Project Draft

 Human Circuits

Week 9 Reading

Article

Hawksley, A. J. (2015). Exploring ratios and sequences with mathematically layered beverages. Bridges Conference Proceedings, 519–524.

Introduction

The article Exploring Ratios and Sequences with Mathematically Layered Beverages by Andrea Hawksley describes a creative, hands‑on workshop that teaches mathematical ideas through the construction of layered drinks. By adjusting the ratios of sugar, water, and flavorings, participants can create liquids of different densities that neatly stack in a glass. This visually striking activity serves as an accessible way to introduce or reinforce concepts such as ratios and fractions, because students must calculate sweetness levels, compare densities, and test their predictions by physically pouring the layers. When calculations are incorrect, the layers mix, giving immediate and intuitive feedback.

Beyond basic ratios, the workshop extends into the exploration of integer sequences, especially those that are monotonic, since each subsequent layer must be less dense than the one below it. Participants can construct drinks reflecting arithmetic sequences, recurrence‑based sequences, or famously, the Fibonacci sequence. In the case of “Fibonacci lemonade,” the proportions of sugar and lemon juice in each layer follow adjacent Fibonacci numbers. As the layers progress, the ratio of these ingredients approaches the golden ratio, allowing students to taste a mathematical limit in action. This multisensory approach deepens the conceptual connection between numerical patterns and real‑world phenomena.

The article also gives practical instructions for building these beverages, emphasizing the importance of beginning with the densest mixture and pouring each layer slowly over ice to keep them distinct. Hawksley highlights the broader educational value of this activity, noting that many students struggle with fractions and benefit from tactile, playful learning experiences. She argues that food is an underutilized medium for mathematics education and suggests that layered beverages, snow cones, popsicles, and even salad dressings can become “mathematical foods” when ingredient ratios are intentionally designed. Overall, the workshop demonstrates how everyday experiences—like drinking lemonade—can become rich mathematical explorations.

 

Reflection

After just finishing the article, I am struck by how unexpectedly delightful the idea of Fibonacci lemonade is. I had never imagined that a mathematical sequence—something I usually encounter in diagrams, equations, or natural patterns—could be expressed through something you can actually taste. The realization that mathematics can live inside a beverage, layered through density and sweetness, feels almost like discovering a new form of art. It makes me rethink how math and creativity can intertwine in playful and surprising ways.

What stands out to me most is how this approach transforms math from something abstract into a sensory experience. Layering drinks based on ratios and sequences is not only visually appealing but also allows for immediate, hands‑on adjustments. You can change a ratio, alter a flavor, modify the density, and instantly see the result. That freedom to experiment reminds me of how artists adjust strokes on a canvas or modify tones in a piece of music. The process feels creative, intuitive, and personal—even though it is grounded in mathematical reasoning.

Reading this also challenged my previous definitions of art. I used to think of art as something you primarily see, hear, or touch. But this example shows that taste can be an artistic medium too. The idea that flavors can encode mathematical patterns expands the boundaries of what counts as artistic expression. It makes me realize that art and math share a deep connection through structure, experimentation, and the emotions they can evoke.

 

Question

How might you reimagine a familiar mathematical idea through a different sense—taste, smell, touch, or sound?

 

Week 8 Reading

 Reading

Karaali, G. (2014). Can zombies write mathematical poetry? Mathematical poetry as a model for humanistic mathematics. Journal of Mathematics and the Arts, 8(1–2), 38–45. https://doi.org/10.1080/17513472.2014.926685

 

Introduction

Gizem Karaali’s (2014) article explores the deep connections between mathematics, poetry, and human creativity, arguing that mathematical poetry can serve as an ideal ambassador for humanistic mathematics—the view that mathematics is fundamentally a human, creative activity.

Karaali begins by reflecting on her personal journey with poetry and mathematics, describing how the two domains once felt separate in her life: poetry lived in her native Turkish, whereas mathematics belonged to her English academic world. Over time, however, she recognized that both mathematics and poetry share the core human traits of cognition, consciousness, and creativity.

The article then introduces the broader concept of humanistic mathematics, which includes both teaching mathematics in a way that values students’ lived experiences and viewing mathematics itself as a cultural, emotional, and creative human endeavor. Karaali recounts the history of this movement, including the Humanistic Mathematics Network and the founding of the Journal of Humanistic Mathematics.

A significant portion of the essay focuses on mathematical poetry as a powerful bridge between the mathematical world and the emotional, artistic world. Karaali describes poetry readings at mathematics conferences, examples of poems published in the Journal of Humanistic Mathematics, and her own development as a writer of mathematical poetry

She also shows how integrating mathematical poetry into the classroom—through reading, writing, and discussion—can help students see mathematics differently. Students in her seminars enjoyed creating mathematical poems, which challenged stereotypes and encouraged them to engage playfully and creatively with mathematical ideas.

Karaali concludes that mathematical poetry can help humanize mathematics for students and the general public. Since poetry is widely recognized as a deeply human art form, pairing it with mathematics invites more people to appreciate the creativity, emotion, and humanity inherent in mathematical practice.

 

Reflection

Karaali’s article deepens our class discussion about viewing both mathematics and art as forms of creative human expression. She argues that mathematics is not just a technical subject but something rooted in the most essential aspects of being human: cognition, consciousness, and creativity. This perspective helped me rethink how mathematics can connect with poetry and other artistic practices.

One of the first places I paused while reading was the statement: “Translation from one natural language to another of mathematical texts may be deceptively simple, but note that mathematics itself is speaking a language of its own” (p. 39). This idea highlights the deep relationship between mathematics and poetry because both are forms of language that rely on structure and pattern. In this sense, they share an identity as systems for expressing meaning. Later, Karaali writes, “My mathematics and my poetry did not play together. They spoke different languages. They were of two different worlds” (p. 40). This made me think about people who are comfortable with mathematics but do not feel confident writing poetry. It raised an interesting question for me: if mathematics is already a complete language, could it be used on its own to create a meaningful poem? The possibility of writing a poem entirely in mathematical symbols feels both challenging and exciting, and it expands my understanding of what mathematical expression could look like.

Another important moment in the article appears when Karaali observes her students creating mathematical poetry: “In the classroom as they wrote and afterwards as they read their work, I could see that my students were engaged and enthusiastic about the ongoing creative process” (p. 43). This illustrates how creativity brings together the three human ingredients she emphasizes. Students enjoyed mathematics more when they could use it as a space for expression rather than as a task where correctness is the only goal. Traditional math learning often becomes exclusive because students do not experience joy until they “master” the material, and many give up before reaching that point.

Overall, the article suggests that humanistic mathematics can challenge traditional views of the subject and make it more inclusive. Through creativity—and especially through mathematical poetry—mathematics becomes a more inviting and human activity.

 

Question

What would happen to our relationship with mathematics if we all began to see it, not as a gatekeeper of correctness, but as a deeply human practice rooted in imagination, interpretation, and creativity?

 

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?