Relational and instrumental understanding

Philip Braham
May 17, 2019 · 4 min read

In 1976 Richard Skemp, a professor of mathematics at Warwick University in the UK, wrote an article on what he called rational and instrumental understanding. He was referring to how we teach mathematics but his ideas have repercussions that extend into how we teach generally and how we understand.

When I was managing a part of a large computer project, I would often employ contractors who had no specific experience in what we were working on. If they were intelligent and had worked on many different systems it was apparent to me that they would pick up what were doing, and generally they proved their abilities. They had a relational understanding of these systems. On the other hand, some other managers only employed people who had specific experience in the actual applications they were using. Often these people proved to be useless and were unable to adapt to a system that they hadn’t been specifically trained in. They had an instrumental understanding in that they knew how to use a particular application but they had no overall understanding of the systems they were working on

In the context of teaching mathematics, someone who only understands the process of how to teach mathematics may know for instance how to do long division without understanding how the process of doing long division actually works. If a student makes a mistake the teacher may only be able to say that the answer is incorrect. Someone who understands mathematics in a wider context would be able to say not just that the answer was wrong but could say something like ‘I can see why you got that answer, here’s what you're doing wrong’.

Richard Skemp wrote his article in the 1970s and points out that what he called ‘understanding’ was relational understanding. Instrumental understanding was not real understanding at all. Nowadays, most education is instrumental understanding.

A professor of physics at a major US university (and I don’t have the link available so I can’t be more specific that that) was addressing a final year group and made the remark that a small, light ball dropped from the university roof would hit the ground at the same time as a large, heavy one. His students laughed and said that it was obvious that the heavier ball would reach the ground first. Now he was shocked that final years physics students could be so ignorant. He went over the equations for gravity and acceleration and they were familiar with them, but were unable to apply that theoretical knowledge to the real world. He subsequently changed his approach to teaching by emphasising practical work and getting students to work out the theory from the experiments (rather than the other way round).

Unfortunately, instrumental understanding (what Richard Skemp referred to as ‘not real understanding’) is what passes as understanding at most schools, colleges and universities.

I heard a woman on the radio talking about her experiences at university in Australia and she said that when she left school many years ago and went to university it was a place of discussion and students were encouraged to question through seminars and tutorials. When she returned to university as a mature student she found that all that had gone. It was a production line where students were expected to answer the questions as prescribed, there were only lectures and group sessions for discussion only existed to fill in the gaps in knowledge so students could pass the exams.

You can’t achieve a relational understanding by using exams which have multiple-choice questions, nor by using a right/wrong, good/bad binary approach to knowledge.

Professors at universities achieve their positions by passing exams that were set by their professors and so on back up the chain. If you don’t agree with the assumptions made by the professors you will fail the exams, so higher education establishments perpetuate a way of thinking. They don’t encourage dissent and consequently don’t promote relational understanding.

The same applies to schools — probably more so. A packed syllabus results in children having information crammed into them without any understanding of context and no ability to analyse the information they have. I have discussed this previously.

We are producing a generation who are unable to really think in the true sense of the word. Instead they seek more and more information because they are unable to use discernment with the information they already have.

It’s worth noting that as a general rule you will find that people with a relational understanding love discussing and dissent as it stretches the mind. People with instrumental understanding refuse to discuss ideas that they don’t agree with because they lack the ability to question the processes that they have been programmed with. It is endemic in our society to suppress ideas that don’t go along with the mainstream of opinion and Universities that should be in forefront of discussion of ideas instead suppress ideas that are not approved of.

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