Pictures, Numbers, and Words

This is a really interesting article  [click HERE] on the importance of using pictures to represent a full understanding of a math concept. When I give my students a math problem to solve, I always tell them to represent their thinking in pictures, numbers, and words. It's astonishing how many will develop an anxiety when they think about all that drawing they'll have to do. 

Question:

I had 57 books on my book shelf. I have read 12 of them. How many more books do I need to read so that they have all been read? 

Do I expect my students to draw me 57 detailed little books, including a little scribble where the title would go? Of course not! We don't have all day. 
I teach them that "Pictures don't always mean pictures". 
At this point there are a lot of seriously confused and concerned faces. 
"Pictures," I tell them, "can be diagrams. They can be quick drawings that represent a number". 
They're a little less frightened, and they're willing to hear me out.
"I can represent 57 books with a few circles. Let me use my Friendly Numbers (1, 5, 10-because these are easy to work with in my head) to represent the parts of 57. This picture here represents 10 books." 

The students begin to relax. They even start looking excited because they know what I have to do next. They decide to help me out. Even though I love the enthusiasm, I have to remind them to raise their hands. There's no room for excess chaos in the math classroom. 

"How can I represent my 57 books?"

Together, we come up with this (note: I put the tens in a dice pattern. It makes it easier to instantly see that there are five):

"Using my picture, how can I show that I have read 12 books?"

The kids begin to cross out the parts. 

It's decided. I have to read 45 more books in order to finish all the books on my bookshelf. 

Sure, I can teach the kids how to line up the numbers and begin subtracting the ones column, then the tens, but I wouldn't be sure they understood why they were doing that. When your students show you their understanding in pictures, you can really see who gets the deeper concepts and who doesn't. 57 is more than just 57. It's 5 groups of tens and 7 groups of ones. Pictures show us this. Numbers alone do not.

Remember to reinforce the point that math pictures are different than art pictures. Mathematicians always try to determine their answers in the most efficient ways. This may take longer than lining up the numbers, but it shows us that our students are able to break apart numbers and see the math question, rather than simply follow a rote process.