Latest / Rounding Up / The How, When & Why of Physical & Digital Manipulatives with Natalie Crist & Patrick Vennebush

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The How, When & Why of Physical & Digital Manipulatives with Natalie Crist & Patrick Vennebush · Rounding Up

• 0:00
  Since our founding in 1976, visual models have been central to the mission
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  of The Math Learning Center. Physical manipulatives and visual context feature
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  prominently in our curriculum materials and in our professional learning.
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  We've also developed a set of free,
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  publicly available digital manipulatives for teachers and for students.
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  That said, there's a great deal of debate around the role physical
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  and digital manipulatives should play in elementary math classrooms.
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  To address this question, I've invited two of my colleagues from
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  The Math Learning Center, Patrick Vennebush and Natalie Crist,
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  to join us on this episode of Rounding Up.
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  Patrick and Natalie, thank you both so much for joining me on Rounding Up.
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  Thanks, glad to be here.
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  Really happy to be here. Thanks for having us,
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  Mike.
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  Before we get started, I'd love for you both to share a bit about your background
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  and also the work you do now at The Math Learning Center.
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  Natalie, would you be willing to go first?
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  Sure. I'll be glad to go, Mike.
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  So my name is Natalie Crist and I am the director of content development
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  at The Math Learning Center. I previously was a curriculum director
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  for mathematics and also a teacher as well as an administrator
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  in various schools across Maryland,
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  ~ and have spent most of my career teaching elementary school.
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  I do also spend some time teaching courses at local universities,
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  both graduate and undergraduate courses in mathematics and math methodology.
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  Awesome. Patrick?
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  Yeah, I'm the chief learning officer at The Math Learning Center.
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  ~ I've been very fortunate in my career.
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  I got to do work developing digital textbooks.
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  I got to work at the National Council of Teachers
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  of Mathematics developing materials for teachers to use.
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  And now I get to oversee a lot of what we do at The Math Learning Center
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  in terms of content development.
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  I get to work very closely with Natalie,
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  our professional learning team,
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  ~ our research and publishing teams and get to oversee all of the work that goes
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  into making a great curriculum product.
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  Awesome. Thank you both.
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  So, for our listeners, we decided that we wanted to do this podcast
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  for many reasons, but a big one was the current debate in education about
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  the impact of digital resources,
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  and especially screen time, and the way that those were impacting student learning.
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  I think I'd like to start by asking you both:
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  Why does it make sense for The Math Learning Center to weigh in on this debate?
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  Yeah, I think there's a couple of things going on.
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  So one is that we're at an interesting moment in ~ math education
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  in the country where a lot of people are reacting to screen time
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  for students and what that looks like.
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  I also think it's important for us to get into this conversation because
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  for 50 years, The Math Learning Center has been promoting the use
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  of visual models and students learning kinesthetically.
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  But we also produce math apps,
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  and we have a lot of different digital manipulatives that students
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  are able to access. So
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  we're working at the intersection of those two things and we wanna be able
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  to talk about, you know, the right times to use either one of them.
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  We also want to think about how those manipulatives really support students
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  in the classroom and the decisions that teachers need to make when choosing whether
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  to use physical or digital manipulatives and really focusing in on what it's going
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  to do to help students with their learning.
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  I think that's probably a really great pivot point,
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  Natalie. And I'd love to have you describe the value of physical materials.
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  What do they offer students as they're trying to make sense of mathematics?
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  Sure. Thanks, Mike. ~
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  Really, those physical materials,
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  they allow students to really engage with the mathematics using their sense
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  of touch as well as sight to create their understanding.
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  So we think about when kids are developing the concepts in the classroom,
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  they're using those manipulatives—such as Unifix cubes,
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  for instance—really ~ looking at counting and touching.
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  And they build things like towers of 10.
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  And they can physically see the count of 10 based on the height,
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  as well as just comparing those towers.
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  So they could have 8 cubes stacked and 10 cubes and be able to see the difference.
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  So part of that is just the sight of it,
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  but also the physical touching and counting.
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  So that physical interaction, that tactile experience that students are having,
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  ~ where, the touching and feeling,
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  their gripping and those gross motor skills are really being developed
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  along with teachers are able to,
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  to watch students and see how they're counting,
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  how they're pointing at those Unifix cubes,
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  for instance. So it really sets kids up to have discussions not only with
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  the teachers, but with one another as well.
