Week 7 Interview & Reading Reflection

 I finished watching the interview and found myself thinking about how much time, effort, and commitment artists like Nick Sayers dedicate to their work. I was really struck by the depth of thought behind his projects, as well as his persistence in seeing complex ideas through to completion. As I watched, I had a few “stop and think” moments along the way.

Around the 6-minute mark, Nick talks about how both he and his mother believed he wasn’t good at math until he got his own computer and started programming. This really resonated with me. I think this is the case for many students who believe they are “not math people.” Often, their mathematical interests or strengths don’t emerge until later, or until they encounter the right context or tool. Personally, I didn’t think I liked math until Grade 11, and I didn’t even consider majoring in mathematics until my second year of university. This made me reflect on how important timing and opportunity are in shaping students’ relationships with math.

Between about 18 and 23 minutes, Nick discusses his mathematical sculptures created from everyday materials such as garbage bags, cola bottles, and plastics. This section reminded me that both art and mathematics can be highly accessible if we pay attention to what’s around us. It made me think about how powerful it could be to invite students to see mathematical potential in ordinary, even recycled, materials.

At around 35 minutes, Nick introduces his work with spirographs, which instantly brought back childhood memories. I used to have a spirograph set and loved drawing the intricate, flower-like patterns, but at the time I had no idea there was mathematics behind them. Seeing this now highlights how mathematical ideas can be present in playful, aesthetic experiences long before we formally name or understand the concepts.

Near the end of the interview (around 1:47–1:50), Nick presents his work with space-filling fractals and empires, connecting them to histories of colonization. This part stood out to me as it showed how art, mathematics, culture, and history can be deeply interconnected, rather than existing as separate domains.

What does this artist’s work offer in terms of understanding math–art connections, and what does it offer me as a math teacher?

As a math teacher, I’m fascinated by the many ways mathematical ideas are embedded in Nick’s artwork. While some of his pieces operate at a level of complexity that may not be easily replicated in everyday classroom practice, they open up powerful possibilities for thinking about math–art connections. His use of accessible, everyday materials especially inspires me to think about how recycled or found objects could be used to create math-related art with students. This feels like a meaningful and worthwhile direction to explore, particularly in helping students see mathematics as creative, connected, and present in the world around them.

===========READING REFLECTION ================

Reading: Writing a Mathematical Art Manifesto by Fumiko Futamura

Summary: 

This reading builds on a workshop from the 2025 Bridges Conference proceedings,and explores the aim of developing a mathematical art manifesto that frames mathematical art as a form of expressive artistic practice rather than as illustration, demonstration, or pedagogy. The article discusses the historical context of manifestos, examines definitions and examples of works that may be considered mathematical art, and outlines the defining characteristics of an art manifesto. It also incorporates reflections and quotations from mathematical artists and identifies “rejected paradigms” that challenge conventional assumptions about both art and mathematics.

Stop 1:

“Doris Schattschneider wrote in her review of the 2005 Bridges Conference [11] that “Mathematics creates art”; “Mathematics is art”; “Mathematics renders artistic images”; “Hidden mathematics can be discovered in art”; “Mathematics analyzes art”; “Mathematical ideas can be taught through art.”

This quote reminded me of an earlier task in this program with Dr. Nicol, where we were asked to identify mathematics in nature. Examples included patterns in sunflowers and leaves, symmetry in butterflies, as well as fractals and spirals. These observations align closely with Doris Schattschneider’s ideas that mathematics can be understood as art and that hidden mathematics can be discovered through artistic and natural forms. At the same time, I believe that natural beauty involves more than just mathematics, even though mathematics often renders what we perceive as artistic images, such as those associated with the golden ratio. Although traditional mathematics textbooks do not always explicitly connect mathematical concepts to art, I agree with Schattschneider that mathematical ideas can be meaningfully taught and explored through artistic contexts.

Stop 2:

“What is Mathematical Art? I will choose work that meets at least one of the following three criteria: The art 

1. is based on a Mathematical phenomenon, or 

2. it is generated by a Mathematical process, or 

3. it is a personal response to Mathematics by the artist. 

