Week 3 - Sustainable mathematics in and with the living world

This week the temperature has ranged from -25 to a balmy -12. While I continue to bundle up and walk outdoors, I found it hard to sketch outside; the imposition of mittens limited my ability to represent detail and taking them off was too uncomfortable. I tried taking a photo of a tree to come back and sketch indoors, but I found it frustrating because I could not see it all - I could zoom in but I couldn't rotate it to see other sides or where things started and ended. I gave up and moved to my indoor hydroponic garden. 

Curly parsley

While I'm glad I'm not being judged on my artistry, I learned some new things about parsley. Each stem has 3 smaller stems and the leaves also seem to separate into smaller groupings but it is hard to count as they are still attached to each other, if that makes sense. Everything on this small plant is interconnected - hard to see where new growth begins (even though my drawing does not capture that element). It definitely represents my observation of it growing in threes though!

Lettuce

Believe it or not, this is a growing leaf of lettuce. I wish I knew more about shading and composition so I could do it justice, but it really is fascinating. Everything seems to be in motion, even though it is still. The symmetry in the veins of the lettuce is something I didn't notice. Imperfect symmetry, as the smaller veins sometimes go in different directions.

When I tried to draw human made objects, I had a bigger challenge. Even though I knew my plant drawings were imperfect, it didn't bother me. If a line went in another direction, it didn't matter - it still worked because nature is perfectly imperfect. It has a plan and a structure but a fascinating uniqueness.

When I tried to draw the floor of my shower, the hexagons seemed impossible to draw - see all the times I erased it? I forced myself to not use a ruler but it was frustrating. The lines and measurements have to be so precise for it to look right. I didn't really learn anything new while doing this. I think if I was doing this with children, I would encourage them to draw like this and then compare it to a drawing where they use a ruler and protractor. What would they learn by making the comparisons? How would this inform their understandings of straight lines and angles? They could also try to make these shapes and angles with their bodies, like in the video. I think the combination of informal and formal tools and representations would help to foster understanding and help make visible the connections between mathematical concepts (such as shape, space, number, and time).

In Williams (2008), the dangers of isolationist thinking are discussed. Traditional education models have historically encouraged the compartmentalization of concepts and ideas as well as the separation of mind/body. Williams (2008) calls for a shift in paradigms, one that recognizes systems thinking and holistic learning as frameworks for sustainable education. There are three tenants posed for thinking about sustainability education:

• The whole must be considered in order to understand its parts - we cannot just take things apart and understand how it functions as a whole
• Living systems are networked systems - we have networks nested within other networks, not hierarchies
• Ecological community relationships are non-linear and involve multiple feedback loops

These ideas are not new to me, but they definitely challenge many of my constructions of science. Much of my science education has focused on hierarchies - food chains, ladders, levels, etc. I am wondering about how this thinking leads some to separate themselves from the natural world - hierarchies are competitive, cold and fixed; where networks are dynamic, connected and complex.

Williams (2008) explains how this can come to life during an interdisciplinary gardening project. Life sciences, social studies, mathematics, health and physical development all become intertwined, with knowledge from one area fuelling another.

My questions for this week:

In your work, what other interdisciplinary projects have you done that would support the principles of sustainability education listed above?

When working with these types of projects, learning is also networked - it is not linear and can be much more circular. What do you see as the strengths and challenges of this and how do you navigate the challenges?

 Week 2: Multisensory mathematics

As I read the introduction, my first 'stop' was to think about all the ways in which society disables people. If we started to truly value multiple ways of knowing and being, how many "disabilities" could we eliminate? What would be the impact of this? I loved reading how new innovations are being created by people with impairments; how they are not simply modifications of existing structures, but a "driver for innovation and creativity, rather than as a deficiency" (Gerofsky, 2022). I also agree that these new tools would benefit all; making space for seeing 'disability' much differently. Last semester I had a student who communicated through ASL. She explained to me that ASL is a 3-D language and sometimes translations do not capture full meaning of concepts. I have since wondered about the potential of ASL to represent spatial concepts; of course for deaf students but also for hearing students. I wonder how learning and understanding spatial concepts may be enhanced with the use of bodily representations, rather than just verbal or written.

This week I explored hexaflexagons (again) and bagel geometry. I had explored hexaflexagons in a previous course, so I assumed I was still an expert and quickly started folding, ignoring the template provided. Obviously I made an error - I did not fold the right kind of triangle! It still led to some interesting discoveries.

In comparison to the row of equilateral triangles, the way the paper curls up is different. It does not spiral.

The comparison of the final products was interesting also - different shapes and configurations, even though the same number of triangles were used.

Playing with the shape to the left was actually more fun than the hexaflexagon to be honest - I think because I didn't know what was going to happen. When I built the hexaflexagon, I followed the instructions and got the desired result - but I'm still not sure I understand why it works the way it does. Guess the relational understanding isn't there for me yet.

My bagel was also a bit of a flop. I only tried once because I can't bring myself to waste food and I couldn't convince anyone in my house to also eat a bagel. I think if I had a few times to practice, it would've worked. and I do find it quite interesting how unexpected shapes and patterns emerge from these types of experiences. I think combining this, with an explanation and/or visual lesson would help make some of the underlying mathematical concepts clearer for me.

