Latest / Rounding Up / Organizing Board Space to Support Reasoning & Connection, Part 1, with Dr. Tutita Casa & Katherine Mack

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Organizing Board Space to Support Reasoning & Connection, Part 1, with Dr. Tutita Casa & Katherine Mack · Rounding Up

• 0:00
  Last season, I spoke with Dr. Tutita Casa and some of her students
• 0:05
  at the University of Connecticut about a strategy for showcasing students' work
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  in elementary classrooms. That conversation got me thinking about the ways
• 0:14
  we think about our classroom space and how those ideas could impact students'
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  learning experiences. I decided to check back in with Tutita,
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  as well as Katherine Mack, a doctoral student at the University
• 0:26
  of Connecticut's Neag School of Education.
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  Welcome to the podcast, Tutita and Katherine.
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  You are our first guests for season 5,
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  and I'm so thankful to you for taking the time to share your work with
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  our listeners.
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  Hi, thank you so much for having us.
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  Thank you for having us.
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  I'd like to start by sharing an analogy that you two shared with me
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  as we were preparing for this podcast.
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  You said just like the way that a layout of a classroom can influence norms
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  and influence culture, the ways that teachers organize and make use of
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  the board has an impact on students' learning experience.
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  And Katherine, I wonder if we could start by having you talk a little
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  bit more about what you mean by that.
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  Yes, absolutely. I think that analogy is helpful because when
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  we hear the phrase "classroom design," we often think first about physical things
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  like how the desks are arranged,
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  what is displayed on the walls,
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  how materials are organized. The instructional board is part
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  of that classroom environment and arguably one of the most public spaces
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  in the classroom.
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  And because of this, the way the board is used and organized sends
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  a certain message to students.
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  If you think about a very traditional image of a classroom,
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  a common visual or symbol is a teacher with chalk or marker
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  in hand standing at the front of the room with their board.
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  And in that familiar scene,
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  the board is the teacher's space.
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  And I think that as math educators,
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  many of us would like to believe that we have moved beyond that teacher-centered
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  image. And of course, in some ways we have.
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  We've moved from chalkboards to whiteboards to smartboards.
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  And presently I am observing a rise in what could be described
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  as slide-deck instruction, where teachers are using that board space in
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  the front of their classrooms
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  to sometimes click through prepared presentations during their math lessons.
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  So the technology has certainly changed,
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  but if the instructional board is being used for teacher-authored examples
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  on teacher-prepared slides, then that space may still be functioning
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  in a teacher-centered way. So I think we have to ask:
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  How far have we really moved from that traditional image if the most
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  public mathematical space in the classroom is still representing
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  the teacher's thinking? In our work with preservice teachers,
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  we're trying to help them reconceptualize the board as a student-centered space.
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  New teachers come into teacher preparation with their own inherited ideas about
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  the board, and so they can struggle to use the board in ways that are not didactic.
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  And our goal is to help preservice teachers move beyond the board as
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  a site of teacher explanation within their practice.
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  OK, there's a lot that you just said.
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  I mean, you have me thinking about my own experience as a child in classrooms
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  and my experience teaching, whether it was (sigh)—gosh,
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  I'm gonna date myself—the chalkboard,
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  the overhead, the whiteboard, the smartboard.
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  It does strike me how much that space was either dominated by my teachers or by me,
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  really, for a large part of my career when I was in the classroom.
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  You've also got me thinking about some of the unspoken norms around the board.
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  And Tutita, I wonder if you could talk about,
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  you know, are there spoken or unspoken norms around the board that motivated
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  the two of you to engage in this work?
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  For me, I also think back to my childhood and I was a very shy individual,
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  but looking back, myself and my peers were not expected to speak up
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  a lot during class. And it was really the teacher telling us how
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  to solve different problems. I don't remember specifically the board because that
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  was a long time ago for me, but I do recall my undergraduate degree being
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  in mathematics
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  and there are some professors who I thought highly of,
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  but I don't know what their face looks like because I often was looking
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  at the back of their heads. So I never questioned that practice.
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  I thought that's just what you did.
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  But then fast-forward both as a teacher as well as a math-teacher educator
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  and methods class, where I started to question,
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  you know, it's not really about my own ideas.
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  I have a degree in math,
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  but what can I do to support young students,
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  as young as kindergarten, really find their own voice?
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  And part of that came from an experience at Project M² that
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  I helped lead, where we were getting young students in,
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  you know, grades K through 2 to write their own ideas.
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  And we really were interested in not just what they knew,
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  but how they knew different mathematical ideas.
