The arbitrary part of mathematics is the language and definitions we use. It is the stuff we invent for the purposes of communication. For example, Hewitt says that we choose to write thirteen as "13," but there are other equally valid ways to write it if you wanted to. 13 is just the widely recognized and accepted form. Arbitrary means it is something that must be memorized in order to be known. Necessary mathematics, on the other hand, can be worked out independently. Students need to be aware enough of to accomplish it though. To decide if something is necessary or arbitrary, we have the think about whether or not you could figure it out yourself, or represent the exact same thing in a different way.
This article made me think about what questions I ask. Am I asking students to regurgitate a arbitrary fact, or am I asking them to work something out logically? When I think about that, then I can provide students with the appropriate question. Hewett's example was a teacher getting a student to "think about" the name of the number that occurs most often, when that question had no thinking involved - either the student remembered it or not.
I really want to implement a lot of inquiry based learning in my math classes, and I think focusing on the necessary questions, where students get to work things out themselves, is definitely my aim.
Good!
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