At about this stage of the term, many teachers will be teaching calculation. Look at most schemes of work or medium term plans and you will probably find roughly equal amounts of time given to covering addition and subtraction. Yet, almost any diagnostic assessment will tell you that most children are far more secure in understanding addition than they are with subtraction. Perhaps we need to make sure we give a little (or maybe even a lot) more weighting to teaching subtraction.
So why does subtraction cause so many problems? Well let’s think about some of the ideas we use when learning about subtraction in school.
- The ‘taking away’ idea – probably the first that children come across. One group of objects is taken away from a larger group and typically we count what’s left.
- The ‘difference’ or ‘counting up’ idea – we count up from the smaller number to the greater and find the difference between them
- The ‘counting back’ idea – we count back from the bigger number by the number of steps in the smaller number
- The ‘inverse of addition’ idea – we work out what must be added to the smaller number to make the greater number
No wonder our children get confused! Our teaching needs to help them make connections between all these ideas and will need to involve lots of practical work and the use of models and images, particularly number lines.
The new curriculum puts much more emphasis on using formal column methods for calculation and on building fluency with these. If these methods are to serve our children well, it’s vital that well before we move onto them we have laid the foundations by building secure conceptual understanding.
This will start in Early Years and KS1 classroom with lots of practical work, wherever possible using real life situations which connect with the children’s experience of life. This is also the stage where it’s important to start building up children’s mathematical vocabulary by lots of careful modelling and opportunity for discussion. Particularly important at this stage is the language of comparison: greater than, less than, more than, fewer than etc. It will also help enormously if children start to get a ‘feel’ for numbers and the way they can be split apart in different ways. Later on, regrouping is going to be needed and children will find this much easier if they are already comfortable with splitting numbers up in multiple ways. My blogs on building number sense and learning number bonds give some ideas which might be useful. After lots of practical experience, children can be taught to record their work using number sentences, but only once they have clear mental pictures to accompany these.
Counting is another important skill that lays the foundations for subtraction, particularly counting backwards and counting over tens and hundreds boundaries.
Once children are becoming confident with manipulating numbers, number lines may be introduced. It’s important however that children are able to connect the counting that they do along a number line with the practical work they have done. It is not obvious initially to many children that, for instance, 12 – 8 can be represented by counting up from 8 to 12 along a number line. One way of visualising this is to scribble out the portion of the number line up to 8 (representing the part taken away) as above.
At first, children may use ready-made number lines but as they grow more confident, they will be able to draw their own number lines to suit each new calculation. As understanding progresses, counting on along the number line can be used for increasingly large numbers and children will count on using larger jumps, usually to the next tens or hundreds number and then counting on in tens or hundreds. This program can be helpful in illustrating this. As the use of the number line becomes increasingly sophisticated, it’s important to keep making connections with other representations of subtraction. How could you represent the calculation with base ten equipment, for instance?
Eventually children will be ready to move onto column methods and at this stage, it’s vital that we don’t just teach them a procedure. We need to show them how it works by using models and images alongside the formal calculation. This program shows how the expanded method works alongside the more formal compact method. I would suggest also using base ten equipment to make what is happening even more clear.
For more calculation ideas, my pinterest calculation board might be useful.