Developing Spatial Reasoning: An Interview With Mark Chubb

Mark Chubb

ark Chubb has worked on and off for 12 years as a K–8 instructional coach for the District School Board of Niagara, specializing predominantly in mathematics. He’s involved in school improvement plans, but his main role is to help teachers find meaningful, fun ways to bring math to their students.

MARIAN: You cover a lot of topics on X (formerly known as Twitter) and in your blog posts, and one that intrigues me because we haven’t covered it a lot in the magazine is spatial reasoning. Most teachers focus on number in their math classes. How do you convince teachers that spatial reasoning is worth their time?

MARK: Our school board’s focus on spatial reasoning has been a long journey. Some of my colleagues and I began by noticing kids struggling with mathematics and wanted to figure out ways to help them. We found that when students struggle, if we spatialize or help them visualize, it helps them understand the math.

Years ago, we planned board-wide professional development around what we call our scope and sequence. We decided to start the year with some units that weren’t number. For example, we started with patterning and then moved to geometry and then measurement. And we thought, you know, if you’re measuring things all the time, how much number comes out of that? If you’re making patterns, how much number comes out of that? We recognized that those other strands were helping our kids visualize the numbers that they were working with. We saw success with that, and we continued to deepen our understanding of spatial reasoning—recognizing that there is so much more to it than we’d previously considered.

Spatial reasoning encompasses a lot. It includes the ability to compose and decompose, the idea that you can see things from different perspectives, that you can enlarge and shrink, and so on. The tasks that we’ve come up with have really inspired teachers to see mathematics as interconnected. We’ve had so many teachers take AQ [Additional Qualifications] courses in my board. We made it a requirement that spatial reasoning be a part of projects and assignments in these AQ courses. And this has now grown seeds in every building in our system—it’s almost become common practice in our board.

MARIAN: You mentioned that spatial reasoning is important for kids who struggle, but I get the feeling you don’t see it only for those who are struggling. Is it helpful for other students as well?

MARK: Spatial reasoning work isn’t just for certain kids. What’s interesting is that the more we spatialize tasks, the more room there is for students who struggle to have access, and the more room there is for students who are ready for it to have advancements.

So, we can give the same problem to a whole class, and we see very different self-differentiated approaches happening among our students. We’ve noticed that when students focus on representations, they are making connections between concepts. Sometimes, we focus on tasks that are strictly viewed as spatial tasks and may seemingly not have connections with other content. Over the years, I’ve promoted a number of these tasks, for example, skyscraper puzzles, and tasks that involve Cuisenaire rods and other manipulatives.

MARIAN: Can you explain for our readers what a skyscraper puzzle is?

MARK: Imagine a small city with skyscrapers in it. If you look at this city from a certain vantage point, you’ll see some of the towers, but some you won’t see because some buildings are taller than others. Each puzzle is almost like a Sudoku puzzle where in each row and column, there is one tower of each size. We wanted our kids to visualize these puzzles, see them as towers, and see the perspectives. The puzzles have clues around the outside that give information showing how many towers you would be able to see from that vantage point. If you use that information, along with logic and spatial reasoning, you’ll be able to see where all the other towers should be.

So, having students work on spatial puzzles has become normal practice in my board, where our kids are engaged in something that really makes them think. They have to hold information in their head, work with that information, and use it while having that feeling of “this is tough. I have to stick with it. I have to use reasoning.”

MARIAN: Give us an example of a reasoning activity with Cuisenaire rods.

MARK: Many teachers think manipulatives like Cuisenaire rods are to be used to help students follow a procedure and come up with the one correct answer that everyone agrees is the correct answer. However, in my experience, Cuisenaire rods can be used in almost every area throughout the curriculum, K to 8, in ways that help students visualize and make sense of the math they are learning. This is what spatial reasoning is all about.

I was involved in a project with some Kindergarten and Grade 1 teachers in my board, and we designed a handful of spatial reasoning activities that led to algebraic reasoning. In one activity, students would spin a spinner, and it would land on one of the Cuisenaire rod colours, and students would have to try to fill a designated space. And what we noticed, because we hadn’t attached any numbers to the Cuisenaire rods, was that the Kindergarten and Grade 1 students would know intuitively which rod would fit in the given spaces that were left. In fitting the rods into the spaces that remained, they started to recognize that this one is bigger than that one or that this one is that much bigger than that one. That spatial piece of the size gradient between the rods started to come about, and that led to more and more games. We noticed that playing games that relied solely on spatial reasoning helped our students when we moved toward number relationships. Spatial reasoning was directly helpful in developing algebraic reasoning and number sense.

We also played a game that involved symmetry, where one student would place a rod on one side of the table and a second student would place two rods that together were identical in size to the first rod. The sides would have to be symmetrical, and we’d turn those into equations by the time we were done Grade 1. So, students would recognize that 5 = 3 + 2 and so on.

Cuisenaire rods are an excellent way to bring reasoning in while letting it be interesting and fun and really deepening our math learning.

MARIAN: So, this would be an example of something that is context-free but still engages kids’ interest and perseverance and so on.

MARK: Exactly.

MARIAN: Do you think it’s made a difference that this initiative is board-wide in terms of getting more teachers on board?

