Response - Dave Hewitt: Arbitrary and Necessary

Arbitrary is something that cannot be worked out. It is in the realm of memory where students must be informed of this information. A square can just as well be named a "syervel" or any other name. There does not seem to be a definite reason why it is named a "square". Such knowledge is arbitrary.

Necessary is something that can be worked out. Students can use their awareness or prior understanding to figure out a mathematical fact. For example, they can determine the position of a quarter-turn and a half-turn in reference to some point of origin.

For a math lesson, the larger part of the time allotment should focus on working out the necessary mathematics. Less time should be spent on practicing the arbitrary. This influences my lesson plans in that the teacher should have a shorter amount of time talking about what is arbitrary: the names, symbols, notations, and conventions, and the students would have the guidance to work out what is necessary: the properties and relationships in mathematics.