Adding meaning to subtracting
Robert Wilne is Deputy Director (Maths) at the Atlas TSA family of seven primary and secondary schools in South East London, where he is leading the development of their 3-19 maths curriculum. He’s also the London Thames Maths Hub Work Group Lead for the ‘Improving Continuity Across the Y5-8 Transition’ Network Collaborative Project. You can hear Robert talking about the Y5-8 Transition Project (in the context of fractions) in our latest podcast. And you can read more about a continuous approach to fractions in KS2/3 in the latest issue of Bespoke, the newsletter of the Maths Hubs programme.
Last May I was teaching a Y11 Higher Tier revision class: 25 or so students with secure mock grades 7+. Then suddenly, in the middle of their confident answering of exam-style questions about the cosine rule, quadratic simultaneous equations and the like, it all crashed to a halt and I found myself drawing a number line and explaining to them about “9 – 5 = 4”!
I’d asked them
- The point P has coordinates (-3, -2) and the point R has coordinates (9, 18). Q is on the line PR, such that PQ is the length of QR. What are the coordinates of Q?
Pretty quickly, a student came to the board, drew a diagram and explained her reasoning:
“9 minus minus 3 is 12 and 18 minus minus 2 is 20, and we want ¼ of each of these so it’s 3 and 5, so we add 3 to minus 3 and 5 to minus 2 and get 0 and 3 for Q’s coordinates.”
Her peers agreed, we sharpened the language (discussed in the footnote at the end of this article) and were about to move on when another student asked
“Why are we minusing? We want to find the difference between -3 and 9, what’s that got to do with subtracting?”
And that question revealed to me a huge gap in his (and in his peers’ as they agreed with him) conceptual understanding of something as fundamental, as bottom-layer-of-the-Jenga-tower, as whole number subtraction: that ‘9 – 5 = 4’ is an abstract representation of the two very different concrete processes of ‘take away’ and ‘difference between’. The same string of symbols represents two very different pictures:
The pictures are different because the models are different: the first one is “subtraction representing reduction” and the second is “subtraction representing comparison”. The different models always give numerically the same answer, so long as we are precise about interpreting “difference between” as “from the subtrahend (the second number in the subtraction) to the minuend (the first)”. This precision, that difference between means difference from the subtrahend to the minuend, becomes necessary when the abstract subtractions become less easy, or less natural, to interpret as ‘take-away’ in the concrete, particularly when the subtrahend is negative:
- Why does 9 subtract 3 = 6? Because the difference between 3 and 9, from 3 to 9, is 6.
- Why does -9 subtract 3 = -12? Because the difference between 3 and -9, from 3 to -9, is -12.
- Why does 9 subtract -3 = 12? Because the difference between -3 and 9, from -3 to 9, is 12.
- Why does -9 subtract -3 = -6? Because the difference between -3 and -9, from -3 to -9, is -6.
A model of temperature change leads to the same numerical answers:
- if the thermometer yesterday recorded 3°C and today records -9°C, then the temperature change from 3°C to -9°C is a fall of 12°C, which we can write as -12°C, hence -9 - 3 = -12;
- if yesterday it recorded -3°C and today it records -9°C, then the temperature change from -3°C to -9°C is a fall of 6°C, which we can write as -6°C, hence -9 - -3 = -6;
- if yesterday it recorded -9°C and today it records -3°C, then the temperature change from -9°C to -3°C is a rise of 6°C, which we can write as 6°C, hence -3 - -9 = 6.
My Y11 students didn’t know, or didn’t grasp, the difference between (pun fully intended) ‘difference between’ and ‘taking away’: why not? They’ve known about subtraction for umpteen years! The operation of subtraction occurs in every key stage: the youngest learners subtract small positive integers from slightly larger ones, and A-level Further Maths students subtract the arguments of complex numbers and wonder what the connection is with logarithms. In the middle, usually in Y7 or Y8, there is the conceptual shift from subtracting ‘numbers of things’ to subtracting ‘abstract numbers’, and then to subtracting ‘abstract expressions’. KS3 teachers should be asking their students to explore what’s the same and what’s different about ‘9 – 3’ and ‘9½ − 3.7’, and ‘9 − -3’ and ‘-9 – 3’, and ‘9x – 3x’ and ‘-9x − -3x’.
If learners are to develop confident, flexible and secure understanding of subtraction, they need to encounter it in a conceptually and procedurally continuous way, in every key stage. That can only happen if teachers in each key stage communicate with each other about how they are teaching subtraction: the procedures their students are using, and the language they are using to describe what those procedures are representing. For example, consider this Y6 SATs question:
The first step is to calculate “£20 subtract £14.96”. In my experience, most secondary teachers see this as a ‘taking away’ problem, and they expect their students to apply the column subtraction algorithm and get the answer “£5.04” (assuming they navigate (twice!) the difficulty of exchanging between adjacent columns). But many primary pupils see this as a ‘difference between’ problem, and they reason “£14.96 add on 4p and then add on £5 is £20, so the difference from £14.96 to £20 is £5.04”. If the secondary teacher presents ‘taking away’ reasoning to a class of pupils who are used to ‘difference between’ reasoning, confusion is inevitable.
