Response to Skemp (1976) article

    Skemp’s (1976) compelling exploration of relational and instrumental understanding in math instruction made me “stop” and reflect on three points: (1) the definitions of the two types, (2) the advantages of relational mathematics, particularly what is not included as an advantage, and (3) the concept of simplicity achieved through synthesis and unification (pp. 4, 13).

    Firstly, the technical definitions separating relational and instrumental understanding, followed by examples, got me to reflect on my own mathematical knowledge – which parts are relational and which are instrumental. Even if I initially learned a concept instrumentally, can I now teach it relationally, and vice versa? For example, my early understanding of trigonometry was purely instrumental and disconnected from my previous knowledge, but learning the unit circle and graphing functions transformed my perspective. Now, I wonder whether I would decide to teach the holistic trigonometry picture with different modalities, representations, and functions first, or follow the sequence in which I was taught. What factors would guide the decision?

    Secondly, not listed as the advantages of relational mathematics are the lack of drills and memorization and “looser” assessments. While relational, big-picture understanding of math fosters retention and the ability to apply skills to novel problems and new environments, it does not justify overlooking rote memorization for skill acquisition and fluency. Likewise, assessments for relational mathematics might differ from those focused on procedures, yet that does not mean they are easier. The choice of assessment method should fit the context, whether it's pen-and-paper, project-based, or inquiry-driven.

    Finally, relational understanding of mathematics is holistic, yet it does not equate to being excessive. In fact, it often requires teachers to synthesize and simplify information in a concise, efficient, and meaningful way. These decisions depend on students’ developmental level and the scope of the topic.

    Regarding my stance on the issue, I strongly believe in the necessity of both types of understanding in math. They are not opposite but complementary, each offering distinct advantages that enhance problem-solving and reasoning, as mentioned in Skemp (1976). My grade school experience, unfortunately, was mostly focused on instrumental understanding, leaving me fluent in procedures without a deep grasp of their stories and logic. It was not until my Pre-Calculus teacher integrated both relational and instrumental approaches that my curiosity and passion for math began to grow. Therefore, I am committed to exposing students to multiple means of representation with the dual goals of developing skill fluency and deeper relational understanding, empowering them to become confident and capable problem-solvers.