We are continuing our 'Getting the basics right' theme this term with a series of posts on calculation. This blog post from Nick Hart looks at strategies for counting up, helping children to move from heavily scaffolded jottings to mental calculation. Make sure you look out for Part 2 of this blog post coming in November!
A subtraction calculation is always a ‘part unknown’ question. Counting up from the known part to the known whole will result in working out the unknown part and is a strategy that is particularly useful when the known part and whole are close in size. In working with children who have found this strategy difficult, it is obvious that the number of steps involved and the basic number knowledge required needs to be worked on systematically if the child is to move from heavily scaffolded jottings to more efficient jottings to mental calculation.
In the example above where children have to subtract a three digit number from another three digit number, the trickiest part in my experience is the first jump from the known part to the next multiple of 100. If the child does not know number bonds to one hundred by heart then this first step can be time consuming and often results in the child forgetting the big picture. All of their thinking capacity will have been taken up dealing with that initial step and they need heavy support from the teacher to get back on track once they have completed that step. To combat this, the child of course needs to work on complements to one hundred. First, they should do this concretely and base ten blocks are perfect. Place a hundred on the table, make the known part with tens and ones, then show children how to fill the spaces with ones, then tens, counting as you go.
With sufficient practice, there will come a time when this strategy can be made more efficient. For example there will no longer be a need to represent the hundred or the known part. Both of those can be held mentally. The child then just needs to use the ones and then the tens to count from the known part up to 100.
At some point, this will also become cumbersome, in which case it is time to introduce a pictorial representation. Instead of counting out ones and tens to get to one hundred, the child can draw ones and tens using dots and lines, counting as they do so up to one hundred.
When the drawing becomes inefficient, the child should be at the stage where they can count mentally or better still the ones and tens simply pop into their head. This last condition of course relies on secure knowledge and quick recall of number bonds to ten. If the child does not have that knowledge, they can acquire it whilst still working on number bonds to one hundred.
Above is every number from one to one hundred - of course children need lots of practice answering questions where we say a number and they work out the complement to one hundred. At this stage, it’s the order in which we ask the questions which can aid or hinder fluency. Consider two possibilities:
Option 1: Work through from 1 to 100, finding each complement
Option 2: Randomly select questions to ask
Option 1 sounds appealing. For the first few numbers, children might work out the complement mathematically soundly. At some point they will realise a pattern though and will take a short cut. Now this is fine if the child can reliably and quickly calculate complements to one hundred but if they can’t, this shortcut is to the detriment of their understanding.
Option 2 is probably the default. However the continual variation could slow children down to an extent that they do not do enough practice to internalise the concept.
There is an option number 3 that deliberately makes use of our innate search for pattern whilst keeping the task focused on the mathematical thinking that we want from children.
By focusing on all the numbers that end in 5 in turn, the child should begin to internalise that if one part ends in 5, so does the other. They can bank the ones digit as a 5 for the unknown part then count up to one hundred accordingly. These numbers that end in 5 need not necessarily be pointed out in order. Randomising these numbers works just fine. The same can be done with other numbers. For example, the child could look at all the numbers that end in 7. They will soon realise that if the known part ends in 7, the unknown part will end in 3. Simply sitting with the child, pointing out numbers in a deliberate order can have a huge impact on their fluency. We could also time how many they can do in 2 minutes (or how long it takes to complete all the complements) which turns this practice task into one that can easily be analysed for progress. Over time they will do more in the allocated time or complete all complements quicker.
In the second part of this post (coming in the New Year!), we will return to the counting up strategy and look in more detail at getting children to think more efficiently, moving from heavy scaffolding with base ten blocks and a number line to working entirely mentally and at speed.
Looking for resources to help support the development of number fluency? Take a look at our Arithmetic Practice Tests and Mental Maths Tests.