Addition is perhaps the most straightforward of the four operations to understand, but that doesn’t necessarily mean it’s always easy to teach. I was speaking to a Year 5 teacher earlier this week who had planned to start this term by spending a couple of weeks revising the four operations with her (lower set) maths group. She’d planned only a couple of days on maths, but after two days still felt there was a lot the children weren’t understanding and decided to keep going for the rest of the week and possibly into next week too.
Having a good understanding of the progression of skills for addition can help when trying to ‘unpick’ why older primary children are having difficulty with it. Are they familiar with the vocabulary around it? Do they have good mental images of what is happening when they add? Are they being let down by a shaky grasp of basic number bonds which leads to mistakes in some steps of longer methods? Is there understanding of place value secure? Have they got a good conceptual understanding of the method they are using rather than trying to remember a ‘trick’ they were taught last term or last year?
Children’s early experiences of addition should be very practical and the idea should be introduced in a meaningful context? They need to understand addition both as combining two sets of objects and as adding more to an existing set. Early Years environments tend to be full of opportunities for them to do this. We have 4 red cars and 3 blue cars, how many cars altogether? Seth has 3 sweets and Tami gives him two more, how many does he have now? After lots of experience with concrete objects, children may be ready to move to using representations. So instead of using actual cars or sweets, cubes or counters might be used to represent them. Another step beyond this is to use pictorial representations, such as drawing circles or dots to represent objects. One very powerful representation for addition is the bar model (as above) which clearly shows two parts making one whole.
After lots of experience of combining and augmenting sets, children will begin to learn some of the number facts, beginning with those within ten. Working with tens frames, Numicon or Cuisenaire can all be helpful when doing this. Tens frames and Numicon particularly reinforce the idea of ten being two lots of five, helping children to build mental images of numbers. This program is useful for demonstrating the use of tens frames. Begin by adding on one more and then two more – if children are used to counting, this should come fairly easily. After a while, they will need to start learning the number bonds to 10 and within 10 – doubles can often be a good place to start with this. For some ideas about learning number bonds, one of my blog posts from last year has some ideas.
Before children start to add larger numbers together, they should be familiar with the idea of place value and have experienced making numbers with Dienes or other base ten equipment. As the totals they add go beyond ten, they can use the Dienes units to model this and start to exchange ten unit cubes for a tens rod. Later, children can start to add single digit numbers to 2-digit numbers using the Deines. It can be helpful to also model how to do this using hundred squares.
When adding 2 digit numbers, again it is helpful to start by using equipment so that children see the way that partitioning and recombining works in practice. This can later be recorded using lines and dots to represent the Dienes rods and cubes. The next step is to record the partitioning and recombining process, eg.
leading later to expanded column addition, eg.
and later still to more formal column addition.
The important thing is not to move children on until they are secure at any particular stage.
The same process can then be applied to increasingly large numbers so that by the time children get to the top end of primary they are able to add numbers with at least 4 digits. When children are familiar with decimal numbers and their place value, the same methods can be used to add decimals too.
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.