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!

Week 1 Mathematics and the Body (Post 2 of 2)

Application of Body measurement, outdoor and in

Reading: "Math and Measurement in the Garden, Body, Spaces, and Technology"

Reflection: This activity brought me back to my past when my girlfriends and I would determine heel heights in our early 20s. We often had a reference that the width of two fingers would approximately equal to 1 inch. So four finger-widths would be approximately 2 inches. I no longer wear heels but tried this technique to measure the width of a notebook (9.25 inches). Based on my past ‘estimation’ technique, the book was approximately 7.5 inches. This method of measurement is not accurate because the width of my four fingers is much larger now than in my 20s (darn weight gain)! It was also likely miscalibrated as my friend had very slim fingers, so the origins of this ‘technique’ were not accurately calibrated to my physical measurements. However, this activity was entertaining to attempt (and reminisce) and can be an excellent way to personalize measurements based on one’s OWN body, rather than someone else’s. This could be engaging for students to design, based on their own values, and to compare with those estimated by others.

Figure 1-4: Estimation for the width of a notebook. 

To PROPERLY calibrate, the width of my CURRENT fingers is now 2.5 inches across. So 2.5 inches x 3.75 lengths = 9.375 inches, which is closer to the actual width of the notebook (9 ¼ inches)!

Extension: I see my husband integrating measurement and estimations every day for critical decision-making in his job as a marine captain. For example, he uses various boat lengths as a reference point to determine the distance between their origins and other points around the Vancouver harbor. A standard tugboat is roughly between 60 ft and 100 ft and will use these as references. Antonsen (2015) mentioned that “patterns can be conveyed as a language”, and math can be viewed in real life through relevant perspectives that are meaningful and significant for each individual. For my husband and those in his marine industry, they have designed an informal language system that is understood by those in their field. Though it is not official, nor necessarily standardized, it is a conventional way of understanding distance and lengths; significant because they have to estimate to look for clearance between vessels. Digital devices and systems are in place but can be delayed, which can severely impact the ability to make immediate critical decisions. The terms they collectively use provide visual references rather than actual distances; they are compared and judged against known, relevant lengths. This is not an exact form of measurement, nor is it precise, but it is an example of how measurement with body or known references can be integrated every day. However, according to my husband, “I don’t do written math”.

    

        Figure 4 (Left): Standard 65ft (20m) tug             Figure 5 (Right): Standard 105ft (322m) tug

Conclusion: This applies to our classroom experiences because so many students don’t see themselves as ‘mathematicians’ but are often kinesthetic and integrate using references that hold significant meaning to them; that’s how they experience the world and make their references. This practice or way of understanding can be used as an integrated aspect to support their learning. It doesn't have to be one or the other but an approach that can be regularly interconnected.

Week 1 Mathematics and the Body (Post 1 of 2)

Application of Gerofsky's (2011) article on graphing and gestures

Figures 1 - 3: J's actions became smaller and smaller as he could not convey accuracy with T (drawer/guesser)

Reading C: Gerofsky (2011) “Seeing the graph and being the graph”

 

Activity: “Function Charades”

Introduction: After reading Gerofsky’s (2011) article, I designed the game “Function Charades'' based on the activities conducted in her research on graphs and gestures. I explored this game with a Grade 6 class, and intentionally re-named it “Function Charades” instead of “Graph Charades” to move away from known math connotations. They were told the game was a combination of "Taboo" and "Charades". During the game, observations were made on how the students moved, the words they chose to use, and how they progressed from observing each other’s technique round to round. 

Students: There were a total of 17 students in this section, participation to play or observe the game was optional. 14 opted to join in. Students had a variety of mathematical experiences, though most in this section are quite confident and comfortable with each other. The students have some prior knowledge of graphing, coordinate systems, the x-axis and y-axis from previous grades, and Science 6,  but have not officially been taught the Cartesian plane, quadrants, or functions. Though the students were aware that it was a math game, they were encouraged to not think about it as a math game. Their objective was to simply convey enough information either verbally (Speaker) or physically (Actor) to one student (Drawer/Guesser) who would then draw the image as close as possible.

 

Materials: The Drawer/Guesser was provided a whiteboard that had grid lines and a whiteboard pen.  The Function Charades PowerPoint was shown on the Loft screen.

 

Observations: 2 rounds were played. Round 1 – Drawer/Guesser (T), Actors (O, J) and Speakers (D, A). Round 2 - Drawer/Guesser (L), Actor (G), Speaker (C).

Other Observations:

• Students did not realize the images were graphs, completely forgot they were doing math! 

• Students were really engaged and enjoyed the game, and wanted to continue playing.

• Great demonstration of learning techniques and prompts from each other, noted that symbols helped with the visualization. 

Video 1: O's first volunteer to act out the graph in Round 1

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Conclusion + Links

This would be a fun activity to trial with students in Math 9 and 10 who have a stronger knowledge of functions and coordinate systems. The students were given more ‘strict’ parameters than Gerofsky’s study but modifications could be made in future observations. Because there was no risk, the Grade 6 students were quite enthusiastic to be guinea pigs for this activity. 

This activity linked well to Antonsen's (2015) TED talk emphasizing the need for students to find patterns, "represent patterns with language" (even if they need to make it up), make assumptions, and try new things. This activity got the students to play around with assumptions and trial techniques to determine which were successful and which perhaps were not. They had to use their imagination to think creatively to meet the objectives of the game and empathize because it was a lot harder to think on the spot than it looked, especially when others were not understanding the language or actions used.

Welcome to my EDCP 553 Blog

 Hello World! I am excited to explore the beautiful connections between mathematics and the arts. I'm looking forward to exploring this journey with you all!

"Aloha" from Hanauma Bay Nature Preserve, Honolulu- Patterns on coral reefs