Response to Hewitt (1999)

Hewitt (1999) prompted me to consider what is “arbitrary” versus “necessary” in math education and how I address these distinctions in my teaching. His ideas resonate with Skemp’s (1976) concepts of “instrumental” versus “relational” understanding: while “arbitrary” knowledge and “instrumental” understanding provide the language and foundational procedures for discussion, “necessary” knowledge and “relational” understanding facilitate higher-order thinking, allowing students to internalize and apply their knowledge. During my recent short practicum, I co-taught a Pre-Calculus 11 unit on radical operations and equations with my SA. We emphasized students working through some processes, such as applying the expansion (A+B)^2 = A^2 + 2AB + B^2, and treated other concepts as “arbitrary”—a practical decision due to time constraints and the fact that these concepts had been covered in earlier grades.

This article has made me more conscious of which ideas students can discover for themselves to enhance retention and understanding. In my lesson and unit planning, I intend to incorporate time for students to explore concepts independently or with guided prompts, allowing them to derive rules through their own reasoning. For arbitrary concepts, I plan to use visual aids and classroom posters to support memorization. Reflecting on Hewitt’s perspective also deepens my empathy for students who struggle to memorize arbitrary details. Building a classroom culture where students feel comfortable exploring, making mistakes, and supporting each other will be essential for fostering this balance between “arbitrary” and “necessary” learning.