Instrumental vs. Relational Understanding

                Skemp’s article solidified some of the ideas that have been vaguely floating around in my brain. While doing my undergrad, some of my favourite math moments came from finally learning where certain formulas came from – for example, when the mystery origins of the ‘area of a circle’ formula were finally proven in calculus class. In fact, proofs were often my favourite parts of my courses, and I didn’t really enjoy ‘rule’ heavy courses (ex. differential equations: “just ‘guess’ this solution!”). When I was in elementary school, I actually hated math because all it was to me was memorizing a bunch of times tables and facts. It wasn’t until I got to high school when I got to do some logical problem solving did I start to enjoy it. Because of this, I related a lot to Skemp’s arguments for relational understanding as a basis for mathematical instruction. I agree that to fully comprehend and appreciate math, having that foundation is crucial. I also believe that, at least in my case, it makes the subject more engaging, making students feel more like a detective and less like a machine. However, I really appreciated his devil’s advocate section, because in some cases, it is still appropriate to use instrumental approaches. And with the time given, it is extremely difficult for a teacher to make sure every student has a relational understanding of all the topics in the jam-packed curriculum.

                Three specific lines stood out to me. One, “well is enemy of better.” If our only focus is for students to perform well on tests, then teaching beyond simple rules seems unnecessary. But if this way of thinking is what is taught, it inhibits further development in the subject, because math builds on itself grade after grade. Without a good foundation, ‘better’ is much more difficult. Two, there can be “a less obvious mismatch … between teacher and text.” I hadn’t thought of this conflict before, but it makes sense. When there is disconnect between a teacher’s style and the text, or other teachers’ styles, this can make it a lot harder for students to understand the material in either relationally or instrumentally! I appreciated this acknowledgment of difficulties of introducing more relational based approaches into instrumental oriented schools. Three, “there are two effectively different subjects being taught under the same name, ‘mathematics.’” This sentence took me a while to muddle through - it is an unusual statement. But it really ties in with the earlier statement about mismatching the two approaches to understanding.