Addressing Difficulties with Subtraction Aims of Research When I first took the role of mathematics subject leader I implemented an action plan to improve the quality of provision for calculation. Calculation refers to approaches used to find the answers to problems in each of the four operations of addition, subtraction, multiplication and division. As part of the action plan staff identified key learning objectives and written methods associated with calculation from The Primary Framework (2006) and ‘Guidance Paper: Calculation’ (DCSF 2006). We matched the objectives and written methods to different year groups and I produced ‘calculation pyramids’ that showed the progression for each operation from Foundation to Year 6 (see Appendix 1 for Subtraction Pyramid). The exercise helped staff to recognise the development in calculation and the resource was used in most classes. However, SATs analysis (see Appendix 2 for analysis of SATs results) indicated that a significant proportion of children continued to experience difficulties with calculation, suggesting that calculation was an area that needed to be improved. Alongside this a Year 6 test showed significant weaknesses within subtraction (see Appendix 3 for Year 6 test analysis). Amongst the children who found it most difficult were those who generally had lower levels of attainment, including those with special educational needs (SEN). This reinforced the aim identified in the School’s Development Plan (SDP) which was to improve the attainment of ‘school action’ SEN children (children who receive extra provision as stated in a Provision Map). As part of the SDP staff needed to develop strategies to support these children. As a result this assignment aims to focus on addressing difficulties with subtraction; especially those experienced by children with lower levels of attainment. Review of Literature There has been much debate over the ‘correct’ (Brown, 2010) way to teach mathematics and about what children need to learn or understand. In the Primary Framework (2006: p.66) it states that mathematics ‘is a combination of concepts, facts, properties, rules, patterns and processes.’ It also states that there should be ‘lessons where the emphasis is on technique and [lessons where] children are steered to discover the rules, patterns or properties of numbers or 1
shapes.’ Such an approach covers two teaching positions: procedural, the ‘accurate use of calculating procedures’; and conceptual, the ‘possession of number sense which underlies the ability to apply [calculation] procedures sensibly.’(Brown in Thompson, 2010: p.3). The Primary Framework’s (2006) ‘emphasis on technique’ can be interpreted as a procedural position and within the strategy there are calculating objectives that call for the use of ‘efficient written methods’ (The Primary Framework, 2006: p.80). Written methods vary from informal methods such as an empty number line that can support children’s understanding to formal or standard written algorithms. Many (Anghileri, 2006; Brown, 2010; Thompson, 2010) argue that the teaching of formal written algorithms is limiting and that a conceptual position is required, developing children’s relational understanding where learners’ knowledge ‘extends from how to do a calculation to why the procedure works’ (Skemp 1976, in Anghileri, 2006 p.8). The Primary Framework also holds conceptual position though as learners are required to ‘understand the underlying ideas’ of calculation and, by year 6, they should be have a variety of mental, written and calculator methods and be able to ‘decide which method is appropriate’ (The Primary Framework, 2006, p.67). This would appear to suggest that a combination of procedural and conceptual understanding is needed in order to achieve success. In the next section I will consider how young children begin to develop an understanding of subtraction. The Ofsted Report ‘Good practice in primary mathematics: evidence from 20 successful schools’ (2011) identifies that to achieve a successful foundation in the four operations there needs to be ‘a clear emphasis on practical, hands-on activities in the Early Years Foundation Stage and Key Stage 1, with a high profile given to developing mathematical language and mental mathematics’. Anghileri (2006) has shown that concrete and practical experiences are at the heart of children’s early development in subtraction and other operations. Children manipulate collections of objects and are taught how to refine the language that they use, such as take away, subtract and difference. Counting experience is also vital as it ‘help[s] children to associate the concrete models with some abstract number patterns’ (Anghileri, 2006: p.50). A similar stance to early number work is taken in the ‘Development Matters in the Early Years Foundation Stage’ (EYFS) non-statutory guidance material, and like the Primary Framework, the EYFS guidance is used in the school where I work.
