Week 3 Reflections - Math Outdoors

 Changing Perspectives

This week I read all three articles. As I read each article, I kept coming to the idea of perspectives. (This idea has been rattling around in my head since watching the Antonsen talk in week one.) 

The article I am discussing is c) 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

The stated purpose of this article is to “apply stress to the grid concept, in order to discover some of its failures, and then propose alternative geometries better suited to our needs in many domains, including education” (Doolittle, 2018, p. 102).

In talking about the grid Doolittle asks, “How we can reflect reality better in our thinking?” (Doolittle, 2018, p. 102). Our reflection of reality comes from the perspective with which we see reality. How does our reflection change when we change our perspective? Are there perspectives that reflect reality in a more authentic way than the “grid” perspective through which we often unconsciously live our lives? Do we see the grid and its structures as being only beneficial, easy, comfortable, familiar, and giving us a sense of control; or do we also consider the possibility of dangers associated with using the grid?

Doolittle proposes failures of the grid, particularly with regards to roads and maps, giving examples of towns/reserves in which he has lived. He brings in Euclidean geometry as an example of a failure of the grid, its value is limited to “small, uniform regions of space” (p.108), essentially ignoring the power of nature. He discusses alternatives to the rectangular grid which include: the hexagon (consider the bees), Penrose tiling, Amman-Beenker tiling, and Riemannian geometry. 

Riemannian geometry gives us another way to consider perspective, taking us outside of the Euclidean geometric idea of grid. Riemannian geometry, Doolittle describes as applying the Copernican Principle that we are not the centre of the universe, to grids. “Accepting all kinds of straight and curved grids as equally valid ways to measure space” (p.111). No grid is better than another, rather we are encouraged to look past the grid to “refocus on the actual underlying geometry of the situation” (Doolittle, p. 111).

Indigenous perspectives are discussed through string figures and the time grids of Indigenous ways of farming. Indigenous perspectives are also brought in as Doolittle describes his approach to the Königsberg Bridges problem. Doolittle asked the question, “what after all this time, haven’t we done that we should have done?” Thus the problem is considered from a different perspective, opening up previously unconsidered possibilities and his answer to this question leads him to a solution.

The following quote, about the grid in relation to agricultural land caused me to stop and consider whether we could apply the same idea to students. “Too often, the specific life, qualities, and character of a particular place become subordinated to the forcefully imposed “evenness” and uniformity of the grid geometry. (Doolittle, p. 104) Are we “forcing” students into grid like conformity that goes against their natural “grain”?

Using a chaotic control analogy, Doolittle brings the discussion back to education. He states, “transformations in the context of education where we have the notion of the teachable moment, in which a few small ‘bursts’ at the appropriate time might accomplish more than months of haranguing… The non-trivial question is how to identify those critical moments, and in which direction to provide the nudge” (p116-117). How does this compare with the control inherent in living within the grid, where we (teachers) are in control, with little space of students?

How does seeing the grid as beneficial or as dangerous change the way we interact with our world?

In describing the path to the source of the river in the Königsberg Bridges problem, Doolittle maintains “the path… would be long and arduous… but it is available to the diligent, the brave, the caring, and the open-minded” (p. 119). It is my hope that as educators, we can be diligent, brave, caring and open-minded.

As I read through the introduction this week, the ideas of roots and connection stood out to me. Maybe it is because I love taking my students into the Delta Watershed and examining the roots of the big trees in the forest. We consider the interconnectedness of the roots, what is happening beneath the soil, the life under the soil, how deep the roots might go in comparison to the height of the tree, the path that the roots take etc… (lots of good mathematics related to roots!). I am drawn to the idea of students understanding the sources and roots of mathematical ideas as well as how this can be explored and discovered by being outside. We explore the fractals that we see everywhere! (Doolittle discussed complexity theory as a superset of fractal geometry.) “What do you notice” and “what do you wonder” questions draw students attention to how mathematics shows itself in the natural world of the forest. How do we move beyond these questions with young students? The land as an entity is a great teacher. 

This week I attempted the activity a few times. My drawings… perhaps I will keep those to myself for now, but… as I was exploring the world through mathematical perspectives, I used my camera to capture images that highlighted for me natural lines and angles – forms that would not fit easily with the grid. The complexity of patterns and lines found in nature are unmatched with human-made things. We only come close when we mimic nature. 

I have not used a lot of whole body movement in teaching mathematics outside, but I am inspired by the Dancing Euclidean proofs to consider ways to allow students to explore their understanding using their bodies.