Week 7: Mathematics and Poetry

Summary and Application of Writing and Reading Multiplicity in the Uni-Verse by Radakovic et al. (2018).

The article showcased a unique style of poetry that reminds me of a “Scale of the Universe” (a really cool video visually showcases proportions and scales). The poem starts at a 1m level and expands to the power of 10. The great thing about this type of poem is that it allows students the opportunity to personalize and offers them a narrative voice. I wrote my own to model the style portrayed in this article.

Within the stretch of a metre,

Is Jax’s soft fur beneath my fingers

Within the span of ten metres,

Are the endless stacks of papers and marking

Within a spread of a hundred metres wide,

Are signs of possibilities in the backyard, my place of zen

Within the distance of one kilometre

I strive to reach from end to end in 6 minutes flat

Within the commute of ten kilometres

Are starting points to a number of serene mountains and trails

Within the leisurely drive of a hundred kilometres,

Are the shores of my favourite camping scene. 

Within a flight of a thousand kilometres,

I am surrounded by the little arms of my niece and nephews. 

Within the journey of ten thousand kilometres,

Lies the origins of my parents, grandparents and generations before

Within the space of a hundred thousand kilometres

Orbit the satellites that source life to devices that are always 0 meters away.

This was quite enjoyable to write, and I love that it allows “students to make real connections between mathematical measurements and their lived experiences” (p.3).  While I wrote this poem, I noticed I had to research some of the actual distances to determine if the values were actually accurate. This is great for students to learn about the accuracy of the information they present. Ultimately, this allows students to create ‘textual writing’ where there is “interpretive space” that link to a reader’s knowledge, experiences and construct meaning through experiences. 

Activity PH4 Poems: 

I followed Susan’s braided PH4 poems. I’m not a poetic person, so this was more about following a pattern, less about evoking strong emotions from my rearrangements of letters. Though I did play around with it visually and thought it would be cool to see it as a pattern or weave…I enjoyed that this process could also be transformed visually, not just poetically.

 

Week 6: Readings (Post 2 of 2)

 Viewings and Readings Vogelstein et al. (2019)

Summary

In this article, the authors conducted exploratory analyses between three cases where groups utilized choreography recordings (videos) from the Rio 2016 Olympics. These groups of four were then asked to make observations and create their own dances, incorporating similar props. The design of this task contributed to embodied mathematical research on multiple levels including foraging, dissecting and reenacting” and ensemble learning.

A scene from the opening ceremony of the Olympic Games in Rio de Janeiro. Credit...Doug Mills/The New York Times

Foraging and Dissection Public Media to Design for Creative Reuse

• By using videos of public performances, it allows educators and students to “dissect” content for their own creative platforms. 

• Useful for cultural performances, where there can be “a mix of styles and ideas and aesthetics” (p. 332)

• A “dissecting environment” can create a meaningful connection between cultural performance and mathematics 

Ensemble Learning

• Large scale performances, where learners are required to work collaboratively. 

• Propose situations where working together is essential for both performance and learning.

• Example: Flipping a sheet to wave, if one participant is not involved, the act cannot be successful

• Links to ‘collective mathematics’ where “social and interaction structures of groups are used in generative ways to produce and explore mathematical structures”

This was a lengthy article but what I took away from this week’s article is the significant role that each student plays in embodied learning. It’s not just about the movement, but also their involvement on a larger scale, being a significant piece, a key element. According to Schaffer and Stern (2012) “we tend to think more effectively with spatial imagery on a larger scale”. As well, the role of a physical prop allows for students to learn from “viewing” and “doing”. To actually construct meaning through trial and error, especially when creating a hybrid that bridges mathematics and dance. This allows them to explore limitations to map possible connections between performance structures and mathematical structures. 

This article is relevant to my own learning (and as a teacher). Stella and I hope to focus on traditional Chinese and Tawainese Ribbon Dance for our final project. Similar to this article, we would integrate the use of a prop (silk ribbon) where students can utilize dance sequences and movement to generate a performance as a group. In this potential activity, elements of foraging and dissecting will come into play. What can they re-enact? What can they not do? What challenges do they predict? How does the prop itself affect the performance (length of silk, type of material, width, etc)? 