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Anghileri (2006) also argues that children need plenty of experience before moving onto the use of symbols such as ‘–‘ as then they may see its use as a particular ‘ritual’ and can misinterpret its meaning. For example, ‘-‘ can be interpreted as ‘taking away’, subtracting a specified number (subtrahend) from an initial quantity (minuend) or ‘finding the difference’, comparing how much greater or smaller two quantities are from one another. Progression from these early practical experiences can be described in terms of subtraction structures. Carpenter and Moser (1983, cited in Thompson, 2010) (see Appendix 4, Levels for Mental Subtraction) identified levels of progression for mental subtraction beginning with ‘count out’ where the child counts out a number of objects, removes the amount to be subtracted and counts the objects remaining. Next children ‘count back from’, and this usually requires children to maintain two counts at the same time. For example, to calculate 9 – 3 start at nine and count back three ‘steps’: 9 – 8, 7, 6. Here one of the counts is backwards (8, 7, 6) while the other is a mental count forwards (1, 2, 3) to keep track. With the same calculation 9 – 3, a ‘count up’ strategy could be used where children count forwards from three to nine: ‘four, five, six, seven, eight, nine’. Whilst doing so they keep a finger tally of how much they have counted up; in this case six. During children’s primary education they continue to develop mental and written strategies. Reports such as the Cockcroft Report (1982, in Thompson, 2010) have argued and identified reasons for the teaching of mental strategies. For example, it supports progression through to written methods and develops children’s ‘number sense’ where they have an improved knowledge of and facility with numbers and operations (MacIellan, 2001 in Thompson, 2010). However, the levels of progression for mental subtraction are not clear (Thompson, 2010), children’s understanding can be underestimated and ‘children themselves may not realise the methods they are using’ (Anghileri, 2006: p.57). Thompson identified other strategies used in mental methods: • Split method - partitioning numbers to carry out the subtraction, usually into tens and ones • Jump method – starting at one number and subtracting chunks of the second number • Split/jump method – a combination of the above • Compensation – subtracting more than is necessary and then adjusting, useful when subtracting near multiples of ten or hundred (Thompson, 2010 p.167) 3
Beishuizen (2010) has identified how split or decomposition strategies can lead to difficulties. For example, with the calculation 56 – 28, the calculation is split into 50 – 20 and 6 – 8 and the calculations are carried out separately. Subtracting the ones/units causes a conflict. Many children try to solve this by subtracting the small number from the larger. So 6 – 8 is often misinterpreted as 6 from 8, giving 2. This can be an over generalisation of the commutative law where children believe they can swap the numbers around and still get the same answer (Lawton, 2005). Ryan and William’s (2007) suggest another reason for the mistake is that children’s first experience of subtraction is to take the smaller from the larger and they may have been taught this explicitly. A jump or sequential strategy provides a ‘less vulnerable and more efficient computation procedure’ (Beishuizen, 2010: p.176). The empty number line (ENL) is often used to support these mental calculation strategies. The learner starts with the first number and shows jumps on the number line as chunks are subtracted, for example, 56 – 28 becomes 56 – 10 – 10 – 8 (see Figure 1a below). Klein et al. (1998 cited in Beishuizen, 2010) identified how the ENL supports informal strategies, leads to a higher level of mental activation, is a natural and transparent model for operations and is flexible. For example, the ENL allows the learner to split the ones/units in complements to (new) decadal tens and to use compensation methods (see Figures 1b and 1c respectively). Figure 1a -8 28
-10 36
-10 46
56
Figure 1b -2 28
-6 30
-20 36
56
4
Figure 1c -30 +2 26
28
56
The ENL features in the ‘Guidance Paper: Calculation’ (DCSF 2006) that was used by the school where I work. In the guidance paper the ENL is used in the first in three stages of written approaches to calculation: the ENL; partitioning; and expanded layout leading to column (or standard) methods. These were to be used for ‘building up to using an efficient method for subtraction of two-digit and three-digit whole numbers by the end of Year 4’ (Guidance Paper: Calculation DCSF 2006: p.8). However, Thompson and Beishuizen explain how there has been a misunderstanding in the purpose of the ENL. It was ‘never envisaged as a link between mental and written strategies, but rather as a tool to support mental calculation’ (Thompson, 2010: p.189). Thompson explores other areas of difficulty outlined in the Guidance Paper (DCSF 2006). One such area is the empty number line. The claim that it is used for ‘counting back’ is not always true. Counting back is ‘reciting backwards as many number names as you are subtracting (the subtrahend), and then giving the last number that you said as your answer to the subtraction’ (Thompson, 2010: p.194). Thompson argues that no counting back happens in the calculation in Figure 2. However, a similar example was used by Beishuizen (2010) and was considered good practice. Figure 2 Guidance Paper – Calculation: The Empty Number Line 74 – 27 = 47 worked by counting back:
