Symbols and Visualization/Impact of Higher Order Cognition and Spatial Ordering

In developing an awareness of mathematical concepts, students must engage their nonverbal thinking skills. Nonverbal thinking involves the use of spatial and visual processes to learn or think about a problem or concept.

Nonverbal thinking may involve the use of symbols. The numerals 6 and 26, for example, are symbols that represent quantities. Students use and manipulate symbols when doing operations ranging from basic addition to algebraic equations.

Nonverbal thinking also may involve visual or spatial representations of math processes and relationships. Students must be able to interpret visual and spatial information (as when looking at a map, graph, or geometric shape), and to form and understand visual and spatial concepts (as when translating graph images into usable mathematical information, or describing attributes of shapes).

Some concepts lend themselves to ‘visualization’, creating a mental image to represent a mathematical relationship. The concept of proportion is a good example. A student may have a difficult time interpreting proportion through words and verbal explanation, but being able to visualize the relationship (e.g., the number of boys to girls in the class, the ratio of eaten slices in a pizza) may greatly enhance his/her understanding of proportion as a concept.

Here are some strategies to help students develop and strengthen their understanding of symbols and their abilities to visualize.

Helpful Hints

  • Integrate hands-on activities and verbal explanations into the learning of spatially based concepts. For example, have students use pattern blocks or geoboards to make geometric shapes, then discuss and write down the characteristics of the shapes, such as number of sides, types of angles, etc.  
  • Use examples of familiar situations, or analogies, to talk and think about math concepts. This helps students link the concepts to a visual image. For example, the concept of ratio may be illustrated by asking students to imagine two brothers sharing a pizza, and the amount of pizza left over after the big brother takes his portion.  
  • Guide students in visualizing patterns. For example, talk students through ‘seeing’ a geometric shape in their minds, ‘picturing’ a math process taking place, such as 1/3 of a pizza being taken away, and 2/3 of the pizza remaining, etc.