Arbitrary and Necessary

This article is one of the finest articles I have read so far about teaching Mathematics. The author says it very clearly that arbitrary is something that students have to be informed by an external mean; whereas, necessary is something that students can become aware of without any effort made by an external source. Arbitrary part of your lesson is that is very definitive and cannot be changed as it has been set by someone a long time ago. For example, teaching students about locating hypotenuse and the opposite side on a triangle is teaching them arbitrary part of the lesson. Necessary part of the lesson would be something that students may discover on their own based on all they knowledge that is given to them. For example, students may figure out on their own that when doing fraction cancellation, one may start with any common factor first and continue cancelling instead of always starting with the biggest common factor.

This article was an eye opener for me in terms of thinking, before teaching or even planning, about what to put in your lesson and what to leave for the students to figure out on their own. My lessons so far have been very heavy on examples. I will  be teaching one Math 8 challenge class which I think will benefit from only learning arbitrary things from my lesson and it would be better if I leave necessary things for them to figure out for themselves. As the author mentions that if a teacher decides to teach something that is necessary then a student may take it as a 'fact' and decide to memorize it. Whereas, if this concept is taught through awareness, which is by letting them figuring it out on their own, it would sit in their minds as a necessary concept that they came up with on their own and they do not need to memorize it.