Week 6 - Critical Reflection - Mathematics & dance, movement, drama, and film

 “We tend to think more effectively with spatial imagery on a larger scale: it’s as if our brains take larger things more seriously and can devote more resources to them.”  William P. Thurston

“Mathematical thinking can be tied to symbols but it is not necessarily.” Erik Stern

    I have a fairly extensive dance background and I do love to dance even though I haven’t been lately (pandemic, bike crash, undiagnosed foot injury (hoping this will change on February 28)). I did ballet from when I was 4 until I was 17 and jazz dance as well for the last 3 years of my teenage dance life. About 6 years ago, one of my ballet school friends recruited me to do adult ballet with her and I did that for about 3 years with the odd drop-in class after that pre-pandemic. My enjoyment from dance comes from moving to music rather than being technically strong. What I am very good at though is remembering steps/choreography (show me once and I’ve got it) and my teenage dance teacher thought this was connected to my strong math skills; I think it is to my love for patterns. In the Dances with Math interviews, Schaffer talks about how mathematics and dance are connected and that “you can’t pull them apart.” I appreciate this idea because it makes a lot of sense in my soul – dance and math live together and thrive together.

(Sorry, the first paragraph is what I addressed in the Zoom meeting and all of my reading group was there! Hope it doesn’t seem repetitive!)

Dancing feet in the
parking garage

I drew on my dance experience and memories this week when I engaged with the Math in Your Feet task. I took the first lesson plan, Math in Your Feet Starter Kit, printed out the dance moves, and recruited my boyfriend to dance along side me. We went downstairs to the parking garage after trying the first one (4 jumps in place) and realized it was not going to be downstairs-neighbour-friendly. After 2 sequences my injured foot was talking to me and asking me not to jump, not even the tiny jumps I was doing. So, I had to adjust my engagement. However, I found that I could feel what the 4 beat sequences would be like if I had been jumping, even with the little bit of bounce I put in my body as I moved. I could imagine how my body would twist in the air to do a 180-degree turn, for example, even though all I did was bend my knees, stretched up and swiveled around, and bent my knees again. Is this an embodied memory? I connected this to the reading I did with the student’s who were blind and engaging with symmetry and how they drew on previous experiences (visual memories or physical memories) to perform a new task. In a comment on my week 3 reading summary, Susan notes that some of our students will have lots of embodied experiences to draw upon (from dance, sports, etc.) and others will not. How might I have experienced these dance sequences with a foot injury if I did not have a dance background?

    In my reading, Reenacting Mathematical Concepts Found in Large-scale Dance Performance can Provide Both Material and Method for Ensemble Learning (Vogelstein et al, 2019), I learned a definition of ensemble learning. As the authors define it, this is “learning that is fundamentally collective and performative, where learners recognize the need to act together” (p.332). This was a “stop” for me and I scribbled “orchestra and dragon boat” next to the authors’ example of marching bands. I taught orchestra grades 4-12 for five years and ensemble learning was integral to the experience. It was interesting to see the grade 4s develop from trying to be the first one finished a song to understanding it sounds much better if everyone finishes at the same time. I was recruited to help coach the high school dragon boat team because many of the team members were either orchestra students (and band students). One year at one of the regattas, one of our boats of all music students had a terrible first race. During the break between races, they got together to figure out how to make their strokes synchronize. One said, “Wait, you know how Ms. Fox and Mr. Reid (band teacher) say we need to sub-divide the beats in our heads to be more accurate? Let’s do that with our counting!” So they pulled together and practiced counting “1 and 2 and 3 and 4 and…” and did that in their next race – winning by more than a boat length! This understanding of dividing beats into smaller pieces, which is mathematical, contributed to the ensemble working and learning together to improve their performance. Again, these students had other experiences of ensemble learning to draw upon and was maybe enhanced by it being the same experience. Like I find dance and mathematics to be inseparable in how I understand choreography, I wonder if mathematics and ensemble learning weave together similarly so each helps the other.

