About Drawing and Geometries of Liberation

In Off the Grid, Doolittle (2018) highlights the omnipresence of the grid in our society, shows the limits we can encounter with straight grids and propose alternatives to it. First, Doolittle shows how 2-dimensional grids are shaping the space and time in our society. This is due to some cultures and human’s nature: grid provides a feeling of control and evenness that many enjoy. A good example is how cities are built, like Montreal below.

 

We can also think about time in a sense that it is divided by years, seasons, months, days, hours, etc. that is dividing our time. Although, this idea of the perfect grid has its limits. For example, what can we do about Mont-Royal? It is a big mountain that cannot be moved. Should we go over it (and risk an elevation?), around it (and risk a curve?), or through it (the keep the evenness of the streets intact). And what if geese come to Schefferville before or after the time allotted in the already planned school calendar for the hunting break? Will we be able to move the break?

Secondly, one of the alternatives to Euclidean geometry (grids) Doolittle presents is Riemannian geometry. Riemannian geometry allows varying the way distances are measured from one location to another, or one direction to another. One of the most famous uses of Riemannian geometry is the Mercator projection (map of the world flat). When presented this way, the decision was made to keep the angles accurate and represent the area of different locations somehow inaccurately (especially in the poles). This link brings you to this explained with a sketch: https://sketchplanations.com/the-mercator-projection

Finally, it is important to note that Doolittle does not argue to get rid of grid systems, but to supplement it with different geometric systems that could allow us to be able to represent the world as it really is.



“We must look through the structures imposed by our minds to the reality that lies below the surface grid we have drawn on top of it (Doolittle, 2018, p.113).”

I decided to try this week activity with my students. I wanted to use branches on trees as models for different types of angles for my next lesson. Below are some of my students’ drawings.

The biggest win of this activity: all of my students really took the time to try to represent the 3D element of the tree. Some of them drew only branches, some of them decided to also include the way the tree looked with the needles. Also, some of them even drew the tree’s roots or the land on which the tree rests.

I was expecting some of them to draw the spruce the way we often see it, in a grid type of way:

Being able to represent the tree with a 3D aspect and sometimes showing its relationship with its environment (with the roots) even if we do not see them show that my students are able to look through the grid structures to represent the true nature around them. I wonder if it is because we live in an environment that is not as impacted by grids as a city or because their way of seeing is not through grids and more through relationships and actions. I cannot imagine how I could ask students in Montreal to let go of their idea of grids to get a sense of the whole picture since they have been omnipresent in their life for so long. Even I had to work hard to try to understand how alternative geometries can work while reading the article.

“Alternative geometries, geometries of liberation, deserve a major role in the future of mathematics education (Doolittle, 2018, p.119)” and “It does not seem very mathematical to me (my students during the drawing activity).”

Because alternative geometries have no space in the curriculum, my students thought that it must not be mathematics. Although, those geometries have an important potential (other than being culturally relevant for my Indigenous students).

After shocking me a little, this comment opened up a discussion about fractals and how fascinating it is that we can find them all around us. As Gerofsky mentions (personal communication, January 27, 2026), this activity offered an opportunity to look into mathematical ideas coming from patterns found in nature and made my students wonder about mathematical idea (angles and fractals). Furthermore, it opened up the discussion about mathematics is a broad field that goes beyond what we see in the curriculum. And after discussions like these, I can see many demonstrating a new (or renewed) interest for mathematics. Thus, those geometries of liberation bring students not only closer to nature, but also to their identity and mathematics.

How would you incorporate or present alternative geometry meaningfully to your students knowing how surprising it may sound to them (or even ourselves!)?

References

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