Aaargh Ruddy BIDMAS
The order of operations is commonly taught using one of the following mnemonics: BIDMAS, BODMAS, BOMDAS, BIMDAS, PEMDAS, PEDMAS or, if you’re Colin Beveridge (@icecolbeveridge) Boodles! (I’m not going to discuss that here, so if you’re interested, do follow the link)
For the first six the letters stand for the following: B: Brackets, P: Parentheses (essentially the same thing) I: Indices, E: Exponents, O: Orders (Again, the same in essence), D: Division, M: Multiplication (Inverse operations) A: Addition, S:Subtraction (Again inverses).
There is a massive problem with these mnemonics, and their use in teaching. I do, however, think it’s down to the teaching rather than the mnemonics themselves, and possibly down to a lack of understanding from some who teach it.
And example of the problem I’m hinting at occurred on Friday with my top set year 8 class. As part of a ten quick questions core skills starter I included the question: 3 – 1 + 2 = . A problem free, easy, question for a class working at level 6-8 one would think, but the uproar when going through the answer was unbelievable. I asked one lad what he had written, he told me zero, I asked if anyone could tell me what he’d done wrong and was met first with blank stares, then with “Sir, it is zero. Have you forgotten about BIDMAS”. The scale of the misconception was enormous. After the first person said it the entire class were up in agreement. I settled them down, jettisoned that days lesson plan and retaught them the order of operations, but properly.
The misconception is heavily tied to the mnemonic. A comes before S in BIDMAS, so Addition comes before Subtraction. But this isn’t the order of operations. When I teach it I make a point of telling them all six mnemonics mentioned above and specifically drawing their attention to the way the D and the M are interchangeable. Discussing that because Division is the inverse of Multiplication they are, in essence, the same operation, certainly “of the same order”, and as such take equal precedence in the order of operations so you read from left to right. I then ensure they know this is true for Addition and Subtraction. If I ever write one of these mnemonics I always write it:
B
I
DM
AS
Or even
B
IR
DM
AS
With the R standing for roots, as this shows the inverse relation for that level too.
The year 8 class now understand the order of operations, despite protests that “(insert name of former colleague here) told us addition ALWAYS comes before Subtraction!” as they all thought this, it is possible. I’ve seen it taught wrong in more that 1 school. I had a real row with another trainee on my PCGE course in a microteaching session when she taught it wrong and tried to say I was wrong. Every year the pupils from one of the feeder primaries get it wrong, so I think their yr6 teacher must teach it wrong. I’ve also been brought in to settle arguments on social media networks that have erupted over those stupid viral questions based around this. All this shows that there is a perpetuation of this misconception in this country, and we need to stamp it out.
cavmaths 
Some teachers in the States have moved to “gems” instead to overcome some of the common issues:
Groups – including (eg) the ‘group’ that is the numerator, or expression in a square root
Exponents – self-explanatory
Mulitiplication – and its inverse
Subtraction – and its inverse
I would love to teach this instead, but worry for the kids trying to revise/practise and finding nothing when they google it. I’ll bite the bullet someday soon!
Sounds like a better mnemonic!
Amen.
I always told Kids that addition & subtraction were like little boys and girls. It doesn’t matter if the boys or girls are first. Whomever is in line first, goes first. Because I’ve taught this concept to mostly 4th & 5th graders, we keep the same ideas of boys and girls, but we all know that 5th graders would go first (as mult/div). Few students of mine forgot that part of order of operations.
That’s quite nice.
I don’t know…I had this problem with my sixth graders. When I taught PEMDAs (which can hardly be avoided as it’s in the textbook, and most of them have heard of it already) I was careful to write M and D together and A and S together. I emphasized that they’re on the same level. I made up a chant with motions to reinforce this point: Parentheses, exponents, multiply AND divide from left to right, add and subtract from left to right. I still had kids who wanted to do all the multiplication before the division, etc. I don’t know why.
Aye, it’s a puzzler!
It boils down to the subject knowledge ultimately, which I know you’ve discussed more recently. I remember when I was an excellent A Level student. I was doing Further Maths A Level as well and struggling mightily (by comparative standards)
Part of that was down to the fact that my Further Maths teacher was in her first year of teaching and was probably still finding her way with regards teaching that level of Maths but the flip side was that my core skills were not polished enough to enable me to manipulate algebra the way I later could.
Mastering collision theory was only made possible by intimately understanding order of operations and what balancing on a level I had never been challenged at. I was solving exam questions through examining style of questioning rather than applying overall knowledge (I hope that makes sense)
I don’t know if I could teach order of operations as effectively without that experience. I go out of my way to let students discover WHY the multiplication and division are interchangeable? WHY the subtraction and addition are interchangeable?
I can then guide them to better forms of notation (such as using fractional notation) instead of reliance on the soon to be never used division symbol. I know if I can do this bit well I can save a lot of heartache when it comes to rearranging and changing the subject etc.
Always lots of food for thought. Cheers
Aye, i agree with you on all points again. I too aim to lose tge division symbol asap and go out of my way to enforce the why.
Thank you so much for this article. I am a Y6 teacher and have had similar arguments about this one. It does come down to how you were taught – badly it would seem for most of us! GlosMaths and Kangaroo Maths have both been trying to rectify this misconception among primary teachers, so hopefully it will eventually filter through to the upper schools and you won’t have a riot on your hands next time you present a similar problem!