Season 2 | Episode 17 – Making Sense of Spatial Reasoning - Guest: Dr. Robyn Pinilla

Rounding Up

Rounding Up

Season 2 | Episode 17 – Spatial Reasoning

Guest: Dr. Robyn Pinilla

Mike Wallus: Spatial reasoning can be a nebulous concept, and it's often hard for many educators to define. In this episode, we're talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas, El Paso. We'll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students.

Mike: Welcome to the podcast, Robyn. I'm really excited to be talking with you about spatial reasoning.

Robyn Pinilla: And I am excited to be here. Mike: Well, let me start with a basic question. So, when we're talking about spatial reasoning, is that just another way of saying that we're going to be talking about ideas that are associated with geometry? Or are we talking about something bigger?

Robyn: It's funny that you say it in that way, Mike, because geometry is definitely the closest mathematical content that we see in curricula, but it is something much bigger. So, I started with the misconception and then I used my own experiences to support that idea that this was just geometry because it was my favorite math course in high school because I could see the concepts modeled and I could make things more tangible. Drawing helped me to visualize some of those concepts that I was learning instead of just using a formula that I didn't necessarily understand. So, at that time, direct instruction really ruled, and I'm unsure what the conceptual understandings of my teachers even were because what I recall is doing numbers 3 through 47 odds in the back of the book and just plugging through these formulas. But spatial reasoning allows us to develop our concepts in a way that lead to deeper conceptual understanding. I liked geometry, and it gave me this vehicle for mathematizing the world. But geometry is really only one strand of spatial reasoning.

Mike: So, you're already kind of poking around the question that I was going to ask next, which is the elevator description of, “What do we mean when we talk about spatial reasoning and why does it matter? Why is it a big deal for students?”

Robyn: So, spatial reasoning is a notoriously hard to define construct that deals with how things move in space. It's individually how they move in space, in relation to one another. A lot of my ideas come from a network analysis that [Cathy] Bruce and colleagues did back in 2017 that looked at the historical framing of what spatial reasoning is and how we talk about it in different fields. Because psychologists look at spatial reasoning. Mathematics educators look at spatial reasoning. There [are] also connections into philosophy, the arts. But when we start moving toward mathematics more specifically, it does deal with how things move in space individually and in relation to one another. So, with geometry, whether the objects are sliding and transforming or we're composing and decomposing to create new shapes, those are the skills in two-dimensional geometry that we do often see in curricula. But the underlying skills are also critical to everyday life, and they can be taught as well. Robyn: And when we're thinking about the everyday constructs that are being built through our interactions with the world, I like to think about the GPS on our car. So, spatial reasoning has a lot of spatial temporal processes that are going on. It's not just thinking about the ways that things move in relation to one another or the connections to mathematics, but also the way that we move through this world, the way that we navigate through it. So, I'll give a little example. Spatial temporal processes have to do with us running errands, perhaps. How long does it take you to get from work to the store to home? And how many things can you purchase in the store knowing how full your fridge currently is? What pots and pans are you going to use to cook the food that you purchase, and what volume of that food are you and your family going to consume? So, all those daily tasks involve conceptions of how much space things take. And we could call it capacity, which situates nicely within the measurement domain of mathematics education. But it's also spatial reasoning, and it extends further than that. Mike: That is helpful. I think you opened up my understanding of what we're actually talking about, and I think the piece that was really interesting is how in that example of “I'm going to the grocery store, how long will it take? How full is my fridge? What are the different tools that I'll use to prepare? What capacity do they have?” I think that really helped me broaden out my own thinking about what spatial reasoning actually is. I wonder if we could shift a bit and you could help unpack for educators who are listening, a few examples of tasks that kids might encounter that could support the development of spatial reasoning. Robyn: Sure. My research and work [are] primarily focused on early childhood and elementary. So, I'm going to focus there but then kind of expand up. Number one, let's play. That's the first thing that I want to walk into a classroom and see: I want to see the kids engaging with blocks, LEGOS, DUPLOS, and building with and without specific intentions. Not everything has to have a preconceived lesson. So, one of the activities I've been doing actually with teachers and professional development sessions lately is a presentation called “Whosits and Whatsits.” I have the teachers create whatsits that do thatsits; meaning, they create something that does something. I don't give them a prompt of what problem they're going to be solving or anything specific for them to build, but rather say, “Here are materials.” We give them large DUPLO blocks, magnet tiles and Magformers, different types of wooden, cardboard and foam blocks, PVC pipes, which are really interesting in the ways that teachers use them. And have them start thinking as though they're the children in the class, and they're trying to build something that takes space and can be used in different ways.

