Week 7 - Mathematics and Poetry

Growing Up

Once

tiny,

now big.

In a blink,

fully formed people.

They’re ready to tackle the world.

The connections between mathematics and emotion is resonating with me this week. I have a daughter who is 13 and a son approaching his 16th birthday. My mind is blown - so many emotions lately - proud and happy but also sad.....how did this happen so quickly? As someone who hasn't written a poem in a really long time, I was surprised that my Fibonacci poem was quite simple to write. I think it's because I have so many feelings and words about the topic. It matters to me. It would be really interesting to write a mathematical poem about their growth, or patterns, or time..... so many possibilities.

In one of the viewings, I watched Mike Naylor recite The Last Crumb. He weaved together an amazing poem, in the style that reminds me of Dr. Seuss. I was smiling the whole time, so fun to listen to the clever rhymes, brought together by interesting visual elements. The story in the poem explored measurement concepts - time, space, and fractions. It made me realize that it's not only the structure of poetry that can be mathematical, but the actual content itself.

In terms of connecting emotion, I was really taken aback by my reaction to Amy Uyematsu's A section from Praise for the Irrational, and Countdown as well as Eveline Pye's The Shapes We Walk In.

In Countdown, Uyematsu's poetry reveals statistics and mathematical connections to school shootings. Very powerful. It reminded me of some of our discussions in Cynthia's course on Data Feminism - how data is not neutral, how it communicates viewpoints and evokes emotion. I was also drawn to Pye's poem about how people physically move about in Covid riddled world - The Shapes We Walk In. I was struck by the imagery evoked by the statement "magnetic repulsion: contact can kill". 

The additional reading I did this week is very applicable to my job, as it is written by teacher educators who decided to use poetry to inspire students' mathematical development and confidence. Radakovic et. al (2018) noticed that many of their students were math averse and/or carried anxiety about teaching mathematics. I can definitely relate on many levels. As a new teacher, I was math anxious. And now that I am contributing to teacher preparation, I find many of my students carrying similar fears. My ECE students often feel they are stronger in languages and/or the arts but I've never really explored the connections between these strengths and mathematics. The paper I read is definitely an a-ha moment. Students explored poetry - both analyzing and creating mathematical poetry about topics that were meaningful to them. Some wrote serious poems, others played with comical, everyday situations such as the poem below (Radakovic et. al., 2018, p. 4).

The author's have inspired me to consider how I could use poetry to create a safe and meaningful place for mathematical explorations. It is a space where everyone can bring their unique personalities, interests, and emotions - things that are not often seen in traditional math classrooms.

 Week 6 - Dance, movement and mathematics

I had a lot of fun and some a-ha moments with this week's module. When I looked at the resources, I thought right away 'I've seen many of those before'. However, I pushed past my 'been there, done that' attitude and enagaged with the viewings. We had watched the TedTalk before, but I found myself understanding aspects of it that I hadn't thought much about before - such as the timing of movements and predicting when various counts will meet back up. I think that after doing a similar movement task myself last week, I could relate better. The rhythm of math program is also not new, I actually shared it on my blog last week. But, after trying the dance sequence today, I rewatched it and again and could really see and feel the possibilities. I guess this is embodied learning in action.

I wanted to dance this week, so I cajoled my 13 year old daughter to help me. Actually, I bribed her with sushi before her volleyball game. We are both quite athletic and co-ordinated in sport, but she has a solid background in TikToking (she would roll her eyes at my use of a verb here). I think it helped....

Here is our set-up (my dog wanted to play too)

We only practiced the sequence once, and here is what we came up with, mistakes and all. She had it - I got lost a few times. The main problem was the sequencing - the pictures went up, down, up, down, rather than straight across. I would definitely cut them out and give students a chance to arrange them in ways that made sense to them, especially when we were backwards - having a direction on the other side would be helpful.

I can't seem to be able to put my video in here so I uploaded it to youtube - https://youtu.be/4aDjW18UNQk

 In the article I read by Campbell and von Renesse (2019), university students explored inquiry-based mathematical explorations as part of a liberal arts program. The teacher's goals included building community, positive dispositions towards mathematics, problem solving and reasoning skills, and mathematical confidence. Students were responsible to collaboratively explore problems and come up with proofs using various representations. The students explored maypoles and were tasked with representing the ribbon patterns to create theorems and prove them. A variety of representations were explored including tree, letter and screen, As they explored, they weighed the pros and cons of each, creating justifications. 

