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Season 3 | Episode 05 - Building Asset-Focused Professional Learning Communities Guests: Summer Pettigrew and Megan Williams 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 STEMs. 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.
Summer: Uh-hm.
Megan: 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 and just not being a 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 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. And 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 flashback to what Megan, what you said earlier about, 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 a hundred percent right. Summer has teachers sometimes taken 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 there's a lot of good conversation that comes from that.
Mike: I mean, in some ways, your PLC design, the word that pops into my head is almost like a “rehearsal” of sorts. Does that analogy seem right?
Summer: It seems right. And just to add on to that, I think, too, again, providing that time within the school day for them to look at the math, to do the math, to think about what they want to ask, is like a mini-rehearsal. Because typically, when teachers are planning outside of school hours, it's by themselves in a silo. But this just gives that opportunity to talk about all the possibilities together, run through the math together, ask questions if they have them. So, I think that's a decent analogy, yeah.
Mike: Yeah. Well, you know what it makes me think about is competitive sports like basketball. As a person who played quite a lot, there are points in time when you start to learn the game that everything feels so fast. And then there are points in time when you've had some experience when you know how to anticipate, where things seem to slow down a little bit. And the analogy is that if you can kind of anticipate what might happen or the meaning of the math that kids are showing you, it gives you a little bit more space in the moment to really think about what you want to do versus just feeling like you have to react.
Summer: And I think, too, it keeps you focused on the math at hand. You're constantly thinking about your next teacher move. And so, if you've got that math in your mind and you do get thrown off, you've had an opportunity, like you said, to have a little informal rehearsal with it, and maybe you're not thrown off as badly. ( laughs )
Mike: Well, one of the things that you’ve both mentioned when we've talked about PLCs is the impact of a program called OGAP. I'm wondering if you can talk about what OGAP is, what it brought to your educators, and how it impacted what’s been happening in PLCs.
Megan: I'll start in terms … OGAP stands for ongoing assessment project. Summer can talk about the specifics, but we rolled it out as a whole school. And I think there was power in that. Everybody in your school taking the same professional development at the same time, speaking the same language, hearing the same things. And for us, it was just a game-changer.
Summer: Yeah, I taught elementary math for 12 years before I knew anything about OGAP, and I had no idea what I was doing until OGAP came into my life. All of the light bulbs that went off with this very complex elementary math that I had no idea was a thing, it was just incredible. And so, I think the way that OGAP plays a role in PLCs is that we're constantly using the evidence in our student work to make decisions about what we do next. We're not just plowing through a curriculum, we're looking at the visual models and strategies that Bridges expects of us in that unit. We're coupling it with the content knowledge that we get from OGAP and how students should and could move along this progression. And we're planning really carefully around that; thinking about, “If we give this task and some of our students are still at a less sophisticated strategy and some of our students are at a more sophisticated strategy, how can we use those two examples to bridge that gap for more kids?” And we're really learning from each other's work. It's not the teacher up there saying, “This is how you'd solve this problem.” But it's a really deep dive into the content. And I think the level of confidence that OGAP has brought our teachers as they've learned to teach Bridges has been like a powerhouse for us.
Mike: Talk a little bit about the confidence that you see from your teachers who have had an OGAP experience and who are now using a curriculum and implementing it. Can you say more about that?
Summer: Yeah. I mean, I think about our PLCs, the collaborative part of it, we're having truly professional conversations. It's centered around the math, truly, and how students think about the math. And so again, not to diminish the need to strategically lesson plan and come up with activities and things, but we're talking really complex stuff in PLCs. And so, when we look at student work and we that work on the OGAP progression, depending on what skill we're teaching that week, we're able to really look at, “Gosh, the kid is, he's doing this, but I'm not sure why.” And then we can talk a little bit about, “Well, maybe he's thinking about this strategy, and he got confused with that part of it.” So, it really, again, is just centered around the student thinking. The evidence is in front of us, and we use that to plan accordingly. And I think it just one-ups a typical PLC because our teachers know what they're talking about. There's no question in, “Why am I teaching how to add on an open number line?” We know the reasoning behind it. We know what comes before that. We know what comes after that, and we know the importance of why we're doing it right now.
Mike: Megan, I wanted to ask you one more question. You are the instructional leader for the building, the position you hold is principal. I know that Summer is a person who does facilitation of the PLCs. What role do you play or what role do you try to play in PLCs as well?
Megan: I try to be present at every single PLC meeting and an active participant. I do all the assessments. I get excited when Summer says we're taking a test. I mean, I do everything that the teachers do. I offer suggestions if I think that I have something valuable to bring to the table. I look at student work. I just do everything with everybody because I like being part of that team.
Mike: What impact do you think that that has on the educators who are in the PLC?
Megan: I mean, I think it makes teachers feel that their time is valuable. We're valuing their time. It's helpful for me, too, when I go into classrooms. I know what I'm looking for. I know which kids I want to work with. Sometimes I'm like, “Ooh, I want to come in and see you do that. That's exciting.” It helps me plan my day, and it helps me know what's going on in the school. And I think it also is just a non-judgmental, non-confrontational time for people to ask me questions. I mean, it's part of me trying to be accessible as well.
Mike: Summer, as the person who’s the facilitator, how do you think about preparing for the kind of PLCs that you've described? What are some of the things that are important to know as a facilitator or to do in preparation?
Summer: So, I typically sort of rehearse myself, if you will, before the PLC kicks off. I will take assessments, I will take screeners. I'll look at screener implementation guides and think about the pieces of that that would be useful for our teachers if they needed to pull some small groups and re-engage those kids prior to a unit. What I really think is important though, is that vertical alignment. So, looking at the standards that are coming up in a module, thinking about what came before it: “What does that standard look like in second grade?” If I'm doing a third grade PLC: “What does that standard look like in fourth grade?” Because teachers don't have time to do that on their own. And I think it's really important for that collective efficacy, like, “We're all doing this together. What you did last year matters. What you're doing next year matters, and this is how they tie together.”
I kind of started that actually this year, wanting to know more myself about how these standards align to each other and how we can think about Bridges as a ladder among grade levels. Because we were going into classrooms, and teachers were seeing older grade levels doing something that they developed, and that was super exciting for them. And so, having an understanding of how our state standards align in that way just helps them to understand the importance of what they're doing and bring about that efficacy that we all really just need our teachers to own. It's so huge. And just making sure that our students are going to the next grade prepared.
Mike: One of the things that I was thinking about as I was listening to you two describe the different facets of this system that you've put together is, how to get started. Everything from scheduling to structure to professional learning. There's a lot that goes into making what you all have built successful. I think my question to you all would be, “If someone were listening to this, and they were thinking to themselves, ‘Wow, that's fascinating!’ What are some of the things that you might encourage them to do if they wanted to start to take up some of the ideas that you shared?”
Megan: It's very easy to crash and burn by trying to take on too much. And so, I think if you have a long-range plan and an end goal, you need to try to break it into chunks. Just making small changes and doing those small changes consistently. And once they become routine practices, then taking on something new.
Mike: Summer, how about you?
Summer: Yeah, I think as an instructional coach, one of the things that I learned through OGAP is that our student work is personal. And if we're looking at student work without the mindset of, “We're learning together,” sometimes we can feel a little bit attacked. And so, one of the first things that we did when we were rolling this out and learning how to analyze student work is, we looked at student work that wasn't necessarily from our class. We asked teachers to save student work samples. I have folders in my office of different student work samples that we can practice sorting and have conversations about. And that's sort of where we started with it. Looking at work that wasn't necessarily our students gave us an opportunity to be a little bit more open about what we wanted to say about it, how we wanted to talk about it. And it really does take some practice to dig into student thinking and figure out, “Where do I need to go from here?” And I think that allowed us to play with it in a way that wasn't threatening necessarily.
Mike: I think that's a great place to stop, Megan and Summer. I want to thank you so much for joining us. It's really been a pleasure talking to both of you.
Megan: Well, thank you for having us.
Summer: Yeah, thanks a lot for having us.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
© 2024 The Math Learning Center | www.mathlearningcenter.org
Season 3 | Episode 4 – Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics
Guest: Beth Hulbert
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 one-tenth. So, think about your numbers in a place value system. In our base 10 system, 1 of 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 10 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 four times. Or your composite unit could be 4, and you're going to repeat or iterate it three times. When I was in school, the teacher wrote 3 times 4 up on the board and she said, “Three 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 four 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? One 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 four things or three things, depending upon what your unit is. If you needed 3 times 8, you could take your 3 times 4 and add 4 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 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 19 times. 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 needed 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 abstract or obscure. I may be thinking of that wrong, but I'm curious how you think the language that you use when you're trying to get kids to think about composite numbers matters.
Beth: Well, I'll say this, that when you draw your 3 circles and count 4 dots in each circle, the result is the same model than if you thought of it as a unit of 3 stamped four times. In the end, the model looks the same, but the physical and mental process you went through is significantly different. So, you thought when you drew every dot, you were thinking about 1, 1, 1, 1, 1, 1, 1. When you thought about your composite unit copied or iterated, you thought about this unit being repeated over and over. And that changes the way you're even thinking about what those numbers mean. And one of those big, significant things that makes addition different than multiplication when you look at equations is, in addition, those numbers mean the same thing. You have 3 things, and you have 4 things, and you're going to put them together. If you had 3 plus 4, and you changed that 4 to a 5, you're going to change one of your quantities by 1, impacting your answer by 1. In multiplication, if you had 3 times 4, and you change that 4 to a 5, your factor increases by 1, but your product increases by the value of your composite unit. So, it's a change of the other factor. And that is significant change in how you think about multiplication, and it allows you to pave the way, essentially, to proportional reasoning, which is that replicating your unit.
Mike: One of the things I'd appreciated about what you said was it's a change in how you're thinking. Because when I think back to Mike Wallus, classroom teacher, I don't know that I understood that as my work. What I thought of my work at that point in time was I need to teach kids how to use an algorithm or how to get an answer. But I think where you're really leading is we really need to be attending to, “What's the thinking that underlies whatever is happening?”
Beth: Yes. And that's what our work is all about, is how do you give teachers a sort of lens into or a look into how kids are thinking and how that impacts whether they can employ more efficient and sophisticated relationships and strategies in their thinking. And it's not enough to know your multiplication facts. And the research is pretty clear on the fact that memorizing is difficult. If you're memorizing a hundred single facts just by memory, the likelihood you're not going to remember some is high. But if you understand the relationship between those numbers, then you can use your 3 times 4 to get your 3 times 5 or your 3 times 8. So, the language that you use is important, and the way you leave kids thinking about something is important. And this idea of the composite unit, it's thematic, right? It goes through fractions and additive and proportional, but it's not the only definition of multiplication. So, you've got to also think of multiplication as scaling that comes later, but you also have to think of multiplication as area and as dimensions. But that first experience with multiplication has to be that composite-unit experience.
Mike: You've got me thinking already about how these ideas around unitizing that students can start to make sense of when they're multiplying whole numbers, that that would have a significant impact when they started to think about fractions or rational numbers. Can you talk a little bit about unitizing in the context of fractions, Beth?
Beth: Sure. The fraction standards have been most difficult for teachers to get their heads around because the way that the standards promote thinking about fractions is significantly different than the way most of us were taught fractions. So, in the standards and in the research, you come across the term “unit fraction,” and you can probably recognize the unitizing piece in the unit fraction. So, a unit fraction is a fraction where 1 is in the numerator, it's one unit of a fraction. So, in the case of three-fourths, you have three of the one-fourths. Now, this is a bit of a shift in how we were taught. Most of us were taught, “Oh, we have three-fourths. It means you have four things, but you only keep three of them,” right? We learned about the name “numerator” and the name “denominator.” And, of course, we know in fractions, in particular, kids really struggle.
Adults really struggle. Fractions are difficult because they seem to be a set of numbers that don't have anything in common with any other numbers. But once you start to think about unitizing and that composite unit, there's a standard in third grade that talks about “decompose any fraction into the sum of unit fractions.” So, in the case of five-sixths, you would identify the unit fraction as one-sixth, and you would have 5 of those one-sixths. So, your unit fraction is one-sixth, and you're going to iterate it or copy it or repeat it five times.
Mike: I can hear the parallels between the way you described this work with whole numbers. I have one-fourth, and I've duplicated or copied that five times, and that's what five-fourths is. It feels really helpful to see the through line between how we think about helping kids think about composite numbers and multiplying with whole numbers, to what you just described with unit fractions.
Beth: Yeah, and even the language that language infractions is similar, too. So, you talk about that 5 one-fourths. You decompose the five-fourths into 5 of the one-fourths, or you recompose those 5 one-fourths. This is a fourth-grade standard. You recompose those 5 one-fourths into 3 one-fourths or three-fourths and 2 one-fourths or two-fourths. So, even reading a fraction like seven-eighths says 7 one-eighths, helps to really understand what that seven-eighths means, and it keeps you from reading it as seven out of eight. Because when you read a fraction as seven out of eight, it sounds like you're talking about a whole number over another whole number. And so again, that connection to the composite unit in multiplication extends to that composite unit or that unit fraction or unitizing in multiplication. And really, even when we talk about multiplying fractions, we talk about multiplying, say, a whole number times a fraction “5 times one-fourth.” That would be the same as saying, “I'm going to repeat one-fourth five times,” as opposed to, we were told, “Put a 1 under the 5 and multiply across the numerator and multiply across the denominator.” But that didn't help kids really understand what was happening.
Mike: So, this progression of ideas that we've talked about from multiplication to fractions, what you've got me thinking about is, what does it mean to think about unitizing with younger kids, particularly perhaps, kids in kindergarten, first or second grade? I'm wondering how or what you think educators could do to draw out the big ideas about units and unitizing with students in those grade levels?
Beth: Well, really we don't expect kindergartners to strictly unitize because it's a relatively abstract idea. The big focus in kindergarten is for a student to understand four means 4, four 1s, and 7 means seven 1s. But where we do unitize is in the use of our models in early grades. In kindergarten, the use of a five-frame or a ten-frame. So, let's use the ten-frame to count by tens: 10, 20, 30. And then, how many ten-frames did it take us to count to 30? It took 3. There's the beginning of your unitizing idea. The idea that we would say, “It took 3 of the ten-frames to make 30” is really starting to plant that idea of unitizing 3 can mean 30. And in first grade, when we start to expose kids to coin values, time, telling time, one of the examples we use is, “Whenever was 1 minus 1, 59?” And that was, “When you read for one hour and your friend read for 1 minute less than you, how long did they read?”
So, all time is really a unitizing idea. So, all measures, measure conversion, time, money, and the big one in first grade is base 10. And first grade and second grade [have] the opportunity to solidify strong base 10 so that when kids enter third grade, they've already developed a concept of unitizing within the base 10 system. In first grade, the idea that in a number like 78, the 7 is actually worth more than the 8, even though at face value, the 7 seems less than the 8. The idea that 7 is greater than the 8 in a number like 78 is unitizing. In second grade, when we have a number like 378, we can unitize that into 307 tens and 8 ones, or 37 tens and 8 ones, and there's your re-unitizing. And that's actually a standard in second grade. Or 378 ones. So, in first and second grade, really what teachers have to commit to is developing really strong, flexible base 10 understanding. Because that's the first place kids have to struggle with this idea of the face value of a number isn't the same as the place value of a number.
Mike: Yeah, yeah. So, my question is, would you describe that as the seeds of unitizing? Like conserving? That's the thing that popped into my head, is maybe that's what I'm actually starting to do when I'm trying to get kids to go from counting each individual 1 and naming the total when they say the last 1.
Beth: So, there are some early number concepts that need to be solidified for kids to be able to unitize, right? So, conservation is certainly one of them. And we work on conservation all throughout elementary school. As numbers get larger, as they have different features to them, they're more complex. Conservation doesn't get fixed in kindergarten. It's just pre-K and K are the places where we start to build that really early understanding with small quantities. There's cardinality, hierarchical inclusion, those are all concepts that we focus on and develop in the earliest grades that feed into a child's ability to unitize. So, the thing about unitizing that happens in the earliest grades is it's pretty informal. In pre-K and K, you might make piles of 10, you might count quantities. Counting collections is something we talk a lot about, and we talk a lot about the importance of counting in early math instruction actually all the way up through, but particularly in early math. And let's say you had a group of kids, and they were counting out piles of, say, 45 things, and they put them in piles of 10 and then a pile of 5, and they were able to go back and say, “Ten, 20, 30, 40 and 5.”
So, there's a lot that's happening there. So, one is, they're able to make those piles of 10, so they could count to 10. But the other one is, they have conservation. And the other one is, they have a rope-count sequence that got developed outside of this use of that rope-count sequence, and now they're applying that. So, there's so many balls in the air when a student can do something like that. The unitizing question would be, “You counted 45 things. How many piles of 10 did you have?” There's your unitizing question. In kindergarten, there are students—even though we say it's not something we work on in kindergarten—there are certainly students who could look at that and say, “Forty-give is 4 piles of 10 and 5 extra.” So, when I say we don't really do it in kindergarten, we have exposure, but it's very relaxed. It becomes a lot more significant in first and second grade.
Mike: You said earlier that teachers in first and second grade really have to commit to building a flexible understanding of base 10. What I wanted to ask you is, how would you describe that? And the reason I ask is, I also think it's possible to build an inflexible understanding of base 10. So, I wonder how you would differentiate between the kind of practices that might lead to a relatively inflexible understanding of base 10 versus the kind of practices that lead to a more flexible understanding.
Beth: So, I think counting collections. I already said we talk a lot about counting collections and the primary training. Having kids count things and make groups of 10, focus on your 10 and your 5. We tell kindergarten teachers that the first month or two of school, the most important number you learn is 5. It's not 10, because our brain likes 5, and we can manage 5 easily. Our hand is very helpful. So, building that unit of 5 toward putting two 5s together to make a 10. I mean, I have a 3-year-old granddaughter, and she knows 5, and she knows that she can hold up both her hands and show me 10. But if she had to show me 7, she would actually start back at 1 and count up to 7. So, taking advantage of those units that are baked in already and focusing on them helps in the earliest grades.
And then really, I like materials to go into kids’ hands where they're doing the building. I feel like second grade is a great time to hand kids base 10 blocks, but first grade is not. And first-grade kids should be snapping cubes together and building their own units, because the more they build their own units of 5 or 10, the more it's meaningful and useful for them. The other thing I'm going to say, and Bridges has this as a tool, which I really like, is they have dark lines at their 5s and 10s on their base 10 blocks. And that helps, even though people are going to say, “Kids can tell you it's a hundred,” they didn't build it. And so, there's a leap of faith there that is an abstraction that we take for granted. So, what we want is kids using those manipulatives in ways that they constructed those groupings, and that helps a lot. Also, no operations for addition and subtraction. You shouldn't be adding and subtracting without using base 10. So, adding and subtracting on a number line helps you practice not just addition and subtraction, but also base 10. So, because base 10’s so important, it could be taught all year long in second grade with everything you do. We call second grade the sweet spot of math because all the most important math can be taught together in second grade.
Mike: One of the things that you made me think about is something that a colleague said, which is this idea that 10 is simultaneously 10 ones and one unit of 10. And I really connect that with what you said about the need for kids to actually, physically build the units in first grade.
Beth: What you just said, that's unitizing. I can call this 10 ones, and I can call this 1, worth 10. And it's more in face in the earliest grades because we often are very comfortable having kids make piles of 10 things or seeing the marks on a base 10 block, say. Or snapping 10 Unifix cubes together, 5 red and 5 yellow Unifix cubes or something to see those two 5s inside that unit of 10. And then also there's your math hand, your fives and your tens and your ten-frames are your fives and your tens. So, we take full advantage of that. But as kids get older, the math that's going to happen is going to rely on kids already coming wired with that concept. And if we don't push it in those early grades by putting your hands on things and building them and sketching what you've just built and transferring it to the pictorial and the abstract in very strategic ways, then you could go a long way and look like you know what you're doing—but don't really. Base 10 is one of those ways we think, because kids can tell you the 7 is in the tens place, they really understand. But the reality is that's a low bar, and it probably isn't an indication a student really understands. There's a lot more to ask.
Mike: Well, I think that's a good place for my next question, which is to ask you what resources OGAP has available, either for someone who might participate in the training, other kinds of resources. Could you just unpack the resources, the training, the other things that OGAP has available, and perhaps how people could learn more about it or be in touch if they were interested in training?
Beth: Sure. Well, if they want to be in touch, they can go to ogapmathllc.com, and that's our website. And there's a link there to send us a message, and we are really good at getting back to people. We've written books on each of our four content strands. The titles of all those books are “A Focus on … .” So, we have “A Focus on Addition and Subtraction,” “A Focus on Multiplication and Division,” “A Focus on Fractions,” “A Focus on Ratios and Proportions,” and you can buy them on Amazon. Our progressions are readily available on our website. You can look around on our website, and all our progressions are there so people can have access to those. We do training all over. We don't do any open training. In other words, we only do training with districts who want to do the work with more than just one person. So, we contract with districts and work with them directly. We help districts use their math program. Some of the follow-up work we've done is help them see the possibilities within their program, help them look at their program and see how they might need to add more. And once people come to training, they have access to all our resources, the item bank, the progressions, the training, the book, all that stuff.
Mike: So, listeners, know that we're going to add links to the resources that Beth is referencing to the show notes for this particular episode. And, Beth, I want to just say thank you so much for this really interesting conversation. I'm so glad we had a chance to talk with you today.
Beth: Well, I'm really happy to talk to you, so it was a good time.
Mike: Fantastic.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
© 2024 The Math Learning Center | www.mathlearningcenter.org
Season 3 | Episode 3 – Choice as a Foundation for Student Engagement
Guest: Drs. Zandra de Araujo and Amber Candela
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 pre-packaged 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. 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 forward. We give kids the problem, and we want them to come up with the answer. 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 seven times eight?” 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 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 with the class.” So, there have been lots of studies done on the types of questions that teachers ask: tend to be closed, answer-focused, like single-calculation kind of questions. So, “What is the answer? Who got this?” You know, that kind of thing? Instead, you can give students choices, and I think a lot of teachers have done something akin to this with sentence starters or things. But you can also just say, instead of a sentence starter to say what your answer is, “I agree with X because of Y.”
