Order of operations (a note from Richard Dunne, to accompany the Dispatches film Kids Don't Count.)
When you look at an expression like ...
2 × 4 – 1 × 4 + 1½ × 2
... the 'stuff' between each addition or subtraction sign (and the beginning and end of the expression), no matter how complicated it might be, is called a 'term', so you can see three terms:
2 × 4 – 1 × 4 + 1½ × 2
The point about this is that you then evaluate each term (in this case: 8, 4, 3) so that
2 × 4 – 1 × 4 + 1½ × 2
has the same value as 8 – 4 + 3 and then you work from left to right to get the answer 7.
Some schools (especially in the past) used the mnemonic BODMAS, which (it is said, but hang on! ...) specifies the order of operations as:
Brackets / Of / Division / Multiplication / Addition / Subtraction.
Unfortunately, this was routinely (and wrongly) recalled [and perhaps taught!] to mean that 'brackets' were followed by dealing with the four operations in that order. What it in fact means that brackets are evaluated first, then division and multiplication are to be done before addition and subtraction, but each of these pairs of operations are to be evaluated from left to right.
You can see that 8 – 4 + 3 is wrongly evaluated as 1 by people who have misunderstood BODMAS, failing to get the correct value of 7.
Because the understanding (or teaching?) of this was so poor, the fashion was introduced of writing my expression
2 × 4 – 1 × 4 + 1½ × 2
like this:
(2 × 4) – (1 × 4) + (1½ × 2)
which would ensure that people did evaluate each term (because each term is now placed in brackets). It seems like a good idea, but it is not!
The trouble lies in the fact that this means that this makes it easy for pupils (when good teaching resists making it 'easy' and instead makes it 'clear'). The 'ease' renders the logic unexamined. This means that when these pupils see 8 – 2 × 3 (which is 8 – 6 = 2) the pupils (in secondary schools) tend to interpret it as 'The brackets are missing. I'll put them in.' and lurch, unthinking, into
(8 – 2) × 3 = 18
18 is the correct answer to (8 – 2) × 3 but this was not what was asked!
I was quite specific and unambiguous in asking 8 – 2 × 3.
It may be interesting to know that The Maths Makes Sense Learning System ensures that the four operations are taught in Key Stage 1 so that this mistake cannot be made. The Learning System, using concrete apparatus, exaggerated actions and special vocabulary, is designed so that this little matter, like so many 'little matters', is resolved through absolute clarity. I really am not interested in allowing people to suffer misconceptions which at some stage have to be 'mended'.
If you would like some indication of why this convention has been adopted I will supply a little bit of algebra which will seal the matter.
Excel spreadsheets are programmed to use the convention and evaluate 8 – 2 × 3 correctly as 2.
'Scientific' calculators (the more expensive calculators) have enough memory to be programmed to give the correct answer. It is only the cheap (ie small memory) calculators which make the mistake of giving the wrong answer 18 for 8 – 2 × 3.
In this case, there is a need to know this and (if the numbers were hard) to use the memory function to store the value of 2 × 3 and then subtract it from 8.
Maths Makes Sense teaches young children to test any calculator they are handed 'to see if it is mathematical or not mathematical'.
