It’s the first day of our study of 3D shapes. I hold up a rectangular prism Geoblock, and I ask my second graders what they notice.
“It’s a rectangle!”
“It’s a 3D rectangle!”
“It’s a rectangle block. It’s like a rectangle, but taller.”
And one student with a great memory for vocabulary adds, “No, it’s not a rectangle. It’s a rectangular prism.”
I ask the class, “Where on this block do you see a rectangle?” One student notes that there is a rectangle on the front face of the block. “How do you know that’s a rectangle?” The student responds, “because it’s just like the rectangle on the paper. It has two long sides and two short sides.” (We’ll get to squares, rectangles, and right angles later.) Another student points out that there are rectangles on the ends of the block. Students continue to volunteer until they’ve identified all six of the Geoblock’s faces as rectangles.
Understanding the difference between a rectangle and a rectangular prism is a difficult job for many second graders, who often label 3D shapes with 2D shape names — or see 3D shapes as merely 3D versions of 2D shapes.
In their article, “Shape Up!” in Teaching Children Mathematics, Christine Oberdorf and Jennifer Taylor-Cox describe how children’s books and tv shows can perpetuate these misconceptions.* Rather than use the more difficult, correct vocabulary, children’s media often labels cubes as squares, spheres as circles, and triangular prisms as triangles. Oberdorf and Taylor-Cox contend that “it is better to introduce new terminology to students rather than subsequently attempt to distinguish between two- and three-dimensional shapes.” They go on to say that even if young students don’t remember terms like rectangular prism, they need to be “given the opportunity to discuss and explore the concept, which will be a building block for future geometric understanding.”
As the teacher of a looping classroom, I see firsthand how experiences in second grade lay a foundation for third grade. Even if my second graders don’t remember all of this terminology next year (experience tells me at least half of them won’t), it’s important that we begin working against these common misconceptions now.
Beyond the lessons in our math program (we use Investigations, which I love), there are a few other tools that I have found really useful in helping students develop their understanding of 3D shapes. All of these tools help students unfold 3D shapes into their 2D nets, and thus, understand how 2D shapes are linked together to form the faces of 3D shapes.
Solids Elementary is an iPad/iPod Touch app that allows students to virtually unfold 3D shapes into different 2D nets. It also allows students to manipulate the shapes in other ways — by changing the color of the faces, marking edges and vertices, and making the shapes translucent — that help them see and understand each shape’s characteristics.
Here, I’ve chosen a right prism with a square base. The shape starts out solid and opaque, but with the swipe of your fingers, can be rotated, enlarged, or reduced.
Using the tools in the menu on the bottom of the screen, you can change the color of the faces, mark the shape’s edges and vertices, and make the shape translucent. This helps students with a common difficulty: understanding 2D representations of 3D shapes.
One of my least favorite test questions is asking students to identify the number of faces, edges, and vertices in a 3D shape represented on paper. If the representation is of an opaque solid, kids don’t count the faces, edges, and vertices they can’t see. If the representation is of a translucent solid, kids get confused by the overlapping lines. This app bridges the gap between what we show on paper and what we can manipulate with our hands when we hold a Geoblock.
Finally, using the slider on the right, you can unfold the shape into one of its 2D nets.
Using another tool from the menu, you can change the net you want to use when you unfold your shape. Here’s the same prism, unfolded another way.
After using Solids Elementary as one of our math workshop activities, I asked the kids what they noticed about 3D shapes. One student explained, “3D shapes are made of 2D shapes that you can fold and unfold in different ways. When you unfold them all the way, they become 2D. They’re flat. But when you fold them back up, they have space inside.” Other students had equally powerful realizations after just one class block. Weeks later, Solids Elementary is still one of their favorite apps to use during Academic Choice Time.
While Solids Elementary allows for virtual manipulations of shapes, GeoFix allows you to actually build 2D nets and 3D shapes with your hands.** GeoFix come in a variety of shapes and colors, and they connect simply by snapping the edges together with your fingertips.
With GeoFix, you can make many kinds of polyhedra — cubes, square pyramids, hexagonal prisms — as well as beautiful imaginative structures. If you are a regular reader, you know I believe learning always starts with play. GeoFix are not only a powerful learning tool, but they are also one of my students’ favorite toys.
Recently, when my class began exploring shapes with GeoFix, one second grader wanted to make a cube, but she was having a hard time. I asked to see what she had come up with so far, and she showed me this structure:
No matter how hard she tried, it just wouldn’t fold up into a cube! She was perplexed. I was intrigued. What a clear demonstration of how kids can perceive 3D shapes as 2D shapes, only “bigger.” To create a cube — in the student’s mind, a 3D square — she connected four squares to make a bigger square. I was struck by the four-ness of the student’s design. Even though, as a class, we had modeled counting the six faces of a cube, the idea of a square (and thus, a cube) being four — four sides, four angles — persisted.
To help her, I took out a Geoblock cube. Together, we counted the faces. Six! Just like on dice. We rolled the cube around on the desk to visualize how the faces might fold up. Then she gave it another try and came up with this:
Together, my student and I investigated why the shape wouldn’t fold up. “Oh, there has to be room for the squares to move,” she said. We borrowed a GeoFix cube from a friend and carefully unfolded it together into its 2D net, then practiced folding it back up. We unfolded it again a different way.
“What do you notice?” I asked. “They’re like walls,” she said, “and this one [pointing to the top face] is like the ceiling. I didn’t think it would look like this when I unfolded it. I thought it would look like a square.”
We set the cube aside, and she tried again, starting with what she called the “walls” of cube.
She attached a “floor,” and through trial and error, figured out where to put the “ceiling.”
One last, simple tool is the humble cardboard box. I love to have students bring in a variety of empty (clean) boxes from home that they can fold and unfold in the classroom. Some package designs are absolutely ingenious! And the nets that result are often surprising to students. They won’t look at cereal boxes the same way again!
It’s a funny thing, being a looping teacher. When you’re in the second year of your loop, and your kids don’t remember something from the year before, you have only one person to blame. “Who was your second grade teacher?” you think to yourself. Oh, right. It was me.
At the same time, you get these clear, daily demonstrations of how early experiences prepare students to take on more complex ideas the following year. Last year, when I taught third grade, I was delighted to find that my students had a strong grasp of how 3D shapes were related to 2D shapes. The investigations they had done as second graders clearly helped them when it came to understanding depth, volume, 2D representations of 3D shapes, and the characteristics of polyhedra.
As teachers, it’s essential that we not shy away from the complexities of math, but expose them to students in gradual, developmentally appropriate ways. It’s ok with me that students are sometimes confused. It’s ok that most of them won’t remember the term hexagonal prism. What’s important is that, after second grade, instead of looking at a cube and seeing a big square, they’ll see how they could unfold it into many squares. They’ll see a picture of a prism and understand that it represents a 3D shape, one with faces in the back as well as the front. Put simply, they’ll have developed critical visual-spatial skills that, with or without the terminology, will serve them well in the future.
*Oberdorf, Christine D. and Jennifer Taylor-Cox. “Shape Up!” Teaching Children Mathematics 5 (February 1999): 340-345.