Latest / Rounding Up / Making Sense of Spatial Reasoning with Dr. Robyn Pinilla

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Making Sense of Spatial Reasoning with Dr. Robyn Pinilla · Rounding Up

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Spatial reasoning can be a nebulous concept,
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and it's often hard for many educators to define. In this episode,
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we're talking about spatial reasoning with Dr.
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Robin Pania from the University of Texas El Paso.
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We'll examine the connections between spatial reasoning and other mathematical
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concepts and explore different ways that educators can cultivate this type of
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reasoning with their students. Welcome to the podcast, Robin.
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I'm really excited to be talking with you about spatial reasoning.
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And I am excited to be here.
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Well, let me start with a basic question.
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So when we're talking about spatial reasoning,
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is that just another way of saying that we're going to be talking about ideas
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that are associated with geometry, or are we talking about something bigger?
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It's funny that you say it in that way, Mike,
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because geometry is definitely the closest mathematical content that we see in
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curricula, but it is something much bigger.
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So I started with the misconception and then I used my own experiences
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to support that idea that this was just geometry because it was my favorite math
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course in high school because I could see the concepts modeled and I could make
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things more tangible.
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Drawing helped me to visualize some of those concepts that I was learning
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instead of just using a formula that I didn't necessarily understand.
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So at that time,
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direct instruction really ruled and I'm unsure what the conceptual
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understandings of my teachers even were because what I recall is doing numbers
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three through 47 odds in the back of the book and just plugging through these
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formulas.
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But spatial reasoning allows us to develop our concepts in a way
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that lead to deeper conceptual understanding.
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I liked geometry and it gave me this vehicle for mathematizing the world,
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but geometry is really only strand of spatial reasoning.
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So you're already kind of poking around the question that I was going to ask
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next,
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which is the elevator description of what do we mean when we talk about spatial
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reasoning and why does it matter? Why is it a big deal for students?
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So spatial reasoning is a notoriously hard to define construct
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that deals with how things move in space.
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It's individually how they move in space in relation to one another.
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A lot of my ideas come from a network analysis that Bruce and
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colleagues did back in 2017 that looked at the historical
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framing of what spatial reasoning is and how we talk about it in different
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fields because psychologists look at spatial reasoning. Mathematics,
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educators look at spatial reasoning.
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There's also connections into philosophy, the arts.
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But when we start moving towards mathematics more specifically,
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it does deal with how things move in space individually and in relation to one
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another. So with geometry,
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whether the objects are sliding and transforming or we're composing and
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decomposing to create new shapes,
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those are the skills in two dimensional geometry that we do often see in
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curricula.
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But the underlying skills are also critical to everyday life and they can be
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taught as well.
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And when we're thinking about the everyday constructs that are being built
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through our interactions with the world,
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I like to think about the GPS on our car.
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So spatial reasoning has a lot of spatial temporal processes that are going on.
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It's not just thinking about the ways that things move in relation to one
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another or the connections to mathematics,
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but also the way that we move through this world,
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the way that we navigate through it. So I'll give a little example.
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Spatial temporal processes have to do with us running errands perhaps.
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How long does it take you to get from work to the store to home,
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and how many things can you purchase in the store knowing how full your fridge
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currently is?
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What pots and pans are you going to use to cook the food that you purchase,
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and what volume of that food are you and your family going to consume?
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So all of those daily tasks involve conceptions of how much space things
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take, and we could call it capacity,
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which situates nicely within the measurement domain of mathematics education,
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but it's also spatial reasoning and it extends further than that.
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That is helpful.
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I think you opened up my understanding of what we're actually talking about,
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and I think the piece that was really interesting is how in that example of I'm
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going to the grocery store, how long will it take? How full is my fridge?
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What are the different tools that I'll use to prepare what capacity do they
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have?
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I think that really helped me broaden out my own thinking about what spatial
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reasoning actually is.
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I wonder if we could shift a bit and you could help unpack for educators who are
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listening,
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a few examples of tasks that kids might encounter that could support the
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development of spatial reasoning.
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Sure.
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My research and work is primarily focused on early childhood and elementary,
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so I'm going to focus there but then kind of expand up.
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Number one, let's play.
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That's the first thing that I want to walk into a classroom and see.
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I want to see the kids engaging with blocks, legos, duplos,
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and building with and without specific intentions.
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Not everything has to have a preconceived lesson.
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One of the activities I've been doing actually with teachers and professional
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development sessions lately is a presentation called Who's Its and what
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sits.
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I have the teachers create what sits that do.
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That sits meaning they create something that does something.
