Young children are often fascinated by comparing and ordering the sizes of things. Perhaps it appeals to their innate sense of justice to determine whose apple is bigger and their equally well developed competitiveness to see who is taller. Early Years teachers build on this by providing lots of opportunities to compare and order things and begin to use non-standard measures to quantify. How many grapes balance an apple? How many cubes high is the toy garage? How many cups of tea can be poured from the teapot? At this stage, it’s important too to give children lots of opportunity to experience and use the language associated with comparison: more, less, fewer, higher, lower, taller, shorter, heavier, lighter etc. I’ve put together a few ideas for activities which support developing comparison language and you can download the document from the link at the bottom of this post.
As children move on in their understanding of measures, we move to using standard units of measure. Children often struggle with estimating length, mass or capacity using standard units and they need lots of practical opportunities to measure familiar things using these units. Wherever possible, opportunities should be found outside of the maths lesson for these activities, perhaps as part of topic work, for instance, to give them a meaningful context. Children can weigh out ingredients for their chocolate snack in technology or find the capacity of a liquid before an evaporation experiment in science, or measure how far they can jump in P.E. Another activity that can support children in becoming more familiar with units of measure is to give regular opportunities for estimating, and use these as opportunities to develop the skill of working out an unknown measure by comparing it with a known one. Estimation 180 is a great source of visuals to support this (I blogged about this site here.)
Another common difficulty for children is remembering just how many grams in a kilogram, centimetres in a metre, millilitres in a litre etc. One activity that can support this is by including counting in measures in daily counting activities, alongside counting in whole numbers, decimals and fractions etc. So, for instance, when children are counting in hundreds, also count in steps of 100 grams. I find a counting stick useful for this. Develop skills progressively. So for instance, you might count up first of all from 0 to 1 kg in steps of 100g, moving backwards and forwards along the counting stick. As children become more familiar with this, use different starting points so that they become familiar with what happens after 1 kg. At this point you have a choice of ways to count: 1100g, 1kg and 100g, 1.1 kg or 11/10 kg, and I’d suggest you use all of these ways alongside each other so that children start to also understand the equivalence of these. Doing this will also help enormously when children begin to convert units of measure.
Children often find reading scales challenging too. Again, there is no substitute for practical experience, and if you are able to have analogue scales, measuring jugs, tape measures etc. continually available in your classroom, this can be helpful in making it easier to pick up on opportunities for measurement that arise in other subject areas – a trip to hunt through the maths cupboard will probably make you less likely to do this! The Measuring Scales ITP and Measuring Cylinder ITP can also both be helpful for focused opportunities to practise measuring scales skills. Again, counting can also be useful in supporting reading scales. Most scales are in intervals of 1, 2, 5, 10, 20, 50, 100, 200, 1000 etc. so regular opportunities to practise counting in these steps will help children to use these skills when reading scales.
One of the main problems with children working with measures, I suspect, is that we move far too quickly to working with abstract measures or with diagrams rather than working practically. I’ve been guilty of this myself – practical work involves finding equipment, it can be messy (particularly when working on capacity). But practical work can also be lots of fun and really help children connect their learning to real life situations, so I’d encourage you to do as much as possible.
There are other ideas and resources for teaching measures on my Measures 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.