Week 8 - Activity - Paper folding: Miura Origami and more

I may have a new hobby!

Well, on Wednesday night, I sure didn't feel this way.  I used the first video Susan linked for us for the Miura Origami but got quite confused when the demonstrator's hands went off the screen.  I made some mis-folds which made things more confusing and the paper was becoming soft and I was ready to give up.  Here is the result of the first attempt:

First attempt at the Miura Origami type folding

I walked away to do dishes and other chores. I couldn't quite let go of the paper folding conundrum so before I went to bed, I decided to watch and work with the second video that Susan posted. Ah...much better.  Fewer vertical folds and bigger "pieces" to push in and out from a similar sized paper. Apparently I needed this scaffolding and the result was crisp and the experience satisfying - I could go to bed feeling accomplished!  Here is what had me feeling this way:

Fewer starting folds was more successful

Well, I didn't actually go to bed - I should have gone to bed.  However, YouTube was suggesting videos to watch and I came across an art teacher (Marquette University High School) who has lessons for his students posted online (starting spring 2020). I went to his YouTube channel and looked at his video library. There was a series of videos called Cut-Fold that caught my eye:

Peter Becks video library on YouTube

I noticed that the Cut/Fold Vol. 1 video image was a design that Cynthia showed me that was her "room" for the Virtual Family Math Fair this year. I browsed through a few videos and discovered that Mr. Beck references some paper artists at the end of Vol. 5.  I had to check a couple of them out online.  I'll talk more about that later.

After staying up too late Wednesday night, I really thought I would be in bed early on Thursday night, especially since I had to be online at 7am Friday morning for a full-day workshop with Katie White. Walking by my work area, my paper folding caught my eye and I decided I needed to try another fold. I first started with the Cut/Fold Vol. 1 and got this:

And I couldn't leave it at that.  I returned back to the original task, this time using the number of initial folds on my first try but employing a larger piece of paper. This experience was so satisfying! I started understanding what I was doing and what would be coming next. I had another video playing in the background and I heard that demonstrator say something along the lines of "The paper will want to do what you want it to, it just needs some prodding." That's how this attempt felt.  At least down to the last "row" which was the most difficult because there was nothing to stabilize it below. That video I was listening to also taught me that paper stretches, especially on the fold lines (but origami paper less so - it is designed to maintain more of its original structure) and that what these folds result in is "tessellating surfaces" - cool term, right? Here is the result of staying awake too late on Thursday night:

When the tessellated surfaces are all folded up, it reminds me of the start of a hexaflexagon!

When I was in my online workshop on Friday, my folded paper designs were next to me. Sometimes, I want to keep my hands a bit busy when I am listening and not note-taking (hexaflexagons have been good for this!) and I took a small piece of paper left from cutting down other paper and started folding. This wasn't successful - my fingers are big and the folds were small and the paper was getting soft as it did on my first attempt. I think it could be really cool to fold a small version but I think I need a "bone" folder to help crease the paper tighter and "punch" the folds in the right direction...or maybe stronger paper?

Tiny paper folding - got too soft to work with

OK, so back to those artists I researched.  Below are two that caught my particular attention.

The first is Li Hongbo who makes sculptures out of folded paper. Some of them look solid but they can be stretched out in all sorts of amusing ways.

from https://www.instagram.com/lihongboart/?hl=en

from https://www.instagram.com/lihongboart/?hl=en

Watch this one-and-a-half minute video if you have the inclination:

The second artist is Matt Shlian. He is a sculptor, too, who works with paper and other materials.  Here are a couple of his paper designs from his website Matt Shlian :

Aren't they incredible?!

And my favourite find - Matt Shlian on Sesame Street making paper sculpture with kids (it's just one-and-a-half minutes - take a look):

I have looked online at DeSerres art supplies - they have cutting mats, knives, and folding tools in stock in Vancouver.  This might be my reward for finishing my reflection post later!

Week 7 - Critical Reflection - Mathematics & poetry and novels

 Before reading further, if you haven’t read my reading summary, please read at least the summary poem “Poems are made by erasing.”

    This may have been my favourite week of the course so far; at least that is what the hours I have spent playing with and writing poetry this week tell me. It might be tied with the week were we re-created art and I got sucked into flow with tessellating trapezoids. My brain (and body) is a bit exhausted from late nights not noticing the clock but still needing to get up in the morning. Please forgive me if my writing is discombobulated in this post.

    When I reflect on my classroom practice with poetry, I see that I connected math at least implicitly, if not explicitly. I admit I was somewhat neglectful of having kids engage with reading poetry (but when I had teaching partners, they were good at that), but loved exploring the writing of poetry with them. I had a book from Scholastic that I used a bunch that taught different forms (haiku, cinquain, limerick, acrostic, ballad, and others) through examining a few examples and then scaffolding the writing (then I had students develop a book of their own poetry with at least one example of each of these and a bunch more of their choice). We were mostly counting and noticing patterns – of lines, syllables, stress, rhythm. So, I can see the mathematics connected to poetry through these experiences but this week opened up my understanding to more ways that math and poetry live together in this world.

