Current practice emphasises the teaching of informal mental calculations to work out division problems, rather than written computation from the outset. But how do children learn to relate to the use of written methods so they can develop effective calculation strategies?
In this study of a sample of Year 5 pupils in ten schools, progress was monitored as taught methods of division were introduced over the course of five months. The author suggests that the use of formal methods may inhibit children’s understanding of mathematical problems unless it is underpinned by sound mental strategies. She found that increase in the use of formal algorithms created more errors, finding that only half the attempts to use them were effective and that efficient but less formal methods of problem-solving produced more successful results.
The author concludes that the use of formal written methods should be deferred until children have the confidence in their own informal methods and are aware of the likelihood of the correct answer.
'Otherwise it is likely to inhibit performance and have a negative effect on the development of mathematical thinking.'
This study makes use of various specialised words and phrases, for definitions of these click here.
Keywords:
United Kingdom; England; primary schools; pupils; curriculum; mathematics; mathematical problems; numeracy; Year 5; classroom teaching; thinking skills; pedagogy; problem solving; learning strategies; arithmetic
The author’s intention was to analyse the strategies that pupils used to solve division problems before and then after they had been taught to do division in Year 5. During the intervening time classroom activities would have involved the direct teaching of division.
The researchers wanted to look at pupils’ written recording of their work in order to:
The informal strategy of ‘High Level Chunking’ was the most effective method in both tests, with mental methods being the second most efficient. Despite this, the standard algorithm was the most frequently used method throughout.
Low level chunking led to long, inefficient working out, successful in only 1 in 4 cases. Working with separate digits and partitioning into hundreds, tens and units were rarely successful in either test.
Test scoresWhat strategies did pupils use?
The researchers identified nine broad strategies that pupils used for attempting division problems:
Year 5 pupils in ten schools in and around a small city were selected by minimum class size of 25 and by their high mathematics test scores in National Curriculum tests. Scores in English and Science were also well above average. The children all had good reading skills, the confidence to tackle problems and ability to show their working out, thus giving valuable insight into the strategies used.
Whole year groups were tested in four schools and a representative half of the pupils in the other six schools. In all, 275 pupils completed two test papers.
These papers containing 10 division problems (of which half were context and half bare problems), were administered to the children in January and June of their Year 5. The questions were selected to enable children to choose appropriate mental strategies and invite the use of number facts that they should already have been familiar with. It was possible to approach all problems without using the standard algorithm, that is, the formal method.
This study is part of a continuing larger study where parallel groups in the Netherlands and England undertake the same tests. This is now reported in the current issue of Educational Studies in Mathematics. Click here for the full reference to the article.
Two practice questions were presented to the class and the pupils were asked to come up with solution strategies. These were written on the board so that at least three different strategies were available to all pupils for each problem.
Pupils then worked individually on the 10 problems, each one being provided with space to show working and an answer. They were encouraged to record ‘the way they think about the problems’ to show their solution strategies.
Some findings were identified which may help teachers to relate to their pupils understanding of division:
Where the context was given there was evidence in the children’s working that it influenced how they selected procedures e.g. some children had done drawings of shopping baskets in a problem that involved sharing apples.
Similarly, in the second test, where the problems were reversed so that bare problems became context problems and vice versa it was notable that the percentage of correct answers increased for the problems that became context problems and decreased for the problems that became bare problems. This suggests that where the child could not model a given context, he or she was more likely to use a procedural approach to doing the problem.
Pupils appeared to be strongly influenced to use the standard algorithm with the ‘bare’ problems. The use of the standard algorithm rose from 38% of all questions in Test 1, to 49% in Test 2. However, half of these attempts led to an incorrect solution. Where the increases in use of the algorithm were largest, there were decreases in the number of correct answers suggesting that it replaced a more successful informal strategy.
In both tests, informal strategies were most often low-level and inefficient for the large numbers involved. The results suggests that more efficiency must be gained without losing the children's intuitive understanding. One way of doing this is to develop the idea of repeated subtraction but introducing progressively larger chunks to subtract.
Gender issuesBoys used more higher-level strategies than girls, including mental methods of computation, but were more likely to resort to guesswork or omit answers.