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  But inevitably I started wondering,
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  like, wait, they just said something like "multiply," but how do I record that
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  on the board? Do I use a drawing?
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  Do I use a symbol? Do I use numerals?
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  What is it? Because I really wanted to honor what they were doing and thinking.
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  And then fast-forward, Katherine joined me in teaching a methods class where
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  for many years we have had our preservice teachers peer teach.
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  And
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  the partners who are peer teaching,
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  they got to decide who is going to take on what role.
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  And one of them would take on the role of writing,
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  but we saw the opportunity do a little bit more and asked them to
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  be mindful that their role was to capture their peers',
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  who are role-playing elementary students,
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  their ideas. So it was, we're trying to take those unspoken norms that
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  the board is a space that traditionally is led by the teacher
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  and how can we open up that space to reflect what our students are thinking?
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  I think one of the things that you have me pondering is:
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  There's not a dispute about whether the board as
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  a public space to do mathematics is something that's valuable.
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  The question we're really asking is:
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  Whose space is the board, and what's its purpose?
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  We know it's a space where mathematics can become public,
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  but
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  who makes it public, and whose ideas are represented,
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  and what happens at the board—those feel like the things that you
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  all are really trying to shine a light on and help us build a
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  set of different norms. Tell me how I did in understanding that.
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  Yeah, I think that's spot on. And I'd like for teachers
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  to understand that the board work that happens during routines such
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  as a number talk—like writing the student name next to the strategy,
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  writing student thinking during that 15-minute routine—these
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  are practices that can extend beyond just that routine,
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  and these are practices
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  that can be used throughout the entire math block.
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  So it's not that the board belongs to the students just during a number talk
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  or just during a number string.
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  How can we bring that board work in?
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  I wanna try to name a norm that you all have got me thinking about.
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  I think you've helped me find language for something that I was trying
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  to create in my classroom but didn't know how to say.
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  And the norm that I've come to is:
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  We want the board to be a place where students expand their reasoning.
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  I wonder if you could talk a little bit about what that norm might look like
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  and sound like in a classroom.
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  I would argue that the board is a shared space between teachers
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  and students where ongoing students' reasoning is captured.
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  And when we say "board," we're really situating this in your traditional classroom
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  board that is meant to be viewed by the entire class at the same time,
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  where traditionally teachers have had command of that.
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  But we're looking to establish a norm where it's an open space.
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  It's both for the teacher to help students transition into
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  the world of mathematics. And the way I think about it is that,
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  as teachers, we know the end goal:
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  In mathematics, we want our students to get the answer right.
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  And sometimes it's also the process and procedures to get there with students'
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  understanding of those procedures.
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  But that's a really messy process.
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  And that messy process to me is why I love what I do.
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  There's such value with that messiness.
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  Like you try a particular approach.
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  Oh, wait, that didn't work. Let me try something else.
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  And that—what I call the residue,
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  that problem solving, the sensemaking,
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  the argumentation, the reasoning,
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  the conjecturing that to me are at the heart of mathematics—how can
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  we make sure that's prominent
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  during the learning process? And I see the board as that space where obviously
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  the students whom we're teaching don't know the content yet—otherwise,
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  we shouldn't be teaching the content.
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  But there are some things that they can figure out and there are other things that
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  they cannot. And I think that's in particular with the language of mathematics.
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  No one knows what a quadrilateral is because it's a specific math term.
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  But it's also with the written record,
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  it's the unique symbols, unique diagrams that we use that over time,
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  as students get older, become more sophisticated and aligned with what professional
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  mathematicians use. So in the elementary space,
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  we might see when they're learning multiplication,
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  circles, and stars. I know professional mathematicians aren't using those diagrams.
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  But the ideas, the concepts are there.
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  So that norm is a shared space where we're trying
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  to help the class understand their peers' ideas and starting
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  to introduce more sophisticated ways of communicating those ideas.
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  So Katherine, I wanna stay on this theme of language and representation because I've
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  heard you say that one of the things that teachers can do is they
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  can consider whether the ways they write and record help students
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  to follow the mathematical story of a lesson.
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  I mean, there's a lot in that statement.
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  Can you talk about what that means and what might make the story easier
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  or harder for students to follow?
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  Yeah, exactly. We want teachers to understand that the methods used
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  to record student ideas on the board can either help students see
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  the reasoning or make the reasoning harder to follow.
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  And that being precise with mathematical writing at the board is just
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  as important as being precise with mathematical language.
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  When teachers model clear,
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  purposeful mathematical writing at the board.
