Week 9 : Traditional and Contemporary Practices of Making and Doing

Week 9: Reading, reflections, and activities

I found this week’s topic engaging because of the use of the word “technology”. I was recently in a meeting discussing the uses and integration of technology in our school, and how often we peg ‘technology’ as simply “electronic devices”, rather than techniques or advancements of tools that date back for thousands and thousands of years. I loved Susan's comment that "ancient technology involves mathematical patterning every bit as much as do contemporary technologies, and the ancient technologies are sophisticated and beautiful."

Article Reflection and Summary; MacKenzie, A. (2021) "'The spirit of the medicine will lead us back: How Avis O'Brien is guiding Elders to weave their first cedar hats."

In MacKenzie's (2021) article, they noted how the power of cedars and weaving have been integrated into the Haida and Kwakwka'wakw cultures. In an interview, the power weaving is seen as "sacred medicine of cedar that led her back to herself, her identify, and her people". Weaving is part of their art but it is also embedded in their livelihoods, " 'we use it when carving masks, poles and making hats. I just think, wow, our Ancestors were so smart in how they thought about all these different uses and ways' ".

This article reminded me of the significance of learning from the previous generation and how so many traditions are often lost nowadays. I wish I took the time to learn traditions from my grandma and my aunties. My mother's family immigrated to Regina, Saskatchewan in the 60s and 70s, and played a large part in the growing Chinese community in that city, especially new immigrants. My grandma and auntie were known for their embroidery and culinary skills. They would handcraft intricate beading for silk garments and traditional wear, as well as produce sought-after bamboo sticky rice wraps (joongs) and other traditional buns (baos). I never took interest in learning and it is something I regret to this day. This article resonated with me as there is a need to go back to traditional practices, and to re-learn attributes that are embedded in family traditions.

Activity - Rope and Twine, 7 Strand Braid

For this week’s activity, I started off with the S and Z twine making using plastic bags. I have seen projects where plastic bags have been used to create weaving projects (Persil UK) , sleeping mats (Lex 18), and other unique projects. I’m notorious for keeping plastic bags on hand (I blame having an Asian mom embed this habit into me growing up), so I thought I’d use a few I had in my home. While watching the video, I learned about how rope was able to stay in place by the tension that is applied in the opposite direction. I thought that it was fascinating that “the strength and stability of rope and yarn are dependent on the geometrical properties of its and helical structure, and is more or less independent of its material properties” (Gerofsky, 2017).

Figure 1 (Left): Twisting strips of plastic bags to create “S” twist.
Figure 2 (Middle): Basic S strand completed

Figure 3 (Right): Twisting it on itself, in the opposite direction, creating a “Z” twist. 

For the 7-stranded braid, I tried, I really really tried. But failed MISERABLY. I think it was the type of string that I was using (“Friendship bracelet string”) that made it challenging to place the right amount of tension, thus I was unable to get the clean braid or a tight sequence. I spent hours trying to sort it out and in the end, I did what every “bad student” would - I gave up and tried my own variation to this task (I see this often in my class when students get frustrated).

My thought went to two things: consistent patterns and repeats. So I used the 7 strands and instead of ‘weaving’, I looped the left string (under, over, under) to the right string (the anchor). I world repeat this process moving through all 7 strands (left to right). So with each rotation, the right string (the anchor) became the next left strong (the looper). There are so many patterns that can be followed for “Friendship Bracelets”. Perhaps this is how new patterns are discovered? Through attempted failures and trial and error?

Figure 1 (Left): Pattern - Blue left strand (“looper” - under anchor, over ancho, under looper), purple right strand (anchor). I repeat the process with the purple strand as the “looper” and the next strand to the right as the “anchor”
Figure 2 (Middle): After about 10 rotations. Tension plays a factor and was something I didn’t keep consistent but the pattern is somewhat visible. 

Figure 3 (Right): As a result, the product tended to curl on itself. 

