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Visuals in Math? Helping Students Share What They See

by Dr. Marian Small

Reasoning is an important element in mathematics learning, whether the learner is in Kindergarten or in Grade 12. Visuals, either with or without accompanying words, can be extremely powerful tools for reasoning and explaining; they are powerful not only because they help us understand more quickly, but also because sometimes they lead us to be more general than when we use specific numerical examples. This can be true at any grade level.


Here is a mathematical concept captured in a visual.











The picture may evoke any of the following ideas:

  •  equal groups of 2 could be created by splitting the icebergs with 4 penguins into two mini-icebergs
  • equal groups of 4 could be created by combining pairs of icebergs with 2 penguins
  • you can show any amount as a multiplication if one factor is 1 (e.g. 32 = 32 x 1 or 1 x 32)


Resist telling students what to look for.Rather, ask simple open questions about the visuals, such as:

When do we use multiplication?

Are all of the groups of penguins the same size? Does that matter?

Could the penguins be rearranged into equal groups?

Is it easier to use multiplication for this picture than if there had been 5 icebergs with 2 penguins on them?

This powerful concept enables students of all ability levels to share what they see.


What math concepts can you think of where the use of an image would help your students understand?


Marian Small is a Canadian mathematics educator and a regular speaker on K–12 mathematics throughout Canada and the US.  Marian is the former Dean of Education at the University of New Brunswick, and has been a classroom teacher and professor of mathematics education for over 30 years.

In her latest resource, Eyes on Math, Dr. Small provides an extendable strategy. Teachers are provided with examples so that they might think of their own simple drawings for other math concepts. For more information on Eyes on Math click here.

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  • Marian Small

    Aviva, thank you for your kind words. I’d suggest another visual idea related to area and perimeter:

    I’d show a very wide, but very thin rectangle and no measurements at all.
    Then I might ask:
    Which measurements of the rectangle are “big” and which are “little”? How do you know?

    Or I might show a shape that has 5 sides, 3 of equal length and another 2 of a different, but equal, length (Imagine a “house” where the bottom is a square and the top is an isosceles triangle).
    Then I’d ask– how many side lengths do you need to know to figure out the perimeter? What other shapes could you draw where you’d need that number of side lengths to be given to figure out the full perimeter?

    • Aviva (@avivaloca)

      Thank you so much for the reply! I love your ideas, and I’m actually thinking of a way that I can use them for our 105 the Hive Radio Show on Tuesday about Area and Perimeter. I think these would be great questions for discussion, and we could always tweet about the questions as well as orally discuss them. You’ve just inspired me to try something new. Thank you!


  • Aviva Dunsiger (@avivaloca)

    Marian, I’m very inspired by your books and your ideas, and this post of yours really has me thinking. I love how you can use these open-ended questions with the visuals to really get students thinking about math. In Grade 6, I have been working on area and perimeter, and to get students thinking about both concepts, I’ve used shapes with measurements. I asked students to tell me what they know about the picture. The math talk has been wonderful. Students are not only getting to the concept of area and perimeter, but they are also connecting the shapes with geometry and spatial sense, and identifying many of the properties of shapes. Today was the best though because one student saw the rectangle, and connected one way that he could figure out the area with arrays and multiplication. The layout of the rectangle reminded him of this, and just having this visual, resulted in some great math thinking. Now I’m considering other ways that I can use visuals in math. What would you suggest?