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Analysing the amended level descriptors also suggests little change is brought by the new curriculum, except using calculators where numbers include several digits is no longer found in level 3.
The message from the National Numeracy Framework with reference to calculator use appears under the strand 'calculations'. The objective being for children to be 'taught to develop calculator skills and use a calculator effectively', with outcomes specified for Year 5 and 6 (ages 9 to 11) children only. The framework expresses the need to teach children the technical skills to use the basic facilities of a calculator. Most of the detail in the outcomes for Years 5 and 6 addresses these technical issues. As would be hoped, the arguably more mathematical skills, involving having a feel for or approximating and estimating the size of answers, checking results using inverses and interpreting decimal and negative number displays and rounding errors, are also included.
Research
Other advice regarding the use of calculators from the National Numeracy Strategy came in the publication 'Teaching Mental Calculation Strategies'. This guidance advises that 'for much work at key stages 1 and 2, a mental approach is most efficient and needs to be taught explicitly.... Calculators can help develop a better sense of number, but should not be used for calculations until [the child] can at least add and subtract any pair of 2-digit numbers in their head.' In the introduction to the NNF, it is suggested that this 'powerful and efficient tool' be used to make use of real data across the curriculum and it states how it can 'offer a unique way of learning about numbers and the number system, place value, properties of numbers and fractions and decimals'. The question remains of how best to do this and hence 'enable more ambitious exploration of numbers to be undertaken' (NC Non-Statutory guidelines)
Long before the introduction of a national curriculum and the National Numeracy Strategy, there were several attempts to list the basic number skills needed to ensure the efficient use of a calculator. Bell (Bell at al. 1978, p.31) concluded that:
'3 basic computational skills seem essential when the calculator is used freely:
facility in single-digit arithmetic
a good understanding of place value, including decimals
ability to estimate and check.
Girling's list (1977) included:
asking whether the answer makes sense,
repeating the calculation using a different operation or procedure,
making a very rough approximation, and
using pattern (for example checking the units' digit).
The need to understand the relative size of numbers.'
Either of these models offers some sensible guidelines which may still be applied in the Primary classroom today, including deciding when to introduce the use of a calculator.
There remain some opposing views surrounding the appropriate and subsequently effective use of calculators as a learning tool. 'It might be thought that possession of a calculator brings with it the ability to perform any ordinary calculation. Far from it. Effective use of a calculator requires an understanding of number.' (McIntosh, 1990) Despite the lack of evidence, the view of many parents and educators remains that children using calculators will become lazy and reliant on calculators to do their work for them.
Further confusion came when research carried out by NIESR (National Institute of Economic and Social Research) (TES 9/96) involving international comparisons, suggested that calculators should be blamed for poor standards in mathematics and that they should be banned from tests with more emphasis being placed on arithmetic. Conversely, the Cockroft Report (1982) acknowledged the issue of fluency in mental and written calculations and the use of calculators and advised that:
'the availability of a calculator in no way reduces the need for mathematical understanding on the part of the person who is using it.'
and that:
'the weight of evidence is strong that the use of calculators has not produced any adverse effect on basic computational ability.'
More recently the CAN (Calculator-Aware Number) Curriculum project reported that Key Stage 1 'CAN children' displayed the same performance as 'non-CAN children', with more children lying at the extremes of achievement. More significantly, at Key Stage 2 the 'CAN children' were found to be more liable to compute mentally and adopt powerful mental strategies (Ruthven, 1998). The findings of this project, which allowed children access to calculators at all times, appear to suggest that free calculator use helps rather than hinders the mathematical thinking strategies and number awareness of young children.
Various studies have shed light on the effect of calculator use on children's attitudes in mathematics. They point towards children having more positive attitudes towards mathematics and improving children's self-esteem across all ability levels. They can bring a positive and powerful message about the relevance of school experiences to the world outside. Some also believe that calculators can act as a good 'leveller' for mixed ability groups, as they enable the less able to do what they would otherwise be unable to do. In such cases the children may often be able to concentrate on the purpose of the activity rather than being side-tracked by 'number crunching'. Calculators have also been recommended for use in extending the mathematical insight of the more able. They can free up memory, cognitive effort and time, enabling children to concentrate on strategic aspects of problem solving.
