Dr. Kim Markworth, Open Tasks
ROUNDING UP: SEASON 1 | EPISODE 12
Mike Wallus: Lately, terms like “rich tasks,” “multiple entry points,” and “low floor, high ceiling” are being used so often in the world of mathematics education that many educators are confused about their meaning. Today we talk with Kim Markworth, director of content development at The Math Learning Center, about what these terms look like in practice and how they support student learning. Welcome to the podcast, Kim. It's great to have you.
Kim Markworth: Thank you, Mike. I'm really honored and excited to be here.
Mike: I would love to start this conversation by talking about what it means for a task to be open-ended and have a low floor and a high ceiling. So, is there a way that you think about these terms that might help educators clarify their meaning?
Kim: That's a great question, Mike. In truth, when we think about these terms, they're really all interconnected. And I don't know that anyone has really settled on meanings. And lately there's been a bit of a transition from thinking about low floor and high ceiling to low floor and no ceiling at all. And so, when we think about the variations across continua, this really contributes to how we look at tasks and what we appreciate about tasks. And so, when I think about open-ended, this might correspond with high ceiling or no ceiling. We could keep going with the task. There's not really a defined ending point, uh, but instead many pathways where we could take this task further. And when I think about multiple entry points, this might correspond with low floor. And there [are] so many different ways to approach or enter a task. And rich task, probably my favorite term of them all, corresponds to what we've maybe called a problem historically, where we don't have an anticipated solution path or maybe we haven't solved something like this before. We might have multiple solutions to the task. We might have multiple solution paths or viable strategies. We could have opportunities for extension or generalization. And I love tasks that have an unexpected twist or a need to think about something in a unique way, like where your first inclination, your intuition, might be wrong. And all of a sudden you're like, “Ooh, this is really something that I wasn't expecting.”
Mike: Can you share an example or a few examples with folks who are listening? There's so much that you said just there about the nature of these things. Are you OK telling us a little bit about one or two of them?
Kim: Yeah. One task I'm really excited about is a new third grade task that we've been working into our new curriculum. And this task is positioned at the beginning of a unit on multiplication. And so, for third graders, this is really an introduction-to-multiplication task. And I'm going to apologize to the audience here because this really does require some mental visualization, but I'm going to describe this image that third graders look at, and it involves a pet store. And so, if you can imagine a pet store and going into a pet store, then we have a dog bone display, and these dog bones are hung on three hooks, and Annie took, there are two packages—so that the back package isn't visible because it's behind the front package. So, three hooks, two packages per hook, and in each package there are eight dog bones. And those eight dog bones are arranged in a four-by-two array.
Kim: And finally, one additional detail here. Each package is labeled $12. And this is a problem-posing situation for kids where we put this image in front of kids and ask them to think about the different mathematical questions that they might ask about this particular image. So ultimately, the question that we ask them to explore is, “How many dog bones are on display?” Once you have that image in your head, I want you to be thinking about that final question. And the numbers right now are kind of irrelevant for this discussion, but it's highly unlikely that it's something that kids will already know.
Kim: So, this is a problem-posing situation for the third graders. It's asking them what mathematical questions could you ask about this particular display? And ultimately, we're going to direct them to how many dog bones are on display, but they could explore other questions with additional time. So, how much for all of these dog bones? How much per dog bone? If you had a given number of dogs, how many dog bones would each get? And so, this to me is a task that is open-ended. We could go multiple ways with this, although we are going to focus in the classroom on a particular question. But it's also rich in that kids are going to connect with it, especially if they've been into shops like this. It has multiple entry points. And so, in a lot of ways it really connects to these different terms that you've brought up earlier.
Mike: It's interesting because as you describe it, particularly the fact that there's a visual component to this, it really comes clear how there are multiple ways that a child could think about the question or attack the question that you asked. Is there any role for number choice in thinking about how to design a rich task?
