Week 4: Mathematics and the Arts (Post 2 of 2)

Reading: Eve Torrance (2019), Bridges 2018, Nexus Journal  

Summary: The article “Bridges Stockholm 2018” by Torrence (2019), it focuses on the variety of presentations, highlighted exhibits, and themes throughout the 2018 conference in Stockholm, Sweden. I chose this as my art inspiration piece by Karen Amanda Harris, which was featured at this particular exhibit. What I took away from this article was the number of creative approaches the artists used to mathematical components. Everything from gardens to card manipulation, to bands (that included instruments made from underwear), were incorporated. I also appreciated the stories of key contributors, such as Marjorie Rice, “a woman with no mathematical training beyond high school” who was able to discover a “new pentagonal tiling” in 1976. At the conference, Colm Mulchahy integrated a magical component through card tricks, all stemming from math, despite having a cast on his arm! 

The article also mentioned some presenters and authors who I would like to further explore (I’ve linked some of their other studies to their names). 

• Hannu Salmi “Bridging the gap between formal education and informal learning”

• Minna Houtilainen “Neuroscience in understanding learning in the STEAM context”

• Paul Moerman “Dancing Math - aesthetic and math literacies intertwined”

Reflection
When I jotted notes along the margin of the paper, I noticed I constantly used the word “different”; “different perspectives”, “different angles”, “different mediums”. Partway through, I wondered, why do I use the word “different”. To be different would assume that they were not aligned to begin with. This week’s activity and reading have prompted me to re-assess my own perspectives and how I gravitate towards viewing mathematics and arts as separate entities or worlds. When the truth is, there are intertwined. The article caused me to pause and reflect on how this approach to mathematical learning compared to the standardized math classroom setting, where the most colour that is integrated is simply from the red ink used for marking. 

Applications

There is a greater acceptance to teaching mathematics while integrating the arts. Platforms such as Desmos offer beautiful renditions of student artwork composed of graphing functions and formulas. These activities prompt students to struggle and emphasize skills in problem-solving, creative thinking and as well, build resilience. There is more than simple repetition and math worksheets, and it is thrilling to see this more often in the classroom setting. 

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Figure 1: Desmos Graphing Art Contests 2022 Student Finalist

References:

Moerman, P. (2016) Dancing Math: Teaching and Learning in the Intersection of Aesthetic and Mathematical Literacy. In: Torrence, E., Torrence, B., Séquin, C., McKenna, D., Fenyvesi, K., Sarhangi, R. (ed.), Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture: Conference Proceedings: Bridges Finland 2016: University of Jyväskylä (pp. 269-276). Phoenix: Tessellations Bridges Conference Proceedings

Mossberg, F. (Red.) (2017). Child & Noise: How does the child percieve the sound environment? (Skrifter från Ljudmiljöcentrum vid Lunds universitet; Vol. 17). Ljudmiljöcentrum vid Lunds universitet.

Thuneberg, H., Salmi, H., Fenyvesi,K. (2017) "Hands-On Math and Art Exhibition Promoting Science Attitudes and Educational Plans", Education Research International, vol. 2017, Article ID 9132791, 13 pages, 2017. https://doi.org/10.1155/2017/9132791

Week 4 Mathematics and the Arts Introduction (Part 1 of 2)

Reflection:

This week’s activity resonated with me because it emphasized the significance of re-merging these two cultures together.  In Susan’s introduction, she notes that our “society often deals in binary”, portraying mathematics and arts as “polar opposites”. For most of my life, I often thought the same way. I loved fine arts as a child, and as a teen considered pursuing an art degree rather than the science/math route. However, cultural and family pressures dictated that one had greater value than the other, one could provide a stable career, so art had to be placed on the backburner. However, it’s intriguing to learn,  “there was not the concept of such a split between the arts and the sciences, even in mainstream Western society and among the academics, up until the early 20th century.” (p.2). My students are often amused to learn that if I wasn’t a math and science teacher, I’d love to be an art teacher. Through this program, and especially this course, I am realizing there is no need to ‘choose’ either but that both worlds can exist in harmony, and better yet, enhance the beauty in each. 

Activity:
For this week’s activity, I was inspired by Karen Amanda Harris’ “Broken Sphere” (Bridges Exhibition, 2018). I admired the symmetry in the outline of her piece and was captivated by her choice of colours and the filled segments she used to break that symmetry. The rays and angles that deviated from various points prompted me to think of so many questions I could ask my students: 

• What angles are formed? 