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  So they can construct their ideas and discover through their senses.
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  Part of what you're making me think about is:
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  There's a level of, kind of, coordination,
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  I guess, is the word that is coming to mind,
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  between what's happening with your eyes,
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  touch, the movement. And to some extent,
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  it feels like that level of: I'm not just seeing,
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  I'm not just touching, but I'm touching and seeing at the same time.
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  And that's part of the value that comes out of physical materials.
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  Yeah, I would agree with that,
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  Mike, that when you are connecting all of your senses,
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  that's how students are making sense of the world.
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  So we know l early childhood students,
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  they engage in the physical world by their senses.
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  And that continues as they move through elementary school.
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  So it's really important that we're allowing kids
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  the time to use their hands and the materials to develop their understanding.
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  Patrick, I think I wanna ask the same question,
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  but also with the lead in, which is:
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  We wouldn't be developing a set of virtual manipulatives if we also didn't think
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  there was value there for kids.
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  So what value, from your perspective,
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  do virtual manipulatives offer to kids?
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  Yeah, so I'll piggyback off of what Natalie said.
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  You know, students have the ability to see models,
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  use them, and experience them.
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  The benefit of a digital manipulative is that students can see multiple models
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  at the same time. And that can happen with physical as well,
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  but it, it takes up a certain amount of space and things like that in a,
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  on a screen, you can sort of do a little bit more.
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  And I guess the example I'd give is we released the Math MultiApp [by]
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  MLC this summer. And in response to that,
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  we got feedback from a teacher who said that their grade does a "number
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  of the day" every day,
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  and it used to be that they were opening various apps or they just wouldn't
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  use apps because ~ they couldn't see all the things at once.
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  Now with the Math MultiApp, students are showing numbers in different ways using
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  multiple models at the same time and they're,
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  and they're being able to see those connections.
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  And her sort of crescendo to this feedback was that we can build number sense
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  in our students much easier. And I think that's really,
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  you know, the big benefit that we're looking for.
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  It's also possible that with an app,
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  and certainly something like the MultiApp,
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  you have access to [an] array of manipulatives for students that
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  may not be available in the classroom.
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  You can have one app or one website as opposed to needing a bunch
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  of boxes of physical manipulatives.
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  There's also the ability for something like the Math MultiApp with accessibility
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  features to be able to help students with disabilities.
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  And it also gives students access to tools 24-7.
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  So when they're not in the classroom,
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  they can still touch these things at home by going to a website or to the app.
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  There's a couple things I want to unpack about that.
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  And the first one that jumped out to me is when you were describing
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  the ability that virtual or digital manipulatives offer,
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  the word that I was thinking about is that it gives you the ability
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  to make it public, right? Like I can project those sets
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  of multiple representations in front of the entire class.
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  And that allows me to take the things that one kid might have understood with
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  a set of manipulatives and another kid might have understood with
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  a set of manipulatives and bring that out to the whole class.
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  I think that's exactly right. The ability to compare the work
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  of two different students and be able to discuss their strategies
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  and what was similar, what was different between them.
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  I think it's also the case for a student to create something maybe using pattern
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  blocks or using an array or using,
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  you know, a different visual model that we have available,
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  a number line, and being able to see what looks similar,
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  what looks different and how those two things connect.
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  So I, I think it's both of those.
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  The other thing that really jumped out is access,
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  right? Both in the sense that you have access to an infinite number
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  of digital or virtual manipulatives at one time that might or might not
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  be in the classroom. And then when you're not in the classroom,
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  they're still available. But I was really taken with what you were saying about
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  offering access to kids who may have a disability.
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  Sure, that's correct. We really focused on accessibility when we built
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  the Math MultiApp. Some of the listeners may be familiar with
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  the single standalone apps that we had previously that were not accessible,
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  and we really wanted to make a push for that.
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  So it's now possible for students to be able to use the keyboard instead
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  of having to use a mouse. So kids who have limited motor skills are able
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  to access the apps and do things with the models that you couldn't before.
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  ~ It's also the case now that the Math MultiApp is accessible to a screen reader
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  so that students will be able to hear the text on the screen read aloud,
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  or there's alt text associated with images,
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  and things like that, so they can participate more fully.
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  So I just want to hold on this for a second.
