Week 2: Multisensory Math

This week found me connecting to things I had done before, learning new things, and being a little (lot?) bit confused while I was excited about what I was learning. This is a long post, sorry! 

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I found I had the most stops when reading the Week 2 introduction particularly in the paragraphs discussing rethinking ideas about disabilities. Gerofsky (2024) states that "in order to minimize disability, the society would need to stop disabling people!" This really stood out to me as I was thinking about the society we live in where those with a disability often are in an "other" category and how we can change this simply by changing the narrative we use towards those with impairments. The introduction says that disabilities result from a restriction of activity that disadvantages and oppresses the person. It seems like a fairly easy fix to stop disabling people by providing everyone with the same opportunities that will support those with impairments. Obviously this is easier said than done and may take a lot to support everyone's impairments but I think it is doable especially if we focus on one classroom at a time. I really liked when Gerofsky (2024) said that "everyone experiences impairments to some degree at different points in their life" thus there is not a clear divide between who is disabled and who is not. We need to work towards moving the barrier between those who are 'able-bodied' and 'disabled' so that everyone feels supported. 

This reminded me of the Universal Design for Learning model where multiple means of engagement, representation, and action & expression. Through this model teachers are providing all the students in the class with the same opportunities to optimize their learning. 

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Reading Reflection 

Kepler, J. 1611/2010. The six-cornered snowflake: A New Year's gift. Paul Dry Books. 

While I was excited about what this book included, I found it hard to read and ended up re-reading many sections to fully try to grasp what was being said. The book explores why shapes are the way they are, why snowflakes always have perfect symmetry with six points. In the section I read for this week, Kepler bounces back and forth exploring the shape of beehives and pomegranate seeds. He also discusses shapes, mainly a rhombus, and how they can create other regular solids and Archimedean solids. Kepler explains that a beehive is composed of a hexagonal plane where each cell is surrounded by six others and each bee has nine neighbors. When explaining pomegranates, the seeds are round at the start but as the pomegranate grows the rind stiffens and the seeds are forced to crowd, forming a rhombic shape. 

In explaining the shapes of pomegranate seeds, Kepler explains how this connects to beads of equal size and material being contained in a vessel. The beads are squeezed into a rhombic shape. There are two arrangements of spheres when placed in a container - either they will align in a triangular or a square arrangement. While playing with my dog and her tennis ball, I decided to try this out myself so that I could connect to the reading a little more (please ignore the hair, I know it's gross but these are the only round objects I have.) 

This is a recreation of Kepler's explanation to show what is occurring with the tennis balls

*Paisley guest appearance because I had her toys* 

While I was reading, I was thinking about how easy it is to make these changes and implement multisensory experiences into elementary schools but am struggling to think of ways that they could be used in an academic high school math class. I can see how they work in science classes with the hands on activities that arise but am stumped thinking of how they may work, for example in a calculus class, with math students who are academically driven and don't necessarily have the time for multisensory experiences. I can see how many of the students I work with would be frustrated that they are doing these activities as opposed to doing math the way that they will be doing it in University. 

Questions: 

• Have you consciously tried to remove the barrier between 'disabled' and 'able-bodied' before? Can you think of any instances where this sort of approach will not work? 
• How can we implement these multisensory experiences into high school math classes that traditionally focus on lectures, assignments and tests in a way that supports all students learning?

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Activity

I completed the activity prior to the reading however after reading for this week I connected what I read to the activity I completed. 

Part 1: The beginning of the activity suggest finding mathematical foods such as a pomegranate, apple, persimmon, zucchini, squash, tomatoes and honeycomb. I struggled with this part of the activity as the less common foods are not available in my grocery store and I did not want to purchase foods that I would likely not use or were quite expensive. I tried to use what I had at home already and supplemented with pictures off the internet. 

Apple: When I cut the apple horizontally I saw the clear "flower" shape in the center of the apple, this had five petals like the flower blossom below. The apple seeds fall into the cavities of this flower pattern but do not appear to be evenly distributed between them (maybe if they all fell to one half they would be even). I have used this cut of apples before with young kids as a paint stamp as the pattern sometimes shows up in the artwork. I have to say, I don't know if all apples are like this but I can't think of any types of apples that do not start with a flower blossom, therefore I assume that most apples start like this and follow the fivefold pattern of the flower. 

