Visualization | All Kinds of Minds

The results of our recent poll are in! You, our readers, expressed a strong interest in hearing about learning challenges related to math … so in response, this week’s blog is about the spatial ordering demands involved in understanding math symbols. Thank you to everyone who participated in our poll. We love the feedback.

In developing an understanding 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.

One mathematical concept that involves nonverbal thinking is the use of symbols, such as numbers. The number 6, for example, is a symbol that represents a quantity. Another common math symbol is “=”, often referred to as an “equals sign,” that represents the concept that quantities on each side of the symbol are the same, or equal (e.g., 3+3 is the same as 6). Students use and manipulate symbols when doing operations ranging from basic addition to algebraic equations.

Understanding and using math symbols taps into a student’s higher order cognition and spatial ordering abilities. In this post, we’re going to focus on the role of spatial ordering.

Neurodevelopmental factors:

Nonverbal thinking involves visual or spatial representations of math processes and relationships. Students must be able to interpret visual and spatial information (as when looking at a graph or geometric shape), and to form and understand visual and spatial concepts (as when interpreting information from a graph 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 possible signs that a student is succeeding with the spatial ordering demands of math:

The student …

understands mathematical symbols and can visualize patterns, math concepts, and the parts of a problem in his/her head
uses visual analogies successfully (e.g., determines how two symbols relate and applies that understanding to link other symbols)
quickly learns new science and math concepts (e.g., place value, perimeter, equations, resistance in a wire)

Here are some possible signs that a student is struggling with the spatial ordering demands of math:

The student …

has trouble associating math symbols with the concepts they represent
is unable to recognize the systematic organization of charts, diagrams, tables, or maps
is slow to master the alphabet and numbers because of difficulty recognizing symbols
has trouble forming concepts and solving problems without substantial use of language

Strategies to help students struggling with understanding and using mathematical symbols:

Integrate hands-on activities and verbal explanations into the learning of spatially based concepts. For example, have students use pattern blocks 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.

We’d love to hear what strategies or activities you’ve used to help promote understanding of math symbols in your classroom. Leave a comment below with your ideas!

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