## What does BODMAS stand for?

The BODMAS acronym stands for brackets, orders, division, multiplication, addition, subtraction.

It is sometimes known as BIDMAS (with ‘Indices’ used instead of ‘Orders’) or the PEMDAS rule in America (with ‘Parenthesis’ and ‘Exponents’).

## BODMAS rule

This mathematical rule dictates the correct order of operations to be followed when you complete a mathematical number sentence question with different operations.

The first step is to do anything in brackets, then orders next (such as square root or indices). Division and multiplication are on the same level, meaning they are given equal priority, and should be done from left to right, rather than all division, then all multiplication. Similarly, addition and subtraction are on a level together, and should be done from left to right.

**I began my teaching career at Highgate School. Youthful, untrained, and not yet balding, I was thrust into the steepest learning curve of my life.**

Weekly meetings with my head of department were vital for discussing pedagogy and I clung faithfully to his instructions: “Never abbreviate Cumulative Frequency”, “We always flip coins and get tails, we never toss coins and get heads”, and of uppermost importance, “We never, ever use BODMAS”.

Not using BODMAS was less easy than you might imagine. Students arrived well-versed in its application.

We had to *un*teach it. We had to persuade rooms full of teenagers that they had to alter the fundamental tenets of their arithmetical belief system. This was difficult because teenagers hate change and they hate adults proselytising.

So why on earth would we bother? What had convinced an entire department that so much effort should be expelled on such a seemingly trivial matter?

BODMAS is wrong. That’s what.

## Wrong answer

Its letters stand for Brackets, Order (meaning powers), Division, Multiplication, Addition, Subtraction. Simplification of any given mathematical expression is thus supposed to occur in this sequence.

For example, to evaluate* 3 + (3 + 3) ^{3} ÷ 3 – 3 x 3* we proceed in the order above:

This would be a really useful algorithm if it worked in every situation but consider the much simpler expression, *1 – 2 + 4* . It contains no brackets, powers, division, or multiplication so we’ll follow BODMAS and do the addition followed by the subtraction:

This is erroneous. The correct value is 3. BODMAS has failed us. Shame on BODMAS!

## Mathematical problems

We can’t have a magic mnemonic that doesn’t work all of the time; suppose it decided not to work at an important moment. Imagine trying to explain to your student that the reason they dropped a grade in an exam was that the thing that you had told them always works didn’t actually work on every occasion and, in fact, one of those occasions happened to have occurred in that GCSE paper.

This is not a new problem. I am not the first person to write about it. Even Wikipedia tackles the issue and suggests some alternatives. Students love Wikipedia! So why is it that BODMAS is still a thing?

There was such stigma surrounding it at Highgate that I have been ridiculed by certain parties for over a decade after a colleague experienced a classroom exchange that went something like this:

*Teacher: How do we simplify this expression?
Student: BODMAS, Sir.
Teacher: We don’t use BODMAS here.
Student: But that’s what Mr Elton taught us last year, Sir. *

Thereafter I was unfairly assigned the moniker “BODMAS” which followed me everywhere. I had no defence; a fee-paying student had made the claim so it must be true. At least one individual (he knows who he is) still calls me BODMAS more frequently than he uses my actual name.

Even though I am absolutely not guilty of sullying innocent students’ minds, I feel duty-bound to make amends so that’s how I’m using this platform. Think of it as community service.

## Correct answer

There’s no point BODMAS-bashing, however, without offering an alternative. The error illustrated above is caused by the fact that addition and subtraction should not necessarily happen in that order. If we have a string of these two operations it is called a sum and we should work from left to right:

Similarly, division is no more important than multiplication. If we have a string of these two operations it is called a product and we will work from left to right again:

We now have the following order: Brackets, Order, Products, Sums.

This gives us BOPS which is a whole syllable shorter than BODMAS and has the considerable advantage of being reliable.

I’m sure that if someone had proposed BOPS before BODMAS then the latter would be consigned to obscurity. Even now, it is not too late to rid ourselves of bi-syllabled arithmetic acronyms.

I call on my colleagues around the world to ban BIDMAS and to purge PEMDAS. Leave no trace of them. Allow BOPS to strike a victory blow for young mathematicians everywhere.

**Owen Elton is a maths teacher, a writer/performer of silly songs and author of matheminutes. You can follow him on Twitter at @owenelton.**