Week 3 - Sustainable mathematics in and with living world outdoors - Critical Reflection

    I noticed that my ‘stops’ this week were not isolated to stopping and thinking/questioning/wondering in the moment when I stopped during reading or viewing or doing but became kernels for my mind to play with across the activities and after the activities were complete while I reflected on bringing things together. Maybe now we have a few readings, viewings, and activities under our belts I can draw upon different resources to make connections and allow my mind to revisit the ideas over and over when new information stimulates ideas.

Nellie's arbour and my noticings of its construction

    One ‘stop’ for me was when viewing the short film, Dancing Euclidean Proofs. Sam, the education student in the film, talks about how dancing the proofs makes them take time, revealing the proof slowly rather than how the proofs are laid out in their entirety all at once on a page in a book (Gerofsky, 2018). This ‘taking time’ made connections for me to last week and to this week’s activity. In the paper by Fernandes and Healy (2013) Multimodality and Mathematical Meaning-Making: Blind Students’ Interactions with Symmetry, the authors note that they observed the students using specific hand movements when they explored cardboard shapes and figures on geoboards that allowed the students to “intentionally see” (Fernandes & Healy, 2013, p. 53). Perhaps the dancing of proofs allowed Sam and Azul (the other student in the dance) to ‘intentionally see’ the proofs, seeing the parts as well as the whole. When I was sitting drawing living and non-living things for the activity this week, I made a connection to this intentional seeing, too. Slowing down and taking time to really look at the subjects of my drawings made me explore their boundaries and the internal structures (line, angles, colour tints and shades, connections, etc.) more than just noticing the object’s or being’s existence in my world allows. I know that the arbour exists in Nellie’s garden next to mine, but I had never looked carefully at how it was constructed. I know the general form of a maple samara with the seed heavy at the bottom and the fragile ‘wing’ full of veins, but I had never noticed before how the veins leave the seed clumped together and then spread over the wing (which was more noticed with the digital microscope but the process of drawing made me curious enough to spend time digging out the microscope to take a closer look). It has come up already in my group’s blogs and responses the difference between rushing through the curriculum and slowing down to offer meaningful experiences to students to explore and embody mathematics; the video and activity showed me that time is important and slowing down and taking time to really look or experience the world mathematically helps mathematical understanding.

Maple samara through the digital microscope

    It was interesting to take the extra step of looking at my living subjects of my drawings with the digital microscope. After drawing both living and non-living subjects, I was ready to declare (in answer to some of Susan’s questions) that humans love straight lines and right angles and that my living things did not exhibit 90-degree angles at all. This was reinforced in my reading when Doolittle references Leopold Kronecker and says, “the Creator gave us shapes, straight lines are the work of man” (Doolittle, 2018, p. 104). However, looking at the mesclum leaf through the digital microscope, I did see what are in the very least approximate right angles and this surprised me and challenged my initial stance. Looking closely, revealed some access to “the grid” within the fractal geometry of the veins, but as Doolittle describes, that grid fails at a larger scale, as when looking at the leaf in its entirety, “zoomed out.”

Noticing right angles

    I connected the grid and the drawing activity together in my experiences (and frustrations, as much as I was self-talking that these were unnecessary) of getting the living and non-living subjects onto paper. I wonder if some of my frustrations with drawing from ‘live’ things, 3D objects/beings, is that shift from 3D to 2D (I do a little better if drawing from a photo but I didn’t take that ‘cheat’ in this week’s activity). In my pre-reqs for education, I did a 300-level geometry course where we wrote computer programs to represent 3D objects on a computer screen (my final project was a Rubik’s cube…that failed after about 3 or 4 turns of parts of the cube). I can see how drawing onto paper imposes the grid onto beings and non-beings that exist beyond two dimensions. I’d be very curious to explore recreating my objects as 3D models to see if my brain welcomes that more than drawing. For example, I could model Nellie’s arbour with popsicle sticks or create the maple samara out of clay. These activities, I believe would still give me the opportunity to look closely and notice lines and angles while still maintaining the 3-dimentional nature and maybe not letting the grid do some of the looking for me in its imposition on the objects or beings.

Screen grab from Global Oneness Project (2018)

    Thinking about the grid, mapping and drawing from 3D to 2D, and the Doolittle article, reminded me of the video we watched in Cynthia’s EDCP 551 course about counter mapping. Zuni Elder (New Mexico) Jim Enote introduces viewers to counter mapping his people have done to represent the land, in ways that “counter and challenge the notion of what maps are” (Global Oneness Project, 2018). Both Enote and Doolittle talk about Indigenous lands being lost through mapping. Not only can the grid fail in representing our 3D world, but it may also be used as a tool of oppression.

Questions to Ponder

Here are some questions that I am working on answering for myself still and I invite others to help me with:

Where can we find room in the curriculum to challenge dominant mathematics through embodied learning? (There is room, I know it, but I want to look for specific opportunities.) How can we discuss dominant culture and oppression in mathematics classrooms, even with younger students?

Is there a binary of non-embodied learning and embodied learning? Where is the boundary between the two? Does a boundary need to exist? How much of the body needs to be involved for learning to be considered embodied? Was using a digital microscope in anyway embodied?

Doolittle says, in reference to Brent Davis, “Because we do not let go of the grid, we are internalizing it. We go from “straight” lines and “right” angles to well-defined “equal” plots, to “rules” governing our own behavior, and finally, to a sense of control and mastery. This progression is due in some degree to culture and to some other degree to innate human nature, and the weight of each cause is subject to debate” (Doolittle, 2018, p. 104). How can we, as teachers, let go of the grid?

 

References

Doolittle, E. (2018). Off the grid. In S. Gerofsky (Ed.) Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7

Fernandes, S. & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind stduents' interactions with symmetry. RIPEM, 3(1), 36-55.

Gerofsky, S. (2018). Dancing Euclidean proofs [Video]. Vimeo. https://vimeo.com/330107264

Global Oneness Project. (2018). Counter mapping [Video]. YouTube. https://www.youtube.com/watch?v=U7DQmTjpFI0&t=1s