Going mental over methods

21st March 1997 at 00:00
Ian Thompson and Jennie Kerwin compare the algorithms taught for subtraction with the mental strategies children use

Read any current document dealing with the teaching of mathematics and you will find a reference to "mental maths". The National Numeracy Project has made mental calculation one of the linchpins of its Framework Document, and the Barking and Dagenham experimental reforms have also stressed the importance of this approach to number work. The inclusion of a mental test in the 1997 key stage 2 standard assessment tasks, and the prominence given to the teaching of mental arithmetic in the Teacher Training Agency's Initial Teacher Training National Curriculum for Primary Mathematics further illustrate the importance being attached to this area of the number curriculum.

However, there are signs that we are rushing into advocating major changes in content and teaching approach without reflecting and thinking things through. A good illustration of this can be seen in the rushes of a video being made on primary maths teaching in Barking and Dagenham schools. The initial film sequence illustrates the emphasis being placed on the teaching of mental calculation. However, the final sequence shows the teacher, followed by a pupil, demonstrating the decomposition method for three-digit subtraction to a class: a method which seems to bear little or no relation to the mental strategies featured in the earlier sequences.

This raises the question of the extent to which the algorithms taught for subtraction actually reflect the mental calculation strategies that children use. For example, it is well known that most people carry out mental calculations by operating from left to right, working with the tens or hundreds first - thereby obtaining successive approximations to the answer. Written methods, on the other hand, operate from right to left and involve the manipulation of symbols as abstract entities.

After considering the algorithms taught now, we shall propose an alternative approach to written subtraction methods.

The standard algorithms for subtraction The two main written methods are "equal additions" and "decomposition". They are illustrated below and are accompanied by a mental commentary on the various stages of the calculation: Equal additions "You cannot take 6 from 2 so borrow a ten and pay back. ..

...6 from 12 leaves 6 7 and 1 is 8. You cannot take 8 from 3, so borrow a ten...

8 from 13 is 5. ..4 and 1 is 5...

5 from 7 leaves 2" Decomposition "You cannot take 6 from 2 so exchange a ten for 10 ones...

...6 from 12 leaves 6...

...You cannot take 7 from 2 so exchange...

7 from 12 is 5... 4 from 6 leaves 2" The debate about written subtraction algorithms over the past few decades has focused narrowly on the relative merits of these two methods. There is evidence that users of the equal additions method are quicker, but that decomposition users are more likely to be accurate.

The main reason for the virtual disappearance of the equal additions algorithm from primary maths schemes was that it was considered to be more difficult to understand. For example, the language associated with the operation is somewhat mystifying: "borrowing a ten" from some unspecified place and "paying it back" to a different place does not make logical sense. Decomposition, on the other hand, was considered to be easier to explain.

It was argued that, by using Dienes base-ten blocks and emphasising the concept of "exchange" - whereby a ten-stick can be exchanged for 10 ones, or a hundred block for 10 ten-sticks - children could be taught to understand the algorithm. However, research showed that children were having difficulty in making connections between their manipulation of the practical apparatus and the related pencil and paper algorithms. It would appear that the blocks constitute an excellent model for clarifying the algorithm to someone who uses it with confidence and has some understanding of the meaning of the operations involved.

As a result of the difficulties involved in teaching this algorithm so that children could understand it, teachers were next advised to introduce children to "expanded notation". This is illustrated below: Standard Expanded notation algorithm 732 700+30+2 600+120+12 -476 -400+70+6 -400+ 70+ 6 256 200+ 50+ 6 The "expanded notation" example has the advantage of working with numbers as whole entities rather than as a collection of digits, but the modification did not appear to make a substantial difference to children's understanding of the method. Based on our experience we would argue that few, if any children are likely to invent this algorithm for themselves - either in compact or expanded form. A further problem is that the plethora of addition symbols used in the expanded notation can detract from the fact that it is a subtraction problem that is being carried out.

An alternative approach We would expect a revised number syllabus built on mental calculation to be based on research into the methods that children have been shown to use for carrying out calculations in their head. We would also anticipate that no vertical written methods for the four basic operations would be taught in key stage 1.

An important aim of the number strand in the mathematics curriculum throughout the primary school would be to develop children's proficiency in mental calculation strategies to the extent that they could add or subtract any pair of two-digit numbers in their heads with confidence and accuracy, resorting occasionally to pencil and paper. To subtract three-digit numbers we would want children to have access to a written algorithm that supported their mental methods. But are there any such algorithms?

A common strategy for the subtraction of a one-digit from a two-digit number, involving "bridging through ten", is illustrated by Mark in his calculation of 24-7: "17. When I've got 24 and I take away 4 it makes 20, and I know that 3 and 17 makes 20, so I have 3 left to make 17". Mark has been influenced by the units digit of the 24 in his decision to subtract four first rather than try and subtract seven in one fell swoop. Doing this leaves him with three more to take away from a nice round multiple of ten.

Research suggests that the most common mental method for two-digit subtraction is "partitioning". Using this method both of the numbers involved in the calculation are partitioned into multiples of ten and separate units: 56-32 would be calculated as "50-30 is 20, 6-2 is 4, so the answer is 24". We feel strongly that giving children a thorough grounding in basic mental work will enable them to learn to cope with more complex subtractions, such as 54-37, in their heads.

During their learning of mental calculation strategies we would expect children to have acquired the ability to do the following: * partition any single-digit number in a variety of ways (7 can be split into a 4 and a 3) * subtract a single-digit number from any multiple of ten (70-6 is 64) * subtract a two-digit multiple of ten from another (70-30 is 40) * subtract two-digit numbers with the same units digit (74-34 is 40) * subtract a two-digit multiple of 100 from another (700-300 is 400).

By using these specific skills, and by extending Mark's method, children should be able to execute the following subtraction algorithm with understanding. A two digit subtraction is shown first, and a "running commentary" is included to illustrate the thinking involved: "60 take 20 is 40 ...5 take 7... If I take 5 away that leaves me with two more to take away. ..

Write down the 2 that I still have to take away."

Working with three-digit numbers is not much more difficult: "700 take 200 is 500 ...

... take away 50 ...

and I've still got 30 more to take away . ... take away 4 and ...

... I still have 2 more to take away ... so the answer is 468."

To illustrate the extent to which this algorithm is generalisable a four-digit calculation is included: "3,000 take 1,000 leaves 2,000...

...take 400 and I still have 200 to take away...

...take 70 and I still have 10 to take away...

...take away 3 and I still have to take another 4...

... so the answer is 1,786" The "partitioning" method provides a more suitable written calculation algorithm for three-digit subtraction than does "decomposition". One reason in its favour is that it operates with quantities that are meaningful to the user while retaining the place value meaning of the digits. It also demands, builds on and extends good mental calculation facility. If teachers succeed in helping their children learn this particular written algorithm for subtracting three or four digit numbers then they will be "working towards" satisfying the requirements of that section of the national curriculum for mathematics which states that children should be taught to "extend mental methods of computation to consolidate a range of non-calculator methods of addition and subtraction of whole numbers".

* Ian Thompson lectures in maths education at the University of Newcastle upon Tyne. Jennie Kerwin is a National Numeracy Project consultant at Birmingham Numeracy Centre

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