# Overview

## What is Visual Math?

Visual math refers to the graphs, charts, and pictures through which math facts, relationships, and ideas are displayed. Some examples of visual math may look familiar to you—either from your own schooling days or from working with students more recently. You may also come across examples of visual math that are completely new to you, especially if you explore the Links for More Information resource in this unit.

## Uses of Visual Math

Methods for presenting math visually tend to evolve more quickly than other topics in school mathematics. They evolve more rapidly because visual math is about clarifying and communicating the math—two processes that have a tendency to change as situations, people, and cultures change. But the fact that visual math adapts well to change is precisely why it used so broadly. Visual math can be used to present large amounts of information concisely, clarify tricky ideas, send persuasive messages, and disclose patterns that lead to important breakthroughs.

## Examples of Visual Math

There are far too many types of visual math to cover every one, so this unit focuses on a few of the main types used in elementary and middle school math classes.

Many types of visual math are based on data. Data are bits of information, in context, that help us understand situations better. Some data can be described with numbers—such as height or shoe size. These are called numerical data. Other data cannot be described in numbers, such as car color or dog breed. These are called categorical data. Notice that any of these numerical data and categorical data sets  (i.e., collections of data) can have several different values. A height could be 20 inches, or 55 inches, or 105 inches. Dog breeds include German Shepherds, Irish Wolfhounds, and Siberian Huskies, among many others. Any piece of data that can vary between values in this way is called a variable.

Variables are at the heart of visual math—especially visual math based on data. Choosing the best type of visual math often depends on the type and number of variables being portrayed. Tables, pie charts, and bar charts are often used to present categorical data.

• table is a grid that lists out the values of variables. The only really “visual” part of a table is the organized way in which the numbers or values are displayed.
• pie chart is a circular chart that is divided into slices, where the area of each slice is determined by the amount of data in the category represented by the slice. Pie charts are a simple way to show how a categorical variable is divided. Pie charts are not as good for more complex situations such as showing a variable with many categories, a variable with only a few data points, or a comparison of how two things are divided up differently. In such situations, a bar chart is better.
• bar chart uses bars to show how much data there is for each category of a variable: the more data, the bigger the bar. As students move into higher grades, there is more focus on numerical variables.
• line graph is a common type of visual math used to present data described in terms of two numerical variables. In a line graph, data points are plotted according to their values on each of the two variables, and then a line is drawn to connect the points.
• Other major types of visual math for numerical variables include histograms and scatterplots.

## Tips for Paraprofessionals

The good news is that you don’t need to memorize all the different types of graphs, charts, and pictures. It’s more important that you develop the ability to tune into the messages being sent by the visual math. As you start to get the hang of reading visual math, then you can work more on creating visual math. The creation process helps with your own learning of math because it gives you a way to make abstract math concepts more concrete. And it gives you a new avenue for sharing messages with others. In your case, some of those “others” may be students. As you work with students, you might want to look for situations that give you opportunities to help students experience visual math in productive ways. Two situations are particularly fruitful:

1. Look for situations where students are required to create or interpret a chart, graph, or picture. Ask questions to help them make sense of the visual math. What does the title mean? What variables are being shown? If there is only one variable, what does the visual math show about that variable? If there are two variables, what does the visual math show about the relationship between the variables? Finally, what idea or message is the visual math suggesting?
2. Look for cases where students might benefit from the use of visual math, even if they are not required to create a visual image. Here are some examples:
1. students who are frustrated that they can’t “see” the math relationships,
2. students who feel overwhelmed at trying to keep track of too many abstract math thoughts in their head at once, and
3. students who can’t quite explain which part of a problem they don’t understand.

These are all situations where the students could benefit from using visual math. You may be able to see a specific way that visualizing the math could get them past particular obstacles. But even if you don’t, you can still ask a question like, “Can you think of a way to draw the problem?” It won’t always work, but it does open a window of opportunity to a fresh way of looking at the problem.