Rounding Up

The Math Learning Center

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EDUCATION

Rounding Up

Welcome to “Rounding Up” with the Math Learning Center. These conversations focus on topics that are important to Bridges teachers, administrators and anyone interested in Bridges in Mathematics. Hosted by Mike Wallus, VP of Educator Support at MLC.

JAN 9

Season 3 | Episode 9 – Breaking the Cycle of Math Trauma - Guest: Dr. Kasi Allen

Dr. Kasi Allen, Breaking the Cycle of Math Trauma ROUNDING UP: SEASON 3 | EPISODE 9 If you are an educator, you’ve likely heard people say things like “I’m a math person.” While this may make you cringe, if you dig a bit deeper, many people can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you are a math person as well. Today we’re talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. BIOGRAPHY Kasi Allen serves as the vice president of learning and impact at The Ford Family Foundation. She holds a PhD degree in educational policy and a bachelor’s degree in mathematics and its history, both from Stanford University. RESOURCES “Jo Boaler Wants Everyone to Love Math” — Stanford Magazine R-RIGHTS Learning to Love Math by Judy Willis TRANSCRIPT Mike Wallus: If you're an educator, I'm almost certain you've heard people say things like, “I am not a math person.” While this may make you cringe, if you dig a bit deeper, many of those folks can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you're actually a math person. Today we're talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. Well, hello, Kasi. Welcome to the podcast. Kasi Allen: Hi, Mike. Thanks for having me. Great to be here. Mike: I wonder if we could start by talking about what drew you to the topic of math trauma in the first place? Kasi: Really good question. You know, I've been curious about this topic for almost as long as I can remember, especially about how people's different relationships with math seem to affect their lives and how that starts at a very early age. I think it was around fourth grade for me probably, that I became aware of how much I liked math and how much my best friend and my sister had an absolutely opposite relationship with it—even though we were attending the same school, same teachers, and so on. And I really wanted to understand why that was happening. And honestly, I think that's what made me want to become a high school math teacher. I was convinced I could do it in a way that maybe wouldn't hurt people as much. Or it might even make them like it and feel like they could do anything that they wanted to do. But it wasn't until many years later, as a professor of education, when I was teaching teachers how to teach math, that this topic really resurfaced for me [in] a whole new way among my family, among my friends. And if you're somebody who's taught math, you're the math emergency person. And so, I had collected over the years stories of people's not-so-awesome experiences with math. But it was when I was asked to teach an algebra for elementary teachers course, that was actually the students’ idea. And the idea of this course was that we'd help preservice elementary teachers get a better window into how the math they were teaching was planting the seeds for how people might access algebra later. On the very first day, the first year I taught this class, there were three sections. I passed out the syllabus; in all three sections, the same thing happened. Somebody either started crying in a way that needed consoling by another peer, or they got up and left, or both. And I was just pretty dismayed. I hadn't spoken a word. The syllabi were just sitting on the table. And it really made me want to go after this in a new way. I mean, something—it just made me feel like something different was happening here. This was not the math anxiety that everybody talked about when I was younger. This was definitely different, and it became my passion project: trying to figure how we disrupt that cycle. Mike: Well, I think that's a good segue because I've heard you say that the term “math anxiety” centers this as a problem that's within the person. And that in fact, this isn't about the person. Instead, it's about the experience, something that's happened to people that's causing this type of reaction. Do I have that right, Kasi? Kasi: One hundred percent. And I think this is really important. When I grew up and when I became a teacher, I think that was an era when there was a lot of focus on math anxiety, the prevalence of math anxiety. Sheila Tobias wrote the famous book Overcoming Math Anxiety. This was especially a problem among women. There were dozens of books. And there were a number of problems with that work at the time, and that most of the research people were citing was taking place outside of math education. The work was all really before the field of neuroscience was actually a thing. Lots of deficit thinking that something is wrong with the person who is suffering this anxiety. And most of these books were very self-helpy. And so, not only is there something wrong with you, but you need to fix it yourself. So, it really centers all these negative emotions around math on the person that's experiencing the pain, that something's wrong with them. Whereas math trauma really shifts the focus to say, “No, no, no. This reaction, this emotional reaction, nobody's born that way.” Right? This came from a place, from an experience. And so, math trauma is saying, “No, there's been some series of events, maybe a set of circumstances, that this individual began to see as harmful or threatening, and that it's having long-lasting adverse effects. And that those long-lasting effects, this kind of triggering that starts to happen, is really beginning to affect that person's functioning, their sense of well-being when they're in the presence, in this case, of mathematics.” And I think the thing about trauma is just that. And I have to say in the early days of my doing this research, I was honestly a little bit hesitant to use that word because I didn't want to devalue some of the horrific experiences that people have experienced in times of war, witnessing the murder of a parent or something. But it's about the brain. It's how the brain is responding to the situation. And what I think we know now, even more than when I started this work, is that there is simply trauma [in] everyday life. There are things that we experience that cause our brains to be triggered. And math is unfortunately this subject in school that we require nearly every year of a young person's life. And there are things about the way it's been taught over time that can be humiliating, ridiculing; that can cause people to have just some really negative experiences that then they carry with them into the next year. And so that's really the shift. The shift is instead of labeling somebody as math anxious—“Oh, you poor thing, you better fix yourself”—it's like, “No, we have some prevalence of math trauma, and we've got to figure out how people's experiences with math are causing this kind of a reaction in their bodies and brains.” Mike: I want to take this a little bit further before we start to talk about causes and solutions. This idea that you mentioned of feeling under threat, it made me think that when we're talking about trauma, we are talking about a physiological response. Something is happening within the brain that's being manifested in the body. And I wonder if you could talk just a little bit about what happens to people experiencing trauma? What does that feel like in their body? Kasi: So, this is really important and our brains have evolved over time. We have this incredible processing capacity, and it's coupled with a very powerful filter called the amygdala. And the amygdala [has been] there from eons ago to protect us. It's the filter that says, “Hey, do not provide access to that powerful processor unless I'm safe, unless my needs are met. Otherwise, I gotta focus on being well over here.” So, we're not going to give access to that higher-order thinking unless we're safe. And this is really important because modern imaging has given us really new insights into how we learn and how our body is reacting when our brain gets fired in this way. And so, when somebody is experiencing math trauma, you know it. They sweat. Their face turns red. They cry. Their body and brain are telling them, “Get out. Get away from this thing. It will hurt you.” And I just feel like that is so important for us to remember because the amygdala also becomes increasingly sensitive to repeat negativity. So, it's one thing that you have a bad day in math, or you maybe have a teacher that makes you feel not great about yourself. But day after day, week after week, year after year, that messaging can start to make the amygdala hypersensitive to these sorts of situations. Is that what you were getting at with your question? Mike: It is. And I think you really hit on something. There's this idea of repeat negativity causing increased sensitivity, I think has real ramifications for classroom culture or the importance of the way that I show up as an educator. It's making me think a lot about culture and norms related to math in schools. I'm starting to wonder about the type of traumatizing traditions that we've had in math education that might contribute to this type of experience. What does that make you think? Kasi: Oh, for sure. Unfortunately, I think the list is a little long of the things that we may have been doing completely inadvertently. Everybody wants their students to have a great experience, and I actually think our practices have evolved. But culturally, I think there are some things about math that contribute to these “traumatizing traditions,” is what I've called them. Before we go there, I do want to say just one other thing about this trauma piece, and that is that we've learned about some things about trauma in childhood. And a lot of the trauma in childhood is about not a single life-altering event. But childhood trauma is often abou

27 min
12/19/2024

Season 3 | Episode 8 – Helping Our Students Build a Meaningful Understanding of Geometry - Guest: Dr. Rebecca Ambrose