• What polygons are formed? 

• Where is the symmetry? Why isn’t it symmetry

Karen Amanda Harris: Broken Sphere

30 x 42 cm

Promarker, gel pen, ink liner, nail varnish and correction fluid on watercolour paper

2016

For my own design, I chose to deviate from her pen and paper medium and challenged myself by constructing a 3D version of her work using a board, nails, and coloured string. I started with a 12 x 12 inch board, nailed a center point, and roughly traced a circle using a plate. With a measuring tape, I marked 4 points, and then divided the difference into thirds, creating a dodecagon. I added nails to the four corners and then two more in between. Mathematically, there are a number of circle theorems, line segments, and tangent points I’m sure I could have discussed. However, I haven’t taught high school math in years so they aren’t popping in right now!

   

Figure 1 & 2: (Left) Nails around the circle template, creating a dodecagon. (Right) Segmenting the quarters into additional thirds by dividing the difference by three. 

I had no idea what I wanted to do with the string but had three different colours to use. There was no set design but I started by simply looping the string between nails. Once I finished with a particular sequence, I would tie off the string, change to a different colour, then proceed with a new pattern. Questions I considered as I attempted this art piece:

• Which points had the most loops of string? 

• What type of polygons am I forming? 

• How do I keep seeing these patterns? 

• Are there patterns or connections that I have missed? Many! But I ran out of string and space on some of the nails

• How much string did I use in total?

• Could this be expressed algebraically? 

Unfortunately, I ran out of two of the strings, so the layers were not as clear. I did realize that as I continued the process, there were even more questions that could be asked, depending on the angle the art was viewed. Perhaps in the future with more mindful planning, I could be strategic with the colours and create a designated colour scheme. 

Figure 3 (left) & 4 (right): Final product based on Karen Amanda Harris' inspiration Broken Sphere. 

Week 3: Reading Summary "Off the Grid" (Doolittle, 2018) (Post 2 of 2)

 Summary of “Off the Grid” (Doolittle, 2018)

“The failures of the grid are due to the phenomenon that at some point the ability to compensate for failures of the preconditions for the grid breaks down, no matter how determined and how powerful you are.” (Doolitte, 2018, p.104)

I completed the activity for this week before reading Doolittle’s (2018) article, and by doing so, it gave me a more meaningful perspective of the initial observations in my drawings. Doolittle emphasizes the failures of the traditional grid system and the troubles of “grid-based spatial and temporal forms (p. 101). He explores the notions of Indigenous ways of knowing, specifically the Rotinonhsonni culture, which does not integrate any lines, curved or grid, in the division of boundaries. Rather, their “territory was defined by the river” (p. 113). In his critique of failed grid systems, he offers “alternative geometries better suited to our needs in many domains, including education” (p. 102). The roots of the grid, according to Doolittle, is a method to organize space and time, and ultimately, gives the illusion of a sense of control. However, it often ignores the “specific life, qualities and characteristics of a particular place” and as a result, “forcefully imposes ‘evenness’...” (p. 104). 

List of failures

Failure #1: Gardening, agricultural plots, grids of rectangles, water may not b evenly distributed, accessibility more difficult inland.

Failure #2: Land division, plots, does not take into consideration elevation or drops

Failure #3: Street Grid Systems, does not account “gentle but insistent curvature of earth” (p. 107), and the north edge is slightly shorter than the south. This requires correction lines 

Failure #4: Calendar, grids, set timetables, rather than adjusting schedules based on the rise of the sun, the length or day, the changes to seasonal temperatures.

Gerofsky (2018) ‘hints’ at the possibility of another form of geometry, Riemannian geometry, where “we can vary the way we measure distance from one location to another, and from one direction to another” (as cited in Abelson & diSessa, 1981). No particular system is better than the other, and “allowing all grids on an equal basis…allow us to look past the particular rid we may be using, to refocus on the actual underlying geometry of the situation” (p. 111).

Many of these notions tie back to the notion of “one size fits all”. Like the grid system, we often impose learning styles or content that is heavily standardized, rather than focusing on the needs of the students. Rather than understand or foster those who are 'off the path', we tend to force or fix them like we do uneven lines on a grid. This notion of fixing broken grids stem from our education systems to other systems as well. Though the author offers many concrete examples of failed grid systems, what it does emphasize is the need to break from static, forced, grid-like means of solving problems. Perhaps rather, allow for the dynamics to play out and influence how we may approach ‘boundaries’ or rather embrace the “chaotic control”. 