Robyn: So, the session we did a couple of weeks ago, some teachers came up with … first, there was a swing that they had put a little frog in that they controlled with magnets. So, they had used the PVC pipe at the top that part of the swing connected over, and then were using the magnets to guide it back and forth without ever having to touch the swing. And I just thought, that was the coolest way for them to be using these materials in really playful, creative ways that could also engender them taking those lessons back into their classroom. I have also recently been reminded of the importance of modeling with fractions. So, are you familiar with the “Which One Doesn't Belong?” tasks?

Mike: Absolutely love them.

Robyn: Yes. There's also a website for fraction talks that children can look at visual representations of fractions and determine which one doesn't belong for some reason. That helps us to see the ways that children are thinking about the fractional spaces and then justifying their reason around them. With that, we can talk about the spatial positioning of the fractional pieces that are colored in. Or the ways that they're separated if those colored pieces are in different places on the figure that's being shown. They open up some nice spaces for us to talk about different concepts and use that language of spatial reasoning that is critical for teachers to engage in to show the ways that students can think about those things.

Mike: So, I want to go back to this notion of play, and what I'm curious about is, why is situating this in play going to help these ideas around spatial reasoning come out as opposed to say, situating it in a more controlled structure?

Robyn: Well, I think by situating spatial reasoning within play, we do allow teachers to respond in the moment rather than having these lesson plans that they are required to plan out from the beginning. A lot of the ideas within spatial reasoning, because it's a nebulous construct and it's learned through our everyday experiences and interactions with the world, they are harder to plan. And so, when children are engaged in play in the classroom, teachers can respond very naturally so that they're incorporating the mathematizing of the world into what the students are already doing. So, if you take, for example, one of my old teachers used to do a treasure hunt—great way to incorporate spatial reasoning with early childhood elementary classrooms—where she would set up a mapping task, is really what it was. But it was introducing the children to the school itself and navigating that environment, which is critical for spatial reasoning skills.

Robyn: And they would play this gingerbread man-type game of, she would read the book and then everybody would be involved with this treasure hunt where the kiddos would start out in the classroom, and they would get a clue to help them navigate toward the cafeteria. When they got to the cafeteria, the gingerbread man would already be gone. He would've already run off. So, they would get their next clue to help them navigate to the playground, so on and so forth. They would go to the nurse's office, the principal, the library, all of the critical places that they would be going through on a daily basis or when they needed to within the school. And it reminds me that there was also a teacher I once interviewed who used orienteering skills with her students. Have you ever heard of orienteering?

Mike: The connection I'm making is to something like geocaching, but I think you should help me understand it.

Robyn: Yeah, that's really similar. So, it's this idea that children would find their way places. Path finding and way finding are also spatial reasoning skills that are applied within our real world. And so, while it may not be as scientific or sophisticated as doing geocaching, it has children with the idea of navigating in our real world, helps them start to learn cardinality and the different ways of thinking about traversing to a different location, which … these are all things that might better relate to social studies or technology, other STEM domains specifically, but that are undergirded by the spatial reasoning, which does have those mathematics connections.

Mike: I think the first thing that occurred is, all of the directional language that could emerge from something like trying to find the gingerbread boy. And then the other piece that you made me think about just now is this opportunity to quantify distance in different ways. And I'm sure there are other things that you could draw out, especially in a play setting where the structure is a little bit looser and it gives you a little bit more space, as you said, to respond to kids rather than feeling like you have to impose the structure.

Robyn: Yeah, absolutely. There's an ability when teachers are engaging in authentic ways with the students, that they're able to support language development, support ideation and creation, without necessarily having kids sit down and fill out a worksheet that says, “Where is the ball? The ball is sitting on top of the shelf.” Instead, we can be on the floor working with students and providing those directions of, “Oh, hey, I need you to get me those materials from the shelf on the other side of the room,” but thinking about, “How can I say that in a way that better supports children understanding the spatial reasoning that's occurring in our room?” So maybe it's, “Find the pencil inside the blue cup on top of the shelf that's behind the pencil sharpener,” getting really specific in the ways that we talk about things so that we're ingraining those ideas in such a way that it becomes part of the way that the kids communicate as well.

