To What Extent Does the Teaching of Mental Calculation Strategies Support the Development of Formal Written Methods?

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TO WHAT EXTENT DOES THE TEACHING OF MENTAL CALCULATION STRATEGIES SUPPORT THE DEVELOPMENT OF FORMAL WRITTEN METHODS?

INTRODUCTION

To answer the question to what extent does the teaching of mental calculation strategies support the development of formal written methods we should first of all consider what is meant by ‘mental calculation’. I would consider it to be a way of finding solutions to problems using a knowledge of number facts and strategies, without using written algorithms or methods. This could be done either by ‘visualising’ the problem or simply ‘just knowing’ it.

BACKGROUND

Prior to the introduction of the National Curriculum, there was very little emphasis placed on teaching mental calculation strategies. The ‘mental’ aspect of mathematics was usually done in the form of recitation of multiplication tables or sometimes as twenty questions at the start of the lesson. These methods could certainly help to embed a body of facts within a child’s mind, but it did not provide strategies or a knowledge of how to apply these facts to other mathematical problems. The Cockroft report (1982) stated that one reason it did not occupy a prominent position within most mathematics teaching was that

‘ It is difficult to find mental questions which are suitable for all the pupils in the classes’,

this was due to the fact that most primary classes contained pupils of a very wide range of abilities. I believe that this statement shows the difference in our definitions of mental calculation, we would not consider it to be a matter of shouting out questions to the class, rather it would involve working on strategies and methods. The report did however recognise that the decline of teaching mental and oral work within mathematics

‘ represents a failure to recognise the central place which working in the head occupies throughout mathematics’.

Investigations of the Numeracy Task Force led to the Implementation of the National Numeracy Strategy. These investigations centred on research into evidence provided by the National Numeracy Project, OFSTED investigations experience of mathematics educators and practice of other countries. The National Numeract Strategy was aimed at achieving a national target that by 2002, 75% of 11 year olds would reach the level expected for a child of their age. According to OFSTED’s review of Primary schools, in 1998 at key stage 2 only 59% of pupils achieved level 4 or above which was well below this target.

The Numeracy Task Force produced a list of recommendations for the National Numeracy Strategy. Of the key recommendations, one feature was that in the daily mathematics lesson

‘ oral and mental work should feature strongly’.

They also recommended that the definition of numeracy underpinning the National Numeracy Strategy should include

‘calculating accurately and efficiently, both mentally and on paper, drawing on a range of calculation strategies’.

Both of these statements confirm the view that mental calculation lies at the heart of being proficient with numeracy.

The National Numeracy Strategy’s Framework for Teaching Mathematics pinpoints how the development of ...

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The Numeracy Task Force produced a list of recommendations for the National Numeracy Strategy. Of the key recommendations, one feature was that in the daily mathematics lesson

‘ oral and mental work should feature strongly’.

They also recommended that the definition of numeracy underpinning the National Numeracy Strategy should include

‘calculating accurately and efficiently, both mentally and on paper, drawing on a range of calculation strategies’.

Both of these statements confirm the view that mental calculation lies at the heart of being proficient with numeracy.

The National Numeracy Strategy’s Framework for Teaching Mathematics pinpoints how the development of mental calculation strategies are to be used as a basis for progression with numeracy and gives guidance on how and when to implement these strategies. The view is held that mental mathematics should address:

Strategies to use and apply number.
Quick recall of facts.
Efficient mental calculations.

MENTAL CALCULATION

Mental calculation begins at a very early age. Pre-school children have the ability to understand concepts such as ‘more or less’ and ‘larger or smaller’ on a visual basis before they have begun to understand the values of numbers or the quantitative properties that they hold. Once they begin school, children then start to understand number as we know and recognise it. By introducing them to certain principles and strategies, they then begin to move away from abstract ideas of number to having a fully developed sense of how and why we use number.

The first principles children are introduced to are the one to one principal, the ordinal principle, the cardinal principle and the order-irrelevance principle. Once a child has gained a good knowledge of these they are then ready to begin using these basic principles to calculate and it is here that the idea of mental calculation strategies becomes relevant.

