Reading 3-Off the Grid- Edward Doolittle

 In this chapter, Edward Doolittle explains that grids—straight lines, squares, clocks, calendars, maps, and schedules—are a very common way people organize space and time. Grids feel comfortable and give us a sense of control, so they are widely used in cities, farming, schools, and mathematics. However, Doolittle argues that grids often fail to match reality, especially nature and human life.

The author shows that the grid is only effective in small, flat areas. In the real world, the land is not flat, the Earth is curved, the real world is complex, and time does not repeat exactly. When we apply the grid to the real world in an unnatural way, harm can be done. Cubical watermelons, dangerous city roads, confusion in areas where two grid systems meet, and the measurement and division of Native American land with simple lines on a map are just a few examples.

Doolittle contrasts grid thinking with Indigenous ways of understanding space and time, which are more flexible and closely connected to nature. Indigenous farming follows natural signs (like plants and insects) rather than fixed calendar dates. Indigenous ideas of territory often follow rivers and watersheds rather than straight boundary lines. These natural boundaries are better described by fractal geometry, which reflects how rivers, streams, and landscapes branch and repeat at many scales.

Another aspect that this chapter introduces is the different geometries from the traditional Euclidean Geometry:

• Riemannian geometry: This is a kind of geometry that embraces curved space and demonstrates that no one grid is special, just like different map projections of the Earth.

• Complexity and chaos theory is the study of systems that change over time such as climate or education systems. These ideas have shown that small well-timed actions may therefore have big effects which is chaotic control.

• Knots, weaving, and string figures, drawn from Indigenous traditions, which emphasize movement, relationships, stories, and three-dimensional thinking rather than rigid shapes .

In the final section, Doolittle revisits the famous Königsberg Bridges problem. While Euler proved it impossible within abstract mathematics, Doolittle adds an Indigenous perspective: before solving problems, we should acknowledge place, relationships, and gratitude—in this case, the river itself. By widening perspective beyond abstraction, new insights become possible .

Overall, the chapter argues that moving “off the grid” helps free our thinking. By learning from Indigenous knowledge and alternative geometries, mathematics and education can become more connected to nature, more just, and more meaningful.

Stop 1:

However, I also stopped at the idea that "a few small 'bursts' at the appropriate time might accomplish more than months of haranguing" in the context of education.

This idea made me reflect, as it runs counter to the prevailing understanding of teaching in most educational settings. These setups often believe that achieving more in learning comes from more explanation, more talking, or more rules being set in place. However, Doolittle’s concept views learning in an alternative fashion, particularly in terms of observations being connected to the true student understanding.

This strongly connects to embodied learning, which emphasizes that thinking is not only in the mind but also shaped by the body, movement, gesture, and interaction with materials. As Sara mentioned in our last class discussion, a small change made a huge difference for a student. That change did not involve more explanation, but rather a shift in how the student could physically engage with the task—through movement, visuals, or hands-on interaction. This helped the student access the idea in a way that words alone had not.

These minor bodily changes—enabling the student to move, gesture, manipulate objects, or change their bodily posture—could channel Doolittle’s “small bursts”: Good teaching can be liberating if timed just so as to unlock student understanding, reducing frustration as a bonus. As a result, this stop was helpful in my reconsideration of what it means to be inclusive. I realized that I was conceptualizing accommodations as more effort, as more permanence, but as more embodiment, I see them as teaching moments, teaching moments that I can provide students in the present, teaching moments that recognize as physical, emotional, and cognitive. I realize now, in a sense, lessons can unfold in response to students, not in response to some structure, not in response to some rubric, but in response to students.

Stop 2:

I stopped at the idea that nature gives us shapes, while straight lines are something humans impose.
This statement made me pause because it captures the core message of Doolittle’s argument in a very simple way. In nature, nothing grows in perfect straight lines—rivers curve, landscapes rise and fall, seasons shift, and living systems develop unevenly. Straight lines, grids, and fixed structures are human tools, not natural truths.

This stop made me reflect on how education often relies on imposed “straight lines”: rigid schedules, uniform pacing, standardized assessments, and expectations that all students should learn in the same way and at the same speed. These structures can work for some students, but they often fail others, particularly students with disabilities or those who learn differently.

Doolittle’s idea helped me see that when students struggle, the problem may not be the learner but the structure imposed on them. Just as forcing straight roads through mountains ignores the land, forcing all learners onto a single pathway ignores the natural diversity of how students think and learn. Inclusion, from this perspective, means reshaping the learning environment to follow students’ natural learning paths rather than asking students to bend themselves to fit rigid systems.

This stop encouraged me to think of teaching as a practice of listening, observing, and adapting, rather than controlling. It reinforced the idea that flexible, responsive teaching is not a lack of structure, but a more respectful and realistic way of supporting learning for all students.

Questions:

1)     Doolittle argues that grids provide a sense of control but often fail to represent reality accurately.
How might this insight help us critically examine the assumptions embedded in curriculum standards and learning outcomes, particularly in mathematics education?

2)     How might a flexible approach to teaching inform educators’ responses to learner diversity and unanticipated classroom situations?