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  You're making me think about how when I taught kindergarten and first grade,
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  if a kid's motor skills were less developed and you looked at their representation,
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  it kind of coded as: Do they understand the same amount as a kid
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  who has better handwriting or is better at drawing?
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  And I think what you're really making me think is that the MultiApp levels that
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  playing field for kids. Am I, am I understanding what you were getting at?
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  I think that's certainly part of it for sure.
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  The, the interpretation of what a student puts down on paper is based a
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  lot by how it looks. I can think of classmates who were penalized
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  by the teacher because their handwriting wasn't as nice or their drawings weren't
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  as clean. And I think it gets rid of some of that for sure.
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  I think there's the flip side to it though,
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  right? So you have students who are academically compliant,
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  let's say, and they want to draw the perfect hexagon.
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  And they spend all their time making the perfect hexagon,
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  and they don't get to think about any of the good math because they spent
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  all their time making sure that every angle was exactly 120 degrees.
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  You free up space for both of those kids to explore the math and think about what
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  we want them to think about.
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  Natalie, where are you at with that?
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  So I definitely agree with Patrick that the idea of using
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  a digital manipulative helps teachers to really uncover what students are able
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  to do because it removes that barrier to being able to represent something
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  on a piece of paper with a pencil.
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  I wanna take it from the other side,
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  however, thinking about hands-on materials.
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  When students are using those manipulatives and,
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  and really building their understanding using those three-dimensional objects,
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  they're moving those pieces. So if they're using whether it's base
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  ten blocks or whether they're using Unifix cubes or the pattern blocks,
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  they're manipulating those objects and using their hands.
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  It's helping to develop some of those motor skills
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  by getting students to turn, rotate,
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  pick up, grip. So using those materials,
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  we can uncover students' understanding as well as develop their hand-eye
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  coordination.
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  Natalie, you made a really strong case for the importance of seeing,
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  being able to touch, teachers being able to watch the direct action
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  of students and their materials in real time as a way to learn what they're thinking
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  about. All that said, are there any caveats to what you said earlier
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  or any limitations that you think it's worth people thinking about when
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  it comes to physical manipulatives?
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  Yeah, absolutely. One of the things you need to think about with physical
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  manipulatives is that they have to be used appropriately in the classroom.
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  So we really want to think about the manipulatives not being used in isolation,
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  but instead within a clear sequenced plan of instruction,
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  including the structures that need to be put in place for students to use them.
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  And so those choices around how they're used
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  is really the most important piece that teachers can consider.
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  ~ We want kids to develop that concrete understanding in order
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  to move towards abstract representation.
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  And so when we think through it,
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  teachers have to really coordinate how they're going to do that
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  and make decisions around when it's appropriate to start shifting students away from
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  those hands-on materials
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  and having them there as a tool in case students need them,
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  but then beginning to shift towards that two-dimensional representation
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  or symbolic representation.
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  One of the things that you made me think about is when you were talking about
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  concrete, representational, and abstract,
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  it's less a limitation and more of a consideration,
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  meaning that physical manipulatives are meant to help kids make sense,
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  right? To, to understand the mathematics,
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  to be able to put their hands on it,
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  to see it, to coordinate that understanding.
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  But I think part of what I'm taking away is there are points in time where
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  we want to start to help kids make that shift
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  to something that's a little bit more abstract.
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  And that's something that a person really has to think about,
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  is: Am I building understanding?
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  And when am I ready to start to nudge kids toward a more abstract
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  way of either understanding or representing their understanding?
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  Yes, ~ absolutely. That is the consideration teachers have to,
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  to make. I mean, we know the standards.
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  They start calling for students to become fluent with things like algorithms,
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  for instance. At that point, they're using symbolic representation,
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  but we've developed it through using hands-on materials.
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  But [at] some point the teacher has made the decision,
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  it's time to start shifting to a different type of model.
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  And it might be that they start sketching and drawing and then go to symbolic,
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  but the teacher has to ultimately make that decision for when
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  the students are ready.
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  Yeah, Patrick, let's go back to you.
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  I think we really made a case that there's some strong value in virtual
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  or digital manipulatives. Any caveats or limitations you think as
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  a teacher you'd want people to think about?
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  Oh, for sure. You know, virtual manipulatives can assume a level
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  of understanding that students have that's not quite there yet.
