Teachers and BIDMAS, BODMAS, PEMDAS, GEMS, etc

Here's one for you:

Do even maths teachers argue about this? by T. Briggs, licensed under CC BY-NC-SA 4.0

But before we get onto that, have a think about this one:

\[8 \div 2 \times (2 + 2) \]

I've had a post on this subject in mind for (quite literally) years, but it's such a contentious topic I haven't dared to commit it to HTML until I was sure it was ready. The problem is that with anything involving mathematics the conversation very quickly descends into complaining about maths teachers and listing what they're doing wrong and what they should be doing right, and how horrible and evil and lazy they all are. So:

Maths teachers are hard-working, knowledgeable gifts to society. This post is not intended in any way to be a criticism of these real-world superheroes. They are an endangered species as it is and I'm sick of seeing ill-informed vitriol directed at them online. Anybody who takes anything in this post and uses it to attack maths teachers hasn't understood it at all. The problems with mathematics in modern Western society are many and varied, and cannot be solved by maths teachers alone. Any proposed solution to these problems that begins with "maths teachers should..." is short-sighted at best.

 Now that's out of the way I can return to my theme.

"Order of operations" is a topic that's taught in maths lessons all around the world, and in many of them it's often oversimplified down to a mnemonic, probably one of those featured in the title of this post, but there are others bandied around. The most popular (probably) are BIDMAS and PEMDAS. The latter is common, for example, in parts of the USA whilst the former might be found in classrooms across the UK. I'll focus on these for the rest of this post, with other examples available that are very similar (and with the exact same purpose).

The letters in these two mnemonics stand for the different mathematical operations that might appear in a calculation, and their order is intended as a reminder of their importance:

Brackets	← Highest priority →	Parentheses
Indices	|	Exponents
Division	|	Multiplication
Multiplication	|	Division
Addition	|	Addition
Subtraction	← Lowest priority →	Subtraction

"Brackets" and "Parentheses" mean the same thing, here, referring to a section of a calculation enclosed by the symbols ( and ). Indices and Exponents also refer to the same thing: the little numbers above and to the right of another number that indicate the calculation of a power, not to be confused with the label for a footnote^[2].The idea is that. for any given calculation, its various operations are not merely completed from left-to-right as you get to them^[1], but strictly in order of a precedence that have been agreed by mathematicians in order to make sure that for any given written calculation any mathematician attempting to perform it will get the same answer. This is analogous to using punctuation and spelling appropriately in written English to make sure that a reader clearly understands the meaning behind what you write down.

People taught to use the BIDMAS mnemonic often perform the calculation \(8 \div 2 \times (2 + 2) \) as follows:

1. Brackets first: \(8 \div 2 \times (2 + 2) = 8 \div 2 \times 4 \)
2. There are no Indices to calculate
3. Division next: \(8 \div 2 \times 4 = 4 \times 4 \)
4. Multiplication next: \( 4 \times 4 = 16 \) 
5. As there's no Addition or Subtraction to calculate, we're finished: the answer is 16.

On the other hand, people taught to use the PEMDAS mnemonic often do it like this:

1. Parentheses first: \(8 \div 2 \times (2 + 2)  = 8 \div 2 \times 4 \)
2. There are no Exponents to calculate.
3. Multiplication next: \(8 \div 2 \times 4 = 8 \div 8 \)
4. Division next: \(8 \div 8 = 1 \)
5. As there's no Addition or Subtraction to calculate, we're finished: the answer is 1.

This is how different people get different results whilst being utterly convinced that they're correct and everyone else is wrong.

So who is wrong?

Well, really, neither of them are wrong. And neither of them are right.

There's a lot of discussion to be had around this, not least the fact that the implication in both methods that there's some order of precedence to Multiplication and Division is wrong: multiplication and division are the same thing. Or, rather, every division can be written as a multiplication (and vice-versa): a division is just multiplication by a fraction.

The key issue, though, and the one that seems to be forgotten by almost everybody who comments on these viral "puzzles" that are OnLy FoR gEnIuS is that the question itself is wrong. It's not a mnemonic that decides which order we're supposed to perform calculations, but context. If we need to write a calculation down then the agreed convention that we refer to as the "order of operations" informs how we write down the calculation in order to communicate to other mathematicians how it is to be performed. In the case of \(8 \div 2 \times (2 + 2) \) we have no knowledge of the original context of the calculation, and the calculation has been written ambiguously^[3]: the only correct response to this problem is that we do not have enough information to be sure that we are performing the calculation correctly, and the answer may be either 16 or 1. It may even be possible to justify other answers.

I'll repeat that, a little louder for those at the back in case anybody's not sure of the correct answer yet: there is no correct answer: the only winning move is not to play.

Here's the rub:

Rather than correctly responding to these clickbait calculations with something like "aha, you little scamp! It's written ambiguously and I'm not falling for that one again!", most commentors throw their chosen integer into the mix, and charge after it, pitchfork held high and screaming "BIDMAS!", "BODMAS!", or another slightly misunderstood mnemonic of choice in yet another internetty battle of Right vs. Wrong.

I don't mean to belittle those who are engaging with these posts here. More, I'm hoping to use this phenomenon as an illustration that the way we're teaching this topic is really quite flawed. So much so that...

Even maths teachers get it wrong...

... sometimes.

