Reflections on Skemp

Skemp presents the idea of faux amis to introduce the problem with different meanings of understanding and extends it to that of mathematics. I would relate the “instrumental understanding” to knowing simply which rule to use and how to compute the answer. “Relational understanding” is knowing the concept which leads to explanation with more than just the equation. One of the examples given was a “teacher reminding a class that the area of a rectangle is given by the area being A = L * B” (area is length times the breadth) (1). I had difficulty with how an elementary school teacher would teach the relational understanding of the concept of area. The only way I could think of was a picture of the rectangle with a grid drawn over it. If one was to be very thorough about it one would have to remind the class that length was a measure of how much space a line took up and that area was like the same concept but in two dimensions. One would also have to help the students understand why one has to use multiplication. One of the gems of Skemp's article is the quote: “Unless the two sides (who have a different interpretation on what was going on) stop and talk about what game they think they are playing at, long enough to gain some mutual understanding, the game will break up in disorder and the two teams will never want to meet again”(2). I would agree it would be quite important for the student and teacher discuss the end goal of the lesson if they are different. The frustration that would mount up for the student would cause them to have problems each lesson and may cause them to steer away from the subject in the future. It would help the student to understand why they were learning such things and for the teacher to understand the needs of the student. As a teacher candidate for physics, I would hope to teach the student relational understanding of the subject. This would help them have the skills to analyse and tackle new problems. If they only understood the subject instrumentally, they would only be able to take the same type of questions they learned and look for some equations that would give them the answer; they would not be able to teach a peer how the equations applied to the physical situation. As a music enthusiast, music taught “as a pencil-and-paper subject” compared with students “taught to associate certain sounds with these marks on paper” (3) was a very strong example depicting the importance relational understanding. The first group would miss the idea of music completely and miss the goal of learning music. They would not know music at all. Skemp states, “it is nice to get a page of right answers, and we must not underrate the importance of the feeling of success which pupils get from this” (4). As a new teacher, I am more enthusiastic on students learning which would leave me not remembering students want to feel some sort of accomplishment. In this education system, marks are perhaps used for students to judge each other and a source of pride and self-confidence. However, I would like to at least empower the students to be at a point where they can say “I can do this”. Skemp states that the difficulty with teaching relational mathematics is “the backwash effect of examinations” (5). In the current education system, measurements of success stem mostly from examinations. This results in students just wanting to cut corners and learn mathematics the quick and fool-proof way (instrumental learning). Physics education groups have been developing questions testing the relational understanding of physics instead of computations. I think students would switch from just learning instrumental mathematics to relational mathematics if relational mathematics was tested. Overall, I learned mathematics and physics relationally and would be frustrated at just an instrumental understanding of these subjects. However, I would have to be mindful and plan carefully how to teach the concepts and problem solving without the students just filtering all the information and learning just the computational part of the lesson.