You can also say, “You can share an incorrect answer that you know is wrong because you did it, and it did not work out. You can also share where you got stuck because that's valuable information for the class to have.” You could also say, “I don't want to really share my thinking, my solution because it's not done, but I'll show you my diagram.” And so, “Let me show you a visual.” And just plop it up on the screen. So, there are a lot of different things you could share a question that you have because you’re not sure, and it's just a related question. Instead of always sharing answers, let kids open up what they may choose to share, and you'll get more kids sharing. Because answers are kind of scary because you're expecting a correct answer often. And so, when you share and open up, then it's not as scary. And everybody has something to offer because they have a choice that speaks to them.
Amber: And kids don't want to be wrong. People don't want to be wrong. “I don't want to give you a wrong answer.” And we went to the University of Georgia together, but Les Steffe always would say, “No child is ever wrong. They're giving you an answer with a purpose behind what that answer is. They don't actually believe that's a wrong answer that they're giving you.” And so, if you open up the space … And teachers say, “We want spaces to be safe, we want kids to want to come in and share.” But are we actually structuring spaces in that way? And so, some of the ideas that we're trying to come up with, we're saying that “We actually do value what you're saying when you choose to give us this. It's your choice of offering it up and you can say whatever it is you want to say around that,” but it's not as evaluative or as high stakes as trying to get the right answer and just like, “Am I right? Did I get it right?” And then what the teacher might say after that.
Zandra: I would add on that kids do like to give wrong answers if that's what you're asking for. They don't like to give wrong answers if you're asking for a right one and they're accidentally wrong. So, I think back to my first suggestion: If you ask for a wrong answer and they know it's wrong, they're likely to chime right in because the right answer is the wrong answer, and there are multiple, infinite numbers of them.
Mike: You know, it makes me think there's this set of ideas that we need to normalize mistakes as being productive things. And I absolutely agree with that. I also think that when you're asking for the right answer, it's really hard to kind of be like, “Oh, my mistake was so productive.” On the other hand, if you ask for an error or a place where someone's stuck, that just feels different. It feels like an invitation to say, “I've actually been thinking about this. I'm not there. I may be partly there. I'm still engaged. This is where I'm struggling.” That just feels different than providing an answer where you're just like, “Ugh.” I'm really struck by that.
Zandra: Yeah, and I think it's a culture thing. So, a lot of teachers say to me that “it's hard to have kids work in groups because they kind of just tell each other the answers.” But they're modeling what they experience as learners in the classroom. “I often get told the answers,” that's the discourse that we have in the classroom. So, if you open up the discourse to include these things like, “Oh, I'm stuck here. I'm not sure where to go here.” They get practice saying, “Oh, I don't know what this is. I don't know how to go from here.” Instead of just going to the answer. And I think it'll spread to the group work as well.
Mike: It feels like there's value for every other student in articulating, “I'm certain that this one is wrong, and here's why I know that.” There's information in there that is important for other kids. And even the idea of “I'm stuck here,” right? That's really a great formative assessment opportunity for the teacher. And it also might validate some of the other places where kids are like, “Yeah. Me, too.”
Zandra: Uh-hm.
Amber: Right, absolutely.
Mike: What's next, my friend?
Amber: I remember very clearly listening to Zandra present about choice, her idea of choice of feedback. And this was very powerful to me. I had never thought about asking my students how they wanted to receive the feedback I'd be giving them on the problems that they solved. And this idea of students being able to turn something in and then say, “This is how I'd like to receive feedback” or “This is the feedback I'd like to receive,” becomes very powerful because now they're the ones in charge of their own learning. And so much of what we do, kids should get to say, “This is how I think that I will grow better, is if you provide this to me.” And so, having that opportunity for students to say, “This is how I'll be a better learner if you give it to me in this way. And I think if you helped me with this part that would help the whole rest of it.” Or “I don't actually want you to tell me the answer. I am stuck here. I just need a little something to get me through. But please don't tell me what the answer is because I still want to figure it out for myself.” And so, allowing kids to advocate for themselves and teaching them how to advocate for themselves to be better learners; how to advocate for themselves to learn and think about “What I need to learn this material and be a student or be a learner in society” will just ultimately help students.
Zandra: Yeah, I think as a student, I don't like to be told the answers. I like to figure things out, and I will puzzle through something for a long time. But sometimes I just want a model or a hint that'll get me on the right path, and that's all I need. But I don't want you to do the problem for me or take over my thinking. If somebody asked me, “What do you want?” I might say like, “Oh, a model problem or something like that.” But I don't think we ask kids a lot. We just do whatever we think as an adult. Which is different, because we're not learning it for the first time. We already know what it is.
Mike: You're making me think about the range of possibilities in a situation like that. One is I could notice a student who is working through something and just jump in and take over and do the problem for them essentially and say, “Here, this is how you do it.” Or I guess just let them go, let them continue to work through it. But potentially there could be some struggle, and there might be some frustration. I am really kind of struck by the fact that I wonder how many of us as teachers have really thought about the kinds of options that exist between those two far ends of the continuum. What are the things that we could offer to students rather than just “Let me take over” or productive struggle, but perhaps it's starting to feel unproductive? Does that make sense?
Zandra: Yeah, I think it does. I mean, there are so many different ways. I would ask teachers to re-center themselves as the learner that's getting feedback. So, if you have a principal or a coach coming into your room, they've watched a lesson, sometimes you're like, “Oh, that didn't go well. I don't need feedback on that. I know it didn't go well, and I could do better.” But I wonder if you have other things that you notice just being able to take away a part that you know didn't go well. And you're like, “Yep, I know that didn't go well. I have ideas for improving it. I don't really want to focus on that. I want to focus on this other thing.” Or “I've been working really hard on discourse. I really want feedback on the student discourse when you come in.” That's really valuable to be able to steer it—not taking away the other things that you might notice, but really focusing in on something that you've been working on is pretty valuable. And I think kids often have these things that maybe they haven't really thought about a lot, but when you ask them, they might think about it. And they might grow this repertoire of things that they're kind of working on personally.
Amber: Yeah, and I just think it's getting at, again, we want students to come out of situations where they can say, “This is how I learn” or “This is how I can grow,” or “This is how I can appreciate math better.” And by allowing them to say, “It'd be really helpful if you just gave me some feedback right here” or “I'm trying to make this argument, and I'm not sure it's coming across clear enough,” or “I'm trying to make this generalization, does it generalize?” We're also maybe talking about some upper-level kids, but I still think we can teach elementary students to advocate for themselves also. Like, “Hey, I try this method all the time. I really want to try this other method. How am I doing with this? I tried it. It didn't really seem to work, but where did I make a mistake? Could you help me out with that? Because I think I want to try this method instead.” And so, I think there are different ways that students can allow for that. And they can say: “I know this answer is wrong. I'm not sure how this answer is wrong. Could you please help me understand my thinking or how could I go back and think about my thinking?”
Zandra: Yeah. And I think when you said upper level, you meant upper grades.
Amber: Yes.
Zandra: I assume.
Amber: Yes.
Zandra: OK, yeah. So, for the lower-grade-level students, too, you can still use this. They still have ideas about how they learn and what you might want to follow up on with them. “Was there an easier way to do this? I did all these hand calculations and stuff. Was there an easier way?” That's a good question to ask. Maybe they've thought about that, and they were like, “That was a lot of work. Maybe there was an easier way that I just didn't see?” That'd be pretty cool if a kid asked you that.
Mike: Or even just hearing a kid say something like, “I feel really OK. I feel like I had a strategy. And then I got to this point, and I was like, ‘Something's not working.’” Just being able to say, “This particular place, can you help me think about this?” That's the kind of problem-solving behavior that we ultimately are trying to build in kids, whether it's math or just life.
Amber: Right, exactly. And I need, if I want kids to be able … because people say, “I sometimes just want a kid to ask a question.” Well, we do need to give them choice of the question they ask. And that's where a lot of this comes from is, what is your goal as a teacher? What do you want kids to have choice in? If I want you to have choice of feedback, I'm going to give you ideas for what that feedback could be, so then you have something to choose from.
Mike: OK, so we've unpacked quite a few ideas in the last bit. I wonder if there are any caveats or any guidance that you would offer to someone who's listening who is maybe thinking about taking up some of these practices in their classroom?
Zandra: Oh, yeah. I have a lot. Kids are not necessarily used to having a lot of choice and autonomy. So, you might have to be gentle building it in because it's overwhelming. And they actually might just say, “Just tell me what to do,” because they're not used to it. It's like when you're get a new teacher and they're really into explaining your thinking, and you've never had to do that. Well, you've had 10 years of schooling or however many years of schooling that didn't involve explaining your thinking, and now, all of a sudden, “I'm supposed to explain my thinking. I don't even know what that means. What does that look like? We never had to do that before.”
So maybe start small and think about some things like, “Oh, you can choose a tool or two that helps you with this problem. So, you can use a multiplication table, or you can use a calculator or something to use. You can choose. There are all these things out. You can choose a couple of tools that might help you.” But start small. And you can give too many choices. There's like choice overload. It's like when I go on Amazon, and there are way too many reviews that I have to read for a product, and I never end up buying anything because I’ve read so many reviews. It's kind of like that. It could get overwhelming. So purposeful, manageable numbers of choices to start out with is a good suggestion.
Amber: And also, just going back to what Zandra said in the beginning, is making sure you have a purpose for the choice. And so, if you just are like, “Oh, I'm having choice for choice's sake.” Well, what is that doing? Is that supporting the learning, the mathematics, the number sense, the conceptual understanding, and all of that? And so, have that purpose going in and making sure that the choices backtrack to that purpose.
Zandra: Yeah. And you could do a little choice inventory. You could be like, “Huh, if I was a student of my own class today, what would I have gotten to choose? If anything? Did I get to choose where I sat, what utensil I used? What type of paper did I use? Which problems that I did?” Because that’s a good one. All these things. And if there's no choice in there, maybe start with one.
Mike: I really love that idea of a “choice inventory.” Because I think there's something about really kind of walking through a particular day or a particular lesson that you're planning or that you've enacted, and really thinking about it from that perspective. That's intriguing.
Zandra: Yeah, because really, I think once you're aware of how little choice kids get in a day … As an adult learner, who has presumably a longer attention span and more tolerance and really likes math, I've spent my whole life studying it. If I got so little choice and options in what I did, I would not be a well-behaved, engaged student. And I think we need to remember that when we're talking about little children.
Mike: So, last question, is there research in the field or researchers who have done work that has informed the kind of thinking that you have about choice?
Zandra: Yeah, I think we're always inspired by people who come before us, so it's probably an amalgamation of different things. I listen to a lot of podcasts, and I read a lot of books on behavioral economics and all kinds of different things. So, I think a lot of those ideas bleed into the work in math education. In terms of math education, in particular, there have been a lot of people who have really influenced me, like Marian Small's work with parallel tasks and things like that. I think that's a beautiful example of choice. You give multiple options for choice of challenge and see which ones the students feel like is appropriate instead of assigning them competence ahead of time. So, that kind of work has really influenced me.
Amber: And then just, our team really coming together; Sam Otten and Zandra and their ideas and collaborating together. And like you mentioned earlier, that Barb Dougherty article on the different types of questions has really been impactful. More about opening up questions, but it does help you think about choice a little bit better.
Mike: I think this is a great place to stop. Zandra, Amber, thank you so much for a really eye-opening conversation.
Zandra: Thank you for having us.
Amber: Thanks for having us.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org
Season 3 | Episode 2 – Responsive Curriculum
Guest: Dr. Corey Drake
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.
Mike: 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.
Mike: 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.
Corey: 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.
Corey: 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 responsibly 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 brings ideas about how children learn and how we learn and teach mathematics. And then, to use that well and responsibly, the teacher then needs to work in ways, make decisions to enact that responsibly. 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 Julia Aguirre and her colleagues.
Mike: I think for me, one of the things that hit home was thinking about this idea that there's a mathematical goal and that goal is actually part of a larger trajectory that the curriculum's designed around. And when I've thought about differentiation in the past, what I was really thinking about was replacement that fundamentally altered the instructional goal. And I think the challenge in this work is to say, “Am I clear on the instructional goal? And do the things that I'm considering actually maintain that for kids or are they really replacing them or changing them in a way that will alter or impact the trajectory?”
Corey: I think that's such a critical point. And it's not easy work. It's not always clear even in materials that have a stated learning goal or learning target for a session. There's still work to do for the teacher to say, “What is the mathematical goal? Not the activity, not the task, but what is the goal? What is the understanding I'm trying to support for my students as they engage in this activity?” And so, you're right. I think the first thing is, teachers have to be super clear about that because all the rest of the decisions flow from understanding, “What is the goal of this activity, what are the understandings that I am trying to develop and support with this session? And then I can make decisions that are enhancing and providing access to that goal, but not replacing it. I'm not changing the goal for any of my students. I'm not changing the goal for my whole group of students. Instead, I'm recognizing that students will need different ways into that mathematics. Students will need different kinds of supports along the way. But all of them are reaching toward or moving toward that mathematical goal.”
Mike: Yeah. When I think about some of the options, like potentially, number choice; if I'm going to try to provide different options in terms of number choice, is that actually maintaining a connection to the mathematical goal, or have I done something that altered it? Another thing that occurs to me is the resources that we share with kids for representation, be it manipulatives or paper, pencil, even having them talk about it—any of those kinds of choices. To what extent do they support the mathematical goal, or do they veer away from it?
Corey: Yeah, absolutely. And there are times when different numbers or different tools or different models will alter the mathematical goal because part of the mathematical goal is to learn about a particular tool or a particular representation. And there are other times when having a different set of numbers or a different set of tools or models will only enhance students' access to that mathematical goal because maybe the goal is understanding something like two-digit addition and developing strategies for two-digit addition. Well then, students could reach that goal in a lot of different ways. And some students will be working just with decade numbers, and some students will be working with decades and ones, and some students will need number pieces, and others will do it mentally. But if the goal is developing strategies, developing your understanding of two-digit addition, then all of those choices make sense, all of those choices stay aligned with the goal.
Corey: But if the goal is to understand how base ten pieces work, then providing a different model or telling students they don't need to use that model would, of course, fundamentally alter the goal. So, this is why it's so critically important that we support teachers in understanding, making sense of the goal, figuring out how do they figure that out. How do you open a set of curriculum materials, look at a particular lesson, and understand what the mathematical goal of that lesson is? And it's not as simple as just looking for the statement of the learning goal and the learning target. But it's really about, “What are the understandings that I think will develop or are intended to develop through this session?”
Mike: I feel like we should talk a little bit about context, because context is such a powerful tool, right? If you alter the context, it might help kids surface some prior knowledge that they have. What I'm thinking about is this task that exists in Bridges where we're having kids look at a pet store where there are arrays of different sorts and kinds of dog foods or dog toys or cat toys. And I remember an educator saying to me, “I wonder if I could shift the context.” And the question that I asked her is, “If you look at this image that we're using to launch the task, what are the particular parts of that image that are critical to maintain if you're going to replace it with something that's more connected to your students?”
Corey: Connecting to your students, using context to help students access the mathematics, is so important and such an empowering thing for teachers and students. But you're asking exactly the right question. And of course, that all relates to, “What's the mathematical goal?” Again. Because if I know that, then I can look for the features of the context that's in the textbook and see the ways in which that context was designed to support students in reaching that mathematical goal. But I can also look at a different context that might be more relevant to my students, that might provide them better access to the mathematics. And I can look at that context through the lens of that mathematical goal and see, “Does this context also present the kinds of features that will help my students understand and make sense of the mathematical goal?” And if the answer is yes, and if that context is also then more relevant to my students or more connected to their lives, then great. That's a wonderful adaptation. That's a great example of responsive curriculum use. If now I'm in a context that's distracting or leading me away from the mathematical goal, that's where we run into adaptations that are less responsive and less productive.
Mike: Well, and to finish the example, the conversation that this led to with this educator was she was talking about looking for bodegas in her neighborhood that her children were familiar with, and we end up talking quite a bit about the extent to which she could find images from the local bodega that had different kinds of arrays. She was really excited. She actually did end up finding an image, and she came back, and she shared that this really had an impact on her kids. They felt connected to it, and the mathematical goal was still preserved.
Corey: I love that. I think that's a great example. And I think the other thing that comes up sometimes when we present these ideas, is maybe you want to find a different context that is more relevant to your students that they know more about. Sometimes you might look at a context that's presented in the textbook and say, “I really love the mathematical features here. I really see how knowing something about this context could help my students reach the mathematical goal, but I'm going to have to do some work ahead of time to help my students understand the context, to provide them some access to that, to provide them some entry points.” So, in your example, maybe we're going to go visit a pet store. Maybe we're going to look at images from different kinds of stores and notice how things are arranged on shelves, and in arrays, and in different combinations. So, I think there are always a couple of choices. One is to change the context. One is to do some work upfront to help your students access the context so that they can then use that context to access the mathematics. But I think in both cases, it's about understanding the goal of the lesson and then understanding how the features of the context relate to that goal.
Mike: Let's shift and just talk about the second notion, this idea of opening up space for students' voice or for sensemaking when you're using curriculum. For me at least, I often try to project ideas for practice into a mental movie of myself in a classroom. And I wonder if we could work to help people imagine what this idea of opening space for voice or sensemaking might look like.
Corey: I think a lot of times those opportunities for opening up voice and choice and sensemaking are not in the direct, action steps or the direct instructions to teachers within the lesson, but they're kind of in the in-between. So, “I know I need to introduce this idea to my students, but how am I going to do that? What is that going to look like? What is that going to sound like? What are students going to be experiencing?” And so, asking yourself that question as the lesson plays out is, I think, where you find those opportunities to open up that space for student voice and choice. It's often about looking at that and saying, “Am I going to tell students this idea? Or am I going to ask them? Are students going to develop their strategy and share it with me or turn it in on a piece of paper? Or are they going to turn and talk to a partner? Are they going to share those ideas with a small group, with a whole group? What are they going to listen for in each other's strategies? How am I going to ask them to make connections across those strategies? What kinds of tools am I going to make available to them? What kinds of choices are they going to have throughout that process?”
Corey: And so, I think it's having that mental movie play through as you read through the lesson and thinking about those questions all the way through. “Where are my students going to have voice? How are they going to have choice? How are students going to be sensemaking?” And often thinking about, “Where can I step back, as the teacher, to open up that space for student voice or student choice?”
Mike: You're making me think about a couple things. The first one that really jumped out was this idea that part of voice is not necessarily always having the conversation flow from teacher to student, but having a turn and talk, or having kids listen to and engage with the ideas that their partners are sharing is a part of that idea that we're creating space for kids to share their ideas, to share their voice, to build their own confidence around the mathematics.
Corey: Absolutely. I think that, to me, is the biggest difference I see when I go into different classrooms. “Whose voice am I hearing most often? And who's thinking do I know about when I've spent 20 minutes in a classroom?” And there are some classrooms where I know a lot about what the teacher's thinking. I don't know a lot about what the students are thinking. And there are other classrooms where I can tell you something about the thinking of every one of the students in that room after 10 minutes in that classroom because they're constantly turning and talking and sharing their ideas. Student voice isn't always out loud either, right? Students might be sharing their ideas in writing, they might be sharing their ideas through gestures or through manipulating models, but the ideas are communicating their mathematical thinking. Really, student communication might be an even better way to talk about that because there are so many different ways in which students can express their ideas.
Mike: Part of what jumped out is this notion of, “What do you notice? What do you wonder?” Every student can notice, every student can wonder. So, if you share a context before you dive right into telling kids what's going to happen, give them some space to actually notice and wonder about what's going on, generate questions, that really feels like something that's actionable for folks.
Corey: I think you could start every activity you did with a, “What do you notice? What do you wonder?” Students always have ideas. Students are always bringing resources and experiences and ideas to any context, to any task, to any situation. And so, we can always begin by accessing those ideas and then figuring out as teachers how we might build on those ideas, where we might go from there. I think even more fundamentally is just this idea that all students are sensemakers. All students bring brilliance to the classroom. And so ,what we need to do is just give them the opportunities to use those ideas to share those ideas, and then we as teachers can build on those ideas.
Mike: Before we close this conversation, I want to spend time talking about responsive curriculum use being a vehicle for opening up space for equity-based practice. Personally, this is something that you've helped me find words for. There were some ideas that I had an intuitive understanding of. But I think helping people name what we mean when we're talking about opening space for equity-based practice is something that we might be able to share with folks right now. Can you share how a teacher might take up this idea of creating space for equity-based practice as they're looking at lessons or even a series of lessons?
Corey: Yeah, absolutely. And I think student voice and choice are maybe outcomes of equity-based practices. And so, in a similar way, I think teachers can begin by looking at a lesson or a series of lessons and thinking about those spaces and those decisions in between the action steps. And again, asking a series of questions. The equity-based practices aren't a series of steps or rules, but really like a lens or a series of questions that as a teacher, you might ask yourself as you prepare for a lesson. So, “Who is being positioned as mathematically capable? Who's being positioned as having mathematically important ideas? Are all of my students being positioned in that way? Are some of my students being marginalized? And if some of my students are being marginalized, then what can I do about that? How could I physically move students around so that they're not marginalized? How can I call attention to or highlight a certain student's ideas without saying that those ideas are the best or only ideas? But saying, ‘Look, this student, who we might not have recognized before as mathematically capable and brilliant, has a really cool idea right now.’”
Corey: You and I have both seen video from classrooms where that's done brilliantly by these small moves that teachers can make to position students as mathematically brilliant, as having important or cool or worthwhile ideas, valuable ideas to contribute. So, I think it's those kinds of decisions that make such a difference. Those decisions to affirm learners' identities. Those aren't big changes in how you teach. Those are how you approach each of those interactions minute by minute in the classroom. How do you help students recognize that they are mathematicians, that they each bring valuable ideas to the classroom? And so, it's more about those in-between moments and those moments of interaction with students where these equity-based practices come to life.