Mike: You have me thinking that there's an intentionality in language choice that can create that, but then I would imagine as a teacher I could also revoice what students are saying and perhaps introduce language in that way as well.

Robyn: Yeah, and now you have me thinking about a really fun routine number talks, of course. And if we do the idea of a dot talk instead of a number talk, thinking about the spatial structuring of the dots that we're seeing and the different ways that you can see those arrangements and describe the quantification of the arrangement. It's a nice way to introduce educators to spatial reasoning because it might be something that they're already doing in the classroom while also providing an avenue for children to see spatial structuring in a way that they're already accustomed to as well, based on the routines that they're receiving from the teacher.

Mike: I think what's really exciting about this, Robyn, is the more that we talk, the more two things jump out. I think one is, my language choices allow me to introduce these ideas in a way that I don't know that I'd thought about as a practitioner. Part two is that we can't really necessarily draw a distinction between work we're doing around numbers and quantity and spatial reasoning; that there are opportunities within our work around number quantity and within math content to inject the language of spatial reasoning and have it become a part of the experience for students.

Robyn: Yeah, and that's important that I have conveyed that without explicitly saying it because that's the very work that I'm doing with teachers in their classrooms at this time. One, as you're talking about language, and I hate to do this, but I'm going to take us a little bit off topic for a moment. I keep seeing this idea on Twitter or whatever we call it at this point, that some people actually don't hear music in their heads. This idea is wild to me because I have songs playing in my head all the time. But at the same time, what if we think about the idea that some people don't also visualize things, they don't imagine those movements continuously that I just see. And so, as teachers, we really need to focus on that same idea that children need opportunities to practice what we think they should be able to hear but also practice what we think they should be able to see.

Robyn: I'm not a cognitive scientist. I can't see inside someone's head. But I am a teacher by trade, so I want to emphasize that teachers can do what's within their locus of control so that children can have opportunities to talk about those tasks. One that I recently saw was a lesson on clocks. So, while I was sitting there watching her teach, she was using a Judy Clock. She was having fun games with the kids to do a little competition where they could read the clock and tell her what time it was. But I was just starting to think about all of the ways that we could talk about the shorter and longer hands, the minute and hour hands, the ways that we could talk about them rotating around that center point. What shape does the hand make as it goes around that center point and what happens if it doesn't rotate fully? Now I'm going back to those fractional ideas from earlier with the “Which One Doesn't Belong?” tasks of having full shapes versus half shapes, and how we see those shapes in our real lives that we can then relate with visualized shapes that some children may or may not be able to see.

Mike: You have me thinking about something. First of all, I'm so glad that you mentioned the role of visualization.

Robyn: Yeah.

Mike: You had me thinking about a conversation I was having with a colleague a while ago, and we had read a text that we were discussing, and the point of conversation came up. I read this and there's a certain image that popped into my head.

Robyn: Uh-hm.

Mike: And the joke we were making is, “I'm pretty certain that the image that I saw in my head having read this text is not the same as what you saw.” What you said that really struck home for me is, I might be making some real assumptions about the pictures that kids see in their head and helping build those internal images, those mental movies. That's a part of our work as well.

Robyn: Absolutely. Because I'm thinking about the way that we have prototypical shapes. So, a few years ago I was working with some assessments, and the children were supposed to be able to recognize an equilateral triangle—whether it was gravity-based or facing another orientation—and there were some children who automatically could see that the triangle was a triangle no matter which direction it was “pointing.” Whereas others only recognize it if a triangle, if it were gravity-based. And so, we need to be teaching the properties of the shapes beyond just that image recognition that oftentimes our younger students come out with. I tend to think of visualization and language as supporting one another with the idea that when we are talking, we're also writing a descriptive essay. Our words are what create the intended picture—can't say that it's always the picture that comes out. But the intended picture for the audience. What we're hopeful for in classrooms is that because we're sharing physical spaces and tangible experiences, that the language used around those experiences could create shared meaning. That's one of the most difficult pieces in talking about spatial reason or quite frankly, anything else, is that oftentimes our words may have different meanings depending on who the speaker and who the listener are. And so, navigating what those differences are can be quite challenging, which is why spatial reasoning is still so hard to define.

Mike: Absolutely. My other follow-up is, if you were to offer people a way to get started, particularly on visualization, is there a kind of task that you imagine might move them along that pathway?