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  And so the foundational learning has to happen with those physical manipulatives.
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  I think, you know, talking about concrete,
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  representational, and abstract,
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  the, the concrete is built with kids having their hands on the manipulatives.
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  They can then progress to the virtual.
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  And I think that's the limitation.
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  It's a, it's a matter of, again,
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  of when you use it, the consideration of how you use it.
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  And it doesn't mean that virtual is bad.
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  It just means they need to be used at the right time.
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  It's also the case that sometimes virtual tools will give too much information
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  to students, and it reduces the cognitive load.
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  So you wanna think about what you're asking students to do with it,
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  how much information is presented there with it.
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  We allow the ability for teachers to create save states of the MultiApp
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  and be able to share those with students and,
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  and how much text you put on the screen,
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  how much background you give to students to make it feel like they're ready
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  can sometimes be a little more than,
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  than they actually need.
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  So Patrick, I want to just make sure that everybody's clear on
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  the language we're using and what it means.
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  Could you break down CRA, what that means,
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  what we're referencing?
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  Yeah, so CRA refers to "concrete,
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  representational, abstract." And the first part of that is the,
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  the C for the concrete understanding,
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  ~ for students to develop that initial level and and foundational understanding
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  of what's going on. So using ~ you know,
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  pattern blocks or the Unifix cubes or whatever pieces they're using
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  to get that understanding. The R then takes that to representations
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  and in particular pictorial representations.
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  So maybe instead of using pattern blocks,
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  it's drawing the pattern blocks or something like that,
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  or instead of creating an array,
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  they draw an array. And then that moves to the abstract,
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  and, and we typically think of that as symbolic,
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  like with algebra or things like that.
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  Or sometimes we'll talk about,
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  like, maybe even an open array.
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  So you don't have to draw in every single square of the grid,
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  but you draw just a big rectangle and sort of label what it looks like.
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  So it's moving from one to the other through that progression.
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  Let me just unpack that for a second for the folks who are listening.
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  So I might be holding seven Unifix cubes in my hand.
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  I also then when I get to representation might be able to,
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  with my pencil or crayon, draw seven markers that represent those.
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  And then when I get down to the ~ abstract,
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  I might write the number 7. ~ So that's a little bit of what that progression from
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  concrete to representation to abstract looks like.
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  Is that, again, Patrick, Natalie,
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  make sure I've got that right.
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  You've got it right, Mike.
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  Awesome. I, I think there's two points here.
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  We started this by saying: We want teachers to understand the value,
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  potentially the limitations, the question that they should ask.
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  And what I heard you say on the front end of this is:
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  It's not necessarily an "if" but a "when" when it comes to virtual manipulatives.
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  And that "when" is generally after kids have had their hands on physical materials.
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  Is that a fair statement?
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  I think that's exactly right. The,
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  the benefit of physical materials,
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  ~ you know, as Natalie said before,
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  we want kids to use all of their senses.
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  So touch is a big one. In a virtual manipulative,
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  you can see it, you can move it,
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  and do things like that. You don't get that feeling in your hands,
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  and that creates some permanence in your mind,
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  and, and we want that to be locked in.
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  You know, one of the things I wanna note,
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  though, is the MultiApp that we've built does actually have sound.
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  So when kids virtually put some stuff together,
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  there's a sound that approximates what you might hear with physical manipulatives,
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  right?
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  Yeah, we were very intentional about that.
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  So our linking cubes, whenever whenever one linking cube connects to another,
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  there's a snapping noise. Whenever the, the die is double-clicked and it rolls,
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  it, it sounds like a die rolling on a table.
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  So absolutely that reinforcement for students that something is happening.
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  You know, y'all have both made several really important points during this
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  conversation. Can one or both of you share what educational research says about
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  the use of physical and virtual manipulatives?
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  Sure, I'll go ahead and hop in there,
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  Mike. So researchers like Jerome Bruner and Piaget,
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  they really focused in on how students develop understanding.
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  And we know that Bruner had three levels similar to CRA,
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  where he had a 3-D understanding where kids were action based
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  or physical and a two-dimensional idea around
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  sketches or that iconic representation and then moving into symbolic.
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  So it's very similar to our approach with the concrete representational
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  and abstract thinking.