Another potentially problematic area is partitioning. Of the two examples in Figure 3 the first is similar to the ENL but in a horizontal format, and the second example is used to partition both minuend and the subtrahend. However, Thompson argues that there is no logical 5
progression from the ENL to partitioning. The ENL cannot be used to partition both numbers, as they are in the second example, when the tens and ones/units numbers are treated separately. Figure 3 Guidance Paper – Calculation: Partitioning
74 – 27 = 74 – 20 – 7 = 54 – 7 = 47 74 – 27 = 70 + 4 – 20 – 7 = 60 + 14 – 20 – 7 = 40 + 7
Other problems are suggested in the progression from expanded layout to compact methods. Here a decomposition or column written method is used (see Figure 4) and the guidance suggests that children will exchange between the split minuend numbers when necessary. Initially children are to use the expanded layout where numbers are split into hundreds, tens and ones. However, Thompson (2010) argues that the use of the addition symbol when children are expecting to subtract is confusing and children are expected to begin with the lowest value digit which goes against children’s previous experience. Thompson states that the method is not a ‘natural’ form of progression for children, especially with the compact method because this step ‘involves a major shift in the way the digits in the numbers are interpreted: a shift from treating them as quantities to treating them as digits in columns’ (Thompson, 2010: p.218). Figure 4 Guidance Paper – Calculation: Expanded layout, leading to compact written method Example 563 – 271 500 + 60 + 3 − 200 + 70 + 1
400 + 160 + 3 − 200 + 70 + 1 200 + 90 + 2
400
160
500 + 60 + 3 − 200 + 70 + 1 200 + 90 + 2
4 16
5 63 − 2 71 2 92
Nevertheless, it is assumed that children will need to use some form of standard written procedure when dealing with larger numbers (Haylock, 2006). Haylock identifies how decomposition as shown above is preferable to other standard written methods such as ‘equal additions’ because ‘it is much easier to understand, in the manipulation of concrete materials, the manipulations of the symbols and the corresponding language’ (Haylock, 2006: p.59). 6
However, others such as Hart (1989 cited in Thompson, 2010) have found that children found it difficult to make connections between the manipulation of practical apparatus and their written methods. Thompson identifies advantages to the counting up or complementary addition method (see Figure 5). Here children record on the ENL or in columns how they jump from the subtrahend to the minuend. Because children can reduce the number of steps they take as their mental strategies improve Thompson (2010) argues that it has built-in progression. Furthermore, it leads to a more formal written notation. Figure 5 Guidance Paper – Calculation: The Counting-up method Example 74 – 27 74 − 27 3 40 4 47
→ 30 → 70 → 74
Now that the problematic nature of progression has been identified I will turn to specific difficulties that might be encountered by less able pupils. It is a general consensus that ‘children should be encouraged to use any method that they are confident fits the requirement’ (Anghileri, 2006: p.115). Unfortunately, research has shown how less able children find it difficult to move away from concrete counting methods (Lawton, 2005). They can struggle to develop more flexible abstract approaches and fail to understand written calculations because of poor place value knowledge. Another difficulty faced by children is recognising the requirement and interpreting the objective of ‘real life’ questions. Haylock has shown how different structures of subtraction: partitioning, reduction, comparison and inverse-of-addition can meet ‘a daunting range of situations’ (Haylock 2006, p.33). For example, if a child was asked how much taller a girl of 167cm is than a boy of 159cm, this requires the subtraction 167 – 159, and a comparison structure could be used.
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Nevertheless evidence has shown that mathematical understanding is ‘developed out of problem-solving rather than learned separately and then applied’ (Hughes, 1986; Beishuizen, 1995: in Anghileri 2006: p.52). Furthermore, real activities and experience help to motivate children and give a familiar situation making mathematics more accessible. Ethics The next sections consider the outcomes of the research events in different Year groups. Permission to use children’s work has been granted from parents/guardians and to preserve children’s anonymity any names have been removed or substituted by letters. Current Position Before I collected data on subtraction, myself and the teaching staff discussed the broader question of how to improve children’s attainment in mathematics; especially the lower attaining children. We analysed the results of Year 6 National Curriculum tests but they were inconclusive as evidence suggested both division and subtraction as areas of weakness. For this reason it was agreed that any intervention should begin by establishing the baseline more clearly, so the first tests, for Year 3 and 4, contained questions on all four operations. The questions were designed so that they were of a suitable level and would expose different types of mistakes readily. It was soon evident that we should focus on subtraction as tests indicated a more significant weakness in this area (see Appendix 5a for Y3 Y4 Tests). For example, Figure 6 shows a typical response by children. In Year 3, where only seven out of twenty one children correctly answered the question.
