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?

 

Week 3 Post

Doolittle, E. (2018). Off the grid. In S. Gerofsky (Ed.), Contemporary environmental and mathematics education: Modelling using new geometric approaches (pp. 101–121). Springer. https://doi.org/10.1007/978-3-319-72523-9_7

Introduction

In “Off the Grid,” Edward Doolittle critiques the widespread reliance on grid-based ways of organizing space and time, arguing that grids—rectangular plots, straight borders, fixed schedules, and Euclidean geometry—create an illusion of order and fairness while often distorting reality. Drawing on Indigenous knowledge (particularly Rotinonhsonni traditions) and advanced mathematics, Doolittle shows that grids work only under ideal conditions and frequently fail in natural, social, and educational contexts.

Through examples from urban planning, land surveying, agriculture, and mapping, Doolittle demonstrates how grids clash with uneven terrain, the curvature of the earth, ecological variation, and lived experience. He links grid thinking to control, ownership, and colonial power, noting how imposed grids have contributed to Indigenous land dispossession and rigid schooling practices. In contrast, Indigenous spatial and temporal frameworks emphasize relationship, responsiveness, and place, such as defining territory by watersheds rather than straight lines, or timing agricultural practices by animal and plant behavior rather than calendars.

Doolittle proposes moving “off the grid” by embracing alternative geometries—including Riemannian geometry, fractals, complexity theory, knot theory, and weaving—which better reflect natural systems and change over time. He concludes that these “geometries of liberation” can help free mathematics education from rigid abstractions and reconnect it to ethics, ecology, and lived reality.

 

Reflection

Reading Edward Doolittle’s Off the Grid made me think deeply about how mapping and geometry differ across countries, cultures, and histories. One contrast that stood out to me is the way political boundaries are drawn. In Canada and the United States, many provincial and state boundaries appear as straight lines, reflecting grid-based surveying systems imposed for administrative efficiency. In contrast, in China (and similarly in Korea and some other regions), boundaries are often nonlinear, shaped by natural features such as rivers and mountain ranges, as well as long-standing political and historical processes.

In addition, I remember that in Chinese history education, even though I am not good at it, there is a commonly referenced natural division between northern and southern China, often defined by a major mountain range and river system. This geographic boundary is not just a line on a map; it shapes people’s sense of belonging. Many people still identify themselves as being “from the north” or “from the south,” and these identities correspond to clear differences in climate, agriculture, food cultures, dialects, and lifestyles. These differences emerged because people adapted to the land and environment around them, rather than forcing the land into a uniform grid. This resonates strongly with Doolittle’s argument about thinking “off the grid,” and it also makes me wonder whether this difference reflects, at least partly, the contrast between immigration countries and non-immigration countries in how land is conceptualized, divided, and governed.

This chapter also reinforced my belief that bringing this kind of knowledge into mathematics education is critical. Geometry should not be limited to straight lines, right angles, and rectangular grids. To me, mathematics is better understood as a discipline of reasoning, pattern recognition, and sense-making. The way people live differently on the two sides of a mountain or river in China is not random—it reflects patterns shaped by geography, climate, and history. These patterns can be analyzed mathematically, but only if mathematics is allowed to connect with geography, history, ecology, and culture.

Doolittle’s work reminds us that mathematics is not separate from lived reality. Teaching students to see math as a tool for understanding human–environment relationships, rather than just a set of abstract rules, opens powerful possibilities for cross-disciplinary learning and more just, meaningful education.

 

Question

Because Off the Grid is primarily an ideological and philosophical piece rather than an empirical study, I find it difficult to pose a highly technical question about it. Instead, could you share an experience, observation, or example—from teaching, learning, mapping, or everyday life—where grid-based thinking failed to capture reality, or where thinking “off the grid” offered a deeper or more just understanding of space, time, or relationships? How does this example help you interpret Doolittle’s argument?

 

Week 2 Post

Reading 2

Fernandes, S. H. A. A., & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind students’ interactions with symmetry. *RIPEM – Revista Internacional de Pesquisa em Educação Matemática*, 3(1), 36–55.

Introduction

Fernandes and Healy investigate how blind students construct mathematical meaning in geometry, focusing on symmetry and reflection, through the lens of embodied cognition. Challenging the assumption that mathematical understanding is primarily visual, the authors argue that cognition is grounded in bodily action and multimodal experience, drawing on phenomenology (Merleau‑Ponty) and neuroscience (e.g., Barsalou, Gallese, Damasio).

The study is based on task-based interviews with two blind secondary‑school students: Edson, who lost his sight in adolescence and retained visual memories, and Lucas, who had been blind since early childhood and had no visual memory. Using tactile tools such as cardboard shapes and a geoboard with elastic bands, the students worked on tasks involving symmetrical figures and reflections.