Susan Happersett, artist [6]‘

This definition has changed how I view art, as it encourages me to actively consider how mathematics may be embedded within or connected to an artwork before identifying it as mathematical art. However, the third criterion prompted me to reflect more deeply on what a “personal response to mathematics” might mean. Does this suggest that the artwork is intentionally created in response to a specific mathematical concept, such as a piece developed for a mathematics-related project? In many cases, these criteria may be difficult to identify unless the artist explicitly articulates their intent or process. Nevertheless, I understand Happersett’s broader point: mathematical art invites viewers to look beyond surface aesthetics and thoughtfully examine the mathematical ideas underlying the artwork.

Week 6 Reading + Activity Reflection

Reading:

Dancing Mathematics and the Mathematics of Dance by Sarah-marie Belcastro & Karl Scaffer

Summary:

In this article, the authors explore the deep connections between dance and mathematics through ideas such as patterns, symmetry, graph theory, and choreography. By providing concrete examples of how mathematical concepts can be embodied and expressed through movement, they show that mathematics is not only present in dance but can also be experienced through it. The article highlights the usefulness and relevance of mathematics in different forms of dance, with the aim of helping readers see mathematics beyond abstract symbols and recognize its beauty and creativity in embodied forms.

Stop 1

“Each dance tradition has its own characteristic way of using mathematical concepts. For example, classical western ballet and Bharatya Natyam both use a strong sense of line” (p. 16)

It’s interesting to see that although there are so many different kinds of dance, they are all connected to mathematics in some way. In other words, there is always math behind the dances we see. When we watch a dance performance, we don’t usually notice the mathematics right away, but it’s fascinating to realize that mathematical concepts are embedded beneath the movements. This also reminds me of how mathematics can be a universal language. While different cultures and traditions have their own unique dance forms, there is still mathematics underlying the movements and choreography. That connection across cultures is really cool.

Stop 2

“He has explored polyhedra in dance using various props: PVC pipes, loops of string, and even fingers. In one dance, he used linked and glow-painted PVC pipes to make both polyhedra and whimsical shapes. This is an example of how mathematical ideas have led to dance movements that then led back to a mathematical question.” (p.19)

In the article, the authors provide many concrete examples of how mathematics is deeply linked to dance. What stood out to me is how simple, everyday materials such as PVC pipes, string, or even our own fingers, can be used to explore complex mathematical ideas like polyhedra. This highlights how mathematical thinking does not always require formal tools or written notation. Instead, it can emerge through movement, manipulation, and play. This makes me stop and feel both curious and amazed at how common mathematics is in our everyday lives, even in places we might not expect, like dance and art. 

Wonder:

How might using simple, everyday materials like string, sticks, or our own bodies change the way students experience and understand mathematical ideas? 

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Activity Reflection:

I tried the star-making activity inspired by Scott Kim and Karl Schaffer with my daughter, and it immediately reminded me of Cat’s Cradle, which seems to be a big hit among elementary students. My daughter loves playing it with her friends, creating increasingly intricate figures until there are no more twists left and the string returns to a simple loop again.I think this would be a great activity for teaching concepts related to geometry, sequences, and spatial reasoning. It can also be connected to ideas from knot theory in an intuitive, hands-on way. Beyond the mathematical concepts, this activity naturally encourages communication and collaboration, as students often need to explain, demonstrate, and problem-solve together. This feels especially suitable for younger grades since the materials are simple and accessible, as you only need a piece of string. Students can also explore and invent their own patterns, which opens up space for creativity and playful mathematical thinking.

My Star

My daughter's star

Our star

Week 5 Reading Reflection

Reading option A:  

Leslie Dietiker: What Mathematics Education Can Learn from Art: The Assumptions, Values, and Vision of Mathematics Education

Summary:

In this article, Dietiker draws on Eisner’s (2002) proposal of applying an artful lens to educational challenges and re-imagines the mathematics curriculum as a form of art. She argues that traditional mathematics instruction often follows predictable, standardized structures that can feel uninspiring and limit imagination. Instead, Dietiker invites educators to view mathematics learning as an art form that can spark creativity, insight, and renewed ways of seeing. She also suggests conceptualizing mathematics lessons as stories, intentionally crafted experiences rather than instructional manuals, to create richer and more engaging learning opportunities for students.

Stop 1: “ Mathematical aesthetic can generally be understood to be an individual's response to a mathematical experience, such as a sense of fit of a possible pattern or insight into an underlying structure of a particular problem.”

