Skemp's Relational vs Instrumental Understanding of Mathematics

There were several points that stuck out to me while reading Skemp's article. They are as follows:

1. Before realizing there are two different kinds of understanding, he thought relational understanding was the only one. Because of this, he would not have considered instrumental understanding to be understanding at all. 

This sticks out for two reasons, both his initial rejection to the notion of instrumental understanding as well as his acceptance after knowing about its existence. Once you come to realize the significance of relational understanding, it almost always is the kind of understanding that you would expect. Instrumental understanding instead could be considered something completely different, outside the realm of understanding. His acceptance of it as a different kind of understanding to me, is honestly remarkable. His classification of it as a form of understanding allows there to be a bridge, or at least indicates that there is a divide and that they need to come together. If he simply dismissed it as “oh, that isn’t real mathematics or a solid understanding” he may not have identified how many people exist with only an instrumental understanding of the math they are learning (or teaching!). 

2. “Take it over to the other side and change the sign” is appalling in hindsight. Sometimes the way we learn or teach mathematics is so lazy, taking every shortcut possible. The equal sign is right there, staring at us from quite literally the center of the equation. How many times were we taught to think of the equal sign as some bridge that once something crosses it becomes the opposite instead of being asked what the equal sign actually means. I can’t off the top of my head come up with how I could direct a student to have a relational understanding of how we solve algebraic equations and I think that points to something important. The initial hump of getting a relational understanding of a math concept requires some investment of both time and effort on both the student and the teacher. The rewards both will reap from this initially difficult road would be immeasurable compared to taking the shortcut of showing a student how to solve algebraic equations with steps. I guess that’s an important distinction, teaching vs showing (relational understanding vs instrumental understanding). 

3. Skemp says, “All they want is some kind of rule for getting the answer”. I think this speaks to something larger that is intrinsically human – short term gratification. Our inclination towards getting to things quicker rather than trudging through the long and difficult road that will undoubtedly be better for us. Skemp also addresses this later when he discusses knowing the answers for the final examinations that will give success and entry into careers/professions. Within the school system this is easy to understand, students just want the quickest route to the answer so that they may “succeed”. This strikes me because what starts out as a problem in the math classroom seems to stretch out as a problem of school – why are kids (and teachers) at school? Is it to learn or is it to get good grades on exams? The former leads to the latter, but the latter does not always come from the former. Maybe we first have to deconstruct the importance of grades and exams.  

To say I wholeheartedly agree with Skemp may be an understatement. In the fourth year of my undergraduate program I took a course titled Math Topics in Education as a senior level math courses necessary for the completion of my degree. One of the first things we read was this very same article written by Skemp and it blew me away. Math was always my favourite subject and I was one of those students who most of the time was sitting on the instrumental understanding of math concepts but those rare times I would have a relational understanding, I knew first-hand just how enjoyable math was. Of course, I was not aware of what I had felt until I read this article and came to know about relational and instrumental understanding in math. This article, and that course played a massive role in me being at UBC today pursuing teacher education so my response here may be a little biased. 

Although I agree with Skemp in his generally favouring towards a relational understanding, I think his breakdown of why teaching an instrumental understanding of mathematics still exists is equally as important. He is accurate in his assessment that it is deeply rooted in textbooks and teaching methods. It requires, as he says, “accommodating (restructuring) our existing schemas”. He also points out how sometimes there isn’t time to teach a relational understanding not just because the curriculum is so tightly packed but because there is material in other courses (science, etc.) that requires some foundational concepts that are first taught in the math classroom. 

Skemp could have focused his whole article on criticizing instrumental understanding of mathematics and in my opinion, he would have been correct in doing so. But he took an approach that is much more effective and far reaching – he diagnosed the issue and carved out potential solutions. I think his understanding on how the curriculum of mathematics needs to be restructured puts into light the seriousness of what is occurring in classrooms (even now!). It is easy to highlight the obvious shortcomings of instrumental understanding against comprehensive relational understanding; someone like me would love to have read an article nodding my head profusely in anger and agreement, “Yes! Finally, someone gets it!” but that would be gratifying only in the short term. Outlining the work necessary to undo the framework of how most math in school is taught is far more important, and it gives future teachers like me and others the awareness of how important our jobs will be; it forces us to teach math consciously.