Sunday, 6 November 2016

Methods of last resort 2 - Order of Operations

Teaching the correct order of operations is possibly one of the most debated topics for maths teachers. In my #mathsconf8 session I was asked 'what is my problem with BIDMAS' and proceeded to outline times when this acronym is redundant (e.g. 4 x 3 ÷ 2) or even downright wrong (4 - 5 + 6 would mistakenly be given as -7 rather than the correct answer as 5). Various diagrams have been mooted as the solution to this, and there are several examples below:

I have several issues with these diagrams, which can be summarised as:

(a) It isn't specific enough for all of the possible functions that can be applied to numbers (even those that include square roots don't involve higher roots, and no mention of sin, cos, tan, log etc)

(b) BRACKETS ARE NOT AN OPERATION (please forgive the shouting). This may seem like semantics but for me it is an important distinction - brackets are used to either alter or clarify the order of operations intended, but are not an operation in themselves (just a note on clarify, an example of this is 12 ÷ (3 x 4) needed clarity as without these brackets the answer would be 16 and not 1). If we are going to teach pupils to understand the maths they are doing then we need to be communicating understanding like this, and not allowing pupils to mistakenly believe that brackets are an operation themselves.

But this post is not about teaching correct order of operations (although that segue has outlined my thoughts on it quite nicely); this is about when you wouldn't want pupils teaching using the correct order of operations in the first place. The example I used in my #mathsconf8 session was:

673 x 405 — 672 x 405

Any mathematician is definitely not applying the correct order of operations in this situation; and is quickly writing down that this is just 405. With the advent of 'teaching for mastery' gaining ground in mathematics education pupils are being increasingly exposed to questions like this when looking at distributive laws, or factorisation but I am yet to see it, or anything like it, thrown into a lesson on Order of Operations as a non-example. There is good evidence out there now to back up the idea that non-examples are important in communicating a concept and so if we are trying to communicate the correct order of operations we should be highlighting cases like this as when applying the correct order of operations is not wrong, but is just wildly inefficient compared to use of the distributive laws (in this case the formal statement would be something like 673 x 405 - 672 x 405 = 405 x (673 - 672) = 405 x 1 = 405).

Some other examples of times when correct order of operations are an inefficient way to solve problems (particularly without a calculator) are:
  • 12 x 345 ÷ 6
  • 182 ÷ 92
  • √128 ÷ √32 (although this one does require some real mathematical understanding)
  • 372 + 845 – 369
I would be exploring all of these questions prior to teaching the correct order of operations, and then including questions like it in the deliberate practice on the correct order of operations to ensure that pupils are recognising when not to apply them alongside when they are absolutely necessary.


  1. This resonates really strongly with how we have been teaching Year 7 this year with lots of these types of egs. I'm slowly coming to a conclusion that we don't need to explicitly teach Order of Operations. By the time they've learnt about associative laws and distributive laws, understood the role of brackets and done some algebra (a=3, what is 4 + 2a?) does it become self - evident that Multiplying and dividing happens before adding and subtracting?

    1. I am coming to the quite firm conclusion that all of Maths should be self-evident, in that it should always be clear how each step is made, but at the same time it should be explicitly taught so that it is shown to be self-evident.

    2. Lets remember that when they take this mathematics forward into other disciplines and professions the confidence they can build upon helps them become efficient with writing formulae and building models e.g. representing a calculation within a cell or cells of a spreadsheet can be done with great economy and confidence if the algebra they employ is as simple and succinct as possible. BIDMAS aids this. I know from my own experience if I can put a single formula into one cell without putting multiple brackets the subsequent debugging if the model doesn't produce what I expect is a clearer swifter process. Matt Dunbar

    3. I definitely agree that efficiency is a crucial requirement, and I think the blog reflects this. I can't see how BIDMAS helps, particularly if pupils are too wedded to it and end up creating problems or working inefficiently because they are following the process without understanding. If they truly understand then they shouldn't need BIDMAS in my view.