Reading Response: Dylan Thomas: Coast Salish artist

Dylan Thomas & Doris Schattschneider (2011) Dylan Thomas: Coast Salish

artist, Journal of Mathematics and the Arts, 5:4, 199-211, DOI: 10.1080/17513472.2011.625346

 https://doi.org/10.1080/17513472.2011.625346

In the article "Dylan Thomas: Coast Salish artist," Dylan Thomas and Doris Schattschneider explore the intersection of traditional Indigenous art and mathematical precision. The text details how Thomas utilizes the fundamental geometric elements of Coast Salish design, the trigon, circle, and crescent, to create complex symmetries that resonate with both cultural heritage and mathematical principles like tiling, rotation, and reflection. By examining Thomas’s creative process alongside M.C. Escher’s influences, the authors demonstrate that Indigenous art is not merely decorative but a sophisticated system of "visual math" that reveals deep relationships between form, space, and spirituality.

Stop 1: My first stop occurred on page 201 when the authors describe the specific geometric "rules" for Coast Salish design. It reminded me of how we often present mathematics as universal and fixed. However, seeing the trigon and crescent used as the "building blocks" of a specific cultural geometry stopped me. It made me realize that my students often think of math as something that belongs only in a textbook. This passage shows that geometry can be a living language used to express identity and history, rather than just a set of abstract theorems to be memorized.

Stop 2: I paused again on page 205, where Thomas discusses his fascination with M.C. Escher. As a secondary teacher, I’ve used Escher to teach tessellations many times, but I’ve rarely connected him to Indigenous art. The stop here was the realization of a shared "neural substrate" between these different cultures, the human drive to find order and beauty in repeating patterns. It made me reflect on how my curriculum often separates "Western" math from "Indigenous" art, when the underlying logic of rotation and reflection is exactly the same.

Stop 3:  Symmetry in these designs isn't just for aesthetics but is used to represent spiritual balance and the "interconnectedness of all things." In our math classrooms, we usually treat symmetry as a cold, mechanical property of a shape. This reading made me consider what is lost when we strip the "soul" away from geometry. It shifted my perspective on how I might introduce symmetry to my students and not just as a transformation on a coordinate plane, but as a way to visualize balance in the world around them.

A Question I Wonder: Thomas describes his process as a blend of traditional constraints and modern mathematical exploration. How can we, as secondary math teachers, design authentic assessments that allow students to demonstrate geometric mastery through cultural or artistic expression without "colonizing" the art form or reducing it to a simple math problem?

Reading Response: Off the grid.

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

Summary

In the chapter "Off the Grid " Edward Doolittle looks at the things that the grid's not good at and the things it does wrong. He writes this from the point of view of a mathematician. Edward Doolittle says that the grid is a tool for people, in the West who do math and for people who run colonies. The grid often does not do a good job of showing how the natural world really is. The natural world is not always straight and simple it is often. Complicated. The grid fails to show these things. Edward Doolittle is talking about the grid. How it does not work well for everything, especially the natural world. Through a series of diverse examples—including the mathematical distortions of map projections, the self-organizing behavior of bees, and the devastating real-world impact of applying flat grid-based surveying to Indigenous treaty territories—Doolittle suggests that a truly comprehensive mathematics education must move beyond rigid Western paradigms. He advocates for "off the grid" thinking that embraces Indigenous traditions, chaos theory, and the anomalies that arise when we try to force a curved reality into a straight-lined system.

Stop 1: My first "stop" occurred on page 102 when Doolittle discusses how the grid is used to "rectify" the world. This term stopped me because it implies that the world is somehow "wrong" or "crooked" until we apply our coordinate systems to it. In my own teaching, I realize I have often presented the Cartesian plane as a neutral, perfect truth rather than a specific cultural tool used to flatten and control space. Doolittle’s analysis "tugs on my sleeve" by forcing me to reconsider the underlying power dynamics of the geometry I teach every day.

Stop 2: I paused again on page 112 at the description of how flowers and bees coordinate their actions without a central master plan or grid. In secondary mathematics, we spend so much time teaching students how to find points on a grid to model behavior. Doolittle reminds us that highly complex "math" happens in nature through proximity, curvature, and organic interaction. This is a "stop" for me because it suggests that our curriculum might be overlooking "emergent" mathematics—systems that organize themselves from the bottom up rather than being imposed from the top down.

Stop 3: The most significant "stop" was on page 107, regarding the miscalculation of Indigenous treaty territories. Doolittle explains that applying a flat, rectangular grid to the curved surface of the Earth isn't just a technical error; it resulted in the physical loss of land for Indigenous people. As a math educator, this moment arrested my habits of engagement. We often treat spherical geometry as an "advanced" elective topic, but Doolittle shows that failing to understand the curvature of the Earth has deep moral and political consequences. It highlights that math is never truly "off the hook" from social justice.

A Question I Wonder

Doolittle argues that the grid eventually "falls apart" when faced with the complexity of the real world. How can we adapt our secondary math pedagogy to teach the "standard" grid-based skills required by the curriculum while simultaneously honoring the "off the grid" Indigenous and organic mathematical perspectives that Doolittle proves are so vital?

