Draft Project Link

Slides for final project

Week 9: Traditional Practices

Apologies for the format - I can't figure out Blogger this week. My input screen looks like html code for some reaason and I'm not a coder lol. Videos: I must say the videos this week were personally inspiring. I had day-dreams of leaving my job and becoming a rope maker or something of the like. From age 12 to 25 my job was a farmer. From 25 to now, 38 I’ve been a teacher. I have a friend who has become a part time blacksmith, which arose from a hobby. He has his own forge and classes come to take part and learn. Dancing braids is definitely something I’d like to do someday with students. Personally, I’d like to investigate the math a bit deeper. Watching the Vest-ed Interest video about sheep wool and making clothing seemed like a nice way to address the environmental impacts of the modern clothing industry. That being one of the math for social justice issues in our previous course. I find it relates to traditional kamik and mitt making in the NWT. From experience handmade mitts are much warmer and flexible than their industry made counterparts. I remember a simple pair of rabbit fur mittens being the most comfortable way to keep my hands warm walking around town in temps that went to the negative forties. So from my experience I tend to use handmade items for winter as practicality and as an added benefit it has less environmental impacts. Activity: I must say, I learned a lot just ripping strips from old fabrics. I figured I’d need to use scissors to get nice straight even widths but I only needed my hands.

I found the braiding process quite soothing. It would be something I could do to let my mind unwind. The act of ripping the strips from the fabric was also a bit of a stress reliever. I thought about how there could be so many science experiments that could come from rope making and braiding. For instance, how does the strength of one fabric strip compare to three strips braided together? Is the relation linear or something else? Other investigable question could be: How does friction between strands contribute to the overall strength of a braided rope? What is the relationship between the diameter of a rope and its load-bearing capacity? What mathematical models can describe the braiding patterns and their impact on rope strength? Article: This week I read the APTN article ‘The spirit of the medicine will lead us back’: How Avis O’Brien is guiding Elders to weave their first cedar hats By Anna McKenzie. It describes Avis ‘Nalaga’ O’Brien’s journey to and experience with teaching We Wai Kai and We Wai Kum elders how to weave their first cedar hats. Twelve years prior Avis was feeling disconnected from her culture but started learning how to weave from her sister. At the start of her weaving journey she still wanted nothing to do with it or her culture. However, the healing power of the cedar began to make her interested in her culture. Today she describes, “It’s like we’re giving back these Elders something that was taken to them, taken from them through our history,” she says. “It’s pretty magical.” One elder, Wasden, “says she believes cedar is “who we are and where we come from. It’s entwined in everything we do — from birth on a cedar mat, to death in a cedar box. It’s used for potlatches, for skirts, wreaths, head pieces. It was used for storage, cooking, blankets, mats, and even personal hygiene.” My question this week is: What science questions do you see coming from exploring these traditional practices like weaving?

Week 8

 This week's activity engaged me in a predictable struggle lol. I have difficulties with orienting the paper correctly. I wish the people in such videos didn’t hold up their paper and turn it all around for us to see until they are done. Simply, fold then flip or rotate in the most efficient way possible. This would leave me guessing less. Although I did get the Miura Ora Origami folds after the third attempt. Interesting math here might be how efficient the new structure is at compacting the surface area. 

I read Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving by Fisher, G. (2015). Gwen’s inspiration for her woven beaded sculptures came from Oscar Reutersvard’s 1934 Impossible Triangle. Impossible triangles are a “two-dimensional drawing that represents three straight beams with square cross sections, and the beams appear to meet at right angles.” (p. 99) Impossible polygons are a natural extension of this optical illusion. 

(Fisher, 2015, p. 99)

Gwen notes that the flexibility of weaving beads removes the optical illusion of the 2D drawings. This is why she refers to her creations as unlikely triangles and polygons instead of impossible. The beading medium allows for techniques like turns and twists or varying sizes and directions. As well as curvature which removes the straight edges of the 2D drawings. Other techniques with this medium include strategic use of bead size, colors, and placements. Bead color can be used to show paths, lines, faces, and vertices. 

