Reading 4

 Dylan Thomas: Coast Salish artist

This article will look at the work of the contemporary Coast Salish artist Dylan Thomas, whose artwork combines the best of Indigenous artistic traditions with the concepts of mathematics, such as symmetry and geometry. Although the work of the contemporary Coast Salish artist Dylan Thomas is rooted in the culture of the Pacific Northwest, it also reflects the influence of traditional art forms such as spindle whorls, houseposts, animal art, as well as contemporary artists and the mathematical art of M.C. Escher. It will also look at the way in which the concepts of symmetry, balance, and the interconnectedness of all things in nature are used in the artwork.

Dylan describes how his early exposure to Escher’s tessellations encouraged him to make his own tessellations with traditional Coast Salish imagery, such as salmon and birds. In his artwork, such as "Sacred Cycle" and "Salmon Spirits," he uses symmetry and tessellations to depict traditional cultural concepts such as the life cycle of salmon and the impact of environmental degradation. Dylan’s artwork is not simply decorative but has significance in terms of mentorship, tradition, and the interrelatedness of the physical and spiritual worlds. He uses geometry to arrange his concepts in a way that honors the fluidity and movement characteristic of Salish design.

The article also points out how Dylan has also explored other forms of symmetry, like reflection and complex rotation. In Horizon, symmetry has come to represent the unity between spiritual and physical worlds. In Mandala, he has designed a cross-cultural work that blends Coast Salish style with Buddhist mandalas, demonstrating the potential of art to transcend different cultural worldviews. In later works like Infinity, he has taken the mathematical theme further, using shrinking patterns to illustrate the limitations of human potential and the limitless possibilities in nature.

In all, the article demonstrates the power of Dylan’s art as an example of the potential intersection of mathematics, culture, and storytelling. His work shows that geometry is not separate from our lives, but can be a living language that speaks to our identities, our cultures, and our place in the natural world.

Stop 1:

I stopped at Dylan Thomas's descriptions of how his art pieces, such as Sacred Cycle and Salmon Spirits, employ repeated salmon shapes arranged according to rotational symmetry to illustrate cycles of life, cultural heritage, and concern for the salmon population.

This experience has altered my perspective on what geometry means and what symmetry does beyond what has traditionally been taught to us: that symmetry is used to make our designs aesthetically pleasing or mathematically correct. Symmetry here has deeper cultural significance. The repeating salmon shapes are not merely a series of shapes repeated according to a pattern; they have a deeper meaning, illustrating cycles of life, survival, and spirituality between the Coast Salish culture and nature. The salmon life cycle has become a visual metaphor for life and mentoring.

 This reminded me of Indian traditional art forms like Warli paintings, in which repeated drawings of humans or animals in circular patterns symbolize community, agricultural activities, or living in harmony with nature. In Warli paintings, repetition is not used to add beauty or rhythm but to emphasize the interconnectedness of all living beings. Similarly, in Dylan’s artwork, repetition of mathematical concepts emphasizes that humans, animals, or nature are all interconnected. Both forms of art emphasize that mathematics can also be a form of cultural expression, rather than being restricted to a school or college curriculum.

 

 

Stop 2:

I stopped again when I read about Thomas’s artwork, titled “Mandala,” which combines elements of Coast Salish design and the inward-drawing quality of Buddhist mandalas. This artwork employs concentric circles and strategically placed symmetrical shapes to draw the eye inward toward the center.

This part stood out to me because it shows how geometry can bring cultures that are far apart, both geographically and historically, closer together. The use of mandalas in Indian and Tibetan cultures represents the universe, balance, and the inward journey toward enlightenment. They are not simply decorations but spiritual tools for reflection and meditation. The way Dylan has combined this structure with Coast Salish symbols and shapes makes it a piece that is both particular to his heritage and universal.

This really resonated with my experience in EDUC 550, where I created a 3 Act Math Task using the concept of pookalam to connect culture and mathematics. The designs created during the Onam festival in the state of Kerala, called pookalam, use concentric circles with symmetry, pattern development, and the use of color in a balanced way. As I was creating the 3 Act Math Task, I realized that the student was not just learning geometry, but also learning about culture, nature, and community through the lens of mathematics.

 

Dylan’s Mandala works in a very similar way. The use of symmetry in the mandala represents harmony, belonging, and connection to the world, nature, and the divine. The use of circles and patterns in mandalas, as in designs from Indian and Malayali cultures, reflects how people view the world and their place in it.

Discussion Question:

I wonder how our current assessment practices might limit students when math is connected to culture, and what new forms of assessment could better recognize creativity, reasoning, and cultural knowledge.

