I’ve had a small number of discussions lately about the order of operations. I wrote about it a little in my book on subject knowledge, but, as with everything, there is always more depth to go to (indeed, that’s one of the main messages of the book – if you think you know it all, you need to think again.)
The two questions that prompted the discussion were the following:
These are the kinds of questions that go viral on Facebook with a million comments and about as many different answers. People find it hard to know what to do in what order. Most know that brackets come first, but then some read from left to right (regardless of the operation), some always do division before multiplication or addition before subtraction (probably because of some variant of BIDMAS).
The order of operations, of course, goes like this:
- brackets (and similar groupings where brackets can be implied, like in the denominator of this fraction: )
- indices (which, of course, includes roots)
- division and multiplication
- addition and subtraction
We maths teachers always try to impress that division and multiplication (or addition and subtraction) have equal priority and should be read from left to right, but acronyms such as BIDMAS/BODMAS/PEMDAS/PEDMAS, especially when introduced at the start of teaching this topic, often cause students to follow the strict order of the letters therein.
People have tried to get around this well-recognised problem with other aides-mémoire like this pyramid on Don Steward’s excellent Median blog (which was, in turn, taken from elsewhere):
or like the acronym GEMS, which Dani Quinn espouses on her blog.
I am of the opinion that the order of operations should never be introduced with an acronym. I have no problem with acronyms per se, but we don’t “teach BIDMAS”. We teach students about operations – how they work, in what order they work – and acronyms can be a useful way of helping with this.
Time should be spent on calculations of varying complexity until students are comfortable with the order before any mnemonic is introduced. If you start with working addition and subtraction from left to right:
and separately with multiplication and addition from left to right:
then once a question such as
is introduced, and you tell students to do the multiplication and division sections first, then the question is reduced to what they are used to, .
There are complexities. They need to be able to identify unbroken strings of multiplication/division within a calculation, such as in
This can be difficult for novices.
After the principles here are understood, then it’s time to move on to indices and brackets, but again there are layers of complexity that need to be addressed, such as the implication of brackets in something like or .
Students also need to understand when brackets are superfluous, such as in
One of my colleagues calls these ‘overzealous’ brackets and his classes are quick to call people out on their overzealousness the moment they spot it.
Arbitrary or necessary?*
Having gone through all of this, which is mainly to set the scene, there is another thing I want to talk about, and it’s something I think is missing from the discussion on the topic.
Why must multiplication and division (or addition and subtraction) be performed from left to right? Is it just because we need a rule, and someone chose that one? Is the order of operations somewhat arbitrary – something once decided simply to fix a rule – or is it necessary – something which cannot be otherwise? Take the following set of calculations, work them from left to right, and consider which give the same answers:
|Set A||Set B|
In both sets, questions 1-3 give the same answer. So they needn’t be performed strictly from left to right, otherwise we might expect different answers to each question. Question 4 in each set does, however, give a different answer. Why is this?
The answer, of course, is to do with commutativity, one of the axioms of arithmetic and a property of both addition and multiplication but not subtraction or addition. That is:
This means we can move the multipliers and multiplicands, or the summands, around as much as we want and the answer will not change. This makes a question like the first from Twitter (above) much clearer:
The point is and must have the same answer due to the fundamental properties of multiplication. I don’t think anyone would quibble about being the same as , but there are many people who are tempted to conclude that is the same as . If that were the case, multiplication would not be commutative.
If our students are confident with the commutativity of addition and multiplication, they don’t need to think about reading from left to right, they just need to think about moving things about to make the calculation easier to work with.
Clarity of mathematical communication
This leads us nicely to the second image from Twitter (above), with the question
If you follow the thread you will see that Bernie Westacott points out that a Casio fx calculator will give the answer as 1, performing the multiplication of 2 by (1+2) before the division. Wolfram Alpha, however, gives an answer of 9, changing the presentation of the calculation to .
I can see why someone would perform the multiplication first – we are so used to expanding brackets that we jump to ‘deal with’ the bracket before anything else – but we need to pay heed to the hidden multiplication sign between the 2 and the bracket, the calculation is actually . My brain is much less tempted to multiply first now.
That can’t be right, that the simple inclusion of an implicit multiplication symbol potentially changes the answer. I would argue that Casio are mistaken and that the only way the answer to this calculation should be 1 is if it is presented .
The real problem here is a lack of clarity in mathematical communication. The obelus, is much less clear than the vinculum (fraction bar) in many of these calculations. If we write either or there is absolutely no ambiguity. We should never write something mathematical in an ambiguous way.