(Fisher, 2015, p. 102)

A stop that came to me was caused by the uniqueness of the medium and the techniques which allowed for a different take on the impossible triangle versus the 2D paper. As well, as when Gwen mentioned how graphic designer M.C. Escher used the impossible triangle in some of his drawings. The contrast between 2D paper and 3D woven bead sculptures cause me to think of Marshall McLuhan's “the medium is the message”. “Each medium, independent of the content it mediates, has its own intrinsic effects which are its unique message.” (McLuhan, E) The beading medium allows for a new take on what paper says are impossible constructions. 

This kind of flows to Gwen’s desire to make a highly unlikely dodecahedron that shows the ten distinct paths (loops) that occur when moving through the twists and turns of the medium. She referred to this as passing through levels of hierarchy. 

(Fisher, 2015, p. 105)

I also was drawn to the uniqueness of the mediums, and the possible messages that come from each, when I watched the videos. In particular Yackel, who said “wallpaper patterns are realized through the dyeing process”. I appreciated the uniqueness of techniques and limitations that come with the medium, 2D paper is not infinite like the patterns it is trying to capture. Her use of the words constraints and resists - the tightness/clamping of the cloth prevents the dye from entering the cloth - both highlight the uniqueness of the medium. It is also fun to think of the folds as allowing a 2D paper to briefly enter the 3D world, which was similar to Gwen’s mention of passing through levels of hierarchy. 

(Yackel, 2020)

References:

McLuhan, E. (2024). Commonly asked questions about McLuhan – the estate of Marshall McLuhan. Ginkgo Press. https://www.marshallmcluhan.com/common-questions/ 

Week 7: Poetry

 Activity: Words per line Fib poems about golfing.

Autumn:

Red

Orange

Leaves fall

Crisp air aroma

Hunt for balls amongst leaves

Early morning frost coats the soon dormant grass

Spring:

Brown

Gray

Leafless trees 

Lingering snow melt

Balls plunge into saturated fairways  

Feet and clubs become covered in wet soil 

Article: Can Zombies write mathematical poetry? Mathematical poetry as a model for humanistic mathematics. Karaali (2014). This article describes Gizem Karaali’s journey to teaching a course called “Can Zombies do Math?” and writing his first piece of mathematical poetry. 

In his early years Gizem wrote non-mathematical poetry in his native language, Turkish. However, he learned mathematics in English. He reflected, at this time “my mathematics and my poetry did not play together. They spoke different languages” (p. 40) 

Early in his career, after browsing works from the Humanistic Mathematics Network Journal (HMNJ), Gizem began to see how math, intrinsic to himself, could be explored and viewed in a humanistic fashion. This referred to either or both, teaching mathematics humanistically (as if students matter) and mathematics is a humanistic discipline (as if math is a human endeavor). He argued math shared three ingredients with what makes us human, cognition, consciousness and creativity. Inspired by HMNJ he and colleagues launched a new journal, Journal of Humanistic Mathematics (JHM). The goal of the publication was to “provide an open forum for both academic and informal discussions on all the various threads of mathematical inquiry.” (p. 42) 

From there came an invitation to participate in a creative writing workshop. This forced Gizem to ponder blending together his creative writing and love of mathematics for the first time in his life. In particular, it sparked a love for poetry which, he argued, shared the same three ingredients of what makes us human. A few years later, he taught a course, Can Zombies do Math? Students engaged with poetry that dealt with personal experiences and feelings with mathematics. 

For example: 

(p.43)

Here is Gizem’s first piece of mathematical poetry, oddly enough he wrote it after teaching the course in mathematical poetry. 

