Relational Understanding and Instrumental Understanding

One idea that made me stop was the example about students being asked to find area, and the students replying that area is always in square centimeters. I thought the example captured, especially with instrumental learning, how students form misconceptions to fill in their knowledge gaps. Another idea that stood out to me was the idea that there are two different subjects being taught under the umbrella of mathematics. This makes our job particularly difficult as math teachers because it is our job to build a large variety of thinking in our students. A final idea that made me stop was during the comparison of music education to math education. I often compare learning math to learning music because “you have to practice consistently to get better”. However, this idea shows that some practice may be “better” than others.

 

I certainly agree with Skemp that both instrumental and relational learning are distinct. Many students that I have taught struggle most with relational understanding, because instrumental learning has comprised most of their mathematics education. I’ve always had the goal to facilitate relational learning, but it seems harder to put into practice. Instrumental work allows students to check work directly against an answer. It is easier for students to follow prescribed steps independently, in contrast with thinking conceptually and asking questions. I think both relational and instrumental understanding are necessary to fully understand mathematics. Relational understanding gives students an idea of “why” they are performing steps in a problem. Instrumental understanding can help students to put ideas into practice. For example, learning the process for how to convert from entire to mixed radicals emphasizes how numbers can be decomposed into prime factors, and gives students an idea of how this decomposition is applied.