Lastly, Schaffer and Stern (2012) also note, that dance is not meant to sugarcoat mathematics, but it is the “connections between that are the heart of the matter”. 

 

Week 6: Mathematics & Dance, Movement, Drama and Film (Post 1 of 2)

Activity "Rope Polygons" 

For this week’s activity, I chose to modify Rosenfeld’s “Rope Polygons”. This fits in nicely with my two units on Geometry (Angles and Polygons; Perimeter, Area, and Volume). It is a kinesthetic way for students to model these shapes and to integrate core competency skills. I have also included questions for students to consider and explore. (Note: I teach at an International Baccalaureate (IB) school, so some of the terminologies stem from their curriculum).

Table 1: Ideas and questions for Group Participation

I tried doing this activity on my own with some string. What I noticed was that it was much more difficult to create regular polygons as I had limitations to the angles and distances between my fingers. I feel if students were asked to do this, there would be a sense of struggle and perhaps more engagement to think creatively. This activity did remind me a bit of the game "Cat's Cradle", where depending on how you play, can result in various string figures.

Week 5: Mathematical Pedagogies - Riley et al (2016) SUMMARY (Post 2 of 2)

 Summary: Riley et al (2016) “Movement-based mathematics: Enjoyment and engagement without compromising learning through the EASY minds program” 

This article showcased how there is a “worldwide decline in interest and achievement in mathematics in young people” (p. 1653). This is especially prominent in middle school students, where a significant number find traditional teacher-centered approaches to teaching as disengaging (as cited in Attard, 2013). Integration of the Encouraging Activity to Stimulate Young Minds (EASY) program, which connects physical-based activities to “enhance learning and engagement in mathematics” (p. 1563) was the focus of this study. Not only would the students’ level of physical activity increase, but a number of other studies also show linked improvements to childrens’ learning outcomes (as cited in Donnelly & Lambourne, 2011). 

Four teachers and 66 students participated and were then interviewed over a 6-week intervention. Students were chosen from Grade 5 to 6 across 8 public schools in New South Wales, Australia. They were either placed in an intervention or control group. Teachers were given a one-day professional training and resources package with some lesson examples to “encourage creativity, autonomy, and ownership of the lesson content” (p. 1657). Three lessons were conducted 3 times weekly over a six-week period.

Some examples of the EASY program included activities that used “physical activity as a platform for the development of procedural fluency of fundamental operations” (Riley et al., p. 1656). It also focused on mathematics from the real-world. 

A focus group methodology was utilized, 66 students in 11 focus groups. Semi-structured discussion questions were designed and administered for the student-focused groups. Findings from the thematic analysis showed that both teachers and students had increased enjoyment and engagement in the mathematical lessons (p. 1660). Though there were different types of physical activities introduced, “rotating activities - hop, skip, jump, recording averages, times table while jumping through ladders” were commonly preferred (p. 1660). Engagement from “expending energy”, being outside, and away from the classroom were also other common themes that emerged. 

Reflection:

What stood out to me in this article was the statement from the students. Some students noted that their teacher seemed less stressed because they weren’t required to control classroom misbehaviour as much and because the students were also excited. Others note that they were weaker in skills like tables and estimation. But through the program, they had to concentrate on the numbers and the physical task as well. This parallels the message by Chase (2012) on dance movements, that it requires greater focus and concentration to conduct both. 

This stood out to me because so many students struggle to sustain focus, especially for a longer duration of time. I appreciate the article's approach of integrating this program into more advanced math applications, which still saw higher levels of engagement. The findings from this EASY program provided a quality learning environment that “is clearly focused on learning and develops positive relationships between teachers and students and among students” (p. 1664.)

Week 5: Developing mathematics pedagogies ACTIVITY (Post 1 of 2)

ACTIVITY: Hasan, Grabowski, and Hawthorne (2017)

(LINK to Google Doc for better quality)

For this week’s activity, I definitely felt ‘anxiety’ when I watched Chase’s video on rhythmic movement to different sequences. Somedays, I feel I can barely clap my two hands together, so for my brain to command different parts of my body to move in different directions, with different combinations, I could feel my heart racing! Though I appreciate her breakdown at the end of video indicating that this type of learning could reduce anxiety because learners become so focused they do not worry about what they look like, I think that may be applicable if the learner has some basis of coordination. For someone like me, I would panic!

