Relational and Instrumental Understanding - Response to Skemp Article

We were given an article by Richard R. Skemp to read outside of class. The name of the article was: Relational Understanding and Instrumental Understanding. The paper was first published in 'Mathematics Teaching', 77, 20-26, (1976). Wow that is old. When he wrote this, Richard R. Skemp was part of the Department of Education at the University of Warwick.

A quick breakdown, Instrumental Understanding is understanding without knowing the why behind whatever you are learning. You are given a tool or equation to solve a problem and then you use that tool to solve it. Relational Understanding is understanding by relating the topic to things you already know. Using reason and deduction to understand, giving the person a tool box of knowledge to solve a problem.

So, the assignment was to read the 15 page paper, select 5 quotes and respond. Here it is, enjoy thoroughly!

As a student, when it came to math, I would categorize myself into the relational understanding group. But it was not as if I would not do well in a topic if it was taught instrumentally. Whenever that occurred, my brain would just store that information somewhere in my head where it could churn until eventually connections would be made and the ‘why’ behind the topic revealed itself on its own. In high school, those connections rarely took much time. However, I realize that not everyone is math orientated and capable of making rational connections on their own and therefore require a teacher to show them.

“…for many pupils and their teachers the possession of such a rule, and the ability to use it, was what they meant by ‘understanding.’ – page 2, lines 11-13

As Skemp implies, memorizing does not equal understanding. In math, a teacher could give out formulas and rules and if a student managed to memorize the two he would probably do well on an exam. Not great, but well. To many, this result would be satisfactory, but this should not be. Pure memorization rarely becomes long term knowledge with the formulas and rules trickling away from the students minds after the course in question has ended. Little do they expect the hole they are digging for themselves. As math progresses and gets more involved each year, more formulas and rules come into play which overlaps what was learned previously. The result: overwhelmed students who dread math.

“I used to think that math teachers were all teaching the same subject, some doing it better than others.” – page 6, lines 17-19

“…there are two effectively different subjects being taught under the same name, ‘mathematics’.” – page 6, lines 20-21

Skemp’s article impressed me since it really made clear the difference between rational and instrumental understanding and reinforced the importance of rational understanding. He made me take another look at myself and realize that lots of my knowledge remains at “the intuitive level” [page 13, line 5] and the importance of taking that extra time to readdress those topics.

“…if people get satisfaction from relational understanding, they may not only try to understand relationally new material which is put before them, but also actively seek out new material and explore new areas…” – page 10, lines 22-26

Instrumental understanding is very one dimensional. Facts just are and there exists no depth to their meaning. Relational understanding fills that depth and lets the student know that there is indeed a purpose or a reason to what is being taught. Understanding often equals interesting and when something is interesting desires to learn more often arise.

“From the marks he makes on paper, it is very hard to make valid inference about the mental processes by which a pupil has been led to make them; hence the difficulty of sound examining in mathematics.” – page 12, lines 4-8

Skemp makes a good point; it is very challenging to discover all of the learning styles of every student in the classroom. Even if you did discover it, could you accommodate every learning style? From reading this article I would conclude that relational teaching should be used since it accommodates both types of students and hopefully the instrumental students would one day discover there is more to math and desire to know the reasons behind the formulas.

“…learning relational mathematics consists of building up a conceptual structure from which its possessor can produce an unlimited number of plans for getting from any starting point within his schema to any finishing point.” – page 15, line 36-page 16, line 1

Relational teaching best benefits the students. In math, if a student understands the reasons and processes, he can more easily apply his knowledge when problems he has not seen before arise. Also, there may be more than one method to solving a problem and relational understanding would allow the student to possibly find those paths and reach the desired solution.