https://docs.google.com/presentation/d/1aKo1NBUWyKRRUJFRtsMAY4ITfhQYytLp/edit?usp=sharing&ouid=110902289334019534432&rtpof=true&sd=true

 

Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving, Gwen L. Fisher

Summary

In her article, Gwen Fisher explores the intersection of geometry, visual perception, and art by transforming 'impossible' optical illusions into physical 3D sculptures. She focuses on the Penrose Triangle, a paradox that cannot exist with straight beams and right angles. Fisher resolves this paradox by using Cubic Right-Angle Weave (CRAW), a flexible bead-weaving technique that allows her to introduce a 90-degree quarter twist into each beam. This structural adjustment substitutes subtle curves for straight lines, creating 'highly unlikely' objects that function as topological equivalents to the illusions. These sculptures exhibit Möbius-like properties, where a single path of beads must travel around the entire structure multiple times (four times in the case of the triangle) to return to its start. Fisher successfully extends these principles to more complex 3D polyhedra, such as tetrahedra and dodecahedra, creating 'strange loops' that challenge our traditional understanding of surfaces and boundaries."

Stop 1

“The twist in the beadwork allows the impossible triangle to be constructed in 3D. The twist in the beaded version also destroys the optical illusion due to the curvature it introduces to the edges that appear straight in the 2D drawing” pg. 100

Fisher’s description of the bead 'destroying the optical illusion' hits on a powerful truth. It makes us see that something only feels impossible because of the 'rules' we’ve trapped ourselves in. For instance, if I tell Grade 7 students to connect four dots in a square with only three lines, they’ll say it’s impossible, but until they can fold the paper or use a thick brush to change the 'straightness' of the line, they may not connect. That means that just like Fisher’s quarter twist, these aren't just tricks; they show that flexibility is a mathematical tool. Another thing that made me pause and reflect is that just like how the beads were used to make the impossible triangle clearer, students can use hand on manipulatives to gain a concrete understanding of abstract ideas.

Stop 2

Constructing a quarter twist in each beam takes some thought. Looking closely at the bottom beam, you will see that I accidentally omitted a quarter twist; the bottom beam is straight.pg 101

One aspect of the article that struck me was Fisher's admission of her 'very first attempt' failure. This connects deeply to what we see in the teaching and learning environment, where students often give up when a problem becomes difficult. Fisher’s mistake wasn't a sign of inability, but a productive struggle that allowed her to see the logic behind the shape better than if she had accidentally gotten it right. As a teacher, I emphasize that the 'understanding of why they failed' is more valuable than the final answer because they should be able to learn from their mistakes. Fisher's omitted twist serves as a perfect mathematical metaphor: sometimes it is our errors that finally reveal the hidden rules of the system we are trying to understand. 

Considering Gwen Fisher’s 'impossible' structures that need a 90-degree twist and four passes to close, how can we expect a linear lesson plan to effectively teach complex math? Have you ever taken shortcuts in your teaching, only to realize that this resulted in students creating flawed mental frameworks that couldn’t support true understanding without hands-on exploration and new perspectives?

 

Can zombies write mathematical poetry? Mathematical poetry as a model for humanistic mathematics by Gizem Karaali's(2014)

Summary

Karaali delves into the rich intersection of mathematics and poetry, highlighting the deeply human aspects of both fields. As a mathematician and a poet, she argues that cognition(rational and logical mind), consciousness (feeling heart), and creativity(Intuitive spirit) intertwine mathematics and poetry, illustrating that both forms of expression rely heavily on language and the engagement of a sentient being.

She reflects on her personal journey, noting the divide between her poetry, written in her native Turkish, and her mathematical studies conducted in English. This distinction shifted when she began to see mathematics not as a rigid set of rules but as a dynamic social phenomenon. This realization led her to co-found the Journal of Humanistic Mathematics in 2011, which emphasizes a philosophical approach to teaching students as if they matter. She emphasizes that both disciplines communicate ideas and emotions, and the creative process in mathematics mirrors that of poetry, requiring intuition and heartfelt expression.

