Geometry in elementary school tends to be a lot about just shapes. Today, I want to talk about why we need to take a look at teaching geometry for elementary students beyond just shapes.

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• Brief overview of Van Hiele Levels

Van Hiele Levels

Now the first thing I want to direct your attention to is the van Hiele levels. These levels are one of the main research that we have around geometric thinking. And there’s a lot of people who don’t know about them. I’m going to give you a brief overview. And, if you want to know more, here’s a link to a great site by Nrich that you can go to to get a little bit more info about these levels.

But basically, kids in elementary school are typically in Level 0 and 1, and some a little bit in Level 2 in the upper grades of elementary. The research of Van Hiele showed that Level 0 is known as Visualization. This is where kids see visuals and it looks like something. So this is where you see kids who will say that a ball is a circle because it looks like a circle. They aren’t looking at attributes, knowing that it’s three-dimensional, and that makes it a different shape. They just say it looks like whatever. An ice cream cone looks like a triangle to them because of the shape. It’s got a point. And that’s all that they’re noticing.

A Level 1 is when they start to get into the Analysis, or what’s known as the Description level. This is where kids will start to notice some properties about something. And they use those properties. But they aren’t starting to see connections between properties. So, they might notice and say how a square is a square, because it has all the same sides, but they don’t know how a square is connected to a rectangle. They don’t see those connections.

That starts to happen in Level 2, where they are able to use properties, they’re able to justify thinking about shapes, they’re able to analyze a little bit, and kind of get informal proofs. This is the stage that’s known as Informal Deduction. And that’s about where elementary kiddos stop.

There are Levels 3 and 4, as kids progress through their understanding of geometry, into middle school, high school, and beyond. And this is where kids start giving formal proofs, they start doing very abstract geometry. We’re not going to go into those levels here. But I wanted you to understand those three levels that most of your students will progress through, throughout elementary school.

We tend to focus a lot on shapes because that’s a lot of the description that happens in those Van Hiele levels. I want to give you three things that will help kids progress through those Van Hiele levels, that isn’t necessarily about describing and identifying shapes.

Use Lots And Lots Of Visuals

Our first one is to use lots and lots of visuals. Now, in geometry, geometry is very visual. And I was a kid, growing up, that I did wonderful in mathematics, until I was in high school, and did geometry. And it was because it was so visual and I had to prove things. And to me, I was like, just give me the procedure and tell me how to do it. I wanted the abstract numbers and the equations to do. I wasn’t used to doing visual things. Yet there were other groups of kids who struggled in mathematics, but then they finally flourished when they got into geometry because it was so visual. So, the first recommendation is to include lots of visuals that aren’t just shapes.

You guys hear a lot from me around number sense, about how we need to build a sense of numbers. That three isn’t just the digit 3, it is three things. And giving kids those visuals in all areas of mathematics will help kids as they’re developing that Level 0, and moving into Level 1.

Let Kids Draw

Now, a thing that will help with Level 1 is our second recommendation here, which is to let kids draw. Not just in geometry, but in all of mathematics. You guys know I’m a big fan of the C-R-A model, which is Concrete, Representational, to Abstract. Kids need to start out with concrete manipulatives. But oftentimes we jump from those concrete manipulatives to the abstract symbol. So they’re seeing three, and then we’re attaching the digit 3 to it.

The in-between phase is a representation. They need to draw three things. Or draw a model of three. Because we all know there is a whole lot that goes into drawing something versus just seeing it, and saying, “Oh yeah, that’s three.” When you have to sit there and draw three squares, or three rectangles, something is starting to build in that kid about what they start to notice about shapes, that they don’t do if they just see three cubes. So get kids to draw, and that will help kids in that Level 1.

Get Kids Explaining & Proving

Alright, my third recommendation is to get kids explaining their thinking and proving their thoughts. Ask them, “Why is this correct?” It’s not just in geometry. But if they figured out 7 + 8 is 15, having them explain it and prove their thinking isn’t just to be mean but to get them to explain why they think that is the answer.

And there comes a point that, yes, we just let kids say, “7 + 8 is 15.” But having them explain their thinking will start to build other ideas. It’s not just about getting to the answer. Remember that it’s not just about creating calculators. It’s about building their math minds and getting kids explaining their thinking. And justifying why they’re right.

Plus, It’s not just about building their math minds, guys, it’s about just common things that we need kids to be able to do in life, right. Not just proving why they’re right about an answer in mathematics, but proving why they think something is correct out there in life. It’s a habit that we want to help develop in all aspects of our children. But mathematics, when we do it in there, it also helps start to build them into Level 2, and get them to start having informal deductions.

So, you can focus on shapes in geometry. But, I want to encourage you to provide lots of visuals outside of shapes and geometry, get kids drawing, and get them explaining and proving their thinking. All of that goes into helping your kiddos progress through the Van Hiele levels and build their geometric thinking.

Alright, I hope that this helped you build your math mind, so you can go build those math minds, and not just create calculators.

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