Thoughts on Relational and Instrumental Understanding

In Richard R. Skemp’s article he gives examples of several faux amis, each one capable of causing confusion. Skemp proposes that understanding and mathematics are both faux amis and that this could be one problem facing mathematics education. Is an instrumental understanding of any subject, truly understanding? In history, does one truly understand the ratification of Canada’s Constitution just by memorizing the date and using a mnemonic device to remember the names of those involved? No, one must contextualize the event in order to understand its place in history. Math should also receive the contextualization it deserves.

In Skemp’s defence of instrumental understanding he writes: “If what is wanted is a page of right answers, instrumental mathematics can provide this more quickly and easily” (8). Is this really a defence? When in life, outside of school, will we ever be faced with a page of problems just waiting to be solved? Practically never, it is a situation that is unique to the school experience, and thus a skill that is of very little practicality.

Continuing Skemp’s (lacklustre) defence of instrumental understanding: “The pupils just won’t want to know all the careful groundwork [the teacher] gives in preparation for whatever is to be learnt next. All they want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest” (4). Why then, would anybody just ask for the answer? Can we motivate students to learn in another way? According to Skemp, yes: “Relational knowledge can be effective as a goal in itself” (10). I find it odd that in other subjects this is taken for granted. In English, and most other arts classes, we write essays or engage in discussions to show that we have some insight in to the subject at hand. What would an equivalent test of relational knowledge be for math?

Perhaps, one problem is that most people view math instrumentally. Skemp suggests a solution for this problem: “One [point] is whether the term ‘mathematics’ ought not to be used for relational mathematics only” (16). I would tend to agree with him, with one reservation: who is to say that we get the term? I agree that there should be some sort of differentiation between instrumental and relational mathematics but, if the common usage of the word mathematics means instrumental mathematics then it is an uphill battle to make people even see that there is such a thing as relational mathematics.

People want to learn “useful math.” Skemp says of instrumental mathematics: Pupils “will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects, and whose lack has recently been the subjects of complaints by teachers of science, employers and others” (6). I would argue that it is instrumental mathematics that is creating these shortcomings. Employers and science teachers want math skills that are adaptable to new situations. It is not useful to only be able to solve a set of problems on a sheet of paper. It is useful to know how to fit these problems in to one’s schema. Thus, “useful math” is actually relational.

I spent a lot of time after reading this article testing my understanding of fundamental concepts in math. It is often the simplest concepts that can be the most difficult to explain, especially to explain properly but, explaining them is what we do. So, why not do it properly?

References
R.R. Skemp: Relational Understanding and Instrumental Understanding (1976)