I can’t tell you how many times this year I’ve made this exact model. In case you can’t tell, it’s a house. There are variations on this single math problem in every book, in every grade, and I keep trying to show friends and colleagues in other schools and here, that they must be teaching it wrong.

Here’s the problem, and keep in mind that there are lots of variations of this:

“Doris is building a playhouse for her sister. The playhouse is built, and now she is trying to paint it. The paint comes in gallon cans that cover 200 square feet. With the dimensions of the various parts of the house shown in this diagram at right… how many gallons of paint must she buy?”

How do I know they’re teaching it wrong?

Because students from all over keep showing me that they DON’T know how to solve it. The diagrams in the book don’t help them; they are confused by the pictures with their helpful lines and measurements in feet or meters, inches or yards… whatever. The information isn’t enough to help them solve the challenge. They cannot see the problem in three dimensions, on a two dimensional surface.

But I show them one of these quick cardstock models, and they get it right away. They even take my blue Sharpie marker, and figure out how to design the model better, or mark it up in order to solve the problem.

Forced perspective or orthographic projections don’t solve all difficulties. Brunelleschi, the famous Renaissance architect, said that “Errors in the sketch are magnified in the model.”

BUT, if we’re teaching kids to build 3-D card stock models of their math problems, 1) we have introduced a whole new way for students to ‘show their work’ and 2) we’ve allowed them to understand a new dimension to the problem presented. For example, in this quick model, the student and I discovered that Doris’s playhouse wouldn’t need nearly as much paint as we assumed — until the student correctly recognized that the whole back wall of the model was missing… Doris’s playhouse wasn’t complete, when constructed exclusively from what was visible in the diagram in the book.

I DON’T want anyone to walk away from this thinking that I know how to teach math (I don’t, and I shouldn’t), or that I think that my colleagues who teach math are dumb or wrong-headed. I think they’re really top-notch people… But I also see a place for teaching visual thinking and three-dimensional awareness going un-used, and a general blind spot on both the teaching side and the learning side that is easily filled with a piece of card stock and a pair of scissors. I’d like to see that change.

*Via Flickr:*

One of my advisees brought me a math problem she couldn’t see properly. Bad diagram in the text book, perhaps, or just not equipped just yet to understand how the image unfolded into length, width, and height.

So we made a model. And she found that she had a more accurate understanding of the problem than most of her classmates.

the video advertisement on my viewing of this post went on at length re how to find blue jeans which precisely fit one’s body. in a way, clothing design confirms the value of 3d representation and manipulation… in my lifetime, jeans have gone from one size fits nobody to a near-infinite array of parameter-driven styles and sizes. still, it’s misleading to act as if 3d design has somehow abandoned the underlying numerics… we wouldn’t have all these sexy wireframes if some engineer didn’t know his arithmetic.

I’m not suggesting that we abandon the underlying arithmetic. I’m saying that I encounter a LOT of students whose grasp of arithmetic becomes tenuous once presented with a complex multi-step problem, like a 3D one.

Great example: I had someone, a math professor, “prove” algebraically that any odd number is the difference of two squares. It was a great exercise… In futility. I didn’t understand his explanation at all. So I made a little doodle on graph paper. Took the same amount of time, but I “proved” all the odd numbers, 1-13.

Professor harrumphed and insisted I hadn’t proved anything at all because I’d used a drawing not numbers. I don’t need sexy wireframes to make a 3D model, either. I just need some scrap paper. It still reveals an underlying understanding of the problem, not just an ability to plough through procedure.

And he’s right. But we’re talking kids here, not Ph.D’s in mathematics. And I say that making a quick 3D model of a math problem is every bit as valid a way of showing your work as writing it out on graph paper.