In connecting some ideas from the math dance and the reading, the written directions of the math task and the representations of the maypole patterns remind me of coding. It would be really interesting to have students create/code their own dance sequences and offer them to other students to try - to see if their sequences make sense. They could try various representations, test them out and have to justify which sequences and codes work the best and why.

After our dance, I asked my daughter how it might connect with math. She said she didn't know, but maybe something about patterns. The wonder I am left with is how to (or even whether to) make the math visible to students. Do we have to say 'this is math because....' or 'this is similar to when we....' or does the value simply lie in the exploration itself? Maybe we don't have to formalize everything.

 Week 5 - Embodied Modalities

I definitely had some fun playing around with movements this week as I explored movement patterns in Sarah Chase's video and made connections to Kelton and Ma's (2018) article entitled Reconfiguring math settings and activity through multi-party, whole-body collaboration.

As I stood in front of the mirror waving my hands in various configurations, I found it to be very awkward and even a little frustrating. It seemed so much more difficult than I thought it would be. I wrongly assumed my gymnastics and agility background would help me, especially in the area of cross-body movements and the task of co-ordinating one side with the other. Cross-body movements have many sport benefits (balance, core engagement) but they also help co-ordinate the left and right hemispheres of the brain, increasing focus and retention of concepts. However, when I tried to partition movements, doing something different on both sides of my body while counting sequences and movements, I lost track and had to start again quite a few times.

While I work with adults now, I still spend time with young children in our lab school, and I wondered how our young learners would be able to navigate these types of movements. I wonder if it would be easier for them, since they are so embodied with everything they do, or whether it would be too much. I suppose I would start simple and ramp up the complexity based on their engagement.

The process reminded me of a program I saw a number of years ago called Rhythm of Math. Has anyone heard of it? This is a video that briefly explains the program - https://www.youtube.com/watch?v=1DAyUaS23lo and here is an example of a lesson - https://www.youtube.com/watch?v=zjPgZIvYXPQ. I think it may be a good example of what Susan was discussing in the module introduction - a way to bring together embodiment with the more formalized elements of mathematics.

I also appreciate the social context of the Rhythm of Math videos, which supports many of the ideas in Kelton and Ma (2018). There must be so many benefits to doing these tasks in a group - you can see your movement patterns in others - almost like a real mirror. When exploring patterns or multiplicative thinking, there is so much possibility within a group, as well as the energy and problem solving that comes from bouncing ideas around.

In Kelton and Ma (2018), they explore two case studies from a contemporary perspective that connects social space, embodied cognition, and mathematical ideas. Where embodied cognition is often studied from an individual perspective, this approach recognizes the social aspects of learning and being in spaces. We create stories as our individual bodies move through  space, but also as we move together with others in activity. 

In the first case study, students worked through activities involving a giant number line; a "walking scale number line" (Kelton & Ma, 2018, p. 183). The physical movement became a tool for meaning making in itself, and seemed to go beyond just performing operations. "Quantitative relationships were understood and talked about as spatial relations between students' home positions and bodies" (Kelton & Ma, 2018, p. 184). The second case study examined students' exploration of pairs task called whole and half, which uses the body as a tool as well as an object of measurement. As students' used their bodies to represent part-whole relationships, the mathematics became dependent on not only their physicality but also their social dynamics; opening the door for multiple problem solving opportunities.

Future directions/lesson ideas

I am thinking about Sarah Chase's video, which was modelled and practiced individually. It would be very interesting to see this unfold as a collaborative experience with young learners (preschool - grade 1). We could explore various patterns and shapes that could be made with our bodies, challenging the children to create patterns with different counts on each side of their body. It may be challenging for some, but the experience would be easy to differentiate. Younger learners could create shapes and patterns with their whole bodies, making comparisons to others, challenging others to try to make their shapes and patterns. I could also photograph their movements. They could look at the photographs and attempt to recreate the patterns and/or make predictions. We could splice them together and play them back to see if the sequences make sense and why. Over time and experience, children could start to apply the more formal aspects - writing corresponding equations and/or drawing representations of the patterns and operations. I am also thinking about the spatial aspects - I know a lot of the young learners have a hard time recognizing that a shape stays the same, even when flipped or rotated. If we took pictures of their movements, they could physically flip and turn them to make comparisons. 

