Angles Alive: Mathematics in Motion at the Duck Pond

 This week's introduction reminded me that mathematics didn't begin in classrooms with worksheets and whiteboards—it arose from observations of the natural world, from patterns in the skies, seasons, and the living earth around us. The idea that we can return mathematics learning to these roots, teaching "in and with the living world," resonated deeply with me, especially the concept of the land or garden as a "co-teacher" rather than just a backdrop for an indoor classroom moved outside. I take my students out on the land weekly—we have sit spots, nature journals, and regular outdoor experiences—but I realized that mathematics is rarely what we focus on during these times. We notice, we wonder, we observe the living world, but I haven't intentionally designed mathematical learning into these outdoor experiences. This week's materials made me wonder: what am I missing by keeping math primarily indoors?

My students regularly use sit spots and nature journals, but we rarely focus on mathematics during these outdoor experiences.

The video of the two UBC students creating a dance to demonstrate Euclidean proofs on the beach beautifully illustrated what embodied mathematics learning could look like. Watching their bodies become the moving parts of a geometric proof—with sticks and rocks as constants while their bodies changed position—showed me how mathematical concepts could be experienced kinesthetically, not just understood abstractly. The beach setting wasn't incidental; it grounded the mathematics in a living place. It made me wonder: what would change for my students if they could feel an angle with their whole body before they ever had to calculate it on paper?

So with these ideas about outdoor, embodied mathematics learning in mind, I headed to the duck pond this morning, curious about what I might discover—and what I might bring back to my students. Angles, I reminded myself as I headed toward the duck pond. I'm here to observe angles. But suddenly I found myself grinning at a scene unfolding across the street: a gorgeous little dog who was spectacularly stubborn. He sat as his owner began walking, refusing to move, and the taut leash formed a perfect angle between stubborn dog and patient human. I'd never have noticed that angle before. There it was—the perfect tension between nature (the dog), human design (the leash), and human intention, all pulling in different directions to create this living geometry. That stubborn dog opened my eyes to seeing that angles were everywhere—not the rigid, paper-and-pencil kind I'd always imagined, but flexible, purposeful, alive. 

All along the way, I noticed how different nature's angles felt—tree branches curved rather than cornered, the path meandered rather than cutting straight, and the juxtaposition of manmade elements against the softness of nature was startling at times. The straight angles of a U-Haul truck seemed out of place next to a group of ducks with their softer curved lines. When I found a spot to sit and sketch, I grounded myself in place and enjoyed the freshness of the air, the sounds of the ducks and birds, and even the feel of the damp around me. At first, the pond was quite empty of ducks. Finally, one came along, a male Barrow’s Goldeneye, and I watched him dive and swim, wondering about the angle he seemed to dive down and come up at; this was hard to see clearly as the water was murky. As I sketched, I noted the angles: beak to chest, forehead to beak. As I was watching this lovely waterfowl, a whole flock of ducks returned to the pond flying in at yet another angle. One mallard came into the goldeneye’s area and quacked a warning, creating a sharp angle with his bill.

Sketching the Barrow's Goldeneye helped me notice the angles: beak to chest, forehead to beak.

The angles in nature were alive and dynamic, in stark contrast to the man-made angles around me. The man-made world was rigid: benches, streetlights, exact rectangles of crosswalks, planks for bridges, fences, the sharp edges of buildings. Nature's angles, by contrast, were softer and more flexible. As I looked around my neighbourhood, I started noticing steps leading up to front doors—some with gentle angles, others steep and hazardous like my grandparents' old house. Last week, we spent time thinking about universal design, and it crept into my thinking as I observed these barriers. Why aren't homes accessible for people with mobility issues? What would we gain as a society if we softened our angles, both literally and philosophically? I also thought about how architecture could take its inspiration from nature, and enjoyed reading about this at https://salaarc.com/blog/natures-rules-of-architecure-engagement/ . I was struck by how rarely we allow nature to inform our building practices, despite its elegant solutions to structural problems. 

The sharp angles of human-made structures stood in stark contrast to nature's softer, curved lines.

Sitting by that pond, watching angles come alive in duck dives and territorial displays, created a shift in me. I moved beyond a rather rigid, Euclidean concept of angles toward a more flexible, embodied experience. This embodied experience deepened my understanding in ways that diagrams on paper never could—angles weren't just abstract concepts but living, functional elements of the world around me. Though I'd understood their function theoretically, even that understanding came more alive through direct observation.

