How has the introduction of the National Numeracy Strategy (NNS) affected the teaching and learning of the process of division in English primary schools? This study explores the variety and relative success of Year 5 pupils’ strategies for tackling problems in division. It compares how these have changed with the findings of a parallel study that took place five years previously, when the National Numeracy Strategy (NNS) was just about to be introduced to English schools.
The new study found a small overall improvement in pupils’ scores on the test of division. Patterns in the way in which pupils tackled the problems varied widely from school to school, leading the researchers to conclude that schools still differed in their approach to teaching division, despite working with the NNS for five years. For example, some schools, but not all, were teaching pupils to set out in a clearly structured, more formal written record, an approach to division based on repeated subtraction (called ‘chunking’ in the study). The relationship between pupils’ use of different strategies for division and pupils’ success was complex. Different approaches seemed to suit different individuals and the study found some differences beginning to emerge between boys and girls. The study found that pupils from schools that did not support students in constructing a structured written record for division were less likely to do well in the tests.
Keywords: England, Primary schools, Numeracy, National Numeracy Strategy, Gender, Teaching and learning.
When the study compared the performance of pupils in 2003 with that of pupils in 1998, it found that:
Pupils in both of the two best performing schools made extensive use of structured written methods but each school used a different approach. In one which performed especially well, (school 7, average score 6.1), pupils used both the chunking and traditional algorithms extensively. In another top-scoring school (school 5, average score 6.5), the pupils used the traditional short division algorithm for over 75% of the items and no-one used the chunking algorithm. This was a surprising finding in the light of the earlier research in which pupils found it hard to use the traditional algorithm successfully.
The 1999 NNS introduced a structured way of recording division calculations based on repeated subtraction of multiples of the divisor (high level chunking). This offered a new algorithm for division that could build on pupils’ informal understanding of division by repeated subtraction of the divisor itself (low level chunking). The study found that not all schools were teaching this new algorithm, as pupils from some schools made no attempt to use it.
Pupils from some schools, (for instance, schools 3, 6 and 8; average scores 4.42, 4.25 and 4.07), made very limited use of any structured written method. They used mainly informal methods to approach the problems. These schools were clustered at the lower end of the range of scores. It seemed that the biggest difference between schools whose pupils scored well on the test of division and those who did less well was that pupils from the former made much use of structured written methods and those from the latter relied heavily on less efficient, informal methods.
Unusually large numbers of individual pupils in the high-scoring school 7 used more than one approach – 86% of pupils at this school used the chunking algorithm for at least one question. Elsewhere, the number of different strategies pupils used varied, but no relationship was found between the number of different strategies that pupils used and their score on the test.
This study found that differences in performance between boys and girls in 2003 had increased when compared with the results of the first study. The 1998 study found no overall differences between boys’ and girls’ performance on the test, but it did find subtle differences of approach. For example, boys were less likely to use written methods than girls, preferring to calculate mentally. (This tendency continued in 2003.) Boys also guessed or omitted answers more often than girls. Girls tended to use more low-level strategies than boys (such as repeated subtraction of the divisor for large numbers) and they improved more when supported in developing a structured written method for approaching division. The 2003 study found that:
Schools varied in their teaching strategies for division
The variety of strategies pupils in different schools used in tackling the division problems suggested that teaching practice also varied from school to school. This apparent variety led the researcher to ask why five years of the NNS had not led to greater similarity in teaching practice. She suggested that teachers need time and a clear understanding of what to do and why to make changes to their classroom practice. She suggested that teachers in the study might not have reached a clear understanding of what they were being asked to do or, just as importantly, the reasons underlying the proposed changes.
Some schools did not teach the new chunking algorithm
In some schools, pupils made extensive use of the new chunking algorithm, but in other schools, no pupils used it. The researcher suggested that although the chunking method, based on repeated subtraction, would seem to fit well with children’s intuitive understanding of division, it would have been new to many English teachers. It was introduced in the NNS as an ‘informal written method’ and this, as well as its unfamiliarity, may have deterred some teachers from using it. Most schools continued to teach the traditional standard written method. However, it is difficult to progress logically from chunking, which works with whole numbers, to the traditional method, which works with separate digits.
Did teaching the chunking algorithm help Dutch pupils?
