Cool, calm calculating
Calculators are powerful tools and children need to learn to use them properly. They are not appropriate for calculations that can be done more quickly and reliably by mental or pencil and paper working. For this reason, calculators should not be used in arithmetical problems until children are 10 or 11 and have grasped some mental and written strategies.
A quite different use of a calculator - usually in junior school - is as a resource for learning about mathematical ideas.
In the key stage 2 national tests, calculator use is allowed in the second written maths paper. This guidance for Year 5 and 6 teachers should help you to help your pupils.
Anyone who looks at past test papers can see that not every question is best solved by using a calculator.
For example, some children will calculate 51 x 13 by working out 51 x 10 and 51 x 3 mentally, noting the answers 510 and 153, then adding them mentally. A calculator would be a hindrance in most questions that involve time. So children need to identify quickly the best approach to each question. Ask them to study a selection of questions and practise picking out those which don't need a calculator, those which do, and those where they can choose.
Most children do better in a test if they are familiar with the different types of questions asked.
There are four categories that you can help children to recognise and learn how to tackle with a calculator.
Questions involving inverse operations, eg 950.4 V by o = 49.5 (1999 Paper B, question 17)
Questions involving the use of "trial and improvement" methods, eg Write the same number in each box. o x o x o = 1331 (1999 Paper B, question 11)
Questions requiring knowledge of the order of operations, eg (4 + o) x o = 100 (1997 Paper B, question 9)
Word problems involving more than one operation, eg. A shop sells sheets of sticky labels. On one sheet there are 36 rows and 18 columns of labels. How many labels are there altogether on 45 sheets?(1999 Paper B, question 16) Children should learn to use a calculator efficiently.
Shortly after last summer's key stage 2 tests, Chris and Chelsea, both 11, described how they had multiplied 36 by 18 by 45 to answer the sticky labels question above.
Chris used his calculator to find 36 x 18, noting the answer of 648. He used his calculator again to multiply 648 by 45, to make 29,160. He checked part of his answer by dividing 29,160 by 45 to get back to 648.
Chelsea used mental, written and calculator methods.
* To multiply 36 x 8, she worked out 36 x 20 in her head, then subtracted 36 x 2, to get 648.
* To multiply 648 by 45, she split 45 into 40 and 5. She did 648 x 40 on her calculator, noting the result of 25920 (though it would have been just as easy to calculate 648 x 45).
* To multiply 648 by 5, she multiplied 648 by 10 and halved the answer, doing this mentally. She wrote the result of 3,240 below her previous result of 25,920.
* Finally, she used conventional column addition to obtain the correct answer of 29,160.
Not surprisingly, Chelsea didn't have time to check her result. Both Chelsea and Chris achieved level 5 in the tests and showed that they could use numbers confidently. But Chelsea chose a high-risk and long-winded strategy, in which she could have easily made a careless slip.
As a minimum, 10- and 11-year-olds should be able to:
* use the clear and clear entry keys, the operation keys and decimal point;
* know that several additions and subtractions (e.g. 928 + 546 - 75), or several multiplications or divisions (e.g. 36 x 18 x 45) can be entered on a calculator as one "string" of calculations, and that you don't have to work out each bit separately;
* know how to enter a calculation such as 46 x (243 + 387) by working out the brackets first and jotting down the answer to this part (more able pupils can be shown how to use the calculator's memory);
* estimate the rough size of an answer and check it by using the inverse operation to get back to the original number.
To use a calculator to tackle a wider range of questions, children need to know how to enter and interpret numbers representing money or measurements, and how to enter simple fractions, such as 34 or 15.
For example, depending on the context, 43.2 in the display might mean pound;43.20, or 43 kilograms and 200 grams, or even 43 minutes and 12 seconds.You will also need to teach children how to interpret the outcome of a division, possibly a complex decimal, and what would be a sensible answer in the context of the question. For example: Lynne wants to raise pound;100. She is sponsored for pound;6.50 for each lap. What is the least number of laps she must do?
(1997 Paper B, question 6) If a calculator is used to divide 100 by 6.5, it will display 15.384615. Conventionally, this result would be rounded down to 15 laps, but then Lynne would raise less than the pound;100 specified. To achieve her target of pound;100, the decimal must be rounded up to 16 laps.
Practising these calculator skills throughout Y5 and 6 will help maximise pupils' performance in the Paper B mathematics test.
Anita Straker is director of the National Numeracy Strategy