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


Floored by Vectors

When I first started using Thinking Routines, I was teaching Trigonometry to my IB grade 11 class in Shanghai. I used the thinking routine Connect-Extend-Challenge to frame the vector activity.

CONNECT: How are the ideas and information presented
CONNECTED to what you already knew?
EXTEND: What new ideas did you get that EXTENDED or pushed your thinking in new directions?
CHALLENGE: What is still CHALLENGING or confusing for you to get your mind around? What questions, wonderings or puzzles do you now have?


  • String cut in varying sizes under 30cm,
  • masking tape,
  • rules,
  • meter sticks,
  • erasable markers
  • 4 pieces of poster board paper if your classroom does not have a floor you can write on with erasable markers.

Activity Instructions:

Part 1: Naming your line segment

  • As each walked in the classroom s/he received a piece of yarn of varying lengths under 30cm with two pieces of tape.
  • Have each student tape his/her string on the floor in a designated area. (At the time I have tiles in my classroom, I used 4 pieces of poster board when the classroom had carpet, or a surface we could not write on).
  • Use erasable markers for students to name their strings (a, x, or Harry… anything they want)

Start the lesson saying that the students already know everything that we are going to go through but in a different form, so our objective for the lesson is to have an overview of vectors, some notation and where we will be going through the unit. After the students label the strings ask them to describe their string to a partner. You can cycle through the CONNECT-EXTEND-CHALLENGE questions. They might measure their string, or talk about a slope. When you bring them back together as a group, point out to them the math vocabulary they were using. Ask the group what would help them describe their line segments more accurately. Most likely they will put an x-axis and a y-axis and decide on a scale. If the students don’t suggest, the coordinate plane do that for them.

Next, have students determine a more precise description of their line segment. Including the end points, and slope. Ask if anything else can be measured with the lines,  distance or magnitude.

Part 2: Direction is important

You can discuss with them the differences between lines, line segments, and rays. Have them make their line segments into rays.  With this discussion, I like to include direction being important to vectors and start talking more about vectors and the notation.

  • Students need to identify (and write on the floor) the starting point of their vector.
  • Introduce the directional vector to the class- have the students write out their directional vectors.

Part 3: Vector addition


Partner up the students that have vectors of different directions. One partner will need to move his/her vector to the terminating point of the vector. It is important to remind students that the direction of the vector needs to be maintained.

Once that is done, you can go through the CONNECT-EXTEND-CHALLENGE questions to bring out concepts. Students will want to complete the triangle, find the angles of the triangle, possibly the area. Cosine Rule, Sine Rule, and the area formula will come up in the discussion.

Depending on the time you have you can continue the discussion. Again this lesson is designed to show the students where the unit is going with vectors, and also allow them to connect with the material in a different way. It is fun to write on the floor. The next lesson we summarize what we did and more formally go through the vector notation. If I did this on the poster board, I hang it on the wall for the rest of the unit.

To wrap up the lesson we revise what we discussed and put up questions for the CHALLENGE part of the lesson, “where and how are we going to use this information?” “Why is it important?”

Normally distributed Groceries


Recently when looking up a grocery store on my phone on google maps I noticed the following graph on popular times.

Screenshot 2015-11-09 11.23.22

Google provides data/predictions for customers about when people typically visit the specific store. I am not sure how the data is collected, so if any one knows please share.

I used these graphs in a statistics lesson after I had introduced normal distributions. We talked about who this information is important to and estimated the standard deviation on a curve that looked normal. We discussed the missing information on the y-axis as well as how a store opened 24 hours would have a different standard deviation and possible shape to the graph. Then we looked at the rest of the week.

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The students were able to recognize patterns in the specific store and relate it to human behaviors in the work week, we estimated the mean for each graph. We also discussed other locations through Washington, DC and if they would have the same types of distributions. Would stores downtown have a different distribution from stores in the suburbs? Will all types of stores have a similar distribution based on location or is it more based on the type of store (for example, grocery, hardware, clothing).

It would be interesting to see how Thanksgiving preparation will affect the graphs.

Statistics lesson: Justin Bieber was the common denominator.

For my grade 12 class we reviewed the five statistical measures with the following activity. I found a website that releases the data for the top selling songs on iTunes for the past 24 hours, called Digital Sales Data.  We did the activity on September 3rd which was after Justin Bieber’s new single What do you mean? was released.

Outline of activity:

  • The students were in groups of 3-4 students and each group was given data of the top 25 selling song from a different country: USA, Finland, Australia, Malaysia, and Nigeria.
  • Each group had to find the 5 statistical measures and we compared as a group.

What was interesting was that Finland and Nigeria had similar types of numbers, The students were a little confused about if the units were in the thousands or just units as some of the data was so small. One student pointed out that Finland owns Spotify so we talked about the economies of the countries. We also talked about why the values were so different discussing the population of the countries and buying power. It would have been helpful if I had the countries’ population and GDP to compare.

Overall it was a good activity that brought in a global perspective and interesting to see what types of music are listened to elsewhere and helped the students stay focused and interested. We found that Justin Bieber was the top selling in all the countries. We also were able to talk about grouped data and deciding on appropriate class grouping ranges depending on the data given and discussed the standard deviation of each set.

Here is and example of the data from the USA and Finland.

iTunes data from September 3, 2015 Screenshot 2015-09-05 09.25.32