Hewitt : Arbitrary and Necessary

I quite enjoyed Hewitt's article as it brought up some new perspectives on why and how we teach mathematics.  In particular, the idea of received wisdom caught my attention and it was explained clearly in the following example,

"If a teacher stated that the angles inside a triangle add up to half a turn rather than offering a task for students to become aware of this, then students are left with having to accept what the teacher says as true.  In this case, it becomes just another 'fact' to be memorised.  I call this received wisdom."  

This particular example stood out to me because I would have to believe that most mathematics is taught this way, or at least it was during my highschool career.  Providing this 'received wisdom' or arbitrary content simply just provokes a battle of who can memorize the best.  As Hewitt explains, there are arbitrary facts that we simply cannot avoid the need for pure memorization, such as the x before y explanation, or names of certain shapes, however when necessary facts are explained in an arbitrary way, it becomes a problem because then we aren't teaching mathematics at all.  While I tend to agree with what Hewitt was saying, I question that it's much easier said then done.  It is easy for me to see how a teacher can assist in memory, but it is hard for me see how a teacher can educate awareness.  It seems to me that in educating awareness, the teacher is barely doing any educating at all, rather they are just facilitating or directing?  The students are simply figuring things out for themselves?  This concept confused me a bit.  

I would like to see more examples of what Hewitt is doing to get a better understanding of the differences between teaching arbitrarily and teaching awareness.  The idea of received wisdom is definitely something I will keep in mind and I hope to find instances in my lessons where I can stray away from it when it is not needed.