Children first encounter the concept of division as “sharing”, or partitive division, e.g. ‘How can six sweets be shared amongst three children’. Later they learn that division also relates to the number of equal groups in a given total, or quotitive division e.g. ‘How many groups of two are there in six?’. In some ways this relates better to multiplication than sharing. The formal mathematical representation of both of these contexts is 6 ÷ 3 = ?.
Research suggests that in the early stages of a child’s mathematical development, division is undertaken by repeated addition or subtraction of numbers, for example to find how many 2s in 6, and that many children do not progress beyond such inefficient methods in primary school. In Year 5 pupils are introduced to a standard written method for division in order to direct and organise the calculation. So we see that the children’s approach to problem solving comes before recording.
What did the researchers set out to learn?
The author’s intention was to analyse the strategies that pupils used to solve division problems before and then after they had been taught to do division in Year 5. During the intervening time classroom activities would have involved the direct teaching of division.
The researchers wanted to look at pupils’ written recording of their work in order to:
In this study the most successful strategies for solving division problems included high level chunking in which pupils used efficient procedures based upon their understanding of number relationships. Pupils showed a greater understanding of problems which were contextualised. However, many 10-year olds still continued to solve division problems by means of low-level strategies that were both inefficient and prone to error.
Evidence showed that even if a completely bizarre answer was achieved by the standard algorithm, the student would record it, suggesting that it was accepted as a valid answer “although number sense would suggest otherwise.” And therefore, “the algorithm replaces more intuitive procedures rather than enhancing them.”
The author suggests that when beginning written methods for division, teachers need to focus on the progressive development of informal approaches so as to achieve improvement without loss of understanding. She goes on to say that teachers need to help pupils structure written recording of informal approaches in preference to teaching the standard algorithm; also, pupils need to be encouraged to develop flexibility in their approaches so that they can choose to solve different problems by different methods.
The author argues that “applying a correct procedure in a ‘mechanical’ way can be highly effective.” However, in all but the simplest cases, the division algorithm involves approximations and checks that are not automatic, even with good mastery of associated multiplication facts. This implies that, for children who need to think carefully about the intermediate steps in this algorithm, the complexity of the whole procedure frequently appears to cause confusion.
The report concludes:
standard algorithm – in this study, the formal way of setting out a division calculation (sometimes described as the ‘bus shelter’ method
context problem – a mathematical problem given a context, e.g. If there are six sweets to share between three children, how many does each child get?
bare problem – the formal mathematical way of presenting a problem, e.g.: 6 ÷ 3 = ?
informal strategy – a way of tackling a numerical problem based on intuitive understanding
high level chunking – a method for decomposing numbers in efficient ways using known relationships e.g. in the sum 432 ÷ 15, 432 can be ‘chunked’ as 300, 60, 60 and 12 to make division by 15 easier.
low level chunking – a method for decomposing numbers in less efficient ways e.g. in the sum 432 ÷ 15 repeatedly subtracting 60
These definitions have been supplied by the digest authors for the purpose of our website audience.
In completing this digest its authors began to ask the following questions about implications for practitioners:
In completing this digest its authors began to ask the following questions about implications for school leaders:
Anghileri, J. and Beishuizen, M. (1998). Counting, chunking and division algorithm, Mathematics in School, Vol. 27, pp. 2-5.
The thinking underlying the introduction of the National Numeracy strategy can be found in Department for Education and Employment (1988). Implementation of the National Numeracy Strategy, final report of the numeracy task force. London: DfEE.
A recent EPPI systematic review (December 2004) examined the implementation of the National Numeracy Strategy and its effects on pupils’ confidence and competence. Kyriacou C, Goulding M (2004) A systematic review of the impact of the Daily Mathematics Lesson in enhancing pupil confidence and competence in early mathematics. In Research Evidence in Education Library. London: EPPI-Centre, Social Science Research Unit, Institute of Education. The study can be found at: http://eppi.ioe.ac.uk/EPPIWeb/home.aspx?page=/reel/review_groups/maths/maths_rv1.htm
A Research of the Month summary of ‘Effective teachers of numeracy’ can be found at http://www.gtce.org.uk/PolicyAndResearch/research/ROMtopics/
The following websites offer a variety of information and resources for mathematics teaching:
Association for Teachers of Mathematics http://www.atm.org.uk/
Mathematical Association http://www.m-a.org.uk
Qualifications and Curriculum Agency http://www.qca.org.uk