W9 - Reading Reflection!

Exploring Ratios and Sequences with Mathematically Layered Beverages

Andrea Johanna Hawksley

This paper describes a hands-on activity that uses the creation of layered beverages to teach ratios and fractions. By adjusting the sugar concentration in different liquids, participants create layers of varying densities that remain distinct when poured correctly. The curriculum extends to integer sequences, demonstrating how flavor intensities can follow the Fibonacci sequence to approximate the golden ratio. This type of activity aims to make abstract numerical relationships tangible (and tastable), moving beyond purely visual mathematical art. The author highlights how experimentation with food can help overcome common learning barriers by engaging the senses in a practical and fun environment.

Stop #1: "Many recipes have fairly strict ratios of ingredients (for example, baked goods), while others, like those in beverages, allow for a great deal of flexibility" (Hawksley, 2015, p.519).

This made me think back to my CEGEP and early university years. We often talked about how the different sciences could compare to the kitchen. Chemistry was more like cooking, you could kind of eyeball some things or use less precise glassware for many experiments, whereas biology was so precise and the perfect amount of each chemical had to be used by using an insanely precise measuring tool. We said how chemistry was like cooking and biology was like baking! Interestingly enough, I studied chemistry in my undergrad, and as I sit here reflecting on the comparison I just made, I am thinking about myself when I cook vs. when I bake. I think I enjoy cooking more, because I love the flexibility of getting inspired by a recipe, then changing it and making it mine by adding whatever I have or love. Whereas when baking, I find it mildly annoying to always go back to the recipe, and then having to run to the store for that 1 missing ingredient, that will ruin your batch if you don't have it. All that to say, I love baked goods, but I love the flexibility of cooking.

Stop #2: "[...], if the bottom layer has 5 teaspoons of simple syrup per unit of volume, and the top had 3 teaspoons of simple syrup per unit of volume, then the sweetness ratio is 3:5" (Hawksley, 2015, p.519). 

As I read this, I was waiting for the author to describe how they took white board markers and marked the ratios on the cups! But it was never said! (Sad face) So, for my stop #2, I am suggesting an extension! If or when this experiment is performed again, perhaps the students could identify their ratios and fraction with white board marks on each glass to be able to increase the practicality and effectiveness of  using these beverages as visuals (and tastables) to gain a better understanding of fractions and ratios and their relationship. This could also perhaps lead to adding or subtracting fractions, using bigger and smaller cups for sums/differences!

Stop 3#: "Many students struggle with understanding fractions even though they are a crucial skill that is generally taught fairly early in the mathematics curriculum as a precursor to later skills. Fractions can seem illogical and hard to conceptualize (Hawksley, 2015, p.523).

This particularly stuck out to me this week. As Amanda and I are wrapping up our project (focusing on fractions) this is just another scholarly article discussing the same problem. Fractions are a topic that is touched across almost all grade levels (most topics do not stretch this far), yet it still seems to be one of the hardest concepts for students to grasp. It is our job as teachers, to find better ways to teach them, to perhaps spend more time on it younger, and to use these types of activities to help build the foundations and grow! 

Hawksley, A. J. (2015). Exploring ratios and sequences with mathematically layered beverages. In Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (pp. 519–524).

Week 9 - Activity Reflection!

 Viewing: Origami Fashion with Uyen Nguyen Part 1 & Origami Fashion with Uyen Nguyen Part 2

As I watched these videos, I was vaguely remembering something I saw on TikTok one day that I thought was genius. It took me a while to really piece together what I was remembering, and then to try and find it was not easy, but I did it! Baby clothes that grow with the kid. Not only is this a great idea for the rapid rate that children grow at, but it also reduces waste! If you want more info, I have added a link under the photo.  

For more info: Click HERE

Here is mine folding! (Help from: Easy Miura-Ori Origami Tutorial)

My Origami! 

