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

 

Reading Portion:

In this week's reading, Doolittle's exploration of the failures of the grid and its implications on spatial and temporal organization, three stops along this intellectual journey stand out and spark curiosity.

First Stop: Seductive Simplicity of the Grid

The journey begins with the seductively simple nature of the grid. This system effortlessly extends into domains like neighborhoods and towns and three-dimensional spaces like buildings and malls. Doolittle's analogy of gardening leading to geometry and the connection between linen and line resonates deeply. With its straight lines and right angles, the grid provides a sense of control, predictability, and mastery. However, the simplicity that makes the grid appealing also raises critical questions about its adaptability to the complexities of real-world applications. The tension in the fiber, which creates linearity, becomes a metaphor for the tension between imposed order and the organic growth and diversity of specific places.

Second Stop: Failures and Control

As the grid exploration progresses, Doolittle astutely observes that failures in the grid system have been apparent since its inception. However, responding to these failures raises essential questions about control and ownership. The grid, deeply ingrained in our thinking, leads to rules governing behavior, shaping our sense of control. The failure to let go of the grid results in the subordination of a place's unique qualities to the grid geometry's imposed uniformity. The grid is not merely a geometric construct but a manifestation of cultural and human tendencies toward order and control.

Third Stop: Giving Thanks and Liberation from the Grid

A fascinating turn in the journey comes with the Indigenous perspective, introducing the idea of giving thanks and offering an alternative solution to the Königsberg Bridges problem. Acknowledging what has been missing from the discussion, recognizing the river as a foundation, and giving thanks through offerings present a profound departure from the rigid confines of the grid. The narrative weaves through philosophy, bringing in Immanuel Kant and his habitual walks, drawing parallels to the mental abstraction imposed by accepting Euler's negative result. The call to widen our perspective, move off the grid, and see the world as it truly is bringing a refreshing perspective to mathematical problem-solving.

Engaging Questions for Reflection:

1. How do the failures of the grid, as highlighted by Doolittle, resonate with your understanding of spatial and temporal organization in various contexts?
2. How can alternative geometries, described as "geometries of liberation," find a place in modern mathematics education, and what challenges might arise in incorporating them into the classroom?

Reference:

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

 

Activity Portion:

My son's drawing                                                               My spouse's drawing.

The activity described in Doolittle's "Off the Grid" gave me a unique perspective on observing and sketching the world around me, even if I could not go outdoors. I was sitting indoors with my spouse and two-and-a-half-year-old son. We looked through the window; we focused on living beings and human-made things, using watercolors and brushes to document my observations. I did the observation, and my spouse and son made the drawings. As I found a spot to sit and observe, I noticed various lines and angles in the natural and human-made things. I observed natural flowing lines like plants, trees, rocks, and the sky. The natural world seemed to express itself through gentle curves and irregular shapes. On the other hand, human-made things, like buildings, roads, fences, and street lights, displayed more structured and geometric lines and angles. Therefore, the contrast between the organic and the structured was apparent.

Reflecting on these observations, it became evident that specific patterns emerged in both living and human-made elements. While living things displayed a sense of irregularity, human-made objects adhered to a more ordered and intentional design. However, there were exceptions, as nature sometimes exhibited symmetrical patterns, and human-made structures occasionally embraced organic forms. The patterns I observed raised questions about why such distinctions exist. I pondered whether these patterns are inherent like things or influenced by human perception and design choices. The contrast in lines and angles reflects a fundamental difference in the creation processes, one guided by nature and the other by human intention.

Considering how this activity could be translated into teaching, close observation, and sketching could be powerful tools for helping students understand lines and angles. By encouraging students to explore the world around them through drawing, educators can foster a deeper connection between their eyes and hands, promoting a more intimate understanding of geometry in their surroundings.

Moreover, I contemplated integrating whole-body movement into the learning experience. For instance, the video of two UBC undergraduate students, Sam Milner and Carolina Azul Duque, demonstrates the  Dancing Euclidean Proofs. Indeed, it suggests one-way students could experience lines and angles through activities that involve exploring the outdoors. I wonder what other ways we can engage students with the living world to offer a holistic understanding of geometry, connecting abstract concepts to tangible, real-world experiences. This approach could make learning more immersive and memorable for students.