So I gravitated towards Hasan, Grabowski, and Hawthorne’s (2017) extension of the Binary Code. I found the video and the description a little overwhelming, so I embraced what a student would feel and ‘made it my own’.

What resulted was a game, because I enjoy the attributes of ‘play’ and the creativity of turning a task into something playful. Keeping in mind the objective of using combinations, colours and arithmetic, I designed a system where students would have to determine ‘targets’ to generate points. The basics are:

Colour	Point Value	Position	Point value
White	1	Outer ring	1
Black	2	Middle ring	2
Red	3	Inner circle	3

The value of the overall target is dependent on a) the colour(s) used and b) their positions.


For example, the lowest possible “target” value is:

Then integrate the next lowest point colour (BLACK) into the outer ring. And calculate the point total.

Third lowest target point total:

If the pattern continued…

The next question that could be asked is, could more points be generated by repeating the same pattern with RED? Or to add black to the next ring layer?

So wait…could there more configurations with 9 points? 8 points? This would be a great task to figure out! While doing this, we can integrate the concept of BEDMAS, communication and patterns…etc.

And if we arrange all those ‘targets’ into a table…this is what we get!

As I was completing this table, I knew I was missing certain targets. But by observing the patterns, such as the rings, the values, I started noting trends in the diagonal direction. From there, I was able to figure out which images I was missing, and could PREDICT their points value even before I did the calculation. Playing with the targets in each column (point values), I’m certain there are other patterns that could come out of this as well.

This took A LOT of concentration. I could see how one could become so immersed in the combinations and arrangements, especially if more colours were introduced, or if a different point system was created, for example, finding the sum of RING + COLOUR instead of the product of RING x COLOUR.

Colouring by hand would be hands-on, but I liked using this format because I could move and arrange with ease. This would be good to do for students as individual sheets, or if collectively, students each created their own and could organize this into groups, etc.

Week 4: Mathematics and the Arts (Post 2 of 2)

Reading: Eve Torrance (2019), Bridges 2018, Nexus Journal  

Summary: The article “Bridges Stockholm 2018” by Torrence (2019), it focuses on the variety of presentations, highlighted exhibits, and themes throughout the 2018 conference in Stockholm, Sweden. I chose this as my art inspiration piece by Karen Amanda Harris, which was featured at this particular exhibit. What I took away from this article was the number of creative approaches the artists used to mathematical components. Everything from gardens to card manipulation, to bands (that included instruments made from underwear), were incorporated. I also appreciated the stories of key contributors, such as Marjorie Rice, “a woman with no mathematical training beyond high school” who was able to discover a “new pentagonal tiling” in 1976. At the conference, Colm Mulchahy integrated a magical component through card tricks, all stemming from math, despite having a cast on his arm! 

The article also mentioned some presenters and authors who I would like to further explore (I’ve linked some of their other studies to their names). 

• Hannu Salmi “Bridging the gap between formal education and informal learning”

• Minna Houtilainen “Neuroscience in understanding learning in the STEAM context”

• Paul Moerman “Dancing Math - aesthetic and math literacies intertwined”

Reflection
When I jotted notes along the margin of the paper, I noticed I constantly used the word “different”; “different perspectives”, “different angles”, “different mediums”. Partway through, I wondered, why do I use the word “different”. To be different would assume that they were not aligned to begin with. This week’s activity and reading have prompted me to re-assess my own perspectives and how I gravitate towards viewing mathematics and arts as separate entities or worlds. When the truth is, there are intertwined. The article caused me to pause and reflect on how this approach to mathematical learning compared to the standardized math classroom setting, where the most colour that is integrated is simply from the red ink used for marking. 

Applications

There is a greater acceptance to teaching mathematics while integrating the arts. Platforms such as Desmos offer beautiful renditions of student artwork composed of graphing functions and formulas. These activities prompt students to struggle and emphasize skills in problem-solving, creative thinking and as well, build resilience. There is more than simple repetition and math worksheets, and it is thrilling to see this more often in the classroom setting. 