Karaali tested her ideas in a seminar titled "Can zombies do maths?" and discovered that incorporating a literary approach into mathematics had a positive impact.

- Students who had previously disliked math found the literary form refreshing

- Engaging with poetry helped bridge the perceived gap between art and science.

- Using poems to discuss mathematical concepts inspired a newfound appreciation for the subject among her students.

Ultimately, Karaali views mathematical poetry as a vital connection between the emotional and intellectual realms and a powerful tool for fostering a deeper understanding and love for mathematics.

Stop 1

"Both poetry and mathematics may, in fact, be conceived of without or before language, but only with words will they become communicable and complete". Pg.39

This quote illustrates how learners develop ideas in poetry and mathematics before having the words to express them. To share these concepts, they must translate them into language or symbols. In my classroom, I often observe that students often visualize symmetry by imagining or mentally folding paper before using math terminology. They intuitively feel the balance and see how the halves align, which demonstrates that mathematical understanding can exist before language. This reliance on intuition allows them to form ideas that language and symbols then help them express.

Stop 2

"All in all, my personal, professional and pedagogical experiences with mathematical poetry have inspired in me the conviction that mathematical poetry can be seen as the perfect ambassador for humanistic mathematics".Pg.44

I had to pause at this concluding statement because it showcases how both mathematics and poetry reflect our humanity. While poetry captures emotions and intellect, mathematics, which is often seen as a dry and dreary subject, causes many to overlook its creative aspects. By positioning poetry as the 'perfect ambassador,' Karaali suggests it can bridge the gap between these two worlds. As educators, highlighting the similarities between these fields activates a student's 'consciousness and creativity' alongside their 'cognition.' This helps students view mathematics as a form of artistic expression rather than just a collection of rules. This shift in perspective replaces 'zombified' learning with a 'humanistic' experience, enhancing their appreciation for both poetry and math."

Question

How can we, as educators, move beyond the 'zombie' model of instruction to celebrate every 'baby step' of a student's creative journey while treating mathematical poetry not as a rigid exercise, but as a humanistic mirror that proves every student truly matters?

 

                                     Zoom Interview between Susan Gerofsky and Nick Sayer 

 

 "I guess when I was little, when I was about, I remember about 6 or 7, I remember numbers just really terrifying, and I felt like I was really bad at it. My mum got quite impatient with my ability to do longer arithmetic. Remembering times tables and all these sorts of things. So, I had this belief set in my mind quite early on that I was bad at maths".00:05:25

"I've used all sorts of materials to make my, including this one, which is a geodesic haircut, I had, a few years ago, actually, when I went to Bridges Maths Art Conference in 2013, I think, and then I did another one". 00:16:59

Listening to Nick's story deeply resonates with my own journey as a mathematician. It highlights the hurdles we face in our relationship with math as we strive to discover our true selves. Like Nick, I also had parents and teachers who played pivotal roles in my growth and success. This reflects how the people we encounter shape who we become. It is important to note that no matter how daunting math may seem to students, change is always possible. I’m thrilled about the new integration of math and art, which is transforming the way we approach the subject as it helps alleviate the fears many students face. Nick’s haircut, with its unique shape patterns, is a beautiful representation of his evolution through struggles. It symbolizes how he has embraced math as part of his identity, turning challenges into something personal and expressive. This connection between math and art is truly inspiring!