Mike: You said a couple things that I'm glad that you brought out, Corey. One of them is this notion of positioning. And the other one that I think is deeply connected is this idea of challenging places where kids might be marginalized. And I think one of the things that I've been grappling with lately is that there's a set of stories or ideas and labels that often follow kids. There are labels that we affix to kids within the school system. There are stories that exist around the communities that kids come from, their families. And then there are also the stories that kids make up about one another, the ideas that carry about, “Who's good at math? Who's not? Who has ideas to share? Who might I listen to, and who might I not?” And positioning, to me, has so much opportunity as a practice to help press back against those stories that might be marginalizing kids.
Corey: I think that's such an important point. And I think, along with that is the recognition that this doesn't mean that you, as the individual teacher, created those stories or believed stories or did anything to perpetuate those stories—except if you didn't act to disrupt them. Because those stories come from all around us. We hear Pam Seda and Julia Aguirre and people like that saying, “They're the air we breathe. They're the smog we live in. Those stories are everywhere. They're in our society, they're in our schools, they're in the stories students tell and make up about each other.” And so, the key to challenging marginality is not to say, “Well, I didn't tell that story, I don't believe that story. But those stories exist, and they affect the children in my classroom, so what am I going to do to disrupt them? What am I going to do? Because I know the stories that are told about certain students, even if I'm not the one telling them, I know what those stories are. So how am I going to disrupt them to show that the student who the story or the labels about that student are, that they are not as capable, or they are behind or struggling or ‘low students.’ What am I going to do to disrupt that and help everyone in our classroom community see the brilliance of that child, understand that that child has as much to contribute as anybody else in the math classroom?” And that's what it means to enact equity-based practices.
Mike: You're making me think about an interview we did earlier this year with Peter Liljedahl, and he talked about this idea. He was talking about it in the context of grouping, but essentially what he was saying is that kids recognize the stories that are being told in a classroom about who's competent and who's not. And so, positioning, in my mind, is really thinking about—and I've heard Julia Aguirre say it this way—“Who needs to shine? Whose ideas can we bring to the center?” Because what I've come to really have a better understanding of, is that the way I feel about myself as a mathematician and the opportunities that exist within a classroom for me to make sense of math, those are really deeply intertwined.
Corey: Yes, yes, absolutely. We are not focusing on marginality or identity just because it makes people feel good, or even just because it's the right thing to do. But actually in the math classroom, your identity and the expectations and the way you're positioned in that classroom fundamentally affect what you have opportunities to learn and the kinds of math you have access to. And so, we will do this because it's the right thing to do and because it supports math learning for all students. And understanding the role of identity and marginality and positioning in student learning is critically important.
Mike: You're making me think about a classroom that we visited earlier this year, and it was a really dynamic math discussion. There was a young man, I'll call him David, and he was in a multilingual classroom. And I'm thinking back on what you said. At one point you said, “I can go into a classroom, and I can have a really clear idea of what the teacher understands, and perhaps less so with the kids.” In this case, I remember leaving thinking, “I really clearly understand that David has a deep conceptual understanding of the mathematics.” And the reason for that was, he generally volunteered to answer every single question. And it was interesting. It's not because the educator in the classroom was directing all of the questions to him, but I really got the sense that the kids, when the question was answered, were to almost turn their bodies because they knew he was going to say something. And it makes me think David is a kid who, over time, not necessarily through intention, but through the way that status works in classrooms, he was positioned as someone who really had some ideas to share, and the kids were listening. The challenge was, not many of them were talking. And so, the question is, “How do we change that? Not because anyone has any ill intent toward those other children, but because we want them to see themselves as mathematicians as well.”
Corey: Yeah, absolutely. And that is part of what's tricky about this is that that's so important is that I think for many years we've talked about opening up the classroom for student talk and student discourse. And we do turn and talks, and we do think pair shares. And we've seen a lot of progress, I think, in seeing those kinds of things in math classrooms. And I think the next step to that is to do those with the kind of intentionality and awareness that you were just demonstrating there; which is to say, “Well, who's talking and how often are they talking? And what sense are people making of the fact that David is talking so much? What sense are they making? What stories are they telling about who David is as a mathematician? But also who they are as mathematicians. And what does it mean to them that even though there are lots of opportunity for students talk in that classroom, it’s dominated by one or maybe two students. And so, we have opened it up for student discourse, but we have more work to do. We have more work to say, “Who's talking, and what sense are they making, and what does that look like over time? And how is mathematical authority distributed? How is participation distributed across the class? And, in particular, with intentionality toward disrupting some of those narratives that have become entrenched in classrooms and schools.”
Mike: I think that's a great place for us to stop. I want to thank you again for joining us, Corey. It was lovely to have you back on the podcast.
Corey: Thanks. It was great to be with you.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
© 2024 The Math Learning Center | www.mathlearningcenter.org
Season 3 | Episode 1 – Grouping Practices That Promote Efficacy and Knowledge Transfer
Guest: Dr. Peter Liljedahl
Mike Wallus: We know from research that student collaboration can have a powerful impact on learning. That said, how we group students for collaboration matters—a lot. Today we're talking with Dr. Peter Liljedahl, author of “Building Thinking Classrooms in Mathematics,” about how educators can form productive, collaborative groups in their classrooms.
Mike: Hello, Peter. Welcome to the podcast.
Peter Liljedahl: Thanks for having me.
Mike: So, to offer our listeners some background, you've written a book, called “Building Thinking Classrooms in Mathematics,” and I think it's fair to say that it's had a pretty profound impact on many educators. In the book, you address 14 different practices. And I'm wondering if you could weigh in on how you weigh the importance of the different practices that you addressed?
Peter: Well, OK, so, first of all, 14 is a big number that publishers don't necessarily like. When we first started talking with Corwin about this, they were very open. But I know if you think about books, if there's going to be a number in the title, the number is usually three, five or seven. It's sometimes eight—but 14 is a ridiculous number. They can't all be that valuable. What's important about the fact that it's 14, is that 14 is the number of core practices that every teacher does. That's not to say that there aren't more or less for some teachers, but these are core routines that we all do. We all use tasks. We all create groups for collaboration. We all have the students work somewhere. We all answer questions. We do homework, we assign notes, we do formative, summative assessment. We do all of these things. We consolidate lessons. We launch lessons.
Peter: These are sort of the building blocks of what makes our teaching. And through a lot of time in classrooms, I deduced this list of 14. Robert Kaplinsky, in one of his blog posts, actually said that he thinks that that list of 14 probably accounts for 95 percent of what happens in classrooms. And my research was specifically about, “How do we enact each of those 14 so that we can maximize student thinking? So, what kind of tasks get students to think, how can we create groups so that more thinking happens? How can we consolidate a lesson so we get more thinking? How can we do formative and summative assessments so the students are thinking more?” So, the book is about responding to those 14 core routines and the research around how to enact each of those to maximize thinking. Your question around which one is, “How do we put weight on each of these?”
Peter: They're all important. But, of course, they're not all equally impactful. Building thinking classrooms is most often recognized visually as the thing where students are standing at whiteboards working. And, of course, that had a huge impact on student engagement and thinking in the classroom, getting them from sitting and working at desks to getting them working at whiteboards. But in my opinion, it's not the most impactful. It is hugely impactful, but the one that actually makes all of thinking classroom function is how we form collaborative groups, which is chapter two. And it seems like that is such an inconsequential thing. “We've been doing groups for forever, and we got this figured out. We know how to do this. But … do we really? Do we really have it figured out?” Because my research really showed that if we want to get students thinking, then the ways we've been doing it aren't working.
Mike: I think that's a great segue. And I want to take a step back, Peter. Before we talk about grouping, I want to ask what might be an obvious question. But I wonder if we can talk about the “why” behind collaboration. How would you describe the value or the potential impact of collaboration on students' learning experiences?
Peter: That's a great question. We've been doing collaborative work for decades. And by and large, we see that it is effective. We have data that shows that it's effective. And when I say “we,” I don't mean me or the people I work with. I mean “we, in education,” know that collaboration is important. But why? What is it about collaboration that makes it effective? There are a lot of different things. It could be as simple as it breaks the monotony of having to sit and listen. But let's get into some really powerful things that collaboration does. Number one, about 25 years ago, we all were talking about metacognition. We know that metacognition is so powerful and so effective, and if we get students thinking about their thinking, then their thinking actually improves. And metacognition has been shown time and time again to be impactful in learning. Some of the listeners might be old enough to remember the days where we were actually trying to teach students to be metacognitive, and the frustration that that created because it is virtually impossible.
Peter: Being reflective about your thinking while you're thinking is incredibly hard to do because it requires you to be both present and reflective at the same time. We're pretty good at being present, and we're pretty good about reflecting on our experiences. But to do both simultaneously is incredibly hard to do. And to teach someone to do it is difficult. But I think we've also all had that experience where a student puts up their hand, and you start walking over to them, and just as you get there, they go, “Never mind.” Or they pick up their book, and they walk over to you, and just as they get to you, they just turn around and walk back. I used to tell my students that they're smarter when they're closer to me. But what's really going on there is, as they’ve got their hand up, or as they're walking across the room toward you as a teacher, they're starting to formulate their thoughts to ask a question.
Peter: They're preparing to externalize their thinking. And that is an incredibly metacognitive process. One of the easiest forms of metacognition, and one of the easiest ways to access metacognition, is just to have students collaborate. Collaborating requires students to talk. It requires them to organize their thoughts. It requires them to prepare their thinking and to think about their thinking for the purposes of externalization. It is an incredibly accessible way of creating metacognition in your classroom, which we already know is effective. So, that's one reason I think collaboration is really, really vital.
Peter: Another one comes from the work on register. So, register is the level of sophistication with which we speak about something. So, if I'm in a classroom, and I'm talking to kindergarten students, I set a register that is accessible to them. When I talk to my undergraduates, I use a different register. My master's students, my Ph.D. students, my colleagues, I'm using different registers. I can be talking about the same thing, but the level of sophistication with which I'm going to talk about those things varies depending on the audience. And as much as possible, we try to vary our register to suit the audience we have. But I think we've also all had that instructor who's completely incapable of varying their register, the one who just talks at you as if you're a third-year undergraduate when you're really a Great Eight student. And the ability to vary our register to a huge degree is going to define what makes us successful as a teacher. Can we meet our learners where they're at? Can we talk to them from the perspective that they're at? Now we can work at it, and very adept teachers are good at it. But even the best teachers are not as good at getting their register to be the same as students.
Peter: So, this is another reason collaboration is so effective. It allows students to talk and be talked to at their register, which is the most accessible form of communication for them. And I think the third reason that collaboration is so important is the difference between what I talk in my book about the difference between absolute and tentative knowledge. So, I'm going to make two statements. You tell me which one is more inviting to add a comment to. So, statement number one is, “This is how to do it, or this is what I did.” That's statement number one. Statement number two is, “I think that one of the ways that we may want to try, I'm wondering if this might work.” Which one is more inviting for you to contribute to?
Mike: Yes, statement number two, for many, many reasons, as I'm sitting here thinking about the impact of those two different language structures.
Peter: So, as teachers, we tend to talk in absolutes. The absolute communication doesn't give us anything to hold onto. It's not engaging. It's not inviting. It doesn't bring us into the conversation. It's got no rough patches—it's just smooth. But when that other statement is full of hedging, it's tentative. It's got so many rough patches, so many things to contribute to, things I want to add to, maybe push back at or push further onto. And that's how students talk to each other. When you put them in collaborative groups, they talk in tentative discourse, whereas teachers, we tend to talk in absolutes. So, students are always talking to each other like that. When we put them in collaborative groups, they're like, “Well, maybe we should try this. I'm wondering if this'll work. Hey, have we thought about this? I wonder if?” And it's so inviting to contribute to.
Mike: That's fascinating. I'm going to move a little bit and start to focus on grouping. So, in the book, you looked really closely at the way that we group students for collaborative problem-solving and how that impacts the way students engage in a collaborative effort. And I'm wondering if you could talk a little bit about the type of things that you were examining.
Peter: OK. So, you don't have to spend a lot of time in classrooms before you see the two dominant paradigms for grouping. So, the first one we tend to see a lot at elementary school. So, that one is called “strategic grouping.” Strategic grouping is where the teacher has a goal, and then they're going to group their students to satisfy that goal. So, maybe my goal is to differentiate, so I'm going to make ability groups. Or maybe my goal is to increase productivity, so I'm going to make mixed-ability groups. Or maybe my goal is to just have peace and quiet, so I'm going to keep those certain students apart. Whatever my goal is, I'm going to create the groups to try to achieve that goal, recognizing that how students behave in the classroom has a lot to do with who they're partnered with. So that's strategic grouping. It is the dominant grouping paradigm we see in elementary school.
Peter: By the time we get to high school, we tend to see more of teachers going, “Work with who you want.” This is called “self-selected groupings.” And this is when students are given the option to group themselves any way they want. And alert: They don't group themselves for academic reasons, they group themselves for social reasons. And I think every listener can relate to both of those forms of grouping. It turns out that both of those are highly ineffective at getting students to think. And ironically, for the exact same reason. We surveyed hundreds of students who were in these types of grouping settings: strategic grouping or self-selected groupings. We asked one question, “If you knew you were going to work in groups today, what is the likelihood you would offer an idea?” That was it. And 80 percent of students said that they were unlikely or highly unlikely to offer an idea, and that was the exact same, whether they were in strategic groupings or self-selected groupings. The data cut the same.
Mike: That's amazing, Peter.
Peter: Yeah, and it's for the same reason it turns out; that whether students were being grouped strategically or self-selected, they already knew what their role was that day. They knew what was expected of them. And for 80 percent of the students, their role is not to think. It's not to lead. Their role is to follow, right? And that's true whether they're grouping themselves socially, where they already know the social hierarchy of this group, or they're being grouped strategically. We interviewed hundreds of students. And after grade 3, every single student could tell us why they were in the group this teacher placed them in. They know. They know what you think of them. You're communicating very clearly what you think their abilities are through the way you group them, and then they live down to that expectation. So, that's what we were seeing in classrooms was that strategic grouping may be great at keeping the peace. And self-selected grouping may be fabulous for getting students to stop whining about collaboration. But neither of them was effective for getting students to think. In fact, they were quite the opposite. They were highly ineffective for getting students to think.
Mike: So, I want to keep going with this. And I think one of the things that stood out for me as I was reading is, this notion that regardless of the rationale that a teacher might have for grouping, there's almost always a mismatch between what the teacher's goals are and what the student's goals are. I wonder if you could just unpack this and maybe explain this a bit more.
Peter: So, when you do strategic grouping, do you really think the students are with the students that they want to be with? One of the things that we saw happening in elementary school was that strategic grouping is difficult. It takes a lot of effort to try to get the balance right. So, what we saw was teachers largely doing strategic grouping once a month. They would put students into a strategic group, and they would keep them in that group for the entire month. And the kids care a lot about who they're with, when you're going to be in a group for a month. And do you think they were happy with everybody that was in that group? If I'm going to be with a group of students for a month, I'd rather pick those students myself. So, they're not happy. You've created strategic groupings. And, by definition, a huge part of strategic grouping is keeping kids who want to be together away from each other.
Peter: They're not happy with that. Self-selected groupings, the students are not grouping themselves for academic reasons. They're just grouping themselves for social reasons so that they can socialize, so they talk, so they can be off topic, and all of these things. And yes, they're not complaining about group work, but they're also not being productive. So, the students are happy. But do you think the teacher's happy? Do you think the teacher looks out across that room and goes, “Yeah, there were some good choices made there.” No, nobody's happy, right? If I'm grouping them strategically, that's not matching their goals. That's not matching their social goals. When they're grouping themselves in self-selected ways, that's matching their social goals but not matching my academic goals for them. So, there's always going to be this mismatch. The teacher, more often than not, has academic goals. The students, more often than not, have social goals. There are some overlaps, right? There are students who are like, “I'm not happy with this group. I know I'm not going to do well in this group. I'm not going to be productive.” And there are some teachers who are going, “I really need this student to come out of the shell, so I need to get them to socialize more.” But other than that, by and large, our goals as teachers are academic in nature. The goals as students are social in nature. Mike: I think one of the biggest takeaways from your work on grouping, for me at least, was the importance of using random groups. And I have to admit, when I read that there was a part of me thinking back to my days as a first-grade teacher that felt a little hesitant. As I read, I came to think about that differently. But I'm wondering if you can talk about why random groups matter, the kind of impact that they have on the collaborative experience and the learning experience for kids.
Peter: Alright, so going back to the previous question. So, we have this mismatch. And we have also that 80 percent of students are not thinking; 80 percent of students are entering into that group, not prepared to offer an idea. So those are the two problems that we're trying to address here. So, random groups … random wasn't good enough. It had to be visibly random. The students had to see the randomness because when we first tried it, we said, “Here's your random groups.” They didn't believe we were being random. They just thought we were being strategic. So, it has to be visibly random, and it turns out it has to be frequent as well. About once every 45 to 75 minutes. See, when students are put into random groups, they don't know what their role is. So, we're solving this problem. They don't know what their role is. When we started doing visibly random groups frequently, within three weeks we were running that same survey.
Peter: “If you know you're going to work in groups today, what is the likelihood you would offer an idea?” Remember the baseline data was that 80 percent of students said that they were unlikely or highly unlikely, and, all of a sudden, we have a hundred percent of students saying that they're likely or highly likely. That was one thing that it solved. It shifted this idea that students were now entering groups willing to offer an idea, and that's despite 50 percent of them saying, “It probably won't lead to a solution, but I'm going to offer an idea.” Now why is that? Because they don't know what their role is. So, right on the surface, what random groups does, is it shatters this idea of preconceived roles and then preconceived behaviors. So, now they enter the groups willing to offer an idea, willing to be a contributor, not thinking that their role is just to follow. But there's a time limit to this because within 45 to 75 minutes, they're going to start to fall into roles.
Peter: In that first 45 minutes, the roles are constantly negotiated. They're dynamic. So, one student is being the leader, and the others are being the follower. And now, someone else is a leader, the others are following. Now everyone is following. They need some help from some external source. Now everyone is leading. We’ve got to resolve that. But there is all of this dynamicism and negotiation going on around the roles. But after 45 to 75 minutes, this sort of stabilizes and now you have sort of a leader and followers, and that's when we need to randomize again so that the roles are dynamic and that the students aren't falling into sort of predefined patterns of non-thinking behavior.
Mike: I think this is fascinating because we've been doing some work internally at MLC around this idea of status or the way that … the stories that kids tell about one another or the labels that kids carry either from school systems or from the community that they come from, and how those things are subtle. They're unspoken, but they often play a role in classroom dynamics in who gets called on. What value kids place on a peer's idea if it is shared. What you're making me think is there's a direct line between this thing that we've been thinking about and what happens in small groups as well.
Peter: Yeah, for sure. So, you mentioned status. I want to add to that identity and self-efficacy and so on and so forth. One of the interesting pieces of data that came out of the research into random groups was, we were interviewing students several weeks into this. And we were asking them questions around this, and the students were saying things like, “Oh, the teacher thinks we're all the same, otherwise they wouldn't do random groups. The teacher thinks we're all capable, otherwise they wouldn't do random groups.” So, what we're actually talking about here is that we're starting—just simply through random groups—to have a positive impact on student self-efficacy. One of the things that came out of this work, that I wrote about in a separate paper, was that we've known for a long time that student self-efficacy has a huge impact on student performance. But how do we increase, how do we improve student self-efficacy?
Peter: There are a whole bunch of different ways. The work of Bandura on this is absolutely instrumental. But it comes down to a couple of things. From a classroom teacher perspective, the first thing, in order for a student to start on this journey from low self-efficacy to high self-efficacy, they have to encounter a teacher who believes in them. Except students don't listen to what we say. They listen to what we do. So, simply telling our students that we have confidence in them doesn't actually have much impact. It's how we show them that we have confidence in them. And it turns out that random groups actually have a huge impact on that. By doing the random groups, we're actually showing the kids that we believe in them and then they start to internalize this. So that's one thing. The work of Bandura about how we can start to shift student self-efficacy through mastery experiences, where they start to, for example, be successful at something. And that starts to have an impact that is amplified when students start to be successful in front of others, when they are the ones who are contributing in a small group. And that group is now successful. And that success is linked in some small or great part to your contributions; that self-efficacy is amplified because not only am I being successful, I'm being successful in a safe environment, but in front of others.
Peter: Now, self-efficacy contributes to identity, and identity has an interesting relationship with status. And you mentioned status. So, self-efficacy is what I think of myself. Status is what others think of me. I can't control my status. I can't shift my status. Status is something that is bestowed on me by others. And, of course, it's affected by their interactions with me in collaborative spaces. So, how they get to see me operate is going to create a status for me, on me, by others. But the status gets to be really nicely evenly distributed in thinking classrooms when we're doing these random groups because everybody gets to be seen as capable. They all get to be someone who can be mathematical and someone who can contribute mathematically.
Mike: I want to shift back for a moment to this idea of visibly random groups. This idea that for kids, they need to believe that it's not just a strategic grouping that I've called random for the sake of the moment. What are some of the ways that you've seen teachers visibly randomize their groups so that kids really could see the proof was right out there in front of them?
Peter: So, we first started with just cards. So, we got 27 kids. We're going to use playing cards, we're going to have three aces, three 2S, three 3s, three 4s, and so on. We would just shuffle the deck, and the kids would come and take a card. And if you're a 4, you would go to the board that has a 4 on it. Or maybe that fourth 4 is there, so to speak. We learned a whole bunch of things. It has to be visible. And however way we do it, the randomization doesn't just tell them what group they're in, it tells them where to go. That's an efficiency thing. You don't want kids walking around the classroom looking for their partners and then spending 5 minutes deciding where they want to work. Take a card, you got a 7, you go to the 7 board. You got an ace, you go to the ace board.
Peter: And that worked incredibly well. Some teachers already had Popsicle sticks in their classroom, so they started using those: Popsicle sticks with students' names. So, they would pull three Popsicle sticks and they would say, “OK, these students are together. These students are together.” At first, we didn't see any problems with that. That seemed to be pretty isomorphic … to using a playing card. Some teachers got frustrated with the cards because with a card, sometimes what happens is that they get ripped or torn or they don't come back. Or they come back, and they're sweaty or they're hot. And it's like, “OK, where were you keeping this card? I don't want to know. It's hot, it's dirty.” They got ink on it. The cards don't come back. The kids are swapping cards. And teachers were frustrated by this. So, they started using digital randomizers, things like Flippity and ClassDojo and Picker Wheel and Team Shake and Team Maker.