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  students begin to take that up in their own work,
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  which is important, especially now that more students are working
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  at nonpermanent vertical spaces than ever before through practices like building
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  thinking classrooms. So if we want students to tell the story
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  of their own reasoning, then we need to help them experience what clear mathematical
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  storytelling looks like.
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  So we're very interested in the kinds of norms that make
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  the mathematical story being told through board work more accessible to students.
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  For example, one simple but powerful norm we are cultivating with
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  our preservice teachers at the board is to anchor a lesson
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  by having preservice teachers ask their students to revoice a mathematical task
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  in their own words
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  and then record that student revoicing on the board.
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  So now the board is helping to position the task as something the class
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  is working to understand together rather than something that the teacher
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  is simply giving. We notice that teachers use visual and spatial choices
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  to support meaning at the board.
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  They use color-coding to distinguish between ideas,
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  organize strategies in different areas of the board.
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  They underline key quantities and close claims.
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  Teachers use arrows strategically to show connections between two strategies.
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  And I think it's important to not expect teachers to just intuitively develop
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  these norms that help tell the mathematical story.
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  I really love that you said that because depending on your background,
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  a person could listen to you describe some of those things and it might be
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  a part of your practice. But that's not necessarily automatic for everyone.
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  And so I appreciate that you were explicit about some of the things that were
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  supportive of student thinking and that you put those out.
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  I wonder if you could say a bit more about the way that color functions as
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  a tool to help bring clarity.
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  Yeah. We noticed that when color-coding was used to distinguish ideas,
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  the board looked very clear and the mathematical story being told
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  was more accessible. Sometimes teachers use color-coding
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  to distinguish ideas within one strategy space
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  or to draw attention to an important digit like a tens place.
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  Tutita, do you want to add anything to that?
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  It's helpful for me to take a step back and notice over the years
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  how different teachers have used the board.
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  And one of the things that occurred to me when engaging in this work
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  is helping the teachers not view the classroom from their stance.
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  It's more about: How is the audience perceiving what it is that you're recording
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  on the board? Like just pretend there's a student that had to be pulled out
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  of class or maybe they are late to school,
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  and they are not privy to the conversation that took place during math class.
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  They came in at the end. How might they interpret what's on the board?
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  What is the story that reflects that conversation?
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  So that was a light-bulb moment for me.
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  And so that idea, for example,
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  of color-coding, as Katherine mentioned,
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  emphasizing a particular place value.
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  It could be emphasizing particular vocabulary terms.
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  So there are all these little nuances depending on
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  the content—what content students actually also bring up and the order that they
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  bring it up. And if we go back to this idea,
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  I love how Katherine is talking about a story to be told because naturally
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  I'm sure like listeners are thinking about an actual book.
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  And young
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  children, when they're first learning to read,
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  they're sitting in their mother,
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  father's, aunts', uncles', grandparents' lap.
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  There's no way to know that the text goes from top to bottom and left to right,
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  and then we turn the page. That's something that you have to learn.
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  I believe, but I'm not certain.
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  I remember a friend from when I was in elementary school,
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  I think he was from Vietnam, where they would write from top to bottom,
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  not left to right.
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  So it just helps me understand that there's certain hidden ways
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  of understanding what's there.
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  So if we think of the board in the same way,
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  is it a top to bottom? Is it left to right?
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  Are the ideas in the center, you know,
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  with the web and they're coming out from there?
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  Is there a reason to put certain ideas in particular orders?
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  Should we group them in certain ways?
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  And when I say "we," it's with the teachers who are thinking of,
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  like: What are
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  my short-term and long-term objectives with helping students?
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  So if it's a number talk—where students might have distinct ways
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  of solving and computing whatever expression is given to them,
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  may or may not be connected with the one just said and the next one—the number
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  string, where teachers are purposely trying to get students to make connections,
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  that might require,
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  or students would benefit from a top-down configuration because they
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  can see that story. Maybe a teacher at the beginning when they're first implementing
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  a number string explicitly says that,
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  "Let's do this next and how is it connected to the other one?" and might decide
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  to put an arrow from the top of the board to the bottom to help students understand
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  or maybe they're connecting arrows from one expression to,
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  or equation to, another.
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  So it's the story matching the objectives that the teacher is trying
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  to accomplish within that particular lesson,
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  but also over time.
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  So you dove very deep in and I have to reveal my inner teacher nerd and say,
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  I love this kind of conversation because it makes me overthink my practice
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  in a way that's really fun. But part of what I took from what you said is:
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  Stories have flow. Stories have kind of clusters of ideas or key points.
