Week 3 Activity + Connections

In nature, I have rarely seen 90 degree angles (a few exceptions are the plants and trees coming straight out of the ground) I suppose this angle provides the path of equal lateral stress on the organism during growth while reaching for the sun, but beyond the main trunk or stem... obtuse or acute angles of every sort. At bifurcations, almost always acute angles to maintain upward trajectory. With spacing angles between lateral branches taking obtuse angles to set up solar panels without self-interference. 

With my puppy, similarly, acute and obtuse angles: corners of eyes, outlines of ears, angles ears make with each other, folds of skin/fur. Also, a commonality in nature are smooth flowing lines, graceful like a dancer. Even the gnarled tree branches resemble the twining eagle arms yoga pose. The structural pieces vary from curved lines to straight lines, but still they have flow. For example, a blackberry stem - strong, thick, flowing and supportive of the intricate blackberry leaves with its acute and obtuse angles, and curved lines of the blackberry itself. Is there a more general pattern at large here? 

Generally straight bones --> Obtuse/Acute flowing Musculo-vascular system --> generally curved lines of adipose tissue & skin/fur/feathers/scales.  

Generally straight stems --> Obtuse/Acute flowing branches/leaf shapes --> generally curved lines of fruits.

How interesting! 

Human-made objects are a different creature: I've noticed many more 90 degree angles. I suppose this pattern probably plays a functional role for efficiency as well as strength. Take the stone table top on this bench at school. To get from the top of this shape to the bottom is 180 degrees total: Let's imagine the edges would be more decorative with a nice 45 degree slope (instead of 90). Now, it's less aggressive on top, but more aggressive (sharper? more dangerous?) on the bottom angle with it now being a 135 degree turn to reach the bottom side (or counting the interior angle - a sharp 45). Perhaps this means we want to make a second cut to lessen the sharpness of the bottom edge with another 45 degree angle cut. Well, now it's less aggressive, but required two cuts - also, if you lean on the edge, you'll have a single line of pressure across your abdomen. Instead, bringing back the single 90 degree cut means less processing and functionally more comfortable to lean against. 

90 degrees seems like a good trade for efficiency of tool power while maintaining structural strength whether it is with compressive... 

...or tensile forces, like in this chain-link fence. I thought I found a break in the pattern, but upon a gentle 45 degree head tilt, we find the 90 degree theme again. 

If we did not need to use tools (or weren't given the luxury), would our human-made world look vastly different in terms of construction? Would we be constrained to the grid? I imagine I could research the first structures made by ancient cultures for shelter and witness the angles and lines used.
Oh! A thought occurred: when constructing survival shelters (ie. lean-to) the angles do look quite different. There is a uniformity to stacking and laying logs and detritus, but in terms of support, unless you're driving support columns into the ground (like a tree would grow naturally), everything more resembles natures angles of acute and obtuse angles. Huh! That's funny!

Source: https://www.outdoorlife.com/wp-content/uploads/2019/01/23/IOT7YTOGKMQYAD2HOAQNDQDML4.jpg?strip=all&quality=85

At what point does the pattern switch? 

At which point does human-creation want to demonstrate otherness in construction?

In history, tool-use is considered advanced... so does that mean 90 degree angles are more advanced than nature? 

With even more advanced tool use, we construct complex structures that yet again resemble nature as inspired by nature (ie. Science world in Vancouver, Bird's nest stadium in Beijing by Ai Weiwei, etc). At what point does it (tool use, building techniques, human-mentality of otherness to nature) come full circle and witness nature as inspiration as master-builder again? 

Source: https://images.spaicelabs.com/images/flus6j8v/production/83ae119450b402c12531174975b44cab84b26d1e-1920x1438.jpg?rect=241%2C0%2C1438%2C1438&w=3840&fm=webp&q=75&fit=max

Source: https://www.re-thinkingthefuture.com/wp-content/uploads/2024/01/3.-Beijing-National-Stadium-Beijing-cover_900_584.jpg?w=999

Perhaps I could teach my students to learn about lines and angles in a hand-out using these same guiding questions provided by Dr. Gerofsky as they led me to some interesting conclusions with good reflection:

1. What kinds of lines and angles did you see in most living things?
2. What kinds of lines and angles did you see in human-made things?
3. Are there typical lines and patterns that show up in living things vs. human-made things?
4. Are there exceptions to your pattern or trend?
   
   
5. Why do you think these trends exist in nature?
6. Why do you think these trends exist in human-made things?
   
   
7. Have you seen human-made things inspired by nature? If so, what makes them "different" than typically human-made things? If not, google it and report back!

Source: https://pyxis.nymag.com/v1/imgs/93a/6e1/da8f39123264bbe2d28b10fa84d6b18278-strat-investigates-snail-mucin-01.rsquare.w400.jpg

Source: https://mindtrip.ai/cdn-cgi/image/format=webp,w=1200/https://images.mindtrip.ai/attractions/241b/d4a6/ca21/b6eb/d79b/5f8f/c8bf/e2d7

Thinking about ways to experience lines and angles through whole-body movement or large body motions outdoors, the first thought that came to mind was to create an interactive lesson hook involving full body yoga. Here's a potential framework for the lesson:

1) Warm-up with yoga in a large space. 