When I did the first row, I thought this was super easy, then i got stumped as soon as I got to row 2. I had unknowingly made a mistake that took me quite a while to resolve! Row 3 was also a challenge and I was nervous as I didn't know how I made that mistake in row 2 and I was afraid to do it again. Once row 3 was done, and done right, the following rows got easier and easier as I got the hand of it. By the end it was actually quite relaxing. Like knitting (when I knew how to knit).  

I have an origami crane sitting on my desk at work, that I worked so hard on to create, which is why I keep it. But, realistically this crane is useless to me. It was really fun to create origami that is in a functional pattern that can actually be used! (obviously not my little sheet - but the pattern itself!). The tutorial video mentions how this fold is used in solar panels, and the viewing shows how it can be used in fashion!

DRAFT - Final Project (due March 9th)

Here is the link to our draft/slides/work in progress: LINK to our Canva Presentation

Partner - Amanda Wheeler

Week 8 - Reading Reflection!

Writing and Reading Multiplicity in the Uni-Verse

Nenad Radakovic, Susan Jagger, and Limin Jao

SUMMARY: This article explores the interdisciplinary connection between mathematics and poetry, focusing on how creative writing can help students engage with complex numerical concepts. By analyzing Nanao Sakaki’s poem "A Love Letter," the authors demonstrate how concentric structures and geometric scales can be used to represent the universe. The text highlights a classroom experiment where students wrote their own poems to explore spatial relationships while making their poems personal. Blending emotional experiences with mathematical themes. While some students struggled with numerical accuracy, the authors argue that the process allows for a multiplicity of meanings and a more embodied understanding of abstract ideas. Ultimately, the research suggests that integrating the arts into STEM education fosters a dynamic environment for collective knowing and deeper conceptual exploration.

Adding a personal thought: All throughout my bachelors and now my masters, I've always had a big interest in relational pedagogy, when math has meaning, students will learn more. When students can feel a connection to math, or a sense of ownership to their work, students will learn more. This article highlights all of these points. 

STOP #1: Nenad's Poetic Response "My Universe" 

I really loved this poem and how it represented each scale factor. This was a beautiful way to open this article and the explanation of the task at hand. The original poem that this was a response to "A Love Letter" by Nanao Sakaki (1996) was also lovely, but I struggled to understand some stanzas and had to google some words. This one by Radakovic was more simple and easy to understand. 

STOP #2: "We interpreted that the “love” mentioned in the title relates to Sakaki speaking to a person (“you”) and that this person is central in his poetic universe: it begins with this person and ending with the circle that encompasses her or him." (Rodakovic et al., 2018, p. 3)

I really enjoyed this sentence as I believe that it summerizes many types of poetry very well. Often, and even in this weeks activities, poetry is about "me" and what surrounds me, whether its legit, or metaphorically. This idea of poetry is perfect when talking about using it as a tool for understanding math, as students are at the center of their own understanding and learning. 

STOP #3: "[...], we challenged the formalist approach to poetry that assumed that all readers decode textual structures in the same and ideal way thus rendering interpretation a predictable process with a singular outcome (Guerin, Labor, Morgan, Reesman, & Willingham 2005)." (Rodakovic et al., 2018, p. 3) 

"[...] the reader is not to be ignored in the process of interpretation. Rather, the reader takes on the role of authorship in producing the text in her very reading of it. [...] A reader is required to bring meaning to those empty signifiers. Barthes proposed that the reading and subsequent interpretation of any text is a writing of a new text. [...]. There are as many texts as there are readers." (Rodakovic et al., 2018, p. 4) 

This really stuck out to me after this week activity. When I was reading the Bridges poems, I reflected on my interpretation of the poems, and even commented that perhaps that is not what the author was intending, but that's what I got out of it, I mentioned how some of this diminishes when the author themselves reads their own poems aloud. 