What do you think? I haven't tried these types of explorations with such young learners. How might this unfold in 'real life'? 

 Week 4 - Mathematics and the Arts

I have always thought there were many similarities between mathematics and the arts. Certainly many forms of art embrace complex notions of shape and space, as well as patterns - the language of mathematics. There are also similarities in how math and arts are perceived. As a young learner, I had challenges with both. The myths about "math people" certainly parallel the conversations about "being artistic". Even as recently as last week, I have continued to perpetuate this thinking by joking about my lack of artistic ability. I will have to think more about how I can embrace artistic ways of being; having more of a growth mindset in this area. 

In Dylan Thomas: Coast Salish Artist by Dylan Thomas and Doris Schattschneider, we learn of some other connections. There has been a recent resurgence of Coast Salish Art, and that while the pieces are often very modern, they carry traditional stories, symbols and techniques. "Their works, although modern, are rooted in traditional forms and employ the distinctive design elements of circle, crescent and trigon to outline and to decorate the interiors of their stylized figures of humans, animals or supernatural figures." (Thomas & Scattschneider, 2011, p.199).

This week I was drawn to Ligia Unanue's The Island - mainly because I was quite interested at how my eyes would fixate on different shapes, then I would notice more shapes - all linear patterns but also circular and 3-D, which is so interesting. The way the piece works as a whole, but also has each segmented part is fascinating. I spent a lot of time just looking at the shapes, and how they seamlessly meld into other shapes and patterns. The author explains "It is perhaps a dome, a mocarab, a mandala, a three-dimensional geometry, a kaleidoscope, a big puzzle or all at once."

I had fun sketching different elements of this art piece, and each process taught me new things. First, sketching something multiple times and from multiple perspectives gives you new insights. I think at first, I was fixated on a small segment of the work, trying to replicate it but not understanding the whole. This reminds me of systems theory and how last week we learned about how the principles are embedded in Sustainability Education. This also reminds me of a puzzle - you may focus on one section at a time, but you always have an eye on the whole.

Last week, my geometric tile drawing was done freehand and I was frustrated by the lack of precision. This week I started with a quick freehand drawing of a singular segment on blank paper. Then I moved to grid paper to see which helped me understand the work better. 

Picture 1 - Initial freehand drawing - lack of awareness of the whole, imprecise

Picture 2 - pulled into the pattern - didn't pay attention to the artwork, just replicated the pattern - this helped me understand the connections between lines, shapes and angles

Picture 3 - Futile attempt at trying to make the pattern more circular. A challenge for the future. I feel defeated looking at this - like it's a disservice to the original work. But, I'm trying to embrace the process rather than the product.

There are also a variety of circular artworks in Thomas & Scattschneider (2011), such as the one below (left). "One of her commissioned works is Flight Spindle Whorl [9], in the arrivals hall of the Vancouver International Airport, a giant 5-m circular contemporary red cedar carving depicting two eagles, two human forms, salmon, and the moon, sun and earth, all representatives of the Coast Salish people. It is mounted against a flowing sheet of water, representative of the rivers plied by the Coast Salish." (p. 200).

On the right, we see a mandala. I am so glad I read this article - the description helped me to further understand some deeper mathematical ideas embedded in the work I tried to replicate this week (such as the geometric placement of circles and squares; down to the way the spaces in between shapes are filled to produce on overall symmetry).

 

 I am drawn to these pieces for many reasons. Similar to the other art I studied this week, they are circular and dynamic - appearing to be in motion. This is a way of seeing the world that I am spending more time thinking about lately. Boundaries and rules can be more fluid and contextual; math and the arts don't have to be binaries. This reminds me of Susan's comment on my post from a few weeks ago - that there is not a perfect pedagogy that will lead to complex mathematical understandings - there are many approaches, many paths and many ways of knowing that can intersect and weave together.