Week 3 Reading and Activity Reflection

Reading: 

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. 

In this article, Gerofsky and Ostertag use the metaphor of the “grid” to describe the rigid, uniform, and often unchangeable structures of the school system. They argue that from the design of school buildings and classroom layouts to desk arrangements, worksheets, and curricular expectations, schooling is organized through grids that shape how learning happens. Working closely with teacher education programs, the authors invite preservice teachers to become aware of these constraints and to explore ways of moving within and beyond them. Through projects such as gardening, dancing, and swinging, they demonstrate how embodied, place-based, and relational practices can interrupt the rigidity of the grid. Rather than rejecting the grid structure entirely, Gerofsky and Ostertag encourage teachers to creatively “dance” or “parkour” through the grid, and to find a balance that allows for more meaningful ways of teaching and learning.

Stop 1: 

 

“It begins with noticing that teachers are in the squared-off boxes of classrooms for most (or all) of their teacher education experiences. That seems to follow logically and seamlessly from other, previous educational experiences, from preschool through to advanced university degrees, in which teachers and learners sit inside rectilinear rooms, at tables, chairs, and desks, and talk about things that are not present, within the straight-line, right-angled grid structures that are always present. ” p. 173

After spending over 20 years in schools, first as a student and now as a teacher, I had never questioned the “grid” of schooling. Classrooms, desks, schedules, and worksheets all felt natural, as if this was simply how learning was supposed to happen. Reading this article, I started picturing a literal square box in my head, with people moving inside it but rarely stepping outside. I think this comes from a kind of fixed mindset that if the system has always worked reasonably well, why question it? Yet Gerofsky and Ostertag’s question,

“Can we absolutely reject the grid (in environmental education and in garden-based learning) when we are, at least in part, implicated and entangled in it — when it is an intimate part of ourselves and our way of being in the world?” (p.175)

made me realize that the issue may not be about abandoning the grid entirely, but about noticing how deeply it shapes our thinking. The grid may have helped schools function efficiently, but this reading made me stop and wonder: how might we interrupt or merge it in ways that allow for richer, more embodied forms of learning?

Stop 2:

“As new teachers, we desire in some way to command attention and control learners (for the purposes of learning, safety, order)— and at the same time, we recognise the illegitimacy of usurping the freedom of others. We are simultaneously within and beside ourselve,s and the persona of ‘teacher’ we are in the process of adopting. Rather than conforming to this persona, what other ways of being teachers might be possible? Can we dance or daydream teachers into being with a garden?”  p.180

This passage really resonated with me, especially as I still consider myself a new teacher with less than two years of classroom experience. I often find myself caught in the tension the authors describe: wanting to maintain attention, order, and safety, while also feeling uncomfortable with the idea of controlling students in ways that limit their freedom. I am constantly thinking about how to improve my teaching, and I have many ideas I would love to try. At the same time, I am aware of very real constraints such as time, resources, curriculum demands, and the level of support available from the school. As a result, much of my energy ends up being spent on classroom management and covering content, rather than experimenting with more imaginative or embodied approaches. While I believe teachers can “dance or daydream” their way into new possibilities, this reading made me reflect on how much institutional support is needed to turn those dreams into reality.

Discussion questions:

• In what ways might we interrupt, bend, or work within the “grid” to support richer and more embodied learning experiences for students?

• What kinds of conditions or supports could schools put in place to make creative risk-taking more possible, especially for early-career teachers?

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Sketching Activity:

** I have no talent in art, so please excuse my limited sketching skills**

I sat on my balcony on the 7th floor and sketched what I could observe from above. For clarity, I intentionally left out the security bars in my drawing so the sketch would better reflect what I saw through my eyes. The living things I sketched included a human, a dog, and plants, while the non-living elements included buildings, pathways, and a water fountain.

In the human-made, non-living objects, I noticed strong patterns and clear structures. The buildings appeared very square and rectilinear, and the water fountain was carefully designed with intentional angles. The pathways were tiled in repeating patterns, including rectangular tiles in some areas and hexagonal tiles in others. In contrast, the living things appeared far less rigid or structured. Their lines were more curved, irregular, and dynamic. This difference makes sense to me, as living things are constantly in motion, while human-made objects are typically static and designed for stability and order.

