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.