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  Stories have sequence. And I was taken with the example that you shared,
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  which is
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  If I walk into a classroom 5 minutes late because Dad had to drop
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  me off today, to what extent is the recording on the board going to tell that story?
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  So again, it may feel very obvious to some folks,
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  but the use of arrows to help students recognize sequence,
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  the use of color in a way that highlights a particular relationship,
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  or a connection between a symbol and language,
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  grouping particular ideas in a particular location or with a particular color.
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  All of these are moves that are storytelling moves that feel maybe intuitive
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  for folks, but in thinking about them at this level,
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  it helps you be more intentional with some of the moves that you
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  may have just developed, but the why behind your doing them was a little unpacked.
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  I'm really taken with the practicality of thinking this deeply about storytelling
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  at the board.
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  And I would add on, I think it's not just story.
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  This is what makes it more complex.
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  It's not just storytelling, but it's in the moment—you can plan ahead,
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  but it's really hard—in the moment,
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  you're trying to tell a story where students who are still figuring
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  out the mathematics are trying to tell.
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  And so oftentimes when we read a book,
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  there are many revisions that it goes through—editing,
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  proofreading—but so many different ideas,
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  flow being one of them, that is checked early on in the writing process.
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  But that's not necessarily the case during instruction.
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  It's mainly when students are learning.
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  Again, it's really messy. So it could be that in the fourth minute of the lesson,
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  there's a nugget there that for whatever reason doesn't get as much play
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  as you'd hoped,
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  or that a student decides in the ~ 42nd minute of the lesson decides
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  to go back to that idea. So the flow,
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  it's not necessarily sequential.
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  So there is a decision that a teacher needs to make at the board that
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  you could either present it sequentially or you can attach that idea
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  to that original one, maybe underneath.
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  It could be an arrow, you can circle it.
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  It's relative to where it is, you could add it on.
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  And so that's an important decision for teachers to make,
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  but it depends so much on the content,
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  the grade level, what's being said,
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  the timing, and literally how much space you have on the board.
• 22:14
  You know, something else that I wanted to probe is the way that spoken language
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  interacts with representation and written language on the board.
• 22:24
  And the thing that you also have me thinking about is the extent to which
• 22:29
  we rely on spoken language to carry ideas and also the extent
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  to which spoken language is invisible and can feel like it evaporates
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  as opposed to something that's recorded,
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  which becomes an artifact for reference.
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  I wonder if you could talk about how written and spoken language
• 22:48
  and representation can complement one another versus over-relying on
• 22:54
  one or the other.
• 22:56
  I think multiplication is a really good example of the challenge with
• 23:02
  the interplay between the oral and written piece.
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  And that's the part that I think is so important for teachers to be aware
• 23:11
  of when they're writing on the board.
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  That's the place where they can showcase and introduce those specific ways that
• 23:20
  mathematicians use to communicate.
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  So if you start off when multiplication is first introduced
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  in the elementary grades, we use the symbol that is identical to an X,
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  which they just learned not too long ago is a letter,
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  but then we don't say "X" when we're talking about multiplication.
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  We say "times" or "multiplied by" or "groups of." So what we say does
• 23:47
  not match the symbol. Then
• 23:51
  we eventually, and very briefly,
• 23:53
  introduce the dot operator. Well when we first learned it,
• 23:57
  it means a period, which means the end,
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  but at the same time, it also is introduced as a decimal point,
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  which does not mean the end, but that decimal point and the period are lower down
• 24:10
  on the line compared to the dot operator,
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  which is kind of in the middle of the top and the bottom of the line.
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  Exact same symbol but relative to the symbols around it means something else.
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  And then eventually we get rid of any symbols,
• 24:23
  written symbols, to connote multiplication.
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  And I'm not sure that we're aware of the complexity of that.
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  And it's not just writing it, but it's students reading these different symbols.
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  And if they don't understand what's there,
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  it might not be that they don't understand multiplication or they don't understand,
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  you know, exponents. It's just literally they cannot interpret what that language
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  means.
• 24:52
  Thanks so much, Tutita. That makes so much sense,
• 24:55
  and thank you, Katherine, for your valuable insight.
• 24:57
  Listeners, we're going to return to this conversation during our next episode
• 25:01
  to dig deeper into ways teachers can call for shared attention,
• 25:05
  bookend their lessons, and position the board as a place of discovery.
• 25:09
  This podcast is brought to you by The Math Learning Center and
• 25:14
  the Maier Math Foundation,
• 25:16
  dedicated to inspiring and enabling all individuals to discover their mathematical
• 25:21
  confidence and ability.