This made me pause to think about what mathematical aesthetics really means. Many people may not naturally associate mathematics with aesthetics or art, so I appreciated this explanation. Framing mathematical aesthetics as an individual’s response to a mathematical experience highlights the personal and experiential nature of learning mathematics. Rather than focusing solely on correct answers or outcomes, this perspective shifts attention toward the process, the moments of insight, and the feelings that arise when a pattern “fits” or a structure becomes clear. I think this approach has the potential to make mathematics more engaging and meaningful for students, especially those who struggle with viewing math as rigid or purely procedural.

Stop 2: “Interpreting mathematics as a story repositions mathematics curriculum from an instruction manual or a collection of facts to a form of art, intentionally crafted to offer aesthetic experiences for a set of students, whether positive or negative.”

The idea of conceptualizing mathematics lessons as stories resonated strongly with me. It reminded me of a course I took on teaching with illustrated materials, where one assignment involved creating a math concept book focused on patterns in nature. The book introduced foundational math ideas in a way that felt natural, engaging, and age-appropriate, particularly for students in Kindergarten to Grade 2. Connecting this experience back to Dietiker’s argument, I agree that integrating storytelling and artistic elements into math instruction can create meaningful aesthetic experiences. These experiences may help students see mathematics as something alive and connected to the world around them, rather than a set of isolated rules or procedures.

Discussion questions:

• What might it look like to intentionally design a mathematics lesson around creating an aesthetic experience, rather than focusing primarily on efficiency or coverage of content?

• In what ways could viewing mathematics as a story or art form challenge traditional expectations of what “good” math teaching looks like?

Week 5 Activity: Sarah Chase's Movements

 To be honest, this week’s activity was a bit challenging for me, as I don’t have much musical or dance background. My body coordination isn’t great, and dancing doesn’t come naturally to me. However, to help myself better understand the video, I quickly sketched Sarah Chase’s 3-against-2 movements. I realized that this could actually be an extension activity itself: having students sketch the movements as a way to deepen their understanding of the patterns and relationships involved.

Another possible extension would be to ask students to add an additional arm movement on the right hand, such as a “spring” motion (I added a set of spring movements in my sketch). Students could physically try out these movements and explore how the patterns change. What if leg movements were added as well? How many different combinations or possibilities could emerge?

For a further extension, I was also thinking about whether Sarah Chase’s idea could be replicated using smaller-scale movements, such as finger motions instead of full-body movements. These are just some initial thoughts, and I think they would be much more meaningful to explore with students and see how they interpret and respond to them in practice.

Curriculum ideas:

• Mathematical patterns exist in time and motion, not only on paper

• Embodied experiences can support conceptual understanding 

• Students often manipulate patterns and ratios symbolically without understanding what they represent

Guiding questions:

• How can we connect our body with math?

• How can mathematical patterns and ratios be represented with body movements?

• How can we translate movement into diagrams or sketches?

Project Outline: 3D Paper Model

 This is the project idea we created (Sunny & Sukie).  We want to focus on the math topic of surface areas, volume and nets of prisms to engage students in designing a template to form a 3D paper model.  We researched our idea from academic papers, websites, and videos (annotated bibliography is attached as a Google Doc).  We wish to try the idea by ourselves first and introduce it to the class --- and of course, if you are willing to join this 3D paper model design challenge, feel free to let us know!  We are happy to hear from you! 

**Complete project outline with annotated bibliography is attached as a Google Doc: Link to complete outline

Contributors: Sunny Hu & Sukie Liu 

Name of the math project: 3D Paper Model: From Nets to Spatial Structures

Grade level: 8 – 10 mathematics, Richmond School District 

Project Idea:

In this project, students will design and draw a 2D template composed of multiple geometric shapes that can be assembled into a 3D paper model of their choice, such as an animal, fruit, or vehicle. Through the process of planning, measuring, and constructing their designs, students will apply their understanding of nets of prisms, surface area, and volume to combine different geometric forms into a coherent structure. The transformation from a flat template to a completed 3D model emphasizes spatial reasoning and embodied learning, as students physically fold, assemble, and refine their designs. By integrating mathematical accuracy with creative choice, this activity positions mathematics as both a problem-solving tool and a medium for artistic expression. 

