Latest / Rounding Up / Instructional Routines That Support Mathematics Identity Development with Dr. Fawnda Norman

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Instructional Routines That Support Mathematics Identity Development with Dr. Fawnda Norman · Rounding Up

0:00
Lately, building students' math identity along with their conceptual understanding
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and fluency has become an important aspiration for many elementary mathematics
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teachers. But recognizing the importance of helping students build
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a positive math identity is not the same as knowing how to make that happen
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on a day-to-day basis. Addressing the question,
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"But what do I do?" is a major challenge.
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To examine this further, I'll be talking with Fawnda Norman
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from the University of Wisconsin-Oshkosh about the ways
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we can integrate identity-supporting practices into daily math routines.
0:37
Welcome to the podcast Fawnda.
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Thank you so much for taking the time to share your work with our listeners.
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Yes, thank you for having me.
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I'm wondering if you could talk about what drew you to the work of using routines
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to support students' math identity.
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Yeah, so we have the five strands of math proficiency,
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which has conceptual understanding,
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procedural fluency, strategic competence,
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adaptive reasoning, and then the productive disposition.
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And I feel like curriculums really address those first four really well.
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But when it comes to productive disposition,
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that's a really hard thing for curriculums to include for everybody.
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So I wanted to have a practical aspect of that since that's one
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of those five strands. Also, I'm in the state of Wisconsin,
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and they recently updated their standards and it included some shifts in thinking.
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And one of them was focusing on positive math identity and agency in students.
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And so I'm working with DPI to help teachers be able to do that.
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And so one of these things was coming out was what routines could we
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do to help teachers focus on this positive math identity?
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You're making me think about something that I guess I've known for a long time that
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this is true, but I want to say it out loud:
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that it's really important—but it's not sufficient—to simply care about
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students' math identities, to really impact identity.
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We need to examine the design of the learning experiences and
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the structures for engagement that students are encountering on a daily basis.
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Totally agree. And this is where a lot of my work comes in is that we
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can have the mindset and have really good intentions.
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But if we're not really intentional about doing the identity work with our students,
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it's not going to change that classroom culture.
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And so having some of these routines,
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which you'll see, some of the routines that we have actually are already embedded.
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And it's just being intentional to focus on that identity piece
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is really important, and it will build this classroom culture where students
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are willing to share more, they're gonna take academic risks,
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they're willing to work with each other more,
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which then really influences and helps with academic progress and
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how students are performing then in math.
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The other thing that is coming to mind is that in addition
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to supporting students' identity formation,
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routines also have a real potential to shape what students think math is about,
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like what's valued in math classrooms.
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Are those perspective shifts that the routines we're about to talk about would
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support?
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Yeah, so some of these perspective changes come from Jo Boaler's work with
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her YouCubed website. And some that I focus on,
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especially in these routines, is that math is about learning and not performing.
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That's really difficult in our culture where we are taking standardized testing
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all the time, and we have to give a grade and parents want to know
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how their students are doing.
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But we really want to shift the culture from performing to learning because that's
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where students are gonna actually take that academic risk.
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The other important one that I think we really need to focus on is math
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is about making connections and communicating with each other.
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It's not just about solving the problem and getting the right answer.
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And so if we can get students to communicate with each other,
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they're gonna learn that they are all mathematicians,
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and we can all do this.
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I am struck by a phrase that you used,
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and I just wanna say it again because I think it's really an important
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one for our audience to ponder:
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this notion of learning, not performing.
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And I wonder if you could just say a few more words about what that means
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and why that's so important, not only for students' academic growth,
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but also for the classroom.
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Yeah, when I say learning and not performing,
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I mean that we are focused on the mathematical thinking and what's happening
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as opposed to just getting a correct answer and moving on.
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And so when students are focused on learning and not performing,
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they're not worried about making a mistake.
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They can think through the process of what's happening,
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and they can see that mistake as a stepping stone to learning the correct
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way to do it.
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And so when we focus on the learning,
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we're focused on how do we get the answer?
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What do we do to do it? How can we improve this answer?
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What's the most efficient way to do it?
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As opposed to did we get all get five?
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Great, let's move on, right? We're not talking about any of that learning that's
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happening. And when we focus on the learning,
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students are gonna engage in that.
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When we focus on the answer, we get one or two students that are really quick
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and only get a couple of students that are engaged.
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What you're also making me think is that focusing on learning is also a means
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to the end that you also described,
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which is recognizing that mathematics is about connections.