    Vogelstein et al (2019) speak to how the props in the dance-based interviews connected participants so ensemble learning had to take place for both the dance and the mathematical thinking. They noted that the quartet of grade 8 students “acted and reasoned as an ensemble to resolve difficult challenges” but in another activity, the girls put down the prop and then “became disconnected and largely worked individually, talking over one another and using incompatible schemes of reasoning” (p. 338). The authors also note that a prop becomes a partner in the ensemble learning in generating mathematical performances. I connect the swords in the longsword dances to these ideas. The dances have the props/swords and by holding the swords, they are connected to one another in creating the mathematical formations. I can imagine that in practicing, they must come to jointly understand the mathematics of the patterns for the dance to work out and that they are connected in this ensemble learning through the swords. The swords are integral to the dances and are partners, just as the sheets of plastic were partners in the Rio Olympic inspired dances in the study.

    My final “stop” and connection for this week is about how movement/embodiment can assist understanding. Vogelstein et al (2019) had participants recreate dance sequences with a prop from a video as a way to understand the choreography and the mathematics that supports it. The authors found themselves getting up and re-enacting a movement that the STEM educators created but then could not recreate; the authors had to experience the movement to try to understand the experience of their participants. When I printed off the Math in Your Feet movement ‘instructions’/challenges, I was first looking at them and imagining what the four-beat sequences would be like. Then I found myself moving my hands, as ballet dancers do to ‘mark’ steps, to think more complicated ones through. Finally, I danced them out and they made the most sense, especially the ones involving 90-degree and 180-degree turns. However, this stop also had me wondering about students with physical disabilities in an inclusive classroom. If movement of the core/spine/etc. is important as a way of understanding, how do we provide access to embodiment with similar learning effects to children who are quadriplegic, for example?

Oh! My notes have me wanting to keep writing but hopefully I can weave in some more of my thoughts in discussion with others!  Lots to notice, wonder, and experience this week!

Questions to ponder

How do we address the variety of embodied experiences backgrounds our students have so the ones who have less experiences can be just as successful in connecting math and dance as students who have lots of experiences (with dance, sports, etc., even if they haven’t been explicitly connected to math)? (I am making a perhaps incorrect assumption that those with experiences say in dance might have an easier time with connecting with math in dance because they might not need to devote as many mental resources to engage with the dance elements.)

What mathematical ideas can be explored through ensemble learning with props? What are some extensions of this idea outside of mathematics to other curricular areas?

How can we provide inclusive embodiment experiences to our whole class so students with physical disabilities have the benefits of embodied learning/embodied ensemble learning?

Week 6 - Reading Summary

Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. Zdm, 51(2), 331-346. https://doi.org/10.1007/s11858-019-01030-2

 

Using processes of foraging and dissecting and of analysis of interactions (interaction analysis (IA)), this qualitative study looked at the interactions of two participant groups (one a quartet of grade 8 students and the other a quartet of STEM educators) as they attempted to recreate choreography from a video recording of the opening ceremonies of the Rio 2016 Olympics that involves a 7’ x 7’ sheet prop. The researchers additionally analyzed their own processes in IA. Data was collected as video-recordings of a 90-minute interview (involving watching and discussing a video of a segment of the opening ceremonies, participating as a quartet in challenges related to the Rio performance, discussing the mathematics, and creating their own performance piece); these interviews were recorded with four cameras. A second source of data was video recordings of the IA by the three authors and three of their doctoral student colleagues. Analysis was conducted using iterative methods, supported by limited transcriptions, “toon strips” (sequences of still images), and agent-based pseudocode.

Results are presented in relation to each of the two participant quartets and the analysis of the researchers’ own IA methods. The results indicate that foraging in cultural performances can contribute to design learning environments and that dissection and re-enactment of these performances can provide an environment that fuses dance and mathematics as a hybrid activity. The use of a prop was seen to support ensemble learning by connecting the participants. The re-enactment and design of the participants’ own performances resulted in application and challenge of mathematical understanding. The researchers connect the importance of the prop to ensemble mathematics. The researchers themselves engaged with re-enactment of the STEM educators’ performance in order to develop understanding, showing that re-enactment and ensemble work support comprehension and the authors consider this a contribution to IA methods.

Here is Youtube video of the Rio Olympics opening ceremony that was used in the study.  The segment that participants viewed and re-created starts at approximate 12:10.

Fibonacci Circles - just because I can

 I have been thinking about circles and had the question the other day about how different number patterns might look if visualized with circles.  What about the Fibonacci sequence?  That question has been in my head for several days and I have resisted playing with it because I really should be prioritizing other things (eh hem, project proposal).  Sunday afternoon I gave in to the curiosity...

