Early in the term is a good time to go over the basics of reading and writing numbers and putting them in order.
Young children need plenty of practice in reading, writing and representing numbers, and this can usefully be part of a ‘Number of the Day’ activity like this one. It’s important that children start developing their number sense alongside this and representing numbers with practical equipment like Dienes, Numicon or ten frames or by drawing tally marks will help them to do this. The Gordons program ‘Dienes and Coins’ is useful for these representations and this site has some nice interactive ten frames. As children get older, they need to learn to understand much bigger numbers. Children are often fascinated by really large numbers, and once they get the hang of how the number system works, more able KS2 children will enjoy trying to read and write multi-digit numbers. This Wikipedia page lists the names up to centillions.
Number tracks and number lines can also be useful in helping children get a sense of the relative size and position of numbers. The Mathsframe site (which I love) has a really useful activity where children put numbers on number lines. There are lots of different levels at which to use this activity and the option of showing divisions on the line or not. Older children need to also get a sense of how decimal numbers work and the Decimal Number Line ITP is very useful for this. The programme allows us to ‘zoom in’ on a portion of the number line and expand it to look at what happens within that portion. Children in KS1 will also be starting to find numbers on hundred squares. One useful activity is for children to cut up a hundred square along the horizontal lines and then lay out the rows end to end to make a 0-99 (or 1-100) number line. This helps them to see the connection between the hundred square and number lines and shows them why we skip to the next row when counting on over a tens barrier. Putting together ‘jigsaw’ pieces of the hundred square can be useful for children in developing their understanding of how these work, and Nrich has a jigsaw activity which can be used either in its interactive version or in printed form.
Ordering numbers is also an important skill. When the NNS first came in, washing lines of numbers were standard in nearly every classroom and these are well worth using still. Having sets of number cards of different sizes means that these can be regularly changed. Children enjoy putting numbers in order on these or spotting numbers which have mysteriously been switched overnight. The Gordons ordering program uses this idea and again Mathsframe has some good ordering activities. Some of the levels on this are only available to subscribers but a subscription is good value in my opinion as there is a wealth of resources on this site.
Reading, writing and ordering numbers links well with work on place value and there are ideas for that at my blog from last year and there are other ideas for teaching number on my Number Pinterest board.
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.