Doolittle, E. (2018). Off the grid. In Gerofsky, S. (Ed.), Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7

Week 3: Math Outdoors Activity (Post 1 of 2)

I took some time and sat by the green space outside my classroom and then some time by our school’s back parking lot. During this time, I noticed the trees, bushes, evergreens, and the design and layout of the crafted or architecture designs by those that were man-made. 

I chose to use simple lined paper in my notebook and a black pen. As I was sketching, I noticed the flow of my lines and how much I required to use the lines on the paper. Here are some points that stood out to me as I completed my drawings.

Curves and Lines

• The items that were natural (bushes, branches, leaves) all had curves; the flow of my wrist and pen constantly changed directions, or was jagged. Rarely did I follow the blue lines on my paper when I was drawing living or aspects that were natural to the environment.

• The thinner the branches (or thinner my drawing lines) the more dense they became

• The distance between the splits became smaller and smaller

• I heavily relied on the blue lines of the paper when I drew items that were human-made or placed by humans.

• The fencing was parallel to the ground, the lines were parallel and sequential

• The lamp post was a perfecting perpendicular to the ground

• The rocks are natural but shaped and used to create levels. I relied on the blue lines for guide

• The fence was straight and even

• Parking lot lines were perpendicular to the curb. 

Angles

• I was captivated by the bare branches and the forms they took to stem from the main trunk

• Angles were larger towards branches lower down, they became smaller and smaller as they moved up the main trunk

• There were various angles, never really two of the same, each degree was unique

• Even items like pine needles or thorns can branch off a stem repeatedly, but they are not (or rarely) in perfect right angles.

• Items human-made were often in right angles (90O) or straight angles. For example: Fencing

Application to students and the classroom

Taking a walk around the school campus would be a great practice for students to apply their understanding. Because it is based on observations, there are an infinite number of possibilities. For example, in Grade 6, we discuss types of angles, angles on a point, complementary and supplementary angles, etc. This would be a great practical skill for students to measure angles or even estimate. Perhaps not necessarily knowing the precise value, but seeing trends, and patterns.

When I introduce lines and angles to my Grade 6s, we often use bodies and sounds to create angles. For example, we channel our Grade 6 Spanish teacher who loves to say in a high pitch “aww, so cute!” when we describe acute angles. We widen our arms like sumo wrestlers and shout out in a low tone “obtuse” to show wider angles. We arch our backs or “stretch” backwards to emphasize “reflex”, going past the standard “straight angle”. 

As a class, we also ask: which angles are easier to create or model? What limitations do we have on our bodies to make certain angles more difficult? We note the center point and how perspective can change, for example, whether an angle is acute or reflex, and how communicating this is important when conveying understanding. For my students, they tend to remember these angles because of sounds and actions in combination, because when they created the angles with their bodies, they had a certain sensation or multi-sensory experience. 

Random Blog - Student Examples for Geometry in Real Life

My grade 6 students completed and submitted their Geometry in Real Life project this week; I was thrilled to see some of their connections to the topics our EDCP 553 class discussed last week (ie; looking at patterns in fruit, etc).

This was a guided inquiry project where they simply had to answer "How does Geometry relate to real life?" They could explore any area, such as nature, architecture, art, sports, etc. I thought it would be beneficial to share some of their work. 

Many thanks to my students Darsh A. and Flora G. for allowing me to showcase their work!

Week 2: Multisensory Mathematics - Summary and Response (post 2 of 2)

 Summary of Kepler’s (2010) reading

This selection reads like poetry, where Kepler seems to describe the wonders of six-sided shapes, particularly the perplexity of snow or “six-corned stars” (p.35). The article highlights the fascination of different 3D structures forming, such as the “keel of cells”. They discuss rhombuses and compare angles, such as obtuse angles, of beehive cells, which on paper or in words, can be difficult to describe. I can see why the tactile practice of forming or building these shapes is so significant as you need to go through trial and error to see which sides can line up and if they are able to meet and conjoin with precision. The orientation can play a significant factor in how the cells appear and the 2D shapes that form the 3D composition (I found this video as an added visual support, I’m a visual learner myself: Matematicasvisuales: Honeycomb)  

The article then goes on to explore pomegranates and the configuration of their seeds. That they too have this rhombic structure as the fruit matures. The article portrays the story of how the seeds begin small, round, but as they grow, there is less sufficient room. And as the “rind stiffens and the seeds continue to grow, they become crowded and pressed together” (p.53). A great question would be for students to explore, where else may they see this phenomenon? How can they compare volume and surface area, like cells? There are so many excellent extensions students can explore: ratio of the cell's surface area to volume, types of cells in the human body, why are we multicellular organisms versus single-celled, what type of cells do plants have, and how does that impact their rigidity? (Sorry, the biology teacher in me is coming out)! But its application of mathematics and 3D configuration can be extended.

