Skemp on two approaches to teaching and learning mathematics- Reading 1 EDCP342

 Reading Skemp’s analysis highlights three key moments that demand a pause. The first is his framing of "understanding" as a faux ami (pg. 1), revealing that teachers and students often speak entirely different languages under the same word. Honestly, I never thought about how it impacts learning before reading this article. 
The second is the striking observation that "‘Well is the enemy of better,’ and if pupils can get the right answers by the kind of thinking they are used to, they will not take kindly to suggestions that they should try for something beyond this" (pg. 5). This underscores the psychological friction teachers face when students resist deeper conceptual thinking because surface-level rules yield immediate success. And I think, even sometimes, teachers lack tools to assess if their understanding is instrumental or relational unless they encounter a problem-solving situation and fail. 
 Lastly, his cognitive map analogy, where instrumental learning consists of "learning of an increasing number of fixed plans" (p. 14) whereas relational learning is "a mental map of the town" (p. 14), sharply illustrates how rule-based learning leaves students stranded the moment a problem strays off path.

On this issue, I sort of align with relational understanding, as true mathematical fluency requires knowing why a method works rather than simply executing a script. However, as Skemp notes in his "Devil’s Advocate" section (pp. 8–10), instrumental rules offer immediate rewards and ease cognitive load in the short term. Therefore, while relational understanding remains the primary long-term goal for adaptable thinking, instrumental strategies can occasionally serve as useful, temporary scaffolding, provided they are eventually connected back to a cohesive "mental map". The area of a triangle is a prime example of when holding off on a full relational breakdown makes sense, allowing younger students time to grasp the core concept first. We cannot always expect younger students to derive formulas immediately, but initiating classroom discussions and inquiry around these ideas remains essential.