Area model is a visual model useful for representing multi-digit multiplication

Visual Models: Strategies for Conceptual Learning

Have you ever noticed that some students immediately love working with visual models, while others resist, or even refuse, to represent their understanding visually?

I remember one student, Ian, who was really frustrated by visual models. On the first day of the year, several students told me that Ian was the “best math student in the grade.” And when it came to calculations, they were right.

Ian knew his times tables inside and out. He could add, subtract, multiply, and divide four-digit numbers. He even knew all the tricks for operating with fractions and converting them to decimals.

But I also noticed that Ian wasn’t as comfortable with conceptual work. Whether it was visual models, word problems, or even brain teasers, he lacked confidence.

And he wasn’t interested in embracing the challenge. Ian had a reputation to protect, and he preferred to focus on his strengths.

So why did his peers think Ian was such a whiz? It turned out that message had been communicated to him (and them) for the last five years.

Sure, they had done some conceptual work at the lower grades. But Ian had learned to hang back during those activities. When it was time for tests and quizzes, it was all about calculation. His strong suit.

I soon realized that visual models were helping students develop number sense. But they were doing something even more important.

As it turns out, visual models were actually redefining the ‘winners’ and ‘losers’ in my 5th grade math class.

How Visual Models Rebalance the Scales

In my class, opting out of visual models wasn’t an option. I like to align assessments with instruction. So once we started using visual representations, I modified my tests and quizzes. Students had to interpret models and make their own. Their grades also incorporated group projects in which they creating presentations and animated visual models.

While many students thrived, Ian sulked. The rules of the game had changed. No longer were “math facts” the only standard of success. Other students began to see the value in their own strengths. Problem solving and visual representations were just as important as arithmetic. Students who used to think they weren’t “math people” began to actively engage in class.

For many of us, a level playing field in our classes is a good thing. But when standards of success change, those who were winners under the “old rules” can feel slighted.

Once Ian began earning “normal” grades (A minuses and B plusses), his parents noticed. They expressed their concern. They wanted to know what was so important about word problems and visual models. This was math class, after all.

But That’s Not How I Learned Math…

My experience with Ian and his family wasn’t unique. Representing math concepts with visual models is central to the Common Core approach. And the internet is littered with anti-Common Core tirades. Many of the complaints target poorly written textbooks, not realizing these books weren’t written by “Common Core.” Others blame Obama for Common Core math. (Never mind that work on the standards began in 2007, two years before Obama’s inauguration).

Classroom Resources and Professional Learning

See All Visual Model Activities

But even some who understand the difference between standards and textbooks have pushed back. The most common complaint is that the “old” way of teaching math was “just fine,” so there’s no need to change.

The reality is that math education in the United States has been anything but ‘just fine.’ According to the latest PISA results (Programme for International Student Assessment), the US ranks 36th out of 70 tested nations. While we beat the Dominican Republic (70th) and Mexico (56th), we placed behind rivals Russia (22nd) and China (1st).

Perhaps the most troublesome indicator is our Algebra 1 failure rate. Inability to succeed in Algebra leads to a host of problems. These include an increased high school dropout rate and reduced future income.

Traditional math focused on arithmetic (calculation) until high school. Many students couldn’t understand why they suddenly struggled when they reached Algebra. Common Core aims to introduce conceptual math earlier. Visual models are a central element of that approach.

How Visual Models Support Conceptual Understanding

To get the most out of visual models, treat them as tools to support understanding. That may seem obvious, but visual models are often treated as just another item in the curriculum. Instead of thinking “now we need to teach times tables and arrays,” use arrays as a tool for demonstrating the meaning of multiplication.

When students memorize “math facts,” they tend to forget them after the test. They also have difficulty applying their understanding to word problems and real-world situations.

Visual models parallel the progression of concepts at each grade level. Here are four “strands,” or domains, of mathematical understanding that visual models support.

Note: These strands focus on scale models. Some definitions of visual models include non-scale models, such as coins, number bonds, and tables. These serve a distinct purpose, but don’t support numeracy (the meaning of numbers and operations).

Counting and Base-10

Counting and Base-10 Visual Models
Counting and Base-10 Visual Models

Before students learn operations, they need to learn about numbers. Counting starts with the idea of number as quantity (distinct objects) and proceeds to number as length. Along the way, students learn how our number system is built on groups of ten.

Organizing numbers by tens means we can access large quantities without counting one-by-one. It also leads to understanding decimals as a way to work with decimal fragments of one.

Additive Thinking

Additive Visual Models Progression
Additive Visual Models Progression

Counting is an additive process. Addition extends that process to include multiple quantities. Starting at 5 and counting up by 3 can be represented as 5+3.

Visual models can represent addition as combining, extending, or comparing (greater than). Subtraction represents the inverses: taking away, cutting, or comparing (less than).

