Drawing Students to Express their Thinking

“The hand brain connection is something deeply wired within us.”

-       Robert Greene, Mastery

Famous Renaissance figures aside, mathematicians and artists are often thought to possess vastly different intelligences.  The former is branded left-brained and analytical while the latter is considered right-brained and creative.  Many educators cite shapes, canvas borders, and symmetry as ways that Math and Art overlap, but these obvious and somewhat generic examples ignore a more fundamental aspect of their intersection.  Drawing pictures and diagrams is an integral part of elementary mathematics that all students should routinely practice.  Creating meaningful math illustrations helps students reenact the subject’s history, improves their thought expression, and conveys their understanding to others.

With a little imagination, it’s not hard to conjecture how mathematics evolved into its current system of figures and symbols.  To communicate a quantity, a hunter-gatherer likely lined up a series of stones to represent something that couldn’t be seen.  As writing technology evolved, people realized that illustrating the stones was a more efficient way to represent information.  In time, they connected the concrete and pictorial and developed symbols to efficiently represent different amounts.  These symbols eventually led to advanced mathematical forms.

Today, first graders are often presented with basic addition story problems, such as:


Jon has 4 red apples and 3 green apples.  How many apples does he have altogether?

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To solve this problem, a six year-old could use small plastic manipulatives. They might also draw four dark circles and three open circles, representing the red and green apples, respectively. After counting their representation, they discover that Jon has seven apples. An elementary mathematics classroom should be a laboratory for recreating the experiences and discoveries of our ancestors. When children create math pictures they connect to the subject’s genesis, using drawings to convey their thinking.

Because Math is an exact science, teachers have traditionally emphasized correct answers and minimized thought expression.  Therefore, when we think of struggling math students, we usually picture children who get a lot of answers wrong and/or lack strategies for solving problems.  A different, seldom identified struggler is the child who frequently answers questions correctly, but is unable to communicate their thinking.  When asked how they solved certain problems, these students often say I just know it or I did it in my head.  Although their challenges are different, both types of struggling students benefit by pictorially representing information.

In this second grade problem, drawings are useful for all students.


Marriory has 8 boxes of red marbles and 3 green marbles.  There are 10 red marbles in each box.  How many marbles does she have altogether?

Many students on or above grade level will instantly know that Marriory has 83 marbles.  Still, it’s important for them to diagram, so that they can later solve more complex problems such as:

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Marriory had 83 marbles.  She lost 26 of them but found 11 new ones.  How many marbles did she have in the end?


With thought and perseverance, all students can use their first drawing as a foundation to solve the two-step problem.  Mathematicians must represent information to organize and explain their thinking.  This helps them access higher-level mathematics and allows their instructor to diagnose their comprehension.

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Elementary math teachers regularly encounter children who say I don’t get it while staring at a blank paper, or I’m done with the correct answer, and nothing else, written.  Neither response informs the teacher of what the students do or do not understand.  The child who says I don’t get it can organize their thoughts by representing parts of the problem that they do know.  The child who says I’m done must communicate their thinking.  Both benefit if they’re given structure and parameters for their illustrations.

When students are first asked to draw pictures of story problems, they almost always create literal depictions.  It’s important that this habit be broken at an early age.  Because mathematics is a symbolic subject, children should be explicitly taught how to represent information.  Four cars, for example, can be symbolized with four dots or capital Cs.

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Writing P above seven lines and three dots can depict 73 peanuts, and crossing out two lines and one dot can represent 21 of them being eaten.  As numbers get larger and operations vary, students need to be introduced to new abstractions.

The Singapore and Eureka Math programs emphasize spacial diagrams as a bridge to many middle school topics.  In the word problem below, the bar model is used as an intermediary pictorial stage that segues into pre-algebra.


Amy is 15 lb heavier than Selena.  Their total weight is 231 lbs.  Find Amy’s weight.

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Drawing/diagramming effectively can open vistas of understanding that gives students opportunities to grasp more complex topics.

Illustrating mathematical scenarios is a skill that all students should practice daily.  Purposeful math drawings help children understand the subject’s evolution, while communicating their thinking, and allowing teachers to best serve their needs.

The great mathematician/artist, Leonardo Da Vinci once said, To draw is to think.  Recognizing their interconnectivity, he understood:

Mathematicians are artists and artists are mathematicians.