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?
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?
I absolutely loved reading your writing—it’s so personable and flows so naturally. Honestly, you should write a book (just a little side note!). The way you share your thinking makes the experience feel lived-in rather than observed, which pulled me right into your walk to the pond.
ReplyDeleteI really resonated with your reflection on how rare it is to truly experience or feel angles. So often, even as teachers, our training pulls us back toward traditional approaches—textbooks, sharp lines, and static diagrams—almost by default. Like you, I felt something come alive when I started noticing angles in the natural world. Your story of first seeing a single duck at the pond, and then later watching a whole flock arrive, was such a powerful reminder that the angles we encounter in nature are living and constantly changing.
In contrast, the angles we typically present in classrooms are frozen on the page. They don’t move, respond, or evolve. But in the living world, angles emerge, soften, sharpen, and disappear as bodies move and relationships shift—a duck diving, a bird landing, a leash pulled taut. I think it’s so important that students are reminded that angles aren’t just one fixed, static thing; they are dynamic and relational.
I was also struck by the contrast you highlighted between sharp, human-made structures and nature’s softer, more flexible angles, especially your connection to universal design. The idea of moving toward gentler, more accessible angles—both literally in how we design our buildings and philosophically in how we design learning experiences—really stayed with me. It made me think not only about math instruction, but about our classrooms and shared spaces as well. What would change if we truly allowed nature to guide both how we build and how we teach?
I loved the visual of the duck diving down and how many pieces of mathematics we can extrapolate at every level! We have physics in distance per second, forming velocity. I see how the entrance angle and exit angle could be used to calculate how far across the pond the duck moved. I see how it may have made a parabola underwater and could be charted with the CAST rule. I can even see how turbulence and fluid dynamics could play a role for a university crowd!
ReplyDeleteI deeply relate to the steep stairs at my grandparents’ house! The stairs up to the attic were so narrow and so steep that they felt like a ladder that could not be used as such! This universal design can benefit so many if everyone is thought of during the design phase.
I love your idea of feeling the angles using the hillside. This is such a great way to bring it into our senses as our inner ear navigates, our balance adjusts, and our minds recalibrate. I wonder if the hill could even be a part of the worksheet? It seems similar to thinking either with or without grids as are inspecting how to find the balance between precision and natural teachings, similar to how parkour transformed the city grid into a much more embodied adventure (Gerofsky & Ostertag, 2018). Have you read Opening the World through Nature Journalling? This book does a great job of giving concrete tasks, connected to content, and inspiring to look at as a teacher. Perhaps nice for you and your class’s sit spots if you wanted more to pull from (Laws et al., 2010)!
Gerofsky, S. & Ostertag, J. (2018). Dancing teachers into being with a garden, or how to swing or parkour the strict grid of schooling. Australian Journal of Environmental Education, 34/2, 172-188.
Laws, J. M., Breunig, E., Lygren, E., & Lopez, C. (2010). Opening the wOrld thrOugh nature jOurnaling. https://www.cnps.org/wp-content/uploads/2018/04/cnps-nature-journaling-curriculum.pdf