Week Three : Post 2 Article, Introduction, and Video

 Williams urges us to move away from a schooling model characterized by discipline silos and mechanistic, technocratic worldviews. She advocates for a shift towards an educational approach that promotes a sustainable world, incorporating systems thinking and holistic learning. For example, she notes that living systems are not hierarchical but are networks nested within other networks, where the whole is more than the sum of its parts. She emphasizes the need for greater ecological literacy, arguing that pedagogy should be grounded in place to better understand patterns and relationships.

Williams then details a program in a Portland, Oregon school district focused on food and garden-based programming. This initiative involved 3,500 K-8 students across eight schools, as well as multiple sites for school-based learning gardens and environmental and nutritional farms. The program fostered numerous partnerships, with university students forming a learning garden committee that included parents, community members, students, and school staff.

The program addressed local issues: the rising number of hungry children in Oregon, increasing rates of obesity and diabetes, urban students' diminishing connection to nature, and the widening achievement gap between white and ethnic minority students.

Learning about patterns and relationships was deeply integrated into seasonal learning, emphasizing care for oneself, the land, and the community. This approach extended beyond traditional curricular math connections like patterns of time, size, and quantity. These insights from deep student engagement is my first stop. I was blown away when a 12 year old had deep insights to the patterns and relationships of caring for land and nature. “It is strange that people can take pride in large lawns and waste their land by simply growing and cutting grass. If we plant gardens instead, we can also grow food, we can bring wildlife and at the same time eat healthy fresh food.” (p. 47)

My second stop is from this week’s video and the first week’s TEd Talk video. Seeing Dancing Proofs reminded me of week one - changing perspective is understanding. Viewing the body in motion, from above, is a change in the perspective of sight. Therefore, understanding of proofs has occurred. From a ground view to an aerial view.

My question for week three extends from my second stop and learning more about engaging the body and all of its senses. We can change perspectives within the sense of sight, ground view to aerial view. So could it be possible to change perspectives within another sense, say smell? And would such a change in perspective also result in understanding as put forth in week one’s TEd Talk? 

I used AI to generate an image of hearing on land and underwater. 

Week Three Post 1 - Activity

 Week Three Post 1 - Activity (article and video post coming on Saturday)

I am currently in Paulatuk NT and as you can see I’m not going to be sitting or standing outside for long. Also, in terms of living things (plants and animals) right now there are only a few ravens, dogs, bushes, and humans. So this week, I am taking some liberty and drawing lettuce chunks from a salad, a banana, and an apple. The outdoor human made structures are being drawn from the comfort of looking out my window. 

In living things there were no right angles whereas human made structures were riddled with them. Living things had curved or arched lines whereas human made structures had straight lines. Human made structures are probably this way for both simplicity's sake and structural strength. I found it interesting that the banana formed a pair and a set of approximately equal angles. Then I Googled banana blossoms to see if I could determine a relationship between lines and the blossom/flower. However, the image of a banana blossom only confused me further!!!!!

****EDIT banana angles 60/60/80/80/80

I think that close observation and drawing would strengthen the understanding of angles and lines as it combines two senses for the same concept. The student would be highly attentive to sight and feel. This would be especially the case if, as this week’s introduction states, “we work to create a balanced program of mathematics” that is both outdoor and indoor (nature and sensory balanced with traditional methods).

 Tactile Construction of Mathematical Meaning: Benefits For Visually Impaired and Sighted Pupils, Stylianidou, A, & Nardi, E. (2019)

The authors’ study explored whether UDL practices in math fostered better inclusion and benefited all students. 29 lessons were observed from four classes of students aged 6 to 10. However, the article focused on the interactions and descriptions with shapes of just two students, “Luke”, visually impaired, and “Zak”, sighted. 

The activity involved Zak and Luke describing a triangle and another shape. The teacher made both shapes from waxed yarn. The other shape was a circle with a small straightened section. Next, the teacher handed out physical circle manipulatives of various sizes to compare to the original “near circle” shape. 

(Image: Stylianidou, A, & Nardi, E. (2019), p. 346)

Highlights from each student: Zak was uncertain of the straight line using his eyes. He was confident in the existence of the straight line through feel. Zak made an interesting comment, “the hidden facts on the shapes.” (p. 348) Luke commented that the circle would roll more and the “near circle” would bob up and down. Luke's action descriptions were vastly different from his peers who used typical textbook vocabulary. 

My first stop was my skepticism of the rigidity of this waxed yarn material to make shapes. Even if I were to notice the small straight section with my eyes or fingers I would have assumed it was an imperfection of a circle caused by the material. That is to say I’d assume the teacher meant to make a circle. 

Another stop occurred when I read “being aware of the characteristics of vision and touch - vision is holistic and touch is gradual, allowing for the exploration of an object from its individual parts to whole.” (p. 346) Stopping and thinking about the pedagogy or characteristics offered by each sense was quite an “Ah Ha!” moment. It fostered memories of my varying fishing experiences. Growing up in Nova Scotia I learned to fish by watching bobbers. However, moving to the Arctic I struggled with jigging and relying on my tactile reflexes to hook a fish. 

 

My question this week is in relation to Zak’s use of multiple senses to see the “hidden facts on the shapes” and the more traditional views of mathematics that using senses are inferior and cannot be trusted. If we were to use multiple senses in a scientific way then would such a practice be seen as more trustworthy or less inferior? That is, analyzing and comparing the data gained from each sense allows us to test, probe, and justify our conclusions and hypotheses, like how Zak was able to see the straight edge after feeling it. 

