Hewitt article: Arbitrary and Necessary

      Hewitt defines arbitrary as something someone could only come to know it to be true by being informed of it by some external means - whether by a teacher, a book, the internet, etc...  If something is arbitrary, then it is arbitrary for all learners, and needs to be memorized to be known.  Necessary in this context refers to aspects of the math curriculum where students do not need to be informed - things which learners can work out for themselves, depending on their respective realm of awareness.
      For any given lesson, if I believe that it will save the students time in the long run, I will teach a topic as arbitrary as opposed to necessary (even though less 'math' appears to be involved at the outset).  As interesting as it would be to have students devise their own nomenclature when learning something new, imagine the challenges they would encounter attempting to communicate with the rest of mathematical culture subsequently.  My justification for this teaching style comes from an experience I had as an undergraduate.  My Number Theory professor had an assigned text for the course (A Classical Introduction to Modern Number Theory) and a recommended text (An Introduction to the Theory of Numbers).  During the course, I learned from the required text only due to time constraints.  Later on, I was keen on learning more so I looked up G.H. Hardy's book.  The issue was that Hardy's text was written over a half-century prior to the one I was familiar with, even though similar topics were contained within.  I found myself doing much less math while reading the Hardy book no due to lack of material contained within, but simply due to notational differences.  I had to create a rudimentary translator so that I could follow along while reading the theorems.  Once naming and symbolic conventions have been established, it's prudent to adhere to them so that you are free to focus more on building mathematical knowledge using those conventions.  If time permits and I am confident that, if my students' realms of awareness are sufficient, I'd devise an activity (perhaps involving appropriate givens) to have them derive a concept (necessary); I would provide guidance afterwards to ensure uniformity.  Either way, students maximize their mathematics in the classroom, either by using conventional terminology to solve problems or by deriving new concepts using their current cognitive abilities.