This experience made me think about how important it is to give our students bigger perspectives on how to think about angles and mathematics. What if students experienced acute and obtuse angles first through their bodies—reaching up at steep angles to touch tree branches, lying at shallow angles on a hillside—before ever seeing them drawn on a worksheet? How could we design lessons that honor both the precision of mathematical angles AND the organic, functional angles found in living systems?

Learning to See Connections: Gardens as Systems Thinking in Action

 In Sustainability Education’s Gift: Learning Patterns and Relationships, Williams (2008) advocates shifting from “mechanistic and technocratic worldviews” toward more holistic approaches grounded in systems thinking. From this perspective, the whole can only be understood through the relationships between its parts, allowing the fragmentation created by mechanistic thinking to fall away. Williams describes this capacity to learn in patterns and relationships as sustainability education’s “gift,” and illustrates it through a case study of Learning Gardens, a garden-based education program in Portland schools. As students engage in this kind of learning, they begin making meaningful connections among spiders, plants, hunger, and community, linking ecological and human systems. The program also addresses broader issues such as obesity, food insecurity, disconnection from nature, and achievement gaps through learning experiences that are multicultural, multidisciplinary, multisensory, and intergenerational.

“Education is the ability to perceive the hidden connections between phenomena,” writes Vaclav Havel. This line, placed immediately after the abstract, resonated deeply with me and feels like an antidote to rote learning and the individualistic ways of thinking that permeate our society. Framing education as learning to see connections raises important questions: Is being educated the same as being able to recognize relationships between things, or is this something schools must intentionally teach? One place where seeing connections matters deeply is in understanding climate change, where individual choices create collective consequences. Yet ecological literacy has not been prioritized in schools, even though systems thinking makes these connections impossible to ignore.

When we continue teaching in isolation—without helping students see connections—I wonder if we also encourage ways of living in isolation. I don’t mean social isolation, but the compartmentalization of subjects that trains us to think in silos. This mindset can dull our sense of humanness, making it easier to dehumanize others or overlook our responsibilities to one another and the planet. Seeing connections between people, systems, and the environment requires more than knowledge—it requires curiosity, reflection, and care. Avoiding the social and ecological consequences of disconnection may leave students unprepared to recognize or respond to the crises they will inevitably face.

The Learning Gardens program shows what becomes possible when this relational worldview guides learning in practice, using food systems as a unifying concept. Williams describes how local food production connects community well-being, health, pride, and learning, while fostering a strong sense of ownership and belonging. Reading about families sharing cultural foods and creating gardens rooted in their traditions made me reflect on my own school garden, which often becomes overgrown or underused. I wonder what might change if gardens were treated as shared community spaces rather than isolated projects.

What possibilities do you envision if you had a school or classroom garden? What might gardens make possible that classrooms alone cannot?

Permission to Play

In this week's introduction, we were asked, "What would mathematics look like if we took multisensory learning seriously?" My first thought came without hesitation: it would be a whole lot messier and a lot more fun.

I suddenly pictured myself as a little girl being opened up to a world of math that I didn't understand existed. I imagined the joy and relaxation that would enter into many learners' lives as they realized they could play with their food and learn math! Creating something is less anxiety inducing for my ELL students than trying to wrestle with new and strange vocabulary such as, "Which digit is in the hundreds place?" or "How else can you represent this number?" How I would have thrived if I felt like math could include more feeling and magic than what I perceived as logic and procedures. This feeling/logic divide shaped my entire math identity. I always felt like my twin bro had the leg up in our math class—he was Mr. Logical and I was Miss Feeling Everything. It took me a lot of years to realize the validity and importance of my way of looking at things. This connects to the Cartesian/Platonic hierarchy that values abstract thought over sensory experience, but what if they don't have to be at odds? As an elementary teacher, I recognize that bringing multisensory learning into math may look different across contexts, and may feel more immediately accessible in elementary classrooms than in secondary settings.

If multisensory math is about permission to play, Vi Hart's videos are the ultimate permission slip—turning Smarties into data sets and tortillas into geometric transformations. I loved how Vi Hart took a simple candy and came up with so many mathematical questions, reminding me once again that math truly lives everywhere. It made me excited to think about how fun it would be to use Rockets to explore graphing, make guesses about what would show up the most, and think about probability. I loved the idea of making little sandwiches and looking at combinations. How fun would it be for students to play with and eat candy during math? What a great attention grabber. I do wonder about the cost of doing such a thing and how many candies I would need to make it a worthwhile experience for each child. There I go doing real life math myself—which is exactly the point. When learning feels like play, we engage with mathematical thinking naturally, even when we're just planning a lesson.

