Reading Response Week 3: Sustainable Mathematics in and with the Living World Outdoors

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.

Summary:

Gerofsky and Ostertag critique the dominance of the “grid” in schools: rigid structures of time, space, curriculum, and assessment that continue to shape schooling. They argue that grids function to sort and order people, ideas, and practices, offering a sense of control and efficiency that is deeply embedded in classroom norms. While familiar and often comforting, these structures regulate bodies, movement, and relationships within schools. The authors assert that the grid is not neutral, but rooted in Western, Enlightenment, and colonial ways of organizing knowledge, space, and power.

The article also acknowledges that educators are deeply entangled with the grid and cannot simply reject it. Institutional constraints such as teacher roles, policy expectations, and school structures require teachers to work within these systems. In response, the authors explore garden-based teacher education as a space where tensions between control and freedom become more visible. Through arts-based, embodied, and outdoor practices, they examine alternative ways of becoming teachers that do not rely solely on traditional, gridded forms of schooling. 

Gerofsky and Ostertag introduce the concept of being “beside” the grid, rather than strictly inside or outside of it. Drawing on metaphors such as swing dancing and parkour, they illustrate how creative movement and play can occur within structured systems by leaning into, around, and against constraints rather than opposing them outright. Ultimately, the authors propose a ludic, playful approach to education that moves beyond binary thinking. By positioning the garden itself as a co-teacher, they emphasize relational and more-than-human forms of learning, suggesting this approach is particularly important for teacher education in times of ecological and social uncertainty.

Stops

1) "Are there senses and sense-making ways that are part of ourselves that we cannot truly reject, but might rebalance and reharmonise with ways of being and knowing suppressed by the dominant culture? Bringing this to the world of teacher education, are there ways to acknowledge our love affair with the grid and its occasional benefits, alongside and in the process of rewilding schooling and taking it outdoors?" (p. 179)

This quotation stood out to me due to its practicality. As a classroom teacher, I acknowledge both the restrictions and benefits of the grid, while also recognizing that it cannot be fully rejected given the nature and structure of modern schools. Garden-based or outdoor education does not need to be positioned as an “escape” from schooling, but rather as a way for teachers to begin loosening the restrictions imposed by the grid on both themselves and their students. Vocabulary such as “rebalance” and “reharmonise” reinforces the authors’ stance that arts-based and embodied practices can coexist alongside the dominant structures present in schools, allowing educators to move beyond the binary of the grid versus what is “other”, and instead navigate tensions rather than attempt to fully resolve them. 

2) "We are simultaneously within and beside ourselves 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 quotation led me to reflect on my practices as a teacher and to question the persona I take on in my school and classroom. While my views of what a “good” teacher is have changed since I began my career, I still conform to many dominant approaches and have little experience in arts-based, embodied, or outdoor practices. This quotation reframes teacher identity as something that is continually formed and changing rather than a fixed role to be mastered, an idea that resonates with me as society and school communities continue to change over time. 

The quotation describes how teachers are simultaneously within institutional expectations and “beside” them, highlighting the emotional and ethical tensions involved in becoming a teacher. By questioning conformity to a dominant teacher persona, the authors open space for alternative forms of teacher education that are embodied, relational, and imaginative. While I initially struggled with the metaphor of “dancing or daydreaming teachers into being with a garden”, I came to see how it reinforces the idea that teacher education can be creative, performative, and co-shaped by place rather than governed solely by rigid institutional norms and expectations.

Questions:

1) What might it look like in everyday classroom practice to work “beside” the grid, rather than within or against it?

2) Is it possible to integrate arts-based, embodied, and outdoor learning without positioning it as an “alternative” or add-on to the “real” learning defined by the grid and other dominant forces?

Reading Response Week 2: Multisensory Math

Reading: Angelika Stylianodou & Elena Nardi (2019), Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils

Summary:

The authors of this article argue that tactile perception is an important and valuable way of engaging with mathematics for all learners – not just visually impaired ones. Rather than treating touch as a special accommodation, they frame it as a shared classroom practice that can support inclusion and challenge ableist assumptions about how mathematical understanding should be developed. Their study aims to show that when tactile approaches are implemented universally, they can benefit both visually impaired and sighted students by further expanding how mathematical ideas are explored and discussed.

The study draws on classroom data from students in Years 1, 3, and 5, aged 6-10, and focuses on a task involving a shape that resembles a circle, but missing a circular segment (as in, the shape contains a short straight segment in place of being continuously round). One sighted student was able to clearly feel the straight-line segment in the shape by observing it only through touch, but struggled to notice it visually – likely because the shape appears so similar to a circle at first glance. This highlights how visual perception can encourage a quick, whole-shape assumption, while tactile exploration encourages a slower, part-to-whole understanding. A visually impaired student also contributed meaningful observations through touch when comparing the new shape to a circle, showing that tactile engagement supported meaningful mathematical thinking across the entire class. By including this activity as a universal design, the mathematics lesson became more inclusive and encouraged students to develop non-ableist perspectives on mathematical learning capacities. 

Stops:

1) "A conjecture that our study explores is whether, and if so how, universally designed mathematical practices lead not only to better inclusion of [visually impaired] pupils but also bring benefits to all pupils" (p. 1).