QUESTION: After this weeks activities and readings and the connection of Fibonacci to my project, I believe that I would love to try Fib Poems with my classes (hopefully time permits to do this before the end of our project). Would you try to incorporate poetry into your teaching, if so, what kind of poems would you use from this weeks activities/readings and why? 

Week 8 - Activity!

Bridges Poets
Britt Kaufmann

https://www.scientificamerican.com/article/poem-midlife-calculus/

I really loved her play on words. She used words that we see in math (and their equations) to describe feelings and emotions. As a math-y person who has always struggled in ELA classes, I enjoyed this!


Iggy McGovern
https://www2.math.uconn.edu/~glaz/Mathematical_Poetry_at_Bridges/Bridges_2026/IggyMcGovern_TheMathematicalBarman.pdf

This was very comical. I loved the scene that was set up, the idea that mathematicians do math just to do math sometimes - calculating the average size of bubbles in a beer, and the soft touch of humor at the end! This was overall a very fun poem.


Jim Wolper
https://www2.math.uconn.edu/~glaz/Mathematical_Poetry_at_Bridges/Bridges_2026/JimWolper_CangesAndDeltas.pdf

This poem was harder to follow and took more time to understand, yet there are still parts I don't understand. I believe that this poem has many very advanced math topics integrated into it, that perhaps those are the ideas I am struggling to comprehend, but overall, I envy the creativity that went into creating such a concrete lengthy math poem. 

Racheli Yovel
https://www2.math.uconn.edu/~glaz/Mathematical_Poetry_at_Bridges/Bridges_2026/RacheliYovel_TheOneShapedByUs.pdf

What I found unique about this poem, was I felt that I the the first little section put me in a place. I could imagine the words that Yovel wrote and I was placing myself by seeing these perpendicular planes. Then as the poem goes on and comes to an end, there is the idea of infinity, but there also seems to be a message of looking for (human?) connections. This could be a stretch, but that was what I interpreted. 

Dan May
https://www2.math.uconn.edu/~glaz/Mathematical_Poetry_at_Bridges/Bridges_2026/DanMay_DivisionByZero.pdf

WOW! What a great 5th poem to choose. I loved this. I loved the idea of just accepting it and moving on, because there are worst things in life, the confusion of dividing by 0 is less than than a understandable topic such as -1 from everything. Sometimes what we don't know, won't hurt us like the things we do know could! 

Mike Naylor's poems were all very fun! Each were so unique in their own way, and I enjoyed listening to the author himself read is own poems. I feel that when we hear a poet read their own poem, the emphasis and tone is one less thing for us as readers to worry about, and perhaps the fact that it is already done, allows us to interpret and appreciate the poem a bit more! 

Fib Poem #1

I am working on a Fibonacci Sequence / Golden Ratio lesson for my final project, so I have been drawing these golden ratio diagrams, and thought that it would be fun to use my poem as the swirl, although i wrote very linearly and not so curvy... but you get the idea! 

Fib Poem #2

March is Chronic Illness Awareness Month, so here ya go! 

Pain.

Smile. 

Don't show. 

Need to sleep, 

But, so much to do. 

Chronic illness, "you don't look sick."

I really enjoyed this type of poem, I have to say, it was not easy to get the exact number of syllables for each line.

Week 7: Reading Reflection

A basketmaker’s approach to structural morphology

Alison G. Martin

 
Alison Martin explores how traditional basketmaking techniques provide structural algorithms for modern computational design. By hand-building models from bamboo and paper, she demonstrates how 2D tessellations can be deformed into spherical or hyperbolic surfaces by varying connectivity.  These woven structures mirror nanostructures and minimal surfaces found in nature, such as sponges. Ultimately, these physical models serve as powerful pedagogical tools for visualizing complex geometry and mathematical principles in architecture and engineering.

