One of my goals as a middle school math teacher is to help my students build or experience a collection of different mental models that they can draw on to deepen their understanding of processes or algorithms. For example, I want my students to have a visual model to understand adding and subtracting negative numbers. I want them to have a picture in their heads when they are performing operations to solve algebraic equations. Just knowing the algorithms is no longer enough.

# Where We Began

So, the question then becomes how we build those models for the students. The first and second years that my colleague, Irene Witt, and I taught the Rational Numbers Module from Eureka Math, we taught the models presented there as written. Painfully. We spent days showing students how to represent the work on a number line with bumpy lines, then curved lines, then vectors, grading formative assessments that required a magnifying glass to determine if students had started the vectors exactly at the right place or not. Did I mention it was painful?

# Rethinking with UDL

When we just couldn’t stand it any more, we used the Guidelines for Universal Design for Learning (UDL) to help guide our choices in redesigning. The Guidelines tell us to provide options and to build in graduated levels of support. Instead of walking the students through the different models in lockstep, we created Stations or Centers, where students worked with the chip model, the number line model, the Integer Card Game, etc. Students who understood a given model were able to move forward more quickly and do Extension work. Students who were confused either took more time or sought teacher help or both.

We did two days of Stations at the beginning of the Rational Numbers unit to build a strong foundation in addition with integers. We taught one day with a mini-lesson on subtraction with integers so that we could emphasize the concept of adding the opposite. From there, we had two more days of a different set of Stations, this time building on and expanding the models into subtraction.

Now that we are headed into the Expressions and Equations module, we have created a set of stations that allow us to introduce students to a wide range of models for equation solving including Hands-On Equations, Geology Rocks from PBS, and the Integer Card Game from the Eureka program. Again, the goal is to introduce students to these models at their own pace, encouraging students to engage in self-reflection and build self-awareness of whether a given model “works” for them or not.

# The Take-Away

Ultimately, my job is to teach the content, not to teach a certain model or a certain way of thinking about how we do negative numbers or how we solve equations. As long as the students are able to accurately add and subtract negative numbers or solve an algebraic equation and show their work in ways that are mathematically appropriate, it doesn’t matter what models they are working with inside their heads.

Again, thanks to Irene Witt for always keeping me on the straight and narrow about method versus content–what are we actually assessing? What are we actually teaching? What do the standards say?