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I don't give them a prompt of what problem they're going to be solving or
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anything specific for them to build, but rather say,
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here are materials. We give them large dulo blocks,
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magnet tiles and magforms,
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different types of wooden cardboard and foam blocks,
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PVC pipes,
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which are really interesting in the ways that teachers use them and have them
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start thinking as though they're the children in the class and they're trying to
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build something that takes space and can be used in different ways.
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So the session we did a couple of weeks ago,
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some teachers came up with First there was a swing that they had put a
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little frog in that they controlled with magnets.
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So they had used the PVC pipe at the top to be the top of the swing
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connected over,
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and then were using the magnets to guide it back and forth without ever having
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to touch the swing,
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and I just thought that was the coolest way for them to be using these materials
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in really playful creative ways that could also engender them taking those
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lessons back into their classroom.
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I have also recently been reminded of the importance of modeling with
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fractions.
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So are you familiar with the which one doesn't belong Tasks?
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Absolutely love them.
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Yes.
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There's also a website for fraction talks that children can look at visual
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representations of fractions and determine which one doesn't belong for some
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reason,
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that helps us to see the ways that children are thinking about the fractional
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spaces and then justifying their reason around them. With that,
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we can talk about the spatial positioning of the fractional pieces that
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are colored in or the ways that they're separated.
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If those colored pieces are in different places on the figure that's being
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shown,
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they open up some nice spaces for us to talk about different concepts and use
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that language of spatial reasoning that is critical for teachers to engage in to
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show the ways that students can think about those things.
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So I want to go back to this notion of play,
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and what I'm curious about is why is situating this in play going to help
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these ideas around spatial reasoning come out as opposed to say,
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situating it in a more controlled structure?
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Well, I think by situating spatial reasoning within play,
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we do allow teachers to respond in the moment rather than having these
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lesson plans that they are required to plan out from the beginning.
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A lot of the ideas within spatial reasoning,
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because it's a nebulous construct and it's learned through our everyday
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experiences and interactions with the world, they are harder to plan.
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And so when children are engaged in play in the classroom,
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teachers can respond very naturally so that they're incorporating
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the mathematizing of the world into what the students are already doing.
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If you take, for example,
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one of my old teachers used to do a treasure hunt,
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great way to incorporate spatial reasoning with early childhood elementary
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classrooms where she would set up a mapping task is really what it was,
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but it was introducing the children to the school itself and navigating that
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environment, which is critical for spatial reasoning skills.
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And they would play this gingerbread man type game of she would read the book
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and then everybody would be involved with this treasure hunt where the kiddos
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would start out in the classroom and they would get a clue to help them navigate
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toward the cafeteria. When they got to the cafeteria,
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the gingerbread man would already be gone. He would've already run off,
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so they would get their next clue to help them navigate to the playground,
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so on and so forth. They would go to the nurse's office, the principal,
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the library,
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all of the critical places that they would be going through on a daily basis or
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when they needed to within the school.
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And it reminds me that there was also a teacher I once interviewed who used
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orienteering skills with her students. Have you ever heard of orienteering?
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The connection I'm making is to something like Geocashing,
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but I think you should help me understand it.
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Yeah, that's really similar.
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So it's this idea that children would find their way,
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places path finding and way finding are also spatial reasoning skills that are
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applied within our real world.
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And so while it may not be as scientific or
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sophisticated as doing geocaching,
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it has children with the idea of navigating in our real world helps them start
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to learn cardinality and the different ways of thinking about traversing to a
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different location,
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which these are all things that might better relate to social studies or
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technology, other STEM domains specifically,
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but that are undergirded by the spatial reasoning,
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which does have those mathematics connections.
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I think the first thing that occurred is all of the directional language that
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could emerge from something like trying to find the J of red boy.
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And then the other piece that you made me think about just now is this
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opportunity to quantify distance in different ways,
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and I'm sure there are other things that you could draw out,
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especially in a play setting where the structure is a little bit looser and it
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gives you a little bit more space, as you said,
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to respond to kids rather than feeling like you have to impose the structure.
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Yeah, absolutely.
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There's an ability when teachers are engaging in authentic ways with the
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students, that they're able to support language development, support,
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ideation and creation without necessarily having kids sit down
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and fill out a worksheet that says, where is the ball?
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The ball is sitting on top of the shelf. Instead,
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we can be on the floor working with students and providing those directions
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of, oh, hey,
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I need you to get me those materials from the shelf on the other side of the
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room,
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but thinking about how can I say that in a way that better supports children
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understanding the spatial reasoning that's occurring in our room.
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So maybe it's find the pencil inside the blue cup on top of
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the shelf that's behind the pencil sharpener,
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getting really specific in the ways that we talk about things so that we're
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ingraining those ideas in such a way that it becomes part of the way that the
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kids communicate as well.