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  And Piaget really focused in on kids between the ages of 7 and
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  11 in a concrete operational stage,
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  where they were developing their understanding
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  through concrete physical objects and how they were positioned within their
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  spaces, and that they could not really ~ move into that abstract
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  or that hypothetical thinking without doing something concretely first,
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  using their touch and using all their senses.
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  So Patrick, I want to ask the same question,
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  but I think as I listened to Natalie describe some of the work that
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  has informed our thinking about physical materials,
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  it strikes me that we've not been in the business—"we" meaning math educators,
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  have not been in the business—of virtual or digital manipulatives
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  for nearly as long. So I'm curious [about] the extent to which there
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  is research and what that research says.
• 19:45
  Yeah, I think, you know, whenever Wi-Fi and the internet became
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  ~ ubiquitous in education—you know,
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  late '90s, early 2000s—there was a lot of work being done.
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  Much of the research that still exists is from the early 2000s and 2010s.
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  So it, there hasn't been a little bit of dearth recently about what it means.
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  But some of the research that came out was that virtual manipulatives help
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  to improve visualization for students and let them see things that sometimes they
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  might not be able to otherwise see.
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  The big thing is it shows that there's a higher level of engagement
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  that when students are using virtual manipulatives,
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  they tend to stay on task longer.
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  They show that they're more motivated.
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  And it opens the space so they can learn at their own rate.
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  And I think that's a really big important piece.
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  We'd love that to happen in the classroom in general.
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  So if the virtual manipulative fosters that,
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  that's a good thing.
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  I think it's also important to note that there's a lot of support
• 20:31
  for special needs students, and there's highly correlated positive results
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  for students with learning disabilities.
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  It often reduces the cognitive load for students with disabilities
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  in terms of precise drawing and resizing.
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  Virtual manipulatives allow to happen a little more easily.
• 20:44
  I mentioned earlier motor skill barriers,
• 20:46
  and if there's the problem there,
• 20:48
  students can get past that with accessibility controls.
• 20:50
  And there's actually a, I think a really big one from an emotional perspective,
• 20:54
  that it decreases the stigma for students who are learning below grade level.
• 20:58
  If they're using a, a virtual manipulative,
• 21:01
  they're not being seen by their peers as using some type of tool from
• 21:04
  an earlier grade that makes it look like they're learning below grade level.
• 21:07
  They can sort of keep it a little bit more to themselves and it doesn't have quite
• 21:10
  a public face to it.
• 21:12
  But the general piece is that research says that students generally learn
• 21:15
  as well with virtual manipulatives as they do with physical objects.
• 21:18
  But every expert will stress that with manipulatives,
• 21:21
  no matter what kind you're using,
• 21:22
  the teacher has the biggest impact.
• 21:24
  It still has to be good instruction going along with it.
• 21:26
  So I think that's a little bit of what we promote
• 21:29
  within our curriculum—how to use them,
• 21:31
  when to use them, when to bring them in,
• 21:32
  and that we want teachers to be prepared to do both of those things.
• 21:36
  How should teachers think about the choice of using physical manipulatives
• 21:40
  or virtual-slash-digital manipulatives?
• 21:42
  And Natalie, I think I want to throw this to you first.
• 21:45
  Sure. ~ So I, I think ~ question that we might want teachers
• 21:52
  to really think about is: What features of that manipulative support
• 21:57
  the concepts or the skills that I want my students to understand?
• 22:01
  So they should be selecting the material that aligns with their stated
• 22:08
  objectives
• 22:09
  as well as the skills of the session.
• 22:11
  And what do I want my kids to come away with from this particular activity,
• 22:16
  and how will the manipulatives support it?
• 22:18
  So we really want it to be connected to what the students
• 22:24
  are learning and how they're learning it as much as:
• 22:28
  What do we want them to take away from that particular lesson?
• 22:32
  It's not always natural with some of the manipulatives for students to know
• 22:38
  how to construct meaning from them.
• 22:40
  And so the teacher's role is to show how to use the materials.
• 22:47
  The other thing I would say is:
• 22:48
  We don't want to think about just the materials from a standpoint
• 22:53
  of the manipulatives they're using.
• 22:55
  We also want to think about the pencil and paper.
• 22:57
  That is a tool for learning as well.
• 23:00
  And so while they're constructing with the physical hands-on materials
• 23:05
  they need to be also showing the two-dimensional ~ sketch
• 23:11
  or the representation, whether that be an abstract symbolic representation or,
• 23:17
  like I said, a sketch or a drawing.
