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!

**Related links:**

**Learn more about our summer series **

- More information and strategies on understanding math concepts
- Related research on spatial ordering (check out the section on Higher Spatial Thinking)
- All Kinds of Minds’ “Thinking Mathematically” podcast
- Mathematics section of the All Kinds of Minds Parent Toolkit
- Interactive spatial ordering activity

I challenge the notion that all nonverbal thinking is visual and spatial. A lot of mathematical thinking is abstract reasoning that is neither verbal nor visual. The notion of equality used as an example here can be illustrated verbally or visually, but is in it’s essence neither one.

Thank you for your interesting blog this week. I also find that students who have been less successful with math concepts, whether spatial or otherwise, have not had enough experience with understanding the concept in real-world, problem-solving contexts BEFORE attempting the abstract/symbolic level (ie., worksheets or textbook exercises).

You mention that, “Some concepts lend themselves to “visualization,” creating a mental image to represent a mathematical relationship.” I would argue that creating visual images/representations with ANY math concept is important for most learners, especially visual learners. Before children can add/subtract, it helps most children who struggle to first be able to visualize ANY number up to ten. There are many activities and representations that one can use to develop these visual concepts of ‘number;’ too many children are forced to use worksheets/textbook exercises before they have developed this ability. When they have developed the ability to easily ‘subitize’ or recognize a visual quantity to 10 without counting each item one-by-one, then work on adding/subtracting becomes easier ( helping children to develop visual models/representations of these concepts as well).

I would love to read what other teachers have found works for them.

Cheers.

Some children have severe visual-perceptual problems, and I have found that this disadvantage hinders math conceptualization. These children have difficulty visualizing because they don’t perceive the models or examples correctly in the first place. They tend to miss the little details, such as plus or minus (positive or negative at the middle and high school levels) signs, or order within the math equation or inequality. They might confuse position in space, such as numerator over denominator, or the size of the exponent of a number in relation to the number itself. I use a lot of highlighters to point out the details that they tend to miss and to help them focus on one step of the multi-step process at a time, such as adding the ones place first in an addition problem. I also give my students pre-arranged worksheets, with all math problems going the same direction (i.e., all vertical or all horizontal vs. a mixture) or templates (e.g., multiplication or division) to help with spatial organization. I use a lot of concrete materials, such as Cuisenaire rods and Base-10 blocks, geometric solids, fraction strips, etc. and work at the concrete level until the concept is understood. Games, such as Katamino, Connect 4, Othello, and Mr. Mighty Mind are good for developing spatial awareness and spatial orientation (and can be played at home as homework or recommended Christmas presents). As a teacher, I can tell what my students are perceiving from what they put on paper or build with models. Our occupational therapist also helps with spatial awareness as a related service.