawing. That was their strategy.' And that's really not enough information to know the mathematical work that that child is doing. Did they use cubes as a way to count on?  Mike: Uh-hm.  Kendra: Are they keeping track of only one of the quantities but using cubes to do so? Are they doing a drawing that actually represents groups of 10? And they're using ideas about place value inside of it, which is different than if they're just drawing by ones, right? So, there's lots of detail inside of those representations that's important to pay attention to.  Mike: Yeah. I'm thinking about one of my former kindergartners. I remember that I had some work that she had done in the fall, and then I had another bit of work that she had done in the spring. And the fall was this ( chuckles ) very detailed drawing of, like, a hundred circles. And then in the spring, she was unitizing, right? She had a bunch of circles and then within [them] had labeled that each of those were 10. And it just struck me, like, ‘Wow, that is a really tangible vision of how she was drawing in both cases.' But her representation told a really different story about what she understood about math, about numbers, about the base 10 system.  Kendra: Right. And those might be very different starting points. As you, the teacher, you walk over and you see those two different kinds of drawings …  Mike: Uh-hm.  Kendra: … your conversation or your prompt for them … your next step for them might be pretty different.  Mike: Absolutely.  Kendra: Even though they're both drawing.  Mike: Yeah. Well, let me ask you this, 'cause I think I struggled with this a little bit when I first started really thinking about CGI. I had gone to a training and left incredibly inspired and was excited. And one of the things that I was trying to reconcile at that time is, like, I do have a curriculum resource that I'm using, and I wonder how many teachers sometimes struggle with that? I've learned these ideas about how children think, how to listen … what are some of the teacher moves I can make? And I'm also trying to integrate that with a tool that I'm using as a part of my school or my district. So, what are your thoughts about that?  Kendra: That makes a lot of sense. And I think that happens a lot of time in professional learning, where we learn a new set of ideas and then we're wrestling with how do they connect with the things I'm already doing? How do I use them in my own classroom? So, I really appreciate that challenge. I guess one way I like to think about it is that the trajectory that CGI helps us know about how children develop ideas over time is a little bit like a roadmap that I can use regardless of the curricular materials that I have in front of me. And it helps me understand what is on the horizon for that child. What's next for them and their learning? Depending on the kind of strategies that they're using and the kinds of problems that we're hoping to be giving them access to in that grade level, I can look at my curricular materials in front of me and use that roadmap to help me navigate it. So, we were talking about number selection. So, I might take that lens as I look at the curriculum in front of me and think about, ‘Are these the right numbers to be using? What will my students do with the problem …  Mike: Uh-hm.  Kendra: … that is suggested in my curricular materials?' To anticipate how my discussion is going to go and what kinds of strategies I might want to highlight in my discussion. So, I really like to think of it as the professional knowledge that teachers need in order to make sense of their curriculum materials and make informed decisions about how to use those really purposefully.  Mike: Yeah. The other thing that strikes me, that I'm connecting to what you said earlier, is that I could also look at the problem and think about, ‘Does the context actually connect with what I know about my children? Can I somehow shift the context in a way that makes it more accessible to them while still maintaining the structure, the problem, the mathematics, and such?'  Kendra: Right. Yeah. Are there small revisions I can make? Because, uh, I don't envy curriculum writers ( chuckles ) at all because there's no way you can write the exact right problem for every day, for every child across the country. So, as teachers, we have to make really smart decisions and make those really manageable. Because teachers are very busy people.  Mike: Sure.  Kendra: But those manageable, kind of, tweaks or revisions to make it really connected to our students lives.  Mike: Yeah. I think the other thing that's hitting me is that, when you've started to make sense of the progression that children go through, it's a little bit like putting on a pair of glasses that allow you to see things slightly differently and understand that skill of noticing. That's universal. It doesn't necessarily come and go with a curriculum. It's something that's important. Knowing your students is always going to be something that's important for teachers, regardless of the curriculum materials they've got.  Kendra: Yep. That's right.  Mike: So, here's my, I think my last question. And it's really, it's a resource one. So, if I'm a listener who's interested in learning more about CGI, if this is really my first go at understanding the ideas, what would you recommend for someone who's just getting started thinking about this and maybe is walking away thinking, ‘Gosh, I'd like to learn more.' Kendra: Sure. Well, I mentioned the whole laundry list of great texts that you can dig into more. So, ‘Children's Mathematics' being the one on whole-number operation across grade levels. I find that, like, preschool through first- or second-grade teachers have found ‘Young Children's Mathematics' incredibly impactful. It helps connect ideas about counting in quantity with these ideas about problem-solving and operation. And then kind of connects them and helps us think about how to support students to develop those really important early ideas.  Mike: Uh-hm.  Kendra: Anybody who I have talked to that has read ‘Extending Children's Mathematics: Fractions and Decimals' has found it incredibly impactful.  Mike: I will add myself to that list, Kendra. It blew my mind.  Kendra: Yeah, us, too! Everybody who read it was like, ‘Ohhh, I see now.' It points out a lot of really practical ways for us to pay attention. It offers a trajectory much like whole-number about how children develop ideas and also kind of suggests some problems that will help us support students as they're developing those ideas. So, [I] definitely recommend those. And then, ‘Thinking Mathematically' is another great text that helps us connect arithmetic and algebra, as we're thinking about how to make sure that students are set up for success as they start thinking more algebraically. And [it] digs into a little bit of—I talked about young children inventing mathematics—I think even further describes the ways that they invent important properties of operation that can be really interesting to read about.  Mike: That's fantastic. Kendra, thank you so much for joining us. It's really been a pleasure talking to you today.  Kendra: Thanks for having me.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2022 The Math Learning Center | www.mathlearningcenter.org