• 23:19
  Those connections are critical for teachers to make and for students to make.
• 23:26
  A student might show with the manipulatives,
• 23:29
  another student might show with a sketch.
• 23:31
  How do we connect those two ideas together?
• 23:34
  That's what teachers need to think through when they're planning
• 23:39
  for the instruction.
• 23:41
  I think one of the things that you got me thinking about is:
• 23:44
  If I'm a teacher, part of the challenge that you're putting in front of me
• 23:48
  is I need to look at whatever physical material that I have and say to myself:
• 23:52
  What is it about this thing, or these things,
• 23:55
  that actually helps kids make sense of a particular concept or idea
• 24:01
  or solve a problem? And then I need to think about:
• 24:04
  How am I going to introduce that to those students and draw attention
• 24:07
  to those features?
• 24:09
  And all of that has to happen before I put those manipulatives in kids' hands
• 24:13
  and imagine that they're gonna use them to solve a problem or find a solution.
• 24:17
  Absolutely. When teachers are planning for instruction,
• 24:21
  they need to be planning for the use of the manipulatives as well.
• 24:25
  And I always encourage teachers as they're doing their planning
• 24:29
  to solve the problems. Solve the problems like you would want students
• 24:33
  to solve the problems, whether that's with a manipulative,
• 24:36
  whether that's on paper, whether that is thinking through what students
• 24:40
  are going to say and saying it.
• 24:42
  That is part of planning for good instruction.
• 24:46
  I think the other thing that you reminded me of is that part of the work when
• 24:50
  we have physical materials is also pairing them with the opportunity to represent.
• 24:55
  I think the thing that flashed in my mind as you were talking earlier is:
• 24:58
  Even something as relatively simple as putting a collection of materials
• 25:03
  in front of kids and having them count them,
• 25:05
  which might mean that they're doing some groupings,
• 25:07
  they're doing some [one-to-one] correspondence work with their hands
• 25:11
  as they're moving things. But part of that task is to say:
• 25:15
  "Here's a pencil and paper. Represent that." That that opportunity
• 25:20
  to represent physical materials is often a really important part
• 25:24
  of what's helping kids consolidate their understanding.
• 25:28
  Absolutely. It, it, it definitely is.
• 25:31
  And sometimes representing those three-dimensional materials can
• 25:36
  be difficult for students. And so they need the opportunity to do
• 25:41
  it often because especially when you're going from a 3-D to a two-dimensional space,
• 25:47
  it looks different for kids. You're going f-
• 25:50
  flat. A flat piece of paper feels very different,
• 25:54
  especially if I'm sketching a cube,
• 25:56
  for instance. So
• 25:58
  students need lots of practice with that.
• 26:02
  I'll jump in there because one of the things we did in the Math MultiApp when
• 26:05
  we developed it was make sure the pieces work together.
• 26:07
  So when you talk about, like, being able to do the representation,
• 26:10
  ~ we could have kids count square tiles in the Math MultiApp,
• 26:14
  and you can put a number line next to it so the correspondence is automatic.
• 26:18
  So that idea of representation is sort of baked in a little bit.
• 26:21
  But again, to Natalie's point,
• 26:23
  like, if a teacher isn't planning for those two things to show up side by side,
• 26:26
  are kids gonna make the connections on their own?
• 26:28
  Probably not.
• 26:29
  So it's one of those things that a teacher has to anticipate what could happen
• 26:32
  in the classroom, how kids might use the tool,
• 26:34
  and then make sure the connections are made explicit.
• 26:36
  That's awesome, Patrick. And y-
• 26:38
  you jumped in and anticipated my question,
• 26:40
  which is: What are some of the other questions you think teachers should
• 26:42
  ask themselves when they're planning on using virtual or digital manipulatives?
• 26:46
  Yeah, I mean I'd go back to the CRA concern.
• 26:48
  ~ Do students have a, have a concrete understanding of what these objects
• 26:52
  are before we attempt to use them for the R and the
• 26:54
  and the two-dimensional representation that happens on the screen?
• 26:57
  And then how do we bring it to the A?
• 27:00
  How do we make the abstract happen?
• 27:01
  So I'm gonna attempt to paint a picture for you here.
