Skemp on Relational Understanding and Instrumental Understanding - Response

While reading, one point that stood out was how Skemp felt the key flaw in his football and soccer analogy was how students cannot simply “refuse to play,” the way the two teams can when experiencing the understanding mismatch. For me, this highlighted the disengagement or hopelessness youth can develop in classrooms when there is both compulsory attendance but also prolonged or unresolved confusion. Skemp’s later discussion of instrumental understanding being counterintuitively less simple because of its multiplicity of rules also made me pause, as it reminded me of an “axiom” I have heard in other disciplines as well: “Simplicity, rather than complexity, is the hallmark of design.” This often refers to the goal of building or solving something effectively but with the fewest steps or parts possible. I still feel that it shares an overarching idea with Skemp’s point, as they both suggest that unnecessary complexity, rules, or steps are a detriment to solving problems, rather than a sign of sophistication or a necessary evil of more advanced math problems. One other point I stopped on was Skemp's suggestion that assessment may contribute to the difficulty of teaching relational understanding. While I agree that speaking to students would likely uncover their style of understanding quickest, I do feel that traditional math assessments, such as tests, have attempted to probe for this. For example, students with only instrumental understanding can struggle with test questions that are even slight permutations of exercises they saw while studying. In contrast, those who have grasped the fundamental principles and their applications are more likely to have success, meaning these questions would have checked for relational understanding, though perhaps not to an ideal extent.

Ultimately, I tend to agree with Skemp’s perspective that relational understanding ought to be targeted in education, due to it being a deeper and more transferable style of understanding which can apply topic to topic or even between disciplines. It seems rooted in a concept-based style of teaching that can add value to students’ learning by helping them make connections throughout their education and be motivated by the content itself. However, I feel that Skemp’s treatment of the topic implies that these two styles of understanding are more separate than they really are. In particular, I feel that the two are largely intertwined in that an instrumental understanding represents the facts or content of math, while relational understanding includes the underlying concepts, which may not even be math specific, such as proportions or change. If we can teach in this intertwined way, where topic-specific content is the tool with which to grasp larger concepts (which will be useful for the next topic or even subject), then we may be able to stimulate relational understanding without giving up the benefit and utility of math “tricks.”