My Hexaflexagon Activity (Note: the only bagels available in my community were partially pre-cut for easy "rippability". So the cuts seen in the third video were impossible) 

Source of template: https://www.uua.org/re/tapestry/multigenerational/miracles/session-5/leader-resource-1 

The first 10 images show the various centers that were discovered. I marked them in unique ways. However, despite my twisting and turning I noticed some potential uncolored centers as seen on the outsides in the second set of images. I’m thinking there are probably 12 unique centers based on the multiple of 3 (triangles) and 6 (hexaflexagon). 

How might the hexaflexagon experience affect the learner's understanding of the mathematical patterns? In the video, Hart amusingly creates her own unique vocabulary based on her tactile actions while also using traditional vocabulary like clockwise and counterclockwise. Much like this week’s introduction mentions a movement vocabulary. 

Such experiences are great practices for students to see, feel and verbalize “actionable” textbook vocabulary. At the same time, students can create their own vocabulary for actions they see in which they may not know the textbook word or perhaps there is no such English word. I feel, this “student speak” would be packed with student understanding and ripe for teacher probing. 

I attempted to create my own action math vocabulary as I was playing with the hexaflexagon. Alas, I think I’m too grown up, I could not get my imagination to engage in making new words. I could only think of already existing words like rotate and the flexibility of the paper as more and more exploration was done. Perhaps, now as I write this post, I could create the word “open-up-ability” for when a squeeze and a pull could reveal a new side. I did use the word "rippability" in reference to easily pulling apart partially pre-cut bagels. I thought I was inventing a word but turns out it's a geological term.

Week One: Part Two

 Part Two 

Introduction, Video, and Activity 

My wonderings from the introduction focused on, can technology be a great equalizer between the mindly abstract mathematics only accessible by a “small priestly caste of mathematicians” who are capable of great mental effort and the rest of us who supposedly cannot. For instance MapleSoft (https://www.maplesoft.com/) can allow anyone to visualize and see mathematical objects in various forms and views. For example, the screenshots below, of the 3D graph and 2D contour map of the same graph. Such perspectives were not possible when I was in high school. Such graphs were quite abstract and hard to imagine, only for those cable of great mental effort.  

Today there are also 3D printers which could allow students to physically hold and play and explore otherwise intangible and abstract mathematical objects. Which leads into Antonsen's TED talk. Antonsen mentioned the key to understanding was being able to change perspectives. During his video he used lots of technology to do so… to see, hear and explore different representations and perspectives of 4/3. Technology like MapleSoft allows for representations and visualizations to be rotated on any axis, zoomed in and out and changed to different representations at the click of a button. Imagine the new understandings that could be gained using technology to create so many perspectives of something like, the plots of x squared plus y squared. 

For the activity I measured the dimensions of storage containers using my handspan. Through sheer coincidence my hand span was exactly 8 inches which made measuring quite easy. The dimensions, in inches, were 32 x 20 x 20. The storage shed was my outside object and I measured the width and length using cubits. In inches the dimensions were, 74.5 x 93.1. 

I must say the lack of sunlight, cold, winter gear, and snow made this a bit of a challenge. This leads me to my indoor outdoor connection. I intended to have placed these bins in storage before the snow fell. However, I can now use my measurements to determine that I can fit two rows of storage containers against the 5 cubit wall. 

This experiential and physical activity would be viewed as primitive and inferior by those classical philosophers and mathematicians of the Enlightenment. However, this embodied activity serves to be both practical and memorable. I will not remember the exact measurements of my shed. However, I will remember it is at least eight handspans in length. So if I am buying something and wondering it it will fit in storage I can easily get a good estimate.

 Week One - Part One (Assigned Reading)

Gesturing Gives Children New Ideas About Math - Goldin-Meadow et al. (2009)

Researchers studied how hand gestures helped children learn. Participants were 81 girls and 47 boys in third and fourth grade who did not answer any of the six pretest questions, of the form 2 + 5 + 10 = ____ + 10, correctly. Pretest gestures were coded. Next, children were randomly assigned to three groups in which experimenters demonstrated different levels of gesturing while modeling problem solving (no gesturing, partially correct gesturing, and correct gesturing). Finally, during the lesson, without gesturing experimenters answered a question and verbalized their reasoning. Then all children were asked to solve a problem while gesturing and verbalizing. This alternated for 12 total questions followed by a six question post test.  

Results showed that all children who were shown gestures during the pre-lesson gestured for all lesson and post test questions. Children with the most correct gestures answered the most questions correctly. 

My first stop was when researchers described their mediator variable, aside from their independent and dependent variables. I had not heard of such a variable before. “Adding grouping in speech” was their mediator variable saying it predicts the dependent variable, post test performance.. See the image below. Students who combined speech with the V gesturing performed the best. 

(Image: Ingmire, J. (2014, March 10). Gesturing with hands is a powerful tool for children’s math learning. University of Chicago News. https://news.uchicago.edu/story/gesturing-hands-powerful-tool-childrens-math-learning)  

Another key takeaway for me was that neither students nor experimenters used speech for the V grouping gesturing. It was not until the post test did some students start speaking about grouping two numbers to maintain equivalency. Gesturing appears to help students gain and verbalize understanding. 

The positive effects of gesturing combined with this week’s introductory reading promotes the question, to what extent could gestures be meaningful in higher levels of math education and how would they be viewed by students, parents, teachers, and society? For instance the introductory reading states society views manipulatives as babyish and primitive if used for higher math. It further states students of higher math are expected to “just think” their way through learning and understanding while remaining still and silent. It now seems contradictory to not allow gesturing, if gesturing promotes thinking and we want students to “just think” their way through higher level math.

Part Two (video, intro reading, and activity) is coming in a second post.

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