Arbitrary and Necessary

Reading Dave Hewitt’s “Arbitrary and Necessary” was an interesting exercise, as it starkly contrasts most of the approaches I’ve been exposed to and have seen others use in the math profession. It offered a divergent perspective on teaching mathematics. One key takeaway is that Hewitt’s distinction between knowledge that must be memorized (arbitrary) and knowledge that can be discovered through reasoning (necessary) aligns well with aspects of Indigenous ways of knowing, especially in connecting learning to place and story.

The necessary method of discovery that prioritizes exploratory learning feels like it could be compelling when connected to the land or a story, drawing on Indigenous ways of knowing where knowledge isn’t isolated in abstract rules but is instead experienced, contextualized, and interwoven with one’s surroundings. For example, if we’re learning about geometric shapes, we might connect these shapes to natural patterns found in local plants or animals or even the symmetries in rock formations, similar to some of the observations we made in the Orchard Garden before the break. These types of explorations allow students to see math as something that exists around them, something that can be discovered and connected to their lives.

In terms of how this might shape my lesson planning, it’s clear that understanding the importance of classifying each concept as arbitrary or necessary is essential. For necessary concepts, I would focus on activities that promote discovery and allow students to experience the underlying principles (probably an associated learning curve). Related to Hewitt’s video tasks, one idea might be to have students explore fractions by seeing how many fractions they can find between two points on a number line, which could also connect to the natural world, such as measuring distances between two trees or observing how natural objects can be divided or combined. While working through Hewitt’s video exercises, I found I was missing the awareness of just how many fractions exist between 5/7 and 3/4​. For students like me, this type of exploration would be helpful in building a greater understanding.

I also want to incorporate stories and place-based learning when working with necessary concepts. For instance, when learning about angles or proportions, we could explore how local Indigenous artwork or cultural artifacts use symmetry and balance. Tying these math concepts to stories from the land might help students see them as part of a larger picture and not just isolated rules.

Assessment of necessary concepts still feels somewhat abstract to me at this point. For necessary concepts, I agree that assessment should emphasize understanding and application, allowing students to demonstrate that they’ve grasped the concept’s logic. However, gaining these types of understandings for myself is probably a learning curve. For arbitrary concepts, where memorization is the goal, assessment would focus more on recall. Above all, it seems crucial to distinguish what’s necessary and arbitrary before starting the lesson plan. This clarity would help guide how I approach each concept, ensuring that my teaching honors logical reasoning and rich, contextual learning.

One final idea that came to me while reflecting on necessary content in Computer Science was related to recursion, a topic I have often seen as a gatekeeping concept in the field. I thought about presenting recursion through Indigenous art, specifically using a totem pole as an example. Recursion, with its self-referential nature, can be likened to how Indigenous art builds narratives from repeated, nested patterns. Totem poles, for instance, are storytellers that layer symbols representing ancestral tales. Each carved figure on a totem pole reflects a smaller part of a larger story, just as a recursive function calls itself to construct a comprehensive solution from simpler sub-problems. Presenting recursion through this visual and cultural lens could make the concept more accessible to students, connecting a complex idea to an art form rich with cultural significance and emphasizing indigeneity. This approach aligns with Hewitt’s emphasis on necessary knowledge that can be discovered through exploration and contextualized within broader, meaningful experiences. By integrating Indigenous stories and the visual structure of totem poles, students might intuitively understand recursion as a process that builds upon itself. Maybe!!!!!