Robyn: I think the first thing to do is really grasp an approximation. I'm not going to say figure out what spatial reasoning is, but just an approximation or a couple of the skills therein that you feel comfortable with. So, spatial reasoning is really the set of skills that undergirds almost all of our daily actions, but it also can be inserted into the lessons that teachers are already teaching. I think that we do have to acknowledge that spatial reasoning is hard to define, but the good news is that we do reason spatially all day every day. If I am in a classroom, I want to look first at the teaching that's happening, the routines that are already there, and see where some spatial reasoning might actually fit in. With our young classes, I like to think about calendar math. Every single kindergarten, first-grade classroom that you walk into, they're going to have that calendar on the wall. So how can you work into the routines that are occurring, that spatial language to describe the different components of the routine?

Robyn: So, as a kiddo is counting on that hundreds chart, talking about the ways in which they're moving the pointer along the numbers … when they're counting by 10s, talk about the ways that they're moving down. When they're finding the patterns that are on the calendar, because all of those little calendar numbers for the day, they wind up having a pattern within them in most of the curricular kits. So, thinking about just the ways that we can use language therein. Now with older students, I think that offering that variety of models or manipulatives for them to use and then encourage them to translate from having a concrete manipulative into those more representational ideas, is great regardless of age or grade. So, students benefit from the modeling when they do diagramming of their models; that is, translating the 3-D model to 2-D, which is another component of spatial reasoning. And that gets me to this sticky point of, I'm not arguing against automaticity or being able to solve equations without physical or visual models. But I'm just acknowledging this idea that offering alternative ways for students to engage with content is really critical because we're no longer at a phase that we need our children to become computers. We have programs for that. We need children who are able to think and solve problems in novel ways because that's the direction that we're moving in problem-solving.

Mike: That's fantastic. My final question before we close things up. If you were to make a recommendation for someone who's listening and they're intrigued and they want to keep learning, are there any particular resources that you'd offer people that they might be able to go to?

Robyn: Yeah, absolutely. So, the first one that I like is the Learning Trajectories website. It's, uh, learning trajectories.org. It's produced by Doug Clements and Julie Sarama. There are wonderful tasks that are associated with spatial reasoning skills from very young children in the infants and toddler stages all the way up until 7 or 8 years old. So, that's a great place to go that will allow you to see how children are performing in different areas of spatial reasoning. There is also a book called “Taking Shape” by Cathy Bruce and colleagues that I believe was produced in 2016. And the grade levels might be a little bit different because it is on the Canadian school system, but it's for K–2 students, and that offers both the tasks and the spatial reasoning skills that are associated with them. For more of the research side, there's a book by Brent Davis and the Spatial Reasoning Study Group called “Spatial Reasoning in the Early Years,” and that volume has been one of my go-tos in understanding both the history of spatial reasoning in our schools and also ways to start thinking about spatializing school mathematics.

Mike: One of the things that I really appreciate about this conversation is you've helped me make a lot more sense of spatial reasoning. But the other thing that you've done for me, at least, is see that there are ways that I can make choices with my planning, with my language … that I could pick up and do tomorrow. There's not a discreet separate bit that is about spatial reasoning. It's really an integrated set of ideas and concepts and skills that I can start to build upon right away whatever curriculum I have.

Robyn: And that's the point. Often in mathematics, we think more explicitly about algebraic or numeric reasoning, but less frequently in classrooms about spatial reasoning. But spatial reasoning supports not only mathematics development, but other stem domains as well, and even skills that crossover into social studies and language arts as we're talking about mapping, as we're talking about language. So, as students have these experiences, they, too, can start to mathematize the world, see spatial connections as they go out to recess, as they go home from school, as they're walking through their neighborhoods, or just around the house. And it's ingrained ideas of measurement that we are looking at on a daily basis, the ways that we plan out our days and plan out our movements, whether it's really a plan or just our reactions to the world that support building these skills over time. And so, there are those really practical applications. But it also comes down to supporting overall mathematics development and then later STEM career interests, which is why I get excited about the work and want to be able to share it with more and more people.

Mike: I think that's a great place to stop. For listeners, we're going to link all of the content that Robyn shared to our show notes. And, Robyn, I'll just say again, thank you so much for joining us. It's really been a pleasure talking with you.

Robyn: Yes, absolutely. Thanks so much.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

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