Response to Skemp's Article - The Balance of Relational and Instrumental Teachings

    This article was published in 1976! I am so amazed that Skemp foresaw the most critical point in the development of mathematics teaching in the current time. Nowadays, relational understanding turns to be more essential due to the rapid development of technologies and an exponential increase in knowledge databases. Students are educated for jobs that have not been created. To add the current situation to the analogy of map of the town, now our students are required to be capable of exploring out of the town all by themselves. However, instrumental teaching is not good-for-nothing. I appreciate that Skemp addressed the advantages of instrumental mathematics (and I like the section was named as devil’s advocate). 

    In my perspective, teachers should apply both relational and instrumental teachings though I did not think this way not long ago. The change happened during my experience of being a private tutor. Students came to me for improving their marks in a limited time. Additionally, their parents were worried about they did not get enough practice and thus asked for more homework. I was expected to teach instrumental mathematics. Painfully, as a believer of pure relational understanding at that time, I had to meet their requirements -- they paid me for these. My blue days ended when a student approached me and claimed of creating a new formula -- the sum of roots to a quadratic function is -b/a. He generated it based on the worksheet for the axis of symmetry of quadratic equations. Although I had to tell him it had already been introduced by Vieta hundreds of years ago, I started to reconsider the role of instrumental teaching. A good analogy of the relationship between instrumental and relational understandings is the regression model: the more the data is collected, the better the fit is found. I believe that to reach a relational understanding, students rely on instrumental practice. Thus, teachers should balance the use of relational and instrumental teachings to help students to become autonomous learners ultimately.