Peter: There were tons of these digital randomizers, and they all work pretty much the same. But there was a bit of a concern that the students may not perceive the randomness as much in these methods. And you can amplify that by, for example, bringing in a fuzzy [die], a big one, and somebody gets to roll it. And if a 5 comes up, they get to come up and hit the randomized button five times. And now there's a greater perception of randomness that's happening. With Flippity, that turns out actually it'd be true. Turns out that the first randomization is not purely random, and the kids spot that pattern. And we thought, “OK, perfect. That's fine. As long as the students perceive it's random, that it is truly random, that the teacher isn't somehow hacking this so that they are able to impose their own bias into this space.” So, it's seemingly random, but not purely random. And everything was running fine until about six to eight months ago. I was spending a lot of time in classrooms. I think in the last 14 months I've been in 144 different classrooms, co-teaching or teaching. So, I was spending a lot of time in classrooms, and for efficiency's sake, a lot of these teachers were using digital randomizers. And then I noticed something. It had always been there, but I hadn't noticed it. This is the nature of research. It's also the nature of just being a fly on the wall, or someone who's observing a classroom or a teacher. There's so much to notice we can't notice it all. So, we notice the things that are obvious. The more time we spend in spaces, the more nuanced things we're able to notice. And about six to eight months ago, I noticed something that, like I said, has always been there, but I had never really noticed it.
Peter: Teacher hits a randomized button, and all the students are standing there watching, waiting for the randomized groups to appear on the screen. And then somebody goes, “Ugh.” It's so small. Or somebody laughs. Or somebody's like, “Nooo.” And it's gone. It's in a moment, it's gone. Sometimes others snicker about it, but it's gone. It's a flash. And it's always been there, and you think it's not a big deal. Turns out it's a huge deal because this is a form of micro-bullying. This is what I call it, “micro-bullying.” Because when somebody goes, “Ugh,” everybody in the room knows who said it. And looking at the screen, they know who they said it about. And this student, themself, knows who said it, and they know that they're saying it about them. And what makes this so much worse than other overt forms of bullying is that they also are keenly aware that everybody in the room just witnessed and saw this happen, including the teacher.
Peter: And it cuts deeply. And the only thing that makes bullying worse is when bullying happens in front of someone who's supposed to protect you, and they don't; not because we're evil, but because it's so short, it's so small, it's over in a flash. We don't really see the magnitude of this. But this has deep psychological effects and emotional effects on these students. Not just that they know that this person doesn't like them. But they know that everybody knows that they don't like them. And then what happens on the second day? The second day, whoever's got that student, that victimized student in their group, when the randomization happens, they also go, “Ugh,” because this has become acceptable now. This is normative. Within a week, this student might be completely ostracized. And it's just absolutely normal to sort of hate on this one student.
Peter: It's just not worth it. It cuts too deeply. Now you can try to stop it. You can try to control it, but good luck, right? I've seen teachers try to say, “OK, that's it. You're not allowed to say anything when the randomization happens. You're not allowed to cheer, you're not allowed to grunt, you're not allowed to groan, you're not allowed to laugh. All you can do is go to your boards.” Then they hit the random, and immediately you hear someone go, “Ugh.” And they'll look at them, and the student will go, “What? That's how I breathe.” Or “I stubbed my toe where I thought of something funny.” It's virtually impossible to shut it down because it's such a minor thing. But seemingly minor. In about 50 percent of elementary classrooms that I'm in, where a teacher uses that digital randomizer, you don't hear it. But 50 percent you do. Almost 100 percent of high school classrooms I'm in you hear some sort of grunt or groan or complaint.
Peter: It's not worth it. Just buy more cards. Go to the casino, get free cards. Go to the dollar store, get them cheap. It's just not worth it. Now, let's get back to the Popsicle stick one. It actually has the same effect. “I'm going to pull three names. I'm going to read out which three names there are, and I'm going to drop them there.” And somebody goes, “Ugh.” But why does this not happen with cards? It doesn't happen with cards because when you take that card, you don't know what group you're in. You don't know who else is in your group. All you know is where to go. You take that card, you don't know who else is in your group. There's no grunting, groaning, laughing, snickering. And then when you do get to the group, there might be someone there that you don't like working with. So, the student might go, “Ugh.” But now there's no audience to amplify this effect. And because there's no audience, more often than not, they don't bother going, “Ugh.” Go back to the cards, people. The digital randomizers are fast and efficient, but they're emotionally really traumatizing.
Mike: I think that's a really subtle but important piece for people who are thinking about doing this for the first time. And I appreciate the way that you described the psychological impact on students and the way that using the cards engineers less of the audience than the randomizer [do].
Peter: Yeah, for sure.
Mike: Well, let's shift a little bit and just talk about your recommendations for group size, particularly students in kindergarten through second grade as opposed to students in third grade through fifth grade. Can you talk about your recommendations and what are the things that led you to them?
Peter: First of all, what led to it? It was just so clear, so obvious. The result was that groups of three were optimal. And that turned out to be true every setting, every grade. There are some caveats to that, and I'll talk about that in a minute. But groups of three were obvious. We saw this in the data almost immediately. Every time we had groups of three, we heard three voices. Every time we heard groups of four, we heard three voices. When we had groups of five, we heard two voices on task, two voices off task, and one voice was silent. Groups of three were just that sort of perfect, perfect group size. It took a long time to understand why. And the reason why comes from something called “complexity theory.” Complexity theory tells us that in order for a group to be productive, it has to have a balance between diversity and redundancy.
Peter: So, redundancy is the things that are the same. We need redundancy. We need things like common language, common notation, common vocabulary, common knowledge. We need to have things in common in order for the collaboration to even start. But if all we have is redundancy, then the group is no better than the individual. We also have to have diversity. Diversity is what every individual brings to the group that's different. And the thing that happens is, when the group sizes get larger, the diversity goes up, but redundancy goes down. And that's bad. And when the group sizes get smaller, the redundancy goes up, but the diversity goes down. And that's bad. Groups of three seem to have this perfect balance of redundancy and diversity. It was just the perfect group size. And if you reflect on groups that you've done in your settings, whatever that setting was, you'll probably start to recognize that groups of three were always more effective than groups of four.
Peter: But we learned some other things. We learned that in K–2, for example, groups of three were still optimal, but we had to start with groups of two. Why? Because very young children don't know how to collaborate yet. They come to school in kindergarten, they're still working in what we call “parallel,” which means that they'll happily stand side by side at a whiteboard with their own marker and work on their own things side by side. They're working in parallel. Eventually, we move them to a state that we call “polite turn-taking.” Polite turn-taking is we can have two students working at a whiteboard sharing one marker, but they're still working independently. So, “It's now your turn and you're working on your thing, and now it's my turn, I'm working on my thing.” Eventually, we get them to a state of collaboration. And collaboration is defined as “when what one student says or does affects what the other student says or does.”
Peter: And now we have collaboration happening. Very young kids don't come to school naturally able to collaborate. I've been in kindergarten classrooms in October where half the groups are polite turn-taking, and half the groups are collaborating. It is possible to accelerate them toward that state. But I've also been in grade 2 classrooms in March where the students are still working in parallel or turn-taking. We need to work actively at improving the collaboration that's actually happening. Once collaboration starts to happen in those settings, we nurtured for a while and then we move to groups of three. So, I can have kindergartens by the end of the year working in groups of three, but I can't assume that grade 2s can do it at the beginning of the year. It has a lot to do with the explicit efforts that have been made to foster collaboration in the classroom. And having students sit side by side and pair desks does not foster collaboration. It fosters parallel play.
Peter: So, we always say that “K–2, start with groups of two, see where their level of collaboration is, nurture that work on it, move toward groups of three.” The other setting that we had to start in groups of two were alternate ed settings. Not because the kids can't collaborate, but because they don't trust yet. They don't trust in the process in the educational setting. We have to nurture that. Once they start to trust in working in groups of two, we can move to groups of three. But the data was clear on this. So, if you have a classroom, and let's say you're teaching grade 6, and you don't have a perfect multiple of three, what do you do? You make some groups of two. So, rather than groups of four, make some groups of two. Keep those groups of two close to each other so that they may start to collaborate together.
Peter: And that was one of the ironies of the research: If I make a group of four, it's a Dumpster fire. If I make two groups of two and put them close to each other, and they start to talk to each other, it works great. You start with groups of two. So, having some extra groups of two is handy if you're teaching in high school or any grade, to be honest. But let's say you have 27 students on your roster, but only 24 are there. There's going to be this temptation to make eight groups of three. Don't do it. Make nine groups, have a couple of groups of two. Because the minute you get up and running, someone's going to walk in late. And then when they walk in late, it's so much easier to plug them into a group of two than to have them waiting for another person to come along so that they can pair them or to make a group of four.
Mike: Yeah, that makes sense. Before we close, Peter, I want to talk about two big ideas that I really wish I would've understood more clearly when I was still in the classroom. What I'm thinking about are the notion of crossing social boundaries and then also the concept of knowledge mobility. And I'm wondering if you could talk about each of them in turn and talk about how they relate to one another.
Peter: Certainly. So, when we make our groups, when we make groups, groups are very discreet. I think this comes from that sort of strategic grouping, or even self-selected groupings where the groups are really separate from each other. There are very well-defined boundaries around this group, and everything that happens, happens inside that group, and nothing happens between groups. In fact, as teachers, we often encourage that, and we're like, “No, do your own work in your group. Don't be talking to the other groups.” Because the whole purpose of doing strategic groups is to keep certain kids away from each other, and that creates a very non-permeable boundary between the groups. But what if we can make these boundaries more porous, and so that knowledge actually starts to flow between the groups. This is what's called “knowledge mobility,” the idea that we don't actually want the knowledge to be fixed only inside of a group.
Peter: The smartest person in the room is the room. We got to get that knowledge moving around the room. It's not groups, it's groups among groups. So, how can we get what one group is achieving and learning to move to another group that's maybe struggling? And this is called “knowledge mobility.” The easiest way to increase this is we have the students working at vertical whiteboards. Working at vertical whiteboards creates a space where passive knowledge mobility is really easy to do. It's really easy to look over your shoulder and see what another group is doing and go, “Oh, let's try that. They made a table of values. Let's make a table of values. Or they've done a graph, or they drew a picture” or whatever. “We'll steal an idea.” And that idea helps us move forward. And that passive can also lead to more active, where it's like, “I wonder what they're doing over there?”
Peter: And then you go and talk to them, and the teacher can encourage this. And both of these things really help with mobilizing knowledge, and that's what we want. We don't want the only source of knowledge to be the teacher. Knowledge is everywhere. Let's get that moving around the room within groups, between groups, between students. And that's not to say that the students are copying. We're not encouraging copying. And if you set the environment up right, they don't copy. They're not going to copy. They'll steal an idea, “Oh, let's organize our stuff into a table of values,” and then it's back to their own board and working on that. And the other way that we help make these boundaries more porous is by breaking down the social barriers that exist within a classroom. All classrooms have social barriers. They could be gender, race. They could be status-based.
Peter: There are so many things that make up the boundaries that exist within classrooms. There are these social structures that exist in schools. And one of the things that random groups does is it breaks down these social barriers because we're putting students together that wouldn't normally be together. And our data really reveals just how much that happens; that after three weeks, the students are coming in, they're socializing with different students, students that hadn't been part of their social structure before. They're sitting together outside of class. I see this at the university where students are coming in, they almost don't know each other at all. Or they're coming in small groups that are in the same class. They know each other from other courses, and within three, four weeks, I'm walking through the hallways at the university and I'm seeing them sitting together, working together, even having lunch together in structures that didn't exist on day one. There are so many social structures, social barriers in classrooms. And if we can just erode those barriers, those group structures are going to become more and more porous, and we're creating more community, and we're reducing the risk that exists within those classrooms.
Mike: I think the other piece that jumps out for me is when I go back to this notion of one random grouping, a random grouping that shifts every 45 to 75 minutes. This idea of breaking those social boundaries—but also, really this idea that knowledge mobility is accelerated jumps out of those two practices. I can really see that in the structure and how that would encourage that kind of change.
Peter: Yeah. And it encourages both passively and actively. Passive in the sense that students can look over the shoulder, active that they can talk to another group. But also passively from the teacher perspective, that random groups does a lot of that heavy lifting. But I can also encourage it actively when a group asks a question. Rather than answering their question, looking around the room going, “You should go talk to the sevens over there.” Or “We're done. What do we do next?” “Go talk to the fours. They know what's next.” That, sort of, “I as a teacher can be passive and let the random groups do a lot of the heavy lifting. But I can also be active and push knowledge around the room. By the way, I respond to students' questions.”
Mike: Well, and I think what also strikes me is you're really distributing the authority mathematically to the kids as well.
Peter: Yeah, so we're displacing status, we're increasing identity. We're doing all sorts of different things that are de-powering the classroom, decentralizing the classroom.
Mike: Well, before we go, Peter, I'm wondering if there are any steps that you'd recommend to an educator who's listening. They want to start to dabble, or they want to take up some of the ideas that we've talked about. Where would you invite people to make a start?
Peter: So, first of all, one of the things we found in our research was small change is no change. When you make small changes, the classroom as a system will resist that. So, go big. In building thinking classrooms, random groups is not a practice that gets enacted on its own. It's enacted with two other practices: thinking tasks, which is chapter one of my book, random groups, which is chapter two. And then, getting the students working at vertical whiteboards. These are transformational changes to the classroom. What we're doing in doing that is we're changing the environment in which we're asking students to behave differently. Asking students to behave differently in exactly the same environment that they behaved a certain way for five years already is almost impossible to do. If you want them to behave differently, if you want them to start to think, you're going to have to create an environment that is more conducive to thinking.
Peter: So, that's part of it. The other thing is, don't do things by half measures. Don't start doing, “Well, we're going to do random groups on Mondays, but we're going to do strategic groups the rest of the days,” or something like this. Because what that communicates to students is that the randomness is something that you don't really value. Go big. We're doing random groups. We're always doing random groups. Have the courage. Yes, there's going to be some combinations that you're going to go, “Uh-oh.” And some of those are going to be really uh-oh combinations. But you're also going to have way more situations where you go and then it turns out to be amazing. So, have that courage. Go with the random groups and do it persistently and consistently. Because there is going to be resistance. The students are going to resist this thing because at least when you're being strategic, you're being thoughtful about it.
Peter: But this feels like too much chance. And they start to attribute, they start to map their emotions around being placed in strategic groups, which were often for a month, into this setting. And what we need to do is, we need to show that this is not that by being consistent, doing it randomly, doing it frequently, so they start to realize that this is different. This is not the kind of grouping structures that have happened in the past. And do it. Do it consistently, persistently. Do it for at least 10 days before you start to really see and really reap those benefits.
Mike: I think that's a really great place to stop. Thank you so much for joining us on the podcast, Peter. It really has been a pleasure chatting with you.
Peter: Thanks so much. It's been a great conversation.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org
Season 2 | Episode 18 – Counting Collections
Guest: Danielle Robinson and Melissa Hedges
Mike Wallus: Earlier this season, we released an episode focused on the complex and interconnected set of concepts that students engage with as they learn to count. In this follow-up episode, we're going to examine a powerful routine called “counting collections.” We'll be talking with Danielle Robinson and Dr. Melissa Hedges from the Milwaukee Public Schools about counting collections and the impact that this routine can have on student thinking.
Mike: Well, welcome to the podcast, Danielle and Melissa. I can't tell you how excited I am to talk with y'all about the practice of counting collections.
Danielle Robinson and Melissa Hedges: Thanks for having us. Yes, we're so excited to be here.
Mike: I want to start this conversation by acknowledging that the two of you are actually part of a larger team of educators who really took this work on counting collections. You introduced it in the Milwaukee Public Schools. And, Melissa, I think I'll start with you. Can you take a moment to recognize the collaborators who have been a part of this work?
Melissa: Absolutely. In addition to Danielle and myself, we are fortunate to work with three other colleagues: Lakesha King, Krista Beal, and Claire Madden. All three are early childhood coaches that actively support this work as well.
Mike: So, Danielle, I wonder for some folks if we can help them see this practice more clearly. Can you spend time unpacking, what does counting collections look like in a classroom? If I walked in, what are some of the things that I might see?
Danielle: Yeah, I think what's really amazing about counting collections is there might be some different ways that you might see counting collections happening in the classroom. When you walk into a classroom, you might see some students all over. Maybe they're sitting at tables, maybe they're on the carpet. And what they're doing is they're actually counting a baggie of objects. And really their job is to answer this question, this very simple but complicated question of, “How many?” And they get to decide how they want to count. Not only do they get to pick what they want to count, but they also get to pick their strategy of how they actually want to count that collection. They can use different tools. They might be using bowls or plates. They might be using 10-frames. They might be using number paths. You might see kiddos who are counting by ones.
Danielle: You might see kids who are making different groupings. At times, you might also see kiddos [who] are in stations, and you might see a small group where a teacher is doing counting collections with a few kiddos. You might see them working with partners. And I think the beautiful piece of this and the unique part of counting collections within Milwaukee Public Schools is that we've been able to actually pair the counting trajectory from Doug Clements and Julie Sarama with counting collections where teachers are able to do an interview with their students, really see where they're at in their counting so that the kids are counting a just right collection for them—something that's not too easy, something that's not too hard, but something that is available for them to really push them in their understanding of counting. So, you're going to see kids counting different sizes. And we always tell the teachers it's a really beautiful moment when you're looking across the classroom and as a teacher, you can actually step back and know that every one of your kids are getting what they need in that moment. Because I think oftentimes, we really don't ever get to feel like that, where we feel like, “Wow, all my kids are getting what they need right now, and I know that I am providing the scaffolds that they need.”
Mike: So, I want to ask you a few follow-ups, if I might, Danielle.
Danielle: Yeah, of course.
Mike: There's a bit of language that you used initially where I'm paraphrasing. And tell me where I get this wrong. You use the language “simple yet complicated,” I think. Am I hearing that right?
Danielle: I did. I did, yeah.
Mike: Tell me about that.
Danielle: I think it's so interesting because a lot of times when we introduce this idea of counting collections with our teachers, they're like, “Wait a minute, so I'm supposed to give this baggie of a bunch of things to my students, and they just get to go decide how they want to count it?” And we're like, “Yeah, that is absolutely what we're asking you to do.” And they feel nervous because this idea of the kids, they're answering how many, but then there's all these beautiful pieces a part of it. Maybe kids are counting by ones, maybe they're deciding that they want to make groups, maybe they're working with a partner, maybe they're using tools. It's kind of opened up this really big, amazing idea of the simple question of how many. But there's just so many things that can happen with it.
Mike: There's two words that kept just flashing in front of my eyes as I was listening to you talk. And the words were access and differentiation. And I think you didn't explicitly say those things, but they really jump out for me in the structure of the task and the way that a teacher could take it up. Can you talk about the way that you think this both creates access and also the places where you see there's possibility for differentiation?
Danielle: For sure. I'm thinking about a couple classrooms that I was in this week and thinking about once we've done the counting trajectory interview with our kiddos, you might have little ones who are still really working with counting to 10. So, they have collections that they can choose that are just at that amount of about 10. We might have some kiddos who are really working kind of in that range of 20 to 40. And so, we have collections that children can choose from there. And we have collections all the way up to about 180 in some cases. So, we kind of have this really nice, natural scaffold within there where children are told, “Hey, you can go get this just right color for you.” We have red collections, blue collections, green and yellow. Within that also, the children get to decide how they want to count.
Danielle: So, if they are still really working on that verbal count sequence, then we allow them to choose to count by ones. We have tools for them, like number paths to help do that. Maybe we've got our kiddos who are starting to really think about this idea of unitizing and making groups of 10s. So, then what they might do is they might take a 10-frame and they might fill their 10-frame and then actually pour that 10-frame into a bowl, so they know that that bowl now is a collection of 10. And so, it's this really nice idea of helping them really start to unitize and to make different groupings. And I think the other beautiful piece, too, is that you can also partner. Students can work together and actually talk about counting together. And we found that that really supports them, too, of just that collaboration piece, too.
Mike: So, you kind of started poking around the question that I was going to ask Melissa.
Danielle and Melissa: ( laugh )
Mike: You said the word “unitizing,” which is the other thing that was really jumping out because I taught kindergarten and first grade for about eight years. And in my head, immediately all of the different trajectories that kids are on when it comes to counting, unitizing, combining … those things start to pop out. But, Melissa, I think what you would say is there is a lot of mathematics that we can build for kids beyond say K–2, and I'm wondering if you could talk a little bit about that.
Melissa: Absolutely. So before I jump to our older kids, I'm just going to step back for a moment with our kindergarten, first- and second-graders. And even our younger ones. So, the mathematics that we know that they need to be able to count collections, that idea of cardinality, one-to-one correspondence, organization—Danielle did a beautiful job explaining how the kids are going to grab a bag, figure out how to count, it's up to them—as well as this idea of producing a set, thinking about how many, being able to name how many. The reason why I wanted to go back and touch on those is that we know that as children get older and they move into third, fourth, and fifth grade, those are understandings that they must carry with them. And sometimes those ideas aren't addressed well in our instructional materials. So, the idea of asking a first- and second-grader to learn how to construct a unit of 10 and know that 10 ones is one 10 is key, because when we look at where place value tends to fall apart in our upper grades. My experience has been it's fifth grade, where all of a sudden we're dealing with big numbers, we're moving into decimals, we're thinking about different size units, we've got fractions. There's all kinds of things happening.
Melissa: So, the idea of counting collections in the early elementary grades helps build kids' number sense, provides them with that confidence of magnitude of number. And then as they move into those either larger collections or different ways to count, we can make beautiful connections to larger place values. So, hundreds, thousands, ten thousands. Sometimes those collections will get big. All those early number relationships also build. So, those early number relationships, part-whole reasoning that numbers are composed and decomposed of parts. And then we've just seen lots really, really fun work about additive and multiplicative thinking. So, in a third-, fourth-, fifth-grade classroom, what I used to do is dump a cup full of lima beans in the middle of the table and say, “How many are there?” And there's a bunch there. So, they can count by ones. It's going to take a long time. And then once they start to figure out, “Oh wait, I can group these.” “Well, how many groups of five do you have?” And how we can extend to that from that additive thinking of five plus five plus five plus five to then thinking about and extending it to multiplicative thinking. So, I think the extensions are numerous.