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Figure 6
I then administered a test for Year 5 children, this time looking only at subtraction (see Appendix 5b for Y5 Test). I also undertook a range of research activities with groups of lower attaining children from Years 1, 2, 3, 4 and 5. In Years 1 and 2 groups of children were given tasks, enabling myself to make observations and engage in discussion with the children to assess their understanding (see Appendix 6 for Year 1 and Year 2 tasks and assessment). Informal discussions were carried out with children in Years 3, 4 and 5 reflecting upon how to answer different test questions (see Appendix 7a, 7b and 7c for Year 3, Year 4 and Year 5 discussions). Such discussions were enlightening and support Ryan and Williams (2010) argument for quality discussion as they helped to unpick the errors and misconceptions children had. Lower attaining Year 1 children performed well at counting down in ones from numbers of twenty or below. However, they did find this considerably more challenging than counting up in ones. When working independently most children were able to subtract numbers up to twenty, achieving the Assessing Pupil Progress (APP) Level 1 statement to ‘understand subtraction as ‘taking away’ objects from a set and finding how many are left’ (Crown Copyright 2009). To do this most used a ‘count out’ strategy (Carpenter and Moser, 1983 cited in Anghileri, 2006). Some average and higher attaining children had progressed onto the ‘count back from’ method, using their fingers. Children found it difficult to explain their methods. For example, child A, who counted back, stated, “I just did it with my fingers.” Only the higher attaining children could count back in numbers above twenty and cross the tens boundary. Furthermore, a significant proportion of Year 1 children were still developing 9
their understanding of symbolizing relations (Anghileri, 2006) and were unsure of how to use the ‘ – ‘ symbol. Figure 7 shows how child B initially wrote ‘ + ‘ symbol and they reverted to addition when the question was subtraction. Figure 7
In the first assessment activity most Year 2 children used a ‘count out’ strategy (see Appendix 6) and chose to use counting resources such as unifix cubes to support them in answering questions. When counting out, typically average attaining children were surprised by the appearance of apparently small numbers such as 37. This indicated a lack of opportunity for creating a visual image for two digit numbers greater than thirty. The class teacher noted how some average ability children had previously used a ‘count back from’ strategy but they did not do so when using counting resources such as unifix cubes. This reflects Carpenter and Moser’s (1982, cited in Maclellan, 1997) argument that some counting aids encourage the count out strategy. This was confirmed in a subsequent activity (see Appendix 6) when the children used a different counting aid, fingers, and then used the ‘count back from’ strategy. Lower attaining Year 2 children were restricted by their lack of place value knowledge as they were unsure of numbers greater than twenty. When counting back both the average and lower ability children found it difficult to cross the tens boundary with numbers greater than twenty. For example, when counting back from 35 they counted 34, 33, 32, 31, 30, 39, 38 and so on. Whilst most average ability children quickly identified that the counting was incorrect the less able children did not and needed extra support.
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Throughout Key Stage 2 (KS2) tests and discussions (Appendices 3, 5a, 5b, 7a, 7b, 7c) showed that lower attaining children experienced similar difficulties with mental strategies. A number of SEN children had not progressed beyond using the ‘count out’ strategy. For example, Figure 8 shows how a Year 6 child drew objects to support their counting. This indicates that the child has not had sufficient experience at counting or using concrete materials such as a bead string to support their visualisation of abstract number patterns (Anghileri, 2006). Figure 8
Nevertheless, most lower attaining KS2 children had progressed to using a ‘count back from’ strategy. They tended to apply a ‘jump’ method when subtracting a single digit number and a ‘split’ method (Thompson, 2010) when subtracting a two digit number. Poor place value knowledge hampered some children’s counting back using the jump method because of difficulties crossing the tens or hundreds boundary (see Appendix 7a, Year 3 discussions, question 1). When subtracting two digit numbers using the ‘split’ method (Thompson, 2010) the lower attaining children made a variety of mistakes. By far the most common error of children of all abilities was to swap the ones around if the subtrahend was greater than the minuend or the ‘taking the smaller from the larger bug’, see Figure 9.
