September 10, 2007
Response for “Relational Understanding and Instrumental Understanding”
In this article Richard R. Skemp examines relational mathematics/understanding and instrumental mathematics/understanding. He defines relational and instrumental understanding and discusses the advantages and disadvantages of both, while drawing on many examples. Overall, I agree with what Skemp argues about these two types of understanding.
Before reading this article, my definition of understanding was knowing how to perform a certain task and why that method worked for performing that task. Therefore, I considered understanding to be relational understanding. I never considered instrumental understanding – just knowing how to do something – as a definition of understanding.
From my experiences as a student and volunteering in high school math classrooms, I have come to the conclusion that instrumental mathematics is the form of understanding mathematics that many high school students use. This form of understanding allows students to answer questions quickly and in many cases perform well on exams – which is the goal of many high school students. Furthermore, many of the questions on exams are very similar to the lecture notes or homework questions the students have worked on; therefore, they just follow the same method from notes and homework assignments to answer test questions. I think Skemp is accurate in saying that students who do not have a relational understanding of certain mathematical concepts have a difficult time answering questions that vary from the questions that follow the “rule” for that particular concept. I found this to be especially true in junior level math classes. Moreover, I think it is accurate to say that relational understanding is a difficult form of understanding to always use because it does take longer to tell students why a certain method works than just telling them how it works. Also, with the amount of topics to cover in a year it is difficult to spend a great deal of time on each one.
Skemp’s article is informative about relational and instrumental understanding and it gives excellent examples about each form of understanding.
In the past the author used to describe instrumental understanding as “rules without reasons” (Skemp, 2).
I liked this quote because in very few words it summarizes basically what instrumental understanding means. This definition also makes it easy to distinguish between relational and instrumental understanding.
“If the teacher asks a question that does not quite fit the rule, of course they will get it wrong” (Skemp, 4).
I thought this is a very good quote about how, unlike relational understanding, instrumental understanding makes it difficult for students to work on and solve questions that vary from the questions that follow the “rule” that they were taught. I think this quote is very true because while volunteering in high schools I experienced many incidents when students had no idea how to do certain questions because they were slightly different from the examples they got during the lecture.
“If pupils can get the right answers by the kind of thinking they are used to, they will not take kindly to suggestions that they should try for something beyond this” (Skemp, 5).
I found this quote to be very accurate because many high school students are mostly concerned about getting answers on assignments and exams correct. They feel that since they are getting answers correct by using the method that they are using then there is no need for them to do extra work and find why the method works or to find another way to solve the problems. Therefore, like this quote indicates relational understanding is not extremely important to all students because in some cases they can get answers correct without it.
“Relational understanding would take too long to achieve, and to be able to use a particular technique is all these pupils are likely to need” (Skemp, 11).
I found this quote to also be very accurate. There are a great deal of topics to cover in a mathematics curriculum; therefore, it is difficult to spend a lot of time on one particular topic – and in many cases it takes a lot of time to explain why a certain method works for a particular problem. Additionally, on in-class and provincial exams students just need to know how to use a certain technique to solve a problem. Therefore, teachers have to teach the techniques and sometimes they run out of time to get the students to understand why the techniques work.
“In view of the importance of examinations for future employment, one can hardly blame pupils if success in these is one of their major aims” (Skemp, 11).
Doing well on exams is very important to students because report cards and entrance into university are based on marks; therefore, getting questions correct is all that matters. In many cases students “cram” for exams and, thus, they only have time to learn the techniques needed to solve the problems – they do not have the time to learn the reason why the technique works for the certain problems. Hence, I think this is a good quote in explaining why it is sometimes difficult to use relational mathematics.