 

Examples of final product

(source: https://www.polypapercraft.com/products/fox-low-poly-papercraft-kit)

Mathematical topics:

-       Nets of 3D shapes and transformations from 2D to 3D 

-       Surface area and volume 

-       Geometric solids (prisms and related polyhedral)

-       Spatial reasoning and visualization

-       Mathematical communication through design and construction

Embodied and arts-based pedagogical approaches:

-       Students design, cut, fold, and assemble 3D paper models to physically experience mathematical structures 

-  Using origami-inspired techniques, students fold and form-making to emphasize precision, symmetry, and geometric relationships 

-       Learning through touch and manipulation of materials support conceptual understanding

-       Combining mathematics, visual design, and craftsmanship encourage creativity

Week 4 Reading and Activity Reflection

Reading: Berezovski, Cheng & Damiano (2016). Spinning Arms in Motion: Exploring Mathematics within the Art of Figure Skating. Bridges Math and Art proceedings. 

Summary

This reading is based on a workshop presented at the Bridges Finland Conference, where the authors explore the mathematics embedded in figure skating, specifically focusing on upright spins and arm movements. Using a bird’s-eye view, the authors introduce two mathematical models that represent the motion of skaters’ arms during spins, with the intention of connecting these movements to middle and secondary school mathematics topics. The workshop highlights how concepts such as proportional relationships, regression, circular geometry, and trigonometry can be explored through the analysis of spinning motion. Rather than presenting detailed results or student responses, the paper primarily serves as an invitation to view artistic and athletic movement as a rich site for mathematical inquiry and classroom activity design.

Stop 1: 

“Some of the mathematical topics used in these activities include proportional relationships, regression, circular geometry, and trigonometry.  “

What stood out to me here was how many mathematical ideas can be embedded in something as brief as a single spin that may only last a second or two. While it is not surprising that movement involves mathematics, I was still amazed by how much mathematical thinking could be unpacked from such a small, focused moment of motion. This made me reflect on how often these mathematical opportunities go unnoticed in everyday life, especially when we observe movement passively rather than analytically. It also reminded me that mathematics does not always have to come from long, structured problems and it is sometimes hidden in fleeting moments that we usually take for granted.

Stop 2:

“Contexts of interest to students include art and sports. Figure skating is a captivating art and fascinating sport. One of the most beautiful required elements in a singles figure skating program is a spin. Being able to perform a spin requires the simultaneous interaction of multiple body parts in order to execute each movement within the spin.”

I strongly agree with the idea that art and sports can be powerful contexts for engaging students, especially during moments of shared excitement such as worldwide events like the Olympics. These contexts feel meaningful and relevant, and they naturally invite curiosity. Although I am not personally a figure skating fan, I have watched competitions during the Winter Olympics and found the sport visually captivating. When watching skaters spin, I often find myself wondering about the mathematics behind their movements like the angles of takeoff, the direction and speed of rotation, and how subtle changes in body position affect balance and motion. This reading encouraged me to think more intentionally about those questions and about how much mathematical reasoning and embodied knowledge must be involved and supported by extensive practice and precision

 

Discussion questions:

• What are some practical ways teachers could adapt activities like the figure skating models described in this article for students who may not have access to specialized sports or physical spaces?
  
  

• How might embodied activities like analyzing motion through dance, sports, or everyday actions support students who struggle with more abstract representations of mathematics?

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Activity

https://gallery.bridgesmathart.org/exhibitions/2016-bridges-conference/dsthompson01

For this week’s activity, I created a hand-drawn art replication inspired by David Thompson’s maze artworks based on the Traveling Salesperson Problem (TSP), originally presented at the Bridges 2016 Conference. Thompson’s original pieces were generated using digital tools such as StippleGen 2, MATLAB, and Microsoft PowerPoint, combining mathematical algorithms with artistic design.

In my own attempt at replication, I chose not to use any software and created the maze entirely by hand. I selected the letter S as the frame for my design, inspired simply by the cup sitting beside my computer. I began by outlining the letter shape and then gradually filled it with continuous, maze-like paths, treating the process as a form of structured doodling.

Although my hand became a bit sore during the process, I found the activity surprisingly calming and enjoyable. My attention was fully absorbed in tracing the paths and maintaining continuity, which allowed my mind to slow down and focus in a different way than it usually does during academic tasks. This experience made me appreciate how mathematical thinking can support artistic creation, even when the math itself is not explicitly visible. At the same time, this activity made me reflect on how deeply technology has transformed both art and mathematics. Tools like software and algorithms not only expand what artists can create but also make complex patterns more accessible, precise, and scalable. This activity reminded me that mathematics and technology work together as powerful creative tools to express ideas and make meaning through art.