(p. 44)

In the Bridges Math Art video, Mike Naylor, spoke in zeros and ones, as if it were a poem. Indeed it rhythmically sounded like a poem as he stated. To me this showed how poems can be structured mathematically, just like the Fib poems (1, 2, 3, 5, 8 words or syllables). In such cases the poem did not need to be about a mathematical theme. Whereas, Gizem’s humanistic mathematical poetry did not need a mathematical structure but rather it needed a mathematical theme (personal experiences and emotions surrounding mathematics).

My questions this week are also my “stops”. Before reading and viewing the materials, what were your thoughts about math and poetry, … poetry that was mathematical structured …. poetry that was mathematical themed? 

What might poetry look like if it were both mathematically structured and themed? Maybe a poem like Gizem’s that is 1, 1, 2, 3, 5, and 8 words?

Math 

Dread

Soul sucking

Numbers cause numbness

Mind feels dumb and lost

Marks, grades, and report cards are the proof

Week 6: Math. Dance and Movement

This week’s introduction hinted at some fun and interesting mathematics, around imagining the infinite and infinitesimal. The article, Dancing Mathematics and the Mathematics of Dancing, by Sarah Belcastro and Karl Schaffer (2011), presented lots of interesting representations of mathematics. Both have Ph.D. 's in Mathematics and have created mathematical themed concerts. Sarah and Karl lay out the math that is in dance. Math such as, dividing music into counts, dance patterns arising from music rhythm, symmetry (of individual movements and movements of the people in a dance group). Symmetries include translations, mirror reflections, 180 degree rotations, and glide reflections. 

My stop this week is the seemingly high level math that Sarah and Karl also present. Math such as graph theory and topology. These math topics can be symbolically daunting. However, stepping away from the symbolic representations, these topics can prove quite interesting and fun. I recall one professor saying a doughnut and coffee mug are topologically equal. This was not a topology class, just conversation that popped up. Essentially, pretend each is made of clay. Preserving all the holes or loops, could you mold one into the other? (The hand grip and doughnut hole)

(Image created by ChatGPT)

We could get students to “act out”, like in drama class, to make themselves topologically equivalent to other objects. We could extend this to making human clay figures that have the child’s pose. They could be molded to whatever the original object was.

In the image above the person (with arms making a loop), the coffee mug, and doughnut are all topologically equal. A biologist would have to ignore the fact the digestive system is an open loop, lol. 

My question is what other objects could children make themselves topologically equal to? How would this look?

Activity - Rope Polygons (Regular)

Adapt the rope and knots to other math concepts:

Pythagorean triples. Example 3,4,5 Use the knots as a way to measure sides.

Three other ropes and knots could be used to make the squares of 3,4, and 5.

Circles - estimating the connection between diameter to circumference.

C = pi X d or circumference of any circle is a bit more than three times its diameter C - 3.14 x d. Let the distance between knots be a diameter then the circumference of the circle would

be three of these lengths plus about 15 percent of the way to the next knot.   

Link to My Final Project Draft

EDCP 553 Draft Final Project Outline

Week 5

 Week 5 Article Summary 

First off apologies to Amanda and Patrick for not replying to your posts last week. Life and work unexpectedly got away from me. 

Movement-based Mathematics: Enjoyment and Engagement without Compromising Learning through the EASY Minds Program. Riley et al. (2017).

Four grade 5 and 6 Australian classrooms participated in a six week physical and mathematical intervention program, Encouraging Activity to Stimulate Young Minds (EASY minds).  Two types of math lessons were used. One used physical activity as the platform to develop procedural fluency and the other focused on moving around and looking at the mathematics around the school. 

Researchers sought both student and teacher perceptions of the intervention. 6 of 66 students went on to participate in a detailed student focus group. Students particularly enjoyed rotating activities like hop-skip-jump and recording and determining averages. Student enjoyment was a common result, student comment “I like doing sport and being active and when you combine that with maths it makes it much more enjoyable” (p. 1660).