Findings show that both students relied heavily on touch, movement, gesture, and imagination to make sense of symmetry. Edson often used visual memories—especially mirrors and folding—to simulate reflections, initially reasoning at an intrafigural level by focusing on congruent parts of single figures. With guidance, he began to coordinate distances and orientations, moving toward interfigural reasoning. Lucas, lacking visual memory, developed strategies grounded in tactile exploration and spatial relationships, often focusing on angles, distances, and invariants across figures. His thinking frequently exhibited interfigural characteristics, sometimes more readily than Edson’s.

A key contribution of the paper is the demonstration that knowing symmetry does not replace feeling symmetry: mathematical concepts were not abstracted away from bodily experience but were constituted through it.

 

Reflection

One meaningful finding from the article is the authors’ observation that both blind students naturally moved their hands in symmetrical ways when first exploring tactile materials. This behaviour was not anticipated or intentionally designed into the tasks, suggesting that symmetry can emerge through embodied interaction rather than only through formal instruction or visual representation. This observation highlights that mathematical ideas may first be sensed and enacted through the body before they are explicitly named, defined, or symbolized.

This finding encouraged me to think differently about classroom learning. It suggests that students’ gestures and physical movements are not incidental or secondary, but may be important expressions of their thinking. Paying closer attention to how students interact physically with learning materials can reveal early forms of understanding that may not yet be verbal or symbolic. In this way, the article challenges teachers to recognize embodied action as a legitimate and valuable form of mathematical reasoning.

Another moment that stood out to me was when the authors argued that blind learners do not necessarily follow the same developmental trajectories proposed for sighted learners, and that mathematical reasoning is strongly shaped by task design and the mediational resources available to students. This idea connects closely to a framework of learning that views cognition as built on experience. I found this particularly compelling because it resonates with my reading from another course on experience‑based learning, in which learning is described as a process of making sense of the world rather than simply acquiring abstract knowledge.

The article helped me reflect on the distinction between experience and knowledge. Experience itself is not equivalent to learning outcomes or finalized understanding; the same learning experience can lead different students to develop different forms of knowledge. Simply providing identical experiences does not guarantee equal learning gains. Students interpret experiences through the lens of their everyday lives and embodied interactions with the world, which helps explain why blind students should not be expected to learn mathematics in the same ways as sighted students.

In this context, the authors’ emphasis on feeling symmetry rather than only viewing it is particularly important. Engaging with symmetry through touch and movement does     not just provides blind students with learning experiences that align with how they make sense of mathematical ideas, but can also be extended and used  to help students from different background with distinct view of the world. These embodied experiences also support cognitive processes that visually oriented instruction alone cannot fully address, enabling deeper and more meaningful understanding.

 

Question

How can the findings of this article inform everyday teaching practices to enhance students’ learning?

 

Week 1 Post

Reading 2

Goldin-Meadow et al (2009). Gesturing gives children new ideas about math.

Summarization

The study by Goldin-Meadow, Cook, and Mitchell investigates how gesturing influences children’s math learning. Researchers compared three groups of third- and fourth-grade students: those instructed to produce correct gestures, partially correct gestures, or no gestures during a math lesson. Results showed that children who used correct gestures learned the most, followed by those using partially correct gestures, and then those who did not gesture. This improvement was mediated by whether children incorporated the grouping strategy—conveyed only through their gestures—into their spoken explanations after the lesson. The findings suggest that gestures do more than aid communication; they help learners extract and create new knowledge by embodying problem-solving strategies. In short, guiding children’s hand movements can enhance conceptual understanding in mathematics.

 

Reflection

One point that really stood out to me while reading the article was the statement: “So, perhaps people gesture for themselves” (p. 267). The author notes that people often gesture even when no one is watching, which made me reflect on my own experience. I realized that I frequently use gestures while talking on the phone. This suggests that gestures are not only for others but also help us organize our thoughts and speech. From my perspective, many gestures don’t carry specific content; sometimes we point when saying “you” or gesture toward ourselves when saying “I,” but often, especially when there is no physical reference like a whiteboard, gestures become random movements. In these cases, I believe gestures function like punctuation: they help structure our ideas, emphasize important points, and convey attitude. This process makes our expression clearer and helps us remember what we want to say.

I’ve also noticed that when someone is simply reading or memorizing a speech, gestures are minimal. For example, in my students’ presentations, when they read directly from slides or recite memorized text, gestures almost disappear. This happens because they don’t need to organize ideas in real time and may lack confidence in the content since it’s not coming from their own thinking. That’s why we often associate gestures with confidence—gestures reflect conviction and engagement with the message.

Another point that caught my attention was the article’s conclusion that gestures can create new knowledge. Compared to the earlier idea of gestures as a sign of confidence and attitude, this finding highlights a different role: gestures combined with meaningful content can be a powerful tool for communication. They help guide the audience’s attention, making it easier to follow the speaker’s ideas. In this way, gestures reduce cognitive load for listeners and enhance clarity. Overall, I think gestures serve two important roles in our lives: they help speakers organize their thoughts and express confidence, and they help audiences understand and retain information more effectively.

 

Question

l   What role does gesture play in your teaching and everyday communication?

l   How do you perceive and respond to gestures when you are in the audience?

 

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First post for 552 2026