Latest / Rounding Up / Organizing Board Space to Support Reasoning & Connection, Part 2, with Dr. Tutita Casa & Katherine Mack

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Organizing Board Space to Support Reasoning & Connection, Part 2, with Dr. Tutita Casa & Katherine Mack · Rounding Up

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On the last episode of Rounding Up,
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we spoke with Dr. Tutita Casa and doctoral student Katherine Mack about
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the impact of a classroom's layout,
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specifically the influence of the board as a resource for classroom communication.
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Because that conversation was so rich,
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we decided to welcome Tutita and Katherine back to dive a little bit deeper.
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So, Katherine, another element of the board-related practice that we've talked about
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is something, I think you've described it as calling for shared attention.
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I wonder what this means and how it might help sustain students' ideas during
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a discussion.
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So we were really looking closely at the patterns that emerged
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as preservice teachers were recording student thinking on the board.
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And one pattern that just kept coming up over and over is that teachers like
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to call for shared attention on the board,
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whether it be an equation or a single digit.
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And so calling for shared attention means that the teacher
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is not only writing something on the board but also helping the class attend
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to it together through underlining,
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color-coding, enclosing. In a discussion,
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student ideas can disappear really quickly,
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but when a teacher records that idea and then calls the class's attention back
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to it, then the idea becomes available once again for continued reasoning.
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Another pattern that we noticed when we started really observing
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the board moves that our preservice teachers were enacting over and over again
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is something we call bookending,
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where they begin the board work by anchoring the task
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in a prominent place on the board,
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and then they end with a whole class takeaway or a shared understanding.
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And we feel that that structure can be really supportive because it gives
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the board work a beginning, a middle,
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and an ending, which kind of circles back to why we're calling this
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a mathematical story. But we hope that by beginning to understand
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how teachers naturally approach board work,
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we as math teacher educators hope to design better supports for them so that these
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board moves aren't expected to be developed intuitively.
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Could you say more about bookending?
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Just perhaps an example of those kinds of steps that you were describing?
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Yeah, absolutely. So whether it be a number talk,
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a number string, a word problem,
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we noticed that at the board, preservice teachers start by anchoring
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a problem prominently, whether that be a rich task or a story problem.
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But we also noticed that sometimes they didn't.
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And we kind of asked ourselves how important it is to give board work
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a beginning, middle,
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and end. And we kind of realized that making sure that
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the task was prominent and then ending with a shared takeaway makes
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the math work and the reasoning more accessible.
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So I'm intrigued by this notion of anchoring the task prominently.
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I'm trying in my own mind to picture what that means or maybe what that looks
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or sounds like.
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We noticed our preservice teachers putting a lot of thought into
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how they began their board work.
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So sometimes the preservice teachers would write the equation at the center
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of the board and draw a large circle around it and then write student strategies
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around the anchor in a web, or sometimes the preservice teachers would write the
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anchor problem very large at the top of the board,
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which we're calling a prominent placement.
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And in terms of a story problem,
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we know that a lot of times students forget that
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a task is asking them why and not just asking for the answer.
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And so we've noticed preservice teachers.
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making sure to write "why?" at the end—sometimes in all caps,
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sometimes with three lines under them—to really draw shared attention that this
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task is not just asking you, the students,
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for the answer, but also asking students to explain their thinking.
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Thank you so much for helping me see that in my mind's eye,
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Katherine. What is really exciting about that is I can imagine
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how those micro moves: putting it in the center and then recording around it,
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placing it up high. I'm really in love with this micro move of,
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if it's a contextual story problem,
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having the word "why" in capital letters large just as a way of cueing
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the parts of the task that are central or cueing the big question that's being
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asked. All of those seem like both super practical,
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but really powerful things that a teacher could do at the board.
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And I think one thing for teachers to think about is taking a step back.
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You know, again, going back to that idea that the board has typically been
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the teacher's domain. And this is one part of the board that we want
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to make sure that the class understands what the discussion is about.
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Without having that anchor
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with the task, like what are we really talking about?
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To me, it's analogous to picking up a book that might be really good,
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but there's no title page. We don't know what it's about.
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And especially with our youngest students—or actually,
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quite frankly, anyone who's learning—you might get attached to
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a particular idea and kind of forget,
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"Wait, what were we really talking about?" And so that anchoring the board with what
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the main task is about
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allow students to continuously read what that might be.
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But also we found with our preservice teachers it's a reminder,
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like, "Wait, what are we really talking about here?" And if they have that "why"
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in caps and underlined, it's not just for the class's benefit,
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it's also for the preservice teachers who are starting to learn really good
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practice.
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And it's not just repeating what the task is.
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So often,
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in my experience, I've noticed that students have these wonderful ideas
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and it just so happens to be not aligned with the task at hand.
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And it's not because they don't know;
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is that because they don't understand the task?
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And so I always feel it's really important part of practice to pause
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and make sure all the students understand what the task really is asking before
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you dive into really the fun problem solving and that figuring it out
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part of it is to make sure that the discussion is aligned with your instructional
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goals. But importantly, it's not just copying that task on the board.
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It overlaps with classroom management.