STOP #1: "The geometry principle of introducing singularities in a mesh - each pentagon in a hexagonal mesh introduces positive convex curvature to a fullerene shell or a basket shape while a heptagon introduces concave curvature - applies equally in molecular structure and woven baskets. The shape of a basket can be defined in mathematical terms; a 3-D representation of hyperbolic functions (saddles) results from adding extra weavers, while subtracting from the numbers of weavers produces closed shapes." (p. 2)

Although I had to read this several times to begin to understand what was being describes, I was hooked after the first read. I really loved the idea of the transition from 2D to 3D and what had to be done to achieve that shape and formation in the basket. You can tell in this excerpt just how mathematical basket weaving really is. This was the first time I truly saw a deep connection between the two. The use of the pictures that go along with this text, really helped solidify my understanding of the concept.

STOP #2: "Porifera (sponges) are simple aquatic invertebrate whose shape seems to depend on nature’s ability to find the simplest algorithm which will generate the most extensive surface with minimal materials. As cells multiply their surface arrangement adds extra area to the organism. This is a matter of life and death for filter feeders like sponges and corals. Evolution is a great optimiser." (p. 6)

As soon as I am introduced something where math is being done in nature, without the intervention of humans, I am so HOOKED! This concept here is beyond cool. In the last 2 years of being in my MEd, I have been expose to a variety of ways that math is being done in nature, every day, by nature itself. Ever since I was introduced to Fractals, I see them everywhere. And now sponges... WHAT!? This lead me to run to my washroom, where I have a natural loofah, and observe its complex net. Although I failed to pin point it's pattern, I am convinced that it is in there somewhere. Of course a loofah is a plant, and not a Porifera, but I thought perhaps there was a connection. 


Martin, A. G. (2015). A basketmaker’s approach to structural morphology. In Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam: Future Visions.

Week 7 - Nick Sayers Interview

STOP #1: I really enjoyed how Nick Sayers spoke about how creating spheres out of recycled materials can be a representation of the end of waste (around the 20 minute mark). Since spheres have an endless surface, than the end of waste can be had. Nick Sayerscreated a type of shelter with these spheres when created at a large scale (like with the signs), but I wonder what other purpose (besides a really neat piece of art with a story) a smaller sphere could have.

Stop #2: Nick Sayers mentioned (around the 29 minute mark) how numbers have meanings and related many numbers to OCD and superstitions. I had to laugh when I heard him say this. I don’t let volumes (radio, TV) be on odd numbers. My partner and parents laugh at me for this, but there is something about odd numbers that feels off, whereas even numbers are just superior in my head. Perhaps because they can be paired? Unsure.

I was curious if this was just a “me” thing, so I did a quick google search. AI’s response was:

People tend to prefer even numbers due to cognitive ease, perceiving them as more stable, balanced, and orderly than odd numbers. This preference is rooted in the ease of dividing, pairing, and processing even numbers—particularly multiples of 2, 5, or 10—which reduces mental effort and feels more "complete" or symmetric.

And that makes sense to me. Even numbers and multiples of 5 really do reduce mental effort when any kind of math is needed. There is also a beautiful degree of symmetry, perhaps relating to my idea of pairing.

Stop #3: The entire idea of pinhole cameras, which was a great section at the end of the video. I loved how there was an extensive discussion around how the image ends up upside down when it comes through the pin hole. I immediately thought about how our eyes work. The science teacher in me found this fascinating and how this would be a great visual, hands-on activity to describe the biology and functionality of our eyes. So cool.

On the topic of the pin holes photos, I really enjoyed this visual (around 1 hour 10). The explanation of the description of seasons shown throughout the photo, with this morse code look representing the sunny and cloudy days, and the description of the differences of light coming through the tree depending on the season, I was so in awe the entire time Nick Sayers was talking about this photo.

Nick Sayers really highlighted how there are so many different forms of art, and that there are millions of entry points into math. I believe that watching this interview reminded me that math does not always need to be formulas and complex tasks, it can include beauty and adapt to different forms.

Video Link: https://vimeo.com/1166172275/3a7a243bce?share=copy&fl=sv&fe=ci