Structures of Subtraction - What's the Difference?
Posted 13th March 2019
As one of the four number operations, subtraction is a concept that is familiar – and indeed possibly routine – to every primary school teacher. Yet, it is perhaps surprisingly complicated. Subtraction has three separate structures. Many pupils’ conceptual misunderstandings of subtraction originate in their lack of awareness of these structures, resulting in pupils at the end of primary school struggling to answer seemingly simple questions such as, “The population of Cambridge is 125,000 and the population of Bedford is 89,000. How many more people live in Cambridge than Bedford?”. Many pupils respond, Pavlovian-esque, to the word ‘more’ and add, calculating the answer as 214,000.
The first subtraction structure is the reduction model, and this is the model that is often mistaken for being the only model of subtraction, leading to such confusions as pupils (and many adults) referring to the subtraction sign as ‘take away’; this is only accurate when the subtraction follows the reduction structure.
Reduction is the most easily understood of the structures, and the one that children encounter most often in everyday life. Reduction concerns starting with a quantity and then this quantity getting smaller (for example, ‘Katy starts with 7 sweets and eats 3, how many sweets does she have left?’). The First….then….now story model works well when teaching young children the reduction model: ‘First Katy had 7 sweets, then she ate 3, now she has how many sweets?’.
These reduction stories need to be carefully considered to ensure that reduction which is being explored. Reduction involving small quantities should be represented concretely and pictorially, with children physically moving objects away and crossing out drawings, before using a single bar model with the subtrahend (the number being taken away) crossed out.
The second model of subtraction is partitioning. As pupils have become comfortable with subtraction being regarded as ‘taking away’, partitioning questions can prove challenging initially; pupils come to conclude that subtraction is only used when a quantity is being taken away. Partitioning questions demonstrate that this is not the case. When a subtraction question requires partitioning, the first number (minuend) is being split into two parts (the subtrahend and the difference). Crucially, nothing is being lost when partitioning occurs, yet subtraction is still necessary to calculate the answer. For example, ‘Katy has 7 sweets, some are marshmallows and the rest are lollipops. If 3 are lollipops, how many are marshmallows?’. No sweets are being eaten, thus 7 is not being reduced in magnitude. Rather, 7 is being split into two groups – lollipops and marshmallows. Concrete resources such as cubes (labelled as lollipops and marshmallows) support pupils’ understanding of this structure, before linking the creation of two groups of manipulatives to a part-whole model.
Thirdly, there is the comparative difference model of subtraction. This is distinct from both reduction and partitioning because with difference problems the ‘whole’ (the whole number of apples, or the whole number of pupils in the class) is not relevant to the situation (indeed, calculating the ‘whole’ is very often unhelpful when answering difference questions). This subtraction structure is therefore considerably removed from the reduction model (where the reduction of the magnitude of the whole is of central importance). A pictorial representation using a comparison bar model is helpful for pupils to use here, to avoid them relying on frequently misleading linguistic ‘clues’ in a difference question (the use of ‘more’ in the population example given above being such a clue). To consider the population question we discussed earlier, being able to represent this as a bar model makes the necessary operation explicit:
The bar model shows clearly that the word ‘more’ in the question refers to the gap or difference between the bars, which each represent the population of a locality. Note that the gap is not shown as a bar but rather with an arrow: when bar modelling, bars are used to represent quantities which exist (so number of apples, pens, or in this case people). Frequent practice of difference questions (and use of the bar model to show what difference is being calculated) is crucial.
In conclusion, as teachers we need to be conscious of the three distinct subtraction structures, and ensure that our pupils have deep experience of exploring all three during their primary Maths education. Reduction is generally taught first, and as pupils progress through primary school, they encounter the partitioning and difference structures. Yet, it is crucial that pupils who are nearing the end of their primary career continue to read, understand and answer reduction questions. As with all topics within Maths, the mathematical language we use (and which we insist on pupils using) in the classroom when teaching subtraction is of critical importance if pupils are to avoid all too common misconceptions; specifically, the mathematical symbol – means ‘subtract’, not ‘take-away’.
Can you match these questions to their subtraction structure (some questions may include more than one structure)?
In a box are 5 cars. Two are Jane’s and the rest are Ian’s. How many are Ian’s?
I have 4 bananas. How many more bananas do I need to buy so that I have 10 in total?
Mount Everest is 8,848 metres high. Ben Nevis is 1,344 metres high. How much higher is Mount Everest than Ben Nevis?
There are 191 pupils in a school. 84 are girls. 23 boys leave the school. How many boys are left in the school?
Sarah has £985.00 in her bank account. She spends £110.00 on Monday. On Tuesday she spends half of what she spent on Monday. On Wednesday she spent £248.00. How much money does she have left?
References:
Beckmann S (2004), Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore, The Mathematics Educator, 14 (1), pp42 - 46
Harris A (2000), Addition and subtraction, St Martin’s College
NCETM (2018), Additive structures: introduction to augmentation and reduction