Here is one activity example (p. 1661): 

I had a pre-stop, before reading the article. Would there be a balance? As Susan Gerofsky says in the introduction,  don’t throw the baby out with the bathwater. After the intervention, do math lessons involve a mix of “regular” learning and learning similar to the EASY minds program? It appears so in some cases. “Many students reported more hands-on activities since involvement in the program, and many reported that their teacher was now more innovative using varied and interesting activities to improve their learning.” (p. 1662) 

Furthermore, engagement in math went up. Teachers were able to harness this engagement  during the more traditional learning. One teacher reported more students now attempting written and explicit work. Many students said they found math easier when they returned to the classroom. 

There were some negative comments, such as when the physical goal of the activity was hard to achieve. Example, throwing a bean bag into a hoop could prove difficult/frustrating and distract from the math. 

Week 5 Activity 

I decided to do base 4 as the extension. Then I wanted to explore perfect squares. First showing 4 and 16 in base 4. Which are 10 and 100 in base 10. The “1s” being red. Is it a coincidence that 4 and 16 are perfect squares and have one solid red ring? What about others?

Finally showing the product of identical factors in base 4 to see what a “perfect square” might look like. 36 and 49 both had a red ring but they also contained a white ring and the other color from the factors. 

Idea - Number can be represented in expanded exponential form. Ex: 342 = 2x100 + 4x101 + 3x102

Guiding Questions - How can we represent this in a more advanced place value chart?

Thousands 103	Hundreds - 102	Tens - 101	Ones - 100
	3	4	2

The Story - A place value chart is a linear/grid method to display a number. It aligns the same way we write numbers.  It shows powers of ten or ones, tens, hundreds. It is kind of arbitrary. We could use nested circles to display place value in base 10. 

Integrating art based activities - Color code the digits 0 to 9. The colors now represent the amount of one, tens, hundreds, etc… Choose one color and make as many numbers as you can. Ex: 2, 22, 222 would all be the same color but different number of shaded circles. 

Possible Extension - Extend this to base 9 (or powers of 9) What do your numbers look like now? Do the same thing for base 11. 

EWeek 4: Post one - Eve, T. (2019). Bridges Stockholm 2018 and this week’s Introduction

 Eve, T. (2019). Bridges Stockholm 2018 and this week’s Introduction

Eve Torrence is a mathematics professor at Randolph-Macon College, in Ashland, VA. Eve incorporates the arts into their teaching and designing mathematical sculptures. Eve provides an overview of the 2018 Bridges Conference in Stockholm. 

The conference takes place at the National Museum of Science and Technology. However, the museum is on a campus that is also shared by the Ethnographic Museum, Maritime Museum, Swedish Sports Museum and the Police Museum. In this particular year the proceedings included 44 papers, 55 short papers, and 21 workshops. 

Eve’s description of the museum makes it sound the perfect place for the conference, especially as day four honoured Nordic Day (marking the signing of the Helsinki Treaty) and Family Day. At the front of the museum is the Mathematical Garden. This playground contains but is not limited to a slide in the shape of a nautilus and a giant xylophone to explore the relation between fractions and tones. 

For Nordic Day 4 lectures on creative education occurred in the morning. Family Day opened in the afternoon with Susan Gerofsky’s world premier of her play “Witches of Agnesi”. Here is a link to a video of “Witches of Agnesi” ,https://vimeo.com/339892315. The play brings together three different mathematicians who all faced obstacles in their careers because they were women. (Maria Agnesi (1718–1799), Sofya Kovalevskaya (1850–1891), and Emmy Noether (1882–1935))

Some of the activities of Family Day reminded me of our virtual Family Math Day we did in Cynthia’s course. However, the virtual aspect lacks some of the personal connections that an in-person event has. Family Day included over 20 workshops featuring - making paper geometric models, spirographs, and origami. Family Day ended with a Music Night and a Bridges Fashion Show. 

