Instrumental and Relational Understanding

Instrumental and relational understanding are two contrasting kinds of mathematical understanding distinguished by the mathematics educator Richard Skemp in a widely cited 1976 article, drawing on terminology he credited to his colleague Stieg Mellin-Olsen. Relational understanding means knowing both what to do to solve a mathematical problem and why that method works, so that a rule or procedure is connected to the broader structure of mathematical ideas it comes from; instrumental understanding, by contrast, means being able to carry out a rule or procedure correctly without necessarily understanding the reasoning behind it, described by Skemp as rules without reasons. Skemp argued that although he personally favored relational understanding as the deeper and more transferable goal of mathematics teaching, instrumental understanding has practical advantages that make it attractive to many teachers and students: within its own limited context it is often easier to grasp, the reward of a correct answer follows more quickly and directly from applying the rule, and because it requires less background knowledge it can be reached more easily and with more reliable results. The distinction has remained a reference point in discussions of mathematics teaching for describing the difference between students who can perform a procedure and students who genuinely understand the mathematics behind it.