Week 3 Reading - Off the Grid

 

The article by Doolittle discusses the ubiquity of grids in organizing society and raises some questions about the illusion of control that comes along with them. Doolittle explores some of the failures of the grid system, writing about some agricultural applications as well as city planning. Within the article, Doolittle suggests some possible alternatives to the grid system that is everywhere today, such as hexagonal patterns. I find this somewhat interesting because I play a lot of modern board games, and I would say that many if not most of them employ some kind of hexagonal grid system. I wonder what makes hexagons an attractive shape for board game designers/players. Perhaps the flexibility of movement, offering 6 possible “connections” rather than just 4. Octagons would offer 8 but would not fit together in an interlocking way. Hexagons are also a reflection of nature, as bumblebees create a hexagonal honeycomb. 

My first “stop” in this week’s reading was the passage about the grid system in Saskatchewan. Doolittle explains that on large maps of the province, the western border appears to be straight, while the eastern border is jagged. “The townships from which provinces are built are not exactly square, because their east and west boundaries are great circles of the earth converging at the North Pole, so the north edge of every township is slightly shorter than the south edge. This inconsistency leads to an incompatibility in the grid along east-west lines called correction lines,” (Doolittle, 2018). This passage highlights the challenge of imposing a grid on a real-world landscape, because of the incompatibility due to the curvature of the earth. Within my elementary mathematics classroom, I could challenge students to find examples of grid applications and their limitations, and the relationship between structure and flexibility. Doolittle (2018) goes on to explain, “no matter how determined we are to extend our grids, we must eventually bow to the gentle but insistent curvature of the earth.”

My second “stop” in the reading was Doolittle’s passage about the moon. He explains, “the path of minimal energy [to travel from Earth to the moon] would use almost no energy at all. The trick is using tiny bursts of fuel at exactly the appropriate time.” This made me think of the control we have when doing high speed sports, and how tiny changes or adjustments in certain scenarios can lead to a larger impact. For instance, if you are a golfer, a 1 mm adjustment to the face of your club at impact can change the destination of the ball by 40 yards left to right. The spin rate also has a huge impact on the result of the ball. These are all things that are impacted by tiny control adjustments that are almost imperceptible to the untrained naked eye. Skiing and skating, too, require tiny micro-adjustments in order to stay on course and optimize speed and energy.

Overall a very interesting week this week, which I hope leads to some fascinating conversations in my classroom.  A question:

- How might the exploration of alternative geometries, like hexagonal patterns, influence the way we approach problem-solving and design in various aspects of our lives?

Week 3 - Sustainable Math in and with the living world outdoors

I did not get a chance to read this week's material until Friday this time, so I was not able to get outside with my students and try this activity, but I will reflect on a previous similar activity we did in September in relation to the reading. About a block from our school we have a small tree nursery that is kept up by the community and is approved for short walking field trips. In mid-September, I had the idea to take students there and observe this nursery near the Autumn equinox, and we would go do so again at the winter solstice, spring equinox, and summer solstice to specifically observe the changes in the trees (part of our Grade 6 Science curriculum). It did not even occur to me at this time the mathematics that could have been present in this work. 

When we sat with our journals, my direction completely ignored any of the man-made things in the area. I think I'd love to go back and only focus on some of the man-made objects in order to pull out some more of the math that is present. We looked specifically at natural objects and spent a few minutes just observing before we sketched. I love the idea of sketching three man-made objects and three living things in the area. The prompting questions on the activity sheet are really great for reflecting on some of the mathematical concepts:

• What kinds of lines and angles did you see in most living things? How about in most human-made things? Are there typical lines and patterns that show up in living things vs. human-made things? Any exceptions to this?

• Why do you think these patterns exist (if you notice patterns, that is!)

• How might you use close observation and drawing or sketching to help your students learn about lines and angles?

• Are there ways to experience lines and angles through whole-body movement or large body motions outdoors? In relationship to the living world?

I think having students consider these questions and some of the man-made objects around them will help them to make some deeper connections to the work we're doing with geometry and start to recognize some of the vocabulary "in the wild" so to speak. When I do this activity again, maybe next week, I will emphasize some of the mathematical connections as well as the fact that they're not being judged by their ability to sketch, the goal is to observe and connect, and hopefully report back on the success.

Week 2 Reading - Multimodality and mathematical meaning-making (Healy & Fernandes, 2013)

Healy & Fernandes (2013) focus on teaching geometry and geometric concepts to blind students. Although geometry is typically a very visual domain, their research aims to interpret how students are perceiving geometry and concepts like symmetry when they aren’t able to see the shapes that they are working with. The study found that in tasks involving symmetry, students developed strategies to understand through tactile experience. The research also challenges the assumption that blind students follow the same learning trajectories as their peers who are not blind. The students who were mentioned in the study showed unique approaches to learning geometry which were specific to their own individual contexts.

Although I don’t have much experience teaching students with such significant needs as the ones within Healy & Fernandes’ study, the application of the methods and practices employed within it are similar to those which teachers in generalized elementary environments employ for their students all the time. Multimodal teaching strategies such as activities that involve touch, movement, and other sensory engagement benefit all students. I try to incorporate hands-on manipulatives and visual aids in order to provide a more holistic learning experience.

My team partners and I are taking a bit of a new (to me) approach to science this year – arranging pre-made modules with specific activities for each concept ahead of actually teaching the concept. In small groups, depending on their interest, students access slides, videos, and have a choice of activity for each learning outcome from the curriculum. For example, a group of students might choose that day to engage with the idea that “air takes up space and has mass”. That group of students will spend some time on their own discussing the topic and watching some videos, and then will have the opportunity to choose which pre-arranged hands-on activity they can engage in to experience the concept in a tactile way. This element of choice and multimodal exposure is an individualized approach that acknowledges students’ differences and preferences for engaging with learning. I act during these blocks as a facilitator and guide, prompting discussion about the concepts that are being explored.