Apple blossom: https://commons.wikimedia.org/wiki/File:Apple_blossom_01.jpg

Pomegranate: I couldn't buy a pomegranate at either grocery store in my town so this is an image from the internet. most of the online images are cut from the top down (vertically). The seeds of the pomegranate appear to be enclosed into patterns, the second image shows how they have a similar "flower" like shape to the apple, with each one having two rows of evenly distributed seeds. The first image does not appear to have much of a pattern except for the fact that the seeds exist in between the pomegranate 'membrane'. I saw that the apple followed the shape of the flower blossom but I realized I didn't actually know how a pomegranate grew - they also start with a flower however they vary from three to five flowers.  

https://pixabay.com/photos/pomegranate-fruit-fruit-kernels-2018968/
https://commons.wikimedia.org/wiki/File:Inside_of_a_pomegranate.jpg

Other: The other fruits / veggies that I looked at were a zucchini and a tomato. The zucchini I had didn't have much of a clear pattern it just looked like a few different colors however I have noticed before that a cucumber does have seeds enclosed in the middle. I cut my tomato the wrong way so had to redo it with a horizontal cut and forgot to take a photo of it this time. I can see in the tomato that they follow a similar pattern of the seeds being enclosed by the outside, harder, edge. 

Honeycomb: the honeycomb has a clear pattern. There is a 6 sided shape (hexagon) connected to another on each side. All of the shapes are the same and share their sides with the neighboring hexagon. I think bees make their hives this way for simplicity reasons and less work, it is easy to create a hive where each wall is shared with another. This appears to take up less space to allow for more honey in the hexagons.  

As I was reading Kepler, I confirmed that the bees create this hexagonal plane because hexagons have a lot of space inside and therefore store more honey than other shapes and because the hexagons share walls it requires less work from the bees. 

https://www.pickpik.com/honeycomb-bees-hexagons-comb-honeycombed-insect-148106

Part 2: It snowed many days this week but was wet snow so I could not find any snowflakes that were photo worthy. I will keep looking as it snows more and see if I can update with my own images. 

Every snowflake has six sides, and resembles a hexagon. When I look quickly on google it states that snowflakes create a hexagon shape because the hydrogen and oxygen (h20) molecules of water combine together more efficiently to form a hexagon. 

Part 3: I made the following shapes using the cut and tape templates and tried to make some of the origami ones but found myself frustrated as they were not working easily for me (I have good origami paper but couldn't find it anywhere) so ended up giving up on those ones for now. 

Paper shapes made with my nephew. We used tape to hold them together but found the larger shapes tricky to make.  

I have also made a lot of these shapes with sonobe unit origami before. I first learned how to make these in my undergrad math education class and have since used them in many classes that I have taught. Here are some images of the sonobe units we learned to make, then the cubic shape that the students have made as well as some larger shapes. 

This site shows how you can make sonobe units and the modular shapes: 

https://mathcraft.wonderhowto.com/how-to/modular-origami-make-cube-octahedron-icosahedron-from-sonobe-units-0131460/ 

Sonobe Units bundled into 6 each for a cube
Students working to make their cubes. 

The students cubes they made on their own! 
Sonobe units, hexahedron, triangular hexahedron, and octahedron

Reflection questions: Cutting up the fruit, looking at images and building the shapes helped me to understand what the text was saying in multiple ways. I found that this helped me to develop a deeper understanding especially since I found the text somewhat confusing to understand. As in our introduction for the week, it says that "multisensory, bodily experiences of mathematical patterns and relationships have potential to serve as meaningful, symbolic or representational resources as students develop the ability to work with generalization and abstraction in mathematics" (Gerofsky, 2024). Through using these multisensory experiences with the fruit and shapes I was able to use these objects as representational resources to help me develop this understanding.

Using 3D objects with shape, texture, smell, taste etc. invites opportunities for all the senses to be used rather than relying only on sight and hearing. These additional opportunities to use the remaining senses can help all students thrive in an educational setting. Not all students are capable of simply sitting, listening and doing work and even if they are they likely would benefit from additional activities. By including multiple modalities to teach students more complex or abstract concepts educators are allowing them to understand the concept in a way that makes sense to them without feeling like they are less capable than others. This also applies to students experiencing sensory impairments, "innovations supporting learning for students with sensory impairments will support learning for all" (Gerofsky, 2024). By implementing multisensory activities all students are supported without needing to feel different than the rest of the students, activities such as this one create opportunities for all senses to be used.