"I feel there are these estate agent advertising boards all over the city, cluttering our sort of visual space. Meanwhile, there’s a homelessness crisis, and I was renting a home at the time. trying to get on the housing ladder, and so I wanted to make a statement about homelessness and sustainable architecture and the housing, you know, market and other things. I guess it went, again, beyond just the mathematics of it".00:21:40

In reflecting on Nick's quote, I appreciate how he uses art to illuminate mathematical concepts and address societal issues. His slot-together cord sculptures not only showcase mathematical patterns but also draw attention to pressing topics like homelessness and the clutter of advertising in urban spaces. By incorporating recycled materials, he engages with ideas of sustainability and social responsibility, showing that mathematics can be a lens through which we understand and respond to real-world challenges. This connection highlights the role of ethnography in mathematics, allowing us to see how our lived experiences can inform and enrich our understanding of mathematical concepts. It inspires me to think about how I can use mathematics in similar ways to engage with pressing issues in my own community.

"This is actually the pattern of the 18 gears that this bicycle spirograph has, and you can actually see that top left is just a circle, because that’s where you‘re struggling up the hill, and you‘re at the speed that your crank speed is as the wheel speed, whereas the bottom right, that’s where your"00:36:38

The bicycle spirograph elegantly illustrates the connection between mathematics and motion through repeated rational motion. Each traced point reflects the simultaneous revolutions of the crank and wheel, represented as periodic functions such as sine and cosine. The relationship between these rotations, particularly the gear ratios, is crucial; a 1:1 ratio produces a simple circle, while ratios like 1:3 or 2:5 create intricate multi-petaled shapes. This interplay reveals how algebra (through ratios) manifests as geometric patterns, highlighting concepts such as symmetry and least common multiples (LCM) in rotation. It's fascinating how cycling can so deeply intertwine with mathematical principles.

"I brought this little mascot, which is my Lego Spaceman from the 1980s, but I also went to the vegetable market there and found that Jupiter is a coconut in its shell. And then Saturn is a coconut with its husk taken. And then these two fruits here, Uranus and Neptune, are about the same sort of size and got this wrong. I started doing workshops, making these little cameras, so, rather than using them again, when I started, I brought in a whole bag full of beer cans into schools. I quickly realized it felt a bit weird, so I made another version using soft drink cans".01:07:38

I really appreciate the fact that in explaining his concepts, Nick uses familiar resources like cans for a camera, bottles for Christmas trees, fruits for the description of the planet, a train ticket for a sphere and lots of other things. This helps to drive home the idea that science and mathematics are all around us and not far away. To further make his point more explicit, he carefully selects items that suit the purpose of what he is trying to portray, like coconuts and other fruits, to explain the planets so the children can relate to the size of each planet in relation to one another.

 

 

What Can We Say About “Math/Art”? By George Hart

Summary

In the article "What Can We Say About 'Math/Art'?", George Hart, an applied mathematician and sculptor, explores the maturing yet ill-defined field where mathematics and art intersect. Hart argues that while a vibrant community exists, the discipline lacks a coherent formal framework or even a rigorous definition. He suggests that much of what is produced by this community may be better categorized as craft, design, models, or visualization rather than traditional "fine art," but he views this not as a deficit, but as a unique cultural expression that shares the "mathematical landscapes" and joys of discovery familiar to mathematicians. Ultimately, Hart encourages mathematicians to engage in artistic creation as a rewarding form of self-expression and a way to communicate the wonders of mathematics to a broader audience.

Stop 1

Quote: “From a human perspective, I find no contradiction, rather a great resonance, in the blending of mathematics with fine art. It is a central part of my life. Yet when attempting any rational introspection into the nature and power of mathematical art, one is immediately stymied by the fact that the subject seems ill-suited to our usual tools of formal analysis. One can’t even define “art” in the rigorous way that elementary mathematical practice would require. And even without a universal definition of “art,” if we agree that a particular object is art, people may still disagree on whether it is also “mathematical art.” Pg.521

Explanation: The author talks about how closely connected mathematics and fine art really are, even though most people think they are completely separate. Growing up in Nigeria, I was taught to see math as a tool for science subjects like physics, chemistry, and accounting. In contrast, students who weren’t good at math were encouraged to focus on history, languages, or the arts. This made me think of the two as completely different worlds.

