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Outline, aims and rationale

 

Aims

This resource is designed to:

show the importance of mental computation in the school curriculum

set out goals for balancing mental, written and calculator computation

explain some common strategies for mental computation

explain how to develop understanding of the written algorithms.

Definitions

In this resource, the term "written computation" refers to computation which is done on paper, following a standard written algorithm. The common written algorithms for subtraction and multiplication are described on the resource.

The term "mental computation" refers to all computation which is not following a set algorithm, mainly done in the head but possibly assisted by quick jottings on paper to support short term memory. (Note: some researchers forbid any jottings in mental, but we allow them)

Why is mental computation important?

Mental computation is important because:

Everyone is faced with situations in daily life where there is a need for quick calculations, often in the absence of paper and pencil and without a calculator. Being able to calculate mentally, and especially to make quick mental estimations, is essential for everyday life

Mental computation is vital to check the reasonableness of calculator answers.

Children do not merely absorb information passively but interpret selectively and construct their own meanings from it. Well-taught children will always use their understanding of place value and their number sense to work mentally with numbers. Hence they will devise variations of the algorithms taught at school and sometimes invent their own methods to suit the question.

By understanding the invented methods that children use for mental computation, the teacher is better able to introduce formal algorithms based on the children's prior understandings.

Why are formal written algorithms important?

A formal written algorithm offers a reliable procedure which can be used to make any calculation efficiently. They are of less importance now that electronic calculators are widely available in everyday life. Until about 1970, ability to perform written algorithms accurately and efficiently was needed on a daily basis for employment in most businesses, including shops and banks. Computer and cash register calculation has replaced this, leaving mainly incidental computation that is often best done mentally.

Subtraction and multiplication

Of the four basic arithmetic operations (addition, subtraction, multiplication and division), this resource focusses on only two: subtraction and multiplication. The principles which are shown for these operations mostly also apply to the other two operations. In particular, there are strong commonalities between strategies for addition and subtraction (a pair of inverse operations) and multiplication and division (a pair of inverse operations). The resource is only about whole numbers, because this is the most important for everyday life - computation with fractions and decimals is not treated here, but needs attention at school.

When do children learn mental computation?

Children should be practising and extending their mental computation before they learn written algorithms, after they learn written algorithms and whilst they are learning written algorithms. It needs emphasis at every level of schooling. Remember that many children can work out answers to arithmetic questions mentally before they are formally taught the procedures at school.

What is missing from this resource? Answer: Real contexts and estimations

All the examples used in this resource are "naked number" computations. In real life, where mental computation is really important, mathematics is usually applied to quantities with a meaning, such as measures or prices. The real world meaning can often help in computation. This resource, however, just shows the general principles behind mental computation. Remember that it will be learned by drawing on real life examples.

Mental computation in everyday life is most often used to get estimates, rather than precise answers. For example, I might need to know that my shopping bill will be about $30, that I will need about 5 litres of chlorine for my swimming pool and that I can sew all of the clown costumes for the school play with 15m of material which is 120 cm wide. As well as using strategies for computation similar to those explained here for exact mental computation, real life estimation often relies on knowing benchmark figures (e.g. my pace is just under 1 metre long).