 

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February 18 addition

My brain was too tired to compose a post for Week 6 after a day of hosting pro-d events...I tried.

But my compass and pencil crayons called out for playtime.
Still did a set of circles with radii growing according to the Fibonacci sequence and still placed the centre of the next circle on the edge of the previous one in the sequence. However, this time I rotated the position on the edge by 90-degrees counter clockwise.
I like the effect of my first play-time better.  What do you think?

Week 5 - Critical Reflection

     I feel like I am rushing to gather my thoughts because I went down so many exploratory paths this week – pursuing more of the tessellating trapezoids from last week (see my Scratch coding addition), thinking I would pursue more of Sarah Chase’s movements than the New Teachers’ Riff on Kandinsky in Binary but playing with that as well, and getting caught up in the possibilities of visualizing with technology.  I hope my ideas come together here.

Scratch cat tessellating trapezoids

    First off, in Susan’s introductory video this week she mentioned that mathematics is part of world heritage and that we owe it to students to bring this heritage into our classrooms. I see this as meaning that we do need to offer ways to explore mathematics that are authentic to the way math is used and engaged with across human endeavours. The arts are a heritage, too, and in our tasks so far in this course, we have seen that math and the arts are linked. We can/should bring heritages together in our classroom because they work together outside our classrooms. Part of the interconnectedness is the sense of play that exploring mathematics through arts brings and I think that is why I keep jumping down twisty-turny rabbit holes – because it is just so much FUN! I get myself into “flow” states and suddenly 2 hours or more have gone by or it is 1:30 in the morning!

Q: How would you manage students that might go down rabbit holes and not pursue the intended outcomes? Does this need managing? Why or why not?

    As part of the data analysis in the EASY Minds study, the authors found that the bit of criticism a few students had toward the program was that sometimes they got frustrated with the physical activity and they commented that this frustration interfered with them learning the mathematics (Riley et al, 2017). Upon reflection, I experienced this, too, in my attempts at the activities, both with the physical coordination and with technology. This makes me think about implementation with students. The EASY Minds study wasn’t looking to refine physical literacy but just to get students active more throughout the day, specifically in math lessons. So, I think it might be important to 1) be sure the focus is on just moving, not refining physical skills and 2) to introduce some of the physical skills needed for an embodied math task separately from this task so students develop competence and confidence in that skill. This involves some planning ahead and predicting what could get in the way of success.  The same would go for using technology to create visual representations; the students might be more successful connecting with and deepening understanding of the mathematics if they already know how to use the app or program or coding platform that will be used for exploration.

Q: How can we mitigate frustration with the embodied, outdoors, or art aspects of our integrated mathematical activities so students can engage deeply with the mathematical concepts?

I don't think 6 and 8 are wrong

When Sarah Chase talked about doing the dance patterns with other people and that they might choose the “wrong” numbers, meaning pairs or groups of numbers that might have a lowest common multiple smaller than their product, I will admit my back went up a bit. In thinking about why, I came around to there being interesting discoveries with what she calls “wrong” numbers and just because her intended outcome is to have a long sequence of movements before they repeat, it does not mean that other people cannot have other outcomes, intended or not. “Happy accidents” can lead to interesting discoveries. I saw the “wrong” numbers as an opportunity to explore multiples and lowest common multiples (which Cassie mentioned in a text chat she is immersed in with her students working with fractions) and that is
what my movements and notations explored. The “wrong” numbers could also lead to an interesting discussion about number choice and how it effects the length of a dance sequence that is choreographed in this way that Chase uses.

Q: How can we as teachers ensure that we are open to ‘happy accidents’, to explore these with students?

iPad piano is providing fun!

If you look back to Week 4 posts and the first image in this post, you’ll notice played with coding tessellating trapezoids on Scratch. I also played with PowerPoint to work with circles and base 5 in the Kandinsky binary activity and to try some illustrating/animating combinatorics of 5, 3, and 2 with clothing. My iPad piano came out this week, too. Working with technology allowed me to make more observations about the mathematics and how it worked. Are coding and working with technology to create visuals embodied activities? I feel like they are – I am doing something to create a visual representation so my sense of sight is used to understand the mathematics. Also, when I am working to figure out how to represent ideas in code or to make them animated on sequential slides, I am doing a heck of a lot of visualization and manipulation in my head. This reminds me of the 2013 Fernandes and Healy paper I read about students who are blind and their experiences working with symmetry; I understood from this paper that manipulating physical representations in the mind is an act of embodiment where the physical and abstract are working together.

