Arbitrary and Necessary

    This article reminded me of the first assigned reading on relational and instrumental teaching written by Skemp.  I believe that arbitrary mathematical content requires memorization on the part of the student and is primarily instrumental teachings on the part of the teacher. Necessary mathematical content, on the other hand, is about student awareness and is related to relational teachings.

    Since I did not study in English-based education before 18, remembering and understanding math conventions have been a challenge for me at times. Therefore, when planning my lessons, I always try to see the content from the students' perspective first, especially when introducing arbitrary knowledge. As mentioned in the article, covering the history of the content is a good strategy; however, some terms are chosen at random when they were first introduced into an English context. In this case, I think it is helpful to introduce history from other cultural backgrounds. Stories like the mistranslation of rational numbers that I shared in the presentation could help students memorize the mathematical inventions or provide at least something fun to listen to in a math classroom or to share with others later on. 

    As for teaching necessary mathematical content, holding activities to assist students in freely exploring or playing around with math is critical. When planning unit plans, I always try to include at least two activities for a topic. During my short practicum, my SA also inspired me a lot. I am fascinated by Grade 8’s explorations of shapes in 3D in the Halloween cubes activity. Here is a picture of their awesome works!

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.