In my last post I proposed a possible introduction to order of operations.  It was by no means a complete planned lesson, and neither shall this post be. What happened after, I posted on Twitter "how do we teach order of operations as a concept, rather than a rule to be memorised?"  My goal was to teach students a reason why multiplication precedes addition, rather than simply saying...

Or is that all we can do? It is a convention right?

The first thing I thought was that the order of operations are not simply a random guess in the dark convention.  I do not imagine the world's greatest mathematicians developing this necessary convention by the toss of a coin.  Rather, I expected that there is some reason the great minds of the past did in fact decide that there was an optimal way to follow the procedure of calculation now infamously known as BEDMAS or PEMDAS depending on the mnemonic device you have learned.

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?

My algbraic understanding moves me to discuss terms.  The standard convention makes sense in my mind when we discuss terms: 6x+6y.  I know that whatever x is, I have 6 of them, same with y, but since I do not know the values of x and y, therefore I cannot complete and simplify this term any further, because I am not given enough information. 6x+6y is necessarily simplified at this moment.

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?"

I think I just heard @fnoschese cheer, but it really comes down to understanding our models of addition and multiplication. When I think of my conceptual understanding of addition, this comes to mind...

That is, addition is largely a combining operation.  I have this many things here, I have this many things there, I bring them together and TADA!  However when I think of multiplication, I think of this...

When I think of multiplication I often think in groupings.  I do not take a quantity and combine it (like addition), but I create a quantity (seemingly from nowhere) via grouping.

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.

This is not an airtight argument for multiplication to necessarily precede addition, but since order of operations is a convention it need not be.  However I think this is my first fruits of grasping why this is a well defined convention, not a coin flip.  If I am uncertain of which order to use when given the problem 3x4+2x5, it makes sense to me that I cannot do any addition until I am guaranteed to have separated values that I can combine, multiplication doesn't have that requirement, so logically I can start with that multiplication.

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...