Section 6

SPECIFIC ISSUES IN MATHEMATICS TEACHING

Calculating

The purpose of teaching algorithms for calculating (step by step procedures for adding, multiplying etc.) is to help pupils to understand number and use it effectively. Written algorithms are only one possibility. Mental algorithms are often more powerful; they are much more varied and need to be recognised, encouraged and discussed. Algorithms can also be devised for the calculator.

When written methods are used, adults often do not make use of standard procedures which they were taught at school. Increasingly, too, they carry out calculations with a calculator. Many everyday calculations are carried out mentally - quick approximations made in our heads when shopping or travelling. Mental calculation is the basis of all estimation and approximation, as well as being the form of calculation commonly used in everyday life. How much is possible, mentally, depends entirely on what one is capable of, and this, for most pupils, will gradually increase with practice and training.

This suggests some changes in emphasis in teaching computation: mental calculation will require more attention, written methods need not always be standardised on a concise but difficult to understand algorithm - we do not need, for example, to set out numbers vertically for real life subtraction - and new skills in using a calculator will need to be learned. Pupils will require the ability to decide which method, mental, written or using a calculator, is appropriate in particular circumstances. These changes in emphasis are reflected in the targets for mathematics.

However, such changes do not imply that calculators can replace the need to learn basic facts. One of the features of the mathematics curriculum will be an increased emphasis on being able to approximate and estimate. To do this, one needs to know what mathematical operation is involved, have easy recall of single digit number facts and sound understanding of place-value. Oral work to encourage mental agility with numbers will be a necessary part of the pupils' experience.

When tackling a calculation problem, it is good practice to have an idea at the outset of the approximate answer. This is often obtained by rounding numbers and calculating mentally. Once the result has been obtained, the most obvious check is by comparison with the initial estimate.

Whenever possible, pupils should make approximations in the first instance in order to "estimate, calculate, check". Pupils' skills in mental computation will support this routine.

There are a number of other useful ways of checking calculations which pupils should learn to adopt. When using a calculator, for example, there is a risk of mis-keying. From the earliest stages of using a calculator, pupils should learn to repeat as a matter of course. Totals can be checked by repeating the addition in a different order. The inverse operation can be used: subtraction by addition (243 - 135 = 108 checked by 108 + 135 = 243), and division by multiplication (56.28 ÷ 6 = 9.38 checked by 9.38 x 6 = 56.28). A quick check on any calculation can be made by looking at the final digit: for example, 14 x 7 could not be 97 because the final digit check would give an 8 (7 x 4 = 28). It is always wise also to check that the result is reasonable in the context (would a brick weigh 2.0 or 20 kg?).