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...

I have never introduced students to order of operations.  I have always received students after their initial introduction.  I always feel like receiving students after their rote memorisation of BEDMAS (or PEDMAS if you call 'em parentheses, or are very apologetic about your kin), means that when I try to let them explore they just default to what they memorised, but they cannot recall that after the lesson.  It makes me feel stuck between a rock and a hard place, but I still wonder what the initial introduction can look like.

One introduction that I have just thought of this morning, I'd like to try.  It's heavy on symbols, which I am afraid of, but I think it is basic enough that it might spawn some conversation and debate.  I just want to share it and see what you all think. I like it because I feel it introduces students to some manner of proof, and following logical principles, and I think it would be very interesting to see students' misconceptions about the order of operations.  Here goes:

Premise 1
Ask the class what is 5+12?  What is 12+5?
Can we agree that 5+12=12+5?

Premise 2 
What is 3x4? Can we agree that 3x4 is another way of writing the number 12?

Conclusion
I am going to write something on the board.  Tell me if you Totally agree, somewhat agree, somewhat disagree, or totally disagree...

5+3x4=3x4+5

Have students get in groups and say, "give me a knock down reason why you think you are right."

The agree-er- This student believes it, and would focus on the fact that 3x4 is another way of writing 12.  This student would be good to play devil's advocate with.

The choose your own adventure - This student says, "If you decide to multiply first it can work, if you work left to right it doesn't."  This is pretty easy to counter, ask them to try both ways on just one side of the equation. You get that 5+3x4≠5+3x4, "That seems ridiculous doesn't it? So which one should we choose?"

The reader - We read a book left to right, so we read math sentences left to right as well. This student gets that we need to have a common way to solve all math problems, but does not quite understand the separate operations.  (This is the student that I need help with!)

This is where I have never been and wonder how worthwhile this exchange would be.  Would students logically piece it together? Would most students be an agree-er, a choose your own adventure, or a reader? How would I encourage each of these groups to prove themselves and come to a consensus?  What questions would you ask your class in this discussion or is this just a bad introduction?

I have been thinking that I would much rather introduce Order of Operations, as terms rather than BEDMAS.  This would prepare them for algebra, and would provide a more conceptual understanding of the process.  I do not know what the progression would look like, but I assume that beginning with multiplication and addition would be the natural place to start.  Where would you go next? Subtraction?  Maybe you are thinking, "Timon you are CRAZY, this is HOW you should do order operations." Please tell me about it; I want to know.

This is easily one of my favourite problems that I have come up with.  Mainly because of the back story.  I handed a student this calculator, and he told me that it didn't work.  The numbers weren't working.  I showed someone else and they decided to throw it out, but I couldn't help but think that something more than broken buttons was the problem.

So ask the students: "What is wrong with this calculator?" or "What is this broken calculator going to give us for '433+233'?" 

Okay full disclosure here, this is not really what I want to give my students, but as a low tech version (and one that you can use as well), I have made these...

What I really want students to do is to explore their own numbers and find patterns on their own.  In order to do this, I want to program a base 5 calculator that kids can use on the school netbooks, BUT I don't know how to program.  If anyone has ideas about how I could put this calculator into my students hands without telling them that it is a different base please put them in the comments.

Update: The awesome Jed Butler programmed this amazing base five calculator on Geogebra. If you are not plugged into the MTBoS, you need to get on that!

The other option is I just put my calculator under the document camera and have students ask and record class wide.  That doesn't help you guys though, so this is what I have started with.  If you think I need some more/better examples please tell me in the comments and I will make them (groups of four look nice).

Update: I have also begun having students exploreJames Tanton's Exploding Dots. This is a great intro into different number bases, and really stretches students, but is not completely out of the range of grade seven students. So much love for this.

This is a pretty pure mathematics WCYDWT so I can only think of standard sequels. (Please give me more ideas in the comments, these are pretty lame).

• How does multiplication work in this number system?  Can you find some easy methods for solving basic multiplication statements?
• Pick a random base (2,7,12,4.5(?), 16), and create some problems, and share them with a partner.  What is different and similar among different bases?
• From @trianglemancsd How would you represent 1/2, 1/4, and 1/10 as a "decimal" number? What does 1.3, 1.021, and 0.033 become as a fraction? (All sorts of headaches happen here, clarify a fraction in base 10 or base 5; what does 1/10 mean?
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