Conventions can still be conceptual
For me the order of operations has always made sense in terms of algebra, and that is exactly what Dr. Math (referenced in paper) claims. Well, great, I want to teach students order of operations conceptually, but they do not know algebra yet, what can be made of this?
Terms are the cornerstone
This is a great way to think of the operations, if you know the basics of algebra, but when we teach order of operations, students do not have a strong (if any) concept of a variable and especially like terms. My struggle over the last few days has been this: "How do I reconcile my algebraic understanding of the order of operations within arithmetic?"
Models, models, models
The point that expresses this for me the most is when we look at the number line. With addition we start at a particular number, and then move onward. With multiplication we start at zero! Multiplication is an ex nihilo process, we get something from nothing. With addition we combine two quantities that are there, whereas with multiplication we simply start making groups. I think this is the "multiplication is a stronger" operation argument, but I just had to put it in my terms.
Putting it all together, and bringing it back
I'd like to hear all of your thoughts on this though. Is there a point to all of this or should we just say, "Bedmas everyone! Look it's bedmas!"? Do you want to get mad at me for my lame models or start the war on multiplication is not repeated addition? Let's talk this through so we can make the order of operations a less daunting, more natural task for students...