Week 6 Activities + Connections

 

This week, we observed a handful of mathematical dancing and dancing mathematics videos. Many of them involved Dr. Karl Shaffer, a dancer and mathematician. Video links are at the end of the post. 

I was fascinated by the body calculator by Miranda Abbott through her program Dancing Digits. 
I wanted to learn this with my children and try to perform calculations of higher numbers to see if my 5 year old and 7 year old children could work out these more advanced concepts with me. 

I learned the movements and absolutely love the idea of using the body as a calculator!! I have been pondering how this was a possibility since watching Sarah Chase's video about dancing different numbers on the the body. 

Mathematical connections: As practice, I danced numbers 1-200 in a time lapse (below) and the repetition really demonstrates this base 10 counting system with 10 movements of the legs (10^0) creating one movement in the arms (10^1), and ten sets of arm movements creating one finger movement(10^2), and 10 sets of finger movements to generate one head movement (10^3). 

So, if leg movement = l 

arm movement = a

finger movement = f

and head movement = h

We could calculate our body number with: l(10^0) + a(10^1) + f(10^2) + h(10^3)

It would be really fun to make assumptions like let l,a,f, and h = s

Then have the students dance out s(10^0) + s(10^1) + s(10^2) + s(10^3) to see what kind of full body movements we would get rotating through 1111, 2222, 3333, 4444, 5555... etc. This might be a fun way to assess whether students understand the idea of substitution with an added embodied layer.

The clock really comes to mind with the cyclical motion of the second hand (12^0) generating one tiny movement in the minute hand (12^1). It also really reminds me of a reduction gear. 

A change to the system: If I were to teach this, I think I would try to use both my legs in the positions instead of a turn out of my feet. That way I could ensure a balanced use of the limbs that move the most. In addition, numbers 5-9 may get mis-read if my students and I don't turn out, or register that our support leg is turned out enough. This also helps if actually creating a piece from the movements as each time you put your support leg down (00, 05) you can move your body with it. 

With family: My children tried this experiment with me and my son really took off with it. He asked me to test him. My favourite part of this dancing activity while we practiced adding numbers is that he really had to think about carry over. At the beginning of the video, I asked him to add 10: he would cycle through the feet positions, and carry over to the arms. Then closer to the end of the video I asked him to add 30, he discovered that you could just move the arm positions independently. He was so proud of himself. 

Here's my general "sketch" for learning using this dancing digits idea:

1. First, I would teach the history of numbers at the beginning of the grade 9 curriculum as a foundational piece giving students perspective and appreciation for the symbols that we take for granted in communicating the concept of numbers.
   
   
2. History would lead to the number system: Natural, Whole, Integers, Rational, Irrational, Real, Imaginary and Complex.
   
   
3. Then, we would get into playing with/as integers. This is where I would introduce the dancing digits concept with call-backs to the history of numbers regarding various bases. In addition, I could ask the students to adapt this system (or invent a new system) for base 5, base 3, and base 2.
   
   When doing so, they should see a pattern: a progressive speed increase between the movements of the legs to arms, arms to fingers, fingers to head. It would also involve a placement calculation that may be trickier for them to mentally calculate: 5^0 + 5^1 + 5^2 + 5^3, for example
   
   
4. Then, I would ask them as a class to do addition or subtraction of 1-4 digit numbers (including negative integers) through the Dancing Digits technique. I would make sure to first use numbers that did not involve any carry over between body parts, so the students could get a feel for movements within the range 

• Examples to dance out:
• Easy level: 1 + 100 + 5 + 20 + 300 + 10 - 3 - 1 + 40 - 500 + 20 + 6 - 1 - 200 + 40. 
• Medium mode: 150 + 1037 + 2411 - 1066 + 23
  
  
• Then progressively, I would ask them to add and subtract numbers that would result in carry-over:  
• Hard mode: 232 + 567 + 9031 - 390 - 411
  
  
• Followed by a Journal Reflection:
• When doing arithmetic with your body, explain your patterns of thinking (or shortcuts) that are allowing you to get to the result efficiently. Provide examples. 

