Reflection to Arbitrary and necessary

After reading Dave Hewitt’s “Arbitrary and Necessary: A Way of Viewing the Mathematics Curriculum,” I started to think more deeply about how students actually learn mathematics and how I should design my lessons. In Hewitt’s framework, arbitrary knowledge refers to things that students can only know by being told, such as names, symbols, and conventions that are decided by people. In contrast, necessary knowledge refers to ideas and relationships that students can figure out through reasoning and awareness.

This distinction makes me reflect on how I want to teach mathematics in my classroom. Usually, math teaching focuses on memorizing formulas or following fixed procedures, which can make students see mathematics as a subject of rules to remember rather than ideas to understand. Hewitt’s argument reminds me that real mathematics begins when students start to reason, question, and discover why something must be true. If I only tell them what to do, I might take away their chances to think mathematically.

In my future lessons, I want to be more intentional about separating what is arbitrary from what is necessary. For the arbitrary parts, such as notation or vocabulary, I will make sure to give clear explanations and support their memory. But for the necessary parts, I want to create meaningful activities that allow students to explore and make sense of patterns on their own. For example, instead of directly telling them that the angles in a triangle add up to 180°, I can guide them to discover it through cutting, folding, or measuring. I believe this approach will help students build a deeper connection with mathematics, where they are not just remembering facts but truly understanding them. As Hewitt says, “If I’m having to remember, then I’m not working on mathematics.”