Visualizing Elementary Examples

There are many different ways to represent numbers and representations can be used to focus student thinking on different concepts found within number sense.







“12 is the number that is 2 past 10.”

“Remarkably, brain researchers know that we “see” a representation of our fingers in our brains, even when we do not use fingers in calculation. …Researchers found that when 6-year-olds improved the quality of their finger representation they improved arithmetic knowledge, particularly skills such as counting and number ordering.” (Boaler & Chen, 2016)
Anytime you say a number word, you can show it with your fingers.
“Show me four fingers. … Now show me four fingers another way. .. And another way.”
  • Neuroimaging has shown that even when people work on a number calculation, such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing.
  • Number-line representation of number quantity has been shown to be particularly important for the development of numerical knowledge, and students’ learning of number lines is believed to be a precursor
    of children’s academic success.
  • Everyone uses visual pathways when we work on math. The problem is it has been presented, for decades, as a subject of numbers and symbols, ignoring the potential of visual math for transforming students’ math experiences and developing important brain pathways.
  • To engage students in productive visual thinking, they should be asked, at regular intervals, how they see mathematical ideas, and to draw what they see.
  • They can be given activities with visual questions and they can be asked to provide visual solutions to questions.

The bead number line model, or commercially known as Rekenrek, supports the development of mental images of benchmark numbers. It serves as a visual model while students explore number relationships and develop mental math strategies such as double, plus or minus one, and making tens.
Show me 1-10: Say a number, or hold up a numeral card (0-10). Ask students to show the given number by moving the beads with one push.
Show me 11 – 20: As above but ask student to show the given number of beads using only 2 pushes.3

Addition and subtraction have a total number (whole) compared to the parts. They could be equal parts (often utilizing skip counting) or unequal parts. Students develop flexibility with numbers when discussing the variety of parts that make up the whole amount.

Multiplication and division use equal groups and fair sharing. Students move to managing groups as single units.

Visualizing in this manner also connects to composing and decomposing numbers. The big ideas are rarely done in isolation of each other and when done in connection strengthen students’ number sense. 

1Jo Boaler, L. Chen. 2016. Why Kids Should Use Their Fingers in Math Class. [online] Available at: <> [Accessed 19 August 2021].

2K-5 Math Teaching Resources. n.d. Rekenrek Activities. [online] Available at: <> [Accessed 19 August 2021].

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