Mike: There's a lot there that you said, and I think I wanted to ask a couple follow-ups. First thing that comes to mind is, we've been interviewing a guest for a different podcast … and this idea that unitizing is kind of a central theme that runs really all the way through elementary mathematics and certainly beyond that. But I really am struck by the way that this idea of unitizing and not only being able to unitize, but I think you can physically touch the units, and you can physically re-unitize when you pour those things into the cup. And it's giving kids a bit more space with the physical materials themselves before you step into something that might be more abstract. I'm wondering if that's something that you see as valuable for kids and maybe how you see that play out?
Melissa: Yes, it's a great question. I will always say when we take a look at our standard base 10 blocks, “The person that really understands the construction of those base 10 blocks is likely the person [who] invented them.” They know that one little cube means one, and that all of a sudden these 10 cubes are fused together and we hold it up and we say, “Everybody, this is 10 ones. Repeat, one 10. What we find is that until kids have multiple experiences and opportunities over time to construct units beyond one, they really won't do it with deep understanding. And again, that's where we see it fall apart when they're in the fourth and fifth grade. And they're struggling just to kind of understand quantity and magnitude. So, the idea and the intentionality behind counting collections and the idea of unitizing is to give kids those opportunities that to be quite honest—and no disrespect to the hardworking curriculum writers out there—it is a tricky, tricky, tricky idea to develop in children through paper and pencil and workbook pages.
Melissa: I think we have found over time that it's the importance of going, grabbing, counting, figuring it out. So, if my collection is bears, does that collection of 10 bears look the same as 10 little sharks look the same as 10 spiders? So, what is this idea of 10? And that they do it over and over and over and over again. And once they crack the code—that's the way I look at it—once our first- and second-graders crack the code of counting collections, they're like, “Oh, this is not hard at all.” And then they start to play with larger units. So, then they'll go, “Oh, wait, I can combine two groups of 10. I just found out that's 20. Can I make more 20s?” So, then we're thinking about counting not just by ones, not just by 10s, but by larger units. And I think that we've seen that pay off in so many tremendous ways. And certainly on the affective side, when kids understand what's happening, there's just this sense of joy and excitement and interest in the work that they do, and I actually think they see themselves learning.
Mike: Danielle, do you want to jump in here?
Danielle: I think to echo that, I just recently was speaking with some teachers. And the principal was finally able to come and actually see counting collections happening. And what was so amazing is these were K–5 kiddos, 5-year-olds who were teaching the principal about what they were doing. This was that example where we want people to come in, and the idea is what are you learning? How do you know you've learned it, thinking about that work of Hattie? And these 5-year-olds were telling him exactly what they were learning and how they were learning it and talking about their strategies. And I just felt so proud of the K–5 teacher who shared that with me because her principal was blown away and was seeing just the beauty that comes from this routine.
Mike: We did an episode earlier this year on place value, and the speaker did a really nice job of unpacking the ideas around it. I think what strikes me, and at this point I might be sounding a bit like a broken record, is the extent to which this practice makes place value feel real. These abstract ideas around unitizing. And I think, Melissa, I'm going back to something you said earlier where you're like, “The ability to do this in an abstract space where you potentially are relying on paper and pencil or even drawing, that's challenging.” Whereas this puts it in kids' hands, and you physically re-unitize something, which is such a massive deal. This idea that one 10 and 10 ones have the same value even though we're looking at them differently, simultaneously. That's such a big deal for kids, and it just really stands out for me as I hear you all talk.
Melissa: I had the pleasure of working with a group of first-grade teachers the other day, and we were looking at student work for a simple task that the kids were asked to do. I think it was 24 plus seven, and so it was just a very quick PLC. Look at this work. Let's think about what they're doing. And many of the children had drawn what the teachers referred to as sticks and circles or sticks and dots. And I said, “Well, what do those sticks and dots mean?” Right? “Well, of course the stick is the 10 and the dot is the one.” And I said, “There's lots of this happening,” I said, “Let's pause for a minute and think, ‘To what degree do you think your children understand that that line means 10 and that dot means one? And that there’s some kind of a connection, meaningful connection for them just in that drawing.’” It got kind of quiet, and they're like, “Well, yep, you're right. You're right. They probably don't understand what that is.” And then one of the teachers very beautifully said, “This is where I see counting collections helping.” It was fantastic.
Mike: Danielle, I want to shift and ask you a little bit about representation. Just talk a bit about the role of representing the collection once the counting process and that work has happened. What do you all ask kids to do in terms of representation and can you talk a little bit about the value of that?
Danielle: Right, absolutely. I think one thing that as we continue to go through in thinking about this routine and the importance of really helping our students make sense and count meaningfully, I think we will always go back to our math teaching framework that's been laid out for us through “Taking Action,” “Principles to Action,” “Catalyzing Change.” And really thinking about the power of using multiple representations. And how, just like you said, we want our students to be able to be physically unitizing, so we have that aspect of working with our actual collections. And then how do we help our students understand that “You have counted your collection. Now what I want you to do is, I want you to actually visually represent this. I want you to draw how you counted.” And so, what we talk about with the kids is, “Hey, how you have counted. If you have counted by ones, I should be able to see that on your paper. I should be able to look at your paper, not see your collection and know exactly how you counted. If you counted by tens, I should be able to see, ‘Oh my gosh, look, that's their bowl. I see their bowls, I see their plates, I see their tens inside of there.’”
Danielle: And to really help them make those connections moving back and forth between those representations. And I think that's also that piece, too, for them that then they can really hang their hat on. “This is how I counted. I can draw a picture of this. I can talk about my strategy. I can share with my friends in my classroom.” And then that's how we like to close with our counting collections routine is really going through and picking a piece of student work and really highlighting a student's particular strategy. Or even just highlighting several and being like, “Look at all this work they did today. Look at all of this mathematical thinking.” So, I think it's a really important and powerful piece, especially with our first- and second-graders, too. We really bring in this idea of equations, too. So, this idea of, “If I've counted 73, and I've got my seven groups of 10, I should have 10 plus 10 plus 10, right? All the way to 70. And then adding my three.” So, I think it's just a continuous idea of having our kids really developing that strong understanding of meaningful counting, diving into place value.
Mike: I'm really struck by the way that you described the protocol where you said you're asking kids to really clearly make sure that what they're doing aligns with their drawing. The other piece about that is it feels like one, that sets kids up to be able to share their thinking in a way where they've got a scaffold that they've created for themself. The other thing that it really makes me think about is how if I'm a teacher and I'm looking at student work, I can really use that to position that student's idea as valuable. Or position that student's thinking as something that's important for other people to notice or attend to. So, you could use this to really raise a student's ideas status or raise the student status as well. Does that actually play out in a reality?
Danielle: It does actually. So, a couple of times what I will do is I will go into a classroom. And oftentimes it can be kind of a parent for which students may just not have the strongest mathematical identity or may not feel that they have a lot of math agency in the space. And so, one thing that I will really intentionally do and work with the teacher to do is, “You know what? We are going to share that little one's work today. We're going to share that work because this is an opportunity to really position that child as a mathematician and to position that child as someone who has something to offer. And the fact that they were able to do this really hard work.” So, that is something that is very near and dear to us to really help our teachers think of these different ways to ensure that this is a routine that is for all of our children, for each and every child that is in that space. So, that is absolutely something that we find power in and seek to help our teachers find as well.
Mike: Well, I would love for each of you to just weigh in on this next question. What has really come to mind is how different this experience of mathematics is from what a lot of adults and unfortunately what a lot of kids might experience in elementary school. I'm wondering if both of you would talk a bit about what does this look like in classrooms? How does this impact the lived experience of kids and their math identities? Can you just talk a little bit about that?
Melissa: I can start. This is Melissa. So, we have four beliefs on our little math team that we anchor our work around every single day. And we believe that mathematics should be humanizing, healing, liberating and joyful. And so, we talk a lot about when you walk into a classroom, how do you know that mathematics instruction is humanizing, which means our children are placed at the center of this work? It's liberating. They see themselves in it. They're able to do it. It's healing. Healing for the teacher as well as for the student. And healing in that the student sees themselves as capable and able to do this, and then joyful that it's just fun and interesting and engaging. I think, over time, what we've seen is it helps us see those four beliefs come to life in every single classroom that's doing it. When that activity is underway and children are engaged and interested, there's a beautiful hum that settles over the room. And sometimes you have to remind the teacher step back, take a look at what is happening.
Melissa: Those guys are all engaged. They're all interested. They're all doing work that matters to them because it's their work, it's their creation. It's not a workbook page, it's not a fill in the blank. It's not a do what I do. It’s, you know what? “We have faith in you. We believe that you can do this,” and they show us time and time again that they can.
Danielle: I'll continue to echo that. Where for Milwaukee Public Schools and in the work that we are seeking to do is really creating these really transformative math spaces for, in particular, our Black and brown children. And really just making sure that they are seeing themselves as mathematicians, that they see themselves within this work, and that they are able to share their thinking and have their brilliance on display. And also, to work through the mathematical processes, too, right? This routine allows you to make mistakes and try a new strategy.
Danielle: I had this one little guy a couple months ago, he was working in a pretty large collection, and I walked by him and he was making groups of two, and I was like, “Oh, what are you working on?” And he's like, “I'm making groups of two.” And I thought to myself, I was like, “Oh boy, that's going to take him a long time” cause they had a really big collection. And I kind of came back around and he had changed it and was making groups of 10. So, it really creates a space where they start to calibrate and they are able to engage in that agency for themselves. I think the last piece I'd like to add is to really come to it from the teacher side as well … is that what Melissa spoke about was those four beliefs. And I think what we've also found is that county collections has been really healing for our teachers, too. We've had teachers who have actually told us that this helped me stay in teaching. I found a passion for mathematics again that I thought I'd lost. And I think that's another piece that really keeps us going is seeing not only is this transformative for our kids, cause they deserve the best, but it's also been really transformative for our teachers as well to see that they can teach math in a different way.
Mike: Absolutely, and I think you really got to this next transition point that I had in mind when I was thinking about this podcast, which is, listening to the two of you, it's clear that this is an experience that can be transformative mathematically and in terms of what a child or even a teacher's lived experience with mathematics is. Can you talk a little bit about what might be some very first steps that educators might take to get started with this?
Danielle: Absolutely. I think one thing, as Melissa and I were kind of thinking about this, is someone who is like, “Oh my gosh, I really want to try this.” I think the first piece is to really take stock of your kiddos. If you're interested in diving into the research of Clements and Sarama and working with the county trajectory, we would love for you to Google that and go to learningtrajectories.org. But I think the other piece is to even just do a short little interview with your kids. Ask each of your little ones, “Count as high as you can for me and jot down what you're noticing.” Give them a collection of 10 of something. It could be counters, it could be pennies. See how they count that group of 10. Are they able to have that one-to-one? Do they have that verbal count sequence? Do they have that cardinality? Can they tell you that there is 10 if you ask them again, “How many?”?
Danielle: If they can do that, then go ahead and give them 31. Give them 31 of something. Have them count and kind of just see the range of kiddos that you have and really see where is that little challenge I might want to give them. I think another really nice piece is once you dive into this work, you are never going to look at the dollar section different. You are always just start gathering things like pattern blocks. I started with noodles. That is how I started counting collections in my classroom. I used a bunch of erasers that I left over from my prize box. I use noodles, I use beads, bobby pins, rocks, twigs. I mean, start kind of just collecting. It doesn't have to be something that you spend your money on. This can be something that you already use, things that you have. I think that's one way that you can kind of get started. Then also, procedures, procedures, procedures, like go slow to go fast. Once you've got your collections, really teach your kids how to respect those collections. Anchor charts are huge. We always say, when I start this with 4-year-olds, our first lesson is, “This is how we open the bag today. This is how we take our collections out.” So, we always recommend go slow to go fast, really help the kids understand how to take care of the collections, and then they'll fly from there.
Mike: So, Melissa, I think this is part two of that question, which is, when you think about the kinds of things that helped you start this work and sustain this work in the Milwaukee Public Schools, do you have any recommendations that you think might help other folks?
Melissa: Yeah. My first entry point into learning about counting collections other than through an incredibly valued colleague [who] learned about it at a conference, was to venture into the TED. I think it's TED, the teacher resource site, and that was where I found some initial resources around how do we do this? We were actually getting ready to teach a course that at the time Danielle was going to be a student in, and we knew that we wanted to do this thing called counting collection. So, it's like, “Well, let's get our act together on this.” So, we spent a lot of time looking at that. There's some lovely resources in there. And since the explosion of the importance of early mathematics has happened in American mathematical culture, which I think is fantastic, wonderful sites have come up. One of our favorites that we were talking about is Dreme. D-R-E-M-E, the Dreme website. Fantastic resources.
Melissa: The other one Danielle mentioned earlier, it's just learningtrajectories.org. That's the Clements and Sarama research, which, 15 years ago, we were charged as math educators to figure out how to get that into the hands of teachers, and so that's one of the ways that they've done that. A couple of books that come to mind is the [“Young Children’s Mathematics: Cognitively Guided Instruction in Early Childhood Education”]. Fantastic. If you don't have it and you're a preschool teacher and you're interested in math, get it. And then of course, the “Choral Counting & Counting Collections” book by Franke, Kazemi, Turrou. Yeah, so I think those are some of the big ones. If you want just kind of snippets of where to go, go to the Dreme, D-R-E-M-E, and you'll get some lovely, lovely hits. There's some very nice videos. Yeah, just watch a kid count ( laughs ).
Mike: I think that's a great place to stop. I can't thank you two enough for joining us. It has really been a pleasure talking with both of you.
Danielle: Thank you so much.
Melissa: Thanks for your interest in our work. We really appreciate it.
Mike: With the close of this episode, we are at the end of season two for Rounding Up, and I want to just thank everyone who's been listening for your support, for the ways that you're taking these ideas up in your own classrooms and schools. We'll be taking the summer off to connect with new speakers, and we'll be back with season three this fall. In the meantime, if you have topics or ideas that you'd like for us to talk about, let us know. You can reach out to us at [email protected]. What are some things you'd like us to talk about in the coming year? Have a great summer. We'll see you all in the fall.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org
Season 2 | Episode 17 – Spatial Reasoning
Guest: Dr. Robyn Pinilla
Mike Wallus: Spatial reasoning can be a nebulous concept, and it's often hard for many educators to define. In this episode, we're talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas, El Paso. We'll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students.
Mike: Welcome to the podcast, Robyn. I'm really excited to be talking with you about spatial reasoning.
Robyn Pinilla: And I am excited to be here. Mike: Well, let me start with a basic question. So, when we're talking about spatial reasoning, is that just another way of saying that we're going to be talking about ideas that are associated with geometry? Or are we talking about something bigger?
Robyn: It's funny that you say it in that way, Mike, because geometry is definitely the closest mathematical content that we see in curricula, but it is something much bigger. So, I started with the misconception and then I used my own experiences to support that idea that this was just geometry because it was my favorite math course in high school because I could see the concepts modeled and I could make things more tangible. Drawing helped me to visualize some of those concepts that I was learning instead of just using a formula that I didn't necessarily understand. So, at that time, direct instruction really ruled, and I'm unsure what the conceptual understandings of my teachers even were because what I recall is doing numbers 3 through 47 odds in the back of the book and just plugging through these formulas. But spatial reasoning allows us to develop our concepts in a way that lead to deeper conceptual understanding. I liked geometry, and it gave me this vehicle for mathematizing the world. But geometry is really only one strand of spatial reasoning.
Mike: So, you're already kind of poking around the question that I was going to ask next, which is the elevator description of, “What do we mean when we talk about spatial reasoning and why does it matter? Why is it a big deal for students?”
Robyn: So, spatial reasoning is a notoriously hard to define construct that deals with how things move in space. It's individually how they move in space, in relation to one another. A lot of my ideas come from a network analysis that [Cathy] Bruce and colleagues did back in 2017 that looked at the historical framing of what spatial reasoning is and how we talk about it in different fields. Because psychologists look at spatial reasoning. Mathematics educators look at spatial reasoning. There [are] also connections into philosophy, the arts. But when we start moving toward mathematics more specifically, it does deal with how things move in space individually and in relation to one another. So, with geometry, whether the objects are sliding and transforming or we're composing and decomposing to create new shapes, those are the skills in two-dimensional geometry that we do often see in curricula. But the underlying skills are also critical to everyday life, and they can be taught as well. Robyn: And when we're thinking about the everyday constructs that are being built through our interactions with the world, I like to think about the GPS on our car. So, spatial reasoning has a lot of spatial temporal processes that are going on. It's not just thinking about the ways that things move in relation to one another or the connections to mathematics, but also the way that we move through this world, the way that we navigate through it. So, I'll give a little example. Spatial temporal processes have to do with us running errands, perhaps. How long does it take you to get from work to the store to home? And how many things can you purchase in the store knowing how full your fridge currently is? What pots and pans are you going to use to cook the food that you purchase, and what volume of that food are you and your family going to consume? So, all those daily tasks involve conceptions of how much space things take. And we could call it capacity, which situates nicely within the measurement domain of mathematics education. But it's also spatial reasoning, and it extends further than that. Mike: That is helpful. I think you opened up my understanding of what we're actually talking about, and I think the piece that was really interesting is how in that example of “I'm going to the grocery store, how long will it take? How full is my fridge? What are the different tools that I'll use to prepare? What capacity do they have?” I think that really helped me broaden out my own thinking about what spatial reasoning actually is. I wonder if we could shift a bit and you could help unpack for educators who are listening, a few examples of tasks that kids might encounter that could support the development of spatial reasoning. Robyn: Sure. My research and work [are] primarily focused on early childhood and elementary. So, I'm going to focus there but then kind of expand up. Number one, let's play. That's the first thing that I want to walk into a classroom and see: I want to see the kids engaging with blocks, LEGOS, DUPLOS, and building with and without specific intentions. Not everything has to have a preconceived lesson. So, one of the activities I've been doing actually with teachers and professional development sessions lately is a presentation called “Whosits and Whatsits.” I have the teachers create whatsits that do thatsits; meaning, they create something that does something. I don't give them a prompt of what problem they're going to be solving or anything specific for them to build, but rather say, “Here are materials.” We give them large DUPLO blocks, magnet tiles and Magformers, different types of wooden, cardboard and foam blocks, PVC pipes, which are really interesting in the ways that teachers use them. And have them start thinking as though they're the children in the class, and they're trying to build something that takes space and can be used in different ways.
Robyn: So, the session we did a couple of weeks ago, some teachers came up with … first, there was a swing that they had put a little frog in that they controlled with magnets. So, they had used the PVC pipe at the top that part of the swing connected over, and then were using the magnets to guide it back and forth without ever having to touch the swing. And I just thought, that was the coolest way for them to be using these materials in really playful, creative ways that could also engender them taking those lessons back into their classroom. I have also recently been reminded of the importance of modeling with fractions. So, are you familiar with the “Which One Doesn't Belong?” tasks?
Mike: Absolutely love them.
Robyn: Yes. There's also a website for fraction talks that children can look at visual representations of fractions and determine which one doesn't belong for some reason. That helps us to see the ways that children are thinking about the fractional spaces and then justifying their reason around them. With that, we can talk about the spatial positioning of the fractional pieces that are colored in. Or the ways that they're separated if those colored pieces are in different places on the figure that's being shown. They open up some nice spaces for us to talk about different concepts and use that language of spatial reasoning that is critical for teachers to engage in to show the ways that students can think about those things.
Mike: So, I want to go back to this notion of play, and what I'm curious about is, why is situating this in play going to help these ideas around spatial reasoning come out as opposed to say, situating it in a more controlled structure?
Robyn: Well, I think by situating spatial reasoning within play, we do allow teachers to respond in the moment rather than having these lesson plans that they are required to plan out from the beginning. A lot of the ideas within spatial reasoning, because it's a nebulous construct and it's learned through our everyday experiences and interactions with the world, they are harder to plan. And so, when children are engaged in play in the classroom, teachers can respond very naturally so that they're incorporating the mathematizing of the world into what the students are already doing. So, if you take, for example, one of my old teachers used to do a treasure hunt—great way to incorporate spatial reasoning with early childhood elementary classrooms—where she would set up a mapping task, is really what it was. But it was introducing the children to the school itself and navigating that environment, which is critical for spatial reasoning skills.
Robyn: And they would play this gingerbread man-type game of, she would read the book and then everybody would be involved with this treasure hunt where the kiddos would start out in the classroom, and they would get a clue to help them navigate toward the cafeteria. When they got to the cafeteria, the gingerbread man would already be gone. He would've already run off. So, they would get their next clue to help them navigate to the playground, so on and so forth. They would go to the nurse's office, the principal, the library, all of the critical places that they would be going through on a daily basis or when they needed to within the school. And it reminds me that there was also a teacher I once interviewed who used orienteering skills with her students. Have you ever heard of orienteering?
Mike: The connection I'm making is to something like geocaching, but I think you should help me understand it.
Robyn: Yeah, that's really similar. So, it's this idea that children would find their way places. Path finding and way finding are also spatial reasoning skills that are applied within our real world. And so, while it may not be as scientific or sophisticated as doing geocaching, it has children with the idea of navigating in our real world, helps them start to learn cardinality and the different ways of thinking about traversing to a different location, which … these are all things that might better relate to social studies or technology, other STEM domains specifically, but that are undergirded by the spatial reasoning, which does have those mathematics connections.
Mike: I think the first thing that occurred is, all of the directional language that could emerge from something like trying to find the gingerbread boy. And then the other piece that you made me think about just now is this opportunity to quantify distance in different ways. And I'm sure there are other things that you could draw out, especially in a play setting where the structure is a little bit looser and it gives you a little bit more space, as you said, to respond to kids rather than feeling like you have to impose the structure.