Kim: Yeah, the numbers are really important. And it's fun to play with different numbers and see how they pan out. And so, in thinking about the dog bone task, we want to keep the numbers accessible. So, when I think about a single peg or a single hook for the dog bone packages, I can think about eight plus eight. And from where kids are coming from, eight plus eight should be accessible. When I think about what I'm seeing with the dog bones, I'm seeing two groups of four in each of the packages. And so, that's a nice way looking at doubles that students might find useful, but I'm also looking at three packages of eight. And so, I could add eight plus eight plus eight, which could bring the teacher very easily to a three-times-eight multiplication expression. One that is manageable and a great way to introduce multiplication, the times symbol where we're going with all this, but one that is also still accessible for kids to be thinking about as repeated addition.
Kim: Ultimately, the numbers get to a final answer of 48 dog bones, and you could think about this in terms of six times eight. But kids aren't going to know six times eight, at least not very commonly in this point of an early introduction to multiplication. But there's various ways that they could get to the eight. We ultimately landed on these numbers: 2, 4, 8, 16, but also this additional number 3, which as a separate prime number really throws some additional mathematical thinking into the mix that elevates the task itself. There's things that are critical to be thinking about as you imagine this task and how it might play out instructionally. It's really important to think about how this stands in the curriculum sequence. So, it is introductory, it's using numbers that the kids are probably not going to know off the top of their heads—related multiplication fact and an answer—but the numbers themselves might elicit different strategies, and the visual might elicit different strategies. And all of this connects to the commutative and associated properties for multiplication and how they might play out with student thinking.
Kim: And so, it's all connected. All these pieces really fit together into what I think is a really interesting and engaging task for students. It's challenging enough, but it's also very accessible simply by counting, kids could count what they see. And the context is engaging for kids as well, because they might be thinking about displays that they've seen and how this corresponds to trips to the store that they've had.
Mike: Part of what you've got me thinking, Kim, is there's the design of the task and then there's the element of how a teacher might go about facilitating it. And I think, I want to come back to that, too, because I heard you not only describe the task—the way that it was designed—but you also described some of the ways that a teacher might introduce it, some of the things that they might pose to kids. I wonder if you'd be willing to talk a little bit about facilitation and some of the things about facilitation that can bring a task to life?
Kim: So, one of the things that I like to think about in implementing tasks, Mike, is the very intentional letting go—that kids need time to think. They need time to explore. I think all too often as teachers, we have this desire to go in and help and direct. And sometimes we just need to back off and let them think about the different ways that they might approach that, those different entry points, and let them explore and let them take the time to do that. And so, one of the things that I always used to describe to pre-service teachers was the walk away. That I would go up and talk to a group or talk to some partners working together and listen to what they were doing and maybe pose an additional question and then I'd walk away. I didn't want to hear their answer right away. I wanted them to talk about that amongst themselves. But if I stood there, they would start talking to me. And so, I would walk away, move on to a different group, come back later and hear what their thinking was. But it creates that space for letting kids explore, think about, and also not feel the pressure to be getting to a particular answer in a particular time frame. And I think that's really important for kids to have that freedom.
Mike: Yeah. It also strikes me that you're reframing your role for kids, too, in the sense that by walking away, you're sending the signal that, “I actually have confidence that you and your partners can think about this and reason about this.”
Kim: Absolutely. It is putting some power, some agency, with the students themselves. “You are capable of doing this; you're capable of thinking about this. You do not need me here to be your sounding board or the mathematical authority.”
Mike: Kim, can you talk a little bit about the idea of entry points? I'm wondering for teachers in the field, how would you actually define an entry point? What does that look like?
Kim: I'm not sure I have a good definition for it, but I do have an analogy, and I would compare it to on-ramps for highways. And when I think about on-ramps, we can all get on the same highway, but we might do it at different places, and we might make choices for where we get on based on our current location or what we know about the on-ramp. But as long as I have a workable vehicle, I can do it. And maybe that's our prior knowledge. But unfortunately, often kids, they don't think that they have a workable vehicle or teachers might even underestimate the child's vehicle that they have. And so, I've probably gone far enough with this analogy, but kids come onto that mathematical highway at different places with particular problems. And I think making sure that we as teachers, as educators, as curriculum designers, that we're thinking about all those different possibilities for getting into a problem and knowing that we can all go to the same place regardless of where we've gotten on.