In the Netherlands, in contrast, the chunking algorithm is the standard method used for division and Dutch teachers rarely use the standard algorithm traditionally used in England. The author noted that Dutch pupils did much better on the original tests than either group of English pupils. She suggested that the standardised use of the chunking algorithm allowed Dutch teachers to build on pupils’ intuitive understanding of division more effectively and helped them to progressively develop pupils’ informal approaches into a structured, written record.
Supporting pupils to use structured written records
Another issue that emerged was that, although the NNS proposed that structured written methods should be introduced in Year 4, it was clear that not all schools in the study had done so before the end of Year 5. Pupils in these schools used neither the chunking algorithm nor the traditional English short division algorithm but relied solely on unstructured written records. These strategies could be correct and show understanding but were cumbersome and subject to error, especially when large numbers were involved. Pupils who relied on them scored less well than pupils who could use a written strategy in the tests. In contrast, pupils in the two best performing schools made extensive use of more structured written methods.
The researcher concluded that it was important to support pupils to develop a structured written record in a timely fashion, rather than leaving them to use informal recording for too long. Such teaching should build on pupils’ informal approaches and their established ideas of division. She particularly highlighted the need to move from the idea of division as sharing to division as repeated subtraction.
The study explored how Year 5 pupils from ten English primary schools tackled a variety of division problems. The fieldwork took place in 2003 and aimed to replicate a study conducted five years earlier in order to examine:
Nine of the ten schools also took part in the original study. The tenth was chosen to be as similar as possible to the school it replaced. The pupils in the study (308 Year 5 pupils, compared with 275 Year 5 pupils in 1998) responded to ten division questions. Half of the questions were presented as word problems and half were presented as ‘bare’, uncontextualised questions, for example, 64:16.
Eight of the division questions in the test were exactly the same as before. Two of the original questions were altered. The original questions involved division of three-digit numbers by ten. They were replaced by two questions involving three-digit numbers to be divided by four. These questions could be tackled through ‘halving and doubling strategies’ highlighted in the National Numeracy Strategy.
In completing this digest the authors began to ask the following questions about implications for practitioners:
In completing this digest the authors began to ask the following questions about implications for school leaders and mathematics coordinators:
You can find a TRIPS digest of the 1998 study of pupils’ methods of division at: http://www.standards.dfes.gov.uk/research/themes/thinkingskills/ThuOct101531302002/
The TRIPS website includes many more digests of research into numeracy and mathematics. Follow these links to find more: http://www.standards.dfes.gov.uk/research/themes/mathematics/?digest=all http://www.standards.dfes.gov.uk/research/themes/num/?digest=all
For a single page summary of large-scale research into helping students overcome difficulties in mathematics, read the article on page 4 of Issue 2 of the NERF Bulletin. Available at: http://www.nerf-uk.org/bulletin/ Or read the full report, Ann Dowker (2004) What works for children with mathematical difficulties? DfES Research Report 554 Available at: www.dfes.gov.uk/research
For a more detailed summary of research into the teaching of mathematics with related teacher case study examples, try these Research of the Month summaries:
‘Effective teachers of numeracy’, based on Askew, M., Brown, M., Rhodes, V., Johnson, D. and Wiliam, D. (1997) Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency London: King’s College, University of London. Available online at: http://www.gtce.org.uk/PolicyAndResearch/research/ROMtopics/numeracy1/
‘Experiencing Secondary School Mathematics’ based on the work of Boaler J. Available online at: http://www.gtce.org.uk/PolicyAndResearch/research/ROMtopics/maths/
Other useful websites include:
Key Stage Three Strategy on the Standards’ Site http://www.standards.dfes.gov.uk/keystage3/subjects/maths/
The Mathematical Association: http://www.m-a.org.uk
These definitions have been supplied by the digest writers for the benefit of our website readers.
algorithm – a succinct and standardised written procedure for solving a numerical problem
traditional algorithm for division – this is sometimes referred to as the ‘bus-shelter method’ as the way of setting out the numbers looks like a cross-section of a bus-shelter. It is an efficient, condensed method of recording the operation of division and is based on partitioning the whole number being divided into hundreds, tens, units and so on and operating on each digit separately.
chunking algorithm – this is a way of systematically recording a method of division based on high level chunking (see below). It is an efficient, condensed method of recording the operation of division and is based on using whole number values throughout the calculations. It was introduced in the 1999 NNS and would have been new to many primary teachers.
informal strategy – a way of tackling a numerical problem based on intuitive understanding. Informal strategies may use idiosyncratic methods of recording the work.
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 15 or 30.