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I call it the intellectual component of classroom management,
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is ask one, two, three students to repeat the task in their
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own words because it allows the rest of the class to really understand
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if the language
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in the original task, which is,
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you know, supposed to be a rich one,
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by definition, it's complex. And so let's make sure we really understand it.
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So by asking the students to rephrase it and write it in their own words,
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as a teacher, you're gaining confidence that the class understands what this
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is about—
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or maybe that you need to tweak that original one and make sure they understand
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before you dive in—but it's also helping everyone else who might
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not understand the original one or that first rewording.
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What is it really asking? And it starts to get your ears going.
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So I want to go to the other end of the bookend.
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And there's another bit of language that I want to try to make real
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for people so they can see it in their mind's eye.
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I love this notion that you end with,
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I think you called it a shared takeaway.
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Talk a little bit about what that might look or sound like.
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Just encouraging teachers to have an end of their board work
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or they pause to understand where their class is at the
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end of the lesson. Whether that means that your class has a partial
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or developing understanding of the task,
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or whether that means that the class is indicating that they
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are
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understanding the, the problem at hand,
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just making it clear to the class that this
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is our shared understanding at this time.
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I think it's the idea that sometimes you have an hour-long lesson
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and you're supposed to just wrap it up at the 59th minute,
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and that's just not the reality.
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And rather than teachers being dictated by the clock,
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we need to really give our students the space to reason themselves.
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Again, that's the residue, that's the really powerful part of learning.
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And they might need just another 11 minutes.
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So it might be the next day, might be the next period,
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it might be, you know, after they come back from recess,
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and all of a sudden there's like this light-bulb moment.
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And that idea of bookending at the end is that conclusion that
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we are giving our class, our individual and collective students,
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the ownership over their ideas,
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and they get to decide if they are valid.
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And that's such an important mathematical disposition to develop.
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It's not that the teacher is validating your answer.
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We need to make sure that we are validating—we as students,
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individuals, anyone doing mathematics—is validating these ideas.
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So by having that bookend on the board symbolically showcases that
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mathematical practice, but it's also honoring students' voices
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and ensuring that they continue to share that space along with the teacher
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who is the one who's deciding that we are now ready.
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Over time, they pick up on these norms and they understand how the board
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is being used, and be like, OK,
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pretty much telling the teacher what to do,
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like, "Record these ideas. We got it." Which is amazing to see.
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But the teacher does need to agree [as] the one who has more
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of that content knowledge that knows that the class is where they need
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to be to solidify that understanding.
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Got you.
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I want to name something that I think is important.
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And you made this crystal clear in the way that you described this.
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I think it would have been possible for someone to hear the language that
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you used to describe this end of the bookend and think:
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"This is where we get to the right answer." And that is not what I hear you saying.
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In fact, I'm gonna use your language,
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Katherine: "It's our current understanding now." Which is important to name.
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It is an ending, it's an ending for now.
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But I think that feels like a really important distinction that it's not like:
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We've been working on this. There are some ideas in the room and like,
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"Well, we're at the end, so now is the point where I as the teacher funnel
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to the answer." That's not what you're saying.
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You're actually saying, "No, let's just take a moment as a class
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to recognize our time on this is closing,
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so let's understand where we're at right now." And that's not only valuable
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for the kids, but it's also helpful as a teacher because it helps
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you think about next steps.
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I agree that emphasizing that partial or developing understandings [belonging]
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on the board is a very important idea and we definitely want to advocate
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for a classroom culture where reasoning does not have to
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be fully developed and there are all kinds of different answers.
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And building off of that, I think another important component of the bookend is:
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It's not just about the answer.
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There's nothing special about the number 54 without context.
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And so if 54 happens to be the correct answer,
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why is that a valid one? So that's an important component of the bookending.
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And there might be different ways to reason to get to 54 as the correct answer.
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But that's the part that we want our students to leave us with and really plant that
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seed so that they can use those strategies and that reasoning repeatedly
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as they continue within learning different mathematical topics.
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You're pointing me toward another part of the conversation we had when
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we were preparing to do this interview.
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I believe what you said is: Part of what's going on because of the way
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the board has historically been positioned is that the way that
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the teacher writes and records on the board,
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to a large extent, it can actually communicate to kids the options that they think
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they have and what's valued.
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I wondered if either of you wanted to say a little bit more about that norm
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or that relationship.
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Actually, I think that's one of the most important norms because so often
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the board has been used as a repository of correct ideas,
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especially if, if particularly when the teacher is in command of the board.
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And the process of reasoning, that problem solving,
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the sensemaking that really defines the discipline of mathematics.
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People are wrong, or it's not as an efficient strategy,
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or they, you know, it just doesn't work within that
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problem, even though it might be a sophisticated way to attempt it.
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And one of the norms that's, I think,
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going to be more challenging for teachers who are interested in using the board
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as a shared space is to record students' incorrect ideas and do
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it with a Switzerland type of stance.
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In other words, students starting in pre-K pick up on these patterns that
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teachers have, where if they
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decide not to write some student ideas on the board,
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they very quickly will start to figure out that's not valued.
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And it's probably not valued because it's wrong.