Coming to UBC and studying EDCP 552 has changed that. I now see mathematics as a way to understand patterns, rules, and relationships that are also at the heart of art, like sculpture, music, and design. The author, being both a mathematician and a sculptor, shows that these worlds can overlap. Even if we recognize something as art, people might still debate whether it is “mathematical art.” The real barrier is not math or art itself, but the assumptions we carry about them.

Stop 2

Quote:” I trust that as society evolves, more and more people will be freed to create art. And as a fundamental humanist expression, the scope of art needs to be enriched by the viewpoint of mathematicians. Those who have journeyed through mathematical lands have unique stories to tell of what they found and how they now see the world.pg 525

Explanation: I am excited to see mathematics moving beyond simple number manipulation to meet the creative and expressive needs of the 21st century. I was particularly inspired by the recorded Zoom meeting between Susan and Nick Sayers, where everyday materials like bottles and sand were used to create patterns that not only expressed ideas but also solved problems, such as the creation of dune fractals to convey messages, much like expressive art would. This approach reflects the ideas of ethnography, showing how mathematical patterns can connect to human experience and culture. I also thought of a colleague who creates tie-dye fabrics. While skilled, he produces relatively few designs because he has not incorporated mathematics into his process. I can imagine how much richer and more expansive his work could become if he embraced the beauty and structure of mathematics. Hart’s article suggests precisely this: that mathematical thinking, when applied creatively, can transform art and human expression, revealing patterns and possibilities that traditional methods alone might miss.

 

Question: If Mathematics describes patterns in everything around us, then why do we often strip it of creativity, movement, and culture when we teach it?

 

Draft of My EDCP 552 Assignment on Arts and Embodied Learning

Topic: Uncovering the Hidden Geometry in Atilogwu dance: Heritage Algorithms and Emplaced Learning.

Name: Clementina Uti

Collaborators: Working individually

Due Date:18^th of February,2026

Description of Project

I am excited to begin a project that creatively merges mathematics with the vibrant cultural heritage of dance. In our Grade 7 math curriculum, we study transformations in geometry, such as translations, rotations, and reflections. I believe that integrating the Atilogwu dance from Nigeria into our lessons will make these concepts more engaging and relatable. I often describe dance as a “heritage algorithm” because much like math, it operates under a set of structured rules and patterns. The Atilogwu dance provides an excellent framework for visualizing and grasping geometric transformations, allowing us to explore mathematical principles in a dynamic way that transcends conventional textbook methods.

My primary aim is to demonstrate that dance consists of intentional movements, enabling students to identify various angles formed by the joints—such as acute, obtuse, and right angles. These are essential for comprehending translations (utilizing vector shifts or congruent images), reflections (synchronized movements between partners), and rotations (turns and jumps measured in degrees, like 90, 180, or 360 degrees). This innovative strategy seeks to connect mathematics with cultural expression, enhancing our understanding of geometry while highlighting its significance in daily life and heritage. Ultimately, I envision this project to refine our math skills while deepening our appreciation for how cultural elements, like dance, can seamlessly integrate with academic concepts. It offers a unique chance to investigate geometry in a lively and relevant context.

Research Plan: Exploring the Interplay Between Dance and Geometry

For my research, I intend to explore the connection between dance and geometry through collaboration with a community dance group called African Friendship Society in Vancouver, British Columbia. This approach is particularly appealing because the Atilogwu dance is known for its energetic acrobatics and complex movements. By partnering with experts in this field, I will be able to document authentic performances related to angle identification and transformation mapping, allowing seventh-grade students to discover the hidden geometry before engaging in semiotic enactment.

Additionally, students will have the opportunity to explore a human-scale coordinate grid outside by using chalk to draw angles on the ground, tangibly reinforcing their learning during our classes. To aid my study, I will record their live dance performances, enabling me to trace the invisible lines created by acrobatic jumps and movements, which I will treat as crucial cultural artifacts. My objective is to meticulously observe and analyze the dancers' performances to uncover the geometric patterns that emerge, focusing on aspects like jumps, rotations, and intentional placements of hands and bodies.