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Figure 9
This mistake was most prevalent in Year 3; for the calculation 82 – 29 nineteen out of twenty five children used a split method incorrectly and only those children who used the less vulnerable sequential or jump strategy answer correctly (Beishuizen, 2010). Children using the split strategy tried to follow the procedure but did not understand the principles of decomposition and over-generalised the commutative law which does not apply to subtraction (Lawton, 2005). Sometimes use of the split strategy led to quite complicated procedures that began to resemble what the DCSF (2006) describe as partitioning, but did not provide a correct solution, see Figure 10. Figure 10
Some ‘split method’ errors were due to carelessness, such as adding the numbers instead of subtracting them, see Figure 11a. Other children continued to subtract numbers when the answers from subtracting the tens and the ones needed to be recombined, see Figure 11b.
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Figure 11a
Figure 11b
In upper KS2 many children, including lower attaining pupils, used a compact written method or decomposition for more complex subtraction questions. Most lower and average attaining children displayed poor procedural understanding. They did not understand how to exchange and made the mistake of swapping numbers so that they were taking the smaller from the larger, see Figure 12. In Year 5 only five children answered the question ‘782 – 367’ correctly.
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Figure 12
Surprisingly, some older children were unfamiliar with the column layout. For example, at first a Year 5 pupil could not answer a question presented in the vertical format, see Figure 13, but after a short discussion with myself he exclaimed “Oh, so it’s eighty nine take away fifty six!” and proceeded to answer the question correctly. Figure 13 89 -56
With larger numbers many upper KS2 children also ‘lost sense of what they were doing and the reasonableness of their result’ (Anghileri, 2006: p.114). For example, when discussing the question 305 – 297, Child A, a Year 5 pupil, was surprised by the correct answer and remarked, “so the answer’s just eight!” Using decomposition had resulted in that child seeing the digits as individual quantities rather than whole numbers. Despite such difficulties the decomposition method was popular because children perceived it as being the most efficient method. For example, a Year 4 child who answered the question 309 – 198 incorrectly using the compact decomposition method stated that it was, “the best method because I don’t have to write as much.” In the test a small number of lower attaining Year 4 children used a counting up or complementary addition method. This method had been taught two weeks before the test but when they were given similar questions a fortnight later they had reverted to using a ‘count back from’ method with partitioning. This confirms Fischbein’s argument that because the 14
‘difference by building up’ interpretation is more demanding we tend, consequently to view subtraction as ‘making fewer’ (1993 cited in Cockburn and Littler, 2008). Lower attaining children’s ability to solve ‘real life’ problems or contextualised mathematics varied considerably. Sometimes the context helped them to understand the process involved but when it was more complex or unfamiliar they struggled. Ryan and Williams (2007) explain how such questions are like a game with rules but many children misunderstand them because they do not have a ‘feel’ for such games. Thus the ‘pedagogy needs to make the rules of the game explicit’ (Ryan and Williams, 2007: p.19). Lower attaining Year 5 children found questions which suggested a comparison structure especially difficult because they could not ascertain what they were required to do. Interestingly, some children did not necessarily interpret the comparison structure as a subtraction question. For example, Figure 11 shows how a Year 5 pupil has used an arrow instead of a + or – symbol for recording the calculation. This supports Thompson’s research (2010) that children in England normally interpret subtraction as ‘taking away’. Figure 14
As well as identifying pupils’ current position I carried out a planning scrutiny, focussing upon subtraction (see Appendix 8 for example). Most teachers used the planning tool provided in the ‘Abacus’ scheme, modifying it to meet the needs of the class. Teachers usually planned for children to use the ‘Abacus’ work or text books. These modelled different jottings to support children’s calculation but did not feature the ENL, helping to explain why there was very little evidence of its use. In KS2 there was a lack of visual representation of calculations, and an emphasis on using complementary addition for mental calculations and decomposition method for written calculations.
