Making Math Visual

Since Differentiation is in the news, I thought it was time for us to investigate best practices in special education math classes.  And as a counterpoint to that article, here is Grant Wiggins’ response.

Differentiation is widely accepted as being at the forefront of best practices in special education.  But, when it comes to best practices in special education mathematics classrooms, not much has been clearly defined.  Here’s one take about inclusion from a Huffington Post article at the beginning of the year:


One of these “best-teaching practices” is including visual cues to accompany any words in a worksheet or presentation of information.  This idea is best represented in mathematics as mathematical models.  The What Works Clearinghouse goes into detail about the importance of incorporating visual models of mathematical concepts into lessons for struggling students.  The recommendation also illustrates how to use the concrete-representational-abstract approach in math instruction.


Concrete – Representational – Abstract

I’d like to explore how we can use this kind of instructional modeling during the problem solving process.  Can we pose problems in the concrete-representational-abstract model for students with disabilities? The concrete stage would be physically representing the problem with a hands-on activity (which is another special education “best practice” and it will get it’s own blog post), but can we pose problems to students in the visual representation stage?

I attempted it and it got complicated when I told my colleagues our class focus for the trimester would be problem solving.

“Word problems?!?” They asked, “That’ll be hard since most of the students struggle with reading.”

“I didn’t say we were studying word problems.” I answered, “I said, we were going to be studying problem solving.”

“Yeah, right.  Word problems.”


When I think of problem solving these are examples of what comes to mind: Martin Gardner, James Tanton, Fawn Nguyen, and The Math Forum.

Not this

Can we create situations where students are solving problems, but not solving “word problems?” Matthew Peterson, from the MIND Research Institute says we can…

Solutions to mathematical problems can also be presented without words.  Many mathematical proofs are accompanied with a visual representation.  One of my favorites is the way Archimedes proved the approximate value of pi.


Proofs and the use of visuals have been written about by many general education teachers like Michael Pershan, Justin Aion and Joe Schwartz, just name a few.

But can my students investigate mathematical situations and stories without learning to decode word problems? Is decoding word problems something that they “need” to be learning? Dan Meyer says that they need to be perplexed

I enjoy perplexing my students and I think much more valuably they enjoy it also.  Most of the time my students are pandered to or underestimated, so if I can provide one moment of mathematical perplexity I feel successful. Visual problems are one way that I can facilitate my students’ mathematical perplexity without the anxiety of also testing their reading abilities as well.

Here are some of the visual problems we worked on:

What do you notice? What do you wonder?

What do you notice? What do you wonder?

Fruit Stand Math N&W

How much?

Fruit Stand Math

(I know the prices are in pounds, but for early elementary this is easily explained as British dollars and solved 1:1, or late elementary and early middle schoolers can investigate exchange rates.)

First Row: What do you notice? What do you wonder?

Second Row: How many sticks?


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