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Right. And it's about being creative and making sense,
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right? That's what we want it to be about.
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And so when we are focused on learning,
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we can be creative, we can think of different ways to do it.
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I can do it one way, you can do it another way.
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And it's both valuable, good mathematics.
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The other thing that comes to mind,
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too, is that there's value in making those connections because it helps
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you see aspects of a relationship or a concept that maybe weren't obvious
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or were slightly obscured in one strategy and then making those explicit,
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right? There's value not only for the individual,
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but there's value for the collective because in looking at multiple strategies
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and focusing on learning and making connections,
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something that might have been
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partially clear to someone can become much more explicit not only to them
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but to the whole group.
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Exactly. It builds that classroom culture up where we're all valuing each other's
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ideas and then that's gonna go into the micro when they're working
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in their groups or their pairs,
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right? They're gonna be able to discuss that more,
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too, which is what we really want to happen.
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I think some people would hear us talking about classroom culture and connection,
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and they might say that's like a soft,
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fluffy objective. But what you're making me think is there's really a case
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to be made for creating that culture because it enables academic growth
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and learning.
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Right, exactly. That is what I want these routines to do is create this culture that
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we're not just doing these things during the routine,
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but we're gonna do and talk and communicate and connect in other parts
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of the classroom as well. So the routines help the students give them structure
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to learn how to do and talk and communicate about math.
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And then we're hoping that they bring that into when they're working
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on the problem or when they're working in their groups.
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And so the routine gives them a structure that we're hoping that they bring that
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structure into other places where they're doing mathematics.
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So I think this is probably a great time to make a transition
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and talk specifically about two routines,
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notice and wonder and worked examples.
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And I wanna start with notice and wonder,
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or as you describe this, notice and wonder plus.
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So what does this routine look and sound like in a classroom?
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So a lot of curriculums already have a lot of notice and wonder
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in their curriculum embedded as a warm-up or an introductory-launch part
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of the curriculum. And so this is something that I think can be added
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to anybody's routines that they already are doing.
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With notice and wonder, we show a picture and we have students look
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at it and basically just say, "What do you notice and what do you wonder?"
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And it really sparks curiosity.
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And what I want my students to be able to do outside the classroom is look
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at some everyday thing and be like,
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"I wonder how much that costs?" or "I wonder what you'd need to know to
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be able to do that?" Right? We want students to be curious about mathematics outside
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the classroom, and notice and wonder sets them up for that.
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What the plus does—and what I'm adding to it—is culturally relevant math
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teaching. And so when they're looking at
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the picture, and they're noticing and they're wondering,
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and then you can say, "What math questions do you have about this?
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What experience to do you have about this?
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Where have you seen this in our community that we could work with this?"
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So for example, one of the notice and wonders that I've done with middle-schoolers
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with probabilities, I just show them a bunch of dice and there's
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a bunch of different colors and all the dice have different numbers on it.
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And I say, "What do you notice and what do you wonder?"
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And they say, "They're different colors.
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And I see there's like five 5s" or whatever.
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They start to talk about it. And then I say,
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"What games have you played that you play with dice?" And they start bringing up,
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you know, Yahtzee or Farkel or Catan,
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right? And they start talking about that.
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And we start talking about, "Well,
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what kind of probability questions could we ask about dice?" And they can draw
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in on their experiences. And suddenly they're using math in their everyday life,
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and they're seeing how it's related to them.
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And
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they are the mathematicians now because they're being really curious about it.
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What is really awesome is when we can take those questions that they ask,
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like, "What is the probability of getting a 7?
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In Catan, they say it's the best one,
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right? "What is the probability of getting a 7?
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And then we can talk about, "OK,
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what would we need? What information would we need to do that?
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What kind of things could we do?" We don't always have time to do that in our class,
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but when they ask the question and then we can find a way to answer it,
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they have then stepped into
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the mathematician role of asking a a mathematical question and then finding
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out the answer.
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I wanna name something that I am taking away from what you said.
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And then I wanna pressure test that and have you talk about the extent
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to which it feels accurate. So part of what's jumping out for me is
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the plus that you describe when you ask students about like,
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"How do you connect to this or where have you encountered this?" You're injecting
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context. But it's not just context,
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it's personal context, right? And that personal context
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brings a different level of meaning that they can apply
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to this mathematical image or manipulative or structure.
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Does that make sense?
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It does. And I think that's where the power of it is because if we just show them
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pictures, they can talk about pictures,
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right? And they're still not connecting with it.