Dr. Rebecca Ambrose, Helping Our Students Build a Meaningful Understanding of Geometry ROUNDING UP: SEASON 3 | EPISODE 8 As a field, mathematics education has come a long way over the past few years in describing the ways students come to understand number, quantity, place value, and even fractions. But when it comes to geometry, particularly concepts involving shape, it’s often less clear how student thinking develops. Today, we’re talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry. BIOGRAPHIES Rebecca Ambrose researches how children solve mathematics problems and works with teachers to apply what she has learned about the informal strategies children employ to differentiate and improve instruction in math. She is currently a professor at the University of California, Davis in the School of Education. RESOURCES Geometry Resources Curated by Dr. Ambrose Seeing What Others Cannot See Opening the Mind's Eye TRANSCRIPT Mike Wallus: As a field, mathematics education has come a long way over the past few years in describing the ways that students come to understand number, place value, and even fractions. But when it comes to geometry, especially concepts involving shape, it's often less clear how student thinking develops. Today, we're talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry. Well, welcome to the podcast, Rebecca. Thank you so much for joining us today. Rebecca Ambrose: It's nice to be here. I appreciate the invitation. Mike: So, I'd like to start by asking: What led you to focus your work on the ways that students build a meaningful understanding of geometry, particularly shape? Rebecca: So, I taught middle school math for 10 years. And the first seven years were in coed classrooms. And I was always struck by especially the girls who were actually very successful in math, but they would tell me, “I like you, Ms. Ambrose, but I don't like math. I'm not going to continue to pursue it.” And I found that troubling, and I also found it troubling that they were not as involved in class discussion. And I went for three years and taught at an all-girls school so I could see what difference it made. And we did have more student voice in those classrooms, but I still had some very successful students who told me the same thing. So, I was really concerned that we were doing something wrong and that led me to graduate school with a focus on gender issues in math education. And I had the blessing of studying with Elizabeth Fennema, who was really the pioneer in studying gender issues in math education. And as I started studying with her, I learned that the one area that females tended to underperform males on aptitude tests—not achievement tests, but aptitude tests—was in the area of spatial reasoning. And you'll remember those are the tests, or items that you may have had where you have one view of a shape and then you have a choice of four other views, and you have to choose the one that is the same shape from a different view. And those particular tasks we see consistent gender differences on. I became convinced it was because we didn't give kids enough opportunity to engage in that kind of activity at school. You either had some strengths there or not, and because of the play activity of boys, that may be why some of them are more successful at that than others. And then the other thing that informed that was when I was teaching middle school, and I did do a few spatial activities, kids would emerge with talents that I was unaware of. So, I remember in particular this [student,] Stacy, who was an eighth-grader who was kind of a good worker and was able to learn along with the rest of the class, but she didn't stand out as particularly interested or gifted in mathematics. And yet, when we started doing these spatial tasks, and I pulled out my spatial puzzles, she was all over it. And she was doing things much more quickly than I could. And I said, “Stacy, wow.” She said, “Oh, I love this stuff, and I do it at home.” And she wasn't the kind of kid to ever draw attention to herself, but when I saw, “Oh, this is a side of Stacy that I didn't know about, and it is very pertinent to mathematics. And she needs to know what doorways could be open to her that would employ these skills that she has and also to help her shine in front of her classmates.” So, that made me really curious about what we could do to provide kids with more opportunities like that little piece that I gave her and her classmates back in the day. So, that's what led me to look at geometry thinking. And the more that I have had my opportunities to dabble with teachers and kids, people have a real appetite for it. There are always a couple of people who go, “Ooh.” But many more who are just so eager to do something in addition to number that we can call mathematics. Mike: You know, I'm thinking about our conversation before we set up and started to record the formal podcast today. And during that conversation you asked me a question that involved kites, and I'm wondering if you might ask that question again for our listeners. Rebecca: I'm going to invite you to do a mental challenge. And the way you think about it might be quite revealing to how you engage in both geometric and spatial reasoning. So, I invite you to picture in your mind's eye a kite and then to describe to me what you're seeing. Mike: So, I see two equilateral triangles that are joined at their bases—although as I say the word “bases,” I realize that could also lead to some follow-up questions. And then I see one wooden line that bisects those two triangles from top to bottom and another wooden line that bisects them along what I would call their bases. Rebecca: OK, I'm trying to imagine with you. So, you have two equilateral triangles that—a different way of saying it might be they share a side? Mike: They do share a side. Yes. Rebecca: OK. And then tell me again about these wooden parts. Mike: So, when I think about the kite, I imagine that there is a point at the top of the kite and a point at the bottom of the kite. And there's a wooden piece that runs from the point at the top down to the point at the bottom. And it cuts right through the middle. So, essentially, if you were thinking about the two triangles forming something that looked like a diamond, there would be a line that cut right from the top to the bottom point. Rebecca: OK. Mike: And then, likewise, there would be another wooden piece running from the point on one side to the point on the other side. So essentially, the triangles would be cut in half, but then there would also be a piece of wood that would essentially separate each triangle from the other along the two sides that they shared. Rebecca: OK. One thing that I noticed was you used a lot of mathematical ideas, and we don't always see that in children. And I hope that the listeners engaged in that activity themselves and maybe even stopped for a moment to sort of picture it before they started trying to process what you said so that they would just kind of play with this challenge of taking what you're seeing in your mind's eye and trying to articulate in words what that looks like. And that's a whole mathematical task in and of itself. And the way that you engaged in it was from a fairly high level of mathematics. And so, one of the things that I hope that task sort of illustrates is how a.) geometry involves these images that we have. And that we are often having to develop that concept image, this way of imagining it in our visual domain, in our brain. And almost everybody has it. And some people call it “the mind's eye.” Three percent of the population apparently don't have it—but the fact that 97 percent do suggests for teachers that they can depend on almost every child being able to at least close their eyes and picture that kite. I was strategic in choosing the kite rather than asking you to picture a rectangle or a hexagon or something like that because the kite is a mathematical idea that some mathematicians talk about, but it's also this real-world thing that we have some experiences with. And so, one of the things that that particular exercise does is highlight how we have these prototypes, these single images that we associate with particular words. And that's our starting point for instruction with children, for helping them to build up their mathematical ideas about these shapes. Having a mental image and then describing the mental image is where we put language to these math ideas. And the prototypes can be very helpful, but sometimes, especially for young children, when they believe that a triangle is an equilateral triangle that's sitting on, you know, the horizontal—one side is basically its base, the word that you used—they've got that mental picture. But that is not associated with any other triangles. So, if something looks more or less like that prototype, they'll say, “Yeah, that's a triangle.” But when we start showing them some things that are very different from that, but that mathematicians would call triangles, they're not always successful at recognizing those as triangles. And then if we also show them something that has curved sides or a jagged side but has that nice 60-degree angle on the top, they'll say, “Oh yeah, that's close enough to my prototype that we'll call that a triangle.” So, part of what we are doing when we are engaging kids in these conversations is helping them to attend to the precision that mathematicians always use. And that's one of our standards. And as I've done more work with talking to kids about these geometric shapes, I realize it's about helping them to be very clear about when they are referring to something, what it is they're referring to. So, I listen very carefully to,

36 min
12/05/2024

Season 3 | Episode 7 – How You Say It Matters: Teacher Language Choices That Support Number Sense - Guest: Dr. James Brickwedde