I was especially drawn to the repeating patterns in the building windows and the tiled pathways. I imagine these patterns exist not only for aesthetic reasons but also for functional purposes, such as organization, construction efficiency, and even guiding movement through space. This type of close observation and sketching activity could be a powerful way to help students learn about lines and angles. It naturally invites questions such as: Why do these patterns exist? Why are certain shapes repeated? How are the lines aligned? Could we calculate how many tiles are needed for a specific area? These questions could lead to rich mathematical explorations involving geometry, measurement, and spatial reasoning.

I also see strong possibilities for experiencing lines and angles through whole-body movement. For example, students could use their bodies to form straight lines, curves, or different angles, either individually or collaboratively with peers. This would not only make geometry more embodied but also more playful and engaging. In relation to the living world, outdoor activities such as observing plant growth, tracing natural paths, or even constructing simple structures could further connect mathematical ideas of lines and angles to real, lived experiences.

A photo of what I sketched. 

Week 2: Reading and Activty Reflection

 Reading:

Healy, L., & Fernandes, S. (2013). Multimodality and mathematical meaning-making: Blind students’ interactions with symmetry.

Summary

This paper examines how two blind students engage with and make sense of the mathematical concept of symmetry. The authors frame the study through theories that connect bodily perception to cognitive processes, drawing on philosophical ideas from Francis Bacon and René Descartes about knowledge, sensation, and reasoning. The participants, Lucas and Edson, were asked to explore line segments and represent shapes formed by elastic bands on geoboards. Lucas had been blind since the age of two, while Edson lost his sight completely at age fifteen. The study highlights how each student relied on different sensory and cognitive resources. Edson, who had prior visual experiences, often drew on visual memory, such as imagining a mirror, to reason about reflection and symmetry. The researchers suggest that this allowed him to access embodied conceptual knowledge rooted in past visual perception. In contrast, Lucas had no visual memories to rely on and instead constructed meaning through tactile exploration, movement, and verbal interaction with the researchers. Overall, the paper demonstrates that mathematical understanding can be developed through multiple sensory pathways and is not dependent on vision alone.

Stop 1

What struck me most while reading this article was how adaptable human learning can be. Geometry is often taught as a highly visual subject, so it is easy to assume that learning it would be extremely limited without sight. However, this study shows that students can still engage deeply with mathematical ideas through touch, movement, and verbal interaction. These sensory experiences can also stimulate imagination and support cognitive understanding. In the absence of sight, other senses often become more active and central to meaning-making.

I found a personal connection to this idea because I recently underwent SMILE eye surgery and have since intentionally reduced my screen time. For readings in this course, I have started using the “read aloud” function instead of reading visually. I noticed that I am often more focused when listening, especially when I take handwritten notes at the same time. Compared to reading with my eyes as I used to, listening feels more engaging and helps me stay present with the ideas. This experience helped me better appreciate the article’s emphasis on multimodality and how learning does not rely on a single dominant sense.

Stop 2

One sentence that made me pause was: “For his mathematics lessons, he was included in the regular classroom, and although Braille translations of the exercises were given for these lessons, he received no extra additional support” (p. 44). This raised questions for me about what kinds of support should be provided for students like Edson, who are visually disabled but fully included in mainstream classrooms. The lack of additional support suggests that inclusion may be more symbolic than practical.

Mathematics already presents challenges for many students, even without visual impairments, and topics such as geometry, graphs, and symmetry can be particularly demanding. This makes me wonder whether schools are truly prepared and adequately resourced to support students with visual disabilities across subject areas. Beyond mathematics, other subjects may also pose significant barriers. It raises concerns about how students and families can advocate for appropriate accommodations and what systemic changes are needed to ensure equitable access to learning.

Discussion Questions

How can mathematics educators design learning experiences that intentionally draw on multiple senses, rather than relying primarily on visual representations? 

In inclusive classrooms, what responsibilities do schools and teachers have to move beyond basic accommodations, such as Braille translations, toward more meaningful multimodal support for students with visual disabilities?

Activity Reflection: Hexaflexagon

This was my first encounter with a hexaflexagon, and I found myself immediately drawn to it. Before starting, I assumed it would be relatively straightforward after watching the Vi Hart videos. Using a scrap strip of paper, I attempted to make one right away. Although I managed to form a hexagon, it would not flex properly, which told me that something had gone wrong. I was not sure whether the issue was how I taped the ends together or how I was holding and manipulating the shape. On my second attempt, I followed the template more carefully, but I still failed to produce a functioning hexaflexagon. It was only on the third attempt that I finally succeeded.