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You have me thinking that there's an intentionality in language choice that can
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create that, but then I would imagine as a teacher,
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I could also revoice what students are saying and perhaps introduce language in
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that way as well.
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Yeah,
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and now you have me thinking about a really fun routine number talks,
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of course.
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And if we do the idea of a.talk instead of a number talk,
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thinking about the spatial structuring of the dots that we're seeing and the
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different ways that you can see those arrangements and
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describe the quantification of the arrangement.
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It's a nice way to introduce educators to spatial reasoning because it might be
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something that they're already doing in the classroom while also providing an
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avenue for children to see spatial structuring in a way that they're already
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accustomed to as well based on the routines that they're receiving from the
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teacher.
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I think what's really exciting about this, Robin, is the more that we talk,
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the more two things jump out.
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I think one is my language choices allow me to introduce these
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ideas in a way that I don't know that I'd thought about as a practitioner.
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Part two is that we can't really necessarily draw a distinction between work
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we're doing around numbers and quantity and spatial reasoning that there are
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opportunities within our work around number quantity and within
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math content to inject the language of spatial reasoning and have it become a
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part of the experience for students.
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Yeah,
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and that's important that I have conveyed that without explicitly
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saying it because that's the very work that I'm doing with teachers in their
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classrooms at this time. One,
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as you're talking about language, and I hate to do this,
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but I'm going to take us a little bit off topic for a moment.
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I keep seeing this idea on Twitter or whatever we call it at this point,
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that some people actually don't hear music in their heads.
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This idea is wild to me because I have songs playing in my head all the time,
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but at the same time,
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what if we think about the idea that some people don't also
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visualize things,
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they don't imagine those movements continuously that I just
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see. And so as teachers,
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we really need to focus on that same idea that children need opportunities
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to practice what we think they should be able to hear,
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but also practice what we think they should be able to see.
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I'm not a cognitive scientist. I can't see inside someone's head,
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but I am a teacher by trade,
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so I want to emphasize that teachers can do what's within their locus of control
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so that children can have opportunities to talk about those tasks.
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One that I recently saw was a lesson on clocks.
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So while I was sitting there watching her teach, she was using a Judy clock.
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She was having fun games with the kids to do a little competition where they
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could read the clock and tell her what time it was.
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But I was just starting to think about all of the ways that we could talk about
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the shorter and longer hands, the minute and our hands,
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the ways that we could talk about them rotating around that center point.
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What shape does the hand make as it goes around that center point and what
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happens if it doesn't rotate fully?
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Now I'm going back to those fractional ideas from earlier with the which one
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doesn't belong tasks of having full shapes versus half shapes,
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and how we see those shapes in our real lives that we can then
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relate with visualized shapes that some children may or may not be able to
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see.
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You have me thinking about something. First of all,
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I'm so glad that you mentioned the role of visualization.
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You had me thinking about a conversation I was having with a colleague a while
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ago,
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and we had read a text that we were discussing and the point of conversation
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came up. I read this and there's a certain image that popped into my head,
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and the joke we were making is I'm pretty certain that the image that I saw in
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my head having read this text is not the same as what you saw.
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What you said that really struck home for me is I might be making some real
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assumptions about the pictures that kids see in their head and helping build
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those internal images, those mental movies. That's a part of our work as well.
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Absolutely. Because I'm thinking about the way that we have prototypical shapes.
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So a few years ago I was working with some assessments and
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the children were supposed to be able to recognize an equilateral triangle,
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whether it was gravity based or facing another orientation,
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and there were some children who automatically could see that the triangle was a
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triangle no matter which direction it was pointing,
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whereas others only recognize it if a triangle, if it were gravity based.
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And so we need to be teaching the properties of the shapes beyond just that
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image recognition that oftentimes our younger students come out with.
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I tend to think of visualization and language as supporting one another with the
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idea that when we are talking, we're also writing a descriptive essay.
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Our words are what create the intended picture.
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Can't say that it's always the picture that comes out,
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but the intended picture for the audience.
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What we're hopeful for in classrooms is that because we're sharing physical
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spaces and tangible experiences that
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the language used around those experiences could create shared meaning.
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That's one of the most difficult pieces in talking about spatial reason or quite
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frankly, anything else,
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is that oftentimes our words may have different meanings depending on who the
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speaker and who the listener are.
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And so navigating what those differences are can be quite challenging,
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which is why spatial reasoning is still so hard to define.
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Absolutely.
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My other follow up is if you were to offer people a way to get started,
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particularly on visualization,
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is there a kind of task that you imagine might move them along that pathway?
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I think the first thing to do is really grasp an approximation.
