Math and M.C. Escher


Math and M.C. Escher
Math and M.C. Escher

During the last week of school my students started to explore topography concepts. Topography usually isn’t the first thing that is thought of when someone mentions the word math. That’s why I find it so interesting.  I truly enjoy teaching this topic because it often brings out the best from my students.  I find that most upper elementary students tend to thrive when given geometric shapes and asked to explore, rotate, translate or even turn them inside out.

I generally introduce the unit with M.C. Escher.  The class learns a bit about the life of Escher and his contributions to the world of art.  Moreover, we discuss how art and math are related. This is often a deeper conversations as students start to expand on the notion that mathematics can be found throughout our world.  Topics like the golden ratio and Pi often get brought up during this time.

After learning about Escher’s life and his influencers, the class looked at his different artistic creations. Usually my students recognize at least a few different creations.  Students seem to gravitate towards his optical illusion pieces or the famous Waterfall work.  As each work of art was discussed the more students found mathematics as an integral part of Escher’s work. After reviewing the different pieces of lithograph art, the class watched a short video on how Escher’s design and math are connected.

After the video the students were asked to have a conversation about how math can be found in most art.  The words symmetry, rotations, slides, translations, reversals, surfaces, and perspective were all brought up during the discussion.  What’s nice is that the vocabulary was brought up naturally as students spoke to one another.   I was able to highlight the words and facilitate the discussion as needed.

Eventually the discussion ended and the class moved to the next activity.  I planned to have the students create their own Escher-like artwork.  The students reviewed how to have “Escher-like eyes” when creating their own pieces.  I was proud of the student responses and the imagination that came forth during this discussion.  The class then reviewed the directions to create their own Escher-like creations.

The students went through the directions and asked questions.  Once the expectations were clear I passed out a 8 inch by 8 inch square to each student.  Students created their own tessellation template.  In the future I’m probably going to cut the square dimensions in half so the patterns become more evident.

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Eventually the students used the template to create an Escher-like creation.  Students showcased their work to the class using the vocabulary mentioned above.  The students were able to bring their work home on the last day of school.  All in all, this is a lesson I’m intending on using next year and a definite #eduwin in my book.


How do you incorporate art and math?





Math and Art

Image by Graur Codrin

I have found that students enjoy and often thrive when presented with a challenging real world problem (often outside of the textbook). This can be observed during a problem based learning activity.  When students come to the conclusion that their isn’t one specific right answer, they are more willing to communicate their ideas and opinions to one another. Many practical problems outside of the K-12 education realm have more than one “right” answer.  When students are faced with problems that have multiple solutions the class community asks questions that often spark additional questions.  Active learning often comes to fruition through these activities.  In fact, the #realmath hashtag provides practical resources / images related to math found in the real world.  I have provided one example of using math through art below.  This idea comes for a PD opportunity that I participated in last year.

1.)   Begin by showing a small section of a picture.  Ask students the questions below or add your own (my questions are based on an elementary classroom).  Attempt to stay away from yes or no questions as students offer their own perspectives.  If you ask a yes or no question, follow it up with a why.  You might want to tell the students that some of the images don’t fit like a puzzle, as some pieces were cropped at different zoom levels – this adds to the complexity of the activity. I keep a separate chart on the board to write down math vocabulary that is used during this class activity.  Keep in mind that each picture is displayed one at a time, generally in a presentation format.  You may want to randomly have students answer the questions below.

Picture one questions:

  • What do you see?
  • Does this picture remind you of anything?
  • Describe the polygons in the picture.  Where are they?
  • What type of math vocabulary can you use to describe this picture?
  • Why is one rectangle in the picture lighter?
  • Where/when do you think this picture was taken?

2.)  Now show a small section of another portion of the picture.  Follow the same guidelines as step one.

Picture two questions:

  • Using math vocabulary, what do you see?
  • Using fractions tell me more about this picture.  Where can fractions be found in this picture?
  • Where/when do you think this picture was taken?
  • What similarities can be found between this picture and the first picture

3.)  Now show another section of the picture.  Follow the same guidelines as step one.

Picture three questions:

  • How does this picture similar to the first picture?
  • Why are some of the flags horizontal?
  • What type of information do you think the builders needed to construct this building?
  • How tall do you think this building is?  Why?
  • Do you think the building in the very front of this picture is the tallest?  Why?
  • What direction do you think the sun is shining?  Why?
  • What part of the picture do you think is missing?
  • Where do you think this picture was taken?

4.)  Reveal the full picture

Picture four questions:

  • How accurate were your initial predictions?
  • What are the differences/similarities between pictures one, two, three and four?
  • What additional math terms can you use to describe items in this picture?
  • What materials would you need to construct a building like the one in the picture?
  • Why is there a reflection on the building on the right?
  • How could you estimate the height of the building in the center of this picture?
  • Optional -Reflect on today’s activity in your journal.  Describe your reaction and what you learned during this activity.
  • Optional –  Similar to this example, students could take pictures around the school and create their own presentations on finding math in art.  Students could be given a rubric and work in collaborative groups and present their findings to the class.

Of course feel free to modify or change any of the steps above to meet the needs of your specific students.  My example is only a general template.  I’ve used this in elementary classrooms to introduce specific topics.  You could use a variety of images for this project, or have students create / take pictures on their own.  What about using Escher’s artwork below? As you can see, there are a lot of possibilities.



update 12/29 – An additional resource – Mathematics Meets Photographs