I'm afraid I'm not expressing myself well. Iris, I think we could have an enjoyable discussion about this if we could talk in person. My limitations in expressing myself through text are clearly coming through, because I am getting frustrated at the fact that what I'm trying to say is apparently not coming across well to you and several others.
I'm going to try one last time, not because I want the last word, but because I want to try to accurately convey what I'm trying to say. Also, WillyNilly please also forgive me for using you as a convenient example. You are obviously an intelligent, well-spoken individual and I don't want to make you feel insulted in any way.
My philosophy of math education is that throughout the years, generations of teachers have presented topics without any real sense of context. They push "rules" and "procedures" without teaching why they work or how they are built up from topics that came beforehand. This leads to students who either don't get it, or forget fairly basic concepts shortly after they're taught. Even in more recent generations, with the push to teach the "why", too many elementary teachers just don't know enough themselves to really lay a solid foundation. Occasionally, teachers even come up with "whys" that are completely wrong (like teaching students that multiplication is repeated addition), which leads to confusion and mental brick walls when students have to build on prior knowledge.
I see my job as a math teacher is to "make it real" to my students, while at the same time laying, or re-laying the necessary foundation for my students to build upon in higher level math. As such, I've spent a lot of time working out real-world analogies for everything from basic arithmetic to Calculus, Linear Algebra and beyond (remember I've also taught at the community college level, and have tutored upperclass students). I feel like a failure if any of my students leave my class thinking that math has all these rules that are arbitrary and don't make sense. Every single rule or algorithm I've ever taught has been accompanied by either a real life example or a build up from something that came before it.
WillyNilly's case in particular frustrates me. Here she is arguing that the "rule" is not intuitive, yet every single example she comes up with shows an intuitive understanding of numerical reasoning. This frustrates me on two levels. First, I don't like seeing an obviously intelligent person thinking she's "bad" at something, when clearly she's not. Somebody, somewhere along the lines failed to connect the dots for her, leaving her with this idea that something doesn't make sense. It's a disconnect between the notation and the application. Why should I care about this? This leads me to my second frustration. In the grander scheme of things, much of the public is of the opinion that math is difficult, mysterious, arbitrary with no real connection to real life. This attitude leads to the idea that being bad at math is acceptable, which leads to even more students giving up or not caring. Basic mathematical literacy should be just as important as basic reading. No sane person would ever make small talk with a person at a party by saying "you know, I just can't read. I've always been bad at reading." We should, as a culture, put the same importance on basic mathematics, including notation. That notation is analogous to the rules of phonics, not more complex grammar. We don't do that though.
Every post I've made in this thread has been an attempt to show that this particular concept, the order of operations, doesn't have to seem arbitrary and confusing. That's all I've wanted to show.