Dr. James Brickwedde, How You Say It Matters: Teacher Language Choices That Support Number Sense ROUNDING UP: SEASON 3 | EPISODE 7 Carry the 1. Add a 0. Cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained that many educators use it without even thinking. But what’s the long-term impact of language like this on the development of our students’ number sense? Today, we’re talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. BIOGRAPHIES James Brickwedde is the director of the Project for Elementary Mathematics. He served on the faculty of Hamline University’s School of Education & Leadership from 2011–2021, supporting teacher candidates in their content and pedagogy coursework in elementary mathematics. RESOURCES The Project for Elementary Mathematics TRANSCRIPT Mike Wallus: Carry the 1, add a 0, cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained, we often use it without even thinking. But what's the long-term impact of language like this on our students’ number sense? Today we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. Welcome to the podcast, James. I'm excited to be talking with you today. James Brickwedde: Glad to be here. Mike: Well, I want to start with something that you said as we were preparing for this podcast. You described how an educator’s language can play a critical role in helping students think in value rather than digits. And I'm wondering if you can start by explaining what you mean when you say that. James: Well, thinking first of primary students—so, kindergarten, second grade, that age bracket—kindergartners, in particular, come to school thinking that numbers are just piles of ones. They're trying to figure out the standard order. They're trying to figure out cardinality. There are a lot of those initial counting principles that lead to strong number sense that they are trying to integrate neurologically. And so, one of the goals of kindergarten, first grade, and above is to build the solid quantity sense—number sense—of how one number is relative to the next number in terms of its size, magnitude, et cetera. And then as you get beyond 10 and you start dealing with the place value components that are inherent behind our multidigit numbers, it's important for teachers to really think carefully of the language that they're using so that, neurologically, students are connecting the value that goes with the quantities that they're after. So, helping the brain to understand that 23 can be thought of not only as that pile of ones, but I can decompose it into a pile of 20 ones and three ones, and eventually that 20 can be organized into two groups of 10. And so, using manipulatives, tracking your language so that when somebody asks, “How do I write 23?” it's not a 2 and a 3 that you put together, which is what a lot of young children think is happening. But rather, they realize that there's the 20 and the 3. Mike: So, you're making me think about the words in the number sequence that we use to describe quantities. And I wonder about the types of tasks or the language that can help children build a meaningful understanding of whole numbers, like say, 11 or 23. James: The English language is not as kind to our learners [laughs] as other languages around the world are when it comes to multidigit numbers. We have in English 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And when we get beyond 10, we have this unique word called “eleven” and another unique word called “twelve.” And so, they really are words capturing collections of ones really then capturing any sort of tens and ones relationship. There's been a lot of wonderful documentation around the Chinese-based languages. So, that would be Chinese, Japanese, Korean, Vietnamese, Hmong follows the similar language patterns, where when they get after 10, it literally translates as “10, 1,” “10, 2.” When they get to 20, it's “2, 10”—”2, 10, 1,” “2, 10, 2.” And so, the place value language is inherent in the words that they are saying to describe the quantities. The teen numbers, when you get to 13, a lot of young children try to write 13 as “3, 1” because they're trying to follow the language patterns of other numbers where you start left to right. And so, they're bringing meaning to something, which of course is not the social convention. So, the teens are all screwed up in terms of English. Spanish does begin to do some regularizing when they get to 16 because of the name “diez y seis,” so “ten, six.” But prior to that you have, again, sort of more unique names that either don't follow the order of how you write the number or they're unique like 11 and 12 is. Somali is another interesting language in that—and I apologize to anybody who is fluent in that language because I'm hoping I'm going to articulate it correctly—I believe that there, when they get into the teens, it's “1 and 10,” “2 and 10,” is the literal translation. So, while it may not be the “10, 1” sort of order, it still is giving … the fact that there's ten-ness there as you go. So, for the classrooms that I have been in and out of—both [in] my own classroom years ago as well as the ones I still go in and out of now—I try to encourage teachers to tap the language assets that are among their students so that they can use them to think about the English numbers, the English language, that can help them wire that brain so that the various representations—the manipulatives, expanded notation cards or dice, the numbers that I write, how I break the numbers apart, say that 23 is equal to 20 plus 3—all of those models that you're using, and the language that you use to back it up with, is consistent so that, neurologically, those pathways are deeply organized. Piaget, in his learning theory, talks about young children—this is sort of the 10 years and younger—can only really think about one attribute at a time. So that if you start operating on multidigit numbers, and I'm using digitized language, I'm asking that kindergartner, first [grader], second grader to think of two things at the same time. I'm, say, moving a 1 while I also mean 10. What you find, therefore, is when I start scratching the surface of kids who were really procedural-bound, that they really are not reflecting on the values of how they've decomposed the numbers or are reconfiguring the numbers. They're just doing digit manipulation. They may be getting a correct answer, they may be very fast with it, but they've lost track of what values they're tracking. There's been a lot of research on kids’ development of multidigit operations, and it's inherent in that research about students following—the students who are more fluid with it talk in values rather than in digits. And that's the piece that has always caught my attention as a teacher and helped transform how I talked with kids with it. And now as a professional development supporter of teachers, I'm trying to encourage them to incorporate in their practice. Mike: So, I want to hang on to this theme that we're starting to talk about. I'm thinking a lot about the very digit-based language that as a child I learned for adding and subtracting multidigit numbers. So, phrases like, “Carry the 1” or “Borrow something from the 6.” Those were really commonplace. And in many ways, they were tied to this standard algorithm, where a number was stacked on top of another number. And they really obscured the meaning of addition and subtraction. I wonder if we can walk through what it might sound like or what other models might draw out some of the value-based language that we want to model for kids and also that we want kids to eventually adopt when they're operating on numbers. James: A task that I give adults, whether they are parents that I’m out doing a family math night with or my teacher candidates that I have worked with, I have them just build 54 and 38, say, with base ten blocks. And then I say, “How would you quickly add them?” And invariably everybody grabs the tens before they move to the ones. Now your upbringing, my upbringing is the same and still in many classrooms: Students are directed only to start with the ones place. And if you get a new 10, you have to borrow and you have to do all of this exchange kinds of things. But the research shows when school gets out of the way [chuckles] and students and adults are operating on more of their natural number sense, people start with the larger and then move to the smaller. And this has been found around the world. This is not just unique to US classrooms that have been working this way. If, in the standard algorithms—which really grew out of accounting procedures that needed to save space in ledger books out of the 18th, 19th centuries—they are efficient, space-saving means to be able to accurately compute. But in today's world, technology takes over a lot of that bookkeeping type of thing. An analogy I like to make is, in today's world, Bob Cratchit out of [A] Christmas Carol, Charles Dickens’s character, doesn't have a job because technology has taken over everything that he was in charge of. So, in order for Bob Cratchit to have a job, [laughs] he does need to know how to compute. But he really needs to think in values. So, what I try to encourage educators to loosen up their practice is to say, “If I'm adding 54 plus 38, so if you keep those two numbers in your mind, [chuckles] if I start with the ones and I add 4 and 8, I can get 12.” There's no reason, if I'm working in a vertical format, to not put 12 fully under the line down below, particularly when k

26 min
11/26/2024

Season 3 | Episode 5 - Building Asset-Focused Professional Learning Communities - Guests: Summer Pettigrew and Megan Williams

