One oddity about working in a school that is well-known / notorious is that we are often asked about how we – that is, the maths department- approach or teach different things. I worry about saying anything, as we don’t have results yet (182 sleeps to 21^{st} May 2019!) and our approaches haven’t been through the crucible of The Exam Process. I don’t want to give the impression of certainty as we are always changing, refining and even chucking whole strategies!

However, there is a pair of approaches we have been taking for several years that we do think are working well and I hope you will find useful too: GEMS and SMEG. I’ll write about GEMS below and its unattractively-named cousin, SMEG, in my next post.

GEMS is an alternative to BIDMAS/BODMAS, and is used more in the USA, although not widely. I can’t remember where I first saw a teacher blog about it; please let me know if it is you, so I can give credit where it is due.

GEMS stands for:

- Groups
- Exponents
- Multiplication (and its inverse, division)
- Subtraction (and its inverse, addition)

Why we use it:

- It forces teachers to make clear to the pupils that multiplication and division are equally important operations
- As above, but for for addition and subtraction
- *Even if* the pupils forget, in the case of addition and subtraction, having subtraction ‘first’ avoids errors such as 10 – 5 + 3 = 2. From what I can see, this type of error can only occur if they think they must add first.

- ‘Groups’ is much more accurate than brackets / parentheses, as it includes expressions within numerators, denominators, roots, etc.
- Exponents is no more difficult a word to teach than indices
- It has a nifty image that can go with it to give a sense of the hierarchy (and add general sparkly specialness 🙂 )

(images from Ms Lopez’s blog )

Disadvantages:

- No revision guide or revision video is using this acronym. You have to explain to them that it is the same kind of mnemonic as BIDMAS, and explain why you are teaching it instead.
- You have to be strict with yourself and with them about saying ‘multiplication…
*and its inverse division’*, as it would be even worse if they had no idea where D and S have gone! - In common with BIDMAS, it doesn’t solve the problem of ‘functions’ (e.g. if calculating 3 x sin40 + 5, GEMS doesn’t make clear that first they should find the sine of 40). We’ve been bit hazy with and let them sort of see them as an exponent without ever saying so…a sin of omission that makes things a bit easier, even if it feels a bit wrong!

If you want to make this change, *please *make it a whole-department change. It is a nightmare if one teacher does it and then pupils or classes move on to someone else. You may decide ‘only for Y7, and then add on a year each time,’ but ensure you have within-group consistency.

Bonus tip: introduce the order of operations **after **you have taught directed numbers. Children can quickly infer unintended patterns if all your examples have positive results. Ideally, early examples would be in forms such as this, to force them to preserve the order that is in question:

4 – 3 x 5

Many pupils, knowing they should ‘multiply first’, will conclude it also means ‘put the result of the multiplication first’ and write

15 – 4

for the next line. If you use only positive examples, such as 4 + 3 x 5, you won’t notice there is a growing habit of them writing 15 + 4, and so on, whereas their little leaps become obvious when negative numbers are involved.

Hi Dani, thank for writing this. It’s the first time I’ve come across the acronym and I fully agree it is more useful than the usual BIDMAS/BEDMAS/PEMDAS. One thought that occurred in reading the section on 3 x sin40 + 5, is that you might argue that sin40 represents a ‘group’ compromising of a division of lengths attached to a specific angle. Alternatively, if that’s too much for students to handle at the time, then at least stating that the only way we can progress with a simplification of an expression is to start by getting the ‘units’ the same, which connects to many other areas of mathematics such as length conversions or simplifying algebraic expressions. Simplifying sin40 to a number allows us to then go ahead and combine terms.

Anyhow, thanks again for writing, I learnt something and it made me think.

That has blown my mind! Incredibly helpful and will definitely help with avoiding errors. Lovely, thank you.

P.S. Sin, cos and tan are functions. Hence FBIDMAS!

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Hi

Roots are covered by indices (power of half?)

How do you process: 3 + 8 – 2?

OK. What of 8 / 4 + 2 * 3 / 7? Or 3 * 8 / 6 * 2 ?