As I was reading the introduction for this week, I stopped to mull over the question, “What would mathematics lessons be like if we took multisensory learning seriously?” The principles of UDL came to mind. Ideas about discovering mathematical relationships and using a variety of representations, supported this proposal that multisensory mathematics lessons embrace the principles of UDL (Universal Design for Learning). UDL has 3 guiding principles which include providing multiple means of: 1) Representation – the “WHAT” of learning 2) Expression – the “HOW” of learning and 3) Engagement – the “WHY” of learning. You can look here for more information. Universal Design for Learning Guidelines
UDL proposes that we start with the belief that all students are capable and aspires to take away stigmas that have separated students with disabilities from the group. Gerofsky wrote in the introduction, “innovations supporting learning for students with sensory impairments will support learning for all” (Gerofsky, 2022). This is UDL in action. How would schools change if we stopped thinking of students as being disabled and instead considered how our environments – social and physical – were disabling. It changes the conversation completely. Can we stop thinking through the binary (false binary) of abled and disabled?
The article I am summarizing this week is c ) Angelika Stylianodou & Elena Nardi (2019), Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. This study’s aim was to “contribute to inclusion and challenge ableism in the mathematics classroom” (Stylianidou & Nardi, 2019, p. 343). The study challenged the effectiveness of using only the typical senses of sight and hearing in the mathematics classroom. Citing the Convention on the Rights of Persons with Disabilities, the paper defined two different types of inclusion: reasonable accommodation and universal design. The study involved a grade 5 classroom with a visually impaired student and drew upon Vygotskian sociocultural theory and the theory of embodied cognition, among others. The study engaged students in exploring shapes through touch first, prior to exploring the shapes through sight. The authors considered the wholistic experience of sight compared to the gradual experience of touch. Results confirmed that both VI and sighted students developed a deeper understanding and more accurate description of the properties of the shape through the sense of touch. A sighted student commented that using touch revealed, “hidden facts on the shapes.” (Stylianodou & Nardi, 2019, p. 348)
Mathematics is probably the subject that teachers struggle the most with implementing the UDL principles, likely due to the Platonic, Cartesian worldview that has been the basis of most mathematics instruction in schools. Multisensory, embodied activities create connections for students, allow students to understand relationships between ideas, and can enable a variety of representations. Using a variety of representation can lead students to see different perspectives and, as Antonsen proposed last week, result in deepening understanding. This may occur through the enhanced metaphors that are develop as students interact in these multimodal, multisensory activities.
These thoughts lead me to consider the difference between knowledge and understanding in mathematics. Has what many would consider to be traditional mathematics instruction, led to knowledge without understanding?
In multisensory mathematics I see the coming together of theories of learning, use of imaginative education tools (see here for a mathematics example Learning Math through Stories) and practical applications that build rich and engaging opportunities for students to make sense of the mathematics they are learning.
Reflection on Activities:
Although I am acutely aware of “playing with” food, when there are food insecurities all around us, I can see the potential for using the sense of taste (and multiple senses) in mathematics. (And if the food used is actually eaten as it was with these activities, no food insecurity issues.)
One of my boys and I tried the Mathematically Correct Breakfast activity of cutting the bagel together. We were both successful on the first try (although my success was directly related to Emmett helping me!). This activity revealed to me the increase in surface area (and thus a greater cream cheese to bread ratio - Mmmm) in a way that would have been difficult for me to perceive without actually experiencing it. It helped me to "see" this and "taste" it and has me considering it in a way I wouldn't have without experiencing with my senses.
Making the hexaflexagon was easier for me this time. I had made hexaflexagons for a previous class, worked my way through the productive struggle and found success came easier this time. Eating the hexamex flexagon burrito was just fun and engaging – ripe with possibilities for conversation and thinking mathematically.
As I considered the Rockets video, I started to brainstorm different kinds of activities with candies. One of my boys is a huge Reese’s Peanut Butter Cup fan. For Christmas, we gave him a box full of all things Reese’s (and there are a lot of Reese's products!). Many activities could be done comparing the different types of cups – the chocolate to peanut butter ratio (it would be interesting to know the difference between the 1/2 pound cup, regular cup and the thin cup), weight to calories etc.
Although we would be unlikely to complete a Reese’s comparison in class, I have used candy to graph and practice addition with Kindergarten and grade one students and they loved it.
