Math teaching and learning expert Jo Boaler says, "when we don't ask students to think visually, we miss an incredible opportunity to increase students' understanding and to enable important brain crossing" (n.d.). The benefits of visualization are not exclusive to math. Asking students to make observations, connect their ideas, identify their thinking, and look for evidence helps to increase thinking strategies across all disciplines (Yenawine, 2013).
Visualizing mathematics helps students to focus on and accentuate the following two practices from the Common Core State Standards (Norris & Schuhl, 2016; Halpern et al., 2015):
- Mathematical Practice #1: Make sense of problems and persevere in solving them.
- Mathematical Practice #4: Model with mathematics and transfer those learned and practiced skills to solve real-world problems.
These skills, in turn, can support learning in other content areas. The visual representation of an abstract concept that cannot be directly observed may provide students with an effective way to categorize and compartmentalize information and relate it to other concepts. Visualizing the work prior to putting it on paper or creating a model ties both practices together.
Making Sense and Persevering
Good thinkers create a picture in their heads as they read a problem. This visualization is an important aspect of determining which details are necessary to find a solution, and which portions may be extraneous. This skill helps students to make sense of the problem in order to solve it. Additionally, having a mental picture of all that a problem entails can encourage students to persevere until they find a solution.
Consider how students might visualize this word problem:
Emily and 4 of her friends went to the skate rink. They spent a total of $48.00 on skate rentals, snacks, and drinks. Emily paid half of the total amount. Sarah paid half of what was left. Amy, Maggie, and Jamie split the remaining amount. How much did Maggie pay?
By visualizing the girls' portions in a pie chart, it is easy to keep track of each paid portion and what is still left to be paid. This picture also can help students ignore the unnecessary information (skate rink, snacks, etc.). Building a visual representation of the problem can help students can make sense of it and stick with it, rather than get bogged down with extra information that could cause frustration.
Figure 1. Visualizing a Word Problem
Constructing a Model
Visualizing a problem is also a good strategy before building an actual model; much like a prewriting can help students approach a draft of an essay. By organizing the necessary components of a problem into categories or locations, it is easy to manipulate items into a coherent picture or model of the problem. Visualization also ensures that all the necessary components are included in the solution.
Consider how visualizing, and then creating a model would help students solve this problem:
Quadrilateral ABCD is rotated 90° counterclockwise about the origin. What are the coordinates of the image A'B'C'D'?
While drawing a pictorial representation certainly helps in finding the coordinates of the copied image, mentally rotating the original figure first will likely lead to fewer mistakes and more efficient use of student time. By visualizing the location and orientation of a rotated figure, students are also better able to guarantee that their answer is reasonable.
Figure 2. Visualizing is the First Step
Without first visualizing where the image should go, students may spend time incorrectly placing new points on the coordinate grid. This will likely lead to frustration when they have to start over to solve the problem. Or worse, it may cause the student to give up on the problem altogether.
Figure 3. Without Visualizing, Errors Are Common
Visualizing Across Content Areas
Visualizing is thinking; drawing information helps children see how facts connect. Visualization strategies, and the learning they generate, transfer across content areas and actually prop up thinking strategies in other subjects. Here's how.
Science: Visualizing 3-D cross sections of the Earth can help students understand the scientific phenomena of plate tectonics and volcanoes. Students can create a model of the visualization.
English/Language Arts: Visualizing (perceived) physical surroundings from the speaker's point of view can help students discern between characters in a story. Forming a mental picture of the various components of an essay can greatly bolster students' writing abilities (much like the pictorial process of creating a bubble map or graphic organizer as a prewriting exercise).
Social Studies: Generating mental images of native peoples or historical actors in a social studies context can allow students to recall information about the culture, habitat, and social practices of a group (i.e. ancient Greeks, Allied soldiers in WWII, or aboriginal Australians).
Visual Arts: Before putting their thoughts to paper, canvas, clay, or digital media, an artist creates a visual model of the anticipated final product. Visualizing the process and intended outcome also supports the artist in staying focused throughout the artistic process.
Music: Creating mental images associated with a song can provide context to singers and musicians, helping them to translate the tone and meaning of a piece for performance.
Physical Education: Athletes can create diagrams of plays that they test in practice and apply to competition. Images transfer to kinesthetic movement when athletes visualize successful completion of a move, a shot, or a play.
Verbalizing Visualization
Visualizing strategies also include the overt thinking that comes with student accountable talk, or the words and thoughts students use to explain their thinking. Visualizing sentence stems that work across content areas include the following:
- I'm picturing ….
- The description of _____ helps me visualize ….
- In my mind, I could see ….
- I can see, feel, taste, smell, touch, hear ….
- I can imagine ….
- Picture this ….
- When it said _____, I could imagine ….
- In my mind, I see ….
Transferring visualizing mathematical strategies into other content areas not only supports learning in those content areas, but also increases the depth of learning in mathematics. With the current focus on deep learning within content areas, visualization strategies, when taught explicitly and practiced intentionally, will support mastery of learning.