Episode 192: A Math Task All Teachers Should be Doing

NCTM book Principles to Actions: Ensuring Mathematical Success for All.

Teaching Student Centered Mathematics by John Van de Walle, et al

Lesh, R. (1979). Mathematical learning disabilities: Considerations for identification, diagnosis, 

and remediation. In R. Lesh, D. Mierkiewicz, & M. Kantowski (Eds.), Applied mathematical problem solving. Columbus, OH: ERIC. 

To access the 2025 Virtual Math Summit, become a member of the Build Math Minds PD site: BuildMathMinds.com/bmm

Welcome fellow Recovering Traditionalists to Episode 192: A Math Task All Teachers Should be Doing

I have a free download for you with this episode, so after you listen go to the show notes to get the download at BuildMathMinds.com/192 …what is the download?? Let’s get into the episode so you can find out.

It was surprising to me how many presenters at the 2025 Virtual Math Summit shared the same image.  Time and again I saw it in sessions, which is awesome because I hope it helped sink in just how important the image is.  This image has been around a long time (1979 in fact), but it has changed a bit since then and has made more of an appearance here lately after it was shared in the 2014 NCTM book Principles to Actions: Ensuring Mathematical Success for All. 

The image is the Lesh Translation Model.  The model is the different ways you can represent a single math idea.  These different representations help students build their understanding of mathematical concepts and help them solve math problems.

Personally, I like this version of the model:

If you are listening and can’t see the model you can go to the show notes later at BuildMathMinds.com/192 The model is basically a pentagon shape and at the “vertices” are the 5 different representations: Manipulatives, Realistic, Pictorial, Language, and Symbolic.  In friendlier terms those 5 are: objects, real life context or stories, drawings, verbal description or words, and the abstract numbers or equations. Then it has arrows connecting all the representations together and this version I’m sharing also has arrows that leave each representation but go back into that same representation, meant to depict the idea that we can have multiple types of each of those representations and we need to help students see the connection between those.

For example, let’s take the numeral 5.  That digit of the 5 would be the Symbolic representation on the Lesh Model. Students would use Manipulatives or Objects to model 5 items.  Asking students to tell where they’ve seen 5 things in their life would be the Realistic/Stories representation.  Having students draw 5 things is the Pictorial representation. And then having them say (or write) ‘five’ is the Language/Words representation.

These different representations give students different ideas of what mathematical concepts are.  If they only have one or two of these representations it’s not a fully developed understanding.

In Teaching Student Centered Mathematics by John Van de Walle, et al, they describe it as a Web of Representations.  The more connections we can help students make around a concept by developing this Web of Representations, the better their understanding will be and the easier it is for their brains to recall information when needed. In Chapter 2 they write:

“Students who have difficulty translating a concept from one representation to another also have difficulty solving problems and understanding computations (Clement, 2004; Lesh, Cramer, Doert, Post, & Zawojewski, 2003; NCTM, 2000). Strengthening the ability to move between and among representations improves students’ understanding and retention of ideas.”

In Chapter 3, Van de Walle, et al they talk about using the Translation Task. You fill out one of the representations and ask students to do the other representations, or you can have them do all the representations for a given topic.

I love the Translation Task and I think it is a math task that all teachers should be including and doing it often throughout the year with different concepts.  All you do is give your students one of the representations and then ask them to show all the others.  They show a template based on the Frayer Model and it combines the Manipulative and Pictorial representation together.  However, I think it’s important to encourage students to use and see the power of manipulatives so I want you to include all 5 areas when you do the Translation Task.

To help you with that, I’ve got a download you can request over on the show notes page BuildMathMinds.com/192.  In the download you will get information about the Lesh Translation Model and the Translation Task along with a template that is basically an empty Lesh Model that you can print off and have students use every time you want to do the Translation Task with them.  The download also provides two examples, the one I shared in this episode using ‘5’ and another one with a fraction addition problem.  Get the download, print off the Translation Task Template, and start using it tomorrow with your students and continue to use it continuously to help them build a web of understanding through representations.  

Until next week, my fellow Recovering Traditionalists, keep letting your students explore math, keep questioning, and most importantly, keep Building Math Minds.

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