Week 5 Reading - Kelton & Ma (2018) Reconfiguring mathematical settings and activity through multi-party, whole-body collaboration

I enjoyed the Kelton & Ma (2018) reading this week as it described a really interesting activity that I am going to take up in my classroom. I think this course has given me a ton of very simple activities to do in my classroom that connect with concepts, and I've really enjoyed that about the readings. The article focuses on a teacher who went away from the traditional math classroom and used the Whole & Half activity as a powerful way to learn about fractions. The activity required students to physically position themselves and engage in hand movements creating a new arrangement of their bodies in relation to each other and the environment. This use of space allowed students movements to become integral to their math understanding.

My first stop in the article was that the teacher had the students physically doing math, rather than learning math. This movement allowed them to explore fractional relationships through bodily movements and get creative with the physical space they occupied. 

At one point the article talks about Katie and Claire, who had trouble with the coordination of their bodies and Claire had to wait for Katie to become half. This breakdown (and I'm sure there were others) demonstrated some of the challenges of working with other people. I enjoyed reading about this case study because we are just about to start fractions in our classroom so I would be curious to try it in my classroom to see how the students do with it. 

A question to ponder: How might activities like "Whole and Half" reshape the traditional classroom setting and give students a new perspective on how math is "done" and how one "does" math? Would students make the deep connections necessary or would they think it's just a fun movement activity. What can you do as a teacher to ensure the conceptual understanding hits home?

Week 5 theme - Embodied, multi-sensory, and arts-based modalities

I found Sarah Chase's activity of dancing prime numbers (I feel like that's an oversimplification) to be really intriguing but I don't have the coordination to try it myself. I'd love to get some of the dancing students in my class up trying something like that though to demonstrate a different understanding of number and patterns. I tried a similar activity with my wife and have not recorded us doing it yet but we've made a few attempts.

Activity: Two partners clap on multiples of two different numbers, creating a rhythmic pattern. For simplicity sake, one partner claps on multiples of 3 and the other on multiples of 2. We experimented with multiples of various numbers and observed how it affected the complexity and rhythm of the clapping pattern.

We talked about the mathematical patterns created by different pairs of numbers and discussed why certain pairs produced different rhythms. 

Possible extension: Explore the impact of using prime numbers between 7 and 23 in the clapping activity and see how that influences the regularity or irregularity of the pattern.

Another possible extension: Collaborate with the music teacher to explore a cross-disciplinary connection with playing notes or some other way of finding common multiples by playing chords.

One more: Connect the clapping activity to traditional mathematical notation, with an effort to link the physical and abstract elements of the concept.

As we are just wrapping up some of our work on primes/multiples/factors now, I feel like this is a perfect time to try some of these specific movement activities with my class. So far they have been very receptive to my ideas and have engaged thoughtfully with them! 

Week 4 Reading - Dylan Thomas: Coast Salish artist

The article about Dylan Thomas traces his journey into mathematical art - sparked by his exposure to M.C. Escher in Grade 11 - which happens to be about the same time I was exposed to M.C. Escher, actually. Through some exploration of Escher's style mixed with traditional Salish designs, Thomas starts to see some similarities in the symmetry and rotation. Thomas' latest work, Infinity, was heavily influenced by M.C. Escher's Smaller and Smaller. It is based on dividing a square into smaller and smaller parts.

Salish art has always fascinated me. I was probably about 16 when I visited a Salish artist's studio in Gastown and learned about his work. My dad bought a series of his wood-carved pieces which still hangs in his living room 21 years later. Looking at Thomas' work through the lens of mathematical art, tessellations, and symmetry now as an experienced middle school math teacher is really captivating - I love the interplay of designs. Horizon on page 207 is such a cool piece of art, I am going to keep a screenshot of it to share with my students when we look at reflections later on this year.

There is limitless potential to combining math with art, and a student's interest in mathematical art can evolve into a rich cultural activity. Thomas' work serves as an example of the power of visualizing math in creative expression. I think his work will help my students to see math as a way to express cultural or other important ideas visually.

I don't really have a prompting question this week, just a consideration that the intersection of math and traditional and cultural art can inspire students to view math as an important part of cultural expression!

Week 4 theme - Mathematics and the Arts

This week's introduction resonated with me based on the challenges of teaching math to middle schoolers who have for so long seen it as a separate and rigid subject area. I've tried to incorporate activities all year long that allow students to see the mathematics in a variety of places and spaces, and see and experience mathematics as an interconnected subject. 

I love the idea of integrating math with artistic and cultural elements so I shared this week's activity of viewing the Bridges galleries and reproducing a piece of "math art" with my Grade 6's. Listening to the conversations and watching students explore the galleries I could tell that it was engaging and intriguing for students - many of them went on to explore some of the other years as well. 

The whole point of activities like these is to approach math through a more holistic and interconnected lens and I think in this case students were able to see something a little bit different. As you can see in picture 3 attached, one student even took the idea from the Bridges gallery and "extended" it, connecting his hexagons all the way to the edge of the page. 

Another student not pictured, because I ended up getting his face in the picture, drew his own version of the "pi maze" and then his friend added the digits of pi around the outside. This gave me a great idea of having one student start a piece of mathematical artwork and then randomly hand it off to someone else to have them extend and complete it. Math-Art Mash-Up Monday, perhaps?

 

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?