The hexaflexagon videos were a different experience entirely. I was fascinated by them—they gave me the feeling that math is indeed sparkly and magical. Watching shapes transform and hidden faces reveal themselves felt like discovering secrets. Math as wonder, not work. Math as play, not procedure. This is exactly what draws me into math and how I wish I always felt about it as a student. The way a hexaflexagon can be decorated, folded, and pinched was really hard for my brain to grasp at first. But seeing a tortilla turn into a hexaflexagon—that changed everything. Suddenly I could see what happens as the guacamole and sour cream went into the middle. The 3D, edible version made visible what the paper version had kept hidden from me. I could finally understand how to fold, where the faces went, why it worked. Interestingly, George Hart's bagel cutting never sparked that same click for me—I felt resistant, like it was too hard and I wasn't excited about it. The tortilla worked; the bagel didn't. This contrast crystallized why multisensory learning matters: it's not that everyone learns the same way, but that everyone needs options. Sometimes one modality doesn't click, but suddenly another does.

Even after the tortilla breakthrough in the video, I was skeptical that making one myself could top the magic I'd already experienced watching. Happily, I noticed that creating a hexaflexagon opened up more wonder in me. As I colored and decorated, I found myself noticing where edges meet, and asking questions that I would never have asked from just watching:

• Will these colors show up somewhere else when I flip?

• Which face is underneath—face three?

• Will my markings matter?

• How does this even work?

• How do you know how many faces you'll get?

The experience felt magical in a different way than the video. Using my hands was quite relaxing. This didn't feel like traditional math—no one was making me quickly pick numbers or find answers to problems I didn't understand. It felt accessible to anybody. I imagine if someone was feeling tension or anxiety around math, this wouldn't trigger that response

I also found myself making design decisions. It's my birthday—I don't want to spend my whole day decorating a hexaflexagon! What shapes and designs would be satisfying but not take too long? I didn't want to be completely intricate, but I also didn't want it to look messy. I used the tools I happened to have on hand.

The more I worked, the more questions emerged: What would happen if I made this out of thicker paper? What if I made it skinnier? This led me to find a video about making six-faced hexaflexagons and another showing the folding process, which really helped when the written instructions were challenging. I appreciated that I didn't have to write numbers on the surface if I didn't want to.

Why couldn't math have been all about these magical moments when I was in school? 

If multisensory math is truly about permission to play, then it cannot become another expectation layered onto an already heavy curriculum. Offering multiple modalities does not mean asking students to engage with all of them at once; rather, it means creating space for choice, curiosity, and entry points that feel safe. Play loses its power when it becomes performative or obligatory. What mattered most in my own experience with the hexaflexagon was not the number of senses involved, but the freedom to explore without urgency, correctness, or evaluation. This raises an important pedagogical question: how do we design multisensory mathematical experiences that invite wonder without overwhelming learners? Perhaps the answer lies not in adding more, but in loosening control—trusting that when learners are given permission to play, mathematical thinking will emerge naturally.

When Touch Reveals What Sight Misses

In this week's article, Stylianidou and Nardi argue that tactile tools designed for visually impaired students can benefit all learners, as part of a broader study challenging "ableism" in mathematics education. The authors make an important distinction between two approaches to inclusion: "reasonable accommodation," which involves modifications that enable people with disabilities to participate equally with their abled peers, and "universal design," which creates environments and tools that can be used by everyone from the outset, without requiring additional accommodations. They examine what happens when sighted students and a visually impaired student named Luke explore the difference between two shapes through the sense of touch. One shape ("Shape X") is made with Wikki Stix (a flexible wax-and-yarn teaching material) and depicts a circle with a circular segment removed, creating a subtle flat edge. The other is a regular circular disk (the yellow circle). When students closed their eyes and used only touch, they easily detected the straight edge on Shape X. However, when viewing Shape X visually, the straight segment was much harder to notice because it occupied such a small portion of the otherwise circular shape. This is an example of how universal design benefits all: sighted students noticed details they would have otherwise missed, while Luke participated as a full member of the mathematical community rather than being singled out for special accommodations. Additionally, Luke's tactile-based insights—describing how a circle "feels like it's gonna roll more" while Shape X would "bob up and down"—enriched the mathematical discussion for everyone, challenging assumptions about what constitutes valid mathematical thinking.

