Visualization: The Fourth Leg of Concrete–Pictorial–Abstract

Many educators are impressed with the curriculum and materials from Singapore that have enabled Singapore’s students to be so successful in mathematics. Two of its key features are the emphasis on problem solving and the concrete to pictorial to abstract approach first described by Jerome Bruner.  What is sometimes overlooked in this is the importance of visualization in developing both conceptual understanding and procedural fluency.  You can’t carry around ten frames or base ten materials, or fraction strips in your pocket—people will laugh—but you can carry them in your “mind’s eye.”   In other words, the visual and pictorial models used in the Singapore material enable students to visualize number, operations, and word problems.

The ability to visualize quantitative relationships is critical to learning basic facts, to understanding complex operations with fractions and ratio, to solving routine and non-routine problems and even solving variable equations

In first grade, students learn to “make ten” on a ten frames to learn facts to 20. To learn how much 8 + 6 is, students think moving two from the 6 to the 8, 8 + 6 = 10 + 4 = 14


Later they record this as

Until finally, when students see the problem, 8 + 6 = ?  they can visualize the ten frame and calculate in their heads.

In the intermediate grades students learn their multiplication facts by visualizing a known fact to determine an unknown one. So for instance, students visualize that 9 x 6 is the same as one less row of 6 than 10 x 6 or 60 – 6.

By visualizing a number line or base ten materials, students can decompose numbers in their head so when presented with the problem 161 ÷ 7, they recognize that 161 is the same as 140 (easily divided by 7) and 21 divided by 7 (3) or 23 in total.

When presented with a problem like: The sum of two numbers is 96. The smaller number is 1/3 the size of the larger number. What is the larger number?

Students will visualize something like this:

Just think of all the important concepts that are made easier by visualization:

  • Finding the difference of 198 and 89 on a number line
  • Recognizing that 8 x 7 is just double 4 x 7
  • Identifying equivalent fractions on a number line or on the multiplication chart
  • Understanding multiplication of fractions as an area problem
  • Adding and subtracting integers
  • Solving ratio problems with bar models
  • Recognizing that an equal sign means the amounts on either side are equivalent amounts and resemble a balance, even when variables are on both sides
  • Visualizing that of all the rectangles with a common area, the square has the smallest perimeter

Visualization enables students to see relationships, to simplify problems, and to change problems into a mathematical form.

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