Week 3: Sustainable mathematics in and with the living world outdoors

Wandering around in nature is one of my favourite things to do so I really enjoyed the theme for this week. I like to take my science classes outdoors whenever possible and look forward to finding more reasons to bring math outside as well. The times I have taken math classes outdoors have been an interesting experience as students seem to think the activities are not real math, even when explicitly linked to the content. As we discussed in week one, I find that students in high school have a fairly narrow view of what math is and have some “unlearning” to do before these activities reach their full potential. Perhaps finding the balance mentioned in this week’s introduction might have to start with adding just one small item to the non-traditional side of the fulcrum. 

 

Learning outdoors also takes more time than traditional classroom learning, but it does feel richer and fuller. Milner spoke of the time it took to choreograph and film Dancing Euclidean Proofs and the way it made him slow down and consider the steps of the proof and the importance of their order. That stood out to me because life feels very rushed at times, especially when I am attempting to cover all the content in upper-level math classes. I wonder how much deeper the learning could be and how many more connections could be made if we had the opportunity to slow down, make interesting observations, and experience the concepts in multiple ways.

For my outdoor excursion, I went to a small park near my house. As I was observing and sketching I noted that many of the human-made things had straight lines and right angles while the living beings had curved lines. Even the straight lines in nature like the Magpie’s tail feathers, Ponderosa Pine needles, and stalks of grass are all slightly curved. In my reading for the week, Doolittle notes the same, “To paraphrase Leopold Kronecker, the Creator gave us shapes;

straight lines are the work of man” (p.104). 

Doolittle’s chapter explored the ideas and limitations of our familiar grid systems. He notes their “common use in the dominant culture leads to familiarity and comfort. The main questions we should face are to what extent those sensations are illusory, and how we can reflect reality better in our thinking” (p. 102). This made me laugh even though it was not intended to be humorous. I was reminded of my dad always claiming he and other men navigate by grid rather than navigating by landmarks like women. He was implying his method was superior, yet he was the one who always got lost. As teenagers, my siblings and I left him in the woods after a disagreement about directions and we arrived home hours before he did. 

Doolittle claims that “Euclidean geometry is often promoted for its practical value; the failures of the grid show that its practical value is limited to small, uniform regions of space-time” (p. 108). The limitations of Euclidean geometry can be found in real life with examples such as map projections, correction lines, and roads following geographical features. You can see this failure in the photo of my sketch of the sign. The camera was not directly above the sketch and parallel. As a result, my once straight lines and right angles are distorted.  

 

 

He also raises the issues of perspective and frames of reference, arguing that different systems and frames of reference may be equally valid since they are simply a way for us to impose our own set of rules on a reality that is independent of those same rules. The limitations of Euclidean geometry within mathematics also increase as things get more complex. Once the shapes you are interested in are no longer rectangular it becomes much easier to describe them with cylindrical or spherical coordinates. I rarely used Cartesian coordinates beyond the first year of university.  

 

It was interesting to note that the playground was updated last summer and all of the new equipment was curved. I think it would be interesting to incorporate the playground equipment into lessons involving the unit circle. The swings alone could be used to teach arc lengths, radians, positive and negative angles, co-terminal angles, and reference angles. You could easily move into sinusoidal functions as well. I have also seen incredible art work created with Desmos, you could even challenge students to sketch what they see using mathematical functions.