WONDERS that I'm working on: 

1. How can I adapt this using multiplication or division? 
2. Can I make this a two player game in which the coming together of individuals resembles a mathematical operation in which the result reveals itself rather than needing to be calculated?

Referenced Videos:

Viewing: (1) Karl Schaffer & Mr. Stern TedX  talk (2012) (10:18), 

(2) Karl Shaffer Math Buffet: Squishahedron and Tetrahedron (2021, Julia Robinson Math Festival) (2:32)

(3) Dances with math: Interviews with Karl Shaffer, Saki and Erik Stern (2021, Julia Robinson Math Festival) (12:51)

(4) Jump into Math! Malke Rosenfeld TedX talk (2013) (12:24)

(5) Keith Terry : Rhythm of Math -- Teaching Mathematics with the Body teaser (2015) (1:23)

(6) Keith Terry Rhythm of Math: Polyrhythms -- 3 against 4 (2015) (2:34)

(7) The geometry of longsword dance locks (Steel Phoenix) (2013) (4:47)

(8) George Hart: Mathematical Impressions: Longsword Dancing (2014) (5:46)

Optional viewing -- a bit of math theatre (two plays about women in math):
(a) Gerofsky, Witches of Agnesi musical math history play (2019/2021) Note: This link takes you to the National Math Festival (US) panel discussion and pre-recorded play. If you want to watch just the play, here is the link to it. (55:00)

(b) Moira Chas, The fictional letters of Alicia Boole (2021). (24:10)

Activity: (Note: Malke Rosenfeld’s website has just disappeared and been replaced by some ecommerce site! Here are some different and definitely worthwhile activities for you to try out and think about!)

Here are some math and movement activities from Karl Schaffer and company:

Making Stars (with Scott Kim) (1:38)


Mathematical Hellos (3:13)


Adrienne Clancy on dancing rotations (including the rotation of the earth on its 23.5º tilted axis..) (15:15)


MIranda Abbott’s In Constant Motion (5:02) (Miranda is a Canadian professional dancer who is now a Grade 2 teacher in Costa Rica, and teaches mathematics through dance…)

Week 6 Readings: Mathematics & dance, movement, drama and film

Reference: Belcastro, S. M., & Schaffer, K., (2011) Dancing Mathematics and the Mathematics of Dance. Math Horizons. 18(3). pp. 16-20. DOI: 10.4169/194762111X12954578042939

Summary: Belcastro and Schaffer highlight examples of how mathematics can be explored using dance, and how dance has been influenced by mathematics. Mathematical concepts that were identified in this article with examples in dance and music are counts for rhythms, pattern sequences, local and global symmetries, combinatorics, polyhedral geometries, game theory and graph theory. The authors show how new notations were introduced into the dance world to symbolize mathematical concepts for simplification of choreographing and describing natural observations of movement. Some detailed examples include John Conways "hop-step-jump" terminology, the Klein four group of symmetries, Laban's kinesphere, and polyrhythm clapping.   

Stop 1: "Bharatya Natyam the dancer’s lines end—they are cut off by abstract or representational mudras made by flexing the hands. This situates the dancer “in the world,” rather than extending beyond it into the “world of the gods.” In contrast, the ballet dancer’s lines extend toward infinity, symbolizing an endless extension over the natural world" (Belcastro & Schaffer, 2011, p. 16).

A few things here that stopped me. The Bharatya Natyam I needed to be able to visualize so I googled a few videos. A few observations upon my first viewing: 

- The vocalizations involve microtones as I believe is traditional in this type of music. 

- The body rests on bent knees for balance with both legwork and armwork done in small repeating patterns, or motifs, of being extended from the body symmetrically or asymmetrically and pulled in, sometimes adding variations for complexity as well as movements repeated in double time to music for even more added complexity. (I've just described "dance" in every culture, haven't I?). 

- As described in the article, the wrists and ankles will purposefully create right- or acute- angles to the forelimbs while simultaneously creating unique hand gestures or finger positions. 