Reading about the binaries of arts and math, from the Introduction, had me thinking about examples of art/math crossovers that I could readily recall. The works of Dutch graphic artist Maurits Cornelis Escher came to mind first. I see from some Bridges entries, his tessellations are inspirations for the work of others. As for science, the Element song came to mind. (https://www.youtube.com/watch?v=U2cfju6GTNs) 

The readings also mentioned how the divide is American/European. My questions this week are: What art/math crossovers first came to your mind? 

Thinking back to your Virtual Family Day activity, did it incorporate any art crossovers? 

Also, as the NWT switches to the BC curriculum, there is a high school course called the History of Mathematics. I’m quite interested in teaching this course and I wonder if doing so would provide both historical and modern  examples of arts/math crossovers from different cultures.

 

Week Three : Post 2 Article, Introduction, and Video

 Williams urges us to move away from a schooling model characterized by discipline silos and mechanistic, technocratic worldviews. She advocates for a shift towards an educational approach that promotes a sustainable world, incorporating systems thinking and holistic learning. For example, she notes that living systems are not hierarchical but are networks nested within other networks, where the whole is more than the sum of its parts. She emphasizes the need for greater ecological literacy, arguing that pedagogy should be grounded in place to better understand patterns and relationships.

Williams then details a program in a Portland, Oregon school district focused on food and garden-based programming. This initiative involved 3,500 K-8 students across eight schools, as well as multiple sites for school-based learning gardens and environmental and nutritional farms. The program fostered numerous partnerships, with university students forming a learning garden committee that included parents, community members, students, and school staff.

The program addressed local issues: the rising number of hungry children in Oregon, increasing rates of obesity and diabetes, urban students' diminishing connection to nature, and the widening achievement gap between white and ethnic minority students.

Learning about patterns and relationships was deeply integrated into seasonal learning, emphasizing care for oneself, the land, and the community. This approach extended beyond traditional curricular math connections like patterns of time, size, and quantity. These insights from deep student engagement is my first stop. I was blown away when a 12 year old had deep insights to the patterns and relationships of caring for land and nature. “It is strange that people can take pride in large lawns and waste their land by simply growing and cutting grass. If we plant gardens instead, we can also grow food, we can bring wildlife and at the same time eat healthy fresh food.” (p. 47)

My second stop is from this week’s video and the first week’s TEd Talk video. Seeing Dancing Proofs reminded me of week one - changing perspective is understanding. Viewing the body in motion, from above, is a change in the perspective of sight. Therefore, understanding of proofs has occurred. From a ground view to an aerial view.

My question for week three extends from my second stop and learning more about engaging the body and all of its senses. We can change perspectives within the sense of sight, ground view to aerial view. So could it be possible to change perspectives within another sense, say smell? And would such a change in perspective also result in understanding as put forth in week one’s TEd Talk? 

I used AI to generate an image of hearing on land and underwater. 

Week Three Post 1 - Activity

 Week Three Post 1 - Activity (article and video post coming on Saturday)

I am currently in Paulatuk NT and as you can see I’m not going to be sitting or standing outside for long. Also, in terms of living things (plants and animals) right now there are only a few ravens, dogs, bushes, and humans. So this week, I am taking some liberty and drawing lettuce chunks from a salad, a banana, and an apple. The outdoor human made structures are being drawn from the comfort of looking out my window. 

In living things there were no right angles whereas human made structures were riddled with them. Living things had curved or arched lines whereas human made structures had straight lines. Human made structures are probably this way for both simplicity's sake and structural strength. I found it interesting that the banana formed a pair and a set of approximately equal angles. Then I Googled banana blossoms to see if I could determine a relationship between lines and the blossom/flower. However, the image of a banana blossom only confused me further!!!!!

****EDIT banana angles 60/60/80/80/80

I think that close observation and drawing would strengthen the understanding of angles and lines as it combines two senses for the same concept. The student would be highly attentive to sight and feel. This would be especially the case if, as this week’s introduction states, “we work to create a balanced program of mathematics” that is both outdoor and indoor (nature and sensory balanced with traditional methods).