Mike: That's really helpful. So, if I'm an educator and I'm designing a task, or even if I'm facilitating a task that comes with my curriculum, what guidance would you offer to folks to ensure that there are entry points for kids?
Kim: I think it really depends on the mathematics and what you're trying to accomplish. And so, with this one in particular—the dog bones visual—I might be asking myself, “Can I do it without multiplication since this is an entry point for multiplication. Can I do it without that, or could I do it with basic counting skills?” And so, are those viable entry paths open for kids if they don't have where we're going with the task already in their toolkit? When I think about ensuring that there's entry points, I like to think about stripping away the expectations for where you want to go with the task, really allowing kids to have that freedom for exploration. And it's the variety of entry points that leads to the multiple strategies. And when you have multiple strategies, you can make connections between and among those representations. And then you've got something really robust. Or I might go back to that term rich task. And so, it's about can they do it without where you're going, that mathematical goal, and however they encounter that or engage with it, does it still connect to other strategies that will bring them to your mathematical goal?
Mike: That is really helpful. What that has me thinking is we have heard in the field about the idea of the five practices and anticipating. But this is a little bit of a twist on that in the sense that you're evaluating the task and saying, “What's possible for a kid to get into this task?” I love the example of multiplication. So, for example, if I only have partial or emergent understanding of multiplication, can I still work my way toward an answer to that? And if the answer is no, then what?
Kim: Right? But you mentioned partial or emergent understanding, whereas I think this task, actually, you can get in with no understanding of multiplication. I can look at three sets of eight dog bones, add eight plus eight plus eight to get to 24, and then the teacher has that to latch onto to say, “We have another way of writing this. I can write three times eight to represent eight being added three times.” And so even that visual structure leads us to something that we can hook on to, to bring forth the connection to multiplication.
Mike: I think that's helpful because it means that there's a value and there's a utility to having ways of doing this that you can ultimately connect to the place where you want to go, right?
Kim: Yeah. And I think there's opportunities for that kind of reasoning or openness throughout math education; imagining what we can do with tasks to really not just ensure that there's entry points, but value those entry points as really important connections to all students’ prior knowledge and where we're going mathematically.
Mike: That totally makes sense.
Kim: If we can't imagine that our kids are capable of problem-solving and engaging in challenging tasks, then they won't be able to imagine that themselves. And so, in a way, we have to pass along the agency by sometimes just believing ourselves that, ‘Yeah, maybe they can do this,’ and giving them that time and seeing what happens.
Mike: Before we close the conversation, I'm wondering if you have any resources that you think would help someone listening to this conversation deepen their understanding of designing or implementing rich tasks?
Kim: I think the best way to really think about task design and build your own facility is to do some rich tasks and just engage with them as a learner. So, one of the resources that I frequently turn to is the NCTM journals, both old and new. They have really good problems in there. And you can sometimes take one of those and change it in a way that is making it more challenging or making it a more generalizable situation. There's various problem-solving publications. I could certainly plug my own books, Problem Solving in All Seasons (PreK–2, 3–5).
Mike: I have read it, Kim.
Mike: I would absolutely recommend it.
Kim: Another resource that I love is NRICH. It's a website—NRICH—which is nrich.maths.org. Those are just incredible tasks that really get you thinking in various ways about, [some] good problem-solving experiences. So, what I would recommend for teachers or other people who are interested in this, is to really do that mathematics and then reflect on what made it interesting for you. What surprised you? Were there twists? Were there things, stuck points where you had to get past? And then also to extend the thinking by asking yourself something like, “So does this always work? Or when does this work?” And how could you apply the mathematics to more broad situations? And finally, I think it's really important for teachers to put themselves in the minds of their students and sense what might excite them or challenge them.
Mike: Thank you so much for joining us, Kim. It's really been a pleasure talking to you.
Kim: Thank you, Mike.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.