MARK: I think the more we have professional discussions, the better. If we have more tools, we can say, “I can use spatial tasks to get at this. I can use Cuisenaire rods to help students understand.” The more we can do this, the more the levels within a system can learn from one another. We need enough people on board to be excited about learning math for it to really take hold. When that happens, it’s powerful.

MARIAN: You mentioned a few examples that were lower and middle grades. How can you convince Grades 6, 7, and 8 teachers that spatial reasoning is an important part of their curriculum? They often move from number to algebra and kind of stop there.

MARK: How do we help Grades 6 to 8 teachers see the importance of spatial reasoning? Sometimes it involves us opening up the curriculum and seeing where spatial reasoning already is. For example, we might easily see that geometry is spatial, but we also want to see measurement as spatial. Measurement is more than algebraic formulas. We want our students to think about composing and decomposing shapes to make sense of formulas and to solve tricky problems. We want to see number and operations spatially. Very large or very small numbers are often looked at through the lens of a place-value chart, but we also want to think of numbers spatially, which can include placing the numbers on a number line. Our curriculum expectations can help us see exactly how to spatialize our math programs.

Spatial puzzles are useful at this level too. You really have to think about what you’re doing when you do a spatial puzzle. You’re holding information in your head; you’re often thinking about equivalence and equality. You’re doing a lot of the same tasks that you might do in algebra, for example. We’ve noticed that when teachers see this and how it translates to other tasks, that’s when the buy-in happens. As soon as they see it’s working, they recognize that it will benefit their students too.

MARIAN: Where could teachers find some good spatial reasoning tasks?

MARK: Spatial reasoning is something that’s already embedded in things. In MathUP, for example, if one were to look through all the grades, they’d see that many of the tasks are spatialized. Most math programs will include spatialized material. But if you want tasks that focus only on spatial reasoning, our province has done a great job of leading this research. There’s a resource called Taking Shape, created by Joan Moss, Catherine Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes, that digs into spatial reasoning in very practical ways. It’s primarily a K to 2 resource, but we’ve had success adapting it all the way to Grade 8.

MARIAN: I’ve read that it’s valuable to start using these ideas when kids are young. Do you find it’s easier to get younger kids hooked, or do you find reasonably good success even with older kids who might not be quite where they should be?

MARK: I think you can start doing spatial tasks at any age. It’s never too late. If you’re spatializing number, though, it’s sometimes harder to go the other way around. For example, if you’ve learned the algorithm and you’re asked to think about the size of the numbers you’re using, some kids might think that’s not necessary. They might think that the algorithm is all that is needed, but in real life, we are more likely to use the operation if we do mental math. And much of mental math relies on being able to visualize so that we know which strategies make the most sense for those numbers. So, going the other way around is sometimes tricky.

MARIAN: That makes sense. Are there other similar things that you’ve experienced with kids where you go in an unusual direction that engages them and makes a difference?

MARK: I think that any time we can turn something into a game or show that there is a purpose behind what students are doing, it’s engaging for them. It’s the idea of practice being dynamic. That can be in a spatial puzzle, but it can also be in something that’s not spatial at all—a game that involves students choosing numbers and explaining why they chose the numbers they did.

Any time we see kids active in their thinking, we see more engagement in the classroom. The more students are engaged, the more voices we hear. When we have a variety of voices, kids who might normally try to hide might be willing to participate. It’s about increasing the level of talk and getting to the deeper pieces of the math, not just the fun.

MARIAN: Yes, make it engaging. Are you still blogging about spatial reasoning these days, or are you on a new path?

MARK: I feel like when COVID hit, there was kind of a brain drain, and there was too much out there. The need shifted to online resources, and there was less need for a blog that pushes the boundaries of teaching. So, I took my foot off the gas for a bit. We needed things that were just ready to go.

Now, it feels like we’re getting back to a place where we’re ready to learn and try new things. I find that as a career choice, people who excel at spatial reasoning often choose careers in math and science and maybe not in education. So, I think spatial reasoning is something teachers need to continually explore.

There is a genuine need for us to see how it all fits together. And because our students are at various levels due to how much they took in over the last few years, all these kinds of unknowns are exaggerated. We need to do tasks that allow kids at all levels to be successful. I think focusing on spatial reasoning is the avenue that’s most helpful for teachers to get at that.

MARIAN: What was your most recent blog post about?

MARK: One of my most recent blog posts was about how not to start the school year. Tracy Zager, who I have great respect for, wrote a blog post entitled “How Not to Start Math Class in the Fall.” Basically, I took her idea and was thinking about post-COVID and asked, “What are the things that we don’t want to do?”

We don’t want to focus on negativity, we don’t want to focus solely on gaps, and we don’t want to focus on what our kids can’t do. The purpose of the blog post was to help us start to build up classrooms, build up a community of learners, and build up our students as capable. We want to have more voices in our classrooms, and sometimes that means including tasks that have lots of entry points and lots of room for discussion. So, the types of tasks that we choose, especially at the beginning of a school year, matter a lot. I think spatial reasoning is a great avenue to achieve that.

MARIAN: Is there anything else you’d like to say?

MARK: I sit on the board for OAME [Ontario Association for Mathematics Education], and I’m excited that our conferences are starting back up and that we’re getting our mini conferences back up. The idea that we are learners together is what matters most. Let’s come together and learn.

MARIAN: That’s a great message. Thanks for your time, Mark.

MARK: Thank you, Marian.