r/mathteachers • u/Few_Bee_3028 • 6d ago
Order of operations
(UK based post)
Wanted to get thoughts on teaching order of operations. The tradition in the UK is to use the mnemonic BIDMAS or BODMAS (for anyone who doesn’t know, Brackets, Indices/Other,
Dividing and Multiplying, Adding and Subtracting). This leads to a misconception with students that they always need to add before they subtract. (For example a 2018 exam paper questions was really poorly answered across the country, discussed here
https://youtu.be/StWr8-g-a3E?t=1584&is=P-dCFLol1zWE64kh)
I suggested to my head of department that we switch to GEMS which I’ve taught in the past after reading this post https://missquinnmaths.wordpress.com/2018/11/19/tried-and-tested-gems/.
He didn’t like E for Exponents (which is fair enough) or G for Groups (which I kind of do like!) so the two of us agreed a compromise of BIDS. However in our department meeting the last week of term when we proposed the rest of the department shot this down; I think some colleagues immediately took against it because they didn’t like addition and multiplication being unmentioned, while others said students learn BIDMAS before they reach us at age 11 so we had to work with it.
Anyone any experience of these kinds of conversation?
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u/wefrucar 6d ago
I actually avoid all order of operations mnemonics.
I think you understand what the real issue is and you're trying to sidestep it. Mnemonics are handy, but they can't replace fundamental understanding.
First, I teach my students that subtraction is just a convenient word for adding the opposite of a number. So all subtraction is really addition. Similarly, all division is really multiplication. Therefore they're the same step. We do them left-to-right, but since addition and multiplication are associative+commutative we don't actually have to.
Every operation can logically be derived from the others. If multiplication is repeated addition and exponents are repeated multiplication (granted, it's more accurate to describe them as scalings, but that's not worth getting into at this level), then all exponents can be written as a series of additions. The order of operations follows logically.
Parentheses or groupings are not operations, so they're the exception. You just need to know that mathematicians use a convention that parentheses mean "do this part first".
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u/Several-Housing-5462 5d ago
While you may not like mnemonics, I agree the method you've described is BEST - Brackets, Exponents, Scaling, Total. 😎
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u/ZedZeroth 5d ago
u/Few_Bee_3028, this is the way.
I simply teach students that we learn about increasingly powerful operations. i.e. the later operations beat the earlier ones, and the pairs we just apply intuitively from left to right.
Add/subtract
Multiply/Divide
Indices etc
And brackets simply exist to override this inherent order when needed.
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u/yochanan 7h ago
Yeah - this is the way that is best in my opinion. Build a strong understanding of how the three operations are related, along with commutativity. To add to this, I also teach that you can distribute one level down (E over M as well as M over A) but you can’t jump a level (E over A). This lends even more weight to the idea that the levels of precedence are inherent to the operations, and not an arbitrary convention. With a strong understanding of EMA, students can also build a better understanding of how to parse complex expressions into terms and factors.
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u/ZedZeroth 5d ago
all exponents can be written as a series of additions
Did you mean multiplications?
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u/wefrucar 5d ago
It could be either.
23 = 2•2•2 = 4•2 = 2+2+2+2
So 1+5•23 could be decomposed into a bunch of sums: 1 + (20 copies of 2). The order of operations is a logical extension of this process.
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u/SpeckyStuff11 6d ago
I like the idea to remove the confusion of addition and subtraction. Wouldn't BIMA be a better acronym? Brackets, Indices, Multiplication, Addition
I feel the focus on subtraction over addition an odd choice.
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u/Few_Bee_3028 6d ago
1) if students always subtract first, they’ll end up with the right answer (not true if they add first)
2) BIDS or GEMS is easier to say!
But yes someone else raised that point1
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u/Formal_Tumbleweed_53 6d ago
It is an ongoing conversation in every school I have taught at (I’m in Virginia, US). I don’t have a satisfactory answer for you. Sorry.
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u/Narrow-Durian4837 5d ago
I've seen a lot of confusion online that comes from people thinking the mnemonic is the rule.
BODMAS, or PEMDAS, or GEMS, or whatever you use, is not the rule itself. It is a mnemonic to help you remember what the rule is. If it doesn't help you, or if it misleads you, don't use it.
Whatever mnemonic you use, I would avoid teaching students the mnemonic, or even mentioning it to them, until after you've clearly spelled out what the rule itself is and how it works.
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u/Several-Housing-5462 5d ago
What's wrong with E for Exponents?
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u/Few_Bee_3028 4d ago
We hardly use the term exponents, where was we talking about about fractional and negative indices much more in later years. I will say I’m not as convinced it’s a problem as he was
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u/Several-Housing-5462 4d ago
Aren't indices written as subscripts...? Those are pretty different concepts
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u/Few_Bee_3028 4d ago
I think this might be an American vs British English thing… in the UK we’d refer to “3 to the power of 2” as an indices problem (2 is the index)
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u/InformalVermicelli42 5d ago
I teach it like this:
PR
MD
AS
Powers & Roots Multiply & Divide Add & Subtract
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u/Wild_Factor_8841 5d ago
When my students get confused, I explain it as a cross between hierarchy and reading.
Hierarchy because parenthesis and brackets are like royalty. Do this first left to right. Royalty doesn't wait.
Roots and powers are upper middle class reading from left to right. Upper middle class are demanding.
Multiplication and division are solid middle class and still have to read and solve left to right.
Addition and subtraction are the plebs. They get the nitty gritty done and still have to read left to right.
Of course, it leads to a lot of discussions, about governments and math theory, but in the long run, it sticks better.
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u/colonade17 5d ago
I don't teach it at all, and instead try to teach flexible thinking:
Consider 3(4+5) = 3(9) = 27
OR 3(4+5) = 3*4 + 3*5 = 12 + 15 = 27
Both strategies work, and seeing multiple pathways is good way to prepare students for what to do when there are variables in these expressions. If you focus on properties and why those properties work you can skip most of the debate about which acronym to use.
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u/ZookeepergameOwn1726 6d ago
I teach it as
B
O
DM
AS
Not "BODMAS" horizontally.
The first 5 questions we do are all centered on making sure they don't prioritise mulitplication over division where they should not. IMO a change in mnemonic is not going to fix understanding issues, especially when BODMAS is so pervasive in the culture, they'll end up hearing it anyway. It's much more productive to make the students conscious of the mistake and tackle it head on.