Hewitt Response

(I'm all over the place in this response, I apologize. I had a hard time maintaining a continuous stream of thought for this one.)

I appreciate that Hewitt makes a distinction between the arbitrary and the necessary. Sometimes, I have the initial reaction as the author did to say, "Because it just is" when asked about an arbitrary point in mathematics, such as positive rotations being counterclockwise. It's another convention decided upon within mathematics, and I remember one of my professors stating it as such. The one practical issue which arises from this article relates to classroom management. My understanding is that Hewitt prefers an inquiry-style of teaching over a direct-instruction one, as direct-teaching tends to create more overlap of teaching both arbitrary and necessary information, whereas inquiry-style learning tends to lean toward using arbitrary information and allows students to determine necessary information for themselves. Initially I wondered about students who find it sufficient to trust a result without proof, and how to use inquiry to engage them. However, I am learning that there is much more to inquiry than previously expected; I have also had the opportunity to see various types of inquiry work, including some IB (International Baccalaureate) assignments. One of the assignments involves calculating a side length based on sine ratio of an equilateral triangle located in a circle, followed by finding the area of the associated triangle, followed by finding the area of two other regular polygons in the circle, then generalizing to n-sided polygons. I would like to design a few assignments like that, just to get a better understanding of inquiry. I have not yet designed a lesson based on inquiry. A lot of my work tends to be teacher-focused; however, I am working on changing this.

Back to Hewitt's article; I also think that distinguishing between arbitrary and necessary avoids misuse of memory aids like, "two negatives make a positive". I've seen multiple abuses of the "rule", but if students were able to follow through with a brief proof, I feel that this could be much better understood. For example,
Dr Math provides an excellent explanation on their website:

Let a and b be any two real numbers. 
  Consider the number x defined by

      x = ab + (-a)(b) + (-a)(-b).

  We can write

      x = ab + (-a)[ (b) + (-b) ]       (factor out -a)
        = ab + (-a)(0)
        = ab + 0
        = ab.

  Also,

      x = [ a + (-a) ]b + (-a)(-b)      (factor out b)
        = 0 * b + (-a)(-b)
        = 0 + (-a)(-b)
        = (-a)(-b).

  So we have

      x = ab
  
  and
  
      x = (-a)(-b)

  Two things that are equal to the same thing are equal
  to each other, so 

      ab = (-a)(-b)

(Source: http://mathforum.org/library/drmath/view/64406.html)

Also, suppose a teacher is instructing part of the lesson, and a student asks why a certain rule applies, perhaps one which would have been covered years earlier. At what point can a teacher, if ever, say that something is "the rule"? I know as an undergrad, I was often extremely frustrated when a professor would say, "You should know this by now", because they rarely ever continued with, "But if you don't, here's where you can find how to do x, y, and z." Because of such experiences as an undergraduate student (whether in math courses or others), I tend to over-explain things when I teach, and I was flagged for it by my school advisor when I conducted my two-week practicum last year. I remember a sea of blank stares in a classroom when teaching part of the rational expressions unit; I had questions arising over why I needed to multiply both the denominator and numerator of a fraction in order to add it to another one (in a grade 11 precalculus class). Although I explained it on the spot, I also wonder about the remainder of my students; if I had waited to answer the question, I imagine the student would've understood how to simplify the remaining part of the question, except for that one line of finding common denominators. What about one student who looked absolutely bored as I was going through the explanation of how to add fractions? Next time, can I open that question to the class? Would that have been a better thing to do? Perhaps it would've maintained attention a bit better. 

Finally, Hewitt states, "A teacher taking a stance of deliberately not informing students of anything which is necessary is aware that developing as a mathematician is about developing awareness rather than collecting and retaining memories. Furthermore, this stance clarifies for the students the way of working which is appropriate for any particular aspect of the curriculum - the arbitrary has to memorized, but what is necessary is about educating their awareness." This makes me wonder: How many students know a single of the (over four hundred) proofs available for the Pythagorean Theorem? Yet how many can use it effectively throughout their high school mathematics career, onwards? Do students need to be able to explain every math result they encounter? I wonder, is there time to do so for every particular topic of study? How early on can you begin with inquiry in mathematics, or to acknowledge conventions? Wouldn't it be confusing to students to say in kindergarten when first learning numbers to say, "We use the Arabic number system because it is the most useful to use in calculations and we can establish place value based on the order of the numbers, but had we been in another country, our numbers would've been based on a placement of horizontal and vertical bars"?