Arbitrary vs. necessary in the math curriculum

According to Hewitt, something is arbitrary “if someone could only come to know it to be true by being informed of it by some external means” such as books and teachers. That is, there is no further reason about why it is the way it is, thus anyone who wants to know it needs to be informed by external means. Arbitrary cases include, for example, naming conventions (calling squares, “square” or using Hindu-Arabic numerals for numbers). The arbitrary conventions are based on choices which have been made at some time in the past and they could have been different. In contrast, “the necessary is dependent upon the awareness students already have”. That is, the student should be able to work it out without the need to be further informed.

For a particular lesson, I will see if students do have the required awareness for the subject I am supposed to teach. If so, I will consider it necessary and introduce tasks that help students to use their awareness to work it out. Otherwise, I will consider it as arbitrary and explain the rules or terms.

Knowing what is necessary and what is arbitrary helps me make math more enjoyable for students.  Helping students to use their awareness to work out the math problems would let the student enjoy doing math (that is the most interesting part of learning math) and would not see math as a boring subject consists of numerous formula that need to be memorized.