Moving Towards a Conceptual Understanding of the Order of Operations

8/16/2012

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?

Conventions can still be conceptual

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?

Terms are the cornerstone

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?"

Models, models, models

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.

Putting it all together, and bringing it back

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...

9 Comments

Mike C

08/16/2012 10:42pm

I'm sure you've seen this, but just in case...

http://mathforum.org/library/drmath/view/52582.html

It traces the development of the consensus. However, I kind of agree with @Trianglemancd. Knowing the etymology of any language is fascinating, but not central to the usage of it. I'm happy enough to know the purpose it serves.

It sort of reminds me of this Ted Talk ( http://www.youtube.com/watch?v=60OVlfAUPJg )
In short, the ability to use != ability to reconstruct

Timon Piccini link

08/17/2012 7:32am

I see what you mean and that is part of the reason why I wrote this post. These are my thoughts but are they necessary? I always need to know the why, but in this case is it necessary for my students to know?

The only reason that I do see a necessity is students still struggle with the order of operations. They either forget BEDMAS altogether or forget to go left to right with multiplication and division. I have just seen that there is a clear gap, and this is my struggle with that gap.

I am intrigued at the thought of teaching students algebra before the order of operations and seeing how that works.

Kelly Holman link

08/16/2012 11:35pm

I don't see the distinction you're making between addition and multiplication. When you add, the things you're combining came out of nowhere. (you drew them, or whatever) When you multiply, you're combining the groups. When you add on a number line, you start on the first number, but that's because you skipped the obvious step of counting from zero to that number.

Using the example of 4 + 2 x 3, the way I would teach this on a number line is to say, start with 4, count 2 more, and the x 3 tells you to do that last action 3 times. You don't do the total 3 times, for the same reason that the running total is irrelevant when you're doing only multiplication on a number line. When I tried this it felt really intuitive, like following any other rule would be artificial and arbitrary. The way students feel when we give them PEMDAS without any reason why.

If I were teaching this with counters, (I use little blocks) I would say that multiplication makes a rectangle, and you have to make the rectangles before you add them together. I would talk about finding the area of multiple rooms in a house: you find the area of each room and then add the areas together.

Honest question: Other than following a curriculum, why do students need order of ops before they learn algebra? It's always seemed to me like something that's just wedged in with no connection to anything Honest question: Other than following a curriculum, why do students need order of ops before they learn algebra? It's always seemed to me like something that's just wedged in with no connection to anything else.

08/16/2012 11:42pm

Now why on earth did the computer say that twice?

08/17/2012 7:41am

Like I said Kelly this post is based on how "I" think about these problems. Yes there is a trivial step of adding from zero but that's not how my brain interprets the operation.

What I mean to talk about in this post is the models that I employ to solve a non-contextual operation. 3x4 vs 3+4. There is no context in my head for these calculations, but my mind makes the context. My mind would never start with zero and then add 3 and then 4. The visual image that my mind conjures is 3 things on the left 4 things on the right and then push them together.

Multiplication is not the same "for me" I literally start with nothing and I add groups of three (or four) until I have the correct number of groups. My image didn't show that well before, but I necessarily start with zero in muliplication. I only play a fancy trick with addition to start with zero.

That all being said; I absolutely agree that algebra is the context for all of this and I have in all honesty never heard of people teaching algebra before this context and that makes me very interested that you don't. Like you said curricular mandates makes this hard to say the least. Perhaps you can give us a post that outlines your teaching of this, so we can all steal it :D

08/17/2012 10:32am

Heh, I haven't actually tried this yet. I thought of it during the one time I taught a class that covered basic algebra. But it took time for the idea to develop, and I was still developing the subversive streak that drives me to throw out the traditional rules and teach the way I think is best. In that class, I did try another one of my risky ideas -- I taught simple equation solving before I taught simplifying expressions. And then I discovered <a href="http://www.shawncornally.com/">Think Thank Thunk</a>, and <a href="http://www.hiddentalentstutoring.com/2011/11/a-great-innovation-in-teaching-algebra/">Algebra Models</a>, and many other things my colleagues roll their eyes at. And my subversion was reinforced, and now I'm incorrigible.

But since you asked, maybe I'll write about how I think I'll teach order of ops, when I have the chance again.

I get what you mean now about how you think about addition vs multiplication. For better or worse, my mental model is built around "multiplication is repeated addition".

08/23/2012 1:42pm

Okay, Timon, here you go: http://www.hiddentalentstutoring.com/2012/08/order-of-operations-is-all-out-of-order/

I think you probably want more detail, but I'm not sure what else to say. Ask me questions, 'kay?

Michael

06/30/2014 8:33pm

I think you will enjoy this article

http://mlmsmedia.edublogs.org/files/2011/03/The-truth-About-Pedmas-14oxnsm.pdf

Scott Wiens

08/31/2014 8:43am

I taught PEDMSA to my 7th grade students. In a class of 30, I had a 97% success rate after one period of learning, and this was a typical classroom. Fundamentally, I compared PEMDAS with PEDMSA and reminded them that in their old PEMDAS, they had to remember an additional rule: In MD and AS, they also had to go left to right. With PEDMSA they only needed one rule.

Before teaching this, I racked my brain for ways that it wasn’t right. Would they need the second rule for anything? I talked with a college math professor who was concerned that eliminating the left to right idea was a BAD IDEA, but couldn't give me more reasons than they would need it later. Then finally, a new acquaintance offered the example 24 / 6 / 2. Here PEDMSA doesn't guide you; you still need the left to right rule.

PEDMSA and PEMDAS are both algorithms with no meaning. I wish I could spend 2 weeks teaching students that exponents are repeated multiplication, division is multiplication, multiplication is repeated addition and subtraction is just adding the opposite. If you taught that, then you could ask students to first reduce everything to addition. In order to do that, they have to start with exponents and move down. Consider 6 x 5^2. Is it 30^2 or 6 x 25? If students understand what exponents MEAN, then they can rewrite 5^2 as 5 x 5 and they won’t make a mistake. After that, they can change it to addition and have 25 sets of +5 to add together.

Context is everything. A landscaper wishes to create a walkway to a square patio. The walkway should be 6 tiles long and the patio will be 5 tiles on a side. How many tiles does the landscaper need? The rules make much more sense when there is context. Naked number problems make learning more difficult!