Through detailed observation and analysis, I aspire to reveal how these movements not only convey significant cultural narratives but also expose underlying geometric structures in the air. This research aims to provide a deeper understanding of the intricate relationship between movement, form, and cultural expression.

Bibliography and Annotations

1. Eglash, R., & Bennett, A. G. (2025). African Interlace as Dynamic Grids: New Heritage Algorithms for Diaspora Design Ecologies. Design and Culture, 1-22.

This article establishes a foundational theoretical framework for my project by framing African cultural practices as "Heritage Algorithms," characterized as logically structured and rule-based systems. It draws on Eglash and Bennett's exploration of "Dynamic Grids" and "3D movement paths" in performances such as Capoeira, providing academic support for viewing the Atilogwu dance as a mathematical artifact, transcending its artistic expression. The authors’ concept of "repetition with revision" is particularly relevant as it explains how synchronized dance steps can be seen as geometric transformations, including translations and reflections.

2. Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020, August). The future of embodied design for mathematics teaching and learning. In Frontiers in Education (Vol. 5, p. 147). Frontiers Media SA.

Their article emphasizes the significance of the learner’s body in developing mathematical intuition, suggesting that physical movement is foundational before formal symbols are introduced. It employs the concept of "semiotic enactment" to bridge students’ visual analysis of the Atilogwu dance with their own creative outputs. By moving students from a "pre-symbolic" state, where they appreciate the dance's cultural and geometric intricacies, to a "symbolic" state, where they translate those movements into coordinates on a grid, the lesson fosters deeper engagement.

Utilizing Abrahamson’s idea of "multimodal synthesis," this pedagogical approach integrates digital video analysis, collaborative discussion, and hands-on chalk activities to create an embodied understanding of geometry. This methodology highlights the use of energetic cultural artifacts to initiate attentional anchors necessary for understanding abstract transformations, which ultimately aims for a holistic learning experience that connects physical movement with mathematical concepts.

3. Gerofsky, S. (2025). Embodied Outdoors Arts-Based Approaches to Mathematical Understanding. Encounters in Theory and History of Education, 26, 56-87.                                           

The author advocates for "emplaced" and "outdoor" learning that utilizes materials available in the immediate environment to break away from the static, industrial nature of traditional classrooms. I apply this by having students transition from the digital analysis of expert video artifacts to a physical, material-based representation on an outdoor floor. Using chalk as an environmental tool, students will collaboratively map the "heritage algorithms" and geometric transformations they observed in the Atilogwu dance. This approach supports Gerzofsky’s argument that mathematical understanding is deepened when it is arts-based and situated in an open-air, embodied space.

4. Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM, 46(3), 349-361.

The article discusses the concept of "sensuous cognition. “A concept that argues that mathematical thinking is influenced by cultural and historical contexts, while drawing on sensory experiences and the material world as a crucial foundation. This idea is applied to illuminate the significance of the Atilogwu dance, a vibrant cultural expression, as an essential tool for understanding mathematics. By having students observe professional dancers and then translate those movements onto the pavement using chalk, the approach fosters a "multimodal sentient form" of learning. It reframes the act of drawing as a complex process that engages the senses and material tools, enabling students to creatively interpret and reshape abstract geometric ideas.

5. Fors, V., Bäckström, Å., & Pink, S. (2013). Multisensory Emplaced Learning: Resituating Situated Learning in a Moving World. Mind, Culture, and Activity, 20(2), 170–183. https://doi.org/10.1080/10749039.2012.719991

This article establishes a critical framework of "emplaced learning" that backs up my project's environmental context. The authors advance beyond conventional "situated learning" by positing that knowledge emerges from a multisensory interaction between the dynamic body and its surrounding environment. I apply this innovative theory as evidence of the transition of my Grade 7 students from a traditional classroom to a large-scale outdoor coordinate grid. By engaging in what Fors et al. term a "sensory ensemble," my students transcend mere observation of geometry. They actively embody it by aligning their visual interpretation of the Atilogwu dance with the tactile experience of drawing and moving in an expansive outdoor space.