• 27:03
  There's an activity that we do with pattern blocks where we have kids take hexagons
• 27:08
  and they create pattern block trains.
• 27:10
  So they connect a hexagon to a hexagon to a hexagon.
• 27:12
  You can imagine a long string of these together.
• 27:15
  And we ask them to think about the perimeter.
• 27:16
  And if you think about the perimeter,
• 27:19
  when there's two hexagons together,
• 27:20
  the perimeter is 10, because two sides are touching.
• 27:22
  They don't count the 5 on each of them.
• 27:24
  When you start to add more hexagons in the middle and you have,
• 27:26
  you know, 5, 10, 15, 20 hexagons in a line,
• 27:29
  the ones on the end still each contribute 5 to the perimeter,
• 27:32
  the ones in the middle contribute 4.
• 27:34
  This is getting to the basics of algebra,
• 27:36
  right? You have 5 on one end plus 5 on the other end plus 4 times however many
• 27:40
  are in the middle. So you know,
• 27:41
  10 plus 4n would be your algebraic equation there.
• 27:44
  But making that explicit, showing what happens with the numbers as you
• 27:47
  add more and more shapes, those are the kinds of explicit connections we want
• 27:51
  to do in the classroom. Virtual manipulatives do it very well.
• 27:53
  You can do it with physical as well.
• 27:55
  I think a teacher has to decide which is gonna work for the moment.
• 27:58
  I think that example is another great pivot point.
• 28:01
  We've been making reference to this tool called the Math MultiApp.
• 28:06
  And ~ in many ways it represents,
• 28:08
  like, our latest vision of: How can we help teachers use virtual manipulatives?
• 28:13
  Patrick, Natalie, you both worked on the development of this tool.
• 28:16
  I wonder if you could talk a little bit about what it is and some
• 28:21
  of the functionality that you think is going to support students
• 28:24
  and also support teachers in supporting learning.
• 28:26
  Yeah, I think the MultiApp does a lot of things,
• 28:28
  and we've talked about multiple times the way you can show various visual models
• 28:31
  simultaneously. So there's a real ability for students to do sensemaking
• 28:35
  and problem solving with the tool.
• 28:37
  I think that's a big piece. We've given teachers the ability to
• 28:40
  use the Math MultiApp to create save states that allow some tools to be available,
• 28:45
  but not all of them, if you don't want to overwhelm a student.
• 28:48
  So we have 13 tools currently available in the app.
• 28:50
  Plus you can add some game materials,
• 28:52
  including dice and spinners and things like that.
• 28:54
  Teachers can decide which of those are shown at any given time.
• 28:57
  So for a very young kid, maybe you want to have a save state that includes
• 29:00
  the linking cubes, number lines,
• 29:02
  and number pieces, and that's where you stop.
• 29:04
  And a teacher can turn off all the other ones so students can sort
• 29:07
  of stay focused on the particular tools you want to.
• 29:09
  So I think those are some of the pieces we added in.
• 29:12
  I also mentioned the accessibility concerns.
• 29:14
  That was a really big piece for us when we started out.
• 29:16
  We wanted to make sure that every student had equal ability to use this.
• 29:19
  And I think that that's going to allow a greater number of students to
• 29:22
  be able to access the models that we're including.
• 29:24
  Yeah, the other thing you made me think about is the ability for teachers
• 29:29
  to not just use this in the hands of kids if kids were trying to solve a problem,
• 29:33
  but also to make public some of the ideas that are coming out.
• 29:36
  So there's a set of annotation tools and ways that teachers can decide when they
• 29:41
  want to show things to students if they're sitting in a whole group.
• 29:43
  I wonder if you could just talk about ~ some of the tools if teachers
• 29:47
  are using this in front of a group of students as a,
• 29:50
  as a way to either
• 29:52
  compare or sequence or explicitly kind of make connections.
• 29:57
  Yeah, I'd be happy to talk about those features,
• 29:58
  Mike. In addition to the models and the game materials,
• 30:01
  the MultiApp includes this ability to create share codes,
• 30:04
  so you could give a code to a student and let them look at it.
• 30:07
  There's the ability to take a screenshot or to save the work that you're doing
• 30:10
  so the student could save their work,
• 30:11
  and that would be a way, as you referenced earlier,
• 30:14
  of showing two students' work,
• 30:15
  side by side potentially. There are all the annotation tools that
• 30:19
  are available there, so you can color things,
• 30:20
  you can draw, you can add text on it,
• 30:23
  and you can be able to explain your thinking and,
• 30:25
  and do all those kinds of things as well.