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Figure 1: Desmos Graphing Art Contests 2022 Student Finalist

References:

Moerman, P. (2016) Dancing Math: Teaching and Learning in the Intersection of Aesthetic and Mathematical Literacy. In: Torrence, E., Torrence, B., Séquin, C., McKenna, D., Fenyvesi, K., Sarhangi, R. (ed.), Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture: Conference Proceedings: Bridges Finland 2016: University of Jyväskylä (pp. 269-276). Phoenix: Tessellations Bridges Conference Proceedings

Mossberg, F. (Red.) (2017). Child & Noise: How does the child percieve the sound environment? (Skrifter från Ljudmiljöcentrum vid Lunds universitet; Vol. 17). Ljudmiljöcentrum vid Lunds universitet.

Thuneberg, H., Salmi, H., Fenyvesi,K. (2017) "Hands-On Math and Art Exhibition Promoting Science Attitudes and Educational Plans", Education Research International, vol. 2017, Article ID 9132791, 13 pages, 2017. https://doi.org/10.1155/2017/9132791

Week 4 Mathematics and the Arts Introduction (Part 1 of 2)

Reflection:

This week’s activity resonated with me because it emphasized the significance of re-merging these two cultures together.  In Susan’s introduction, she notes that our “society often deals in binary”, portraying mathematics and arts as “polar opposites”. For most of my life, I often thought the same way. I loved fine arts as a child, and as a teen considered pursuing an art degree rather than the science/math route. However, cultural and family pressures dictated that one had greater value than the other, one could provide a stable career, so art had to be placed on the backburner. However, it’s intriguing to learn,  “there was not the concept of such a split between the arts and the sciences, even in mainstream Western society and among the academics, up until the early 20th century.” (p.2). My students are often amused to learn that if I wasn’t a math and science teacher, I’d love to be an art teacher. Through this program, and especially this course, I am realizing there is no need to ‘choose’ either but that both worlds can exist in harmony, and better yet, enhance the beauty in each. 

Activity:
For this week’s activity, I was inspired by Karen Amanda Harris’ “Broken Sphere” (Bridges Exhibition, 2018). I admired the symmetry in the outline of her piece and was captivated by her choice of colours and the filled segments she used to break that symmetry. The rays and angles that deviated from various points prompted me to think of so many questions I could ask my students: 

• What angles are formed? 

• What polygons are formed? 

• Where is the symmetry? Why isn’t it symmetry

Karen Amanda Harris: Broken Sphere

30 x 42 cm

Promarker, gel pen, ink liner, nail varnish and correction fluid on watercolour paper

2016

For my own design, I chose to deviate from her pen and paper medium and challenged myself by constructing a 3D version of her work using a board, nails, and coloured string. I started with a 12 x 12 inch board, nailed a center point, and roughly traced a circle using a plate. With a measuring tape, I marked 4 points, and then divided the difference into thirds, creating a dodecagon. I added nails to the four corners and then two more in between. Mathematically, there are a number of circle theorems, line segments, and tangent points I’m sure I could have discussed. However, I haven’t taught high school math in years so they aren’t popping in right now!

   

Figure 1 & 2: (Left) Nails around the circle template, creating a dodecagon. (Right) Segmenting the quarters into additional thirds by dividing the difference by three. 

I had no idea what I wanted to do with the string but had three different colours to use. There was no set design but I started by simply looping the string between nails. Once I finished with a particular sequence, I would tie off the string, change to a different colour, then proceed with a new pattern. Questions I considered as I attempted this art piece:

• Which points had the most loops of string? 

• What type of polygons am I forming? 

• How do I keep seeing these patterns? 

• Are there patterns or connections that I have missed? Many! But I ran out of string and space on some of the nails

• How much string did I use in total?

• Could this be expressed algebraically? 

Unfortunately, I ran out of two of the strings, so the layers were not as clear. I did realize that as I continued the process, there were even more questions that could be asked, depending on the angle the art was viewed. Perhaps in the future with more mindful planning, I could be strategic with the colours and create a designated colour scheme. 

Figure 3 (left) & 4 (right): Final product based on Karen Amanda Harris' inspiration Broken Sphere.