 

Dancing Mathematics and the Mathematics of Dance by Sarah-Marie Belcastro and Karl Schaffer

Summary

The article offers an innovative viewpoint on teaching mathematics through movement and dance, challenging the notion that math is solely about symbols, digits, and written calculations. It portrays a classroom environment where students actively engage with mathematical ideas such as patterns, symmetry, rotation, and structure by using their bodies. This method transforms the learning experience into a blend of cognitive and physical activity, making abstract concepts more tangible and relatable through gestures and dance. By connecting mathematics with movement, students discover that this subject can be lively and imaginative, exploring rhythm and structure through choreography and bodily expression.

A key takeaway from the article is the transition from passive to active engagement in learning. Rather than merely receiving information from an instructor, learners take initiative in their educational journey, collaborating, investigating, and understanding concepts via their physical experiences. This approach confronts the conventional "banking model" of education that typically prioritizes memorization, instead promoting dialogue, interaction, and student empowerment. Additionally, the authors emphasize that this embodied method is inclusive, enabling students of varying abilities and language proficiencies to participate fully. Reflection plays a vital role in this process, bridging physical movements with formal mathematical principles, thus making learning both experiential and intellectually robust. In conclusion, the article highlights how integrating dance and movement can transform mathematics classrooms into lively, creative, and inclusive environments for every learner.

Stop 1

Quote: If you're not a dancer, and even if you are, you may be wondering how on earth mathematics and dance are related. Page 1

Explanation: Before exploring the connection between mathematics and dance through Karl Schaffer’s YouTube video shared in class and Susan’s teaching, I had never imagined any real link between the two fields. Growing up, I led a choreographic group of teenagers at my local church and later became a mathematics teacher, yet I still saw these as two completely separate worlds. As Mathematics, to me, was confined to numbers, formulas, and written methods, and my goal was to teach it in the usual traditional setting to cover the syllabus, while dance belonged only to artistic expression, mainly for relaxation and fun, but watching how dance illustrated concepts such as symmetry and patterns transformed my understanding. I was amazed that a creative and enjoyable medium could communicate deep mathematical ideas. The movements made abstract concepts more visible and accessible in ways that symbols alone could not. Reflecting on the video and the ideas from the article, I realized that learning takes place not only in the mind but also through the body. Most importantly, I came to see that this embodied approach could reduce fear and tension around mathematics, making it more lively, meaningful, and approachable.

Stop 2

Quote: “Many mathematical ideas pervade dance and, we would argue, are intrinsic to dance. For example, we divide music into counts and use counting to mark the times at which movements are done”. Page 1

Explanation: When you watch a dancer glide across the floor, it’s easy to be captivated by the beauty and emotion of their movements. But beneath that mesmerizing surface lies a fascinating world of mathematics. Every beat of the music counts out a rhythm that guides them every step, and I believe this connection can be a powerful tool in my teaching practice. For instance, when children are doing a presentation, they often count the steps and movements they make to ensure everyone displays a similar pattern, resulting in a visually stunning performance. I can incorporate this idea by integrating rhythm and movement into math lessons, allowing students to physically experience concepts like patterns and sequences. Imagine the joy of perfectly timed movements, stepping forward, withdrawing, spinning, and clapping, all synchronized with the melody, as a way to explore fractions, ratios, and angles. By encouraging students to create their own dance sequences based on mathematical principles, I can foster a deeper understanding of math as they discover how it plays a vital role in creating beauty and harmony, much like in dance, helping to make learning more engaging and interactive.

Question: How would our mathematics teaching change if we recognized that learning happens not only in the mind but also through the body? What practical steps could we take to integrate this approach in classrooms still dominated by traditional assessment-focused practices?