My question is: 

1) In what ways could you take some of these ideas about multisensory mathematics and incorporate them into a Physical Education class? Or within your teaching context, how could you take a physical activity and have students engage in the mathematics surrounding it? 

Week 2 - Multisensory Math

This week’s readings and theme surfaced the importance of multisensory learning – in mathematics specifically – but the idea of multisensory learning extends into other areas as well. As I am writing through the lens of an elementary generalist, I try to incorporate some of these ideas across subject areas. I thought about different ways the senses can play a part in our understanding of mathematics and my mind immediately went to food. Taste and texture, combined with ingredient measurements and time spent cooking, can play a large part in our understanding of mathematical relationships. Did I overcook that dish? Does it need a little more oil, less sugar? Taste and smell are just as important in this scenario, if not more, as sight is.

The shift in pedagogical approach toward multisensory experiences is an interesting one. I feel like as teachers we are always challenging ourselves to make the content engaging and relevant for our students. In Grade 6, I find I sometimes contend with a lot of groans before we actually get started, then as students start to make connections and see value in the activity or task, that energy shifts toward excitement. 

This week I attempted hexaflexagons with my students and talked about experiencing symmetry and geometry in a sensory way could enhance our understanding of the concepts that we’ve been exploring. Students started off a little bit unsure, but luckily, Vi Hart’s enthusiasm and encouragement got most of them at least attempting one. It didn’t take long until the excitement was reaching a fever pitch and I had students wanting to create more complicated shapes and structures. One student even took it upon themselves to attempt to build a dodecahexaflexagon. 

The difference between looking at a hexagon and making these hexaflexagons and giving students a hands-on activity hopefully led to the students experiencing the polygon in a different way. When I showed them a picture of Taco Bell’s Crunchwrap and asked them if they could make any connections between the hexaflexagon and the Crunchwrap their minds were blown. Who knew you could eat math?

Week 1 Reading - Foundations of Embodied Learning excerpts (M. Nathan)

There are two excerpts this week from Nathan’s (2021) Foundations for Embodied Learning. In the introduction, the author identifies the central problem of a lack of a coherent, evidence-based educational theory of learning, which leads to ineffective education systems. Nathan argues that learning is a gradual process which occurs across different time scales and that current practices too heavily emphasize and rely on abstract concepts and undervalue embodied forms of learning. The second excerpt focuses on the embodiment of mathematics education and focuses on the role of metaphors in making abstract concepts accessible to more people.

The first 'stop' in the second excerpt for me was about the concept of children coming to school with an understanding that numbers are a place along a path, and the fact that schools assume children know this so they don’t explicitly teach it. It’s easy to forget that this is something that needs to be learned through cultural participation opportunities such as games. On Friday afternoons at my school we hold “Collaborative Response” meetings focused on numeracy in our grades and one of the Grade 1 teachers this week brought forward the issue that one of her students, let’s call her B, has a very poor number sense and can’t “count on” from a number or recognize the conservation of number (Piaget). Nathan explains that many children in Western cultures play these games at home and bring this concept of counting with them when they arrive at school, but there are children who don’t have the same opportunities. When I revisit B with the Grade 1 teacher at next week’s meeting, I’m going to bring forward the idea of games where a piece progresses along a track in order to hopefully provide another strategy to move B’s understanding of number forward.

The second 'stopping point' for me, probably because I am in the midst of geometry exploration right now in my classroom, was Nathan’s section on Geometry. Nathan writes about a student who makes connections to obtuse angles by moving his arms and making connections between the contortion of his body and the angles that he was trying to represent. I have students struggling right now to make the connection between the differing classifications of angles, and want to try to approach this challenge through a more embodied experience. Nathan concludes that grounded and embodied learning experiences can offload some cognitive resources so that students can dedicate more mental energy to the concepts they are trying to learn. 

Questions to consider:

1. How might embodied learning impact your teaching strategies, particularly with students who may not have had the same background cultural participation opportunities?

2. How have you or might you incorporate embodied learning experiences to help students struggling with geometric skills and allow them to free up cognitive space to access more abstract concepts?

Week 1 - Mathematics and the body

The introduction to the weekly readings along with Roger Antonsen’s TED Talk highlight the importance of changing perspectives to understand mathematics in a different way. Antonsen’s representation through different perspectives of the fraction 4/3rds is inspiring and reminded me as I listened to him speak that mathematics can be truly enjoyed and learned through play and through manipulating simple objects. Antonsen eloquently emphasizes the role of imagination in deep understanding of mathematics and science.

It wasn’t in a garden, but this week in my Grade 6 classroom we were exploring the concepts of symmetry and congruence between geometric shapes. After briefly reading the introduction of the week’s theme I saw an opportunity to have students use their bodies to further explore and represent these ideas. My students’ challenge this week was to use their bodies to demonstrate symmetry and/or congruence. After some initial hesitation, I found them in bizarre places trying to contort themselves in order to get their point across.

Students ended up on the floor mapping out their limbs with masking tape (think the chalk outline of a dead person in a murder mystery). Others chose to choreograph mirrored mime-like movement routines. One reluctant group eventually came around and devised a cleverly timed handshake. I’ve included a photograph of students using the sliding double-doors in my classroom to attempt a reflection of a shape. 

I can see some potential in the handout on Math and Measurement in the Garden and how I could use it to extend some of the activities the students engaged in already this week. I can imagine some rich discussions about standardizing measurements, as I’ve already heard students using some of this language as they struggled to find congruence and symmetry with their bodies already.

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