This isn't directly related to teaching, but I bet I can find something relevant to say about it.
As of the past two weeks, I've been taking a free, online college course called "Principles of Economics for Scientists." I'd somehow managed to never take any sort of Econ course before, and this one is a great response to my hesitation about the subject relying too much on discussion of human nature and the sociology aspect of things. (While I do respect both of those approaches as valid, neither of them would have helped me make intuitive sense of this new subject.) As it happens, the "for Scientists" part of the course title roughly translates to, "for people who know and understand Calculus", which squarely includes me.
(Ooh, I figured out the connection to teaching and learning!)
In this class (at least so far), the professor is using Calculus as the lens through which he is teaching the fundamentals of economics. Therefore, for the intended population of students, there is a basic level of math needed in order to comprehend the material of the course. We are not doing math just for math's sake; we are using it as a tool for something else. Without a conceptual and practical understanding of calculus, the professor would have to take a more convoluted approach to teaching the exact same topics, probably resulting in students appreciating the material less.
This is equivalent to the pedagogical idea of "learning to read vs. reading to learn," the transition that all Language Arts teachers encounter in their respective grade levels. Once students can physically read a text out loud, what do they get out of it? How will the skill of reading translate into a truly functional literacy?
These same questions can be asked of math: "learning to do math vs. doing math to learn." Many secondary math classes spend all of their time teaching and learning sequences of math skills that build upon each other, with the promise that students "will need to know this later on." But for students that have no desire to continue in math or science classes, that later time never comes. I see myself as a perpetual student of math, but I've realized that while some math is universally necessary in participating in adult life, and other math is elegant and interesting in its own right, some topics are useful not just for the sake of doing math, but rather as a tool for understanding other topics, spread far and wide across the human quest for knowledge.
Every math teacher gets asked the question, "Why do we have to know this? When am I ever going to use this?" In response, think further than just the upcoming test, and even further than the next few courses in the school's sequence. Math is pervasive, and we need to convince our students that there is a bigger picture to the subject, that stretches far beyond the classroom.

Life update, since the last post:
- Completed my Math major.
- Started and finished my student teaching (MS and HS), and my Secondary Education major.
- Graduated from college =)
- Became a certified Secondary Mathematics teacher in two states so far, working on the third.
- Taught my two summer classes for the 2nd year, and got asked back for my 3rd 2013 summer.
- Currently working as a full-time math tutor at a HS in urban Massachusetts, and job-searching for next year.

Hoo boy!

What's been on my mind lately:
- Do I want to start my classroom teaching career in an urban, more difficult-to-manage school, or in an "easier" suburban environment?
- How will I measure the success of my students mid-year and at year-end?
- What can I do to increase my students' retention of math content from week to week, other than just in basic skills?
- What else can I do this school year to improve myself as an educator and thinker, and make myself a more marketable teacher candidate?