9x9square poem by JoAnne Growney
cited in Glaz, 2019

    I appreciated how Susan, in the introduction this week, laid out three ways that “mathematics inspires poetry and novels” (p. 1): they might be about mathematics or mathematicians, they might connect mathematical ideas/concepts and human experiences (analogies, metaphor), or mathematics might be used to create/shape the writing or to analyze it. Without this guidance, I may have needed more time to figure out what mathematical poetry (my focus, I didn’t examine other literature this week) is or is not. I am not sure I have it completely figured out but Susan’s ideas helped with listening to and reading (and creating) poetry this week. I found some examples of these in my reading/viewing/creating this week. The understanding that the subject matter is what can make a poem mathematical is demonstrated in Sarah Glaz’s poems on the Bridges Math and Art 2021 Virtual Poetry Reading website. The two poems Death of Euclid and Archimedes are about the title mathematicians. (I suspect there is something mathematical about the structure of these poems as well, but I didn’t examine them that way.) Amy Uyematsu’s poem Countdown brings together counting and numbers to ‘deal with more serious political issues’, specifically shootings in the U.S. To me this is connecting mathematics and human experiences. I think the final idea, that of mathematics informing the structure of poems, is what I enjoyed exploring most through my own poetry but also finding it in others (when it was obvious). Daniel May used the Fibonacci sequence to structure his poem A Fibonacci Poem for the After where the poem grew in syllables and then contracted again based on that mathematical pattern. In the interview by Sarah Glaz of JoAnne Growney, there are a few square poems where the number of syllables on each line is matched by the total number of lines (e.g., 5x5square or 9x9square). I enjoyed creating Fibs and played with having a pattern of Pascal’s triangle structure a poem of my own creating. When creating found poems as my reading summary, I tired a 9x9square poem for the Romania section (as the poem in that section was of the same type) and it was really difficult (I had the added constraint of using someone else’s words) and time constraints had me abandon it. I am sure if I had dedicated more time to examine other poems, I would have uncovered intriguing ways mathematics structure them.

My poem structured on Pascal's Triangle

    It seems like we can define mathematical poetry pretty broadly (though some people want more constraints than others – I felt JoAnne Growney and Sarah Glaz differ on what is and isn’t mathematical poetry). So, to that end, my question is:

What is NOT mathematical poetry?

Notes for the 'stop' at 
"Poems are made by erasing" in Glaz 2019

    One of my biggest stops, big enough to involve a whole page of stops and scribbles, was when I read the section in Glaz’s interview with Growney titled “Poems are made by erasing.”  Before I had read more than the title I had scribbled “I erased so much or scratched out --> connection to math – draft solutions, back up, try something different – then make elegant.” The poem this section is framed around is called How Did It Come to This? and references Picasso’s art as “notions of a bull…in the end a few lines” (Glaz, 2019, p. 251). In the interview, Growney talks of a writing prof who shared her own draft in class and said, “’That poem has one-third too many words’” (p. 251). Here I scribbled, “Like Mozart – Eugene Oschady.” Eugene was a cellist in the VSO and a member of a local string quartet. At a quartet performance, he talked to the audience about how Mozart’s music was criticized for too many notes but Eugene argued that Mozart’s genius was knowing when he had the exact right number of notes. He played a motif from Mozart’s Symphony No. 40 and added just one extra note in the middle and end of the motif and it made the audience chuckle because sounded silly. Ironically, when I composed the found poem for this section of Glaz’s interview, I had way too many words and was going to add more. I stopped, thought about it, and scratched things down to the minimum that would get the message across.

Editing down "Poems are made by erasing" found poem

    The connection to math education here is the idea of math drafts.  We allow students first, second, third, or more drafts in writing. When do we do that in math class? I was introduced to the idea of math drafts in an NCTM webinar that I only caught the end of and haven’t gone back to watch the whole thing. However, the part I did see gave me a good idea about the concept. The idea is that students take a break from a problem and circulate the room looking at other students’ incomplete work, offering feedback and gathering inspiration for their own work. They may go back to their own work and scratch things out, start again, or have a new idea that breaks them out of being stuck.

    We can also give feedback to students on being more elegant or efficient. I don’t think elegance or efficiency should be the first goal when students are learning a concept, but as they are ready, they can develop this. For an example, I had a student who understood division using the partial quotient strategy but would only divide by 2, 5, or 10 (as I remember it – anyway, the numbers were small) even if dividing by 100 or 200 or 500 might have been more efficient. She didn’t mind working through long columns to get to her answer. I could tell she understood the concept and was able to slowly nudge her to increasing her initial numbers to be more efficient. I think this is like erasing words in writing poetry – getting rid of the extraneous to get at the essence. (If I had the inclination, I’d edit this blog post back by 1/3!)

