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?
Exhibit A
Exhibit B
Exhibit C
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.
Overthinking my teaching 
Nice distinction.
Could it have been this poster?
http://cgi.ebay.com/PROBLEM-SOLVING-STRATEGIES-II-NCTM-Math-Poster-NEW-/370088527031
It doesn’t look like NCTM sells that poster any more.
Nice find, gasstationwithoutpumps! But that one doesn’t look familiar. I do wish I could read everything on the poster.
The one I remember had a cartoony Einsteinish guy with thought bubbles. But I may be misremembering.
Perhaps this one then:
http://www.teachingsupplystore.com/What_Good_Mathematicians_Do_Colossal_Concept_Poster-p-43235.html?source=googlebase
When I talk about similar ideas with my students, I tell them to “draw a portrait of the thing” rather than its name or its symbol. I really like your terminology of diagrams vs. decorations!
This is lovely and poetic, Maria.
I will have to try this formulation with my students. “Draw a portrait of multiplication.” Very very interesting way to think about it.
Nope, gasstation, that’s not it either.
I must say, though, that I am stunned by how perfectly your find matches my cartoony Einsteinish guy. So now I’m wondering if that’s because there are 8000 different problem-solving posters out there; all featuring the cEg.
And may I say that I both admire and appreciate the effort. I am imagining that someone still has this poster in their room (Larry and Stu, perhaps?) and will shoot a photo for me.
Google image search makes finding posters that match your somewhat vague description fairly easy. Finding the one you really meant, though, will require a more precise description (if it is on the web at all—a lot of copyrighted stuff is not).
Nice distinction. So the follow up question is, how do you teach people to draw diagrams instead of drawings?
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