Skemp's two approaches to teaching and learning math

Skemp’s paper on comparing relational and instrumental understanding in mathematics was philosophical and interesting, however there were a few moments that made me stop and think when reading. Skemp defines instrumental understanding as learning “rules without reasons” while relational understanding is more knowing what to do and why.

The first moment that made me stop and reflect was when Skemp mentions one of the mismatches in mathematical learning, where a student’s goal is to understand instrumentally when the teacher wants them to learn relationally. This is extremely evident throughout my undergrad studies of mathematics, where the prof would spend a long time going over the proof and theory behind an equation, but it would be completely overlooked as the proof is more complicated than the equation. When I taught students math, they also only cared more about the equation as it would lead to the right answer quicker than understanding why the equation works; an advantage of instrumental understanding as mentioned by Skemp. The second moment which made me stop and think was when Skemp listed the advantages of relational mathematics, several of which were it being more adaptive to new tasks and being easier to remember. I noticed that with relational understanding, students draw conclusions on new (but similar) questions based on their previous learning as they see similarities in the questions. For example, I had students use the area of a triangular prism to solve for the area of a pyramid. Both shapes are similar but are different enough to require a different equation to solve for the area. The third moment I stopped while reading was when Skemp discussed the “over-burdened syllabi” where a single line of math contains many concentrated ideas. For example, Pythagorean Theorem is the sum of two small squares equals the big one. A student would have to draw out two squares, square their lengths, add them, and take the square root of that to find that diagonal line. In instrumental understanding, that would be the equation a²+b²=c²_.

Skemp argues that relational understanding would take too long to achieve and may be too difficult depending on the topic. I agree, however believe the advantages of relational understanding outweigh the disadvantages. Teaching students the theory behind an equation and why it works can get them to ask questions and think outside the box; which are important aspects to learning and critical thinking.

Paper: Skemp, R. Relational understanding and instrumental learning. Mathematics Teaching, 77, 20-26, (1976)