This quotation truly spoke volumes to me. Over the last few years as a science educator, I have been working to make my courses more accessible to all learners, not only those with learning plans that outline specific accommodations. Inclusion in schools is often framed as catering only to students with identified exceptionalities, but to me, inclusion means designing classroom experiences where all students benefit from purposeful supports. This approach benefits students as well as teachers, as it shifts accommodations away from being seen as something only certain students “need” or as extra work to be added on, and instead positions them as opportunities to enrich and improve instruction overall. Unfortunately, I have encountered many teachers who struggle to implement inclusive practices (often due to limited training, resources, and capacity related to workload). I believe that universal approaches offer a practical way to support the needs of as many students as possible within a shared classroom environment. 

2) "In any case, we have argued that inviting the entire class to explore mathematics through touch could possibly lead to broadening everyone’s perspectives on what constitute valid mathematical practices" (p.4).

This quotation prompted me to reflect on what is meant by “valid mathematical practices” and challenged the assumption that mathematics is primarily visual or symbolic. Traditionally, mathematical practices are often defined through diagrams on paper, visual representations, or written explanations. While I try to switch up my teaching strategies and incorporate different activities to support student learning, this paper helped me see how embodied mathematics through touch can allow students to move beyond their initial assumptions and attend more carefully to parts of an object rather than viewing it only as a whole. In my own experience, activities like these are often considered “extra”, less serious, or even invalid, yet the findings in this study challenge that perspective. They demonstrate the value of non-traditional practices and invite reflection on whose ways of knowing are privileged in classrooms and whose are dismissed. 

Questions:

1) What do you think counts as “valid” mathematical practice in classrooms? Who or what tends to define this, and how does that influence which students may be seen as “successful”?

2) Is there a clear line between what is considered an accommodation and what is simply good teaching practice? If universal design is implemented, in what contexts might individual accommodations still be needed, if at all?

3) If some students learn mathematics most effectively through visual or symbolic representations, should tactile approaches be expected of them in the name of inclusion? Where might choice or flexibility fit into this tension?

Reading Response Week 1: Mathematics and the Body

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

Summary:

This reading introduced embodied learning in mathematics education. The first section discussed challenges within educational systems that rely on teachers who often lack formal preparation in designing learning experiences that foster genuine learning to support a wide range of learners – content and pedagogical knowledge alone are not enough. There is little attention to how people actually learn, yet educational programs and policies are frequently implemented based on assumptions and generalizations about learning. The reading defines learning as a lasting change in behaviour and emphasizes that many different types of learning exist, all of which must be addressed in classrooms. It argues that natural ways of thinking, teaching, and learning are embodied as people use body-based resources to make meaning and connect new ideas to prior experiences. However, classroom design often restricts physical and social interaction, therefor limiting access to embodied resources and impeding learning. Research shows that learning can be improved by physically engaging one’s body in the learning process. The second section builds on this argument by asserting that mathematical ideas are fundamentally grounded in conceptual metaphors of embodiment. Metaphors such as collecting, movement, construction, measurement, and spatial orientation support mathematical meaning making. Concepts like considering numbers as quantities of objects or positions along a path, and arithmetic as object manipulation, provide a meaningful basis for mathematics rooted in physical action and perception rather than disembodied abstraction. The reading suggests that mathematics is difficult for many learners not because the ideas themselves are too challenging, but because students struggle to connect formal notation to these underlying embodied meanings. Embodied and grounded instruction supports mathematical understanding by connecting symbols to body actions and spatial experiences, helping students internalize mathematical structures. These approaches are especially important for learners who have not had prior exposure to dominant Western mathematical metaphors through their cultural or everyday experiences. Overall, the reading rejects the idea of “math people” and critiques traditional instructional practices that are systematically inaccessible to many learners.

Stops:

1) In the section discussing learning as a long-lasting change in behaviour, I was reminded of the idea of conceptual change in science education, in which learners’ internal conceptions of scientific phenomena shift toward more scientifically accepted explanations. From this perspective, learning is less about memorizing information and more about lasting changes in how individuals understand and interpret explanations. The processes through which students undergo conceptual change are highly individual, shaped by prior knowledge, experiences, and motivations, which suggests that no singular instructional approach can effectively support all learners. While we are focusing on mathematics education and grounding metaphors, I see a clear parallel between these ideas. Researchers in both areas emphasize that meaningful learning involves a fundamental shift in how learners understand and engage with concepts, rather than a passive or disconnected transmission of knowledge. 

2) As the reading discussed the ways in which learning can be improved by engaging the body in the learning process, I was reminded of the research by one of my professors, Grant Williams, during my BEd at St.Thomas University on Kinulations (Kinesthetic Simulations). Kinulations are “movement-based learning activities in which students take on the roles of key elements of natural systems in order to act out or kinesthetically simulate particular scientific phenomena” (kinulations.com). While I would love to incorporate more of these types of activities into my classroom, I find it challenging to do so in small, cramped spaces or when working with more advanced scientific content. That being said, I am grateful that I was exposed to this approach before beginning my teaching career, as it has had a lasting influence on how I think about teaching and learning. I have already incorporated elements of this type of instruction into some of my classes, such as using physical simulations to help students understand phase changes.  

Questions to Consider:

1) How might embodied learning disrupt patterns in which traditional instructional approaches privilege certain learners over others? In what ways could it support more equitable and accessible access to meaningful learning? 

2) How has this reading influenced your view of your role as a teacher or educator, particularly as a designer of learning experiences? 

3) From a practical standpoint, what challenges or limitations do you see in implementing embodied learning approaches in classroom settings?


Kinulation Resources

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