Connections to Stylianidou & Nardi (2019)

• Tactile explorations seem to benefit both blind and sighted students. Students who do not always see characteristics with their eyes, sometimes feel these through touch (e.g. a sighted student did not identify a straight line segment when looking at a shape, but did identify it when feeling it)
• Mathematical constructions and concepts are embodied - perhaps looking with only the eyes is too wholistic; feeling leads us to understand the different parts that make up a whole
• The blind student, who explores through touch, used embodied imagination as well as practical explanations of shapes (e.g. a circle is going to roll), more so than sighted students

When exploring the hexaflexagons, as well as the bagel, I did gain more insights than just watching the videos - without a doubt. Being able to change the orientation of the objects - flipping, rotating - helped me to see it from different perspectives. The 'mistakes' I made also added new insights. However, I still feel my understandings are more procedural, or instrumental. 

My questions for today:

I feel I would have more relational understanding if I had a refresher on the mathematical concepts embedded in the activities I did. At what point would we introduce more formal mathematical concepts into these types of activities - before, during or after? Or would we at all? What strategies would you use?

 Embodied Mathematics - Week 1

In measuring and comparing my bodily proportions, I was reminded of a measurement experience a wonderful colleague (credit to Susan Hislop) shared with me many years ago. This project took place in a grade one classroom when the school was under construction. A new part of the school was being built, and the children's playground was to be relocated. Susan explained to the children that their butterfly garden would need to move, and since the new playground was going to be concrete and rubber, the plants would need to go into planter boxes. She gave them a challenge - to figure out how many and how big the planter boxes would need to be. First the children counted the plants and drew their plans. 

They quickly realized that counting was not an effective strategy as it didn't show the plants "bigness". They needed to measure. 

They also realized, in trying out a number of strategies, that measuring a plant is not easy. And, measuring different types of plants requires different tools - for example, a tall plant could be measured and compared with their heights, but a creeping plant needed needed something else (they ended up covering it with tiles). But then they had to grapple with the dilemma of combining and quantifying different measuring units.

Anyway, I could go on and on about this amazing project but the general gist is that these young students, by using their bodies, their imagination and a little persistence, uncovered sophisticated understandings of measurement, much beyond what any curriculum could dictate.

 When I was recording the measurements for my own body, I noticed a few themes. I found myself looking at the already completed chart to see if I was "right" or perhaps "normal". I guess looking for patterns to be sure I was on the "right track". This may reflect some of my notions of mathematics - while I try to see it as imaginative and dynamic, my habits of thinking are more linear. I want to be like Roger Antonsen in the TedTalk, or like the grade one students, who seem to use mathematics as a tool for understanding, rather than a set of procedures. I am thinking a lot about representations now - how multiple representations, such as the drawings above, or our bodies, or standard tools, unlock the potential for seeing concepts in different ways.

In Gerofsky (2011), Seeing the graph and being the graph, I also gained new insights on how bodily representations can offer us valuable information about students' understandings of mathematical concepts. Gestures often act as bodily metaphors, a language in which to demonstrate and explain understandings. It's interesting how in the Ted Talk, Antonsen said that when we don't have a way of expressing something, we often invent language for it. In her multi-year study, Gerofsky (2011) explored the variations in secondary students' gestures when they were asked to describe graphs and their related functions. What I found fascinating about this study is that the participants' gestures and ways of representing fell into three categories, which correlated with their mathematical abilities. 

Before reading the results below - which category do you fall into? If I ask you to use gesture to demonstrate y = 4 - how do you do it?

In the study, the first group of students (11 of 22) used an "arm's length visual model" of the graph, where they generally used finger movements, seemingly to trace a small graph in front of their upper body. They seemed to "see" the graph. The second group of students (9 of 22) seemed to "become" the graph - using whole body movements and metaphors - both through gesture and verbal explanations. The third group (2 of 22) had difficulties producing gestures and they often did not correspond accurately with information in the graphs. 

When this information was cross-referenced with teacher reports on student ability, the students in the first category were often hard working, precise and followed rules, but dependant more on memorization than engaging fully with math concepts. Students in the second group were the top mathematics students, and were both accurate and imaginative in solving problems. Students in group three were the struggling or at risk students. It is argued that more embodiment is needed to fully understand and represent advanced mathematical concepts.

I find it interesting that when I was reading the article (before seeing the results of the study), I engaged in some visualization about how I would represent the sample equation y = 4, using gesture. I fell into the first category, which corresponds with my observations of the measurement activity I did this week - I seem to want to know I'm "correct" while following the rules. I hope, over time, this can change.

Which category did you fall into and do you agree with the classifications?

Gerofsky (2011) explains that "until recently, the norm for high school mathematics classes was to seat students in rows at individual desks and to encourage them to sit quietly, copying down the teacher's lecture notes, answering the teacher's questions and working silently and individually on examples and homework questions. In fact, despite recent reform efforts, the great majority of North American and other mathematics classes continue to operate within these norms." (p. 254).

How have you seen these norms challenged? Besides the example of graphing, what other concepts lend themselves well to embodied mathematics, especially at the secondary level?

 Welcome to my blog. I'm excited for a new semester of mathematical adventures!