Reflection on “Relational Understanding and Instrumental Understanding” by Richard Skemp.

What a remarkable article!

Skemp (1976, 2) shows his new perspective of teaching mathematics and teaching methods to students. The two names of understanding clearly demonstrate how mathematics is conceptualized in students’ hearts: “relational”, and “instrumental”.

            Susan’s example of how babies learn things, from my perspective, can be an example of how instrumental and relational understandings happen. Assume a baby is told that an apple (imaging a real apple) is pronounced as APPLE (imaging the sound of apple). This 1-to-1 relationship is so simple that the baby does not need to understand what apple is. By this the baby can simply apply the rule that things with some similarities (even though s/he may not be able to tell what similarities are) are apples. This is instrumental understanding. Then when s/he becomes an adult, s/he may realize, through relational understanding, that apple is a kind of fruit with a sweet (maybe) taste and a red (maybe green or yellow, depending on species) color. The adult then realizes that apple is a general name of a kind of fruit species.

            Then here comes a question: is relational understanding superior than instrumental understanding? Well, I would say it depends. Skemp (1976, 11) summarizes four reasons of why mathematics is hard to teach, which in my understanding demonstrate a balance of educational resources and students/teachers’ needs. A costly teaching strategy may not be the best option for students with “different goals”.

For these goals, Skemp (1976, 4) summarizes them as of two kinds: (a) to understand mathematics instrumentally, and (b) to understand mathematics relationally. Here the word instrumentally means to understand HOW to finish a pencil-and-paper task, or to become competitive in exams. This situation normally happens when a test is right at the corner while students do not have enough time to carefully sit down and explore the mathematical knowledge. In this case, their driven hearts push them to find some “quick, efficient, and accurate” (according to Kevin in EDCP 550 class) ways to deal with tests.

Similar situation happened in my tutoring experience. Several years ago, I was a math tutor teaching a Grade 12 student whose math score was less than 50%. With a math test right at the corner, the student was so anxious to find some way to pass the test. It is imaginable that with a lower-than-average knowledge base he almost had no time to fully master all the test content through a relational understanding approach.

Skemp, R. R. (1976/2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Originally published in Mathematics Teaching.