Skemp’s Theory for Teaching Maths
I found this really interesting from as part of my Teaching Advanced Mathematics Masters Course at Warwick University this academic year. For me the theory is more important than ever with the advent of more demanding problem solving elements in the GCSE and particularly the forthcoming A level changes in maths which are to demand a greater aptitude and flexibility in solving issues arising in questions.
Skemps theory basically highlights the difference between and instrumental and relational learning in the maths classroom.
I have to say this resonated with my own views on the teaching of the subject.
It is easier and tempting to teach techniques in a nonrelational instrumental manner which even if successful in helping a student pass an exam will will soon be discarded after the exam and not provide a firm foundation for a students to build on in their further education.
A summary of the theory appears as as follows:
Skemp’s theory
Richard Skemp was a mathematician who later studied psychology^{1}. He drew on both these disciplines to explain learning in mathematics. The main ‘thrust’ of his argument is that learners construct schemata to link what they already know with new learning. According to Skemp, mathematics involves an extensive hierarchy of concepts – we cannot form any particular concept until we have formed all the subsidiary ones upon which it is depends. Skemp also suggested that emotions play a dominant part in the way in which we learn.
Skemp suggested that there are two kinds of learning in mathematics:
Instrumental understanding^{2}: a mechanical, rote or ‘learn the rule/method/algorithm’ kind of learning (which gives quicker results for the teacher in the short term), e.g. writing 10 would be understood as “This is how we write 10” in instrumental terms.
Relational understanding^{2}: a more meaningful learning in which the pupil is able to understand the links and relationships which give mathematics its structure (which is more beneficial in the long term and aids motivation), e.g. writing 10 would be understood as “This is why we write 10 like this (in terms of place value)” in relational terms.
Both are deemed important for mathematics.
Relational Understanding and Instrumental Understanding
The article Relational Understanding and Instrumental Understanding was written by Richard Skemp and originally published in the December 1976 issue of Mathematics Teaching. The article was reprinted in the November 1978 issue of Arithmetic Teacher and in the September 2006 issue of Mathematics Teaching in the Middle School. The article is available from NCTM at http://www.nctm.org/publications/article.aspx?id=20558 and from JSTOR at http://www.jstor.org/stable/41182357.
Abstract
In this article, the author defines relational and instrumental understanding. He then explains the impact he feels these two disparate goals have on the attitudes and understanding of students. We believe the reader will find his ideas about the teaching and learning of mathematics remarkably contemporary and thoughtprovoking.
Summary of Relational Understanding and Instrumental Understanding
In Relational Understanding and Instrumental Understanding Skemp contrasts two perspectives of mathematics. Using the terms relational and instrumental from Stieg MellinOlsen, Skemp introduces relational understanding as “knowing both what to do and why” (p. 89) and instrumental understanding as the ability to execute mathematical rules and procedures. Skemp asks:
If it is accepted that these two categories are both wellfitted, by those pupils and teachers whose goals are respectively relational and instrumental understanding (by the pupil), two questions arise. First, does this matter? And second, is one kind better than the other? (p. 89)
Skemp admits his longstanding assumptions that relational understanding is better, but questions then when so many mathematics teachers and texts focus on instrumental understanding. Concerned about conflicts between the two views, Skemp hypothesizes two mismatches:

Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally.

The other way about. (p. 90)
Skemp sees the first mismatch as a shortterm problem while the second is much more serious, as a student focused on relational understanding will get little or no assistance from the teaching. Similarly, a mismatch between teacher and text could also lead to conflicts. With the perspectives and conflict made clear, Skemp admits that “I used to think that maths teachers were all teaching the same subject, some doing it better than others. I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’ (p. 91).
Despite his preference of relational understanding, Skemp proposes three advantages of instrumental mathematics that make it preferred amongst many mathematics teachers: (a) within its own context, instrumental mathematics is often easier to understand; (b) the rewards for following a procedure and getting a correct answer are more immediate; and (c) because less knowledge is involved, it’s often correct answers come more easily and reliably. In contrast, Skemp identifies four advantages to relational mathematics: (a) it is more adaptable to new tasks; (b) it is easier to remember, (c) relational knowledge can be effective as a goal in itself, and (d) relational schemas are organic in quality.
The preference or use of instrumental mathematics by teachers are many, say Skemp. In some cases, instrumental understanding comes more quickly, and for a particular calculation in a class or on an exam it may be all a student needs. The expectations about the nature and amount of content presented in a course can also influence a teacher’s use of instrumental mathematics if they feel pressure to cover many topics. Teachers may also prefer instrumental understanding because it is what they themselves possess, or they have difficulty recognizing relational understanding in their students’ thinking and written work.
To set up a theoretical explanation, Skemp proposes an analogy. He imagines visiting a town for the first time and needing to navigate to particular locations. He compares two strategies: (a) learning specific routes that take him from where he is staying to his destinations or (b) exploring the town in a way that allows him to form a mental map of significant landmarks and features. For getting from point A to B the first strategy will be efficient but limited to that route. The second strategy might result in a longer trip from A to B, but he will be more prepared to find other destinations and is less likely to get lost. Skemp relates the first strategy to instrumental understanding, where learning to navigate consists of learning an increasing number of fixed plans. Relational understanding is like the second strategy, where learning “consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point” (p. 95). Skemp therefore distinguishes relational understanding from instrumental understanding in the following ways:

The means have become independent of particular ends to be reached thereby.

Building up a schema within a given area of knowledge becomes an intrinsically satisfying goal in itself.

The more complete a pupil’s schema, the greater his feeling of confidence in his own ability to find new ways of ‘getting there’ without outside help.

But a schema is never complete. As our schemas enlarge, so our awareness of possibilities is thereby enlarged. Thus the process often becomes selfcontinuing, and (by virtue of 3) selfrewarding. (p. 95)
I would argue as maths teachers that although instrumental teaching has its place we should always be thinking about maximising relational learning opportunities for our students.
I hope you enjoyed thinking about this as much as I have done these last few months and would love to hear any thoughts you have on the subject.
E.Bailey
Deputy Head of Maths
Able Gifted and Talented Coordinator