Q: What do you think of the idea that work on a screen with coding or other applications being representative of embodied learning?

Week 5 - Activity - Piano, Movement, Sketching, Curriculum Idea

    I couldn’t resist playing around with both activities but settled on Sarah Chase’s movements to do the required activities.

    But first – let me quickly tell you how I explored the New Teachers’ Riff on Kandinsky in Binary…

    I jumped right into extending the activities in the video. Ali commented that he wasn’t much of a musician but still came up with a way using 3 notes to add sound to the animation. I wondered if there was a way to play around with some notes and a different base. I chose Base 5 because I could work with a pentatonic scale – one without semitones that is supposed to eliminate discordant sounds but the one I chose has a major 2^nd in it (C D E G A) which I find discordant. Anyway, I did play around with circles and colours on PowerPoint slides but I didn’t save my work and it disappeared…not even a “do you want to recover this file?” notification next time I went into PPT. Sigh. What I did on my iPad keyboard was zoom out so I could get 2 full octaves and another 6^th. I set C as 0, D as 1, E as 2, G as 3, and A as 4. I used the upper most octave as the ones place, the middle octave (starting at middle C) as the 5s place, and the lower octave as the 25s place (is my elementary teacher terminology making people cringe?). I assumed a 0 in the 5s and 25s from the beginning – I wrestled with that and went with that...having a sound for 0. I played up to (255)₅ .  Here is the result with a few slips (wish I had a real piano to play on):

    I had to pause that to go do life things and came back to find my PowerPoint gone so didn’t pursue this further.  I wanted to put the music to the PowerPoint slides of coloured concentric circles that were like Ali and Colin’s template.

    So I moved onto Sarah Chases’ movement.

    I have a physical memory of her 2 and 3 movement – my jazz dance teacher when I was a teenager used to make us do that as part of our warm up. It was interesting to see it used it for combinatorics.

    I did 3 and 4 and 4 and 5.  My introvert didn’t want to be filmed or photographed today so after doing some choreography in my living room (many tries to get through the 4 and 5 but practicing each separately helped), I notated them on paper.

 

Notation of movements for 3 and 4 - common multiples circled

Notation of movements for 4 and 5

I thought this is another way of extending the activity – 3D movement into a 2D representation.

    I took exception to Chase saying that numbers like 6 and 8 together would be wrong but she might let people try it and see that they were wrong. It seemed they were “wrong” because they have a common multiple that is less than their product. I’ll expand on this more in my reflection.

I wondered if 6 and 8 could be notated in a circle some how to find common multiples, lowest or not.  This is what I came up with:

 

The blue lines divide the circle in 8ths and the orange lines 6ths. Then I used 2 colours (green and pink) to ‘count’ around the circle, noting when I wrote the same number on the last 6^th and 8^th number which would be a common multiple – they would start again together on the vertical radius.

I also tried some combinatorics on PowerPoint – 5 shirts, 3 pants, 2 hats. I’m having trouble recording it as an animation.  I might come back to it…I remembered to save!

 

Curriculum Idea Sketch

Issues – common multiples, combinatorics

Guiding Questions – How can you use body movements representing two (or more numbers) to understand how their cycles coincide? How can you notate this? How does this relate to common multiples? How is using strictly prime numbers different than including composite numbers? What kind of problems can we solve with these movements?

The Story – watch part of Sarah Chase’s video (opening and 2 and 3 movements); students try 2 and 3 and then explore other combinations of two or more numbers; explore ways of notating; how can pare down (e.g., write down only the numbers that end the cycle for each number – leads to listing multiples and finding least common); apply to combinatorics – e.g., clothing combinations, lunch combinations, etc;

Integrating embodied learning and other learning – move from 3D to 2D with notation, problem solving

Possible extensions - find another way of notating (e.g, on circle); coding, shuffling on PPT slides, stop motion videos

Week 5 - Reading Summary

 Riley, N. et al. (2017). Movement-based mathematics: Enjoyment and engagement without compromising learning through the EASY Minds Program. EURASIA Journal of Mathematics Science and Technology Education. https://doi.org/10.12973/eurasia.2017.00690a

     This article is part of a larger study on the effectiveness of the Encouraging Activity to Stimulate Young Minds (EASY Minds) program in New South Wales, Australia. The research presented here focuses on the teacher and students perceptions of the program after a 6-week experience. Interviews were conducted with 66 upper primary (elementary) students and 4 teachers.

