Section 1:
The inescapable use of partitioning and its associated dangers
Partitioning is fundamental to calculation in primary school. Whilst in the early years children develop calculation based on “knowing and recalling” number facts, as they move into Key stage
2 the numbers become of a size where, recall is incongruous. No child (nor adult for that matter) would “recall” or “know” 2374 + 4736. The numbers need to be broken down so as they can be brought back into the arena where the key question ; “Can I do this in my head?” can come into play. Thus partitioning is fundamental to primary school mathematics.
However… there are inherent dangers when this approach becomes the core element of the
Maths curriculum.
1. It allows all sums to be reduced to a series of single calculations based on number bonds to 10. Whilst this has great power in allowing children to access a mental approach it creates a strategy base that is detrimental to holistic mathematical understanding.
2. It prevents children developi ng a true “feel for number” in terms of their size, properties, relationships within the number system and pattern structures. This is because children see 324 + 245 as a 3 part calculation involving 3+2; 2+4 and 4+5. This leaves opportunity for them to develop little appreciation for the place value of number.
3. At its worse partitioning allows children to wander through the labyrinth of Mathematical learning without the need to engage with “real” numbers and secure calculations.
Children can leave Key Stage 2 with a shallow understanding of calculation that has reduced all sums to as a series of unrelated tens and units calculations that sometimes are straightforward e.g. 242+121 and at other times involve little tricks; e.g. “carrying one into the next col umn” (246+186) and “borrowing one” (214-189). The “little tricks” are viewed by children as irritating hurdles to get over, or get round in their major goal which is to continue on their narrow “number bonds to 10” journey. Whereas in truth the “trick” is a major block of mathematical understanding with which they should engage.
4. As one Year 5 girl given the sum 253,254,246 + 132,432,732 noted upon observing that none of the numbers caused issues of “carrying”; “… there is no Maths in that calculation at all it is just a set of simple sums”. By the same token the children in the same lesson that calculated 467+288 as 61,415 showed that although they could reduce the sum to a series of simplistic calculations their associated knowledge of place value was woefully lacking.
International Research to Support the local school view i Recent research from Holland shows clearly that children have a natural propensity to revert to partitioning as a strategy. The reasons for this are not given but I would have thought were fairly obvious even to the most casual observer that the children are seeking to simplify these complex calculations into an arena where the answer can come within the range of their “known facts”. (Thompson,1999b) ii Scrutiny of calculations that re ly primarily on partitioning suggests that there is “no evidence of what is normally understood by place value in their methods” (Ruthven, 1998). iii This is especially true where the column method is adopted. When children are encouraged to use mental calcu lations their “strategies use what has been described as the quantity value aspect of place value (56 seen as 50 and 6), whereas standard written algorithms draw on the column aspect (56 seen as 5 tens and 6 units) (Thompson,1999a)
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Section 2:
Common Misconceptions in calculation
The lack of understanding is evidenced clearly through the “standard” mistakes children make in their calculations. These provide a window into the mathematical areas of misunderstanding that exist in the mind of the child. These need to be addressed not just as individual conceptual errors, but to look at the deeper mathematical principles that lie behind them so that these may be backfilled as well. For instance issues of place value may relate to simple semantic column confusion but more probably will be an indication that the children sees little relationship between numbers in base 10. They may not have a “feel” for how big 1000 is, nor how it fits in the greater number system. It behoves us therefore whenever we enter a calculation lesson to ensure that the method taught seeks to underpin these higher level Mathematical concepts as well as sharing with the children a particular strategy for a given calculation.
Misconceptions in Four Rules Calculations
1. Addition a) Place Value (column method) 2 3 4
+ 2 1 6
5 4 0
1
2 4
+ 1 6
3 1 0 b) Place Value (non column based) 3.2 + 17 = 49 or 4.9 c) Relationship of Number 16 + 10 = 25 d) Relationship within calculations 30 + 20 = 50 therefore… 50 – 20 = x
2. Subtraction a) Place Value (Decomposition) 2 5 4
- 1 3 7
1 5 14
1 3 7
2 7 b) Number c) Relationship of Number
4 5 6 – 2 6 4 =
87
400
50
6
– 20 = 66
– 200 = 200
– 60 = 10
– 4 = 2 d) Relation of Calculation 50 – 20 = 30 therefore … 30 + 20 = x
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3. Multiplication a) Computational Understanding b) Place value/ Computational Understanding
24 x 12 = 20 x 10 = 200
4 x 2 = 8
208
145 x 30 = 145,000 c) Place value 2 4 x 1 2
8
4 0
4 0
2 0
1 0 8
4. Division a) Place Value/Feel for Number 242 ÷ 14 242 ÷ 10 = 24.2
241 ÷ 2 = 85.5
= 109.7 b) Place Value 600 ÷ 30 = 2 c) Computational Understanding 540 ÷ 20 = 52 d) Relationship within calculations 3 x 5 = 15 therefore… 5 ÷ 15 = 3
5. Estimation within calculation
Alongside all these calculation errors is the fact that many children in the upper reaches of
Key Stage 2 struggle with attaining a reasonable estimation for a calculation and will often produce answers that show a complete lack of unders tanding of the “feel for the numbers”.