The meaning of the hand and ankle bends in grounding the Bharatya Natyam dancer in reality instead of extending to infinity really caught my attention. Is this a function of culture? (of course) Which part? Is it more beauty-focused? or spirituality-focused? How did this cultural choice develop and become tradition?

Can we see the values of the culture through the lens of their dance form? If so, what does it say about the values of the culture? Conversely, why does ballet extend to infinity? Can we see values of the culture through ballet? If so, what does it say about the values of the culture?

Do both cultures attempt to embody the divine? Or higher ideals? 
In the quote, ballet's arm, leg and body lines symbolized the endless extension over the natural world. I am taking this to represent colonialism.  

Stop 2: "Using just four symmetries - translation, mirror, reflection, 180-degree rotation, and glide reflection - we can create what is called the Klein four group, or Z2+Z2, by combining the symmetries pairwise" (Belcastro & Schaffer, 2011, p. 17).

The combination of these creating a table in which we could complete as an activity was very interesting. In any flashy dance piece (let's say, Jabbawockeez's), you will find each of these symmetries. Here's a video to check out: 

As a dancer and choreographer, I have found myself in situations where I need dancers to traverse the stage a particular number of steps and end up in the same position as others. Working backwards from the endpoint, it tells me which foot they need to start moving on to achieve this and on which beat. Working this out mathematically was very interesting! When choreographing, I try to develop movement ideas that have not been seen before - so unique music, unique movements (cross-cultural is best which the style of contemporary dance lends itself to very well), and unique patterns. I rarely thought of dance as a mathematical activity (this was many years ago when I started teaching as a Drama/Dance/EAL specialist). Looking back, I was problem solving everything from negative space, to timing, to symmetries, to symbolism. Here are a few pieces I've choreographed that show this:

Doing translations and transformations on the Cartesian coordinate system is not very interesting and students can usually see what has happened the moment I display a problem for them to engage with. However, take this three-dimensional posing, then engage in the two processes and figure out how the outcome has differed from the start was unique and challenging!

 

Wonders: So, funny story. I tried the mini lesson that I developed on LCM (from week 5) with much excitement in my heart for a new way to connect with students over what has been an eye-rolling lesson in the past. It did NOT go as well as I expected. Students were struggling to (1) do the movements and (2) find meaning in what we were doing. I thought this would connect with all the students, but some sneakily decided to disengage and do other homework, go chat in the corner in groups, or pretend they were working on figuring this out with no real tangible outcome. 

A few students reported to me that several kids are feeling like we're doing anything but math in our class because they haven't been exposed to a math-class like this before so apparently anything goes. However, the silver lining here is that there are a few that can really see the connections between these embodied concepts and the pen-and-paper ideas in class. In addition, they love it. 

A few things that I'm sure could've made this better:
(1) actually following my paper's recommendations (Kelton & Ma, 2018) and go to a space that my students know represents movement - they gym, the MPR, the foyer, OUTSIDE. I think I am understanding the idea of the "setting" more now from this negative example. 

(2) "Don't throw the baby out with the bathwater" - Dr. Gerofsky. I had a good set-up from the past that I abandoned for something I felt was innovative. The energy and lack-of-buy-in really off-set the innovation so the end of the lesson was a flop. I will add this component into my previous lesson's structure as a body-break before exploring LCM next time instead of a 20-30 minute movement exploration that everyone could see was LCM quite quickly. 

I continue to wonder about how to increase buy-in for movement or artful mathematics?
(I thought the novelty of not doing pen-and-paper work would be enough... but not really)

Students are self-conscious of body movements on a good day, let alone ones that engage the spine and move the internal organs (viscera). I'm noticing a pattern of "the older the age group, the less the buy-in for visceral movements." So, absolutely YES, there are so many mathematical concepts that are related to every form of dance as described in this article. How can we get the students to explore? 

Do we present the math first? Or do we show them the fun activity and connect the math second?
Should the math be front-loaded? Or discovered? Is there a right way to do this?