Reading 2
Fernandes, S. H. A. A., & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind students’ interactions with symmetry. *RIPEM – Revista Internacional de Pesquisa em Educação Matemática*, 3(1), 36–55.
Introduction
Fernandes and Healy investigate how blind students construct
mathematical meaning in geometry, focusing on symmetry and reflection, through the
lens of embodied cognition. Challenging the assumption that mathematical
understanding is primarily visual, the authors argue that cognition is grounded
in bodily action and multimodal experience, drawing on phenomenology (Merleau‑Ponty)
and neuroscience (e.g., Barsalou, Gallese, Damasio).
The study is based on task-based interviews with two blind secondary‑school
students: Edson, who lost his sight in adolescence and retained visual
memories, and Lucas, who had been blind since early childhood and had no visual
memory. Using tactile tools such as cardboard shapes and a geoboard with
elastic bands, the students worked on tasks involving symmetrical figures and
reflections.
Findings show that both students relied heavily on touch, movement,
gesture, and imagination to make sense of symmetry. Edson often used visual
memories—especially mirrors and folding—to simulate reflections, initially
reasoning at an intrafigural level by focusing on congruent parts of single
figures. With guidance, he began to coordinate distances and orientations,
moving toward interfigural reasoning. Lucas, lacking visual memory, developed
strategies grounded in tactile exploration and spatial relationships, often
focusing on angles, distances, and invariants across figures. His thinking
frequently exhibited interfigural characteristics, sometimes more readily than
Edson’s.
A key contribution of the paper is the demonstration that knowing
symmetry does not replace feeling symmetry: mathematical concepts were not
abstracted away from bodily experience but were constituted through it.
Reflection
One meaningful finding from the article is the authors’ observation
that both blind students naturally moved their hands in symmetrical ways when
first exploring tactile materials. This behaviour was not anticipated or
intentionally designed into the tasks, suggesting that symmetry can emerge
through embodied interaction rather than only through formal instruction or
visual representation. This observation highlights that mathematical ideas may
first be sensed and enacted through the body before they are explicitly
named, defined, or symbolized.
This finding encouraged me to think differently about classroom
learning. It suggests that students’ gestures and physical movements are not
incidental or secondary, but may be important expressions of their thinking.
Paying closer attention to how students interact physically with learning
materials can reveal early forms of understanding that may not yet be verbal or
symbolic. In this way, the article challenges teachers to recognize embodied
action as a legitimate and valuable form of mathematical reasoning.
Another moment that stood out to me was when the authors argued that
blind learners do not necessarily follow the same developmental trajectories
proposed for sighted learners, and that mathematical reasoning is strongly
shaped by task design and the mediational resources available to students. This
idea connects closely to a framework of learning that views cognition as built
on experience. I found this particularly compelling because it resonates with my
reading from another course on experience‑based learning, in which learning is
described as a process of making sense of the world rather than simply
acquiring abstract knowledge.
The article helped me reflect on the distinction between experience
and knowledge. Experience itself is not equivalent to learning outcomes or
finalized understanding; the same learning experience can lead different
students to develop different forms of knowledge. Simply providing identical
experiences does not guarantee equal learning gains. Students interpret
experiences through the lens of their everyday lives and embodied interactions
with the world, which helps explain why blind students should not be expected
to learn mathematics in the same ways as sighted students.
In this context, the authors’ emphasis on feeling symmetry
rather than only viewing it is particularly important. Engaging with symmetry
through touch and movement does not just provides blind students with learning experiences
that align with how they make sense of mathematical ideas, but can also be extended and used to help students from different background with distinct view of the world. These embodied
experiences also support cognitive processes that visually oriented instruction
alone cannot fully address, enabling deeper and more meaningful understanding.
Question
How can
the findings of this article inform everyday teaching practices to enhance
students’ learning?
Hi Li, thanks for your summary and reflection—I am just reading this now after having read your response to Amy’s paper. In that response, you suggest that vision remains the dominant mode of learning mathematics, and that embodied and tactile approaches can reveal aspects of mathematics that purely visual approaches cannot. In your post here, you discuss the embodiment of symmetry and how this can lead to deeper and more meaningful understanding. This made me wonder: do you think embodied learning must always be accompanied by visual learning when possible? If so, does one inform the other, or do they operate as a simultaneous and intertwined process?
ReplyDeleteIn response to your broad question, this course has made me more aware of how modern classrooms often strip away embodied experiences, despite the fact that learning through movement is intrinsically natural. We were not made to learn as idle statues, but to engage through embodied gestures and interaction. The findings from this article reinforce the necessity of embodied learning experiences, pushing back against the mainstream prioritization of visual learning. At the same time, I find myself hesitant, particularly when considering post-secondary contexts. If students choose to pursue higher education, they often encounter massive lecture halls and increasingly large class sizes. In such spaces, I am left wondering: what role can embodied learning realistically play in higher education classrooms?
As someone who has long relied on vision as both a student and a teacher, encountering embodied and tactile approaches to learning feels refreshing and opens up possibilities for making mathematics more engaging and inclusive for students with different abilities, backgrounds, and ways of thinking. At the same time, like Anna mentioned above, I find myself questioning how realistic these approaches are in everyday teaching practice. I remember my own university experience: professors often rushed through dense content, while I copied notes as quickly as possible without fully understanding them in the moment, only to spend a long time trying to make sense of the material afterward. From a post-secondary instructor’s perspective, visual explanation is often the fastest and most efficient way to cover required content within limited time, and in that sense, its dominance is understandable—even though efficiency does not necessarily mean quality of learning. Implementing tactile or embodied learning would require additional preparation and materials, and in large lecture-based courses, this may not always be practical or realistic.
ReplyDeleteWhen I think about how these approaches might be applied in secondary mathematics teaching, the possibilities seem highly context-dependent. In Korea, strict standardized testing makes it particularly difficult to implement embodied or tactile approaches, as instruction is often driven by exam preparation. On top of this, a relatively homogeneous educational culture can make it harder to recognize and value different ways of thinking and learning, with a strong emphasis on arriving at correct answers efficiently rather than engaging in exploration. In contrast, in BC, where the curriculum allows for greater flexibility, there may be more room to meaningfully integrate embodied and tactile practices into everyday teaching—although I hesitate to make strong claims, as I do not yet have enough teaching experience in this context to say so with confidence.
Thank you all for struggling with these new ideas -- ideas that challenge many of the established, unquestioned norms of many classrooms. It IS possible to do embodied math learning in post-secondary math classes; I've experienced it in an abstract algebra course at SFU, and perhaps we will try some of the activities that the prof, Harvey Gerber, brought into that class. I think it would be harder to do this in a large lecture hall class (and why do we still have such classes?), but it is not impossible!
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