Robyn: Yeah, absolutely. There's an ability when teachers are engaging in authentic ways with the students, that they're able to support language development, support ideation and creation, without necessarily having kids sit down and fill out a worksheet that says, “Where is the ball? The ball is sitting on top of the shelf.” Instead, we can be on the floor working with students and providing those directions of, “Oh, hey, I need you to get me those materials from the shelf on the other side of the room,” but thinking about, “How can I say that in a way that better supports children understanding the spatial reasoning that's occurring in our room?” So maybe it's, “Find the pencil inside the blue cup on top of the shelf that's behind the pencil sharpener,” getting really specific in the ways that we talk about things so that we're ingraining those ideas in such a way that it becomes part of the way that the kids communicate as well.
Mike: You have me thinking that there's an intentionality in language choice that can create that, but then I would imagine as a teacher I could also revoice what students are saying and perhaps introduce language in that way as well.
Robyn: Yeah, and now you have me thinking about a really fun routine number talks, of course. And if we do the idea of a dot talk instead of a number talk, thinking about the spatial structuring of the dots that we're seeing and the different ways that you can see those arrangements and describe the quantification of the arrangement. It's a nice way to introduce educators to spatial reasoning because it might be something that they're already doing in the classroom while also providing an avenue for children to see spatial structuring in a way that they're already accustomed to as well, based on the routines that they're receiving from the teacher.
Mike: I think what's really exciting about this, Robyn, is the more that we talk, the more two things jump out. I think one is, my language choices allow me to introduce these ideas in a way that I don't know that I'd thought about as a practitioner. Part two is that we can't really necessarily draw a distinction between work we're doing around numbers and quantity and spatial reasoning; that there are opportunities within our work around number quantity and within math content to inject the language of spatial reasoning and have it become a part of the experience for students.
Robyn: Yeah, and that's important that I have conveyed that without explicitly saying it because that's the very work that I'm doing with teachers in their classrooms at this time. One, as you're talking about language, and I hate to do this, but I'm going to take us a little bit off topic for a moment. I keep seeing this idea on Twitter or whatever we call it at this point, that some people actually don't hear music in their heads. This idea is wild to me because I have songs playing in my head all the time. But at the same time, what if we think about the idea that some people don't also visualize things, they don't imagine those movements continuously that I just see. And so, as teachers, we really need to focus on that same idea that children need opportunities to practice what we think they should be able to hear but also practice what we think they should be able to see.
Robyn: I'm not a cognitive scientist. I can't see inside someone's head. But I am a teacher by trade, so I want to emphasize that teachers can do what's within their locus of control so that children can have opportunities to talk about those tasks. One that I recently saw was a lesson on clocks. So, while I was sitting there watching her teach, she was using a Judy Clock. She was having fun games with the kids to do a little competition where they could read the clock and tell her what time it was. But I was just starting to think about all of the ways that we could talk about the shorter and longer hands, the minute and hour hands, the ways that we could talk about them rotating around that center point. What shape does the hand make as it goes around that center point and what happens if it doesn't rotate fully? Now I'm going back to those fractional ideas from earlier with the “Which One Doesn't Belong?” tasks of having full shapes versus half shapes, and how we see those shapes in our real lives that we can then relate with visualized shapes that some children may or may not be able to see.
Mike: You have me thinking about something. First of all, I'm so glad that you mentioned the role of visualization.
Robyn: Yeah.
Mike: You had me thinking about a conversation I was having with a colleague a while ago, and we had read a text that we were discussing, and the point of conversation came up. I read this and there's a certain image that popped into my head.
Robyn: Uh-hm.
Mike: And the joke we were making is, “I'm pretty certain that the image that I saw in my head having read this text is not the same as what you saw.” What you said that really struck home for me is, I might be making some real assumptions about the pictures that kids see in their head and helping build those internal images, those mental movies. That's a part of our work as well.
Robyn: Absolutely. Because I'm thinking about the way that we have prototypical shapes. So, a few years ago I was working with some assessments, and the children were supposed to be able to recognize an equilateral triangle—whether it was gravity-based or facing another orientation—and there were some children who automatically could see that the triangle was a triangle no matter which direction it was “pointing.” Whereas others only recognize it if a triangle, if it were gravity-based. And so, we need to be teaching the properties of the shapes beyond just that image recognition that oftentimes our younger students come out with. I tend to think of visualization and language as supporting one another with the idea that when we are talking, we're also writing a descriptive essay. Our words are what create the intended picture—can't say that it's always the picture that comes out. But the intended picture for the audience. What we're hopeful for in classrooms is that because we're sharing physical spaces and tangible experiences, that the language used around those experiences could create shared meaning. That's one of the most difficult pieces in talking about spatial reason or quite frankly, anything else, is that oftentimes our words may have different meanings depending on who the speaker and who the listener are. And so, navigating what those differences are can be quite challenging, which is why spatial reasoning is still so hard to define.
Mike: Absolutely. My other follow-up is, if you were to offer people a way to get started, particularly on visualization, is there a kind of task that you imagine might move them along that pathway?
Robyn: I think the first thing to do is really grasp an approximation. I'm not going to say figure out what spatial reasoning is, but just an approximation or a couple of the skills therein that you feel comfortable with. So, spatial reasoning is really the set of skills that undergirds almost all of our daily actions, but it also can be inserted into the lessons that teachers are already teaching. I think that we do have to acknowledge that spatial reasoning is hard to define, but the good news is that we do reason spatially all day every day. If I am in a classroom, I want to look first at the teaching that's happening, the routines that are already there, and see where some spatial reasoning might actually fit in. With our young classes, I like to think about calendar math. Every single kindergarten, first-grade classroom that you walk into, they're going to have that calendar on the wall. So how can you work into the routines that are occurring, that spatial language to describe the different components of the routine?
Robyn: So, as a kiddo is counting on that hundreds chart, talking about the ways in which they're moving the pointer along the numbers … when they're counting by 10s, talk about the ways that they're moving down. When they're finding the patterns that are on the calendar, because all of those little calendar numbers for the day, they wind up having a pattern within them in most of the curricular kits. So, thinking about just the ways that we can use language therein. Now with older students, I think that offering that variety of models or manipulatives for them to use and then encourage them to translate from having a concrete manipulative into those more representational ideas, is great regardless of age or grade. So, students benefit from the modeling when they do diagramming of their models; that is, translating the 3-D model to 2-D, which is another component of spatial reasoning. And that gets me to this sticky point of, I'm not arguing against automaticity or being able to solve equations without physical or visual models. But I'm just acknowledging this idea that offering alternative ways for students to engage with content is really critical because we're no longer at a phase that we need our children to become computers. We have programs for that. We need children who are able to think and solve problems in novel ways because that's the direction that we're moving in problem-solving.
Mike: That's fantastic. My final question before we close things up. If you were to make a recommendation for someone who's listening and they're intrigued and they want to keep learning, are there any particular resources that you'd offer people that they might be able to go to?
Robyn: Yeah, absolutely. So, the first one that I like is the Learning Trajectories website. It's, uh, learning trajectories.org. It's produced by Doug Clements and Julie Sarama. There are wonderful tasks that are associated with spatial reasoning skills from very young children in the infants and toddler stages all the way up until 7 or 8 years old. So, that's a great place to go that will allow you to see how children are performing in different areas of spatial reasoning. There is also a book called “Taking Shape” by Cathy Bruce and colleagues that I believe was produced in 2016. And the grade levels might be a little bit different because it is on the Canadian school system, but it's for K–2 students, and that offers both the tasks and the spatial reasoning skills that are associated with them. For more of the research side, there's a book by Brent Davis and the Spatial Reasoning Study Group called “Spatial Reasoning in the Early Years,” and that volume has been one of my go-tos in understanding both the history of spatial reasoning in our schools and also ways to start thinking about spatializing school mathematics.
Mike: One of the things that I really appreciate about this conversation is you've helped me make a lot more sense of spatial reasoning. But the other thing that you've done for me, at least, is see that there are ways that I can make choices with my planning, with my language … that I could pick up and do tomorrow. There's not a discreet separate bit that is about spatial reasoning. It's really an integrated set of ideas and concepts and skills that I can start to build upon right away whatever curriculum I have.
Robyn: And that's the point. Often in mathematics, we think more explicitly about algebraic or numeric reasoning, but less frequently in classrooms about spatial reasoning. But spatial reasoning supports not only mathematics development, but other stem domains as well, and even skills that crossover into social studies and language arts as we're talking about mapping, as we're talking about language. So, as students have these experiences, they, too, can start to mathematize the world, see spatial connections as they go out to recess, as they go home from school, as they're walking through their neighborhoods, or just around the house. And it's ingrained ideas of measurement that we are looking at on a daily basis, the ways that we plan out our days and plan out our movements, whether it's really a plan or just our reactions to the world that support building these skills over time. And so, there are those really practical applications. But it also comes down to supporting overall mathematics development and then later STEM career interests, which is why I get excited about the work and want to be able to share it with more and more people.
Mike: I think that's a great place to stop. For listeners, we're going to link all of the content that Robyn shared to our show notes. And, Robyn, I'll just say again, thank you so much for joining us. It's really been a pleasure talking with you.
Robyn: Yes, absolutely. Thanks so much.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
© 2024 The Math Learning Center | www.mathlearningcenter.org
Rounding Up
Season 2 | Episode 16 – Strengthening Tasks Through Student Talk
Guests: Dr. Amber Candela and Dr. Melissa Boston
Mike Wallus: One of the goals I had in mind when we first began recording Rounding Up was to bring to life the best practices that we aspire to in math education and to offer entry points so that educators would feel comfortable trying them out in their classrooms. Today, we're talking with Drs. Amber Candela and Melissa Boston about powerful but practical strategies for supporting student talk in the elementary math classroom.
Welcome to the podcast, Amber and Melissa. We're really excited to be talking with you today.
Amber Candela: Thank you for having us.
Melissa Boston: Yes, thank you.
Mike: So we've done previous episodes on the importance of offering kids rich tasks, but one of the things that you two would likely argue is that rich tasks are necessary, but they're not necessarily sufficient, and that talk is actually what makes the learning experience really blossom. Is that a fair representation of where you all are at?
Melissa: Yes. I think that sums it up very well. In our work, which we've built on great ideas from Smith and Stein, about tasks, and the importance of cognitively challenging tasks and work on the importance of talk in the classroom. Historically, it was often referred to as “talk moves.” We've taken up the term “discourse actions” to think about how do the actions a teacher takes around asking questions and positioning students in the classroom—and particularly these talk moves or discourse actions that we've named “linking” and “press”—how those support student learning while students are engaging with a challenging task.
Mike: So I wonder if we could take each of the practices separately and talk through them and then talk a little bit about how they work in tandem. And Melissa, I'm wondering if you could start unpacking this whole practice of linking. How would you describe linking and the purpose it plays for someone who, the term is new for them?
Melissa: I think as mathematics teachers, when we hear linking, we immediately think about the mathematics and linking representations or linking strategies. But we’re using it very specifically here as a discourse action to refer to how a teacher links student talk in the classroom and the explicit moves a teacher makes to link students' ideas.
Sometimes a linking move is signaled by the teacher using a student's name, so referring to a strategy or an idea that a student might've offered. Sometimes linking might happen if a teacher revoices a student's idea and puts it back out there for the class to consider. The idea is in the way that we're using linking, that it's links within the learning community, so links between people in the classroom and the ideas offered by those people, of course. But the important thing here that we're looking for is how the links between people are established in the verbal, the explicit talk moves or discourse actions that the teacher's making.
Mike: What might that sound like?
Melissa: So that might sound like, “Oh, I noticed that Amber used a table. Amber, tell us how you used a table.” And then after Amber would explain her table, I might say, “Mike, can you tell me what this line of Amber's table means?” or “How is her table different from the table you created?”
Mike: You're making me think about those two aspects, Melissa, this idea that there's mathematical value for the class, but there's also this connectivity that happens when you're doing linking. And I wonder how you think about the value that that has in a classroom.
Melissa: We definitely have talked about that in our work as well. I’m thinking about how a teacher can elevate a student's status in mathematics by using their name or using their idea, just marking or identifying something that the student said is mathematically important that's worthy of the class considering further. Creating these opportunities for student-to-student talk by asking students to compare their strategies or if they have something to add on to what another student said. Sometimes just asking them to repeat what another student said so that there's a different accountability for listening to your peers. If you can count on the teacher to revoice everything, you could tune out what your peers are saying, but if you might be asked to restate what one of your classmates had just said, now there's a bit more of an investment in really listening and understanding and making sense.
Mike: Yeah, I really appreciate this idea that there's a way in which that conversation can elevate a student's ideas, but also to raise a student's status by naming their idea and positioning it as important.
Melissa: I have a good example from a high school classroom where a student [...] was able to solve the contextual problem about systems of equations, so two equations, and it was important for the story when the two equations or the two lines intersected. And so one student was able to do that very symbolically. They created a graph, they solved the system of equations where another student said, “Oh, I see what you did. You found the difference in the cost per minute, and you also found the difference in the starting point, and then one had to catch up to the other.”
And so the way that the teacher kind of positioned those two strategies, one had used a sensemaking approach based really in the context. The other had used their knowledge of algebra. And by positioning them together, it was actually the student who had used the algebra had higher academic status, but the student who had reasoned through it had made this breakthrough that was really the aha moment for the class.
Mike: That is super cool.
Amber, can we shift to press and ask you to talk a little bit about what press looks like?
Amber: Absolutely. So how Melissa was talking about linking is holding students accountable to the community; press is more around holding students accountable to the mathematics.
And so the questions the teacher is going to ask is going to be more related specifically to the mathematics. So, “Can you explain your reasoning?” “How did you get that answer?” “What does this x mean?” “What does that intersection point mean?” And so the questions are more targeted at keeping the math conversation in the public space longer.
Mike: I thought it was really helpful to just hear the example that Melissa shared. I'm wondering if there's an example that comes to mind that might shed some light on this.
Amber: So when I'm in elementary classrooms and teachers are asking their kids about different problems, and kids will be like, “I got 2.” OK, “How did you get 2?” “What operation did you use?” “Why did you use addition when you could have used something else?”
So it's really pressing at the, “Yes, you got the answer, but how did you get the answer?” “How does it make sense to you?”, so that you're making the kids rather than the teacher justify the mathematics that's involved. And they're the ones validating their answers and saying, “Yes, this is why I did this because…”
Mike: I think there was a point when I was listening to the two of you speak about this where, and forgive me if I paraphrase this a little bit, but you had an example where a teacher was interacting with a student and the student said something to the effect of, “I get it” or “I understand.” And the teacher came back and she said, “And what do you understand?” And it was really interesting because it threw the justification back to the student.
Amber: Right. Really what the linking and press does, it keeps the math actionable longer to all of the peers in the room. So it's having this discussion out loud publicly. So if you didn't get the problem fully all the way, you can hear your peers through the press moves, talk about the mathematics, and then you can use the linking moves to think through, “Well, maybe if Mike didn't understand, if he revoices Melissa's comment, he has the opportunity to practice this mathematics speaking it.” And then you might be able to take that and be like, “Oh, wait, I think I know how to finish solving the problem now.”
Mike: I think the part that I want to pull back and linger on a little bit is [that] part of the purpose of press is to keep the conversation about the mathematics in the space longer for kids to be able to have access to those ideas. I want y'all to unpack that just a little bit.
Amber: Having linking and press at the end is holding the conversation longer in the classroom. And so the teacher is using the press moves to get at the mathematics so the kids can access it more. And then by linking, you're bringing in the community to that space and inviting them to add: “What do you agree [with]?” “Do you disagree?” “Can you revoice what someone said?” “Do you have any questions about what's happening?”
Melissa: So when we talk about discourse actions, the initial discourse action would be the questions that the teacher asks. So there's a good task to start with. Students have worked on this task and produced some solution strategies. Now we're ready to discuss them. The teacher asks some questions so that students start to present or share their work and then it's after students' response [that] linking and press come in as these follow-up moves to do what Amber said: to have the mathematics stay in the public space longer, to pull more kids into the public space longer.
So we're hoping that by spending more time on the mathematics, and having more kids access the mathematics, that we're bringing more kids along for the ride with whatever mathematics it is that we're learning.
Mike: You're putting language to something that I don't know that I had before, which is this idea that the longer we can keep the conversation about the ideas publicly bouncing around—there are some kids who may need to hear an idea or a strategy or a concept articulated in multiple different ways to piece together their understanding.
Amber: And like Melissa was saying earlier, the thing that's great about linking is oftentimes in a classroom space, teachers ask a question, kids answer, the teacher moves on. The engagement does drop. But by keeping the conversation going longer, the linking piece of it, you might get called on to revoice, so you need to be actively paying attention to your peers because it's on the kids now. The math authority has been shared, so the kids are the ones also making sense of what's happening. But it's on me to listen to my peers because if I disagree, there's an expectation that I'll say that. Or if I agree or I might want to add on to what someone else is saying.
So oftentimes I feel like this pattern of teacher-student-teacher-student-teacher-student happens, and then what can start to happen is teacher-student-student-student-teacher. And so it kind of creates this space where it's not just back and forth, it kind of popcorns more around with the kids.
Mike: You are starting to touch on something that I did want to talk about, though, because I think when I came into this conversation, what was in my head is, like, how this supports kids in terms of their mathematical thinking. And I think where you two have started to go is: What happens to kids who are in a classroom where link and press are a common practice? And what happens to classrooms where you see this being enacted on a consistent basis? What does it mean for kids? What changes about their mathematical learning experience?
Melissa: You know, we observe a lot of classrooms, and it's really interesting when you see even primary grade students give an answer and immediately say, you know, “I think it's 5 because …,” and they provide their justification just as naturally as they provide their answer or they're listening to their peers and they're very eager to say, “I agree with you; I disagree with you, and here's why” or “I did something similar” or “Here's how my diagram is slightly different.”
So to hear children and students taking that up is really great. And it just—a big shift in the amount of time that you hear the teacher talking versus the amount of time you hear children talking and what you're able to take away as the teacher or the educator formatively about what they know and understand based on what you're hearing them say. And so [in] classrooms where this has become the norm, you see fewer instances where the teacher has to use linking and press because students are picking this up naturally.
Mike: As we were sitting here and I was listening to y'all talk, Amber, the thing that I wanted to come back to is [that] I started reflecting on my own practice and how often, even if I was orchestrating or trying to sequence, it was teacher-student-teacher-student-teacher-student. It bounced back to me, and I'm really kind of intrigued by this idea, teacher-student-student-student-teacher—that the discourse, it's moving from a back and forth between one teacher, one student, rinse and repeat, and more students actually taking up the discourse. Am I getting that right?
Amber: Yes. And I think really the thought is we always want to talk about the mathematics, but we also have to have something for the community. And that's why the linking is there because we also need to hold kids accountable to the community that they're in as much as we need to hold them accountable to the mathematics.
Mike: So, Amber, I want to think about what does it look like to take this practice up? If you were going to give an educator a little nudge or maybe even just a starting point where teachers could take up linking and press, what might that look like? If you imagined kind of that first nudge or that first starting point that starts to build this practice?
Amber: We have some checklists with sentence stems in [them], and I think it's taking those sentence stems and thinking about when I ask questions like, “How did you get that?” and “How do you know this about that answer?”, that's when you're asking about the mathematics. And then when you start to ask, “Do you agree with what so-and-so said? Can you revoice what they said in your own words?”, that's holding kids accountable to the community and just really thinking about the purpose of asking this question. Do I want to know about the math or do I want to build the conversation between the students? And then once you realize what you want that to be, you have the stem for the question that you want to ask.
Mike: Same question, Melissa.
Melissa: I think if you have the teacher who is using good tasks and asking those good initial questions that encourage thinking, reasoning, explanations, even starting by having them try out, once a student gives you a response, asking, “How do you know?” or “How did you get that?” and listening to what the student has to say. And then as the next follow-up, thinking about that linking move coming after that. So even a very formulaic approach where a student gives a response, you use a press move, hear what the student has to say, and then maybe put it back out to the class with a linking move. You know, “Would someone like to repeat what Amber just said?” or “Can someone restate that in their own words?” or whatever the linking move might be.
Mike: So if these two practices are new to someone who's listening, are there any particular resources or recommendations that you'd share with someone who wants to keep learning?
Amber: We absolutely have resources. We wrote an article for the NCTM’s MTLT [Mathematics Teacher: Learning and Teaching PK-12] called “Discourse Actions to Promote Student Access .” And there are some vignettes in there that you can read through and then there [are] checklists with sentence stems for each of the linking and press moves.
Melissa: Also, along with that article, we've used a lot of the resources from NCTM’s Principles to Actions [Professional Learning] Toolkit. that's online, and some of the resources are free and accessible to everyone.
Amber: And if you wanted to dig in a bit more, we do have a book called Making Sense of Mathematics to Inform Instructional Quality. And that goes in-depth with all of our rubrics and has other scenarios and videos around the linking and press moves along with other parts of the rubrics that we were talking about earlier.
Mike: That's awesome. We will link all of that in our show notes.
Thank you both so much for joining us. It was a real pleasure talking with you.
Amber: Thanks for having us.
Melissa: Thank you.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org
References and Resources:
NCTM: https://pubs.nctm.org/view/journals/mtlt/113/4/article-p266.xml#:~:text=Discourse%20actions%20provide%20access%20to,up%20on%20contributions%20from%20students
ERIC: https://eric.ed.gov/?id=EJ1275372
https://www.nctm.org/PtAToolkit/
https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Annual_Meetings/LosAngeles2022/Campaigns/12-21_PtA_Toolkit.pdf?utm_source=nctm&utm_medium=web&utm_campaign=LA2022&utm_content=PtA+Toolkit
Season 2 | Episode 15 – Making Sense of Story Problems
Resources:
Schema-Mediated Vocabulary in Math Word Problems
Guest: Drs. Aina Appova and Julia Hagge
Mike Wallus: Story problems are an important tool that educators use to bring mathematics to life for their students. That said, navigating the meaning and language found in story problems is a challenge for many students. Today we're talking with Drs. Aina Appova and Julia Hagge from [The] Ohio State University about strategies to help students engage with and make sense of story problems.