It is accepted that all calculations include some element of mental calculation. Even if we are working out an addition algorithm we will add up the individual components ‘in our head’. It is because of this that a large emphasis of the curriculum is placed on teaching and developing mental strategies. This is to provide the child with all the ‘tools’ they will need to develop and be successful with formal written methods.

The approach to mental calculation strategies begins with identifying the individual child’s views and beliefs of how the number system works. It is irrelevant to try to teach children strategies for addition and subtraction if they lack a basic understanding of the principles of our number system. The key issues which we need to address are the basic principles of the base ten system and place values. Once a child has a firm grasp of these, they can provide a sound base for development of both mental and written calculations.

Mental strategies which are fundamental to providing a sound base from which a child can progress are :

Counting on and counting backwards between 1 and 10.
Establishing number bonds and patterns between 1 and 10.
Doubling and halving.
Counting on in tens.

These basic principles can then be expanded and applied to any place value which will secure a basis for the child from which they can progress. Thompson further identified the key strategies for young children as being:

Partitioning single digit numbers ( 7 = 5+2 or 4+3 )
Compensation ( adding 9 by adding 10 and subtracting 1 )

His reasons for these strategies was because of their usefulness in later 2 digit work. This again points us to the conclusion that mental mathematics is not only committing certain facts to memory, but also giving children basic ideas and strategies which they can develop themselves to successfully apply to all four operations.

One of the main purposes of teaching mental calculation strategies is to show the relationships between the four operations. Once a child has become competent in the above strategies, these facts can then be used to introduce the concepts of multiplication as repeated addition and division as repeated subtraction.

FORMAL WRITTEN METHODS

As a child becomes more competent with the use of mental calculation strategies they will naturally begin to move towards formal written methods. One of the ways in which this happens is that they will begin to use ‘jottings’ to help them. These should be encouraged as although they are not done ‘in the head’, they will begin to provide the basis for algorithms and will also help to cement facts and number patterns in their own mind.

The question of when to start introducing children to formal written algorithms is one which has caused debate. The Numeracy Task Force states that progression towards this with mental work is crucial

‘because they are based on steps which are done mentally and need to be secured first’.

If a child is given an algorithm without having a sound knowledge of mental strategies then they can apply the standard written method to it incorrectly and may not recognise that it is incorrect.

One of the main problems of introducing formal written methods to early is that the computational order of an algorithm is different to everything a child knows. When we learn to read we read from left to right, when we read numbers we read from left to right, even when we are calculating mentally we often work from left to right, however when we calculate an algorithm we work from right to left. This can prove extremely confusing, especially when we encounter a subtraction algorithm which involves decomposition. If a child has a secure mental understanding of our number system, they can then understand how the calculation works rather that falling back onto a ‘rule’ they have learnt that they may not apply correctly. A common mistake made here is to subtract the smaller number from the larger.

The move on to formal written methods relies heavily on mental strategies. One of the first addition and subtraction methods is that of expanded notation or breaking down the number into individual place values, for example:

200 + 40 + 8

125 - = 100 + 20 + 5 –

123 100 + 20 + 3

This requires a sound knowledge of place values and also of how the base 10 system works. This is also the case for multiplication where the grid method may be used, for example:

24 x 16 =

The answer then is 200 + 120 + 40 + 24 =384

When we move onto division one of the most common strategies is that of repeated subtraction. This can only work if the child already has a sound knowledge of mental methods of subtraction.

CONCLUSION

It is now widely accepted that children need to have a secure knowledge of number facts and mental strategies in order for them to develop fully with number and be able to work competently with written calculations. This has now been exemplified by the fact that the Framework for Teaching Mathematics does not introduce formal written methods until the age of. We have to remember that there are many different strategies that can be used and that children can be extremely inventive. If we ensure we know which strategies children are using then we have a good indication of their level of ability or development and can help point out any inaccuracies. As Thompson said

‘If teachers are to be successful in teaching mental strategies then they need, as a minimum requirement, to be familiar with the different methods that children use’