Skemp’s instrumental versus relational understanding of mathematics

The following material was taken from M2 – Models of Mathematics Learning Links to an external site. within the AustMS unit. While the unit also discusses Schoenfeld’s approach to mathematical problem solving (Kearsley, n.d.), we will focus on Skemp’s work here (Skemp, 1976).

Richard Skemp draws the distinction between learning and teaching using a relational approach, and that which uses an instrumental approach. Relational understanding refers to knowing both what to do and why – an understanding of all of the parts, how they relate, and why they are applied in the manner they are. On the other hand, instrumental understanding refers merely to being able to apply a series of steps without knowing why they are being applied in that way, or what they mean – 'rules without reasons'. Quite often when students learn mathematics they experience (perhaps encouraged by the approaches of their teachers) temporary success by forming an instrumental understanding of the topic. However, this is at the detriment of their long-term success compared to their potential if they had formed a relational understanding.

For instance, when learning differentiation, students might easily learn 'the rules' of calculating derivatives but have little concept of when to use each rule and what the derivative means in practice. The consequence of forming this instrumental rather than relational understanding is that they cannot sense the utility of differential calculus, and they are unable to solve problems with it.