I recall a poster in math classrooms of my youth that implored me to “draw a picture” as part of the problem-solving process. A useful strategy, to be sure. But it turns out that it’s a learned skill.
Pictures that are useful for demonstrating or examining the mathematical structure of a problem are special. We don’t all make them intuitively. I have asked the future elementary teachers in my courses to draw a picture that might be helpful in solving the following word problem:
Tabitha has five baskets of apples. Each basket has eight apples. How many apples does Tabitha have altogether?
I frequently get back something of this form:
I refer to this as a decoration, and I contrast it with diagram. A diagram demonstrates mathematical structure; it represents mathematical ideas differently from a symbolic form. A decoration makes the symbols look prettier or more contextual, but does not on its own represent the underlying mathematical relationships.
We can decorate our diagrams. The inclusion of an apple in the picture does not preclude it being a diagram. But it doesn’t necessarily make the picture useful for solving problems either.
With that in mind, which of the following are diagrams and which are purely decorations?
P.S. Extra credit to anyone who can find or take a photograph of the “draw a picture” poster. I seem to recall it being one in a set of five or so problem-solving posters.