Summer Pettigrew and Megan Williams, Building Asset-Focused Professional Learning Communities ROUNDING UP: SEASON 3 | EPISODE 5 Professional learning communities have been around for a long time, in many different iterations. But what does it look like to schedule and structure professional learning communities that help educators understand and respond to their students' thinking in meaningful ways? Today we’re talking with Summer Pettigrew and Megan Williams from the Charleston County Public Schools about building asset-focused professional learning communities. BIOGRAPHIES Summer Pettigrew serves as an instructional coach at Springfield Elementary School in Charleston, South Carolina. Megan Williams serves as principal at Springfield Elementary School in Charleston, South Carolina. RESOURCES OGAP website TRANSCRIPT Mike Wallus: Professional learning communities have been around for a long time and in many different iterations. But what does it look like to schedule and structure professional learning communities that actually help educators understand and respond to their students’ thinking in meaningful ways? Today we're talking with Summer Pettigrew and Megan Williams from the Charleston Public Schools about building asset-focused professional learning communities. Hello, Summer and Megan. Welcome to the podcast. I am excited to be talking with you all today about PLCs. Megan Williams: Hi! Summer Pettigrew: Thanks for having us. We're excited to be here. Mike: I'd like to start this conversation in a very practical place: scheduling. So, Megan, I wonder if you could talk just a bit about when and how you schedule PLCs at your building. Megan: Sure. I think it's a great place to start too, because I think without the structure of PLCs in place, you can't really have fabulous PLC meetings. And so, we used to do our PLC meetings once a week during teacher planning periods, and the teachers were having to give up their planning period during the day to come to the PLC meeting. And so, we created a master schedule that gives an hour for PLC each morning. So, we meet with one grade level a day, and then the teachers still have their regular planning period throughout the day. So, we were able to do that by building a time for clubs in the schedule. So, first thing in the morning, depending on your day, so if it's Monday and that's third grade, then the related arts teachers—and that for us is art, music, P.E., guidance, our special areas—they go to the third-grade teachers’ classrooms. The teachers are released to go to PLC, and then the students choose a club. And so, those range from basketball to gardening to fashion to [STEM]. We've had Spanish Club before. So, they participate with the related arts teacher in their chosen club, and then the teachers go to their PLC meeting. And then once that hour is up, then the teachers come back to class. The related arts teachers are released to go get ready for their day. So, everybody still has their planning period, per se, throughout the day. Mike: I think that feels really important, and I just want to linger a little bit longer on it. One of the things that stands out is that you're preserving the planning time on a regular basis. They have that, and they have PLC time in addition to it. Megan: Mm-hmm, correct. And that I think is key because planning time in the middle of the day is critical for making copies, calling parents, calling your doctor to schedule an appointment, using the restroom—those kind of things that people have to do throughout the day. And so, when you have PLC during their planning time, one or the other is not occurring. Either a teacher is not taking care of those things that need to be taken care of on the planning period or they're not engaged in the PLC because they're worried about something else that they've got to do. So, building that time in, it's just like a game changer. Mike: Summer, as a person who’s playing the role of an instructional coach, what impact do you think this way of scheduling has had on educators who are participating in the PLCs that you're facilitating? Summer: Well, it's huge. I have experienced going to a PLC on our planning [period] and just not being one-hundred-percent engaged. And so, I think having the opportunity to provide the time and the space for that during the school day allows the teachers to be more present. And I think that the rate at which we're growing as a staff is expedited because we're able to drill into what we need to drill into without worrying about all the other things that need to happen. So, I think that the scheduling piece has been one of the biggest reasons we've been so successful with our PLCs. Mike: Yeah, I can totally relate to that experience of feeling like I want to be here, present in this moment, and I have 15 things that I need to do to get ready for the next chunk of my day. So, taking away that “if-then,” and instead having an “and” when it comes to PLCs, really just feels like a game changer. Megan: And we were worried at first about the instructional time that was going to be lost from the classroom doing the PLC like this. We really were because we needed to make sure instructional time was maximized and we weren't losing any time. And so, this really was about an hour a week, right, where the teachers aren't directly instructing the kids. But it has not been anything negative at all. Our scores have gone up, our teachers have grown. They love—the kids love going to their clubs. I mean, even the attendance on the grade-level club day is so much better because they love coming in. They start the day really getting that SEL instruction. I mean, that's really a lot of what they're getting in clubs. They're hanging out with each other. They're doing something they love. Mike: Maybe this is a good place to shift and talk a little bit about the structure of the PLCs that are happening. So, I've heard you say that PLCs, as they're designed and functioning right now, they're not for planning; they're instead for teacher collaboration. So, what does that mean? Megan: Well, there's a significant amount of planning that does happen in PLC, but it's not a teacher writing his or her lesson plans for the upcoming week. So, there's planning, but not necessarily specific lesson planning, like, “On Monday I'm doing this; on Tuesday I'm doing this.” It's more looking at the standards, looking at the important skills that are being taught, discussing with each other ways that you do this. “How can I help kids that are struggling? How can I push kids that are higher?” So, teachers are collaborating and planning, but they're not really producing written lesson plans. Mike: Yeah. One of the pieces that you all talked about when we were getting ready for this interview, was this idea that you always start your PLCs with a recognition of the celebrations that are happening in classrooms. I'm wondering if you can talk about what that looks like and the impact it has on the PLCs and the educators who are a part of them. Summer: Yeah. I think our teachers are doing some great things in their classrooms, and I think having the time to share those great things with their colleagues is really important. Just starting the meeting on that positive note tends to lead us in a more productive direction. Mike: You two have also talked to me about the impact of having an opportunity for educators to engage in the math that their students will be doing or looking at common examples of student work and how it shows up in the classroom. I wonder if you could talk about what you see in classrooms and how you think that loops back into the experiences that are happening in PLCs. Summer: Yeah. One of the things that we start off with in our PLCs is looking at student work. And so, teachers are bringing common work examples to the table, and we're looking to see, “What are our students coming with? What's a good starting point for us to build skills, to develop these skills a little bit further to help them be more successful?” And I think a huge part of that is actually doing the work that our students are doing. And so, prior to giving a task to a student, we all saw that together in a couple of different ways. And that's going to give us that opportunity to think about what misconceptions might show up, what questions we might want to ask if we want to push students further, reign them back in a little bit. Just that pre-planning piece with the student math, I think has been very important for us. And so, when we go into classrooms, I'll smile because they kind of look like little miniature PLCs going on. The teacher’s facilitating, the students are looking at strategies of their classmates and having conversations about what's similar, what's different. I think the teachers are modeling with their students that productive practice of looking at the evidence and the student work and talking about how we go about thinking through these problems. Mike: I think the more that I hear you talk about that, I flash back to, Megan, what you said earlier about [how] there is planning that's happening, and there's collaboration. They're planning the questions that they might ask. They're anticipating the things that might come from students. So, while it's not, “I'm writing my lesson for Tuesday,” there is a lot of planning that's coming. It’s just perhaps not as specific as, “This is what we'll do on this particular day.” Am I getting that right? Megan: Yes. You're getting that one-hundred-percent right. Summer has teachers sometimes [take] the assessment at the beginning of a unit. We'll go ahead and take the end-of-unit assessment and the information that you gain from that, just with having the teachers take it and knowing how the kids are going to be assessed, then just in turn makes them better planners for the unit. And

18 min
11/21/2024

Season 3 | Episode 6 – Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades - Guests: Drs. Jody Guarino and Chepina Rumsey