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You can talk about a picture and not connect with it.
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But as soon as you can bring it into what's your experience with this thing,
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right? Then you're having students connect with it,
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which is what we want them to do in mathematics.
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We want them to connect with the mathematics.
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What's lovely about what you're describing,
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too, is the plus is applicable immediately.
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I can ask that follow-up plus question,
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and I can do that tomorrow, right?
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And then you have the the benefit of kids connecting their lived experiences,
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their background knowledge to those notice and wonders in a way that informs their
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thinking and that you as a teacher can use to build conversations
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and build understanding.
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Right. It's building it both ways,
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right? Like the students are connecting to the math.
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But then also me as a teacher,
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I'm connecting with the students.
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The students are then connecting with me as a teacher.
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And it brings in all sorts of positive math identity aspects to it.
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Cause they can see, ~ "math is real and in my life," right?
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"I can talk to my peers about it.
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I can talk to adults about it.
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I can ask questions about it. I can problem solve and get the answer
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to the questions that I ask about it."
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Well, and this also brings me back to something else you said early,
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which is you see the potential for helping kids kind of redefine what math is,
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that math is, as you said, in their world.
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This thing that didn't feel like math has suddenly become connected.
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And it's like, that's math. I have experience with that.
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That matters. That feels so simple,
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but also so significant for kids who are on a learning journey to make sense
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of math and also see themselves as mathematicians.
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Right. I always say this, and my my own children plus my students are...
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get sick of it. But I'm like, "Math is everywhere."
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Right. So my wife and I have a shower that has a leaky faucet.
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And I'll be like at the sink brushing my teeth.
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And I know that there's a 3-second interval where nothing happens.
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And then I know that there's gonna be:
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blip, blip,blip,blip,blip—five drops—and there's another 3-second interval when
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there's nothing. And then there's five drops,
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right?
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Right.
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I might have overlooked that at different points in life,
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but I'm like,
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"There's mathematics behind that,
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right?" Like there's a predictable pattern that has a time increment,
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and there's a certain amount of water that builds up,
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and there's a certain amount of drips and I could predict that.
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And we have a bucket underneath it because we're like,
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"We might as well water the plants with this water..."
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Right.
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...we're not gonna tear the plumbing out." But I could probably figure
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out how long it would take to fill that bucket based on this this background
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information.
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I mean, you just like made like the best math problem right now.
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"Right!?!"
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This is mathematics, right? And it's everywhere around us.
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And that's what's amazing about what you're trying to do with notice
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and wonder plus. It's a super cool way to have people
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see the mathematical relationships and structure in the world around them.
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Yeah, I really want my students to be curious about things and be able
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to ask questions and then be like,
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"Well, what would I need to find that?" "Well,
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I need to find the measurements." Or "I need to find the price." Cause very rarely
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in life do you get a problem that's like,
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"Juan has a garden that's 6 feet wide and 7 feet long and he has to put dirt in it.
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How much dirt does he have to put in?" Right?
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Like very rarely do we get all of the information that we need and
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all of the things. And here's the area like formula that you need to find
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it length times with, right?
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But when we got them up to real problems,
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we're like, "Hey, I wonder how long the bucket would be.
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Like what kind of information would I need to find that out?"
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OK, so for the audience, I have two pieces of homework.
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One, I would love for you to reach out and let me know what information would
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you need to know in order to figure out how long it will take for my bucket
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to be filled. And then two, I would encourage you to look around the world that
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you live in. Think about some things in your life that you may
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not have considered to be mathematics,
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but in fact, after hearing this conversation with Fawnda,
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might actually be mathematical.
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They might lend themselves to the kind of notice wonder plus that we're talking
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about.
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So Fawnda, let's make a shift.
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I'd love to talk a little bit about this other routine called worked examples.
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And I wonder if you could start by just painting a picture of this routine
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for the listeners.
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Yeah, so with this routine and this routine I see...
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sneakily snuck in two routines because there's two parts to or two different ways
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to do work examples that I share.
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And one of them is through showing...
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I call it, honoring mistakes. Like,
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Worked Examples:Honoring Mistakes.
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And so you would show a worked problem;
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that's fully worked out and it includes pretty common mistakes that students would
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make during that problem.
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And you would ask students to look at the problem.
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And you want them first to notice what did the students do correctly
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in this problem first? Where did it start to go off the rails?
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Where did they start to make the mistake?
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And how would you help them get back on track?
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And so again, it's not fix the mistake,
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right? It's how would you help them get back on track?
