Week 3 Reflections

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

Summary

“We must acknowledge that any grid, straight or curved, is an imposition of our own invention for our own custom or convenience” (p. 111). The article exemplifies various examples of how human beings use grids to develop civilization in hegemonic ways, and we can also discover, through scenes of everyday life, how the grids reflect changes imposed on indigenous people and the environment by colonialists. This is achieved by equalizing, subordinating, and imposing a uniform grid geometry on the unique life, quality, and character of specific places. In contrast, if grids represent a human's sense of mastery, non-Euclidean geometry provides an alternative approach to the problem. Riemannian geometry offers a perspective, suggesting that none of those grids is inherently better than the others. Additionally, following the Copernican Principle, all existence in the world is not considered special because none of us is at the center of the universe. Therefore, to elucidate the specific abstraction, I will provide concise explanations for some of the examples discussed in the research, addressing the limitations of grids and exploring alternative approaches to overcome them.

Land planning and Map drawing： The author provides an example from his hometown, Hamilton, Ontario, illustrating how the arrangement of grids on a flat map may overlook the actual variations in terrain elevation, potentially causing vehicle drivers to descend abruptly from a 100-meter escarpment. 

Moreover, due to the limitations of geology, topography, and the curvature of the Earth, dividing grids can easily intersect with two or more different grid systems, resulting in incompatibility and even causing individuals to lose their sense of direction. 

                                

From the description, it is elaborated that the basic elements of straight lines, circles, and grids in flat Euclidean geometry are suitable for small-scale and uniform regions but are not appropriate when shifting or extending to an environment as vast as the actual Earth. For example, the Mercator projection, while careful about maintaining angles, becomes increasingly distorted in higher latitudes, and even the space around the poles disappears. Therefore, using a different curved grid such as a polar projection can remove this obstacle (go take a look in ◉◡◉╭ New York Times Daily 360). 

Date and time: Humans have divided time into different periods using grids and arranged themselves or others to live by it. For example, regardless of the sun's position, the opening times of markets are still dictated by calendars, while farmers, insects, and plants continue to follow the natural rhythms of time for their activities.

Territories: Colonists approach land differently from indigenous people. They measure, survey, and delineate the land's edges in straight lines, and represent them on maps. In contrast, indigenous people probably viewed the territory in terms of the entire drainage basin. The natural shapes of territories are better described by fractal geometry rather than Euclidean geometry. Moreover, within the realm of the territory, there is no need for the skills of measurement and conveyance; one can simply follow the path of the river to find the desired direction. 

Navajo weavers and string figures: Without tension, the fiber behaves in an unruly manner, retaining only the memory of organic growth. However, when tension is applied, the fiber is pulled tight, creating a linear configuration with pleasing symmetry and a sense of fairness (p.102). The geometry of curved strings, knots, linear lines, etc., in the indigenous fabric, is related to the ancient Indigenous tradition of string figures, cultural heritage, personal style, racial history, and values. These indigenous craftsmanship, stories, and activities have been a part of Indigenous education, but they have also been flattened and condensed into books. The development of modern media, such as the web, YouTube, and virtual reality, will help reintroduce three-dimensional space and time into general education.

In conclusion, grids establish rules and discipline in the world of civilization. However, it does not imply that all cultures adopt grids; some prefer recursive organizing principles, while others are comfortable accepting the structures provided by nature. This research introduces a new perspective, prompting readers to reconsider the metaphorical meanings behind grids. I concur with the ingenuity of Navajo weavers who intentionally incorporate defects into their weaving, guided by the belief that "only the Great Spirit is perfect" (Bagel, 2003). This perspective suggests that imperfection is acceptable, encouraging us to embrace and appreciate these imperfections along with their authentic aspects.

Stops

“The connection between gardening and education is made quite explicit in the concept of kindergarten, but is implicit in much of the rest of the education system.” (p. 104)

When reading this paragraph, I found it resonating with my current situation. We often adopt a mentality similar to the specific developmental stage of kindergarten children, emphasizing the need to nurture practical living abilities and explore the environment around us. As I grew older and became capable of managing my everyday life, my interest in exploring the environment diminished, and I developed a preference for abstract thinking or completing tasks with technology. Moreover, what resonated with me was... throughout the reading materials over these two weeks, it was my first time encountering the English terms for pomegranate (which I find difficult to memorize so far), pomelo, and string figures. Interestingly, these may be vocabulary that we had already learned in kindergarten. In contrast, as a graduate student, we are more familiar with words like patriarchy or capitalism, which are less fascinating and lack the allure of the everyday living experience.

“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.” (p. 104)

I used to believe that maintaining regularity and order in all things was the correct approach, making it convenient for us to manage everything. For instance, when arranging class seats, even in group discussion mode, I preferred to arrange the seats in straight lines or grids, as it benefits teachers in overseeing kids' behavior. This inclination is likely influenced by my intrinsic personality and the establishment of our social and cultural institutions. However, it has led me to reconsider the notion of mastering and controlling behavior in everyday life, exposing a metaphorical mindset that suggests we fear children deviating from the framework we set. Perhaps, the truth lies in the fact that we were raised within a framework that makes it challenging for us to imagine the consequences of their departure from these structured grids. 

Question

Typically, we acquire knowledge about grids and their applications during the elementary to high school stages. This research offers a new perspective on challenging the conventional 'grid' mindset. As a math teacher, do you consider incorporating a de-grids and de-colonized perspective into the formal math curriculum?