Drs. Jody Guarino and Chepina Rumsey, Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades ROUNDING UP: SEASON 3 | EPISODE 6 Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergartners, first-, or second graders. What would it look like to encourage these practices with our youngest learners? Today, we’ll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2. BIOGRAPHIES Chepina Rumsey, PhD, is an associate professor of mathematics education at the University of Northern Iowa (UNI). Jody Guarino is currently a mathematics coordinator at the Orange County Department of Education and a lecturer at the University of California, Irvine. RESOURCES Nurturing Math Curiosity with Learners in Grades K–2 Nurturing Math Curiosity on X/Twitter Tools to support K–2 students in mathematical argumentation. Teaching Children Mathematics, 25(4), 208–217. TRANSCRIPT Mike Wallus: Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergarten, first-, and second graders. What would it look like to encourage these practices with our youngest learners? Today, we'll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2. Welcome to the podcast, Chepina and Jody. Thank you so much for joining us today. Jody Guarino: Thank you for having us. Chepina Rumsey: Yeah, thank you. Mike: So, I'm wondering if we can start by talking about the genesis of your work, particularly for students in grades K–2. Jody: Sure. Chepina had written a paper about argumentation, and her paper was situated in a fourth grade class. At the time, I read the article and was so inspired, and I wanted to use it in an upcoming professional learning that I was going to be doing. And I got some pushback with people saying, “Well, how is this relevant to K–2 teachers?” And it really hit me that there was this belief that K–2 students couldn't engage in argumentation. Like, “OK, this paper's great for older kids, but we're not really sure about the young students.” And at the time, there wasn't a lot written on argumentation in primary grades. So, we thought, “Well, let's try some things and really think about, ‘What does it look like in primary grades?’ And let's find some people to learn with.” So, I approached some of my recent graduates from my teacher ed program who were working in primary classrooms and a principal that employed quite a few of them with this idea of, “Could we learn some things together? Could we come and work with your teachers and work with you and just kind of get a sense of what could students do in kindergarten to second grade?” So, we worked with three amazing teachers—Bethany, Rachael, and Christina—in their first years of teaching, and we worked with them monthly for two years. We wanted to learn, “What does it look like in K–2 classrooms?” And each time we met with them, we would learn more and get more and more excited. Little kids are brilliant, but also their teachers were brilliant, taking risks and trying things. I met with one of the teachers last week, and the original students that were part of the book that we've written now are actually in high school. So, it was just such a great learning opportunity for us. Mike: Well, I'll say this, there are many things that I appreciated about the book, about Nurturing Math Curiosity with Learners in Grades K–2, and I think one of the first things was the word “with” that was found in the title. So why “with” learners? What were y'all trying to communicate? Chepina: I'm so glad you asked that, Mike, because that was something really important to us when we were coming up with the title and the theme of the book, the message. So, we think it's really important to nurture curiosity with our students, meaning we can't expect to grow it in them if we're not also growing it in ourselves. So, we see that children are naturally curious and bring these ideas to the classroom. So, the word “with” was important because we want everyone in the classroom to grow more curious together. So, teachers nurturing their own math curiosity along with their students is important to us. One unique opportunity we tried to include in the book is for teachers who are reading it to have opportunities to think about the math and have spaces in the book where they can write their own responses and think deeply along with the vignettes to show them that this is something they can carry to their classroom. Mike: I love that. I wonder if we could talk a little bit about the meaning and the importance of argumentation? In the book, you describe four layers: noticing and wondering, conjecture, justification, and extending ideas. Could you share a brief explanation of those layers? Jody: Absolutely. So, as we started working with teachers, we'd noticed these themes or trends across, or within, all of the classrooms. So, we think about noticing and wondering as a space for students to make observations and ask curious questions. So, as teachers would do whatever activity or do games, they would always ask kids, “What are you noticing?” So, it really gave kids opportunities to just pause and observe things, which then led to questions as well. And when we think about students conjecturing, we think about when they make general statements about observations. So, an example of this could be a child who notices that 3 plus 7 is 10 and 7 plus 3 is 10. So, the child might think, “Oh wait, the order of the addends doesn't matter when adding. And maybe that would even work with other numbers.” So, forming a conjecture like, “This is what I believe to be true.” The next phase is justification, where a student can explain either verbally or with writing or with tools to prove the conjecture. So, in the case of the example that I brought up, 3 plus 7 and 7 plus 3, maybe a student even uses their fingers, where they're saying, “Oh, I have these 3 fingers and these 7 fingers, and whichever fingers I look at first, or whichever number I start with, it doesn't matter. The sum is going to be the same.” So, they would justify in ways like that. I've seen students use counters, just explaining it. Oftentimes, they use language and hand motions and all kinds of things to try to prove what they're saying works. Or sometimes they'll find, just, really look for, “Can I find an example where that doesn't work?” So, just testing their conjecture would be justifying. And then the final stage, extending ideas, could be extending that idea to all numbers. So, in the idea of addition in the commutative property, and they come to discover that they might realize, “Wait a minute, it also works for 1 plus 9 and 9 plus 1.” They could also think, “Does it work for other operations? So, not just with addition, but maybe I can subtract like that too. Does that make a difference if I'm subtracting 5 take away 2 versus 2 take away 5.” So, just this idea of, “Now [that] I've made sense of something, what else does it work with or how can I extend that thinking?” Mike: So, the question that I was wondering about as you were talking is, “How do you think about the relationship between a conjecture and students’ justification?” Jody: I've seen a lot of kids—so, sometimes they make conjectures that they don't even realize are conjectures, and they're like, “Oh, wait a minute, this pattern's happening, and I think I see something.” And so often they're like, “OK, I think that every time you add two numbers together, the sum is greater than the two numbers.” And so, then this whole idea of justifying, we often ask them, “How could you convince someone that that's true?” Or, “Is that always true?” And now they actually have to take and study it and think about, “Is it true? Does it always work?” Which, Mike, in your question, often leads back to another conjecture or refining their conjecture. It's kind of this cyclical process. Mike: That totally makes sense. I was going to use the words “virtuous cycle,” but that absolutely helps me understand that. I wonder if we can go back to the language of conjecture, because that feels really important to get clear on and to both understand and start to build a picture of. So, I wonder if you could offer a definition of conjecture for someone who’s unfamiliar with the term or talk about how students understand conjecture. Chepina: Yeah. So, a conjecture is based on our exploration with the patterns and observations. So, through that exploration, we might have an idea that we believe to be true. We are starting to notice things and some language that students start to use—things like, “Oh, that's always going to work” or “Sometimes we can do that.” So, there starts to be this shift toward an idea that they believe is going to be true. It's often a work in progress, so it needs to be explored more in order to have evidence to justify why that's going to be true. And through that process, we can modify our conjecture, or we might have an idea, like this working idea of a conjecture, that then when we go to justify it, we realize, “Oh, it's not always true the way we thought, so we have to make a change.” So, the conjecture is something that we believe to be true, and then we try to convince other people. So, once we introduce that with young mathematicians, they tend to latch on to that idea that it's this really neat thing to come up with a conjecture. And so, then they often start to come up with them even when we're not asking and get e

24 min
10/24/2024

Season 3 | Episode 4 - Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics - Guest: Beth Hulbert