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Then you would have students think for themselves,
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pair, and then share out their ideas.
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So do something pair share
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and and have a class discussion about what you would do.
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I often ask, "Hey, is there a visual that maybe would have helped them to
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be able to do this? Is there a different strategy that maybe would
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be a more effective way to do this and what they're trying to do
16:05
in this situation?" And it really helps them think about the math thinking
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and not about the correct answer,
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right? For focusing on the learning and not the performing.
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So this is one of those routines that really helps us do that.
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It also really helps them when they are working with a partner and a partner isn't
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maybe doing the problem the correct way.
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If you have them first focus on what did they do correct,
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they can say, "Hey, I can see that you did this,
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but would you have thought about this?" Right?
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And so it gives them some language and some practice working with students,
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their partners who maybe didn't get it right in a kind,
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gentle, respectful way. So that helps that classroom culture out again.
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The other worked example we have is where they compare two correct strategies.
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You show a problem being solved in two different ways.
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And the students talk about what's similar and what's different about
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the ways that they were solved.
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And then if they could come up with a third way to solve the problem.
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Now, what this routine helps with is it takes the mental load again
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off of finding the correct answer and really focuses on the strategy and
17:08
on the math thinking on what's happening.
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What else I usually do with this is I usually do a pretty traditional
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standard algorithm way and then a more visual way.
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And what I want the students to see is the connection between
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the standard algorithm and the visual way and how really the math is
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the same that's happening. It's just a different way of organizing work.
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And so when students can value different ways of organizing work and
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not just having the standard algorithm be the golden way to do it,
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they have more options
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to figure out their math problems,
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and they can see value in that.
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And by coming up with a third way,
17:43
right, is that creative way, they can think about it.
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And then you can have them journal or talk to their partner about which
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way would you solve it? "Which way works best for you?
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And [in] what situation does this strategy work the best?" And again,
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you're focusing on the strategy and not the answer.
17:58
Now, are they gonna get the right answer?
17:59
Yeah. But we're really focusing on what the math thinking is in that situation.
18:04
There's so much in what you just said.
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And so I'm gonna highlight a couple things that stood out.
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The first one is the extent to which the structure itself in having
18:13
a completely finished strategy with an answer,
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be it correct or incorrect...
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it's a really brilliant way of taking answer getting off the table.
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Whenever you have a problem and the answer is in question,
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it feels like the gravitational pull of finding the answer
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can sometimes overwhelm the kind of discussion you want to have.
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So I think I'm taken with the fact that by having a clear answer
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in both iterations of this routine,
18:46
you've kind of taken that answer-getting inclination that seems to persist...
18:51
Right.
18:52
...and you've sidelined it. And so it's a really fascinating way to keep
18:56
the focus consistently on the mathematics,
18:58
the structure, the thinking, the reasoning—
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in a way that's often really difficult,
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even with the best of intentions.
19:05
Yeah, I think, you know, the pressure to get the correct answer is
19:08
a lot for students. Again, it's that mental load of like,
19:11
"Did I get it right? Did I get it wrong?" Right?
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In these worked examples, you tell them it's right or it's wrong,
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right? You know, they know right away,
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and they're not trying to figure out the answer.
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They're trying to talk about the strategy and why it works and what's working
19:24
and what's similar and what's you know,
19:25
right? And you're actually getting to higher-level thinking skills
19:28
on the Bloom's Taxonomy,
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right? We're not just recalling information and getting the correct answer.
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We are actually analyzing, and we're comparing,
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and we're, you know, seeing which one works the best,
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right? And we're maybe—even if we're coming up with a third way—we're creating,
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right? And so we're getting really into those upper-level thinking skills
19:46
in a low-pressure situation where students aren't worried about getting
19:49
the correct answer.
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The other thing that jumped out for me is the potential for language development.
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Because in sidelining the answer and focusing on reasoning,
20:02
particularly in the case where you're saying,
20:04
"How can we help this person get back on track?" That's really
20:07
an invitation to think about how I would name these things,
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how would I explain these things?
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As you said, like what kind of visual might I create or what tool might I use.
20:17
All of those have language attached to them.
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So there's this productive space for language to emerge and then to connect that
20:24
to either the visuals or the procedures or what have you.
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But there's something about the way that language can emerge from this task that
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seems really powerful.
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It is. And I think as we think about language,
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oftentimes with the worked examples,
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you know, depending on the age of your students,
20:39
you might have sentence starters to help them,
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right? I found, you know, they did this correct.