2) Afterwards, breakout into small groups with the new repertoire of yoga flow and analyze the movements that they attempted to perform. 

Perhaps a categorization of movements with a question set like this: 

• Which positions would you associate with acute angles?
• Which positions would you associate with obtuse angles? 
• Are there any positions that resemble human-made things as discussed previously?

3) Construct their own positions/movements inspired by nature's creations. Chain-them into a flow. 
Or using the poses from the example flow, create themed flow sets for exploring angles from smallest to largest. 

That's all for now!

Week 3 Readings: Sustainable mathematics in and with the living world outdoors

Reference: Doolittle, E. (2018). Off the grid. In Contemporary environmental and mathematics education modelling using new geometric approaches: Geometries of liberation (pp. 101-121). Cham: Springer International Publishing.

Summary: Doolittle analyses the system of control and order: the grid. He explains its failures in terms of societal implementation trying to force order upon the flowing irregularity of nature, and how it overly complicates gardens, streets, borders of territories and even education. Doolittle then proposes some alternatives to forced grid-like structures of Euclidean geometry for large-scale application; one such solution may be Riemannian geometry (elliptical geometry) that accepts all forms of surfaces (curved or linear) as equally valid. Doolittle's argument leads to the ideas of following the natural timings (that fluctuate) of plants and insects for farming, instead of following the grid-like structure of the market hours, and calendar months; and generating territory boundaries utilizing drainage basins which follow fractal geometry. By embracing Chaos theory and other alternative geometries, we would be following the path of Indigenous cultures again and thus better able to model our world.

Stop 1: "Too often, the specific life, qualities, and character of a particular place become subordinated to the forcefully imposed "evenness" and uniformity of the grid geometry... We see how the grid is connected to notions of control and ownership" (Doolittle, 2018, p. 104).

I didn't really see this point before. I usually am thankful when I enter a new city and find that navigation is made either when grid structures and sequential labelling is used. It allows for quick rough prediction of direction and distance. An example of this would be Surrey, BC. They follow numerical streets with numerical avenue cross-roads. Although, this system makes it easy to start navigation for newcomers, the numbers don't feel like an appropriate representation of the places - in my mind, they lack personality of street names in Vancouver, BC, some parallel streets of which are named after trees.  Harder for newcomers, but with the names are attached personality and more easily brings images to mind.   

Stop 2: "Euclidean geometry is often promoted for its practical value; the failures of the grid show that its practical value is limited to small, uniform regions of space time" (Doolittle, 2018, p. 108).

I connect with this concept of Euclidean geometry being great for modeling on a small scale in its application of observing the slope on a curve modeled by Mary Everest Boole with sewing cards. If we zoom into a single point, that moment has a uniform slope, but move positions and the slope in that new instant will be different. This becomes the foundation for the understanding of calculus.

Source: https://www.geogebra.org/m/epw5fryv

Source: https://upload.wikimedia.org/wikipedia/commons/e/e1/Mary_Everest_Boole.jpg

Stop 3: "Complexity theory and chaotic dynamics are some of the most flourishing areas of geometrical thought in our time, and they provide an extraordinary alternative to the old notion with which we started, the path of least resistance, through the new concept of chaotic control" (Doolittle, 2018, p. 116).

This concept fascinates me. For those who didn't read the paper, Doolittle goes on to supply examples of using a miniscule amount of fuel to travel to the moon by applying effort (thrust) at the equilibrium point between the orbits of the Earth and Moon. The connection to education blew my mind as this concept has been talked about time-after-time: "teachable moments". This connects to our previous class in which we've discussed butterfly power in association with Chaos Theory (Renert, 2011). Renert  (2011) discussed the flaws in strictly linear thinking (very grid-like) and how this type of modeling to produce a projected solution may miss the natural non-linear (ie. chaotically dynamic) potential future outcome.

In our social justice course, there were many ways to increase affect which would, in turn, increase engagement. To maximize impact, we would need to elevate emotion but, more often than not, the lesson a teacher has provided doesn't touch on the same issue that their students care for - the example of Mahima Lamba's Kindgarten wagon-walks to determine plowing routes which turned into wheel-chair accessibility parking spots. Mahima's example is one where she remained flexible enough to abandon her prepared efforts for one that elevated the children's emotions using a teachable moment to develop into an entire unit with community action. Moreover, "[t]he non-trivial question is how to identify those critical moments, and in which direction to provide the nudge. The theory of chaotic dynamical systems provides a framework within which we can at least begin to approach such questions" (Doolittle, 2018, p. 117).