    The EASY Minds program was designed with two purposes: to increase physical activity through the school day with a specific focus on mathematics lessons and to increase student engagement with mathematics tasks in primary school. Prior to the study, the program underwent a successful efficacy trial. The study upon which this article is based was done in a randomized controlled trial. Teachers that were to implement the program received one day of professional development prior to implementing lessons. The activities in the program are designed to use physical activity to promote fluency of number operations and to use places in and around the school to look for mathematics. The program is based on a pedagogical model from the New South Wales Quality Teaching model in which lessons entail intellectual quality (IQ), teachers provide a quality learning environment (QLE), and have meaningful lessons for students or “generate significance (SIG)” (p. 1656).

    The larger study included 8 public schools in New South Wales, Australia using grade 5/6 students in intervention (6) and control (4) groups. In addition to the professional development mentioned above, teachers received a packet of resources and equipment to support lessons. Data was collected using a focus group methodology for the 66 students and interviews for the four teachers and both types of events were digitally recorded and transcribed. An inductive qualitative analysis approach was used by a researcher independent of the program and data collection.

    The analysis identified themes to student and teacher perceptions of the EASY Minds program. Both groups indicated that the enjoyment of mathematics and engagement had increased. Students also noted that the activities felt freeing and that the expenditure of energy helped them learn. Higher performing students noted they were able to take mathematics to new levels and students who tend to struggle found the hands-on format allowed them to understand concepts. Additionally, students noted that they felt their teacher enjoyed teaching mathematics more than before the program was implemented. Teachers observed that students were not only more engaged during mathematics lessons but also after the lessons. The focus group and teacher interview data revealed that the program met the required components of the NSW Quality Teaching Model for IQ, QLE, and SIG. Students’ statements reflected aspects of higher order thinking and deep understanding and any criticism was related to struggles with the physical activities (like hitting a target with a bean bag) distracting from mathematics content. In regard to QLE, data showed that students identified components of self-direction and social support. The lessons also supported SIG through a feeling of connectedness between the activities and the real world.

    Some specific teacher observations were noted about the movement-based lessons. Teachers recognized that the lessons took longer to prepare than other formats but they felt they might become more fluent at preparing these kinds of activities with practice. Additionally, teachers noted that they had to be “more creative and forward thinking” (p. 1666) and felt they would continue with the program after the study. The idea of extending the program to other subjects and to the whole school was communicated.

    In discussion and conclusion, the authors note that this study confirms and explains findings of the larger study of increased engagement in mathematics lessons. They connect previous studies’ findings that teacher attitude in regard to mathematics is a major predictor of students’ attitudes to how the teachers and students in this study found increased enjoyment of mathematics. The authors also note that the day of professional development contributed in part to the success of the EASY Minds program. Limitations such as cost and availability of equipment are discussed. They conclude by stating that quality teaching and learning is supported by movement-based activities being embedded across the school day.

Week 4 Follow Up - Obsessed with Tessellating Trapezoids

I continue to be fascinated with tessellating trapezoids to the point that I see them inside my eyelids when I go to bed and close my eyes!

Then, this week I did a 2 part Scratch coding workshop through the Edith Lando Virtual Learning Centre (Carol was there in part 1, too!).  The presenter showed quite a few examples of coding geometry concepts and other math concepts.

I had caffeine on "today" (really yesterday, it's 1:30am now) (which I have become so sensitive to for some reason that even decaf can give me a slight buzz) and after going for a late night walk to get steps in for the Meandering Math Teachers Walk for Joy team (Week 1 - 43rd out of 129 teams) and finishing doing my reading, I didn't go to bed but played on Scratch.

So - next rabbit hole - could I get the sprite to draw tessellating trapezoids?

Quilting the trapezoids last week helped with this a bit because I had some insight into how the thread or pen could move continuously to draw a couple trapezoids and some sketching got me to how to code 6 sides and the remaining open side would be closed with a repeat.

So here is my late night ala-Mozart creative time product: