Week 7 Reading

 Reading

Futamura, F. (2025). Writing a mathematical art manifesto. Bridges 2025 Conference Proceedings, 589–594.

Introduction

The article explains a workshop designed to help people think about what mathematical art really is and how to write a mathematical art manifesto. A manifesto is a bold statement that challenges old ideas and proposes new ones, and the author looks at how past art movements used manifestos to break away from tradition. The workshop encourages participants to define mathematical art not as a teaching tool or a branch of mathematics, but as a form of creative expression inspired by mathematical ideas. The article gives a short history of how math has influenced modern art—from artists like Duchamp, Dalí, and Escher to computergenerated art—and shows that math has long shaped artistic creativity. It also presents different artworks for discussion and quotes from artists who use mathematics in their work. Finally, the article explains the typical structure of a manifesto: identify an old artistic belief, criticize it, and propose a new vision. Overall, the article invites the mathematical art community to reflect on its values and possibly create one or more manifestos that express what mathematical art means today.

Reflection

One of the most striking moments in the article comes early, when Futamura writes, “We will attempt to do this from the perspective of creative human expression, apart from its usefulness as demonstration, illustration or pedagogy” (p. 589). This sentence immediately reframes how many of us—especially those who identify as teachers—tend to approach the relationship between mathematics and art. We often default to thinking about how art can help students understand math concepts or how mathematical ideas can be visualized for instructional purposes. But this line reminds us that mathematical art does not need to justify itself through education. It can stand independently as art. It can exist simply because it is beautiful, expressive, or meaningful. This shift in perspective frees us from the constraints of our professional identities and invites us to appreciate mathematical art the same way we appreciate any other artistic form: as something created to evoke emotion, curiosity, or wonder.

The article also emphasizes that many artistic theories and movements have historically drawn from mathematical ideas. When we recognize that artists like Duchamp, Dalí, and Escher were inspired by geometry, topology, and higher dimensions, the boundary between “mathematical art” and “art” becomes much less rigid. Mathematical art is not a niche or secondary category—it is part of a long artistic lineage. Understanding this helps us see that connecting mathematics and art is not an educational strategy but a natural continuation of artistic exploration.

Another sentence that resonated deeply with me is George Hart’s statement: “The art that is mathematical art must bring to mind a landscape of mathematical pleasure” (p. 592). This description beautifully captures the emotional goal behind linking math and art. The purpose is not necessarily academic; it is experiential. Mathematical art invites us into moments of recognition, surprise, or delight—those small “aha” moments that feel both intellectual and aesthetic. Even if these moments do not immediately translate into learning outcomes, they plant seeds. They create memories, associations, and a sense of familiarity with mathematical ideas. Later, when someone encounters a similar pattern, number, or formula, that earlier spark may return. In that moment, learning happens naturally and authentically.

In this way, mathematical art becomes a quiet but powerful form of education—not through direct instruction, but through cultivating the capacity to notice beauty, structure, and meaning. It reminds us that mathematics is not only something to be solved but something to be felt.

Question

If mathematical art is allowed to exist purely as creative expression—without needing to teach, explain, or justify anything—how might that freedom change the way you see mathematics, art, or even your own identity as someone who engages with both? Do you have any experience like that?

Week 7 Interview Reflection

Watching the interview with mathematical and STEAM artist Nick Sayers broadened my understanding of how mathematics and art can intersect in ways that are both intellectually rich and visually compelling. One moment that particularly resonated with me occurred around the 38‑minute mark, when Sayers demonstrated his drawing machine. At first glance, the device seems simple—just a set of rotating parts powered by a pedal—but once it begins tracing lines, the underlying mathematical structure becomes beautifully visible. The machine repeats patterns over and over, and through this repetition, an intricate artwork emerges.

What fascinated me most is how the machine essentially acts as a physical embodiment of a mathematical formula. Changing the structure of the machine is like rearranging mathematical symbols—addition, subtraction, rotation—and observing how the same input produces a different output. When Sayers adjusts the speed of the pedal, it becomes a metaphor for altering a variable. Suddenly, mathematics is not abstract or distant; it is something you can see, hear, and even feel.

This experience pushed me to rethink what “embodied mathematics” can mean. We often talk about embodiment in terms of gestures or physical space, but Sayers expands the idea by showing that mathematical concepts themselves can be made physical. Instead of writing formulas in a textbook, we can build devices that perform the formulas and generate artwork as the result. Because almost anyone can appreciate visual beauty, this becomes a powerful way to make mathematics more inclusive. People who might feel intimidated by symbols or equations can still understand the underlying patterns when they watch them unfold artistically. It becomes a way to invite more people into mathematical thinking without requiring them to first master formal notation.

Another moment that deepened my thinking came around 1:10:00, when Sayers described using natural light to create images on photo paper. This idea challenged my assumptions about what counts as a mathematical “input.” Inputs do not always have to be numerical. Light, time, and motion—natural phenomena—can act as variables that shape an artistic outcome. In this sense, nature itself becomes a collaborator in the mathematical process. Instead of artificially feeding numbers into a formula, Sayers allows the world to provide the input, and the resulting artwork becomes a record of natural patterns. This approach gives mathematics a kind of warmth and “temperature,” transforming it from something cold and symbolic into something alive, dynamic, and observable.

As a math or science teacher, Sayers’s work also connects deeply with the idea of experience‑based education. We often assume that “experiencing” mathematics means applying formulas to real‑world problems, but Sayers shows that experience can also mean making the formula itself into a physical event. His approach reframes what mathematical experience can look like and challenges the traditional boundaries of how students encounter mathematical ideas.

His machines, his light‑based artworks, and his process‑driven creations all demonstrate that mathematical ideas can be encountered through the body, through materials, and through sensory engagement—not just through symbolic manipulation. This expands what it means for students to “do” mathematics. Instead of only solving equations on paper, they can build devices that enact those equations, or use natural phenomena as inputs that reveal mathematical structure. In this way, the experience becomes the mathematics. Students are not just learning about patterns; they are watching patterns emerge. They are not just calculating relationships; they are physically generating them. This shift transforms mathematics from something abstract into something lived and observable.

This approach also gives mathematics a new kind of accessibility and emotional resonance. It allows learners to feel the rhythm of repetition, to see the geometry of motion, and to witness how natural forces like light and time can become variables in a creative process. These sensory experiences help students build intuition and curiosity before they ever encounter the formal symbols. Teachers should be encouraged to imagine a classroom where mathematical ideas are not confined to symbols but are lived, constructed, and observed—where students can experience formulas as actions, patterns, and artworks. Mathematics should not only a tool for describing the world but also a medium for experiencing it directly.

Project Outline

https://docs.google.com/document/d/1agKTDdxMUNeWQfD-SaPVz-xN88vwCd95yhJuY0UAJ9I/edit?tab=t.0

Week 6 Reading

belcastro, s.-m., & Schaffer, K. (2011). Dancing mathematics and the mathematics of dance. Math Horizons, 18(3), 16–20. https://www.jstor.org/stable/10.4169/194762111x12954578042939

 

Introduction

In Dancing Mathematics and the Mathematics of Dance, belcastro and Schaffer explore the deep and often surprising connections between mathematical ideas and artistic dance. Although dance may appear purely expressive, the authors argue that mathematical structures are “intrinsic to dance,” appearing in rhythm, symmetry, spatial patterns, and choreographic design. They describe how dancers naturally use counting, geometric lines, and repeated patterns, while choreographers rely on transformations such as reflection, rotation, and translation to shape movement phrases. The article highlights both local symmetries within a single dancer’s body and global symmetries among groups of dancers.

The authors also discuss the influence of Rudolf Laban, whose movement theories use polyhedra to classify direction, effort, and spatial intention. For example, Laban mapped eight movement qualities—such as press, glide, flick, and punch—to the vertices of a cube. This blending of geometry and kinesthetic experience illustrates how mathematical models can guide artistic expression.

Beyond theory, belcastro and Schaffer describe their own mathematically inspired choreographic works. These include dances based on group theory, graph theory, rhythmic star polygons, and even game theory. One example involves dancers embodying the internal angles of a triangle to demonstrate that they sum to 180∘. Their creative process often moves fluidly between mathematics inspiring dance and dance inspiring new mathematical questions.

Ultimately, the authors emphasize that dance is multifaceted—artistic, social, expressive—and mathematics is just one lens through which it can be explored. Yet the interplay between the two fields enriches both, offering new ways to understand movement, structure, and creativity.

Reflection

Reading Dancing Mathematics and the Mathematics of Dance opened up an unexpectedly rich way of thinking about movement, beauty, and mathematical structure. What struck me most were the concrete dancing examples the authors used to reveal patterns that, on the surface, seem purely artistic but are deeply mathematical. The symmetry exercises, the hopstepjump patterns, and especially the permutationbased choreography immediately resonated with me. When I saw the composition table of symmetries, I was surprised by how similar it was to the examples I’ve shown my own students when teaching permutations and combinations. I had never considered that a dancer switching positions or reversing a phrase could be viewed as a permutation in motion. That realization made me rethink how dance can serve as a living, breathing model for mathematical ideas.

This connection feels like an exciting opportunity for the classroom. Integrating dancebased patterns into lessons could help students see that mathematics is not confined to numbers on a page—it is a language that describes structure, rhythm, and form in the world around them. Using movement as an example might spark curiosity and help students appreciate mathematics as a kind of art, one that underlies many artistic traditions they already enjoy.

What deepened my appreciation even further was the authors’ thoughtful discussion of the relationship between mathematics and art. They write, “To be clear, we don’t view dance entirely through the lens of mathematics—or vice versa… Dance is many things, sometimes all at the same time” (p. 20). This perspective is powerful because it acknowledges that while mathematics can illuminate dance, it should not overshadow the emotional, cultural, and expressive dimensions that make dance meaningful. Instead, the goal is interdisciplinary dialogue—using math to analyze art and using art to reinterpret math.

This idea pushes me to think more broadly about what we are really doing when we bring mathematics into conversations about dance, music, or other creative fields. We are not trying to claim that math is the foundation of everything, even if mathematically inclined people sometimes feel that way. Rather, we are opening doors between disciplines that are too often kept separate. When students see math interacting with art, they gain a new lens through which to understand both. They may begin to view mathematical symbols not as abstract obstacles but as tools for describing patterns they already appreciate intuitively.

Ultimately, this interdisciplinary approach can help redefine what it means to be a “mathematician.” If math is allowed to be expressive, embodied, rhythmic, and creative, then more students may feel welcomed into the field. By showing that mathematics connects to the arts they love, we can cultivate a more inclusive and positive mathematical identity—one that invites curiosity rather than fear.

Question

What opportunities do you see for using dance, music, or visual art to help students understand abstract mathematical ideas?

Week 5 Post

Reading

Kelton, M. L., & Ma, J. Y. (2018). Reconfiguring mathematical settings and activity through multi-party, whole-body collaboration. Educational Studies in Mathematics, 98(2), 177–196. https://doi.org/10.1007/s10649-018-9805-8

 

Introduction

Kelton and Ma (2018) explore how multi‑party, whole‑body collaboration can transform mathematical learning by reconfiguring both activity and space. Grounded in theories of embodied cognition and the social production of space, the authors argue that mathematics learning is inherently bodily, interactive, and situated. Rather than viewing classrooms as neutral containers or cognition as primarily mental, they conceptualize learning spaces as dynamically produced through collective bodily action.

The study presents a comparative analysis of two instructional cases using micro‑ethnographic and multimodal methods. In the first case, Walking Scale Number Lines (WSNL), students work in a gymnasium where a large taped number line allows them to physically enact numerical positions and operations such as scaling and opposites. Students’ bodies function as mathematical objects, and coordination challenges—such as avoiding collisions while moving simultaneously—become central to mathematical reasoning. The fixed spatial structure of the number line interacts with students’ movement to shape emerging mathematical practices.

The second case, Whole and Half (W + H), takes place in a fifth‑grade classroom and focuses on ratio and part–whole relationships. Students work in pairs to create and respond to bodily intervals using hands, arms, and full‑body positioning. As students move away from desks and incorporate features like floors, walls, and boards, the classroom space is reconfigured. Mathematical objects in this case are highly mobile, shifting as bodies move and enabling creative extensions of the task.

Across both cases, the authors show that interactional breakdowns are simultaneously social and mathematical, revealing the inseparability of coordination, space, and meaning‑making. Kelton and Ma conclude that intentionally designing for whole‑body, collaborative activity can expand participation, reposition learners as active components of mathematical objects, and open new possibilities for collective mathematical sense‑making.

 

Reflection

One idea that particularly stood out to me in Kelton and Ma (2018) is their claim that “the physical spaces of learning should not be treated as static boxes waiting to be filled with human activity, but instead as complex, historically constituted, dynamically experienced, and socially produced settings” (p. 178). I find this perspective compelling because space itself is deeply mathematical. Many areas of mathematics—such as geometry, vectors, transformations, and even advanced calculus—are fundamentally built on spatial concepts. From this standpoint, it feels limiting to treat space merely as a neutral backdrop for learning. Instead, space should be understood as an active component that shapes how students experience, interpret, and engage with mathematics.

I strongly agree that learning spaces can provide students with different sensations, orientations, and perspectives that support mathematical thinking both mentally and physically. Even when students’ movements in space are not directly aligned with specific curricular content, those movements can still support mathematical processing. Walking, repositioning, and orienting oneself spatially can serve as ways of organizing thought, testing ideas, and developing intuition. Learning, therefore, does not only happen when students are seated and focused on formal representations; it also happens through bodily engagement with space.

A second passage that resonated with me states that “participants creatively leveraged new possibilities for—and constraints on—physical movement in relation to the environment in order to make innovations and elaborations on the mathematical task” (p. 192). This idea is clearly illustrated in the classroom activities examined in the study. For example, in the Walking Scale Number Line activity, students’ bodies functioned as points on a number line, and the need to move simultaneously without colliding led them to invent new mathematical strategies, such as rotating their bodies together to represent opposites. What began as a practical problem of physical coordination became a meaningful way to experience multiplication by −1. Similarly, in the Whole and Half activity, students used their bodies to create and respond to intervals, creatively incorporating classroom features like the floor, walls, and Smart Board to extend the task.

