Skemp on two approaches to teaching and learning mathematics

There are few ideas that catch my eyes while reding the Skemp’s paper “Relational Understanding and Instrumental Understanding.” When I first saw the French term, faux amis in his article, not only I did not fully understand his intention using that term, but also I had never realized that there are two different approaches to both teaching and learning mathematics. The first thing that made me stop reading the paper was the Mr. Peter Burney’s story. When he asked students to find the area of a field with two different units, they did not convert one unit into the other and got the wrong answer. This is one of the issues that I, as a math tutor, have seen over and over; I have encountered many students forgetting to change both dimensions in the same unit. Of course, there are few students doing excellent job on solving problems due to the repetitive work. However, when they are given other problems with a twist, most of them looked at them closely for a few seconds and then gave up. It is because they are used to and “trained” to solve particular types of problems with the mathematical techniques that they memorize – that is instrumental learning. Here, the issue is that students don’t truly understand how to apply the technique they learned in the past to new types of problems because they don’t know how and why it works.

The second thing that captured my attention was the idea that “[relational mathematics] is easier to remember.” This is very true and I strongly agree with his statement. Although instrumental mathematics give students more immediate and apparent results, knowing how other mathematical concepts are inter-related allows students to remember them as parts of a connected whole. This is easier for students to remember many mathematical concepts since they don’t have to derive them afresh every time, and eventually they can save time for other extra-curricular activities.

The last point that struck me was the following teachers’ objection on relational mathematics : “… the pupils still need it for examination reasons.” I personally believe that teachers, as educators, need to start thinking about the purpose of examination. Is it for the students or is it for teachers themselves? In other words, is it really for students’ leaning or is it for teachers who just want to give their students grades because this is what they do for living? I believe that teachers should encourage students to learn and to participate actively in classrooms rather than simply having them solving similar questions over and over. Although it may take longer time for students absorbing knowledge through relational learning, it is teachers’ job for their students to enjoy learning in classrooms. Of course, it would take longer time for students to learn the same mathematical concepts compared to “traditional” or instrumental learning. However, as mentioned earlier, students would benefit from relational learning since it allows them to understand and to remember many concepts as parts of a connected whole, saving more time eventually.