I spent some time with a wonderful kindergarten teacher a while back. The two of us were chatting when a little boy, probably 4 or 5 years old walked up to us with a inquisitive look and asked if it was home time yet. Since it was only about 10 in the morning I smiled at the question. But the teacher in a very calm and candid manner spread his arms out wider than shoulder width and said, “**There’s still THIS much time left.**“

The child, very satisfied with the response, happily headed over to another activity and continued to play. I, however, was quite stunned at what just transpired. The little boy wasn’t given an answer… or was he?

What was it about the gesture given that helped this child understand that there was still lots of time left in the day? I found myself quite interested in the above conversation for some time until I started thinking about the types of experiences our students have with time as young children.

Clocks like those below don’t have any meaning for most 4 or 5 year-olds. There are far too many numbers or symbols… different ways to read… for many to understand here!

But they have likely had many experiences looking at things like these:

Let’s think about these items for a minute. Children by age 4 or 5 have likely been exposed to a cell phone, a tablet or some type of electronic device that has a battery meter. Without any understanding of percent or without even being able to recognize large numbers, children instinctively know when the device is charged or when it is nearly “dead”. When they watch movies, they can determine if we are currently near the beginning or end based on the progression meter (when they hit pause of fast-forward)… or when downloading, they can tell by how much space is left before they can access the app or game… Even when waiting in a line-up, children can tell if it will be a short time or not without having to count the people in front of them.

Each of the pictures above show our students’ natural ability to determine information based on the relative space provided. This is a great example of how we use Spatial Reasoning to make sense of things.

There is a lot of research about Spatial Reasoning and it’s relationship with mathematics. Much of this idea was first brought to my attention through the article: Paying Attention to Spatial Reasoning. While you probably haven’t read this, I think it is really important to read through the whole article. Really… it will explain a lot better than I can.

**If our students come into our math classes with an intuitive understanding of the spatial relationships behind concepts, shouldn’t we provide experiences that allows our students to access this form of reasoning BEFORE we start with formal rules, procedures, notations…?**

For instance, how might a simple graph like this bring about a conversation of subtraction?

How might a number line be used so our students can visually see how subtraction works? And how subtraction is related to addition?

It is blurry, but the 132-94 was moved to show 138-100. How might this visual help us with future problems?

Or, if we know Mr. Stadel’s height is 193cm. How might that help us think about estimating Mrs. Stadel’s height?

How did you think about this? What spatial cues did you use? Can you see how others might see this as a subtraction or addition problem?

Spatial reasoning isn’t just for subtraction though. If we wanted to know how many chocolates were here what might students do that are just starting to learn about multiplication?

How many different ways can you think of solving this question?

##### An example that really illustrates the need to focus on Spatial Reasoning:

At the end of the year, the grade 7 students throughout our school board were given a year-end test. One of the questions our students did poorest on was this problem:

While the vast majority of students got this problem wrong because of issues with converting units, several students instead took a path that allowed them to make sense of the problem. Below is a diagram we saw a few students draw that represented what was going on in the problem quite nicely. What did they do here? What do you notice?

I notice dimensions that make sense. I notice a piece of cloth being cut up, much like what we would actually do if this were a real-life problem. I notice the student thinking about the space required to cut the bandanas.

So I am left wondering why did only a few students attempt to make a drawing? Why did so many make errors with converting between units? Why did so many students make errors that were not even close to an acceptable answer? What experiences did they have… and didn’t they have that led them to attempt this problem the way they did?

While many teachers might tell students that there are 100 x 100 cm in a square meter, and expect students to understand and remember… or show conversion charts… or offer a page of conversion questions… I think we might be missing a big piece of the learning process. We haven’t tapped into our students’ spatial reasoning at all!

After seeing so many questions like this attempted by students in such procedural ways, I think next year we will start to think more about the reasoning our students already come into our classes with. And continue to think about how we can support the development of our students Spatial Reasoning is in helping them make sense of things!

** The National Research Council describes the current situation as a “major blind spot” ****in education and maintains that, without explicit attention to spatial thinking, the concepts, tools and processes that underpin it “will remain locked in a curious educational twilight zone: extensively relied on across the K–12 curriculum but not explicitly and systematically instructed in any part of the curriculum” (p. 7). **

I can see this being important for my sped students. I spend a lot of time insisting they draw pictures or a diagram before they start solving. Creating that visual makes all the difference.

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Just two thoughts about Marlene’s problem:

1. 25cm won’t go round a grade 7 kid.

2. What is 1.25Msquared? The actual dimensions are crucial.

and for a bonus solve two simultaneous equations.

Good stuff, if “training” is not the important aspect.

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Thanks for pointing out that this problem is kind of a pseudo-context. You are correct that 25cm would not be enough for a bandana.

The dimensions of the original cloth were left out – on purpose- to see what our students would do.

Hopefully we can see that this task was a summative task to see how our students understood the relationship between cm2 and m2.

Thanks for the comment

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