The primary pupils’ ‘difference between’ reasoning is deeper and more powerful than might at first appear: they are reasoning, probably without realising, that
The pupils, therefore, are switching between models: they are working out a ‘take away’ by thinking of it as a ‘difference between, from subtrahend to minuend’. Similarly, we can reason that:
58 ‘take away’ 45
≡ 53 ‘take away’ 40
≡ 43 ‘take away’ 30
without working out that each subtraction equals 13:
This is The Principle of Constant Difference: in a subtraction, if the minuend and the subtrahend both increase or decrease by the same amount, the difference between them (from the subtrahend to the minuend) stays the same. This Y7 student’s number line shows the ‘difference between’ reasoning clearly. She has drawn arrows of the same length to show that the difference from the subtrahend to the minuend is constant when they both increase by 10:
To return to Amina’s stamps: the Principle of Constant Difference is an efficient way to calculate
- £20.00 – £14.96 ‘take away’ has the same numerical answer as ‘difference between’
- ≡ £20.01 – £14.97 because the minuend and the subtrahend both increase by 0.01
- ≡ £20.02 – £14.98 because the minuend and the subtrahend both increase by 0.01
- ≡ £20.03 – £14.99 because the minuend and the subtrahend both increase by 0.01
- ≡ £20.04 – £15.00 which is easy to work out: a ‘nasty’ subtraction has become ‘nice’
- = £5.04
Pupils can use the Principle to reason about subtraction throughout KS2 and KS3:
- in Y5
- and in Y7
Notice that the Y7 students are reasoning about the subtractions, rather than working them out explicitly: this activity is developing their conceptual understanding rather than their procedural fluency. Similarly in Y5:
The power of the Principle is that it extends naturally to subtractions that are procedurally more demanding and/or conceptually more challenging, in particular subtractions with negative subtrahends, and then also those with algebraic terms in the minuend, the subtrahend, or both:
- 5.3 – 2.7 take away’ has the same numerical answer as ‘difference between’
- ≡ 5.6 – 3 because the minuend and the subtrahend both increase by 0.3
- = 2.6 which is easy to work out: a ‘nasty’ subtraction has become ‘nice’
- 8 – -2 take away’ has the same numerical answer as ‘difference between’
- ≡ 9 – -1 because the minuend and the subtrahend both increase by 1
- ≡ 10 – 0 because the minuend and the subtrahend both increase by 1
- = 10 which is easy to work out: a ‘nasty’ subtraction has become ‘nice’
- -5 – -13 ‘take away’ has the same numerical answer as ‘difference between’
- ≡ -2 – -10 because the minuend and the subtrahend both increase by 3
- ≡ 0 – -8 because the minuend and the subtrahend both increase by 2
- 8 – 0 because the minuend and the subtrahend both increase by 8
- = 8 which is easy to work out: a ‘nasty’ subtraction has become ‘nice’
- -3.8 – -7.39
- ≡ -4.0 – -7.59 because the minuend and the subtrahend both decrease by 0.2
- ≡ 0 – -3.59 because the minuend and the subtrahend both increase by 4
- ≡ 3.59 – 0 because the minuend and the subtrahend both increase by 3.59
- = 3.59 ‘nasty’ has become ‘nice’
and then, later still
- -9x – -3x
- ≡ -8x – -2x because the minuend and the subtrahend both increase by x
- ≡ -7x – -x because the minuend and the subtrahend both increase by x
- ≡ -6x – 0 because the minuend and the subtrahend both increase by x
- = -6x ‘nasty’ has become ‘nice’
- (3x + 5) – (x + 4)
- ≡ (3x + 1) – (x) because the minuend and the subtrahend both decrease by 4
- = 2x + 1 ‘nasty’ has become ‘nice’
- (7x – 3) – (2x – 5)
- ≡ (7x + 2) – (2x) because the minuend and the subtrahend both increase by 5
- = 5x + 2 ‘nasty’ has become ‘nice’
- (7x – 3) – (2x – 5)
- ≡ (5x – 3) – (-5) because the minuend and the subtrahend both decrease by 2x
- ≡ (5x + 2) – (0) because the minuend and the subtrahend both increase by 5
- = 5x + 2 ‘nasty’ has become ‘nice’
The Principle of Constant Difference is procedurally powerful and conceptually rich, and is simple enough that it can be – I would say it should be – explored and grappled with and used throughout the ‘middle years’, i.e. from Y5 to Y8. This would ensure that pupils do indeed encounter subtraction in a conceptually and procedurally continuous way from primary to secondary. For this to happen, though, primary and secondary teachers need to be
- confident themselves with justifying and using the Principle;
- confident that their ‘feeder primary’ or ‘destination secondary’ colleagues are justifying and using it too;
and for this to happen, there needs to be regular cross-phase communication between subject leaders – or, even better, cross-phase professional development – so that primary and secondary teachers come together and learn together, as recommended by the Education Endowment Foundation:
Creating the opportunity for, and the culture of, dialogue and development between primary and secondary teachers is one of the key aims of the Maths Hub project ‘Improving Continuity Between Primary and Secondary School’. To find out more and get involved, contact your local Maths Hub directly, which you can do via www.mathshubs.org.uk.
“9 minus minus 3” is woolly because one word (“minus”) is being used to describe two operations: the binary operation of subtraction (“take away”) and also the unary operation of negation. Better, therefore, is to say “9 subtract negative 3” or “9 take away negative 3”, or even “9 minus negative 3” or “9 subtract minus 3” – just avoid using the same word “minus” to describe different operations.