Reading Response: Six-Cornered Snowflake

Kepler, J. (2010). The six-cornered snowflake: A New Year’s gift (C. C. Rodrigáñez, Illus.). (Original work published 1611).

In his 1611 essay, Six-Cornered Snowflake, Johannes Kepler embarks on a scientific quest to find a "New Year’s Gift" for his patron that is "next to nothing." This leads him to a rigorous mathematical investigation into why snowflakes always exhibit six-sided symmetry. Kepler explores the efficiency of hexagonal shapes in nature- comparing them to honeycombs and the packing of pomegranate seeds- to determine if this geometry is a result of physical necessity or a deeper "formative faculty" of the Earth. Ultimately, Kepler’s work acts as a precursor to crystallography and atomic theory, though he concludes with a humble admission that the ultimate "why" of the snowflake’s six corners remains a beautiful mystery.

Stop 1: My first stop occurred right at the beginning when Kepler describes searching for a gift and mentions "prodigiously puny creatures." This immediately brought me back to my Grade 4 classroom. When we made those 3D symmetrical snowflakes, the children weren't just doing a craft; they were holding a mathematical marvel. Kepler’s focus on the "puny" is a stop for me because it challenges our academic habit of only looking for "big" ideas. As a math educator, it reminds me that the most profound truths are often hidden in things we usually walk right over. We often move too fast past the small things in our curriculum that actually deserve our attention.

Stop 2: I stopped again on page 65, where Kepler distinguishes between the "material necessity" that shapes pomegranate seeds (physical pressure) and the "formative" nature of flowers and snowflakes. This was a dense section that forced me to slow down. In math education, we often teach the how - how to measure a shape or identify a pattern, but we rarely stop to ask why nature chooses one geometry over another. Kepler’s struggle to find a "different kind of thinking-cap" for this anomaly is exactly the kind of intellectual "itch" I want my students to feel. It’s a reminder that true mathematical thinking happens when the simple explanations stop working.

Stop 3:  The final stop was Kepler’s "mortification" at arriving without a gift, which actually sparked his entire inquiry. As teachers, we are often terrified of not having the perfect answer or lesson ready for our students.  Kepler’s lack of a physical gift led to a gift of intellectual discovery. This changed my perspective: I realized that sometimes the best thing I can bring to my students is an unanswered question and the vulnerability of being "empty-handed" alongside them.

A Question I Wonder: Kepler spent this entire treatise trying to solve the mystery of "Why six?" and eventually admitted he hadn't fully answered it, yet his curiosity laid the groundwork for modern science. How can we, as educators, create "stops" in our math lessons that prioritize this kind of unresolved wondering over the quick, correct answer, especially when we are under pressure to move through the curriculum at a continuous pace?

Reading Response: Foundations of Embodied Learning

Nathan, M. (2021) Excerpt from Foundations of Embodied Learning pp. 3-7 and 147-151.

Summary Mitchell Nathan frames a core problem in education. Despite its importance, teaching is not guided by a unified, evidence-based theory of how people actually learn. He contrasts the natural, embodied learning of infants, who intuitively grasp balance, language, and facial recognition, with the difficulty of programming these abilities into machines. Nathan argues that humans are “learning creatures” who naturally think and make meaning through bodily experiences. Yet typical classroom design actively restricts physical and social interaction, treating the body as separate from the mind. This “mentalistic” approach is rooted in historical mind-body dualism. It distances students from their own sensory and motor resources, often making learning more abstract, less effective, and unfairly difficult to assess. The reading introduces “embodied learning” as a necessary paradigm shift. It illustrates its power through early examples in algebra, where students naturally use gesture and movement to reason about equations and unknown quantities long before formal instruction. Stop 1: When Learning Loses Its Body I stopped at the idea of "mentalistic education." Nathan describes how schools often treat the body as a distraction. This made me think of my own math students. Some really struggle with equations when they are just symbols on a page. But when they get to act out the problem, like pretending to be a balance scale, the question makes sense to them. It made me sad to realize how many kids might think they are bad at a subject, when really they just learn differently. Schools don't provide enough space and time to allow our first and most natural way of understanding the world is through touch and movement. Stop 2: What We Assume Is Obvious The discussion about grounding metaphors was a stop for me. The book explained that even a simple idea, like numbers living on a line, is something we learn. It is not automatic. The story about the RightStart program for immigrant children was powerful. These students did not grow up with games that taught the number line. So when they got to school, they were already behind. It made me question what other "basic" ideas I assume everyone knows. This part of the book showed me that embodied learning is not just a fun activity. It can be a matter of fairness, giving every student a chance to build the same mental tools. Stop 3: The Wrong Kind of Test Nathan points out that we often test the wrong thing. We test whether students can use abstract symbols, not whether they truly understand the concept. The example of young children who could add things by sight and sound, but failed when asked to do it with numbers, really stuck with me. It means a child could be brilliant and still get a bad grade. That feels like a big failure in the system. We might be measuring their ability to follow school rules, not their ability to think. If embodied learning is so natural and effective, why do you think it remains marginalized in so many schools and what would it take to make movement, gesture, and sensory experience central to teaching in your own classroom or discipline?

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