Reading 3-Off the Grid- Edward Doolittle

 In this chapter, Edward Doolittle explains that grids—straight lines, squares, clocks, calendars, maps, and schedules—are a very common way people organize space and time. Grids feel comfortable and give us a sense of control, so they are widely used in cities, farming, schools, and mathematics. However, Doolittle argues that grids often fail to match reality, especially nature and human life.

The author shows that the grid is only effective in small, flat areas. In the real world, the land is not flat, the Earth is curved, the real world is complex, and time does not repeat exactly. When we apply the grid to the real world in an unnatural way, harm can be done. Cubical watermelons, dangerous city roads, confusion in areas where two grid systems meet, and the measurement and division of Native American land with simple lines on a map are just a few examples.

Doolittle contrasts grid thinking with Indigenous ways of understanding space and time, which are more flexible and closely connected to nature. Indigenous farming follows natural signs (like plants and insects) rather than fixed calendar dates. Indigenous ideas of territory often follow rivers and watersheds rather than straight boundary lines. These natural boundaries are better described by fractal geometry, which reflects how rivers, streams, and landscapes branch and repeat at many scales.

Another aspect that this chapter introduces is the different geometries from the traditional Euclidean Geometry:

• Riemannian geometry: This is a kind of geometry that embraces curved space and demonstrates that no one grid is special, just like different map projections of the Earth.

• Complexity and chaos theory is the study of systems that change over time such as climate or education systems. These ideas have shown that small well-timed actions may therefore have big effects which is chaotic control.

• Knots, weaving, and string figures, drawn from Indigenous traditions, which emphasize movement, relationships, stories, and three-dimensional thinking rather than rigid shapes .

In the final section, Doolittle revisits the famous Königsberg Bridges problem. While Euler proved it impossible within abstract mathematics, Doolittle adds an Indigenous perspective: before solving problems, we should acknowledge place, relationships, and gratitude—in this case, the river itself. By widening perspective beyond abstraction, new insights become possible .

Overall, the chapter argues that moving “off the grid” helps free our thinking. By learning from Indigenous knowledge and alternative geometries, mathematics and education can become more connected to nature, more just, and more meaningful.

Stop 1:

However, I also stopped at the idea that "a few small 'bursts' at the appropriate time might accomplish more than months of haranguing" in the context of education.

This idea made me reflect, as it runs counter to the prevailing understanding of teaching in most educational settings. These setups often believe that achieving more in learning comes from more explanation, more talking, or more rules being set in place. However, Doolittle’s concept views learning in an alternative fashion, particularly in terms of observations being connected to the true student understanding.

This strongly connects to embodied learning, which emphasizes that thinking is not only in the mind but also shaped by the body, movement, gesture, and interaction with materials. As Sara mentioned in our last class discussion, a small change made a huge difference for a student. That change did not involve more explanation, but rather a shift in how the student could physically engage with the task—through movement, visuals, or hands-on interaction. This helped the student access the idea in a way that words alone had not.

These minor bodily changes—enabling the student to move, gesture, manipulate objects, or change their bodily posture—could channel Doolittle’s “small bursts”: Good teaching can be liberating if timed just so as to unlock student understanding, reducing frustration as a bonus. As a result, this stop was helpful in my reconsideration of what it means to be inclusive. I realized that I was conceptualizing accommodations as more effort, as more permanence, but as more embodiment, I see them as teaching moments, teaching moments that I can provide students in the present, teaching moments that recognize as physical, emotional, and cognitive. I realize now, in a sense, lessons can unfold in response to students, not in response to some structure, not in response to some rubric, but in response to students.

Stop 2:

I stopped at the idea that nature gives us shapes, while straight lines are something humans impose.
This statement made me pause because it captures the core message of Doolittle’s argument in a very simple way. In nature, nothing grows in perfect straight lines—rivers curve, landscapes rise and fall, seasons shift, and living systems develop unevenly. Straight lines, grids, and fixed structures are human tools, not natural truths.

This stop made me reflect on how education often relies on imposed “straight lines”: rigid schedules, uniform pacing, standardized assessments, and expectations that all students should learn in the same way and at the same speed. These structures can work for some students, but they often fail others, particularly students with disabilities or those who learn differently.

Doolittle’s idea helped me see that when students struggle, the problem may not be the learner but the structure imposed on them. Just as forcing straight roads through mountains ignores the land, forcing all learners onto a single pathway ignores the natural diversity of how students think and learn. Inclusion, from this perspective, means reshaping the learning environment to follow students’ natural learning paths rather than asking students to bend themselves to fit rigid systems.

This stop encouraged me to think of teaching as a practice of listening, observing, and adapting, rather than controlling. It reinforced the idea that flexible, responsive teaching is not a lack of structure, but a more respectful and realistic way of supporting learning for all students.

Questions:

1)     Doolittle argues that grids provide a sense of control but often fail to represent reality accurately.
How might this insight help us critically examine the assumptions embedded in curriculum standards and learning outcomes, particularly in mathematics education?