What surprised me most was how challenging the hands-on process turned out to be. Watching the video made the construction look simple, and I did not expect to struggle, especially since I am fairly confident with origami. This experience highlighted a clear difference between watching a mathematical activity and physically engaging with it. The hands-on experimentation required coordination between my eyes, hands, and thinking, and each failed attempt helped me notice details that I had overlooked before. Although frustrating at times, the process was engaging and memorable in a way that passive viewing would not have been.

This experience also helped me think about how children might learn differently when interacting with real 3-D objects rather than 2-D images. Physical objects allow learners to explore shape, structure, movement, and texture, which can support deeper understanding. When students manipulate objects directly, learning becomes active and embodied rather than abstract and distant. The trial-and-error process, while challenging, can also foster persistence and curiosity.

For students with sensory impairments, such hands-on activities may be even more significant. Thoughtfully designed tactile and manipulable materials can provide access to mathematical ideas that are often presented visually. As suggested in the readings, learning does not depend on a single sensory pathway. By engaging multiple senses, especially touch and movement, activities like creating a hexaflexagon can support inclusive learning and offer alternative ways for students to build meaning.

Week 1 Reflection: Gesturing Gives Children New Ideas About Math

Gesturing Gives Children New Ideas About Math

By: Susan Goldin-Meadow, Susan Wagner Cook, and Zachary A.Mitchell

In this study, the researchers investigate how gestures and bodily movements support children’s mathematical learning. They argue that students are sensitive to the movements they themselves produce and that these movements can influence cognitive processes. The experiment involved 128 third- and fourth-grade students, ages 9 to 10, including 81 girls and 47 boys. Students completed a pretest and were then randomly assigned to one of three conditions: correct gesture, partially correct gesture, or no gesture. Each group was taught a specific type of gesture prior to the math lesson and was asked to produce these gestures before and after solving problems. Following the lesson, students completed a posttest. The results showed that students who used correct gestures solved more problems correctly than those who used partially correct gestures, who in turn outperformed students who used no gestures. Overall, the findings suggest that producing gestures can positively influence mathematical understanding.

One aspect of the study that made me pause was how simple the gestures used in the experiment were. When the authors described the gestures, such as pointing with one or two fingers, I initially wondered whether these movements were too minimal to meaningfully affect learning. I found myself questioning whether such basic actions could really help students solve math problems. As I continued reading, the authors addressed this concern by explaining that the movements were modeled after gestures that children who already know how to solve these types of problems typically produce. As they state, “the movements that we asked children to produce were modeled after gestures that children who know how to solve problems of this type typically produce” (p. 271). This explanation helped me rethink what counts as a gesture and recognize that even simple hand movements can carry cognitive value.

Another moment of surprise came from how effective these simple gestures turned out to be. The results made me think about similar practices I have observed in schools in China, where teachers often require students in Grades 1 and 2 to use finger tracking while reading. By physically pointing to the words, students are better able to follow along and maintain focus, which supports their reading development. This connection helped me see how gestures may support learning across different subject areas. Just as finger tracking supports early literacy, purposeful gestures may also support mathematical thinking by grounding abstract ideas in physical action.

This reading also left me with lingering questions about the relationship between gesture and thought. When children point at math problems, what is happening cognitively? Does the gesture prompt new thinking, or is it mainly an outward expression of thinking that is already taking place? In addition, while the results of this study are convincing, the math problems and gestures used were relatively simple. This makes me wonder how gestures might function when students engage with more complex mathematical concepts. I wonder how teachers could intentionally design or encourage gestures in mathematics lessons to support deeper conceptual understanding, rather than just procedural success. 

Week 1 Body Measurement Activity

Body Measurement Activity 

I used my hand span of 15cm to measure the width of my door. The width of my door is approximately 5 hand spans, so 15 x 5 = 75cm (approx.) I then measured the actual width of the door with a measuring tape, and it measured approximately 82cm. 

If I were to buy a new piece of furniture and it needs to go through the door, but I don’t have any measuring tools, I could use my hand span to guess. If the furniture is less than 5 hand spans, then it for sure could get through the door. 👍

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