The thing that struck me the most while reading this article was the difference between "reasonable accommodation" and "universal design." Though I have given some thought to these ideas in the past, they really gave me pause as I read this article. I am wrestling with understanding and wanting to ground myself in the idea of universal design. However, we live in a world that is deeply ableist. I see it in systems where students who are supposed to be included in the classroom spend their days wandering the halls or playing in the resource room. I see it in our classrooms as we emphasize reading over listening to books. It is everywhere, and I have been complicit. I tried to strike up a conversation with my husband about this idea—he spent 22 years working with blind students—and was surprised at his response. He immediately began explaining that not only were mainstream teachers resistant to learning how to accommodate, but VI teachers themselves were very protective and rigid about their established methods. Those working with visually impaired students one-on-one, classroom teachers, and even parents all seem to default to accommodation rather than embracing universal design. In other words, ableism is so ubiquitous that even those dedicated to supporting disabled students may unknowingly perpetuate it. The design and delivery of mathematical materials provides a clear illustration of this problem.

Mathematical thinking is typically constructed through textbooks designed for sighted students. Most of us visually imagine a circle based on our experiences of seeing circles everywhere. However, as Stylianidou and Nardi note, accommodations that simply translate visual materials into tactile ones—like Braille math textbooks—often have significant limitations. I've seen how the Braillist is frequently behind schedule, and how Braille doesn't translate directly from visual mathematics. The Braillist must deeply understand the math to translate it accurately, yet misinterpretations are common. This illustrates why the authors argue that reasonable accommodation, while well-intentioned, is insufficient. Universal design—where all students engage with the same tactile or multimodal materials from the start—eliminates these translation problems and the delays that leave VI students waiting for access while their sighted peers move ahead.

What makes this study significant is its focus on a modality typically associated only with visually impaired students, with the premise that it would benefit all learners. It offers an important change in perspective, which Roger Antonsen argues we need to truly understand something. The study’s approach also aligns with Sfard’s commognitive framework (Sfard, 2007), which defines thinking as individualized communication with oneself that emerges through social interaction. This tactile modality opens new pathways for students to engage in mathematical discourse together. When students describe shapes without using sight, they must articulate spatial relationships differently, leading to fresh mathematical insights. As Luke demonstrated, describing a circle as something that "feels like it's gonna roll" offers a perspective that enriches everyone's understanding. This raises important questions: What new mathematical understandings can we arrive at when we deliberately shift our sensory perspective? How might universal design not only include more students, but actually deepen mathematical thinking for all?

Finally, I have one last burning question for us to ponder. In what ways have you seen ableism manifest in your own teaching or learning experiences, even when intentions were good?

References:

Antonsen, R. (2016). Math is the hidden secret to understanding the world [Video]. TED Conferences. https://www.ted.com/talks/roger_antonsen_math_is_the_hidden_secret_to_understanding_the_world 

Sfard, A. (2007). When the Rules of Discourse Change, but Nobody Tells You: Making Sense of Mathematics Learning from a Commognitive Standpoint. The Journal of the Learning Sciences, 16(4), 565–613. http://www.jstor.org/stable/27736715

Stylianidou, A., & Nardi, E. (2019). Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. In M. Graven, H. Venkat, A. Essien, & P. Vale (Eds.), Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 343–350). PME.

Measuring Using Our Bodies = Full Participation and Lots of Energy

 

Interestingly, I have always had a minor bone to pick with Plato and now I feel even more justified. When I first read Plato's Republic, I detested the idea that in his utopia, people would be born knowing exactly where they belonged and the occupations they would hold. For me, the joy in life is through discovery. Logic alone hasn't held my interest as a human; rather, I long to approach most things from a feeling perspective. In fact, when I decide to become more logical, I come across as cold and that's not my authentic self. This week's introduction helped me understand why Plato's philosophy has always bothered me—his view that the physical world is inferior and that only 'pure thought' leads to truth excludes people like me who learn through feeling and doing. The course reading explains how this binary thinking has made mathematics seem like 'an otherworldly realm' accessible only to an elite few, when in reality, math comes from and connects back to our bodily experiences.