Beth Hulbert, Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics ROUNDING UP: SEASON 3 | EPISODE 4 During their elementary years, students grapple with many topics that involve relationships between different units. In fact, unitizing serves as a foundation for much of the mathematics students encounter during their elementary years. Today, we’re talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms. BIOGRAPHY Beth Hulbert is an independent consultant focused on mathematics curriculum, instruction, and assessment at the K–8 level. She has been involved in all aspects of the Ongoing Assessment Project (OGAP) since its inception. Beth is coauthor of A Focus on Multiplication and Division: Bringing Research to the Classroom. The book was written to communicate how students develop their understanding of the concepts of multiplication and division. RESOURCES OGAP Website OGAP Framework Documents TRANSCRIPT Mike Wallus: During their elementary years, students grapple with many topics that involve relationships between different units. This concept, called “unitizing,” serves as a foundation for much of the mathematics that students encounter during their elementary years. Today, we're talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms. Welcome to the podcast, Beth. We are really excited to talk with you today. Beth Hulbert: Thanks. I'm really excited to be here. Mike: I'm wondering if we can start with a fairly basic question: Can you explain OGAP and the mission of the organization? Beth: Sure. So, OGAP stands for the Ongoing Assessment Project, and it started with a grant from the National Science Foundation to develop tools and resources for teachers to use in their classroom during math that were formative in nature. And we began with fractions. And the primary goal was to read, distill, and make the research accessible to classroom teachers, and at the same time develop tools and strategies that we could share with teachers that they could use to enhance whatever math program materials they were using. Essentially, we started by developing materials, but it turned into professional development because we realized teachers didn't have a lot of opportunity to think deeply about the content at the level they teach. The more we dug into that content, the more it became clear to us that content was complicated. It was complicated to understand, it was complicated to teach, and it was complicated to learn. So, we started with fractions, and we expanded to do work in multiplicative reasoning and then additive reasoning and proportional reasoning. And those cover the vast majority of the critical content in K–8. And our professional development is really focused on helping teachers understand how to use formative assessment effectively in their classroom. But also, our other goals are to give teachers a deep understanding of the content and an understanding of the math ed research, and then some support and strategies for using whatever program materials they want to use. And we say all the time that we're a program blind — we don't have any skin in the game about what program people are using. We are more interested in making people really effective users of their math program. Mike: I want to ask a quick follow-up to that. When you think about the lived experience that educators have when they go through OGAP’s training, what are the features that you think have an impact on teachers when they go back into their classrooms? Beth: Well, we have learning progressions in each of those four content strands. And learning progressions are maps of how students acquire the concepts related to, say, multiplicative reasoning or additive reasoning. And we use those to sort, analyze, and decide how we're going to respond to evidence in student work. They're really maps for equity and access, and they help teachers understand that there are multiple right ways to do some mathematics, but they're not all equal in efficiency and sophistication. Another piece they take away of significant value is we have an item bank full of hundreds of short tasks that are meant to add value to, say, a lesson you taught in your math program. So, you teach a lesson, and you decide, “What is the primary goal of this lesson?” And we all know, no matter what the program is you're using, that every lesson has multiple goals, and they're all in varying degrees of importance. So partly, picking an item in our item bank is about helping yourself think about what was the most critical piece of that lesson that I want to know about that's critical for my students to understand for success tomorrow. Mike: So, one big idea that runs through your work with teachers is this concept called “unitizing.” And it struck me that whether we're talking about addition, subtraction, multiplication, fractions, that this idea just keeps coming back and keeps coming up. I'm wondering if you could offer a brief definition of unitizing for folks who may not have heard that term before. Beth: Sure. It became really clear as we read the research and thought about where the struggles kids have, that unitizing is at the core of a lot of struggles that students have. So, unitizing is the ability to call something “1,” say, but know it's worth maybe 1 or 100 or a 1,000, or even 1/10. So, think about your numbers in a place value system. In our base ten system, when a 1 is in the tenths place, it's not worth 1 anymore; it's worth 1 of 10. And so that idea that the 1 isn't the value of its face value, but it's the value of its place in that system. So, base ten is one of the first big ways that kids have to understand unitizing. Another kind of unitizing would be money. Money's a really nice example of unitizing. So, I can see one thing, it's called a nickel, but it's worth 5. And I can see one thing that's smaller, and it's called a dime, and it's worth 10. And so, the idea that 1 would be worth 5 and 1 would be worth 10, that's unitizing. And it's an abstract idea, but it provides the foundation for pretty much everything kids are going to learn from first grade on. And when you hear that kids are struggling, say, in third and fourth grade, I promise you that one of their fundamental struggles is a unitizing struggle. Mike: Well, let's start where you all started when you began this work in OGAP. Let's start with multiplication. Can you talk a little bit about how this notion of unitizing plays out in the context of multiplication? Beth: Sure. In multiplication, one of the first ways you think about unitizing is, say, in the example of 3 times 4. One of those numbers is a unit or a composite unit, and the other number is how many times you copy or iterate that unit. So, your composite unit in that case could be 3, and you're going to repeat or iterate it 4 times. Or your composite unit could be 4, and you're going to repeat or iterate it 3 times. When I was in school, the teacher wrote “3 times 4” up on the board and she said, “3 tells you how many groups you have, and 4 tells you how many you put in each group.” But if you think about the process you go through when you draw that in that definition, you draw 1, 2, 3 circles, then you go 1, 2, 3, 4; 1, 2, 3, 4; 1, 2, 3, 4; 1, 2, 3, 4. And in creating that model, you never once thought about a unit, you thought about single items in a group. So, you counted 1, 2, 3, 4 three times, and there was never really any thought about the unit. In a composite unit way of thinking about it, you would say, “I have a composite unit of 3, and I'm going to replicate it 4 times.” And in that case, every time, say, you stamped that — you had this stamp that was 3 — every time you stamped it, that one action would mean 3, right? 1 to 3, 1 to 3, 1 to 3, 1 to 3. So, in really early number work, kids think 1 to 1. When little kids are counting a small quantity, they'll count 1, 2, 3, 4. But what we want them to think about in multiplication is a many-to-1 action. When each of those quantities happens, it's not one thing. Even though you make one action, it's 4 things or 3 things, depending upon what your unit is. If you needed 3 times 8, you could take your 3-times-4 and add four more 3-times-4s to that. So, you have your four 3s and now you need four more 3s. And that allows you to use a fact to get a fact you don't know because you've got that unit and that understanding that it's not by 1, but by a unit. When it gets to larger multiplication, we don't really want to be working by drawing by 1s, and we don't even want to be stamping 27 nineteen times, right? But it's a first step into multiplication, this idea that you have a composite unit, and in the case of 3 times 4 and 3 times 7, seeing that 3 is common. So, there's your common composite unit. You [need] four of them for 3 times 4, and you need seven of them for 3 times 7. So, it allows you to see those relationships, which if you look at the standards, the relationships are the glue. So, it's not enough to memorize your multiplication facts. If you don't have a strong relationship understanding there, it does fall short of a depth of understanding. Mike: I think it was interesting to hear you talk about that, Beth, because one of the things that struck me is some of the language that you used, and I was comparing it in my head to some of the language that I've used in the past. So, I know I've talked about 3 times 4, but I thought it was really interesting how you used “iterations of” or “duplicated” … Beth: “Copies.” Mike: … or “copies,” right? What you make me think is that those language choices are a little bit clearer. I can visualize them in a way that “3 times 4” is a little bit more a

30 min
10/10/2024

Season 3 | Episode 3 - Choice as a Foundation for Student Engagement - Guest: Drs. Zandra De Arajuo and Amber Candela