Overall, I appreciate how this article shows the fascination of these platonic structures and how their composition is derived from 2D polygons or shapes.

Reflection of Kepler’s reading and assigned videos:

There is a difference in observing real-life objects and feeling them rather than simply observing them in images or photos. For example, when peeling an orange, you start to notice that some of the pieces may not necessarily be the exact size, which can affect the symmetry if you were to have cross-sectioned the fruit. When my student (who has verbal challenges) tried making the icosahedron, she struggled to connect some of the sides. If she had more time, she may have been able to complete the puzzle but oftentimes, we need to rotate, maneuver or manipulate the pieces to gain a better understanding of the configuration. This can be shown through an image but not everyone has the visual processing skills to grasp these mentally. For many students, hearing to feeling items or sounds enhances the comprehension of a concept. Like young babies, they learn through sense through their hands and mouth, and this contributes to a way to grasp their world. Wortman (1988) notes: “The function of the brain is to sort out the information gained from the senses into meaningful learning.” This is especially important as theories from Piaget note that we don’t all learn math at the same rate. Having additional information inputs can provide an enriched learning environment. Why should this not continue as children progress into adolescence or adults as mathematical learners? 

Week 2: Multi-sensory Math Activities (Post 1 of 2)

 Week 2: Multisensory Math Activities - RECAP of activities with students

(Option A #1)

I was incredibly excited reading about patterns in fruit and nature, as well as 2D and 3D configurations for shapes and models. My Grade 6s are currently completing a guided inquiry project on Geometry in Real Life and they are encouraged to explore any area that integrates geometric shapes and/or geometric patterns. A large number of them discovered and shared patterns such as the rotation in pineapples, the behaviour of honeybees, and how they rotate in patterns of exactly 120O to create these symmetrical, regular hexagons. Finding Geometry in Nature (Children’s Discovery Museum) is a fantastic source that encourages students to find patterns outdoors, in their kitchen, etc. One student also talked about the mathematics in pasta and the configuration of different types and shapes of pasta (Pasta by Design). 

For their activity, they were asked to find 3-5 powerful images that captured the beauty in geometry. Below are some of the snapshots or images that my student found that intrigued them.

Figure 1: Exploring patterns of hexagons on pineapples

Q: What did you notice?

A: pieces of pineapple were in hexagons, there are different lines of hexagons, the angle of the spirals go in different directions

Figure 2: Honeybees and honeycombs

Q: What do you notice?

A: They are all made up of the same shape, they repeat, they are hexagons, they are regular, they are made up of triangles, what happens if the pattern is no longer hexagon, is something wrong with the honeybees?

 (Option A #3)

One of my students rarely talks, and in fact, up until this year, there were concerns about her ability to speak (her past teachers contemplated if she was mute or had verbal ‘disabilities’). Coming into middle school, she has improved her communication skills, but minimally. Because we are preparing for math presentations, I knew this would be a difficult task, especially if the topic was not engaging for her. Over Christmas, I learned this student was incredibly talented with origami and paper folding. She created these beautiful pieces- snowflakes, stars, boxes, etc, all from paper folding. Her creative ability to create 3D objects was astounding and expressive!. I offered her the option to create origami pieces for her project (instead of simply researching geometric art or artists). She also helped me with Activity #3 and provided observations when she assembled the platonic solids. In our discussion, I asked her what she noticed. Some of her responses and observation:

• We started with equilateral triangles (I was surprised to hear her integrate specific terminology and to verbalize this out loud)

• The smaller shape created a pyramid

• The more triangles, the more sides the final shape has (tetrahedron, octahedron, icosahedron)

• The triangles are getting smaller and smaller, shape getting more round

It was intriguing to see her level of engagement improve as tactile activities were much more comforting for her. Her application of polygons was more apparent and gave me a better snapshot of her retention and understanding of the content. This parallels the findings from Boaler's (2014) video about how as educators, we can better gauge our students' comprehension if we allowed for the content to be expressed or communicated in ways outside of the traditional math styles. So many students solidify their knowledge through tactile and sensory learning.

I didn’t get a chance to take any pictures because she wanted to keep the shapes that were created!