Roger Antonsen's TED talk resonated with me for several reasons. I embraced the idea of perspective taking-especially the way he demonstrated 4/3 in a variety of ways. I was particularly tickled  by his four triangles. It really uses the imagination and demonstrates deep understanding. It also reminded me that embodiment doesn't equal understanding and how imperative it is to help students connect ideas to how and why they are using manipulatives and so on.

This week, we were learning about standard form and expanded form. I decided to throw in some movement in the hopes of aiding student understanding. For standard form, students went from sitting on the carpet to standing with their arms straight up and overhead. For expanded form, students were asked to spread their arms wide. The physical contrast between 'compact and contained' versus 'spread out' matched the mathematical concept perfectly. In the spirit of using our bodies, my class also tackled traditional measurements of length. For many students, this was a chance to use a ruler in a low-stakes way as it was fun, and they all enjoyed seeing what their personal body measurements calibrated to. Collaborating with partners was part of the experience and the room was alive.

What I really noticed about this exercise was that students were genuinely excited and enthusiastic. I had zero students sitting to the side—it was 100% participation. We extended this to a practical application, using our body measurements to figure out spacing for salmon cutouts along our fence. However, we ran out of time after measuring our salmon template, so we'll continue calculating fence spacing next week. This activity challenged the Bourbaki idea that mathematics should be abstract and divorced from the physical world. When math is grounded in embodied experience, every student can access it.

EDCP 553-26 Week One Reading-Excerpt from the Foundations of Embodied Learning

In Foundations of Embodied Learning, Nathan argues that educational systems fundamentally misunderstand how humans learn by restricting students' physical movement and hands-on experiences. He presents embodied learning—the idea that we learn by engaging our physical bodies and sensory experiences to understand new concepts—as an evidence-based framework. Drawing on Lakoff and Núñez's grounding metaphors, Nathan shows how mathematical concepts emerge from embodied experiences like collecting objects, measuring with sticks, and moving along paths. Through examples ranging from kindergartners learning number lines through movement games to students understanding geometric angles by forming them with their arms, he demonstrates that when schools force students to sit still and manipulate abstract symbols, they cut learners off from their most powerful cognitive resources—with consequences that can shape students' identities and close off entire career paths.

Do you ever find yourself reading something and thinking, “Well that’s obvious! Why haven’t I made those connections before?” On the first page of this week’s reading I found myself doing precisely that. Nathan defines learning as lasting changes in our behaviour. If people can behave and learn in so many different ways, he argues, then our schools need to honor and support that diversity rather than narrowing it down. Roger Antonsen described learning as a change in perspective. Learning and change seem to go hand in hand and should fundamentally shape how we look at students and their abilities. If teachers are looking at test scores as an indicator of learning, we are missing out on noticing all of the ways our students can show us what they know. The very measurements we have in place lack meaningful frameworks for capturing what learning actually is.

This realization made me reflect on my own teaching context. The timeframe I have with my students is a quick snapshot in time and it feels daunting to think I should do anything other than be a quiet 'noticer' of how my students are growing as learners. Perhaps suspending judgement is the best approach so we don't risk missing what our students know as the education system we work within narrows and restricts how students show what they know.

Nathan's discussion of the Math Worlds program stopped me in my tracks. The researchers discovered that some Portuguese immigrant children in Toronto hadn't learned to see numbers along a number line—a conceptual metaphor that seems so basic that schools don't even explicitly teach it. They assumed children already knew it.

Wow, I had NEVER thought about this before. It reminds me of those moments when I think a concept is going to be simple for students and then it isn't. I have to remember that what feels like second nature to me is brand new to them. How many other foundational concepts am I assuming my students already have? And how many students are we labeling as struggling when they simply haven't been given the grounding experiences they need?

All of this leaves me wondering: How do we decide what's valued in the education system? And more importantly, who gets to decide?

References

Nathan, M. J. (2021). Foundations of Embodied Learning: A Paradigm for Education (1st ed.). Routledge. https://doi.org/10.4324/9780429329098

Hello World!

 This is my first ever blog post, and I am crossing my fingers that I've figured out how to do this correctly. I think it seems fairly intuitive and I hope it's fun to play around with over the course of EDCP 553. I included a photo I took in July 2023 because I love this image. The Maasai are artistic people and create gorgeous beaded items and carvings among other things.