Initial reaction to Skemp's "Relational Understanding and Instrumental Understanding"

 The terms “relational understanding” and “instrumental understanding” describe phenomena I have recognized, and I suspect most educators have. In my math teaching, I have called what Skemp terms relational understanding as “deep” or “fundamental” understanding, and “instrumental” as “shallow” or “utilitarian”. For example, there are specific tools used to solve equations, the choice of which can depend on the relationship between the variables in these equations. These tools can feel disconnected. Teaching factoring in the context of polynomials, I like to emphasize to students that there are certain advantages to having expressions represented as products. Perhaps this is an example of relational understanding. However, often students don’t care about this.  Many want the specific tool needed to solve the problem on the page, and then the next tool to solve the next problem. Some are interested in relational understanding, or at least curious about it, but they seem much fewer in number, which makes me think the math mismatches described on page 4 will be a major issue and that the mismatch will be heavily weighted toward students who prefer the instrumental approach. 

Skemp’s argument on page 9 that relational understanding is “easier to remember” first struck me as a surprising, as students may argue that they would rather memorize tricks than spend the time required to get comfortable with broader, potentially more abstract principles. However, the latter approach--few fundamental points of knowledge and understanding how to vary them--requires less cognitive storage than memorization of highly specific tools. I found the argument number six on page ten interesting; I did not think it would be easy to measure the level to which relational and instrumental knowledge can be thought of “goals in [themselves]”. I’m curious about the studies in question, and although I don’t have any specific reasons to doubt whether it’s applicable to math teaching, I would like to see whether there have been studies conducted on that topic since the publication of the text.