The wonder I have connect with this is: Where is this resistance that we are trying to avoid? I disagree that we are trying to find an alternative to the path of least resistance. I believe that embracing fractal geometry and dynamic chaos theory is messy and ALSO the path of least resistance. I've been in situations in my Math class where I'm figuratively just trying to PULL my students up a hill of understanding and care and effort - all of these things are actually out of my power/control - but I've invested time and effort into developing a lesson in the little time I do have to provide a learning experience for the kids... that they detest. At no point am I present enough to see if they express interest in an alternative form of the same concept because I'm still trying to salvage the rest of the lesson. Time and space are resources I would need to trade for positive reflection and moments of clarity, which aren't really an option while instructing. At the end of the day, embracing the chaos may be the path of least resistance (mentally and emotionally) to obtain teachable moments that could better impact children's emotion--> attitude--> belief about math class. 

Summary of Stop 3: A rocket ship blasting off the planet in a straight line fighting all the forces to get to the moon is like bull-headedly trying to finish a lesson you invested time into regardless of learning impact. Following the natural orbits and forces of the planet requires very little energy and can get you to the moon similarly to following the expressed interests and energy in the classroom to maximize learning impact.

Wonder: Pieces of a bigger pictures are coming together for me. A few general points I've found through this program:
(1) Stories/narration allow for increased student engagement - this is through relational understanding between ideas and people
(2) Emotional connection is essential for impact
(3) Nature and the outdoors can provide all the opportunities we need to connect students with math (as we are constantly trying to use math to model nature and its many phenomena!)
(4) Alternative geometries (ie. non-linear) in thinking and doing can better approximate life and change mentalities from one of victimization to action.
(5) Indigenous perspectives to learning and understanding encompass all of the above!

Although we have a framework that more closely resembles "the world as it is, not the way we might imagine it to be" (Doolittle, 2018, p. 119), in what ways can we teach that more closely follows this path?

In other words, harnessing the butterfly power of Renert (2011), what daily changes can we make to the classroom trajectory in general that could optimize learning needs and outcomes?

References:
Renert, M. (2011). Mathematics for Life: Sustainable Mathematics Education for the Learning of Mathematics, 31(1), 20-26. https://go.exlibris.link/m4ZZHhJC 

Week 2 Activities + Connections

A highlight this week has been thinking about disability in a new way: a cultural/societal problem, not an individual one. In the introduction of the weekly topic, Dr. Gerofsky made a very compelling argument that perhaps the social and built environment is what is disabling. When we design more inclusive lessons that target multiple senses - it benefits everyone! 

Watching Vi Hart this week helped to remind me of the math that can be found in everyday life. 
I managed to document ways to take Rocket candies and demonstrate concepts of permutations + combinations, statistics, graphing, graph theory, and geometry.

Then, after viewing the hexaflexagon video, I constructed my own transforming snowflake. I find this use of the hexaflexagon the most intriguing. 

The mathematical fruit yielded some interesting patterns:

The horizontal apple slices yielded five point radial symmetry. This is very similar in structure to the apple blossom (below) mimicking the five-point radial symmetry, and similar to the symmetry of Phylum Echinodermata ("Spiny-skin" like starfish). 

I wondered what the dots above each peak and trough of the star represented, so cutting vertically I found they connected the stem to where the flower would have had its petals at the base of the apple creating a ten-bundle cage around the central star-shape containing the seeds. I would imagine that all apples are constructed in a similar fashion, but I might be over-reliant on my one sense of vision to determine this. 

Source: https://chelanranch.com/cdn/shop/articles/apple_blossom.jpg?v=1697583102

Also cutting into the stem of lettuce, I noticed an entirely different pattern as this isn't a fruit, but a shortened stem with large leaves. 

This arrangement of leaves reminded me of the golden ratio video from Vi Hart, and her explanation on 137.5 degrees as the optimal angle for leaf arrangement to maximize sun exposure without self-sabotage! Down a small rabbit hole on phyllotaxis I stumbled upon this website that teaches you how to use the golden ratio to draw flowers using R ("an open-source programming language and environment designed specifically for statistical computing, data analysis, and graphics," says Google).

Reflection questions: 

I did not construct the paper shapes when reading about Kepler, it was tremendously difficult to visualize what he was trying to communicate. I feel quite confident in my 3D spatial orientation capacity, but I still felt lost here. Instead, I built his descriptions of space-economy using playdoh to facilitate my own understanding of his concepts and it was a night-&-day difference. 

In his book, The Six-Cornered Snow Flake: A New Year's Gift, Kepler (2010) described rotating a pyramid of balls such that the side rather than the apex was uppermost to reveal a special orientation... I couldn't visualize this, then I made it out of playdoh and STILL I struggled (my wife documented this process), then when I correctly oriented the pyramid and removed the sphere revealing the four balls below, I was shrieking with joy (my wife documented this, too). 

The difference between knowing and understanding can be minimized by learning from 3D living things and/or objects with shape, texture, smell, taste, etc. instead of a 2D printed image. 

For individuals with sensory impairment, this might be the only accessible way for students to experience an analogous first-order experience of a phenomena. As stated in our weekly introduction by Dr. Gerofsky, "if we reframe 'disability' as a driver for innovation and creativity, rather than as a deficiency, there is the potential to benefit the population at large, in education as in business."

This coming semester, I know I have students with severe dyslexia, test anxiety and the like. I am going to see if I can approach all subjects with this new frame of mind and model the accessibility that I want to see in our society and built environment.