Mike: A note to our listeners. This podcast was recorded outside of our normal recording studio, so you may notice some sound quality differences from our regular podcast. Mike: Welcome to the podcast, Aina and Julia. We're excited to be talking to both of you.
Aina Appova: Thank you so much for having us. We are very excited as well.
Julia Hagge: Yes, thank you. We're looking forward to talking with you today.
Mike: So, this is a conversation that I've been looking forward to for quite a while, partly because the nature of your collaboration is a little bit unique in ways that I think we'll get into. But I think it's fair to describe your work as multidisciplinary, given your fields of study.
Aina: Yes, I would say so. It's kind of a wonderful opportunity to work with a colleague who is in literacy research and helping teachers teach mathematics through reading story problems.
Mike: Well, I wonder if you can start by telling us the story of how you all came to work together. And describe the work you're doing around helping students make sense of word problems.
Aina: I think the work started with me working with fifth-grade teachers, for two years now, and the conversations have been around story problems. There's a lot of issues from teaching story problems that teachers are noticing. And so, this was a very interesting experience. One of the professional development sessions that we had, teachers were saying, “Can we talk about story problems? It's very difficult.” And so, we just looked at a story problem. And the story problem, it was actually a coordinate plane story problem. It included a balance beam, and you're supposed to read the story problem and locate where this balance beam would be. And I had no idea what the balance beam would be. So, when I read the story, I thought, “Oh, it must be from the remodeling that I did in my kitchen, and I had to put in a beam, which was structural.”
Aina: So, I'm assuming it's balancing the load. And even that didn't help me. I kept rereading the problem and thinking, “I'm not sure this is on the ceiling, but the teachers told me it's gymnastics.” And so even telling me that it was gymnastics didn't really help me because I couldn't think, in the moment, while I was already in a different context of having the beam, a load-bearing beam. It was very interesting that—and I know I'm an ELL, so English is not my first language—in thinking about a context that you're familiar with by reading a word or this term, “balance beam.” And even if people tell you, “Oh, it's related to gymnastics”—and I've never done gymnastics; I never had gymnastics in my class or in my school where I was. It didn't help. And that's where we started talking about underlying keywords that didn't really help either because it was a coordinate plane problem. So, I had to reach out to Julia and say, “I think there's something going on here that is related to reading comprehension. Can you help me?” And that's how this all started. ( chuckles )
Julia: Well, so Aina came to me regarding her experience. In fact, she sent me the math problem. She says, “Look at this.” And we talked about that. And then she shared frustration of the educators that she had been working with that despite teaching strategies that are promoted as part of instructional practice, like identifying mathematical keywords and then also reading strategies have been emphasized, like summarizing or asking questions while you're reading story problems. So, her teachers had been using strategies, mathematical and also reading, and their students were still struggling to make sense of and solve mathematical problems. Aina’s experience with this word problem really opened up this thought about the words that are in mathematical story problems. And we came to realize that when we think about making sense of story problems, there are a lot of words that require schema. And schema is the background knowledge that we bring to the text that we interact with.
Julia: For example, I taught for years in Florida. And we would have students that had never experienced snow. So, as an educator, I would need to do read alouds and provide that schema for my students so that they had some understanding of snow. So, when we think about math story problems, all words matter—not just the mathematical terms, but also the words that require schema. And then when we think about English learners, the implications are especially profound because we know that, that vocabulary is one of the biggest challenges for English learners. So, when we consider schema-mediated vocabulary and story problems, this really becomes problematic. And so, Aina and I analyzed the story problems in the curriculum that Aina’s teachers were using, and we had an amazing discovery.
Aina: Just the range of contexts that we came across from construction materials or nuts and bolts and MP3 players—that children don't really have anymore, a lot of them have a phone—to making smoothies and blenders, which some households may not have. In addition to that, we started looking at the words that are in the story problems. And like Julia said, there are actually mathematics teachers who are being trained on these strategies that come from literacy research. One of them was rereading the problem. And it didn't matter how many times I reread the problem or somebody reread it to me about the balance beam. I had no kind of understanding of what's going on in the problem. The second one is summarizing. And again, just because you summarized something that I don't understand or read it louder to me, it doesn't help, right? And I think the fundamental difference that we solve problems or the story problems … In the literacy, the purpose of reading a story is very different. In mathematics, the purpose of reading a story is to solve it, making sense of problems for the purpose of solving them. The three different categories of vocabulary we found from reading story problems and analyzing them is there's “technical,” there's “sub-technical” and there’s “non-technical.” I was very good at recognizing technical words because that's the strategy that for mathematics teachers, we underline the parallelogram, we underline the integer, we underline the eight or the square root, even some of the keywords we teach, right? Total means some or more means addition.
Mike: So technical, they're the language that we would kind of normally associate with the mathematics that are being addressed in the problem. Let's talk about sub-technical because I remember from our pre-podcast conversation, this is where some light bulbs really started to go off, and you all started to really think about the impact of sub-technical language.
Julia: Sub-technical includes words that have multiple meanings that intersect mathematically and other contexts. So, for example, “yard.” Yard can be a unit of measurement. However, I have a patio in my backyard. So, it's those words that have that duality. And then when we put that in the context of making sense of a story problem, it's understanding what is the context for that word and which meaning applies to that? Other examples of sub-technical would be table or volume. And so, it's important when making sense of a story problem to understand which meaning is being applied here. And then we have non-technical, which is words that are used in everyday language that are necessary for making sense of or solving problems. So, for example, “more.” More is more. So, more has that mathematical implication. However, it would be considered non-technical because it doesn't have dual meanings.
Julia: So, by categorizing vocabulary into these three different types, [that] helped us to be able to analyze the word problems. So, we worked together to categorize. And then Aina was really helpful in understanding which words were integral to solving those math problems. And what we discovered is that often words that made the difference in the mathematical process were falling within the sub-technical and non-technical. And that was really eye-opening for us.
Mike: So, Aina, this is fascinating to me. And what I'm thinking about right now is the story that you told at the very beginning of this podcast, where you described your own experience with the word problem that contained the language “balance.” And I'm wondering if you applied the analysis that you all just described with technical and sub-technical and the non-technical, when you view your own experience with that story problem through that lens, what jumps out? What was happening for you that aligns or doesn't align with your analysis?
Aina: I think one of the things that was eye-opening to me is, we have been doing it wrong. That's how I felt. And the teachers felt the same way. They're saying, “Well, we always underline the math words because we assume those are the words that are confusing to them. And then we underline the words that would help them solve the problem.” So, it was a very good conversation with teachers to really, completely think about story problems differently. It's all about the context; it's all about the schema. And my teachers realize that I, as an adult who engages in mathematics regularly, have this issue with schema. I don't understand the context of the problem, so therefore I cannot move forward in solving it. And we started looking at math problems very differently from the language perspective, from the schema perspective, from the context perspective, rather than from underlining the technical and mathematical words first. That was very eye-opening to me.
Mike: How do you think their process or their perspective on the problems changed either when they were preparing to teach them or in the process of working with children?
Aina: I know the teachers reread a problem out loud and then typically ask for a volunteer to read the problem. And it was very interesting; some of the conversations were how different the reading is. When the teacher reads the problem, there is where you put the emotion, where the certain specific things in the problem are. Prosody?
Julia: Yes, prosody is reading with appropriate expression, intonation, phrasing.
Aina: So, when the teacher reads the problem, the prosody is present in that reading. When the child is reading the problems, it's very interesting how it sounds. It just sounds the word and the next word and the next word and the next word, right? So that was kind of a discussion, too. The next strategy the math teachers are being taught is summarizing. I guess discussing the problem and then summarizing the problem. So, we kind of went through that. And once they helped me to understand in gymnastics what it is, looking up the picture, what it looks like, how long it is, and where it typically is located and there's a mat next to it, that was very helpful. And then I could then summarize, or they could summarize, the problem. But even [the] summarizing piece is now me interpreting it and telling you how I understand the context and the mathematics in the problem by doing the summary. So, even that process is very different. And the teacher said that's very different. We never really experience that.
Mike: Julia, do you want to jump in?
Julia: And another area where math and reading intersect is the use of visualization. So, visualization is a reading strategy, and I've noticed that visualization has become a really strong strategy to teach for mathematics, as well. We encourage students to draw pictures as part of that solving process. However, if we go back to the gymnastics example, visualizing and drawing is not going to be helpful for that problem because you are needing a schema to be able to understand how a balance beam would situate within that context and whether that's relevant to solving that word problem. So, even though we are encouraging educators to use these strategies, when we think about schema-mediated vocabulary, we need to take that a step further to consider how schema comes into play and who has access to the schema needed, and who needs that additional support to be able to negotiate that schema-mediated vocabulary.
Mike: I was thinking the same thing, how we often take for granted that everyone has the same schema. The picture I see in my head when we talk about balance is the same as the picture you see in your head around balance. And that's the part where, when I think about some of those sub-technical words, we really have to kind of take a step back and say, “Is there the opportunity here for someone to be profoundly confused because their schema is different than mine?” And I keep thinking about that lived experience that you had where, in my head I can see a balance beam, but in your head you're seeing the structural beam that sits on the top of your ceiling or runs across the top of your ceiling.
Aina: Oh yeah. And at first, I thought the word “beam” typically, in my mind for some reason, is vertical.
Mike: Yeah.
Aina: It's not horizontal. And then when I looked at the word balance, I thought, “Well, it could balance vertically.” And immediately what I think about is, you have a porch, then you see a lot of porches that balance the roof, and so they have the two beams …
Mike: Yes!
Aina: … or sometimes more than that. So, at no point did I think about gymnastics. But that's because of my lack of experience in gymnastics, and my school didn't have the program. As a math person, you start thinking about it and you think, “If it's vertically, this doesn't make any sense because we're on a coordinate plane.” So, I started thinking about [it] mathematically and then I thought, “Oh, maybe they did renovations to the gymnasium, and they needed a balance beam.” So, I guess that's the beam that carries the load.
Aina: So, that's how I flipped, in my mind, the image of the beam to be horizontal. Then the teachers, when they told me it's gymnastics, that really threw me off, and it didn't help. And I totally agree with Julia. You know when we do mathematics with children, we tell them, “Can you draw me a picture?”
Mike: Uh-hm. Aina: And what we mean is, “Can you draw me a mathematical picture to support your problem-solving or the strategies you used?” But the piece that was missing for me is an actual picture of what the balance beam is in gymnastics and how it's located, how long it is. So yeah, yeah, that was eye-opening to me.
Mike: It's almost like you put on a different pair of glasses that allow you to see the language of story problems differently, and how that was starting to play out with teachers. I wonder, could you talk about some of the things that they started to do when they were actually with kids in the moment that you looked at and you were like, “Gosh, this is actually accounting for some of the understanding we have about schema and the different types of words.”
Aina: So, the teacher would read a problem, which I think is a good strategy. But then it was very open-ended. “How do you understand what I just read to you? What's going on in the story problem? Turn to your partner, can you envision? Can you think of it? Do you have a picture in your mind?” So, we don't jump into mathematics anymore. We kind of talk about the context, the schema. “Can you position yourself in it? Do you understand what's going on? Can you retell the story to your partner the way you understand it?” And then, we talk about, “So how can we solve this problem? What do you think is happening?” based on their understanding. That really helped, I think, a lot of teachers also to see that sometimes interpretations lead to different solutions, and children pay attention to certain words that may take them to a different mathematical solution. It became really about how language affects our thinking, our schema, our image in the head, and then based on all of that, where do we go mathematically in terms of solving the problem?
Mike: So, there are two pieces that really stuck out for me in what you said. I want to come back to both of them. The first one was, you were describing that set of choices that teachers made about being really open-ended about asking kids, “How do you understand this? Talk to your neighbor about your understanding about this.” And it strikes me that the point you made earlier when you said context has really become an important part of some of the mathematics tasks and the problems we create. This is a strategy that has value not solely for multilingual learners, but really for all learners because context and schema matter a lot.
Aina: Yes.
Mike: Yeah. And I think the other thing that really hits me, Aina, is when you said, “We don't immediately go to the mathematics, we actually try to help kids situate and make sense of the problem.” There's something about that that seems really obvious. When I think back to my own practice as a teacher, I often wonder how I was trying to quickly get kids into the mathematics without giving kids enough time to really make meaning of the situation or the context that we were going to delve into.
Aina: Exactly. Mike, to go back to your question, what teachers can do, because it was such an eye-opening experience that, it's really about the language; don't jump into mathematics. The mathematics and the problem actually is situated around the schema, around the context. And so, children have to understand that first before they get into math. I have a couple of examples if you don't mind, just to kind of help the teachers who are listening to this podcast to have an idea of what we're talking about. One of the things that Julie and I were thinking about is, when you start with a story problem, you have three different categories of vocabulary. You have technical, sub-technical, non-technical. If you have a story problem, how do you parse it apart? OK, in the math story problems we teach to children, it's typically a number and operations.
Aina: Let's say we have a story problem like this: “Mrs. Tatum needs to share 3 grams of glitter equally among 8 art students. How many grams of glitter will each student get?” So, if the teacher is looking at this, technical would definitely be grams: 3, 8, and that is it. Sub-technical, we said “equally,” because equally has that kind of meaning here. It's very precise, it has to be exact amount. But a lot of children sometimes say, “Well that's equally interesting.” That means it's similarly or kind of, or like, but not exact. So, sub-technical might qualify as “equally.” Everything else in the story problem is non-technical: sharing and glitter, art students, each student, how much they would get. I want the teachers to go through and ask a few questions here that we have. So, for example, the teacher can think about starting with sub-technical and non-technical, right?
Aina: Do students understand the meaning of each of these words? Which of these could be confusing to them? And get them to think about the story, the context and the problem. And then see if they understand what the grams are, and 8 and 3. And what's happening. And what do those words mean in this context? Once you have done all this work with children, children are now in this context. They have situated themselves in this. “Oh, there's glitter, there's an art class, there's a teacher, they're going to do a project.” And so, they've discussed this context. Stay with it as a teacher and give them another problem that is the same context. Use as many words from the first problem as you can and change it up a little bit in terms of mathematical implication or mathematical solution. For example, I can change the same problem to be, “Mrs. Tatum needs to buy 3 grams of glitter for each of her 8 art students. How many grams of glitter does she need to buy?” So, the first problem was [a] division problem now becomes a multiplication problem. The context is the same. Children understand the context, especially children like myself, who are ELL, who took the time to process to learn new words, to understand new context, and now they're in this context. Let's use it. Let's now use it for the second piece. So, Mike, you've been talking about two things going on. There's a context, and then there's problem-solving or mathematical problem-solving. So, I believe posing the same question or kind of the same story problem with different mathematical implications gets at the second piece. So, first we make sense of the problem of the context schema. The second is, we make sense of that problem for the purpose of solving it.
Aina: And the purpose of solving it is where these two problems that sound so familiar and situate in the same context but have different mathematical implications for problem-solving. This is where the powerful piece, I think, is missing. If I give them a division problem, they can create a multiplication problem with the same Mrs. Tatum, the art students, the glitter. But what I'd like for them to do and what we've been discussing is how are these two problems similar?
Mike: Uh-hm.
Aina: This kind of gets at children identifying some of the technical. So, the 3 is still there, the 8 is still there, you know, grams are still there. But then, “How are these two story problems different?” This is really schema-mediated vocabulary in the context where they now have to get into sub-technical and non-technical. “Oh, well there there's 3, but it's 3 per student. And this, there were 8 students, and they have to all share the 3 grams of glitter.”
Aina: So, children now get into this context and difference in context and how this is impacting the problem-solving strategies. I'd love for the teachers to then build on that and say, “How would you solve the first problem? What specifically is in the story problem [to] help you solve it, help you decide how to solve it, what strategies, what operations?” And do exactly the same thing for the second problem as well. “Would you solve it the same way? Are the two problems the same? Will they have the same solutions or different? How would you know? What tells you in the story? What helps you decide?” So, that really helps children to now become problem-solvers. The fun is the mathematical variations. So, for example, we can give them a third problem and say, “I have a challenge for you.” For example, “Mrs. Tatum needs to buy 3 grams of glitter for each of her 8 art students for a project, but she only has money today to buy 8 grams of glitter. How much more glitter does she need to still buy for her students to be able to complete their art project?” Again, it's art, it's glitter, it's 8 students, there's 3, the 8. I didn't change the numbers, I didn't change the context, but I did change the mathematical implication for their story problem. I think this is where Julia and I got very excited with how we can use schema-mediated vocabulary and schema in context to help children understand the story, but then really have mathematical discussions about solutions.
Mike: What's interesting about what you're saying is the practices that you all are advocating and describing in the podcast, to me, they strike me as good practice helping kids make meaning and understand and not jumping into the mathematics and recognizing how important that is. That feels like good practice, and it feels particularly important in light of what you're saying.
Julia: I agree. It's good practice. However, what we found when we reviewed literature, because one of the first steps that we took was what does the literature say? We found that focusing instructional practice on teaching children to look for key mathematical terms tends to lead to frequent errors.
Mike: Yep.
Julia: The mathematical vocabulary tends to be privileged when teaching children how to make sense of and solve word problems. We want to draw attention to the sub-technical and non-technical vocabulary, which we found to be influential in making sense of. And as in the examples Aina shared, it was the non-technical words that were the key players, if you will, in solving that problem.
Mike: I'm really glad you brought up that particular point about the challenges that come out of attempting to help kids mark certain keywords and their meanings. Because certainly, as a person who's worked in kindergarten, first grade, second grade, I have absolutely seen that happen. There was a point where I was doing that, and I thought I was doing something that was supporting kids, and I was consistently surprised that it was often like, that doesn't seem to be helping.
Julia: I also used that practice when I was teaching second grade. The first step was circle the keywords. And I would get frustrated because students would still be confused in the research that we found. When you focus on the keywords, which tend to be mathematical terms, then those other words that are integral to making sense of and solving the story problem get left behind.
Mike: The question I wanted to ask both of you before we close is, are there practices that you would say like, “Here's a way that you can take this up in your classroom tomorrow and start to take steps that are supportive of children making sense of word problems”?
Julia: I think the first step is adding in that additional lens. So, when previewing story problems, consider what schema or background knowledge is required to understand this word, these words, and then what students would find additional schema helpful. So, thinking about your specific students, what students would benefit from additional schema and how can I support that schema construction?
Mike: Aina, how about for you?
Aina: Yeah, I have to say I agree with Julia. Schema seemed to be everything. If children don't understand the context and don't make sense of the problem, it's very hard to actually think about solving it. To build on that first step, I don't want teachers to stop there. I want teachers to then go one step further. Present a similar problem or problem that includes [the] same language, same words, as many as you can, maybe even same numbers, definitely same schema and context, but has a different mathematical implication for solving it. So maybe now it's a multiplication problem or addition problem. And really have children talk about how different or similar the problems are. What are the similarities, what are the differences, how their solutions are the same or different? Why that is. So really unpack that mathematical problem-solving piece. Now, after you have made sense of the context and the schema … as an ELL student myself, the more I talked as a child and was able to speak to others and explain my thinking and describe how I understand certain things and be able to ask questions, that was really, really helpful in learning English and then being successful with solving mathematical problems. I think it really opens up so many avenues and to just go beyond helping teachers teach mathematics.
Mike: I know you all have created a resource to help educators make sense of this. Can you talk about it, Julia?
Julia: Absolutely. Aina and I have created a PDF to explain and provide some background knowledge regarding the three types of vocabulary. And Aina has created some story problem examples that help to demonstrate the ways in which sub-technical and non-technical words can influence the mathematical process that's needed. So, this resource will be available for educators wanting to learn more about schema-mediated vocabulary in mathematical story problems.
Mike: That's fantastic. And for listeners, we're going to add this directly to our show notes. I think that's a great place for us to stop. Aina and Julia, I want to thank both of you so much for joining us. It has absolutely been a pleasure talking to both of you.
Aina: Thank you.
Julia: Thank you.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org
Season 2 | Episode 14 – Three Resources to Support Multilingual Learners
Guest: Dr. Erin Turner
Mike Wallus: Many resources for supporting multilingual learners are included with curriculum materials. What's too often missing though is clear guidance for how to use them. In this episode, we're going to talk with Dr. Erin Turner about three resources that are often recommended for supporting multilingual learners. We'll unpack the purpose for each resource and offer a vision for how to put them to good use with your students.
Mike: Well, welcome to the podcast, Erin. We are excited to be chatting with you today.
Erin Turner: Thank you so much for inviting me.
Mike: So, for our listeners, the starting point for this episode was a conversation that you and I had not too long ago, and we were talking about the difference between having a set of resources which might come with a curriculum and having a sense of how to use them. And in this case, we were talking about resources designed to support multilingual learners. So, today we're going to talk through three resources that are often recommended for supporting multilingual learners, and we're going to really dig in and try to unpack the purpose and offer a vision for how to put them to use with students. What do you think? Are you ready to get started, Erin?
Erin: I am.
Mike: Well, one of the resources that often shows up in curriculum are what are often referred to as sentence frames or sentence stems. So, let's start by talking about what these resources are and what purpose they might serve for multilingual learners.
Erin: Great. So, a sentence stem, or sometimes it's called a sentence starter, this is a phrase that gives students a starting place for an explanation. So, often it includes three or four words that are the beginning part of a sentence, and it's followed by a blank that students can complete with their own ideas. And a sentence frame is really similar. A sentence frame just typically is a complete sentence that includes one or more blanks that again, students can fill in with their ideas. And in both cases, these resources are most effective for all students who are working on explaining their ideas, when they're flexible and open-ended. So, you always want to ensure that a sentence stem or a sentence frame has multiple possible ways that students could insert their own ideas, their own phrasing, their own solutions to complete the sentence. The goal is always for the sentence frame to be generative and to support students' production and use of language—and never to be constraining.
Erin: So, students shouldn't feel like there's one word or one answer or one correct or even intended way to complete the frame. It should always feel more open-ended and flexible and generative. For multilingual learners, one of the goals of sentence stems is that the tool puts into place for students some of the grammatical and linguistic structures that can get them started in their talk so that students don't have to worry so much about, “What do I say first?” or “What grammatical structures should I use?” And they can focus more on the content of the idea that they want to communicate. So, the sentence starter is just getting the child talking. It gives them the first three words that they can use to start explaining their idea, and then they can finish using their own insights, their own strategies, their own retellings of a solution, for example.