2)     How might a flexible approach to teaching inform educators’ responses to learner diversity and unanticipated classroom situations?

Reading 2

 Reading 2: Excerpts from Johannes Kepler (1611/ 2010) On the Six-pointed Snowflake: A New Year's Gift.

In The Six-Cornered Snowflake, Johannes Kepler takes a common observation-the snowflakes falling on his coat-and elevates it into an acute inquiry about pattern, form, and necessity in nature. He asks why snowflakes always turn out to be six-cornered and pursues similar patterns in honeycombs and pomegranate seeds. Instead of attributing such shapes to either chance or beauty alone, Kepler argues that often form is really the result of material constraints, efficiency, and spatial organization. The following article demonstrates how, quite independent of any conscious design, mathematical order can arise naturally from physical interaction, long before formal equations or symbols are at stake.

Stop1:

I paused when Kepler asserts that it is not possible for snowflakes to have six sides by chance. “If six were not a necessary number, there would be snowflakes with five sides and with seven sides—but there are not,” he argues. I did not initially think much about this question, but Kepler’s insistence that consistency demands explanation made me stop and reflect. What struck me here is how Kepler treats observation as serious knowledge. He does not rush to formulas or authority. Rather, his method involves observing, recognizing a pattern, and seeking a motive for that pattern. This particular stop taught me that great ideas in mathematics lie in recognizing something quite basic.

Stop 2:

The second pause came when Kepler explains why bees build hexagonal honeycombs. I realized that I had never fully noticed this before reading the article: hexagons are the most efficient shape for filling a flat surface without leaving any gaps, allowing bees to store the maximum amount of honey using the least amount of wax. This realization altered the way I thought about the honeycomb pattern. In the past, I thought the honeycomb pattern was either a natural, beautiful concept, but it is, in fact, a useful and necessary concept. The honeycomb pattern isn’t something bees intentionally make because they understand the concept of mathematics. Instead, the honeycomb pattern simply becomes a natural byproduct because it’s the most effective pattern in the physical realm. This topic is vastly related to embodied mathematics. Topics like efficiency, tessellation, and optimization are far more applicable in a hands-on setting, where one can grasp how shapes fit into space. This stop caused me to realize how many situations in mathematics become understandable when learned through physical engagement.

Question:

Kepler valued the process of questioning as much as the answer itself. In what ways might our current approaches to math instruction unintentionally limit students’ opportunities to engage in this kind of open-ended mathematical wondering?

Reading 1

 Seeing the graph vs. being the graph Gesture, engagement and awareness in school mathematics

-Susan Gerofsky

This chapter discusses the gestures employed by students in understanding mathematical graphs and the meaning behind these gestures. The study revealed three major ways in which students gesture graphs: seeing the graph at arm’s length, being the graph by using the whole body, and having difficulty in identifying significant parts of the graph. Students who used whole-body gestures demonstrated high levels of comprehension and creativity, whereas students who used precise gestures demonstrated high levels of accuracy but less meaning. The research suggests that encouraging students to physically “be the graph” can improve engagement and understanding in mathematics learning.

Stop 1

Being the graph in a fully-embodied way fosters engagement and attentiveness far more than merely seeing the graph.(p.254)

I paused on this quote, as it made me think deeply about my own experience with teaching and learning. When I began my teaching career, my attention in the classroom was mainly on teaching and covering the curriculum.The use of gestures in teaching occurred to me in terms of explaining graphs, but it never occurred to me to use gestures in teaching, let alone use them as a method in teaching. I often wonder why we do not think about or intentionally add gestures while teaching, even though they are simple and natural. This reading reminded me of the graph activity with Susan, which was very engaging and educational. That activity helped me understand how simple gestures and embodiment can make a huge difference in learning, especially for a topic like graphs, which I never thought could be taught in this way.

Stop 2

Gestures produced by mathematics teachers and learners provide a rich source of data, comparable in scope to that provided by language, which can be read in terms of bodily metaphors, object development in the formation of mathematical concepts, and the relationships among mathematical concepts.(p.245)

I paused on this quote as it challenged me to think about how understanding is typically assessed within mathematics. We know that understanding within mathematics is often assessed through oral or written means, but the quote indicates that gestures can also hold equal meaning. It challenged me to reflect on how often students understand a concept, yet struggle with expressing it verbally, all while the gestures have demonstrated understanding. It also led me to question what it means for educators. If gestures have equal meaning as words, why do we as educators not incorporate or emphasize them within our classrooms?

Discussion Question

Some students may feel uncomfortable using large or expressive gestures due to cultural norms, personality, or classroom climate. How can teachers create inclusive environments where embodied learning feels safe and accessible to all students?

Hello Everyone.........

Welcome to my blog....