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And so what we want to do is ensure that everything is put up there.
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It's just out there to be investigated.
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And it might be that the class decides not to go with it.
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I recall a first grader I was working with who
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changed his own mind. I had already written his idea on the board,
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and it was incorrect. He changed his own mind.
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And I remember struggling with like,
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"Well, do I erase it? Do I cross it out?
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What do I do?" And I was really mindful that I wanted to honor his voice,
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especially because he was a student with diagnosed special needs,
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and I did not want to diminish his wonderful contribution.
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And I struggled, but then eventually I just put "no" in parentheses there.
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And the class agreed, and they were fine with it.
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Let's move on. But it was still there.
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It was still captured. Then we were able to focus in on the ideas that they wanted
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to talk about where they were starting to get to the correct ideas.
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And so using something, I remember circling the ideas we were still talking about.
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So that's where you really want to just take a step back and it's both:
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"What's the story you want to tell?" but also,
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"What is it that you are having your students read?," which actually starts before
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you even write something. You have to make that decision
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to capture everything that's part of the discussion.
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So Katherine, I wanna ask you about some of the language that your colleague used.
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She used the language, "You wanna take a Switzerland-like approach to that."
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And I wonder if you can help unpack what Tutita was driving at with that.
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Wish that I had heard that being used years ago when I was a teacher,
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because I do remember being an early teacher and standing in front of
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the board and hearing a wrong answer and thinking,
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you know, as a novice, "I can't write that on the board.
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They're all going to think that that is correct.
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It's, for the rest of their life they're going to understand that wrong answer
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is correct." And that's just not true.
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And
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what do you do in that moment as a new teacher when you have
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a young learner excitedly telling you an answer that is incorrect?
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And you just, you write it on the board and you build off of it.
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And I think students learn to look for those cues on teachers' faces.
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It you know, "Is what I'm saying correct or incorrect?" And we just really want
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to encourage students to reason,
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and not be afraid to start somewhere and build off of it.
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And I think acting as Switzerland is a very effective way to do that
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for the students, but it's also giving novice teachers language to know what
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to do in those moments so they don't panic when it happens in those first years.
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So we've reached the point of the podcast where I will typically ask guests
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to share one or two resources,
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one or two small steps, one or two questions that educators might
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ask themselves if they're thinking about where they go with the ideas we've been
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discussing. So I want to offer the two of you that opportunity as well.
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Are there one or two small steps that a listener might take?
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Are there one or two questions that you imagine a teacher might
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ask themselves as they're thinking about their work at the board?
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Tell me a little bit about what you're thinking.
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I think a great first step is becoming aware of your own conception of the board.
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As we've been talking about, the board has been so deeply encultured as
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a space for teacher thinking. It can take time to step back
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and notice how we are actually using it.
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And I think a simple yet powerful instructional shift is
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to flip the classic
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I do, we do, you do model. If we want to promote reasoning,
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we might start with you do. Students reason first.
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Then move to we do. The class examines and connects that reasoning.
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And then finally, lastly, I do.
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The teacher clarifies or formalizes the mathematics only after students have
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first had that chance to think.
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And that's sending a message to the class that student reasoning matters first.
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There's so many benefits to that.
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It gives teachers formative assessment information about where students
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are in their thinking. And as we were talking about earlier,
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it really shows students that those early and developing ideas are just as valuable.
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And importantly, it helps students build their reasoning muscles
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by positioning their
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own thinking as something worth recording in that most prominent space,
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that is the classroom board.
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And jumping off of what Katherine mentioned,
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there are two things that I would recommend.
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The first is to be OK with losing control.
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I think as math teachers, we so want our students to get
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the answer right that sometimes I think we try to speed through the process
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to get there.
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And second, and related to that,
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is to lead with curiosity. There might be
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some ways that you expect students will solve a problem,
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but be genuinely curious about what it is that they have to share.
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And then how can the board acknowledge that
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and reinforce where they are coming from?
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And that's where looping back to the control,
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that's where it's more appropriate because you as a teacher are the conduit
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to welcoming students with open arms to the mathematics communities.
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In other words, "What is it that the students can't figure out on their
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own that you can showcase on the board?" And be explicit with that.
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You know, say things like, "I'm going to write all your ideas down
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and these are just ideas. And don't worry about being right or wrong because we're
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going to figure this out together."
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When we're ready, then we'll close these ideas.
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And so it's having more of a mindset instead of thinking like,
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there's a right way and a wrong way to write on the board because if
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you enter the space where you're trying to acknowledge students' reasoning,
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your practice will evolve over time.
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I think that's a great place to stop.
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Katherine and Tutita, it has been such a pleasure chatting with you today.
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Thank you so much for joining us on the podcast.
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Thank you.
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Thank you for having us.
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And thank you, our audience, for listening.
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We'd love to hear what you all think.
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For the classroom teachers listening,
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what are some ways you've adapted the structure of your classroom
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to help your students engage with one another?
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How has your use of the board shifted to support students' reasoning?
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Getting in touch with us is easy.
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Just head over to mathlearningcenter.org/podcast and fill out the contact form.
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dedicated to inspiring and enabling all individuals to discover their mathematical
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