Drs. Zandra De Arajuo and Amber Candela, Choice as a Foundation for Student Engagement ROUNDING UP: SEASON 3 | EPISODE 3 As educators, we know offering students choice has a big impact on their engagement, identity, and sense of autonomy. That said, it's not always clear how to design choice into activities, especially when using curriculum materials. Today, we’re talking with Drs. Zandra De Araujo and Amber Candela about some of the ways educators can design choice into their students’ learning experiences. BIOGRAPHY Zandra de Araujo serves as the mathematics principal at the Lastinger Center for Learning. Her research examines teachers’ instruction in algebra with students who are primarily English learners. Amber Candela is an assistant professor of mathematics education at the University of Missouri–St. Louis (UMSL). She teaches mathematics methods classes for prospective elementary, middle, and high school teachers in the teacher education program at UMSL. TRANSCRIPT Mike Wallus: As an educator, I know that offering my students choice has a big impact on their engagement, their identity, and their sense of autonomy. That said, I've not always been sure how to design choice into the activities in my classroom, especially when I'm using curriculum. Today, we're talking with Drs. Zandra de Araujo and Amber Candela about some of the ways educators can design choice into their students' learning experiences. Welcome back to the podcast, Zandra and Amber. It is really exciting to have you all with us today. Zandra de Araujo: Glad to be back. Amber Candela: Very excited to be here. Mike: So, I've heard you both talk at length about the importance of choice in students' learning experiences, and I wonder if we can start there. Before we talk about the ways you think teachers can design choice in a learning experience, can we just talk about the “why”? How would you describe the impact that choice has on students' learning experiences? Zandra: So, if you think about your own life, how fun would it be to never have a choice in what you get to do during a day? So, you don't get to choose what chores to do, where to go, what order to do things, who to work with, who to talk to. Schools are a very low-choice environment, and those tend to be punitive when you have a low-choice environment. And so, we don't want schools to be that way. We want them to be very free and open and empowering places. Amber: And a lot of times, especially in mathematics, students don't always enjoy being in that space. So, you can get more enjoyment, engagement, and if you have choice with how to engage with the content, you'll have more opportunity to be more curious and joyful and have hopefully better experiences in math. Zandra: And if you think about being able to choose things in your day makes you better able to make choices. And so, I think we want students to be smart consumers and users and creators of mathematics. And if you're never given choice or opportunity to kind of own it, I think that you're at a deficit. Amber: Also, if we want problem-solving people engaged in mathematics, it needs to be something that you view as something you were able to do. And so often we teach math like it's this prepackaged thing, and it's just your role to memorize this thing that I give you. You don't feel like it's yours to play with. Choice offers more of those opportunities for kids. Zandra: Yeah, it feels like you're a consumer of something that's already made rather than somebody who's empowered to create and use and drive the mathematics that you're using, which would make it a lot more fun. Mike: Yeah. You all are hitting on something that really clicked for me as I was listening to you talk. This idea that school, as it's designed oftentimes, is low choice. But math, in particular, where historically it has really been, “Let me show you what to do. Let me have you practice the way I showed you how to do it,” rinse and repeat. It's particularly important in math, it feels like, to break out and build a sense of choice for kids. Zandra: Absolutely. Mike: Well, one of the things that I appreciate about the work that both of you do is the way that you advocate for practices that are both really, really impactful and also eminently practical. And I'm wondering if we can dive right in and have you all share some of the ways that you think about designing choice into learning experiences. Amber: I feel like I want “eminently practical” on a sticker for my laptop. (Zandra laughs) Because I find that is a very satisfying and positive way to describe the work that I do because I do want it to be practical and doable within the constraints of schooling as it currently is, not as we wish it to be. Which, we do want it to be better and more empowering for students and teachers. But also, there are a lot of constraints that we have to work within. So, I appreciate that. Zandra: I think that choice is meant to be a way of empowering students, but the goal for the instruction should come first. So, I'm going to talk about what I would want from my students in my classroom and then how we can build choice in. Because choice is kind of like the secondary component. So, first you have your learning goals, your aims as a teacher. And then, “How do we empower students with choice in service of that goal?” So, I'll start with number sense because that's a hot topic. I'm sure you all hear a lot about it at the MLC. Mike: We absolutely do. Zandra: So, one of the things I think about when teachers say, “Hey, can you help me think about number sense?” It's like, “Yes, I absolutely can.” So, our goal is number sense. So, let's think about what that means for students and how do we build some choice and autonomy into that. So, one of my favorite things is something like, “Give me an estimate, and we can Goldilocks it,” for example. So, it could be a word problem or just a symbolic problem, and say, “OK, give me something that you know is either wildly high, wildly low, kind of close, kind of almost close but not right. So, give me an estimate, and it could be a wrong estimate or a close estimate, but you have to explain why.” So, it takes a lot of number sense to be able to do that. You have infinitely many options for an answer, but you have to avoid the one correct answer. So, you have to actually think about the one correct answer to give an estimate. Or if you're trying to give a close estimate, you're kind of using a lot of number sense to estimate the relationships between the numbers ahead of time. The choice comes in because you get to choose what kind of estimate you want. It's totally up to you. You just have to rationalize your idea. Mike: That's awesome. Amber, your turn. Amber: Yep. So related to that is a lot of math goes forwards. We give kids the problem, and we want them to come up with the answer, right? And so, a lot of the work that we've been doing is, “OK, if I give you the answer, can you undo the problem?” I'll go multiplication. So, we do a lot with, “What's 7 times 8?” And there's one answer, and then kids are done. And you look for that answer as the teacher, and once that answer has been given, you're kind of like, “OK, here. I'm done with what I'm doing.” But instead, you could say, “Find me numbers whose product is 24.” Now you've opened up what it comes to. There's more access for students. They can come up with more than one solution, but it also gets kids to realize that math doesn't just go one way. It's not, “Here's the problem, find the answer.” It’s “Here’s the answer, find the problem.” And that also goes to the number sense. Because if students are able to go both ways, they have a better sense in their head around what they're doing and undoing. And you can do it with a lot of different problems. Zandra: And I'll just add in that that's not specific to us. Barb Dougherty had a really nice article in, I think, Teaching Children Mathematics, about reversals at some point. And other people have shown this idea as well. So, we're really taking ideas that are really high uptake, we think, and sharing them again with teachers to make sure that they've seen ways that they can do it in their own classroom. Mike: What strikes me about both of these is, the structure is really interesting. But I also think about what the output looks like when you offer these kinds of choices. You're going to have a lot of kids doing things like justifying or using language to help make sense of the “why.” “Why is this one totally wrong?” and “Why is this one kind of right?” and “Why is this close, but maybe not exact?” And to go to the piece where you're like, “Give me some numbers that I can multiply together to get to 24.” There's more of a conversation that comes out of that. There's a back and forth that starts to develop, and you can imagine that back and forth bouncing around with different kids rather than just kind of kid says, teacher validates, and then you're done. Zandra: Yeah, I think one of the cool things about choice is giving kids choice means that there's more variety and diversity of ideas coming in. And that's way more interesting to talk about and rationalize and justify and make sense of than a single correct answer or everybody's doing the same thing. So, I think, not only does it give kids more ownership, it has more access. But also, it just gives you way more interesting math to think and talk about. Mike: Let's keep going. Zandra: Awesome. So, I think another one, a lot of my work is with multilingual students. I really want them to talk. I want everybody to talk about math. So, this goes right to what you were just saying. So, one of the ways that we can easily say, “OK, we want more talk, so how do we build that in through choice?” is to say, “Let's open up what you choose to share wi

23 min
09/19/2024

Season 3 | Episode 2 - Principles for Responsive Curriculum Use - Guest: Dr. Corey Drake

