A commentary on ``Relational Understanding and Instrumental Understanding'' by Richard R. Skemp

A commentary on
``Relational Understanding and Instrumental Understanding'' by Richard R. Skemp

In his paper titled ``Relational Understanding and Instrumental Understanding'', Skemp identifies two faux amis in the context of mathematics: 'understanding', where 'understanding' could take on either of the two meanings: 'relational understanding' and 'instrumental understanding', and the word 'mathematics' itself where the 'mathematics' could refer to two different subjects: 'relational mathematics' and 'instrumental mathematics'.

Skemp's statements are based on his observations as to how the subject matter is being taught and learned in schools. His viewpoint is definitely in favor of relational mathematics, where a student is taught not just what to do to get to an answer or how to solve a problem but also why. This is in contrast to instrumental mathematics where the student is taught a set of rules or formulas without reasons. He backs up his viewpoint by listing four advantages for each method of teaching/learning but at the end, he still votes in favor of relational mathematics, saying: ``If the above is anything like a fair presentation of the cases for the two sides, it would appear that while a case might exist for instrumental mathematics short-term and within a limited context, long-term and in the context of a child's whole education
it does not.'' He also goes on to further say: ``So nothing else but relational understanding can ever be adequate for a teacher.'' I fully agree with his arguments and support his conclusions. For individuals to become 'agents of growth', they must develop an understanding of the basic underlying principles. It is through this understanding that they can dig deeper, explore further, make connections between two seemingly different concepts, and become involved in the creative process. Although Skemp uses mathematics as a particular subject area to discuss, I believe the concept is easily carried to other subject areas such as 'physics', 'chemistry', and even 'history'. Its through critical thinking, teaching, learning, and having a content-centered approach to teaching and learning that any of the above subjects can go from being an instrumental subject to a relational one.

Skemp also discusses the possible 'kinds of mis-matches that can occur exist in mathematical teaching/learning'': Instrumental learner and relational teacher, relational learner and instrumental teacher, and instrumental teacher and rational book. I actually could think of another possibility that he has left out, namely, relational teacher and instrumental book. Of all the above mis-matches, I believe the ones with instrumental teachers are the most serious; Skemp call is more ``damaging''. If a teacher's intention, beliefs, knowledge and skills favor relational teaching style, he or she is able to see beyond the limitations of the required text book and reach the few students who really want to understand rationally.

I loved the distinction Skemp makes on the two kinds of simplicity: ``that of naivety; and that which, by penetrating beyond superficial differences, brings simplicity by unifying. It is the second kind which is a good theory has to offer, and this is harder to achieve.'' I would say 'relational teaching' leads to the latter kind of simplicity. i.e. the ability to connect and find the unifying theme. This is what I call ``the joy of understanding'', when the ``Ah-ha!'
' moments happen.