These real 3D multisensory objects offer multiple ways for students to experience the mathematics in the world.
The activities this week have given me lots to think about!
Hi Joy,
ReplyDeleteThank you for your post which has an excellent summary of the article, and much more. While the authors briefly mention what “Universal Design” denotes, they do not elaborate on the topic of UDL. Your explanation of why the principles of UDL are relevant to teaching students with disabilities, and providing the UDL Guidelines link, are helpful.
It is common practice to use concrete math manipulatives for teaching in classrooms. But the approach in this study to ask all the students to close their eyes and describe the shapes, before the sighted students could see the shapes, was new to me and I found it interesting. I do not disagree with their statement that “tactile perception may lead not only to better inclusion of VI pupils but can also bring benefits to sighted pupils, too.” Their analysis as reported in this study had focused on two students, one visually impaired (VI) and one sighted, sitting at the same table. I wonder how the rest of the class responded while doing this task. The authors may have reported their observation of the entire class’s learning experience elsewhere.
Having policies for inclusion and educational equity are absolutely necessary. Successful implementation of these policies, especially teaching math to visually impaired students, would require professional development courses for teachers to be knowledgeable about the relevant issues. I am thankful for having been introduced to this topic. But I need to learn a lot more, e.g. about developing lesson plans that promote multisensory approaches to teaching and learning.
Thanks for sharing your thoughts and questions, Zaman. This study used manipulatives in a different way than we typically would consider using them. The ideas I found most interesting were the idea the wholistic nature of sight - that when we look, we see the whole object (or situation...) and when we touch, we are gradually introduced to the object and thus are able to discover characteristics of the object that might not be noticeable when we see the object as a whole. This pushes back on our primary reliance on sight and causes me to consider ways that we can acknowledge the value of other senses and incorporate them in our classrooms. It also pushes back on the idea that because we see, we have better understanding of the world around us.
ReplyDeleteI also agree with your questions regarding the way the study was conducted. It would be interesting to know how the other students in the class responded. This paper was a small part of a larger study, so perhaps you are right that this is addressed in a more comprehensive paper on the study.
Professional development is the key, isn't it. If we don't have a chance to consider these ideas as teachers, it is not possible for us to implement them in our classrooms. And changing our thoughts and then actions can be complicated, time consuming and hard, which I am discovering in my new role supporting teachers in my district, but that doesn't mean it isn't worthwhile. As I learn, I am considering how to share what I am learning with others. The constant of being a teacher is that we are always learning and adapting our practice to best support student success - at least that is my hope.
Thank you for these very interesting reflections and connections, Joy! I'm so glad that you were successful in making (and eating) the Mathematically Correct bagel rings AND the flexmex burrito. (Not everyone had such a happy outcome!) Your connections with the article and with Zaman's comments really captured the idea of the ways that tactile representations can enhance mathematical learning.
ReplyDeleteZaman, I have taught whole secondary school classes using the Wikisticks tactile representations of graphs, in classes where there was a student with VI and about 24 others with full vision. The tactile and movement-oriented activities did benefit all the learners, and it was not difficult to bring everyone into these activities.
I think when we are thinking about teacher professional development, it is important to realize that all of us (having done graduate degrees in mathematics education) will be the ones to actually LEAD this professional development based on our broader and deeper knowledge. We will not only be passive recipients of 'training', but will need to take responsibility as teacher leaders, thinkers and doers to be innovative, deeply thoughtful and integrative of new ideas to improve teaching and learning. It is not enough to demand support and training -- we need to be ready to problem-solve and offer support to ourselves and our peers.
Joy, I love that using food catches the interest of everyone -- not just students! It is so much fun to "play" with food, especially when that is discouraged by "adults", so I enjoyed seeing your connections to mathematics.
ReplyDeleteYour question "Has what many would consider to be traditional mathematics instruction, led to knowledge without understanding?" really caught my attention. I struggle regularly when students tell me to "just do ______", yet can't explain to me why that is the move to make. Whether it is cross-multiplying, factoring, rearranging, collecting like terms...the list goes on. Providing opportunities for students to work in a hands-on method to understand these concepts will, I'm sure, solidifying the understanding, while will stick with them for a long time.
Thanks for sharing!!
Thanks Fiona. This is a thought-provoking reflection! Glad to see you getting caught up.
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