REFLECTION: (A bit of a tangent) This week's topic also brought me back to my social justice project where my focus was on the fast fashion industry and how microfibres are becoming a growing concern. Fabrics like polyester and acrylic are often made from recycled plastic that has been turned into fibers and weaved into fabric patterns. While this was initially thought to be a great way to recycle and repurpose plastic existing in our environmental system, there are implications as it produces tiny microscopic fibers that cannot be broken down. This is an example though, of how ancient technology, such as weaving, was a potential solution to a problem, such as plastic accumulation.

 

Week 8: Fibre Arts, Culinary Arts and Fashion Art

Summary, Applications and Overall Reflection:

Hawkley, A. (2015) Exploring ratio and sequences with mathematically layered beverages.

Thank you, Maria, for allowing us to switch. =D I loved reading this article as I teach density to the students and building density towers is one of my favourite labs! In my classroom I just finished a cooking class that looked at fractions and ratios, and I thought this paper was a brilliant way to approach the lesson using drinkable products rather than food!

Adjusting the ratio of sugar to water changes the density of the liquid. The more sugar (greater sweetness), the more dense the liquid.  In Hawksley (2015) article, she explores the idea of the Fibonacci-like sequences, like Lucas numbers, by adjusting three key ingredients of lemonade: lemon juice, simply syrup and water. Essentially, the intensity of the drink would increase exponentially (p. 521) as you go down the layers of the drink. 

For a more simplistic approach, this could also work with fractions sequences for younger grades (I work with grade 6, so this would be a fantastic link to our unit in math and science). By creating a total volume of 1/2 cup, you could create different concentrations of solutions by adjusting the lemon juice, sugar and water. Note, when making the drink, always start with the sweetest or more dense layer!

I tried making it but I forgot about buying food colouring, I only had three. And they weren't such great colours! So I couldn't complete the entire Fibonacci sequence. But, the layers do show up!

Applications:
(Sorry - I may go into a scientific rant here) There are other ways to integrate mathematical concepts. 

• Comparing decimal values: Varying densities and concentration levels could allow students to make observations about how decimals work. For example, starting with pure water at 1.00g/mL and comparing that to other syrups such as honey (1.44g/ml), corn syrup, maple syrup etc. 

• Graphing: This could be a great data set to teach students how to graph, for example compare the % of sugar to water ratio and density (g/mL). This is actually an application to canning goods because the concentration of sugar affects the quality of the fruits (Vitz et al., 2021)

• Temperature also affects density! Water is most dense at 4OC. So changing temperature and determining density would be another way to explore this concept. 

ACTIVITY (Shoelaces):

I didn’t have any clean shoes so tried doing the lace activity using a holed pencil container. The issue was that there were 6 holes that were aligned, and I had to improvise and use 2 interior holes as the ‘7th’. My first go made me realize how important it is to align the laces upward or downward. In the video, this wasn’t as apparent, because you can only see the lines connecting the dots. I realized depending on the direction it made a significant differences in how the final look would appear.

Figure 1 (left): Lace attempt 1, where the starting strand pointed inwards and the resulting longer points ended up being inwards.
Figure 2 (right) Lace attempt 2, where the starting strand pointed outwards, resulting in the longer points directed outwards.

Figure 3 (left): I thought I did a different design than Attempt 2, but it ended up looking very similar! Again, because you can’t see the inside design, the outward design looked the same.
Figure 4 (right): I tried one of the more crazy designs and found it hard to follow. 

One of the issues was also to figure out how to end up with the strands outward.

Overall Reflection:

From my engagement, I was drawn to the links with culinary foods. For the orbifold patterns, I was surprised that there were 17 different types of wallpaper patterns. I love interior design, so looking at sequencing and design from an aesthetic perspective would be engaging! 

Additional Sources

Vitz. E., Moore, Justin Shorb, Xavier Prat-Resina, Tim Wendorff, E. V., John W., & Hahn, A. (2021, July 12). Sugar Solution Density. Chemical Education Digital Library (ChemEd DL). https://chem.libretexts.org/@go/page/49969 

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.