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The evidence provided a baseline for the ‘bookend’ evidence. It showed that children did develop more sophisticated mental and written strategies as they progressed through the school but many children, especially lower attaining children, had a poor procedural and conceptual understanding of subtraction. Action Taken I considered the findings from the literature review and the evidence gathered at my school to identify the change of teaching practice that was needed to achieve the intended outcome of addressing difficulties with subtraction; especially for lower attaining children. The teaching practice should: give children more regular practice with counting; provide concrete examples to support visualisation; provide children with a clearer path of progression; develop a more secure understanding of mental strategies and written methods; provide children with a better understanding of the properties of subtraction and develop a clearer understanding of the structure of different subtraction word problems. In order to develop teaching practice I coached staff using the GROW model (Brockbank and McGill, 2006). With staff I shared the goal (G) of addressing difficulties with subtraction; the reality (R) that was exposed by tests and other forms of assessment; considered options (O) for individual teachers and the class they taught and identified how teachers were willing (W) to carry out the actions. I provided ‘expert coaching’ where I had the expertise in the skill being developed (Ofsted, 2010). Initially staff meetings were used to develop teachers’ subject knowledge and pedagogy. Extra support was given where it was most needed: Years 2, 3 and 4. For teachers of those classes I led demonstration lessons, supported colleagues with their planning and helped to identify the ‘next steps’ for children. The degree and style of support varied from teacher to teacher depending on the colleagues’ and the class’ requirements. Analysis and impact of action taken At the first staff meeting I presented an activity where staff analysed subtraction calculations, methods and objectives. Staff sorted them into appropriate year groups and identified difficulties that children experienced with them (see Appendix 9 for example). I used the opportunity to engage in informal discussions with staff, analysing their perceptions of the 16
subject area and assessing their subject knowledge. Some members of staff felt insecure with certain subtraction procedures and this was an area I would address in subsequent meetings. Together we identified that a number of different methods were being taught for subtraction and that some methods, most noticeably the ENL, were hardly used. Teachers evaluated the Abacus scheme textbooks (2007) and found that the expanded written method was introduced in Year 3 and, like the Guidance Paper, used confusing addition symbols when children are expecting to subtract, see Figure 15a (Thompson, 2010). We agreed to remove the addition symbol and leave a gap between the partitioned numbers instead. Unclear jottings were used to support mental methods instead of the ENL, see Figure 15b. We agreed to use the ENL to support mental methods and revisit subtraction as counting back/ taking away as KS2 children were not having sufficient experience at working with subtraction as ‘making fewer’ (Fischbein, 1993 cited in Cockburn and Littler, 2008). At the next meeting we agreed to bring evidence of children using the ENL. Figure 15a
Figure 15b
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When teachers shared evidence of children using the ENL we identified how most of the KS2 children were able to use a ‘count back from’ strategy successfully. The method allowed for differentiation as the counting back ranged from younger lower attaining children using a jump method, counting back in ones to older children partitioning the subtrahend into thousands, hundreds, tens and ones, see Figure 16. We also noticed how some children had split the ones so that they could jump to a multiple of ten. Figure 16
After referring to Carpenter and Moser’s ‘Levels for Mental Subtraction’ (1984 cited in Thompson, 2010) we recognised that KS2 classes should also develop the ‘count up’ method for subtraction using the ENL. We progressed onto an interesting discussion where we identified how this method related to comparison and complementary addition structure questions (see Appendix 10 for Staff meeting notes). To promote teaching practice that would develop conceptual understanding (Cockburn and Littler, 2008) I shared Haylock and Cockburn’s (1989) connective model of learning 18
mathematics, see Figure 17. It suggests that different mathematical elements need to be experienced and connected to create full understanding. We agreed that our teaching should include the different elements as it provided a useful model for mathematics teaching (see Appendix 10, staff meeting notes). We also agreed to make contextualised questions more accessible to lower attaining children by providing examples that were close to the child’s experience and interests (Cockburn and Littler, 2008). Figure 17
After the two staff meetings, I met informally with the Year 2, 3 and 4 teachers to discuss further options available in the development of their teaching practice. One teacher had recently changed year group and another wanted to be shown how to use new methods and resources so I led demonstration lessons. This helped to develop my own and the teachers’ awareness of the needs of the class. However, to make a greater impact upon teaching practice we moved onto using a ‘self reflection-joint experimentation-feedback’ approach, see Figure 18, based on a model by Showers, Joyce and Bennet (1988). We continued to look at how the children progressed and identify next steps. Once this approach was embedded I found that the teachers or ‘coachees’ acquired the desired skills, took ownership of the lessons and were putting assimilation into practice, modifying their own practices accordingly (Rhodes, Stokes, Hampton, 2004). They were enthusiastic about the lessons and enjoyed discussing the outcomes which were often enlightening (see Appendix 11 for Lesson Plans and Assessment).