Going back to the calculations in Set A in the table above, by using commutativity and the vinculum (and the fact that dividing by 3 and by 4 is the same as dividing by 12) we can write all of questions 1 to 3 as
with multipliers in the numerator and divisors in the denominator. No ambiguity and no risk of breaking the commutative law like Q4 in the table did.
Similarly, looking at the calculations in Set B in the table, we can group the numbers to be added and the numbers to be subtracted (and use the fact that subtracting 5 and subtracting 2 is the same as subtracting 7) to write all of questions 1 to 3 as
summands first and subtrahends second, again without risk of breaking the law of commutativity as Q4 did.
Questions for the classroom
Ignoring brackets, everything else in the order of operations must be the way it is (brackets are introduced to break that order). Can we design sequences of learning so that the order is self-evident? For instance, using the definition of a positive index we can see that , not 36. To force the multiplication first, we introduce a bracket to give .
Similarly, we can demonstrate that multiplication and division have a higher priority than addition and subtraction if we think in the context of multiplication as repeated addition. For instance, , not 20. Thinking carefully about this, do we stop the order being an arbitrary rule to be learnt?
I’m not naïve – I know that many students lack the number sense to really make meaning of these justifications and may well have greater success by simply learning the order, but we must always question what we do and whether there is a better way.
I’m particularly curious about teaching that multiplication/division or addition/subtraction should be performed from left to right. When we say this, do we stop our students from accessing a greater level of fluency in calculation? It works, but it’s not strictly true, after all. I want my students to be able to move numbers around in calculations with ease, I want them to be adept in their workings with numbers. If I spent more time on these ideas, would the part of my order of operations lessons where I tell them about ‘left to right’ become completely redundant and would they have a greater grasp of calculation as a result?
Your thoughts, as always, are very welcome.
* The idea that some things in mathematics are arbitrary, such as the names of shapes, or our choice of symbols, and that many more are necessary, that is they must be so, such as the formula for or the order of operations, comes from Dave Hewitt. You can see his three papers on the topic here, here and here. Hewitt proposes that the way we approach the arbitrary and the necessary in the classroom must be different.
Hi Jemma, thank you for sharing your thoughts, which I both enjoyed and concur. May I add some of my own?
I also was prompted to consider my understanding of order of operations by the a x a divided by b x b question when I saw it on Twitter. It occured to me that the ambiguity could be removed by thinking of division as multiplication by the reciprocal, i.e. as a x a x 1/b x b. As you say, the commutative law (do we need to invoke the associative law, too?) tells us we can order the individual terms any way we wish, but the result will always be the same, a^2 x 1. I find this will work for all the other examples you have included above. Lil’Maths Girl’s example becomes 6 x 1/2 x (1+2), as implied in one of your representations, while the first three examples in Set A are all rearrangements of 12 x 5 x 2 x 1/3 x 1/4 (the fourth, of course, replaces 1/3 with 3).
I find that most learners are comfortable with the idea that division by 2 is the same as multiplication by (or at least “calculating”) one-half. Obviously we wish to generalise this to division by n.
As for addition and subtraction, I have found it can help to turn the calculations into stories. Your set B questions could be stories about receiving sums of 6, 3 and 7 (sweets or pound coins, for example) and spending/eating/giving away amounts of 2 and 5. The order in which these events occur will make no difference to the final amount. Tracking running totals as the stories unfold can help reinforce the mathematical ideas. Certain scenarios will result in negative amounts appearing, but I’m sure we could turn these to our advantage!
Some good extra points, Gerard. Thanks!
I think the amount of time available is key here. Teaching left to right works and avoids the misconception. It is also a quick correction. When teaching the order of operations, or if more time is available, linking the commutative principle would be desirable but it is now clearly an objective.
The danger with demanding greater understanding of core ideas is that, while obviously desirable, it seems to prejudice conceptual understanding over memory. We need both for a full understanding and there is no real reason to believe one needs to always come before the other. When we accept this we can simply switch between the two descriptions over and over again.
The conceptual links reinforce other learning and maximize the retrieval strength while repetition of simple rules limits cognitive load and enhances our ability to recognize simple solutions. The why and what are ironically commutative themselves.
Thanks Michael – I agree with you about the commutativity of why and what. By introducing order of operations in a slow and “drip-feed” manner I am trying to limit undesirable cognitive load, allowing my pupils to grasp elements of the order before introducing layers of complexity. For me, it’s important that my understanding as teacher is always being improved, and this post forms part of my refining of understanding. The extent to which my pupils are invited to depth of understanding is governed by many factors, not least available time as you point out.