Pluses and minuses of mental maths
One of the great success stories of the National Numeracy Strategy has been the steady increase in children's ability to calculate mentally, and in their confidence to do so. It is quite common nowadays to observe a child in reception class finding out "how many beads there are when two beads are added to the three on the table", by counting "3, 4, 5"; or a Year 1 child finding the sum of 6 + 5 to say: "5 and 5 makes 10, so it's 11".
I propose a different calculation strategy, with a view to improving the way I have seen it taught in several schools.
In the National Numeracy Strategy framework, this particular strategy is first introduced in Year 1 and developed throughout key stage 1 via the objective: "Add or subtract 9, 19, 29I or 11, 21, 31I by adding or subtracting 10, 20, 30I and adjusting by 1."
This develops in KS2 into: "Add or subtract the nearest multiple of 10 and adjust", only to reappear, renamed, in the context of written calculations as compensation.
Several years ago, before the National Numeracy Strategy was developed, I interviewed 144 seven-year-olds about their mental calculation methods.
Only one of those children actually added 9 by first adding 10 and then taking one away. This led me to conclude that the strategy involved was not a "natural" strategy, by which I mean "unlikely to be invented by children".
My findings suggested that teaching this strategy, with understanding, might prove more difficult than we think. What I particularly want to question about it is the wisdom of working on adding 9 by "Adding 10 and subtracting 1", at the same time as adding 11 by "Adding 10 and adding 1", as I have seen suggested in National Numeracy Strategy materials.
At first glance, it might seem a sensible way of proceeding, given the ease with which both procedures can be demonstrated, or modelled on the 100-square (see Figure 1). But the result is that many children have difficulty in deciding which way to move in the second part of the calculation, often resulting in an incorrect answer.
Also, when the subtraction of 9 and 11 is introduced later via "Subtracting 10 and adjust", this makes the situation even more complicated.
The problem arises because the National Numeracy Strategy equates the strategies involved in adding 9 and in adding 11, whereas they are fundamentally different.
Adding 11 involves adding 10 and then 1 - exactly the same strategy involved in adding 12, 13, 14 and so on. But adding numbers ending in 9 involves modifying what you have been asked to add - say, 29 - Jand adding 30 instead, ensuring that you remember to compensate by taking away one at the end of the calculation.
I would argue that children need a great deal of confidence to do this, and that it makes far more sense to link adding 9 with subtracting 9, particularly if you model the procedures on the empty number line, although you would no doubt introduce addition first and then relate this to subtraction at a later date (see Figure 2).
Working in this way enables teacher and pupil to focus on the compensation aspect of the strategy. After the initial 10s jump, the child has to compensate for "over-jumping" by jumping one unit in the opposite direction. This process is the same whether the operation in question is addition or subtraction.
Also, treating adding and subtracting 9, 19, 29I (and even 8, 18, 28I) as a different strategy from working with numbers ending in the digits from 1 to 7 means that children can be introduced to the correct terminology for the procedure, thus providing continuity with work in KS2, where a written version of compensation is introduced.