Dr. Nicki Newton and Ann Elise Record are back! They each have their own lists of the best ways for students to practice problem solving and they collaborate to create one essential top ten list. Dr. Nicki and Ann Elise are so fun and knowledgeable. They share amazing resources for math teachers.  

We've got another one of our favorite routines for you today! In this episode Pam and Kim demonstrate the Relational Thinking routine, a routine that helps students understand equivalence and become more comfortable with relational strategies. Thanks to the  Cognitively Guided Instruction group for introducing us to this routine!Resources:Find examples of Relational Thinking problems here: https://www.mathisfigureoutable.com/relational-thinkingFind our Developing Mathematical Reasoning info graphics here:  https://www.mathisfigureoutable.com/blog/development-4

In this week's episode, we are talking about CGI Math which stands for Cognitively Guided Instruction. Cognitively Guided Instruction helps us learn how to facilitate the development of kids' math understanding by listening first, then making instructional decisions based upon what you hear. Go to buildmathminds.com/107 to get links to any resources mentioned in this episode.

Welcome fellow Recovering Traditionalists to Episode 63.  Today I am Admitting When I’m Wrong.  Last week over on my vlog TheRecoveringTraditionalist.com, I talked about Teaching After The Coronavirus Shutdown.  One of the things I recommended in that video was not using your textbook when you start teaching next year.  Anytime I say things about not using a textbook I always get pushback.  Yes there is research to show that fidelity to textbooks breeds good test scores, but it also breeds kids who hate math and whole list of things that I could go on and on about.   However, I need to admit that I am wrong because it really isn’t the textbook that I don’t like and that is causing all those things….it’s how we use that textbook.  We’ve all seen it.  Maybe you’ve even been that teacher.  I know I was.  I think back to how I taught before I learned about Cognitively Guided Instruction and how kids develop their math understandings.  I approached and used my textbook way differently before that than how I used my textbook after The textbook didn’t change. My knowledge about how to build students’ understanding did and that impacted how I used the activities that were in my textbook. Get all the resources mentioned in this episode at buildmathminds.com/63

Welcome fellow Recovering Traditionalists to Episode 27.  Today we are looking at Using Kids’ Intuitive Mathematical Reasoning If you follow my blog, The Recovering Traditionalist, then you know that I’m doing a series of videos to kick off the school year all about how and why to teach math without a textbook. One of the biggest influences on my belief about teaching math without a textbook is the research of Cognitively Guided Instruction.  One of the very first episodes of this podcast was about the book Children’s Mathematics: Cognitively Guided Instruction.  For this episode I want to direct you to a research article by some of the same authors. Plus, I’ve got a special download for you to help you teach elementary math without a textbook.  Get all the info here: buildmathminds.com/27 

Welcome fellow Recovering Traditionalists to Episode 2 where we will explore the Three-Phase Lesson Format that is essential to having a student-centered math classroom.   In Episode 1, I shared about Cognitively Guided Instruction.  A big part of CGI is teaching through story problems.  However, when you do that it’s helpful to have a way to structure your lesson and the Three-Phase format is what I recommend.

Welcome fellow Recovering Traditionalists to Episode 1. Today we are looking at an insight I got from one of the books that forever changed my life: Children’s Mathematics: Cognitively Guided Instruction. This book was a tipping point for me. It was th   Get full show notes, transcript and more info here: https://buildmathminds.com/1

Eliz Fennema, emeritus professor from the University of Wisconsin, discusses her career in mathematics education, her research on gender-differences in mathematics, and her role on the Cognitively Guided Instruction project.Guest host: Susan EmpsonSee the comments for references mentioned during the interviewComplete list of past episodes

Renee Smith, Math Consultant and professional learning guru, is in the studio for this week’s episode of Practicing PBL. Renee[...] The post PBL Meets Math through Cognitively Guided Instruction appeared first on Remarkable Chatter.

Renee Smith, Math Consultant and professional learning guru, is in the studio for this week’s episode of Practicing PBL. Renee[...] The post PBL Meets Math through Cognitively Guided Instruction appeared first on Remarkable Chatter.

http://traffic.libsyn.com/remarkablechatter/btn_6.mp3 Renee’ talks with 4th grade teacher, Kris Styes, who shares the power of professional learning for teachers of math.  Kris shares how[...] The post Kris Styes – Cognitively Guided Instruction (By The Numbers #6) appeared first on Remarkable Chatter.

In part two of their conversation, Renee’ and Heidi Harris discuss the decision to train in Cognitively Guided Instruction (CGI). [...] The post Cognitively Guided Instruction – CGI (By The Numbers #4) appeared first on Remarkable Chatter.