Multiplicative Thinking

Multiplicative Visual Models Progression
Multiplicative Visual Models Progression

Multiplication builds on additive thinking in several ways. First, students apply skip counting to repeated addition and then ‘counting groups.’

Arrays introduce the idea of multiplication as a ‘cartesian product.’ By arranging objects in rows and columns, we introduce 2-dimensional space. This lays the foundation for the concept of area. Area models allow us to work with large, multi-digit numbers, as well as fractions and decimals.

Later, students encounter multiplicative comparison. This is the idea that something can be 3 times as big or 1/4th as large as another number. Double number lines and bar models illustrate the concept. Multiplicative comparison is central to middle school ratio and proportion standards.

As with additive reasoning, each aspect of multiplication applies to division, its inverse.

Patterns and Functions


Patterns and functions are severely underrepresented in K-8 math textbooks. Many educators interpret the standards such that functions begin in 8th grade. 8th graders learn “the vertical line test” and other tricks to tell the difference between functions and non-functions.

This does help some of them pass state tests. But many students miss the central idea of a function: a relationship between two numbers. Something is done to an input (independent variable), producing an output (dependent variable).

One way to build this understanding is to start with patterns. A pattern such as ‘3, 6, 9, 12…’ can be thought of as ‘multiply by 3.’ In fact, the times tables can be treated as functions. Take the 7’s times table. Input 2, and out comes 14. Input 6, and out comes 42.

Treating mathematical operations as functions is incredibly useful. It helps students use algebraic reasoning to generalize arithmetic. Students who understand functions adjust much more easily to middle school math. Function understanding supports concepts such as variables, slope, and rates of change.

Using Visual Models in Your Classroom

Understanding the progressions is an important step in making visual models work in your classroom. The purpose of visual models is to make math easier to understand. But many textbooks make visual models more confusing than the concepts behind them.

Here are some tips to make sure your students get the benefits of working with visual models.

Begin With Explicit Instruction

I’m a big believer in limiting lecture and learning through inquiry. But students will never “discover” what an array is. Nor will they discover rows and columns. Before students work with visual models, explain the vocabulary, the rules, and the purpose of each model.

Start with Scale Models

There are many useful visual models that are not drawn to scale. Coins can introduce students to decimals. Tables can help them organize numbers and find patterns.

But these don’t support numeracy – an understanding of the relative sizes of numbers and the meanings of operations. For example, many students can get confused by the fact that nickels are larger than dimes.

When I first asked my students to draw their own visual models, I gave them a story problem. In it, people bring pies to a party and divide them equally. Most of my students drew the house and the people carrying the pies.

Now, I try to use the term “scale model” to emphasize what I’m looking for in a good model. That helps students understand that visual models aren’t just decorative art. (No offense to the art teachers out there!)

Once students understand scale, feel free to explore non-scale models. Even then, make sure they can distinguish between scale models and non-scale models.

Use Visual Models as a Communication Tool

Part of the beauty of visual models is that there is more than one correct way to make make them. Unfortunately, this also makes it difficult to tell whether a student-made model is correct.

I apply the ‘good communication’ standard. A model is accurate if it can be understood by a third party. If one student makes a model to represent 4 x ⅓, another should be able to translate it back into expression or word form.

Effective models use labels and colors to make communication clear. For models of operations, make sure that the starting value, operation, and result are all represented. Creating a model of the number 12 is not the same as modeling 4 x 3 = 12.

Assessing Visual Models

Visual models can be extremely useful for assessing student understanding. Calculating with an algorithm shows that a student can repeat a process. But they don’t really understand the concepts behind the algorithm if they can’t draw it. A single student-created model can be a more effective formative assessment than ten pages of exercises.

Including visual models in assessments also validates visual models as legitimate math. Students determine what’s important by what is ‘on the test’. If you skip the visual models, you’ll soon find your students less engaged in activities involving models.

Use Graphic Organizers

Graphic organizers help students organize their thoughts and work in multiple steps. When students are first learning to model operations, I give them a visual model organizer to structure their thinking. (It also makes their work easier to read and assess).

Other organizers help students use visual models to solve word problems. Still others help them represent functions in multiple forms.

Creativity, Collaboration, and Conceptual Learning

Inquiry-based math activities can increase achievement and engagement on your math classes. But it can be challenging to plan and execute this type of lesson day-after-day. Textbooks aren’t designed for hands-on learning, which can make it hard for educators to find the resources and professional development that make hands-on, standards-aligned learning work in the classroom.

Room to Discover’s resources and professional development are designed to address this exact challenge. We currently offer workshops that help teachers bring engaging learning experiences to the classroom, day after day.

Subscribe for our Educator’s Newsletter to receive education inspiration to your inbox, every week. We share stories and strategies to help make  your classroom creative, collaborative, and conceptual.



Share on facebook
Share on pinterest
Share on twitter
Share on linkedin
Share on reddit
Share on email


Share on facebook
Share on pinterest
Share on twitter
Share on linkedin
Share on reddit
Share on email