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I'm not going to say figure out what spatial reasoning is,
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but just an approximation or a couple of the skills therein that
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you feel comfortable with.
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So spatial reasoning is really the set of skills that undergirds almost all of
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our daily actions,
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but it also can be inserted into the lessons that teachers are already teaching.
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I think that we do have to acknowledge that spatial reasoning is hard to define,
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but the good news is that we do reason spatially all day every day.
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If I am in a classroom,
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I want to look first at the teaching that's happening,
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the routines that are already there,
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and see where some spatial reasoning might actually fit in with our young
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classes. I like to think about calendar math.
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Every single kindergarten, first grade classroom that you walk into,
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they're going to have that calendar on the wall,
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so how can you work into the routines that are occurring,
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that spatial language to describe the different components of the routine?
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So as a kiddo is counting on that hundreds chart talking about the ways
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in which they're moving the pointer along the numbers.
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When they're counting by tens,
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talk about the ways that they're moving down when they're finding the patterns
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that are on the calendar,
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because all of those little calendar numbers for the day,
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they wind up having a pattern within them in most of the curricular kits.
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So thinking about just the ways that we can use language they're in now with
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older students,
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I think that offering that variety of models or manipulatives for them to use
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and then encourage them to translate from having a concrete manipulative
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into those more representational ideas is great regardless of age or
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grade.
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So students benefit from the modeling when they do diagramming of their
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models.
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That is translating the 3D model to 2D,
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which is another component of spatial reasoning.
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And that gets me to this sticky point of I'm not
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arguing against automaticity or being able to solve equations without physical
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or visual models,
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but I'm just acknowledging this idea that offering alternative ways for students
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to engage with content is really critical because we're no longer at
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a phase that we need our children to become computers.
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We have programs for that.
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We need children who are able to think and solve problems in novel
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ways because that's the direction that we're moving in problem solving.
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That's fantastic. My final question before we close things up.
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If you were to make a recommendation for someone who's listening and they're
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intrigued and they want to keep learning,
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are there any particular resources that you'd offer people that they might be
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able to go to?
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Yeah, absolutely.
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So the first one that I like is the Learning Trajectories website.
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It's a learning trajectories.org. It's produced by Doug Clements and Julie Sama.
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There are wonderful tasks that are associated with
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spatial reasoning skills from very young children in the
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infants and toddler stages all the way up until seven or eight years old.
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So that's a great place to go that will allow you to see how children are
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performing in different areas of spatial reasoning.
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There is also a book called Taking Shape by Kathy Bruin,
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colleagues that I believe was produced in 2016,
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and the grade levels might be a little bit different because it is on the
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Canadian school system, but it's for K through two students,
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and that offers both the tasks and the spatial reasoning skills that are
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associated with them. For more of the research side,
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there's a book by Brent Davis and the Spatial Reasoning study group
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called Spatial Reasoning in the early Years,
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and that volume has been one of my go-tos
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in understanding both the history of spatial reasoning in our schools and also
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ways to start thinking about spatializing school mathematics.
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One of the things that I really appreciate about this conversation is you've
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helped me make a lot more sense of spatial reasoning,
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but the other thing that you've done for me at least,
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is see that there are ways that I can make choices with my planning,
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with my language that I could pick up and do tomorrow.
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There's not a discreet separate bit that is about spatial reasoning.
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It's really an integrated set of ideas and concepts and skills that I can
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start to build upon right away whatever curriculum I have.
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And that's the point. Often in mathematics,
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we think more explicitly about algebraic or numeric
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reasoning, but less frequently in classrooms about spatial reasoning.
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But spatial reasoning supports not only mathematics development,
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but other stem domains as well,
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and even skills that crossover into social studies and language arts as we're
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talking about mapping, as we're talking about language.
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So as students have these experiences,
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they too can start to mathematize the world,
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see spatial connections as they go out to recess,
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as they go home from school,
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as they're walking through their neighborhoods or just around the house.
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And it's ingrained ideas of measurement that we are looking at on a daily
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basis, the ways that we plan out our days and plan out our movements,
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whether it's really a plan or just our reactions to the world that
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support building these skills over time.
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And so there are those really practical applications,
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but it also comes to supporting overall mathematics development and then later
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STEM career interests,
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which is why I get excited about the work and want to be able to share it with
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more and more people.
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I think that's a great place to stop for listeners.
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We're going to link all of the content that Robin shared to our show notes.
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And Robin, I'll just say again, thank you so much for joining us.
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It's really been a pleasure talking with you.
24:06
Yes, absolutely. Thanks so much.
24:11
This podcast is brought to you by the Math Learning Center and the Meyer Math
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