Exploring “How Many?” at the Hirshhorn

Yayoi Kusama is a Tokyo based artist that explores the concept of infinity in her artwork. Her process tends to be is repetitive, incorporates polka dots of various sizes and is known for her infinity mirror rooms. Last year she installed the exhibit Infinity Mirrors at the Hirshhorn Museum in Washington, DC and now one of her pumpkins is on permanent display in the scPumpkin by Yayoi Kusamaulpture garden. While her work explores infinity, there are a finite amount of dots in this pumpkin.

Christopher Danielson‘s book, How Many? encourages and helps facilitate mathematical dialogs with kids that move beyond just counting objects to developing curiosity through describing methods behind pattern making.

On Twitter, I posted three photos of the Pumpkin with the question “How Many?” To use these photos in the classroom, I would combine revealing the images through the Project Zero Thinking RoutineZoom-In.

Reveal the First Photo:

Ask students: How many dots? What would help you answer this question? What do you think this is a photo of?

Let the students dialog about the photo; some will want to count the dots, some will make an estimate. Make sure to ask the students to explain their reasoning by asking “What makes you say that?”


Reveal the Second Photo:

Ask students: How many dots do you think there are now? What would help you answer this question? How does this photo change your hypothesis or thinking?


Reveal the Last Photo:

Ask students: How many dots cover the pumpkin? What other information would help you answer this question? Why do you think the artist, Yayoi Kusama, chose to make a pumpkin sculpture and why did she use dots? This piece is part of the exhibit “Infinity Mirrors,” what is she saying about numbers?

Pumpkin by Yayoi Kusama

If you use this in the classroom, please share with me how it went and what you altered. I would love to know how it goes and how to improve the questions.

Thank you for stopping by my blog.

Islamic Art Exploration

My goal as an educator this year is to intentionally focus on the cultural side of mathematics through authentic applications.  During the geometry unit in pre-algebra, my students explore the intersection of geometry and Islamic art. After studying quadrilaterals, we started playing with 5 and 7 overlapping circles through activities outlined in the book Islamic art and Geometric Designs. Screen Shot 2018-02-01 at 6.37.56 PM.png The activities expose students to the regular shapes that can be made from circles. At first, we worked with compass and rulers, and I gave students the option to work on GeoGebra.

Two videos that explain the process and significance:

The students worked on their project over a week. We first looked at developing an appreciation of the creation process, then started analyzing photos. I used some designs from Eric Broug’s book to help students explore the patterns in the designs and deconstruct designs.

Here is a link to analyzing a pattern tile, Complex Islamic Geometry. Some imagines are taken from Eric Broug’s book and a Thinking Routine from Agency by Design (Parts Purposes, Complexities).

After going through examples and making connections between geometry properties and overlapping circles, we started on the project. Instructions for students Islamic Art Project.

Students used GeoGebra or compasses to create different pattern tiles. We used about 2-3 class periods for students to create a design then one lesson to work on the write-up. I did not assign homework through the process as I wanted to make sure to be available to answer their questions and provide technical help. Upon completion as hosted a Gallery walk to observe student work. I invited fellow teachers, administrators and Grade 2 students joined us since they were kicking off their Geometry unit. It was beautiful to see the students give the second graders a guided tour.IMG_3988.jpg

Overall the students improved describing their process, their use of mathematical vocabulary, comfort level with GeoGebra, and learned about some of the cultural significance of geometry to Islamic Cultures.



Student work 2
This student highlighted different parts of her design to explain her process



The newly renovated Renwick Gallery in Washington, DC is a perfect place to start off 2016 by seeing the exhibit Wonder. The exhibit features nine artists’ installations that give the visitor a sense of wonder. I absolutely loved walking through the beautifully restored building to observe thoughtful pieces created with care and precision. Through the exhibit I kept rerunning the Project Zero Thinking Routine I see… I think… I wonder….. 

Through the various rooms there are quotations about wonder. One that I found thought provoking “It is not understanding that destroys wonder, it is familiarity”  John Stuart Mill, 1865.

The symmetry in Jennifer Angus’s installation of unaltered insects was fascinating, playful and made your skin crawl a little while taking in the details and features of the insects.

The Gabriel Dawe installation of Plexus A1 is just breath taking. I would love to see if I could do a lesson on vectors and intersecting planes for the color refractions he creates through the use of string. If any of you math educators have ideas on this type of lesson please share 🙂 I will see what I can come up this semester.

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At the risk of being to familiar with the art piece, I played around with Geogebra‘s 3D modeling to construct the basic shape and see of the intersecting thread. Figure 1 is the first model with the lines on the floor perpendicular to the lines on the ceiling. The figures are not to scale.

Figure 1

After looking at the photos again I notice that the lines on the ceiling were more angled. The angles are shown in Figure 2. I am interested to know how the Dawe planned the strings line of the intersection. Also, how can you mathematical maximize the intersection string line? When thinking of this from color theory point of view, you can also look at each intersection points of the individuals lines as color mixtures.

Figure 2

I wonder how the tightness of the string effects the intersection of the planes, also how to describe the planes mathematically? In a way it reminds me of the mathematical models that May Ray used in his Human Equations exhibit at the Philips Collection last spring.

If you have an idea or insight on this installation please share. It definitely brings out a lot of wonder even as I become more familiar with the problem.