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Figure 18
Teachers’ feedback and assessment (see Appendix 13) showed that children’s attainment was improving. For example, activities helped lower attaining Year 2 children to understand the ‘logical structure of numbers’ (Anghileri, 2006: p11) with most independently counting back from any two digit number, crossing the tens boundary. Year 3 children had a more secure understanding of mental strategies and most lower attaining children used an ENL successfully. Similarly, in Year 4 children developed a more secure understanding of mental methods with all children being able to use the ENL to subtract a two digit number from another two digit number. In Year 5 lower attaining children had progressed onto subtracting three and four digit numbers. Use of the ENL helped children to become more efficient and effective, and teachers were able to identify and work with children’s misunderstandings. Furthermore, children were using it for other operations. For example, Figure 19 shows how an ENL has been applied by a lower attaining Year 4 pupil for division. Figure 19
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Such work helped children to compare, relate and understand the effect of different operations. Other activities made explicit links between operations and Figure 20 shows how children have demonstrated an understanding of the inverse relationship between addition and subtraction by exploring a concrete example. This understanding helps children to develop more flexible approaches (Anghileri, 2006) and have an awareness of different methods (Askew in Thompson, 2010). Figure 20
In Years 4 and 5 the decomposition method was taught in the expanded or compact form. To address the ‘taking the smaller from the larger number bug’ teaching strategies made explicit reference to the ‘bug’ as ‘awareness of the nature and impact of intuitions is a first step towards resisting them’ (Fischbein 1987 cited Cockburn and Littler, 2008: p.67). Indeed evidence showed that most children used the method more successfully and knew how and 21
when to exchange. However, some lower attaining children still found decomposition difficult especially if there was a zero in the middle of the minuend and they were required to exchange, see Figure 21. Figure 21
Haylock (2010) describes this as a ‘slight problem’ that is ‘easily understood if related strongly to concrete materials and the appropriate language of exchange’. All children used the recommended concrete base ten materials and interactive websites (see Figure 22) to develop their understanding. However, they found it difficult to make connections between the practical apparatus and their written methods which echoes research by Hart (1989; cited in Thompson, 2010). Figure 22 – Interactive programme using Dienes apparatus
I took such findings into consideration when producing a revised subtraction calculation pyramid (see Appendix 12). In order to provide a clearer path of progression the use of the 22
ENL was extended into Year 3 and decomposition delayed until Year 4. Previously children had been introduced to the decomposition algorithm earlier without necessarily exchanging and this led to overgeneralisations (Ryan &William, 2007). It is also intended that, like successful schools identified by Ofsted (2011), the new pyramid with omitted and reduced methods will be less confusing for lower attaining pupils and allow higher attaining pupils to move more swiftly onto more efficient methods used in higher Year groups. An area that continued to be a weakness was lower attaining children’s understanding of comparison or complementary addition structured questions. This supports Anghileri’s (2006: p.61) argument that ‘some expressions will be more difficult to verbalize than others’. More work needs to be done in providing concrete examples and pictures/images for the children to use. Figure 23 shows how one Teaching Assistant (TA) supported a lower attaining Year 5 pupil by drawing a picture of the problem. Support such as this needs to be used more readily by teachers and TAs. Figure 23
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Conclusion From research carried out it was clear that lower attaining children were experiencing difficulties with subtraction. There were weaknesses in both their procedural and conceptual understanding, and this led to a range of difficulties with mental and written methods. As a result of action taken most problems were resolved. For example, younger lower attaining children improved their counting skills and afterwards many were able to count down from a two digit number and cross the tens boundary. When subtracting smaller numbers most children had progressed onto using a ‘count down from’ strategy. Throughout the school the greatest success was the use of the empty number line (ENL) to support mental calculation. Teachers found it easier to identify children’s misunderstandings and children developed more secure mental methods. Children also displayed better ‘number sense’ or feel for number as they built upon skills they had already acquired, linking new information to their existing knowledge (Anghileri, 2006). The ENL also helped teachers to progress children onto using more challenging numbers; to use different subtraction strategies successfully such as ‘count back’ and complementary addition; and to develop children’s conceptual understanding of subtraction and other operations. Older lower attaining children had improved their ability at using a standard written method but some children continued to find decomposition difficult; swapping numbers around to ‘take the smaller from the larger’, and not properly understanding when or how to exchange. I believe the teaching of procedural skills has improved but it needs to continue to build on pupils’ knowledge of place value, using practical equipment and concrete activities. The study also provided a means for me to develop my own and other teachers’ subject knowledge and together we improved the mathematics curriculum. All teachers have participated enthusiastically, gained confidence at teaching subtraction and improved their ability at teaching mental and written methods. Already teaching staff are looking forward to building upon the progress made by lower attaining children so they continue to develop their conceptual and procedural understanding of mathematics.