These movements were not distractions but central to students’ mathematical reasoning, as they adjusted their actions in response to spatial constraints and opportunities. Together, these examples demonstrate experience‑based learning in action: students adapt to changing physical conditions, negotiate meaning collaboratively, and deepen their understanding by making mathematics tangible through movement and space.

As a future teacher, this reinforces for me the importance of designing conditions rather than tightly controlling outcomes. When teachers intentionally create environments that invite movement, exploration, and collaboration, they can reduce conceptual gaps and misunderstandings. More importantly, such environments empower students to think critically, challenge themselves, and develop deeper, more flexible understandings of mathematics through embodied experience.

 

Question

Have you had experiences in school‑based learning where physical space—such as seating arrangements, classroom layout, or opportunities to move—shaped how you understood or engaged with the lesson? How did those spatial arrangements support or limit your learning?

 

Week 4 Post

Capozucca, A., & Fermani, M. (2019). Make music visible, play mathematics. In Proceedings of the Bridges 2019 Conference (pp. 647–650). Bridges Organization.

Introduction

The article Make Music Visible, Play Mathematics by Andrea Capozucca and Marco Fermani presents an interdisciplinary, hands-on workshop that connects mathematics and music through geometry, with the goal of making mathematics audible and music visible through playful, multisensory learning experiences. The authors argue that music and mathematics share deep structural relationships that go beyond counting and ratios, and that these connections become especially clear when musical harmony is explored visually and spatially. Using the chromatic scale arranged as a circle, where the twelve notes are evenly spaced and musical intervals correspond to angles, the workshop provides a concrete geometric framework for understanding music theory.

The workshop follows a five-part discovery-based structure that emphasizes active participation. Participants first explore which musical intervals sound pleasant through embodied listening activities, then construct geometric segments and triangles representing these intervals using simple materials. Through this process, they discover that only four types of triangles can be perfectly inscribed in the chromatic circle, corresponding to the four fundamental chord types: major, minor, diminished, and augmented. By rotating and reflecting these triangles within the circle, participants experience musical transposition and transformation, learning that geometric rotation preserves a chord’s identity while symmetry changes it.

In the later stages, participants apply their geometric understanding to analyze the harmonic structure of familiar songs and collaboratively compose original music using geometric “recipes.” The authors conclude that this inquiry-based, playful approach increases engagement, confidence, and creativity in both mathematics and music learning. They argue that such interdisciplinary workshops support open-ended problem solving, authentic collaboration, and positive attitudes toward learning, and that geometry can function as a shared language that allows mathematics and music to mutually enrich one another across educational contexts.

Reflection

This article is illustrative but not comprehensive; it is a short piece that offers a glimpse into how mathematics and music can be meaningfully connected through geometry and hands-on learning. While it does not aim to fully map the theoretical foundations of either discipline, it succeeds in opening a creative window into interdisciplinary thinking and shows how mathematical ideas can be experienced in ways that are intuitive, sensory, and playful.

One point that made me stop while reading was the authors’ statement that “mathematics is about structure and pattern.” I really appreciate this way of describing mathematics because it captures what feels most fundamental about the subject. I think this is exactly why mathematics can serve as a root for so many different areas and be investigated across disciplines. In basic terms, much of the work we do in mathematics, science, and even the arts involves identifying patterns, finding ways to describe those patterns clearly and systematically, and then creating or building something new based on them. Seen from this perspective, mathematics is not just a school subject but a way of organizing and making sense of the world.

The connection the article builds between music and mathematics is also especially interesting, particularly the idea that “music is the sensation of counting without being aware you were counting.” This insight resonated with me, although I also think music goes beyond counting alone. Sound itself can be understood numerically through properties such as frequency measured in hertz, even if I have not formally studied music from this perspective. Still, the idea that musical elements like chords can be analyzed mathematically—just as the authors do through geometric representations—suggests a rich and enjoyable way to explore music. This approach makes me curious about how much musical structure, including chord types and harmony, might be better understood by uncovering the patterns and numbers behind what we hear.

Overall, it has brought me insight into a way to analyze art. I think it invites a broader view of mathematics as a deeply human and artistic activity rather than a purely technical or procedural one. By using geometry to explore musical harmony, the authors show how mathematics can function as a creative language for interpreting sound, beauty, and structure. Mathematics here is not presented as a set of rules to be memorized, but as a way of seeing, shaping, and creating meaning—much like art and music themselves. I guess that’s why many artworks can be demonstrated mathematically.

Question

Do you have any experience investigating a form of art—such as music, visual art, or dance—through mathematics, or noticing mathematical patterns while creating or experiencing art?