Arbitrary and Necessary

I found this to be a very interesting read. The distinction between the arbitrary and necessary was never really something I thought about before and being made aware of the realms into which these fall, I feel that I am better prepared to teach mathematics.

Being able to determine what is necessary allows me to direct to appropriate topics activities geared towards students using their awareness to understand.

In my experience, I was able to take what Hewitt calls received wisdom and using my awareness, transform this into necessary fact. I fully understand that this is not the case for all or even most students so I must be careful to give activities that open the students’ minds to the concepts and builds on their current awareness.

I must also note that if i choose to take an approach and accompany the received wisdom with an explanation, it is not productive to base my explanation on anything that falls into my awareness but not theirs. As Hewitt states, “A teachers’s explanation is often based upon the teacher’s awareness, and so may use things which students do not find evident - things which are not in the students’ awareness - and so the explanation will not be one which will help those students to educate their own awareness.” As I mentioned in an earlier post, “As I work to become a teacher, I must not forget how a student thinks. Put myself in their shoes and remember what it is like to not know.”

A question I have is whether informing students of necessary mathematics content (received wisdom) can be a preferred method over the alternative in which the teacher works within the realm of awareness.