Some areas where calculators may impact positively on teaching and learning can be explored. Two uses of the calculator which may be considered together are that of working with large numbers and that of working with real data. Calculators may be an appropriate tool for dealing with otherwise complicated and tedious calculations with large numbers (for example, how many days is a million seconds) and for exploring number patterns with larger numbers (for example, in mathematical investigations, where calculators may indeed encourage further exploration). Calculators may be used as a labour-saving device allowing children to manipulate data generated in other subjects, like figures collected in a local study in geography.
More specifically, calculators have been used to help children to understand the central importance of place value in the structure of number. It is also a powerful tool when used in the acquisition and reinforcement of the concepts of negative numbers and decimals. Children's ability to estimate, round and approximate quantities can be greatly enhanced. They need a confident concept of number to enable them to do this.
A less obvious way of using a calculator as a learning tool is if children use calculators to check their own work. This practice not only offers them immediate feedback on their progress, but it can also act as positive reinforcement for the child and may increase confidence. Moreover, it may deter a child from continuing to respond to problems in a way that yields incorrect answers by detecting mistakes which in turn may prevent them from reinforcing the incorrect method by continued practice. Its usefulness for a child checking their work can also be extended if they are developing their own non-standard methods of completing problems, for example, long division.
For the teacher, children's work with calculators may be used to assist their diagnostic assessment. These opportunities may be heightened if children are encouraged to think carefully about how to record their work. If the calculator is doing the 'straight-forward' working out, the teacher may see whether or not the child has grasped the underlying mathematical principle or principles. Such work will also shed light on whether a child has the ability to 'translate' from a real-life situation, to the needed calculation, and to apply the answer back into the situation.
Baroness Blatch (Minister of State for Education, February 1993) expressed her fear that 'instead of learning how to add up, children are being taught how to use a calculator .' With the Numeracy Strategy, children are not only learning how to add up, but are encouraged to utilise a full range of mental strategies. The appropriate use of calculators alongside, not in place of oral and mental work, should encourage greater numeracy still, the calculator being used as a tool for investigating number and not as a substitute for skills and strategies.
Calculations can be presented to children orally for them to key in as a check of their understanding of place value .
For example, How many is 102 - 60?
You can give the children a list of numbers to take one away from without a calculator, for example
97, 101, 200, 379, 1010, 1111, 10 000.
When they have calculated the answer mentally, they can reinforce their understanding by checking the answers with a calculator and marking the work themselves. This can be extended to subtracting 10 from a list of numbers, for example,
110, 290, 1000, 2101, 1111, 7654,
and further extended by subtracting 100, 1000 .... from other lists of numbers. Emerging patterns can then be discussed and explored further as a group.
Estimating skills can be practised whilst at the same time teaching the use of the constant function. Give the children grids with sets of numbers in, as below.
39 | 84 | 15 |
135 | 111 | 72 |
57 | 9 | 102 |
27 | 150 | 42 |
The children set the calculator constant function to 'multiply by 3'. Then they choose a number to multiply by. If it appears on the grid they can
colour the square and try to then finish colouring their whole grid,
play it as a game for 2 players like noughts and crosses, set in any relevant context for the class at the time, or
try to get a straight line of 4 squares with the least possible number of tries, perhaps trying to beat an earlier target.
This activity will also strengthen the concept of multiplication being the inverse of division.
Estimating the likely answer to addition and subtraction questions is also a fundamental skill to be practised. Give the children a list of sums with a choice of estimates for the answers. For example,
47 + 22 Estimates 60 70 80 90
74 - 46 Estimates 20 30 40 50
95 + 36 Estimates 110 120 130 140
The children can choose the estimate they think is closest to the true answer. They then check their answer using a calculator. Rounding and approximating skills are practised. This can also be developed into a game for two players or extended for practising multiplication and division.