 Tactile Construction of Mathematical Meaning: Benefits For Visually Impaired and Sighted Pupils, Stylianidou, A, & Nardi, E. (2019)

The authors’ study explored whether UDL practices in math fostered better inclusion and benefited all students. 29 lessons were observed from four classes of students aged 6 to 10. However, the article focused on the interactions and descriptions with shapes of just two students, “Luke”, visually impaired, and “Zak”, sighted. 

The activity involved Zak and Luke describing a triangle and another shape. The teacher made both shapes from waxed yarn. The other shape was a circle with a small straightened section. Next, the teacher handed out physical circle manipulatives of various sizes to compare to the original “near circle” shape. 

(Image: Stylianidou, A, & Nardi, E. (2019), p. 346)

Highlights from each student: Zak was uncertain of the straight line using his eyes. He was confident in the existence of the straight line through feel. Zak made an interesting comment, “the hidden facts on the shapes.” (p. 348) Luke commented that the circle would roll more and the “near circle” would bob up and down. Luke's action descriptions were vastly different from his peers who used typical textbook vocabulary. 

My first stop was my skepticism of the rigidity of this waxed yarn material to make shapes. Even if I were to notice the small straight section with my eyes or fingers I would have assumed it was an imperfection of a circle caused by the material. That is to say I’d assume the teacher meant to make a circle. 

Another stop occurred when I read “being aware of the characteristics of vision and touch - vision is holistic and touch is gradual, allowing for the exploration of an object from its individual parts to whole.” (p. 346) Stopping and thinking about the pedagogy or characteristics offered by each sense was quite an “Ah Ha!” moment. It fostered memories of my varying fishing experiences. Growing up in Nova Scotia I learned to fish by watching bobbers. However, moving to the Arctic I struggled with jigging and relying on my tactile reflexes to hook a fish. 

 

My question this week is in relation to Zak’s use of multiple senses to see the “hidden facts on the shapes” and the more traditional views of mathematics that using senses are inferior and cannot be trusted. If we were to use multiple senses in a scientific way then would such a practice be seen as more trustworthy or less inferior? That is, analyzing and comparing the data gained from each sense allows us to test, probe, and justify our conclusions and hypotheses, like how Zak was able to see the straight edge after feeling it. 

My Hexaflexagon Activity (Note: the only bagels available in my community were partially pre-cut for easy "rippability". So the cuts seen in the third video were impossible) 

Source of template: https://www.uua.org/re/tapestry/multigenerational/miracles/session-5/leader-resource-1 

The first 10 images show the various centers that were discovered. I marked them in unique ways. However, despite my twisting and turning I noticed some potential uncolored centers as seen on the outsides in the second set of images. I’m thinking there are probably 12 unique centers based on the multiple of 3 (triangles) and 6 (hexaflexagon). 

How might the hexaflexagon experience affect the learner's understanding of the mathematical patterns? In the video, Hart amusingly creates her own unique vocabulary based on her tactile actions while also using traditional vocabulary like clockwise and counterclockwise. Much like this week’s introduction mentions a movement vocabulary. 

Such experiences are great practices for students to see, feel and verbalize “actionable” textbook vocabulary. At the same time, students can create their own vocabulary for actions they see in which they may not know the textbook word or perhaps there is no such English word. I feel, this “student speak” would be packed with student understanding and ripe for teacher probing. 

I attempted to create my own action math vocabulary as I was playing with the hexaflexagon. Alas, I think I’m too grown up, I could not get my imagination to engage in making new words. I could only think of already existing words like rotate and the flexibility of the paper as more and more exploration was done. Perhaps, now as I write this post, I could create the word “open-up-ability” for when a squeeze and a pull could reveal a new side. I did use the word "rippability" in reference to easily pulling apart partially pre-cut bagels. I thought I was inventing a word but turns out it's a geological term.