Dr. Corey Drake, Principles for Responsive Curriculum Use ROUNDING UP: SEASON 3 | EPISODE 2 When it comes to curriculum, educators are often told to implement with fidelity. But what does fidelity mean, and where does that leave educators who want to be responsive to the students in their classrooms? Today we’re talking with Dr. Corey Drake about principles for responsive curriculum use that invite educators to respond to the students in their classrooms while still implementing curriculum with integrity. BIOGRAPHY Corey Drake is the senior director for professional learning at The Math Learning Center. She began her career in education as a middle school special education teacher in the Chicago Public Schools. Corey is committed to supporting teachers in using curriculum materials to teach diverse groups of students in equitable ways. RESOURCES Responding to Student Thinking Via Curriculum Materials (Handout) TRANSCRIPT Mike Wallus: When it comes to curriculum, educators are often told to implement with fidelity. But what does fidelity mean? And where does that leave educators who want to be responsive to students in their classrooms? Today we're talking with Dr. Corey Drake about principles for responsive curriculum use that invite educators to respond to the students in their classrooms while still implementing curriculum with integrity. One of the age-old questions that educators grapple with is how to implement a curriculum in ways that are responsive to the students in their classroom. It's a question I thought a lot about during my years as a classroom teacher, and it's one that I continue to discuss with my colleague at MLC, Dr. Corey Drake. As a former classroom teacher and a former teacher educator who only recently began working for an organization that publishes curriculum, Corey and I have been trying to carve out a set of recommendations that we hope will help teachers navigate this question. Today on the podcast, we'll talk about this question of responsive curriculum use and offer some recommendations to support teachers in the field. Welcome back to the podcast, Corey. I'm excited to have you with us again. Corey Drake: It's great to be with you again. Mike: So, I've been excited about this conversation for a while because this question of, “What does it mean to be responsive to students and use a curriculum?” is something that teachers have been grappling with for so long. And you and I often hear phrases like “implementation with fidelity” used when folks are trying to describe their expectations when a curriculum's adopted. Corey: Yeah, I mean, I think this is a question teachers grapple with. It's a question I've been grappling with for my whole career, from different points of view from when I was a classroom teacher and a teacher educator and now working at The Math Learning Center. But I think this is the fundamental tension: “How do you use a set of published curriculum materials while also being responsive to your students?” And I think ideas like implementation with fidelity didn't really account for the responsive-to-your-students piece. Fidelity has often been taken up as meaning “following curriculum materials page by page, word for word, task for task.” We know that's not actually possible. You have to make decisions, you have to make adaptations as you move from a written page to an enacted curriculum. But still the idea of fidelity was to be as close as possible to the written page, whereas ideas like implementation with integrity or responsive curriculum use are starting with what's written on the page, staying consistent with the key ideas of what's on the page, but doing it in a way that's responsive to the students who are sitting in front of you. And that's really kind of the art and science of curriculum use. Mike: Yeah, I think one of the things that I used to think was that it was really a binary choice between something like fidelity, where you were following things in what I would've described as a lockstep fashion. Or the alternative, which would be, “I'm going to make everything up.” And you've helped me think, first of all, about what might be some baseline expectations from a large-scale curriculum. What are we actually expecting from curriculum around design, around the audience that it's written for? I wonder if you could share with the audience some of the things that we've talked about when it comes to the assets and also the limitations of a large-scale curriculum. Corey: Yeah, absolutely. And I will say, when you and I were first teachers probably, and definitely when we were students, the conversation was very different. We had different curriculum materials available. There was a very common idea that good teachers were teachers who made up their own curriculum materials, who developed all of their own materials. But there weren't the kinds of materials out there that we have now. And now we have materials that do provide a lot of assets, can be rich tools for teachers, particularly if we release this expectation of fidelity and instead think about integrity. So, some of the assets that a high-quality curriculum can bring are the progression of ideas, the sequence of ideas and tasks that underlies almost any set of curriculum materials; that really looks at, “How does student thinking develop across the course of a school year?” And what kinds of tasks, in what order, can support that development of that thinking. That's a really important thing that individual teachers or even teams of teachers working on their own, that would be very hard for them to put together in that kind of coherent, sequential way. So, that's really important. A lot of curriculum materials also bring in many ideas that we've learned over the last decades about how children learn mathematics: the kinds of strategies children use, the different ways of thinking that children bring. And so, there's a lot that both teachers and students can learn from using curriculum materials. At the same time, any published set of large-scale curriculum materials are, by definition, designed for a generic group of students, a generic teacher in a generic classroom, in a generic community. That's what it means to be large scale. That's what it means to be published ahead of time. So, those materials are not written for any specific student or teacher or classroom or community. And so, that's the real limitation. It doesn't mean that the materials are bad. The materials are very good. But they can't be written for those specific children in that specific classroom and community. That's where this idea that responsive curriculum use and equitable instruction always have to happen in the interactions between materials, teachers, and students. Materials by themselves cannot be responsive. Teachers by themselves cannot responsively develop the kinds of ideas in the ways that curriculum can, the ways they can when using curriculum as a tool. And, of course, students are a key part of that interaction. And so, it's really thinking about those interactions among teachers, students, and materials and thinking about, “What are the strengths the materials bring? What are the strengths the teacher brings?” The teacher brings their knowledge of the students. The teacher brings their knowledge of the context. And the students bring, of course, their engagement and their interaction with those materials. And so, it's thinking about the strengths they each bring to that interaction, and it's in those interactions that equitable and responsive curriculum use happens. Mike: One of the things that jumps out from what you said is this notion that we're not actually attempting to fix “bad curriculum.” We're taking the position that curriculum has a set of assets, but it also has a set of limitations, and that's true regardless of the curriculum materials that you're using. Corey: Absolutely. This is not at all about curriculum being bad or not doing what it's supposed to do. This assumes that you're using a high-quality curriculum that does the things we just talked about that has that progression of learning, those sequences of tasks that [bring] ideas about how children learn and how we learn and teach mathematics. And then, to use that well and responsively, the teacher then needs to work in ways, make decisions to enact that responsively. It's not about fixing the curriculum. It's about using the curriculum in the most productive and responsive ways possible. Mike: I think that's good context, and I also think it's a good segue to talk about the three recommendations that we want educators to consider when they're thinking about, “What does it mean to be responsive when you're using curriculum?” So, just to begin with, why don't we just lay them out? Could you unpack them, Corey? Corey: Yeah, absolutely. But I will say that this is work you and I have developed together and looking at the work of others in the field. And we've really come up with, I think, three key criteria for thinking about responsive curriculum use. One is that it maintains the goals of the curriculum. So again, recognizing that one of the strengths of curriculum is that it's built on this progression of ideas and that it moves in a sequential way from the beginning of the year to the end of the year. We want teachers to be aware of, to understand what the goals are of any particular session or unit or year, and to stay true to those goals, to stay aligned with those goals. But at the same time, doing that in ways that open up opportunities for voice and choice and sensemaking for the specific students who are in front of them in that classroom. And then the last is, we're really concerned with and interested in supporting equitable practice. And so, we think about responsive curriculum use as curriculum use that reflects the equity-based practices that were developed by J

30 min

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Rounding Up - Opener

Welcome to Rounding Up, the professional learning podcast brought to you by The Math Learning Center. Two things have always been true in education: Ongoing professional learning is essential, and teachers are extremely busy people. Rounding Up is a podcast designed to provide meaningful, bite-sized professional learning for busy educators and instructional leaders. I'm Mike Wallus, vice president for educator support at The Math Learning Center and host of the show. In each episode, we'll explore topics important to teachers, instructional leaders, and anyone interested in elementary mathematics education. Topics such as posing purposeful questions, effectively recording student thinking, cultivating students' math identity, and designing asset-based instruction from multilingual learners. Don't miss out! Subscribe now wherever you get your podcasts. Each episode will also be published on the Bridges Educator Site. We hope you'll give Rounding Up a try, and that the ideas we discuss have a positive impact on your teaching and your students' learning. © 2022 The Math Learning Center | www.mathlearningcenter.org

S1

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14 Ratings

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11/04/2022

Jill Wittman

I’m loving this podcast!! Thank you for the thoughtful episodes. The length is perfect and the content is impactful. Looking forward to each new episode!

Welcome to “Rounding Up” with the Math Learning Center. These conversations focus on topics that are important to Bridges teachers, administrators and anyone interested in Bridges in Mathematics. Hosted by Mike Wallus, VP of Educator Support at MLC.

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