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Reference List Anghileri, J. (2006) Teaching Number Sense London: Continuum International Publishing Group Askew, M. (2010) ‘It ain’t (just) what you do: Effective Teachers of Numeracy’ in Thompson, I. (ed) Issues in Teaching Numeracy in Primary Schools Maidenhead: Open University Press Askew, M., Brown, M., Rhodes, V., Johnson, D. & Wiliam, D. (1997) Effective Teachers of Numeracy London: King’s College London Beishuizen, M. (2010) ‘The Empty Number Line’ in Thompson, I. (ed) Issues in Teaching Numeracy Primary Schools Maidenhead: Open University Press Brockbank, A. & McGill, I. (2006) Facilitating Reflective Learning Through Mentoring and Coaching London: Kogan Page Brown, M. (2010) ‘Swings and roundabouts’ in Thompson, I. (ed) Issues in Teaching Numeracy in Primary Schools Maidenhead: Open University Press Cockburn, A. & Littler, G (2008) Mathematical Misconceptions London: SAGE Publications DCSF (Department for Children, Schools and Families), Crown Copyright (2006) Primary Framework for literacy and mathematic, Primary National Strategy DCSF (Department for Children, Schools and Families) (2006) Primary Framework for Literacy and Mathematics – Guidance Paper – Calculation http://nationalstrategies.standards.dcsf.gov.uk/ (accessed March 2010) Early Education: The British Association for Early Childhood Education. Crown Copyright (2012) Development Matters in the Early Years Foundation Stage: Non-statutory Guidance
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Haylock, D. (2006) Mathematics Explained for Primary Teachers (3rd Edition) London, Sage Publications Ltd Lawton, F. (2005) ‘Number’ in Hansen, A, (ed) Children’s Errors in Mathematics Exeter: Learning Matters Maclellan, E. (1997) ‘The importance of counting’ in Thompson, I. (ed) Teaching and Learning Early Number Buckingham: Open University Press Mertens, R. & Kirkby, D. (2007) Abacus Evolve Framework Edition, Year 3, Textbook 3 Oxford: Ginn, Harcourt Ltd. Ofsted (The Office for Standards in Education, Children’s Services and Skills), (2011) Good practice in primary mathematics: evidence from 20 successful schools Manchester: Crown Copyright Ofsted (The Office for Standards in Education, Children’s Services and Skills), (2010) Good professional development in schools Manchester: Crown Copyright Rhodes, C., Stokes, M. & Hampton, G. (2004) A practical guide to mentoring, coaching and peer-networking: teacher professional development in schools and colleges London Routledge Falmer Ryan, J & Williams, J. (2007) Children’s Mathematics 4 – 15 Maidenhead: Open University Press Ryan, J. & Williams, J. (2010) ‘Children’s mathematical understanding as a work in progress: learning from errors and misconceptions’ in Thompson, I. (ed) Issues in Teaching Numeracy in Primary Schools Maidenhead: Open University Press Showers, B., Joyce, B., and Bennet, B. (1987) Synthesis of Research on Staff Development: A framework for Future Study and a State-of-the-Art Analysis Educational Leadership November 1987 p77-87
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Thompson, I. (2010) ‘Getting Your Head Around Mental Calculation’ in Thompson, I. (ed) Issues in Teaching Numeracy in Primary Schools Maidenhead: Open University Press Thompson, I. (2010) ‘Written Calculation: Addition and Subtraction’ in Thompson, I. (ed) Issues in Teaching Numeracy in Primary Schools Maidenhead: Open University Press, Maidenhead
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