HMI Report

TEACHING OF CALCULATION HMI REPORT 461 APRIL 2002

PROBLEM SOLVING

26. Pupils in Years 1 to 3 usually solve one-step problems mentally. They identify correctly which operation to use, draw upon their knowledge of number bonds and multiplication facts and are able to explain their reasoning clearly. Pupils who record the processes they have gone through invariably use the horizontal format. For example, in Year 3, in response to the question: A spider has 8 legs. How many legs do 5 spiders have? Pupils who recorded their thinking wrote 8 x 5 = 40.

27. Pupils find word problems involving two or more steps more difficult. Most pupils are unable to identify the key information or questions involved. Few pupils approach such problems systematically or attempt to record their calculations through personal jottings. By contrast, Year 3 pupils used successfully a variety of methods to solve the problem: A set of felt-tips costs £3. Marie saves 20p a week. How many weeks will she take to save up for the felt-tips? One pupil calculated that it would take 5 weeks to save £1 (5 x 20p), thus 15 weeks to save £3 (5 weeks x 3). Another wrote: 20, 40, 60, 80, 100 = £1,300 = £3, 5 x 3 = 15 weeks. While they approach the problem logically and record the information systematically, pupils often omit to note the units of measurement involved. A good example of a useful personal jotting, but one which omitted the units, was when one boy wrote: `5 is one pound and 5 x 3 = 15'.

28. By Year 6, many pupils know and use a systematic approach to solving word problems and teachers often provide pupils with good strategies for tackling them. For example, Year 6 pupils were asked how to: find the number of marbles in only three bags when 960 are divided equally into 16 bags. They recognised the task as a two-step problem and were able to identify the steps to find an answer. One school used a problem frame with pupils to help them to analyse what type of word problem it might be and how to solve it. The teacher highlighted the key words and numbers with the pupils and then considered how the problem and the solution could be written mathematically.

29. Although systematic approaches to problem-solving are taught, too little attention is given to encouraging pupils to predict a sensible answer, using personal jottings and drawing upon mental strategies as good starting-points. Estimation is not a sufficiently established feature of pupils' approaches to problem-solving, nor is the use of checking to decide if the answer arrived at is a sensible one. As a result, pupils rely too much on the mechanical process of written calculation. In some cases, they are confident enough to work out word problems in their heads, but few use an appropriate combination of mental skills, personal jottings and standard written methods.

30. Pupils' anxiety to solve the problem and `get it right' often leads them to use immediately what hey feel is the security of a standard written method rather than a range of strategies related to the nature of the task. For example, in a group of four Year 5 pupils (2 boys and 2 girls) only one was able to calculate accurately the length of a train journey from 11.50 to 15.45. Two of the pupils made inappropriate use of a vertical format:

 15.45
-11.50

The one pupil in the group who answered correctly used his knowledge of number lines and time to count on from 11.50 - 12.00; he then recognised that three hours and 45 minutes remained to which he could add the ten minutes from 11.50 to 12.00.

Points for action: from mental to written calculation and problem-solving

31. To improve the quality of teaching of calculation and pupils' standards of attainment, schools need to:

• Give more emphasis in the teaching of mathematics to the use of jottings as an aid to calculation, particularly when pupils are using commercial materials,
• Clarify the links between repeated addition and multiplication, especially for younger pupils, and between repeated subtraction and division,
• Give more attention, at Key Stage 1, pupils' understanding of the inverse relationship between them
• Give more emphasis to practising the recall of division facts
• Ensure that, when pupils' errors and misconceptions are identified, time is taken to remedy them, particularly in the main teaching activity or during the plenary
• Give attention to pupils' accuracy on the vertical layout of calculations involving decimal numbers
• Give more attention towards the end of Key Stage 2 to the short division of numbers involving decimals
• Help pupils to develop strategies for solving word problems through the use of a combination of methods that include mental strategies, personal jottings, estimation and checking
• Strengthen the teaching of place value and the accuracy of setting out operations where calculations involving decimal fractions are concerned
• Ensure that mathematics policies include guidance, for both key stages, on when to move from mental calculation to using informal jottings and, finally, to using formal written methods.