Questions to ponder

What is NOT mathematical poetry?

What similarities do you notice between the creation of poetry or other writing and working with mathematical problems and solutions?

I’m curious to know what possibilities primary teachers see in writing mathematical poetry with students who have, generally, a smaller lexicon to draw from and relatively limited experience with math patterns and concepts compared to older students.

Week 7 - Reading Summary

Glaz, S. (2019). Artist interview: JoAnne Growney. Journal of Mathematics and the Arts, 13(3), 243-260. DOI: 10.1080/17513472.2018.1532869

 

This reading is an interview conducted by Sarah Glaz with JoAnne Growney at the 2017 Bridges conference and through follow-up conversations. It is framed around 10 of Growney’s poems in 9 sections.

Growney talks about exercise and a good night sleep both being ways she clears her mind when writing poetry.  When I was part way through reading the article, I needed to get out for a walk, despite the quite chilly evening; my body was tired, my brain was tired, and I needed steps for our Meandering Math Teachers Walk for Joy challenge team!  The walk worked.  Maybe too well. Not only did I finish the reading, I decided to do my summary. And not just a quick summary, I had to follow a creative impulse to summarize each of the 9 sections in a found poem using text from the article (and now it is 12:45am and I am just typing this up from my hand-writing). I tried to use Growney’s words as much as possible but sometimes Glaz’s slipped in.  The introduction is made from purely Glaz’s words.

I consider the poems drafts. Some were re-worked but remain drafts. Others are what developed on the first try. You’ll understand why I point this out when you read my reflection.

 

Introduction

Mathematics and poetry:
complimentary, supporting,
conflicting.
C. P. Snow described and lamented
the division.
Many of us seem to have crossed
the great divide -
the creation of a community
of both poets and mathematicians.

 

On the farm

One hundred forty acres,
a white frame house,
a good woman worth three good women.
There isn’t really any history
of advanced math in my family
or of poetry.
A career to fall back on –
I had gotten too much education.

Four teachers

Flamboyant, silver, fearless woman who ahd gone to Columbia.
The geometer had three daughters – kind hearted man.
The one who loves Old English and late-night TV – each thing is our teacher.
A stressful teacher made errors, demanded we write perfect mathematics in perfect English sentences.

 

 Everything connects

Everything connects is the way my mind works
The history of the vacuum cleaner, Roe v. Wade, and Billie Jean King
I put them all together.

Nervous energy, I need to clear my head
Exercise, a good night sleep
The clutter gets cleared.

 

Poems are made by erasing

 Early drafts have far too many words.

I read a poem aloud,
I feel how many words
need to be eliminated.

  

Romania

A tiny organization
'Teachers for Tomorrow’
for teachers to help students
learn English
in Indian and Romania.

I mostly presented poetry.
We read,
Puzzled over,
Discussed
English translations
Of Romanian poetry.

Found a poem
Related to mathematics
By Nichita Stanescu, a particular favourite.
Delighted!

  

Math and poetry – the perfect blend

Play with both meanings,
mathematical and non-mathematical.
Six is a perfect number –
the sum of all its proper divisors.
Six is a perfect number
of husbands.

  

Counting the women

Mathematics -
a man’s world.
Learn how to live in it
in terms acceptable to men.

Look around the room.
If you can’t count how many women
in one glance,
smile.

Intersections – poetry with mathematics

Collecting math-related poems
Tried the blog
It just kept on going
Makes mathematical poetry
More welcome
Visible
Threads of a mathematical idea
Pattern
Structure
Counting
Recursion
Permutations
Sometimes silly
Sometimes evocative
A back-and-forth
Between my poems
And the poems
Of others
A meeting place

Week 7 - My Mathematical Poetry (activity)

 I started my mathematical poetry exploration with Fibs.  I am playing around with PH4 and a couple other ideas; playing around enough that dishes are unwashed and tomorrow's lunch is yet to be made.

I'll start with two of my Fibs and edit the post should other poems come fully to life.

*Edit - added another Fib later the same night so now there's three!

Can you figure out the inspiration for this Fib?

*Edit - Last night after posting the Fibs, I played around with structuring a poem on Pascal's triangle:

I decided to pattern the syllables of a poem on the first 5 rows of Pascal's triangle.  It was a very interesting exercise, not in the least for having to find a 6-syllable word that worked with the subject!  The inspiration was the weather I saw when I looked up from playing with mathematical poetry and peered out my window!  I finished the poem tonight after giving my brain some incubation time to play with words and to evacuate that earworm that was invading all my thoughts!

Here is my Pascal Poem:

(I'll finish my reading sometime - just not tonight!)

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...

 

********************************

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?