If your Facebook newsfeed is anything like mine, you occasionally find wince-worthy pictures like this going viral. That classmate you were never really friends with anyway is suddenly arguing with internet strangers about the order of operations.
Many of the arguments over these things stem from the massive problem with BIDMAS/BODMAS/PEDMAS – that they imply division comes before multiplication and that addition comes before subtraction. They are, in fact, on the same level.
If you believe addition comes first, you’d answer 1 – 0 + 1 as follows:
1 – (0 + 1) = 1 – 1 = 0
When since they’re on the same level, the answer is 2.
Keen to avoid this misconception, and with the freedom afforded by designing my own curriculum, I decided to seize upon an alternative mnemonic: GEMS.
In GEMS, the order is as follows:
Groupings (numerator or denominator of a fraction, under a square root sign, and brackets)
Exponents and roots
Multiplication and division
Subtraction and addition.
My teaching sequence
1. How to manipulate strings of addition and subtraction signs, and multiplication and division signs.
I drilled and drilled the rule that we work left to right with a string of addition and subtraction signs. We can’t randomly start in the middle with a sum like 40 – 4 + 6. The temptation to start in the middle is particularly strong when you see a nice number bond like 4 + 6, but that doesn’t stop it being wrong!
The children were aware that you could make long strings easier to calculate by doing some shuffling about of the numbers. I’d spotted from work earlier in the year that many weren’t conscious of the limits of this, though, and would often shuffle things about incorrectly.
I find this idea of teaching the precise limitations of how you can and can’t manipulate expressions fascinating to think about. Over- and under-generalisation is a big cause of misconceptions. I wanted to give a 100% foolproof rule that was snappy enough for them to quote and remember.
I taught that we always move the number with the sign in front and then work left to right. So 8 – 3 + 2 can become 8 + 2 – 3, because you’ve moved the +2 forward. Then you’d do 8+2 is 10, and 10-3 is 7. I didn’t want to get into the idea that they could jump around the calculation, going back and forth. We’ll look at that idea more carefully when we do negative numbers.
For similar reasons, I didn’t address the issue of how you could move the number right at the front. This whole thing is easier if you can conceptualise subtraction as adding a negative; I haven’t covered negatives yet and it wasn’t something I wanted to get into at this point in the year. However, next year, I am considering teaching the idea that the number at the front has a hidden plus sign in front of it. Thus if you move the first number back, it takes a plus sign with it. I’m not sure if that would be better left until negatives too.
The main emphasis through all of this was at no point do we randomly decide to start in the middle. If we only have addition and subtraction, we are always rearranging the numbers to a nicer order and then still working left to right.
We did the same with multiplication and division.
2. M then S
Multiplication and division comes before addition and subtraction.
Plenty of drill of this very common problem type with a heavy emphasis on the correct layout and use of the equals sign. We did plenty of choral cloze of “equals means…” “the same as!”
I taught pupils to underline every string of multiplication and division on the first line.
Then they were to copy the expression on the next line, calculating the parts which they had underlined. In other words: “underline, copy, calculate” – something we’ve been chanting all week.
On the next line, only addition and subtraction should remain. If not, something’s gone wrong. All that’s left is to evaluate this to get our final answer.
3. E then M then S
We added in exponents and roots into the mix. No trouble with this step. I think vocabulary is the usual stumbling block, but we’d already learnt earlier in the year that, quote unquote, two other words for power are exponent and index.
Explaining that we don’t use power because “GPMS” just doesn’t roll off the tongue as easily as GEMS get a cheap chuckle as well as helping them to remember exponents are powers.
All that changes is that we now have 4 lines: one where we underline all the powers and roots; one where we underline all the multiplication and division; one with just addition and subtraction; and one with the final answer.
4. G then E then M then S
Groups! We chanted the three different types of groups: brackets, numerator or denominator of the fraction, and under a root.
I taught the rule that we can only go onto E, M and S when our group has become one single number.
Once it has become a single number, you don’t have to write the brackets anymore.
5. E,M and S inside Gs
Next up is the idea that EMS works inside G as well. Practice with groups with complicated strings comes next.
A point that I need to remember to clarify upfront next year is the difference between (3 + 2)² and (3+2²).
6. Gs inside Gs
Finally, how to deal with lots of groups.
GEMS works inside a group. So if we have a group inside a group, that must come first. In other words, we start with the inside group and work outwards.
The pupils had some trouble when there was more than one open bracket in a row (e.g. 4 + 2 x ((8 +3) x 2 -3)). Letting the pupils highlight the matching open and closed brackets in different colours really helped them to see which group was inside which.
I said groups that were not inside each other could be done in either order. The key was to take them one at a time and NEVER PANIC :).
7. Inserting brackets/numbers/operations
Once they were superstars at evaluating any expression you could throw at them, we looked at problems where you have to add in brackets to give a particular value, or put the right numbers in boxes to get a particular value, or put the operations in between numbers to get a particular value.
I notice more and more areas where those with excellent number bonds and times tables knowledge benefit. This is one of them. For example:
(_ – _) x _ = 12 (Fill in using 3 numbers from 1,2,3,4 or 5)
Knowing factor pairs of 12 makes all the difference. It frees up working memory. You can quickly dismiss 12×1 and 6×2 as possibilities, and focus on 3×4. You can train your energy on ways of subtracting to make 3 or 4. Knowing fact families involving 3 and 4 is the next crucial step: it means you can work through the options efficiently to find a subtraction that works. Makes all the difference.
This lesson coincided with the launch of the 2015 challenge which has gone down an absolute treat. Highly recommended.
8. Some problems to make your brain hurt
We polished off the week with some problems to make your brain hurt (in the best possible way).
The top end particularly enjoyed working out how we could use 8, 8, 3, and 3 to make 24. Not easy! I let them graft for a good while, giving some clues about the structure. I was really pleased with how this went. Nudging different pupils in the right direction at different times meant everyone spent time struggling in a positive, productive way.
How it went
I was very pleased with how the pupils picked it up. I thought I’d encounter pupils wanting to use BI/ODMAS from primary, but none did. Perhaps my sales pitch about the benefits of GEMS was effective.
I was also worried that they’d be confused when encountering BIDMAS elsewhere. Thankfully, IXL, which we use for homework, is clear that multiplication and division/addition and subtraction are on the same level. The only switch children have to make is between “brackets” and “groups”. Since I banged on about GEMS being better precisely for the reason that it encompasses more than just brackets for G, they’ve taken this in their stride.
It’s been glorious not battling the “A before S” misconception.
It’s been wonderful to see some of the very weakest pupils successfully evaluating really big, nasty expressions with lots of roots, fractions and brackets. It’s a big confidence boost for them too.
I love being able to teach the nitty gritty of topics like this, rather than skimming over it in one lesson like the Scheme of Work at my old school demanded. Doing so can be daunting. You realise that what people can gloss over as “silly mistakes” are often misconceptions or generalisations that really take some time to unpick. But it’s so much more satisfying to teach like this.
It’s been a topic that seems to have hit the right notes across the attainment spectrum. The top end really enjoyed the challenging problems that were chunky enough to get them thinking for extended periods of time. The weaker kids have felt so proud of being able to tackle something that look massive, long and nasty that they would previously have shied away from.
Overall, I’d recommend teaching GEMS if you’re in a position to do so. In most schools, an individual teacher would be wasting their time in doing so. The spiral curriculum in the years previous and after would presumably be taught by a different teacher using the more common mnemonics, so I suspect the only effect would be that they’d be seen as “that weird teacher who hated BIDMAS”. But were you to be lucky enough to have the choice, I would plump for GEMS every time.