I picked a calculation that avoided the issue of ambiguity as much as possible:

\[ 5 - 3 + 2 \]

Simple, eh? Before reading any further, write down what you think is the value of \(5 - 3 + 2\) so that you can compare it with the correct value later. Don't worry about being "right" or "wrong": that's not the point of this.

In this case, there is an unambiguously correct response to the calculation as written (even though we are still not privy to any contextual information).

Also in this case, BIDMAS and PEMDAS agree with each other on the order of the only two operations that are involved: Addition and Subtraction. People who are religiously following either of these mnemonics would perform the calculation like this:

• Addition: \(5 - 3 + 2 = 5 - 5\)
• Subtraction: \(5 - 5 = 0\)

Both camps would obtain a final answer of 0. And they would be wrong.

Just like multiplication and division being (in some sense) the same thing, addition and subtraction are also two sides of the same coin: any subtraction can be written as an addition as a subtraction is just the addition of a negative number. There is no precedence between addition and subtraction as the calculation written as \(5 - 3 + 2\) is just asking us to find the sum of three numbers: five, negative three, and two (which is four).

The correct answer to \(5 - 3 + 2\), then, is 4.

What answer do maths teachers get?

I sent out a survey to be completed by maths teachers. The survey included the instruction to work out an answer to \(5 - 3 + 2\), a response section in which they could choose either 0 or 4, and a free text box in which they were instructed to explain their choice. 510 maths teachers submitted a response.

At this point I need to reiterate my statement from earlier: this should not be taken as either an attempt or a reason to have a moan about maths teachers. That is not the point here, and if you decide to take the opportunity to do anything of the sort you're very much lacking an understanding of what's going on.

• Of those 510 maths teachers, around 95% (485) answered correctly (4) and the rest (25) answered incorrectly (0).

I don't think this is too bad, really: I would expect a certain percentage of incorrect responses, even from maths teachers, due to things like accidentally clicking the wrong button, misreading the question, or just having a bad day. One of the reasons I picked maths teachers for this survey is that I already know that vast numbers of ordinary members of society get this kind of thing wrong (just look at the comments in one of those viral calculation posts on your social media platform of choice). Another reason is that a common statement that accompanies those wrong answers is "that's how my math[s] teacher taught me to do it when I was at school", which, if true, implies that a lot of maths teachers get this stuff wrong. According to my survey, they don't. 

• Of those 510 maths teachers, almost 13% (66) provided an unhelpful response as their 'explanation'.

By "unhelpful" I mean that they simply stated one of the mnemonics discussed earlier with no further explanation, or they wrote, literally, "order of operations". This statistic does not include the ~10 nonsense submissions for this field ( which included single characters such as letters and punctuation marks; "Ooo"; and "No"): I'm counting these as declining the invitation to explain their response^[4].

Whilst 13% of the total respondents simply cited a common mnemonic as their 'explanation', if we compare those who got the answer correct with those who got it wrong we find that:

• Of the people who submitted an incorrect response, 60% provided one of the explanations described as "unhelpful" above, and
• Of the people who submitted a correct response, around 11% provided an "unhelpful" explanation.

This strikes me as potentially noteworthy: most of the respondents who got the answer wrong demonstrated a belief that "BIDMAS" (or equivalent) was enough to explain their entire process. In comparison, almost 90% of those who chose the correct response provided an explanation that either included "BIDMAS" (etc) but expanded upon that, or didn't mention "BIDMAS" (etc) at all.

There's plenty of other observations to be made from the data that was submitted, but this post is getting quite long now and I'm about ready to make my final point. Before I do, I'd be happy to write a more in-depth discussion of the survey results if enough people thought that might be interesting.

And now, to...

My final point

If even a subset of maths teachers - experts in both mathematics up to a minimum expected level and how to get it across to others - are getting calculations involving the order of operations wrong and invoking those mystical incantation of BIDMAS, BODMAS, PEMDAS, GEMS, etc, in support of those misconceptions, are they really the most effective ways to teach this fundamental topic that underlies much of the rest of maths education?

In many a school's scheme of work - including my own - there's a lesson each year that's called "BIDMAS" and almost every time students in my classes need to work out the answer to a written calculation I hear a chorus of "do we need to use BIDMAS?"

It's useful as a mnemonic, but only if it's recognised as one of many wrong but useful mathematical memory-aids rather than, as is unfortunately more often assumed by those who encounter it in their educational journey^[5], a hallowed piece of mathematics lore hammered into stone at the dawn of the universe.

In my opinion, our reliance on BIDMAS (etc) as a teaching tool represents a critical weakness in maths education and it's about time that we gave its use a thorough rethink.

Footnotes:

1. "But why not?" I hear quite often. "After, all, we read from left-to-right!". I take issue with this:
   Consider this sentence!
   And then, if you will,
   Consider this sentence?
   The way that these two otherwise identical statements are read and understood is changed by a single character at their rightmost ends: we do not read exclusively left-to-right.
   A more mathematical argument might point out that it can sometimes be difficult to arrange a set of calculations to be completed strictly from left-to-right whilst resulting in the answer that we need. [back]
2. Such as \(3^2 \), which indicates that you should multiply 3 by itself: \(3^2 = 3 \times 3 = 9\). [back]
3. This is partly down to the use of the obelus sign (÷) for division: some ambiguity can be avoided by writing a division as a fraction, and it is because of this that the "division sign" all but disappears as students progress in mathematics. [back]
4. Why they bothered responding at all in this case is beyond me. [back]
5. And then, in some cases, become teachers themselves. [back]