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:
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?
Olly, I did not realize you had a dance background - your choreographies are awesome! As someone quite foreign to the dance world, I appreciate the idea of notation linking mathematics and choreography. The ideas that some lines would be meant to ground in reality vs. extend into infinity, or as an endless extension over the natural world are powerful and could spark some very meaningful conversations. Between your summary and my article this week, I am quite excited about bringing culture into the math classroom via dance.
ReplyDeleteYour notice that “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!” aligns with some of my experiences lately. We are working with patterns, and students can “see” the ones in the text book questions but when looking outside they struggled. For example, one student told me that moss grows in a pattern, but then could not explain any repeating segments (they sketched it; it looked like a camo outline). We had to go back to the definition of a pattern again. It is interesting that this switch in setting, as Dietiker (2015) had us discussing last week, can make a difference in depth of understanding.
In going outside to look for and sketch patterns many students just threw something on paper and didn’t really get into it. So, your thoughts of how to get them engaging resonate as well. It is a tricky balance that we continue to play with. Let’s keep in touch about our successes! I had one this week with the star finger patterns activity. It was more successful than my outdoor pattern hunt and sketch. Maybe because of the ensemble learning aspect?
So many great insights here -- about cultural values and dance, the math inherent in choreography, the issues in getting buy-in from students as we begin to integrate math and embodied arts! On this last one: yes, the place DOES matter a lot! It also helps to have a warm-up (in a circle?) that uses everyday gestures that have been expanded to medium and then large scale, something I learned from collaboration with my choreographer/ dance colleague, Kathryn Ricketts. (And Kathryn is back in Vancouver again, and might be willing to do some collaboration too?) It's also really important to have mathematical questions and wonderings in mind as you start working with movement, and to build the kids' confidence that these movement-oriented lessons are actually ways of exploring mathematical questions, that they can then write down, symbolize and notate. It's a process that takes a bit of time with each group of students, but perhaps start with the group you think will be easiest to engage, and build out from there. If it starts to become a regular feature of mathematical explorations with your classes, most kids will engage. It could be combined with Building Thinking Classrooms, where the space is more open and students are getting used to being on their feet and working collaboratively on exploration and problem solving, integrated with teacher mini lessons and guidance.
ReplyDeleteOlly, I forgot that you once taught dance and have such an artistic side. After looking at both your posts, I was fascinated how you took on math dance with such ease and enthusiasm, but now I am putting the dots together. I am envy of your depth of knowledge in both domains to be able to work on bringing them together. I also truly appreciate your honestly in your reflections of your lessons, and your ability to not get discouraged and see the silver linings. I am positive that this is how and why these lessons EVENTUALLY work. We need to test them then fix them, then test them again. They are PROTOTYPES! (yay, science brain!) We are in the early stages of our mathematical art prototypes, and we need to work on them, use them, reflect, fix, and try again! This was a fabulous blog post!
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