All quotes are from “Relational Understanding and Instrumental Understanding” by Richard R. Skemp.
Page 8, “These children need success to restore their self-confidence, and it can be argued that they can achieve this more quickly and easily in instrumental mathematics than in relational.”
I feel this is very true and that the instrumental method is a way of getting the quick start or kick start that is needed to capture student interest from a generation of students brought up in the “instant pudding” way of doing things. It is then up to the teacher to show that there can be limitations and that another way to find an understanding is to see that there is a deeper relational way to understand mathematics that can be carried over into everyday situations. It is up to the teacher to relate the mathematics to the relevant with examples of how this can be used in our future lives and jobs. This method is like climbing stairs, that is the instrumental gets us started, and the relational takes advantage of the momentum to keep interest and further enquiry as to how this can work for the student.
Page 11, All of these imply, as does the phrase ‘make a reasoned choice’, that he is able to consider the alternative goals of instrumental and relational understanding on their merits and in relation to a particular situation. To make an informed choice of this kind implies awareness of the distinction, and relational understanding of the mathematics itself.
A cycling of instrumental and relational can be beneficial if the instrumental eventually favours using the firmly grounded rules of mathematics more effectively. In our example of the dividing by a fraction, “invert and multiply”, I would use the rules of Identity to show how powerful it can be in solving this problem in a more relational way, that when learned will make solving many types of mathematical problems simpler. The instrumental may need to be used to get students started but they should be helped to see the path that relational understanding through the implementation of effective use of the Laws of Mathematics. There should be some dwell time to allow students to see that they can go deeper into long lasting understanding compared to the shorter term dissipative instrumental rules that may be forgotten. These can be memory jarring stepwise ways to remember the path from instrumental to relational understanding. I think that there could possibly be a third element that will make this teeter totter into a triad of understanding and that is the usefulness of the information to the learner. This means that the teacher must show that learning this information will have an effect on the potential occupations available to them, thus life and living goals can be achieved.
Page 14, And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’. I agree with this statement and find that it does not conform to my ideals of a good teacher, which is one that has the student learn how to learn so well that you are no longer needed for that material level anymore. The student becomes independent and the teacher is validated with a new student that needs the learning understanding to help them from instrumental to relational understanding and finally to understanding the value of knowing how to apply the knowledge they have learned and how to continue the process on their own.
Page 15, “but what constitutes mathematics is not the subject matter, but a particular kind of knowledge about it.” This statement needs to help the student to see that there are many more ways to look at the understanding of mathematics and these may be helped by new and still to come technology that allows students the time to view the visual interpretations of mathematics. By introducing the senses of visual, 2D and 3D as well as new interpretations of visual and audio, representations of what mathematics is and how to understand it and what it may be used for will expand the knowledge of mathematics in many new ways in the future.
Page 16, “One of these is the relationship between the goals of the teacher and those of the pupil. Another is the implications for a mathematical curriculum.
I agree that there needs to be some time devoted to the attainment of goals and career so that the teacher may show the pupil that a good foundation in mathematics learning will help in any future goals.
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