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.
ReplyDelete“Geometry is often taught as a highly visual subject, so it is easy to assume that learning it would be extremely limited without sight.” --- I have to say I really agreed with your thought when I read it. I guess one reason is that we were all taught that geometry is about the shape of an object, which means, obviously, you must see it first. I guess school experiences will shape one’s thinking, as you mentioned. If there are more ways to engage other senses in learning geometry, future students probably won’t think of geometry as a purely visual subject. They will step out of their comfort zone and start to explore more parts of it. The paper I read also discussed how sighted students and visually impaired (VI) students experienced the same geometric shape, that VI students used imagination to predict the feature of the shape in difficult scenarios (i.e. can it roll on the table) while sighted students focused on the shape itself (i.e. how many sides are there). I would argue this is a process of transferring mathematics from the theory to practice/application. Being able to investigate the problem with multiple senses can offer a broader view of learning.
Thanks for showing the Hexaflexagon activity! I believe that is a great experience in interacting with 3D rather than 2D models from their textbook. It reminds me that in math class, we always let students calculate surface areas and volumes from the given shapes on the handout. It may be challenging for some students because it can be hard to imagine what those shapes “really” look like. However, if we can let students build them by themselves, they will have a better idea of what surface areas and volumes.
You bring up a really important point about how schools tend to shape and guide students’ ways of thinking. Many teachers introduce geometry primarily through images of shapes, which can create a preconceived notion that geometry is mainly about visual representations. As a result, when students think about geometry, they often picture shapes in their minds first. If teachers were to introduce geometry through other senses, such as touch, students might instead imagine how the shapes feel rather than how they look. This shift could open up different ways of understanding geometric concepts and help students build meaning through embodied and sensory experiences, rather than relying only on visual cues.
DeleteI’d love to try and suggest an answer to your discussion question: "How can mathematics educators design learning experiences that intentionally draw on multiple senses, rather than relying primarily on visual representations?"
ReplyDeleteThe first thing that came to mind was a black box experiment in sticking with the theme of visual limitations. Put the solid shapes into an opaque container (shoebox with hand-holes) and have the students feel them out first. Maybe putting one object in at a time can help with isolating in the lesson, perhaps like “count the number of corners!” and the students would feel a tetrahedron, then replace with a ball and stick model of a methane molecule and ask for similarities and differences. Then up the difficulty by replacing with a dodecahedron that will be tricky to need to come up with a physical strategy for counting corners without vision. That might be interesting.
More generally, I wonder if there could be a set of foundational mathematical activities to which almost every concept could be applied. For example, Rhythm/music. When talking about integers, we drum whole beats, rational numbers are portions of the whole beats. Then shift to drawing. Integers are pixel art on a grid. Rational numbers become shaded portions of a grid cell – combine enough together and you get visual illusions.
Can everything relate back to Rhythm/Music (patterns), and Visual art? What else could be a foundational mathematical activity?
Thanks for the thoughts!
Thanks for this very thoughtful conversation! Sukie, I love that you have made connections with your own experiences after laser eye surgery, and that you showed such persistence and patience in getting the hexaflexagon to work! So many great ideas in this discussion... for example, the idea that geometric shapes could be introduced/ worked with using actual 2D and 3D tangible shapes, rather than only with printed perspective versions on a page. Great work!
ReplyDelete