Making Thinking Visible

Making Thinking Visible
Making Thinking Visible

Arts Integration

Arts Integration
Arts Integration

Mondrian Math: Fractions, Decimals, Percents & Blurred Lines

"The power of mathematics is often to change one thing into another,
to change geometry into language."
 Marcus du Sautoy

One of my goals this year is for my students to ask mathematical questions, just as a scientist asks questions before beginning an experiment.  I want them to engage in investigative process. I've been pushing toward this goal by implementing visible thinking routines and using student self-evaluation checklists.  One of the first sections in Making Thinking Visible (Ritchart, Church & Morrison) that made me sit up and take notice was this simple exercise:

Choosing one subject that you teach, make a list of actions and activities that your students are engaged in.

Now take that list and use it to create three new lists that define these activities like so:
  1. What actions and activities account for 75% of what student do in your class on a regular basis?
  2. What actions and activities are real things that real scientists, writers, artists, and so on actually do as they work?
  3. What actions and activities do you remember yourself doing when you were engaged in new learning

  I've been hanging out with number 2 on this list the entire school year, and it has been life-changing.

I like to blur the lines between subjects and types of learning experiences in my classroom.  So I began modeling mathematical questioning with a project that blurs the lines. I began like this:

Day 1
"Friends, I was online the other night looking at art, and I rediscovered an old favorite of mine, Piet Mondrian."

I put up two Mondrian examples on the Smart Board for viewing, "Broadway Boogie Woogie" and "Composition with Large Red Plane, Yellow, Black, Gray, and Blue."  I showed these one at a time, and I led my students through the "See-Think-Wonder" routine.  Then, I posed the question, "I wonder how much of "Composition with Large Red Plane..." is yellow?  You know, what is the fractional amount?"  I showed more examples of his art that I found online. And, we completed more "See-Think-Wonder" rounds.  Then I mused, "I wonder if there are any fractional color similarities between his paintings?" 

Students began estimating and defending their estimates (insert wicked smug math teacher smile here).  They began talking about their own questions as well.  We discussed ways we could figure this out together.  I expected that tons of conversation would be needed around this problem, but I was pleasantly surprised. Two of my students brought up using a 100s grid..."It's too bad we couldn't find a way to use a 100s grid and put it on top of the paintings."  Yes, this math teacher swallowed the canary that morning. We finished this session by reading and discussing a mini-biography I had written about Piet Mondrian.

Day 2

We reviewed our math questions from the prior day, and then I whipped out some 100 grid overlays I had created by copying a grid on transparency film.  I had also printed some examples of Mondrian's art in color.  Students spent part of this session estimating and measuring using the grid.  I modeled my discovery that Mondrian pieces I liked best had more yellow in them than any other color; in fact, they all had 30/100 or more of yellow.
I also gave students some Mondrian-inspired art that I had made on the computer.  They practiced using the grid overlay with this. It was an easier activity, because I had consciously lined up rectangles and colors to work with a grid overlay.  Students practiced measuring and writing the fractional amounts.  Then, I asked them to turn the fractions into decimal numbers.  I modeled, using what we knew about place value. Students recorded the values on their recording sheets. We ended by reviewing our new learning.

Days 3, 4, 5
"Hey! I wonder how many different ways we could represent the same fractional color amounts by creating some Mondrian-inspired art?"  That was the all-consuming question for day 3 of our investigation.

We began by working backwards which is what my students called our strategy.  We decided on the desired fractional amounts for each color.  Then students used a 100 grid printed on paper. They designed their Mondrian-inspired piece using the fraction guidelines we had agreed on. 
The first copy was their blueprint, and the other became their template.  Students cut the second copy into parts an used these to trace onto red, yellow, blue, or gray construction paper. 

 After they had gotten all of their templates cut, they used their blueprints to arrange the construction paper shapes on an 8 1/2" X 11" piece of white construction paper.  After they had followed their blueprints and arranged their pieces, they glued them down on the white construction paper.
The final step was to add the black strip borders. 

During the See-Think-Wonder exercises that we had done earlier, students noticed that Mondrian used the black to border all colors.  In other words, he didn't allow colors to bleed into each other.  I cut black construction paper into narrow strips.  They used these for the borders.  We discovered that placement of the strips could totally change the appearance and feeling of an art piece.  Students played with this arrangement a great deal.


Day 6
After finishing our Mondrian-inspired art pieces, students converted the agreed-upon fractional amounts into decimal numbers.  Some of my students (these are 4th grade students) who needed more enrichment turned the decimal numbers into percentages.

I wanted my students to write about this experience and the learning goals.  We unpacked our thinking together, and then they wrote. 

Day 7
One of our other learning goals is that students will learn to use and navigate MSWord.  On our final day of this project, my kids typed their writing and learned how to use several features on MSWord.

You can see our finished investigation in the pictures below:

One of the skills we have set to the side in our push for data-driven results is that students don't get to use their hands to create something, not often, anyway.  This project was a challenge for my students in many ways:
  • The visualization needed to create the blueprints and templates and transfer them to an non-grid surface.
  • The fine motor skills necessary to cut and glue.
  • The use of a transparent overlay.
  • The ability to translate the same numbers from fractions to decimals, and for some, percents.
  • The problem-solving skills needed to rotate, translate, and reflect shapes and later the black borders to get the desired effect.
  • The keyboarding skills and technological prowess with software.
  • The reflecting, thinking, and observing necessary to understand, translate, and create a Mondrian-inspired piece.
I could go on and on. And, my students are begging for more...which makes me grin from ear to ear. 

"Mathematics has beauty and romance. It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there."
Marcus du Sautoy

We just began a measurement unit, so we're exploring area and perimeter. Guess what we're going to do when we take down our Mondrian art?  You got it! We're breaking out the grids and measuring our hearts out!

If you're interested in trying this project yourself, its applications are varied...fractions, decimals, area, perimeter, percents, geometry, data and probability, etc.  I've created a product to make this project more accessible for any 4th or 5th grade classroom.  Click on the picture below.

You can try some of the visible thinking routines I write about in this post by clicking below!

You might also be interested in reading more about visible thinking routines... Check these links out!


                                                        Until next time my friends, teach on!
                                                                                                    Tracy @


  1. What an interesting way to integrate art into mathematics. This is a very cool project.

    1. Thanks for stopping by to read! This is still one of our favorite projects to conquer together.

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