Mike: Can you share an example of a sentence frame or a sentence stem to help people understand them if this is new to folks?
Erin: Absolutely. So, let's say that we're doing number talks with young children, and in this particular number talk, children are adding two-digit numbers. And so, they're describing the different strategies that they might use to do either a mental math addition of two-digit numbers, or perhaps they've done a strategy on paper. You might think about the potential strategies that students would want to explain and think about sentence frames that would mirror or support the language that children might use. So, a frame that includes blanks might be something like, “I broke apart (blank) into (blank) and (blank).” If you think students are using 10s and 1s strategies, where they're decomposing numbers into 10s and 1s. Or if you think students might be working with open number lines and making jumps, you might offer a frame like, “I started at (blank), then I (blank),” which is a really flexible frame and could allow children to describe ways that they counted on on a number line or made jumps of a particular increment or something else. The idea again is for the sentence frame to be as flexible as possible. You can even have more flexible frames that imply a sequence of steps but don't necessarily frame a specific strategy. So, something like, “First I (blank), then I (blank)” or “I got my answer by (blank).” Those can be frames that children can use for all different kinds of operations or work with tools or representations.
Mike: OK, that sets up my next question. What I think is interesting about what you shared is there might be some created sentence frames or sentence stems that show up with the curricular materials I have, but as an educator, I could actually create my own sentence frames or sentence stems that align with either the strategies that my kids are investigating or would support some of the ideas that I'm trying to draw out in the work that we're doing. Am I making sense of that correctly?
Erin: Absolutely. So many curricula do include sample sentence frames, and they may support your students. But you can always create your own. And one place that I really like to start is by listening to the language that children are already using in the classroom because you want the sentence starters or the sentence frames to feel familiar to students. And by that, I mean you want them to be able to see their own ideas populating the sentence frames so that they can own the language and start to take it up as part of the repertoire of how they speak and communicate their ideas. So, if you have a practice in your classroom, for example, where children share ideas and maybe on chart paper or on the whiteboard you note down phrases from their explanations—perhaps labeled with their name so that we can keep track of who's sharing which idea—you could look across those notations and just start to notice the language that children are already using to explain their strategies and take that as a starting point for the sentence frames that you create. And that really honors children's contributions. It honors their natural ways of talking, and it makes it more likely that children will take up the frames as a tool or a resource.
Mike: Again, I just want to say, I'm so glad you mentioned this. In my mind, a sentence frame or a sentence stem was a tool that came to me with my curriculum materials, and I don't know that I understood that I have agency and that I could listen to kids’ thinking and use that to help design my own sentence frames. One question that comes to mind is, do you have any guardrails or cautions in terms of creating them that would either support kids' language or that could inadvertently make it more challenging?
Erin: So, I'll start with some cautions. One way that I really like to think about sentence frames is that they are resources that we offer children, and I'm using “offer” here really strategically. They're designed to support children's use of language. And when they're not supportive, when children feel like it's harder to use the frame to explain their idea because the way they want to communicate something, the way they want to phrase something doesn't fit into the frame that we've offered, then it's not a useful support. And then it can become a frustrating experience for the child as the child's trying to morph or shape their ideas, which makes sense to them, into a structure that may not make sense. And so, I really think we want to take this idea of offering and not requiring frames really seriously.
Erin: The other caution that I would offer is that frames are not overly complex. And by that, I mean if we start to construct frames with multiple blanks where it becomes more about trying to figure out the teacher's intention and children are thinking, “What word would I put here? What should I insert into this blank?” Then we've lost the purpose. The purpose is to support generative language and to help children communicate their ideas, not to play guessing games with children where they're trying to figure out what we intend for them to fill in. This isn't necessarily a caution, but maybe just a strategy for thinking about whether or not sentence frames could be productive for students in your classroom— particularly for multilingual learners—is to think about multiple ways that they might complete the sentence stem or that they might fill in the sentence frame. And if as a teacher we can't readily come up with four or five different ways that they could populate that frame, chances are it's too constraining and it's not open-ended enough.
Erin: And you might want to take a step back toward a more open-ended or flexible frame. Because you want it to be something that the children can readily complete in varied ways using a range of ideas or strategies. So, something that I think can be really powerful about sentence frames is the way that they position students. For example, when we offer frames like, “I discovered that…” or “I knew my answer was reasonable because…” or “A connection I can make is… .” Those are all sentence starters. The language in those sentence starters communicates something really powerful to multilingual learners and to any student in our classroom. And that's that we assume as a teacher that they're capable of making connections, that they're capable of deciding for themselves if their answer is reasonable, that they're capable of making discoveries. So, the verbs we choose in our sentence frames are really important because of how they position children as competent, as mathematical thinkers, as people with mathematical agency.
Erin: So, sometimes we want to be really purposeful in the language that we choose because of the way that it positions students. Another kind of positioning to think about is that multilingual learners may have questions about things in math class. They may not have clarity about the meaning of a phrase or the meaning of a concept, and that's really true of all students. But we can use sentence frames to normalize those moments of uncertainty or struggle for students. So, at the end of a number talk or at the end of a strategy sharing session, we can offer a sentence frame like, “I had a question about…” or “Something I'm still not sure of is … .” And we can invite children to turn and talk to a partner and to finish that sentence frame.
Erin: That's offering students language to talk about things that they might have questions about, that they might be uncertain about. And it's communicating to all kids that that's an important part of mathematics learning—that everyone has questions. It's not just particular students in the classroom. Everyone has moments of uncertainty. And so, I think it's really important that when we offer these frames to students in our classrooms, they're not positioned as something that some students might need, but they're positioned as tools and resources that all students can benefit from. We all can benefit from an example of a reflection. We all can learn new ways to talk about our ideas. We all can learn new ways to talk about our confusions, and that's not limited to the children that are learning the language of instruction. Otherwise, sentence frames become something that has low status in the classroom or is associated with students [who] might need extra help. And they aren't taken up by children if they're positioned in that way, at least not as effectively.
Mike: The comparison that comes to mind is the ways that in the past manipulatives have been positioned as something that's lower status, right? If you're using them, it means something. Typically, at least in the past, it was something not good. Whereas I hope as a field we've gotten to the place where we think about manipulatives as a tool for kids to help express their thinking and understand and make meaning, and that we're communicating that in our classrooms as well. So, I'm wondering if you can spend just a few minutes, Erin, talking about how an educator might introduce sentence frames or sentence stems and perhaps a little bit about the types of routines that keep them alive in the classroom.
Erin: Yes, thanks for this question. One thing that I found to be really flexible is to start with open-ended sentence frames or sentence stems that can be useful as an attachment or as an enhancement to a routine that children already know. So, just as an example, many teachers use an “I notice, I wonder”- or “We know, we wonder”-type of routine. Those naturally lend themselves to sentence starters. “I notice (blank), I wonder (blank).” Similarly, teachers may be already using a same and different routine in their classroom. You can add or layer a sentence frame onto that routine, and then that frame becomes a tool that can support students' communication in that routine. So, “These are the same because… .” “These are different because… .” And once students are comfortable and they're using sentence frames in those sorts of familiar routines, a next step can be introducing sentence frames that allow children to explain their own thinking or their own strategies.
Erin: And so, we can introduce sentence frames that map onto the strategies that children might use in number talks. We can introduce sentence frames that can support communication around problem-solving strategies. And those can be either really open-ended like, “First…, then I…”-type frames or frames that sort of reflect or represent particular strategies. In every case, it's really important that the teacher introduces the frame or the sentence starter in a whole group. And this can be done in a couple of ways. You can [chorally] read the frame so that all children have a chance to hear what it sounds like to say that frame. And as a teacher, you can model using the frame to describe a particular idea. One thing that I've seen teachers do really effectively is when children are sharing their strategy, teachers often revoice or restate children's strategies sometimes just to amplify it for the rest of the class or to clarify a particular idea.
Erin: As part of that revoicing, as teachers we can model using a sentence frame to describe the idea. So, we could say something like, “Oh, Julio just told us that he decomposed (blank) into two 10s and three 1s,” and we can reference the sentence frame on the board or in another visible place in the classroom so that children are connecting that mathematical idea to potential language that might help them communicate that idea. And that may or may not benefit Julio, the child [who] just shared. But it can benefit other children in the classroom [who] might have solved the problem or have thought about the problem in a similar way but may not yet be connecting their strategy with possible language to describe their strategy. So, by modeling those connections as a teacher, we can help children see how their own ideas might fit into some of these sentence frames.
Erin: We also can pose sentence frames as tool to practice in a partner conversation. So, for example, if children are turning and talking during a number talk and they're sharing their strategy, we can invite children to practice using one of two sentence frames to explain their ideas to a partner. And after that turn and talk moment, we can have a couple of children in the class volunteer their possible ways to complete the sentence frame for the whole group. So, it just gives us examples of what a sentence frame might sound like in relation to an authentic activity. In this case, explaining our thinking about a number talk. And that sort of partner practice or partner rehearsal is really, really important because it gives children the chance to try out a new frame or a new sentence starter in a really low-stress context, just sharing their idea with one other peer, before they might try that out in a whole-class discussion.
Mike: That's really helpful, Erin. I think one of the things that jumps out for me is, when you initially started talking about this, you talked about attaching it to a routine that kids already have a sense of like, “I notice” or “I wonder” or “What's the same?” or “What's different?” And what strikes me is that those are routines that all kids participate in. So again, we're not positioning the resource or the tool of the sentence frame or the sentence starter as only for a particular group of children. They actually benefit all kids. It's positioned as a normal practice that makes sense for everybody to take up.
Erin: Absolutely. And I think we need to position them as ways to enhance things in classrooms for all students. And partner talk is another good example. We often send students off to talk with a partner and give them instructions like, “Go tell your partner how you solve the problem.” And many children aren't quite sure what that conversation looks like or sounds like, even children for whom English is their first language. And so, when we offer sentence frames to guide those interactions, we're offering a support or a potential support for all students. So, for partner talk, we often not only ask kids to explain their thinking, but we say things like, “Oh, and ask your partner questions.” “Find out more about your partner's ideas.” And that can be challenging for 7- and 8-year-olds. So, if we offer sentence frames that are in the form of questions, we can help scaffold those conversations.
Erin: So, things like, “Can you say more about?” or “I have a question about?” or “How did you know to?” If we want children asking each other questions, we need to often offer them supports or give them tools to support that conversation. And that helps them to learn from each other. It helps them to listen to each other, which we know benefits them in multiple ways. And I just want to share one final example about sentence frames that I think is so powerful. There are really different purposes for frames. They can be about reflection. They can be about asking questions of partners. We can use sentence frames to agree and disagree, to compare and contrast.
Erin: One teacher that I've worked with uses sentence frames to guide end-of-lesson reflections. And after children have talked to a partner or shared ideas with a partner, she asked them to complete sentence frames that sound like this: “One thing I learned from my partner today is (blank).” Or “A new idea I got from my partner today was (blank).” And what I love about this is, it positions all kids as having valuable ideas, valuable contributions to offer the class. And if I'm in a partnership with a multilingual learner, I'm thinking deeply about what I learned from that partner, and I'm sharing with the teacher orally or in writing and sharing with the class what I learned from that child. And so, the sentence frame helps me because it gives me a support to think about that idea and to express that idea, but it really helps elevate other children in the classroom who might not always be seen in that way by their peers. So, I think there are just really powerful ways that, that we can use these tools.
Mike: I love that. I'm wondering if we can shift now to a different kind of resource, and this one might be a little less obvious. What we were talking about when we had this conversation earlier was the use of a repeated context across a series of lessons and the extent to which that, in itself, can actually be really supportive of multilingual learners. So, I'm wondering if we can talk about an example and share the ways that this might offer support to students?
Erin: Perfect. So, repeated contexts are wonderful because they offer a rich, really complex space for students to start thinking mathematically, for them to pose questions of their own, and for them to make mathematical observations and solve problems. And the benefit of a repeated context or a context that sort of returns over a sequence of lessons or even across a sequence of units, is that children can start to inhabit the story in the context. They start to learn who the characters are. They learn about the important features of the context, perhaps locations or objects in the context or relationships or key quantities. And every time that that context is reintroduced, the sense-making that they've done previously is a really powerful starting point for the new mathematical ideas that they can explore. And these repeated contexts are especially powerful when they're introduced with multiple supports. So, for multilingual learners, if we can introduce context with narrative stories, with pictures or images, with videos, with physical artifacts, whatever we can do to give children a sense of this, in most cases, imaginary worlds that we're creating, we support their sense-making.
Erin: And this is really different from curricula or programs that offer a new context with each word problem. Or perhaps with each page in a student book, there's a new context introduced. And for multilingual learners and really for all students, every time we introduce a new context, we have to make sense of what's happening in this story. What's happening in the situation? Who are the people? What does this new word mean that I haven't encountered before? And so, we limit our time to really think deeply about the mathematical ideas because we're repeating this space of sense-making around the context. And in classrooms we often don't have that time to unpack context. And so, what happens when we use new contexts every time is that we tend to fast track the sense-making, and children can start to develop all sorts of unproductive ways to dig into problems like looking for a particular word that they think means a particular operation because we just don't give them the time and space to really make sense of the story. And so, because we have this limited time in classrooms, when we can reintroduce context, it really offers that space to students.
Mike: Do you have an example that might help illustrate the point?
Erin: Absolutely. So, in second-grade curriculum that I have reviewed, there's a context around a character, “Jesse and [the] Beanstalk.” It's sort of an adaptation of the classic tale of “Jack and the Beanstalk.” And in this story, Jesse has beans that he gets from an interaction or a sale in a farmers market. And these beans, of course, grow into a giant beanstalk that has a friendly giant that lives at the top. And this beanstalk produces large, giant beans, which have all kinds of seeds inside of them. And this context is used over a series of units. It actually spans most of the school year to give children an opportunity to explore multiple mathematical ideas. So, they make representations of these giant beans with strips of paper, and they use cubes to measure the beans. So, they're looking at linear measurement concepts. They compare the length of different beans, so they're doing addition and subtraction to compare quantities. They find out how many seeds are inside of the beans, and they add those quantities together. So, they're doing all sorts of multi-digit operations, adding the beans.
Erin: And then the context further develops into making bracelets with the seeds that are inside these bean pods, and they group these seeds in groups of 10. So, they have the chance to think about, “How many 10s can we make out of a larger quantity?” Later on, their bracelet-making business expands, and they have to think about how to package these seeds into 100s, 10s, and ones. So, it's a really rich context that develops over time. And children begin to learn about the people in the story, about the activities and the practices that they engage in, and they have the chance to ask their own questions about their story and to make their own connections, which is really powerful. As the story develops, you can see how children develop a sense of curiosity about what's happening, and they become invested in these stories, which really supports the mathematical work.
Mike: So, I want to walk back to our friend Jesse, and I'm glad to hear it's a friendly giant in this particular case. What you were making me think about as you were talking is the way that we introduce the context probably is really important. Could you shed some light on how you think about introducing a context?
Erin: So, asking children to share connections that they can make is really important. When we introduce context with different representations, it's really important to ask children to make connections as a place to start. So, we want to ask them what they already know about this context in particular, or similar context. What connections can they make to their own experiences? We want to ask them to share what they wonder about the context, what they're curious about, what they notice, what observations they can make. And when we have different representations like a story and a picture or a video or an artifact, we give children more possibilities for making those kinds of connections. One thing that we can also do to really support children's connections to the context is, as a context develops over time, we can create anchor charts or other written records with children that represent their perspectives on the key features of the context.
Erin: So, for example, if we go back to “Jesse and the Beanstalk,” after solving a couple of problems about “Jesse and the Beanstalk” and being introduced to that story, we can pause and talk with children about what they see as key aspects of this story. What are things they want to remember? When we come back to Jesse in a few weeks, who are the people [who] we want to remember in this story? What are some important quantities in this story? What are some other important features of this story? And this is not an anchor chart that we create ahead of time as teachers. It's really important that children own these ideas and that they get to start to identify the key quantities, the key features of the situation from their perspective, because then that can become a resource for their thinking later on. We don't have to re-explain the context completely every time. We can refer to these written records that we've co-created with children.
Mike: Well, let's close by talking about one more resource that educators will often find in their curriculum materials. Things like lists of academic vocabulary, or perhaps even cards with vocabulary words printed on them. I wonder how you think educators should understand the value of these particular resources.
Erin: These vocabulary cards can take the form of cards that can be inserted into a chart or even anchor charts themselves. And one thing that I think that's really important, especially when we're thinking about using this tool with multilingual learners, is that these include multiple representations of a concept. We always need to make sure that the cards include a picture or a diagram or a visual image of the term, in addition to an example of how the term can be used. So, that might be a phrase, it might be a symbolic representation of the term. It might be a whole sentence that uses the word to give children an idea of how to use the language in context, which is really important. And one thing that I've seen teachers do really effectively is to create large vocabulary cards with blank space, so that as these cards are introduced in the context of a lesson or activity when they would be relevant, children have the opportunity to share their own ideas about the term.
Erin: And that blank space on the card can be filled with connections that children make. So, children might know that term in another language. That can be added to the vocabulary card. Children might connect that term to another similar idea mathematically or a similar idea in daily life. So, they might know another meaning of the word. That can be added to this blank space so that it becomes a shared and collaboratively generated artifact and not just a static card on the wall of the classroom that is beautiful, but that children may not really use to support their sense-making. So, co-creating these cards with children I've seen to be really powerful, especially if we want them to be used by children and owned by children. And that leaving blank space can help with that.
Mike: So, you're taking this conversation to a place I hoped we might go, which is just to help paint a picture of what it might look like for a teacher to introduce this resource, but then also sustain it, how to bring it to life in the classroom. What does that look like? Or maybe what does that sound like, Erin?
Erin: So, I think when vocabulary card is first introduced, just like with many things in math classrooms, we want children to share what they already know. So, what does this word make you think of? Where do you see this word? Where have you heard this word? What are some other things we've done together in our mathematical work that relate to this word? You want children to share versions of the word in other languages. You want them to share real-world context, connections, anything that they can to connect to their experiences. And it's important that we introduce small sets of words at a time. So, if we're working on a unit on multiplication, we might have words related to “factor” and “multiple” and “products” that become additions to our word wall or to our anchor charts. And that we encourage children to use those words in particular activities in those units.
Erin: So, for example, if we're doing a number talk as part of a unit on multiplication, we might remind children of particular words that have been introduced in prior lessons and encourage them to try to take up those words in their explanations. “See if you can use the word ‘factor’ today as you're sharing your strategy with a partner.” “See if you can use the word ‘product’ today.” And then invite children to share examples of what that sounded like in their partner talk. Or what that looked like when they were writing about their explanation. And it's these constant invitations or repeated invitations that really make these words come to life in classrooms so that they don't just live on the wall. It's also really important that these words are highly visible and accessible for students. So, oftentimes teachers will display key vocabulary words alongside a whiteboard or underneath a whiteboard. Or children might have their own copy of a small set of keywords that they're working on to paste inside of their notebooks. If they're not highly accessible, it really limits children's opportunities to use them on a regular basis.
Erin: So, another way to introduce new vocabulary terms or to support students to use them in context is to connect specific words to a routine that is already in place in your classroom. So, one of my favorites for its potential to help kids use new vocabulary is a routine like, “Which One Doesn't Belong?” But same and different routines that many teachers use also work great for this reason. So, just as an example, if we're doing a “Which One Doesn't Belong?” routine, and the four images that we're using are geometric shapes, we might be able to come up with a list of vocabulary words that would help children describe their decisions about which of the shapes doesn't belong. And we could locate those words alongside our whiteboard or in a visible place in the classroom and just invite children as you're deciding which shape doesn't belong, as you're thinking about how you're going to explain your decision to your partner, think about how you could use some of these words. So, we could have words that describe different kinds of angles or other properties of shapes depending on what we're working on in the curriculum. But that's a way to show children the relevance of particular terms in a routine that they're familiar with and that they're engaged in in the classroom. And that's a way to keep these terms alive.
Mike: That's the thing that I really appreciate about what you just shared, Erin. If I'm autobiographical and I think back to my own practice, I recognize the value, and I aspired for these things to be useful. What you did just now is help paint a picture of what it looks like, not just to introduce the language or support the language, but also to keep these alive in classroom practice. I did have a question that occurred to me. Similar to sentence frames and sentence stems, is there any kind of caution that you would offer when people think about using these?
Erin: Definitely. For me, the most important caution is to not overemphasize formal mathematical vocabulary in classrooms, particularly for multilingual learners. Obviously, we want children to be developing mathematical language, and that's something we want for all children. But if we overemphasize the use of formal terminology, that can constrain communication for students who are developing the language. And we never want students' lack of familiarity or their lack of comfort with a particular vocabulary term to stop their communication or to hinder their communication. We would much rather have children explaining their ideas using all sorts of informal language and gestures and reference to physical models. The important thing is the idea and that children have the opportunity to communicate those ideas. And this formal mathematical language that might be represented in vocabulary cards or on anchor charts will come. It's part of the process, but they most importantly need opportunities to communicate their ideas.
Mike: Well, this has been a really enlightening conversation, Erin, and I'm wondering if before we go, if you have any particular recommendations for educators who are looking to build on what they heard today and continue to take up new ideas of how to support their multilingual learners?
Erin: There's a wonderful set of resources out of the Understanding Language Project at Stanford University, and they have a number of math language routines designed to support multilingual students. Some of them are related to introducing context, which we talked about today. They have a version of a “Three Reads” routine for introducing new contexts that people might find useful. But there's a whole collection of language routines on their website that teachers might find really useful. I always go to TODOS as one of my most meaningful resources for thinking deeply and critically about supporting multilingual learners.
Erin: So, I think that site and all of the books and the journals and the conferences that they develop should definitely be included. And many of the other colleagues that you've had on the podcast have wonderful resources to share, too. So, I think I would start with those two.
Mike: Well, thank you so much for joining us, Erin. It really has been a pleasure talking with you.
Erin: Oh, it's been a pleasure. Thank you again for inviting me.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
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