• 30:27
  One of the things that I was thinking about with the Math MultiApp
• 30:31
  is how teachers can capture the thinking of students on the screen.
• 30:37
  So if they're working with a particular manipulative on a problem,
• 30:42
  teachers can use a share code and share it with students.
• 30:47
  It becomes a digital record of the work for the day.
• 30:50
  It can get shared with parents.
• 30:53
  Parents can open up that share code and look at it.
• 30:56
  They can help their student at home.
• 30:58
  They can see what's happening in the classroom.
• 31:00
  So it's a great way to get parents involved as well because that app
• 31:06
  can be used anywhere by anyone.
• 31:09
  And I think that is one of the most powerful pieces,
• 31:14
  is to be able to bring the family into the math classroom.
• 31:18
  So I'm so glad we had this conversation because I think you've helped
• 31:22
  me clarify some of the things that I might not have had language for—one
• 31:26
  of which is: There's a "when" question that we have to think about when
• 31:30
  it comes to physical manipulatives and virtual manipulatives.
• 31:34
  I also really have to think about the learning goal.
• 31:37
  What am I trying to do? Does the learning goal indicate that physical manipulatives
• 31:42
  really make sense? Or has the learning goal gotten to a place where a
• 31:46
  set of virtual materials
• 31:47
  might be perfectly appropriate?
• 31:49
  Or really, is there a place where these things are working in concert with
• 31:53
  one another to support student learning?
• 31:55
  I think those ideas really for me feel like something we can leave with teachers
• 32:01
  that will help them make some judgments and make some decisions.
• 32:04
  I agree, Mike. I think that you are exactly right.
• 32:08
  Teachers really need to consider where students are in that learning progression
• 32:14
  and what their goal is ~ for that particular lesson.
• 32:19
  If someone wants to try out or learn more about the Math MultiApp,
• 32:22
  how can they find it? What can they do with it?
• 32:26
  Yeah, well finding it's pretty easy.
• 32:27
  If you go to The Math Learning Center website at mathlearningcenter.org,
• 32:31
  there's a link to the MultiApp at the top of the page,
• 32:33
  and within the Free Resources section.
• 32:35
  But you can also just enter the URL,
• 32:37
  which is multiapp.mathlearningcenter.org.
• 32:40
  For teachers who are Bridges [in Mathematics] users,
• 32:42
  there are also links as well within the curriculum to be able to get to it as well.
• 32:46
  And Natalie, any restriction on the ability of a teacher to use this depending
• 32:50
  on whether they use Bridges as a curriculum or whether they're using it
• 32:54
  on the open web?
• 32:56
  Absolutely not. Anyone can access this.
• 32:58
  It can be a teacher, a parent,
• 33:00
  someone at a university, students,
• 33:03
  anyone in the public can access the MultiApp.
• 33:07
  And if I'm not mistaken, Mike,
• 33:08
  ~ I think that links to all the things that we're talking about
• 33:11
  are gonna be included with the podcast episode.
• 33:13
  So folks will be able to get to those directly.
• 33:15
  I should also point out that The Math Learning Center is gonna offer
• 33:18
  a free webinar about the MultiApp and all of its functionality on October 8th,
• 33:21
  ~ and a link to the recording is gonna appear on our website.
• 33:25
  Patrick and Natalie, thank you both so much for joining us.
• 33:27
  It has been an absolute pleasure talking to you both.
• 33:30
  It's been a really fun conversation,
• 33:31
  Mike. Thanks.
• 33:32
  Thanks so much, Mike. It was great being here.
• 33:35
  And I also want to thank you, our listeners,
• 33:37
  for tuning in. Have you tried the Math MultiApp by MLC yet?
• 33:41
  If so, what are some of the ways you've incorporated it into your classroom
• 33:45
  practice? Getting in touch with us is easy.
• 33:48
  Just head over to mathlearningcenter.org/podcast and fill out the contact form.
• 33:55
  Rounding up is brought to you by The Math Learning Center and
• 33:58
  the Maier Math Foundation,
• 34:00
  dedicated to inspiring and enabling all individuals to discover their mathematical
• 34:05
  confidence and ability.