Response to Skemp, "Relational Understanding and Instrumental Understanding"

What is mathematics? Or, what does the word "mathematics" mean to a high school teacher and to their students? Like Skemp himself, the meaning of the term is not something I had ever seriously considered. So I found myself stopping early in the article when Skemp discusses the difference between "relational understanding" and "instrumental understanding." He writes (p. 2) that instrumental understanding "is what I have in the past described as 'rules without reasons', without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was what they meant by 'understanding'." This is the first time I have ever considered that there are many teachers for whom this is the case. I suppose it speaks to the quality of my own secondary math teachers, but I have always assumed that while there will be (many) students who just want to know how to get the right answer, such an approach to "mathematics" would never be the goal or approach of the teacher! Skemp has helped me realize my assumption is incorrect. In discussing why a teacher might "make a reasoned choice to teach for instrumental understanding," Skemp writes that "to make an informed choice of this kind implies awareness of the distinction, and relational understanding of the mathematics itself...One has to face the fact that this is absent in many who teach mathematics; perhaps even a majority" (p. 11). I would not have thought this possible.

Having been made aware of the distinction between relational and instrumental mathematics, another point at which I found myself stopping was when Skemp writes "The most important thing about an activity is its goal" (p. 14). I found this statement to be profound in thinking about the doing and teaching of mathematics, because this stretches me even further to consider what the "goal" of my own teaching will be. I realized at that moment that even though I was taught to understand mathematics relationally, there was still a sense in which the "goal" was quite focused on getting the right answer rather than "building up a conceptual structure (schema)," as Skemp puts it. As a teacher, what is my goal for my math students? I am challenged to more carefully consider this question.

I certainly find myself siding with Skemp as he argues (politely) that "mathematics" ought to be used for relational mathematics only. I cannot imagine myself teaching purely instrumental mathematics as Skemp has described it. And yet, I do have a further wondering. In calculus, we usually follow the pattern of learning how to "do the math" before we then take analysis to more fully grasp "why the math works." Is there a place for a thoughtful approach to teaching some math content more "instrumentally" as a way to build up a sense of familiarity with the material, after which the relational understanding can be better grasped when taught? In other words, I want to think more about whether there might be an "instrumental-relational" option that would work well for the high school classroom. Sometimes answering the question "Why does what you just learned how to do work?" would seem to be an effective approach.