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They did this incorrect, right?
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Like you can have some sentence starters that's gonna help your language learners,
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that's gonna help your regular ed students have really high academic language.
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And so having these routines are gonna build that math identity and also build that
21:00
math fluency as well.
21:02
We talked about this when we were getting ready for this interview.
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And it's been with me for a while,
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but it really came back in listening to you talk today,
21:08
is kids often, because of, you know,
21:12
like sometimes it's my dad showed me how to do this algorithm,
21:16
right? Like some kids often feel like they're kind of predisposed
21:18
to privilege the standard algorithm as,
21:21
"that's the best way to do it." Or "that's the way we should arrive at."
21:24
And so I'm taken with being able to take a standard algorithm
21:29
and a strategy that's more visual or that unveils the structure of
21:34
the number system more clearly and compare and contrast those because one,
21:41
it takes away that, "But this is the way you're supposed to do it,
21:44
the standard algorithm." And it allows you to look at that in a different way.
21:49
Because I think
21:50
with those algorithms, oftentimes the mathematics is there.
21:53
It's just obscured.
21:54
Right.
21:55
It's difficult to see in the same way that you can see it in a more visual model.
21:59
And so you end up with kids who ultimately do start using the standard algorithm,
22:05
but they've had a different kind of experience where they've really been able
22:08
to understand the mathematics of it in a way that,
22:12
I think, we probably would both agree is really significant.
22:15
Yeah, there is efficiency to a lot of the standard algorithms that are out there,
22:19
right? Like that's why we use them is because they're efficient.
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But if we are only taught them as a procedure and not as
22:26
a conceptual understanding of how the mathematics is working and why
22:29
it works that way, then we're just memorizing.
22:33
We're not really thinking through it.
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And so when we can show what's happening with the math,
22:39
right? We have the visual that shows what's happening with the math.
22:42
it oftentimes helps the students use those standard algorithms in
22:46
a better way because they understand the math is happening and where it's showing.
22:49
What I really want students to understand with algorithms is that it's
22:53
a way to organize our mathematical thinking.
22:56
And we want that to be efficient,
22:58
and we also want it to be understandable.
23:00
And so the standard algorithm is one way to do that,
23:03
right? But number lines work very well to do that.
23:06
Area models work really well to do that,
23:08
and they also have the same mathematics.
23:10
And so just 'cause you're drawing an area model doesn't mean that you're
23:13
not doing real math. That's what I want my students to walk away from.
23:16
"I'm doing real math here." "I'm doing real math here." And they're both showing
23:20
the same thing.
23:21
I think that's really powerful.
23:22
I really appreciate you unpacking that a little bit more.
23:24
One thing that I was going to mention,
23:26
particularly if we have people who are using Bridges,
23:28
this really, for me, connects a lot to opportunities that teachers have
23:34
in grades three, four, and five,
23:35
where Bridges has math forums.
23:36
And I think, Fawnda, what you're talking about with worked examples
23:39
to me feels super applicable to that structure.
23:42
So if you're a Bridges listener and you're teaching third,
23:45
fourth, and fifth grade, I think this is probably a real opportunity to
23:48
try out some of these ideas that Fawnda's talking about.
23:51
Fawnda, before we finish the conversation,
23:53
I wanna ask you a question that I ask every guest.
23:55
If someone wants to learn more about the routines and the practices you've shared
24:00
today, what are the resources you'd recommend that they seek out?
24:03
So I've been working with DPI in Wisconsin,
24:06
which is their department of education,
24:07
and we are releasing these routines,
24:10
and they will be on it's WISE Learn—W-I-S-E-L-E-A-R-N—with through
24:16
the DPI of Wisconsin. And it's the Math Identity Project.
24:20
And there's five routines, and so we've looked at two of them today.
24:24
So there's three more that you can find more about.
24:26
They'll have resources for K–2,
24:29
3–5,
24:31
6–8, and 9–12. So it goes all the way up to high school,
24:35
the resources.
24:36
That's fantastic. For listeners,
24:38
we'll put a link to those in our show notes.
24:39
Fawnda, thank you so much for joining us.
24:42
It's really been a pleasure talking with you.
24:43
It's been great to be here. Thank you so much.
24:45
And thank you, our audience, for listening.
24:47
We'd love to hear what you all think.
24:50
What's an aha moment you had as a teacher that inspired you to change
24:55
the culture in your classroom?
24:56
And just like my leaky faucet story,
24:59
what's an experience you had where math unexpectedly entered your life?
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