These two examples are from this year’s More Able group of Year 6 children a) 427x27 = 149,286 b) 376 x 26 = 64,523
Within the same group of children were some who struggled to give accurate estimates for calculations such as 36 x 27 when this was set as an isolated task. They all struggled to estimate answers that contained decimals.
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Section 3:
Why do children keep returning to the partitioning method?
No surprises in the reasons given, firstly the children use partitioning to make calculations more accessible
 “When you split it up the numbers look quite nice and small” (Daniel)
 You can make them into small sums and they look easy” (Harry)
There may be little we can do about this for as international research shows “children have a natural propensity to revert to partitioning” (Thompson,1999b)
However the second reason is more pertinent to our deliberations in that the methods we present to children, especially in the first instance have a profound impact on their later preferred method
 “Partitioning was the first method we were taught so we keep returning to it” (Gillian)
Whilst it is clear that children gain much success from the use of the number line recent research
“indicates that after some initial success in using the model to develop mental methods, most children reverted to formal methods within two months. This may change if its introduction
precedes formal methods” (Rousham, 1997)
What the research shows is that the first form of calculation introduced to children has a powerful impact on their later development. This makes it key that full professional consideration is given to the methods children are introduced, and that this is embedded in a clear framework of progression at a whole school level.
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Section 4:
Where do we go from here?
In the light of the above I would suggest that the two key factors that need to be addressed are: i) The children’s lack of feel for number ii) The lack of true mathematical understanding of what is occurring within the process of a given calculation the following:
I would therefore propose the following as whole school policy: i) The method we teach children is the one outlined in the school’s Calculation Document.
This has a heavy focus on what the Year 5 children came to call “Half Partitioning”; where one of the numbers in any calculation remains as a whole and only the subsequent numbers are partitioned. i.e. 245 + 352 = becomes 245 + 300 + 50 + 2 as opposed to a series of partitioned calculations focusing on the single digits of 2+3, 4+5 and 5+2
This allows children to focus on the “quantity value aspect of place value” where 56 is seen as 50 and 6. Whereas standard written algorithms draw n the column aspect and 56 is seen as 5 tens and 6 units. (Thompson 1999a) ii) To facilitate this more effectively the use of the number line should become the primary tool for calculation. It is both visible and, in conjunction with the approach outlined above, allows children to engage with the “real” numbers as opposed to a set of single digit calculations out of context. iii) The number line should be the standard accepted model throughout the school. The children should be presented with this as a base model and, more importantly, as the first model that they encounter when tackling new calculations. As we know the initial introduction of any calculation method is one of the greatest determining factors in its future use by children. iv) The use of the column method should not be introduced until Year 6 (except for addition which will be taught in year 5). By this time children will have a secure understanding of the operations of the four rules. I fear that earlier introduction of these methods adds little to mathematical understanding and has the potential to be used by children as a route to bypass true mathematical cognition. Its introduction in Year 6 is to be seen as simply an additional strategy for children to use when appropriate rather the definitive, all encompassing strategy to be used for all calculations.
“Children tend to use algorithms as mechanical and where they do not understand the procedures the research evidence suggests that they are unable to reconstruct the processes involved” (Anghileri 2001) v) The school should regularly undertake a thorough review of its Maths resources. If we feel that children are losing a “feel for number” then the use of these materials may become a key feature in addressing this. Whilst Piaget informs us that in the preoperational stage (ages 2-7) the child cannot conceptualise abstractly, it would appear that our children, in the supposed “concrete” phase (ages 7-11), would similarly benefit from such support. Deines blocks, the decimal and fractions blocks etc., were all once standard fare in classrooms but current teaching would appear to have become a little more abstract. Certainly those children in the lower end of the academic spectrum at KS2 who, by definition, will be pre-operational would benefit greatly for a more kinaesthetically based approach.
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Section 5:
Related Issues and FAQ
1. Doesn’t this seeming didactic approach negate our emergent philosophy?
Not at all; the key is that in allowing children to explore and play around with numbers we are
“scaffolding” them towards a given solution that we know will enhance their understanding.
To illustrate here are a series of lessons taught to Year 4/5 children on subtraction:


In the opening lesson the children were presented with a calculation such as 246 -158.
Their initial solutions may include a series of errors including the classical 40 – 50 = 10, there were others that partitioned and then added the resultant calculation as opposed to subtracting from the original number.
 The strategies were logged on the board. They were then analysed for their effectiveness. The majority had chosen a partitioned method and could see the downfalls
 within it.
I then suggested I could provide a method that incorporated their desire to partition with a


 means of alleviating the issues that partitioning both numbers created.
The childr en bought into it because it made “connections” with issues they had encountered for themselves. It is this “buying in” process which lies at the heart of the emergent approach. The key is that children seek their own solutions to their own problems, preferably (with teacher scaffolding) from themselves or if not modelled by the teacher.
The children consolidated the strategy over the next few lessons
They then applied their new found concept of “Half partitioning” to other mathematical areas such as addition.
2. What about SEN children who are not ready for the Number Line?
“There is general agreement that … children progress through a sequence of count all, count on from the first number, count on from the largest number, use known facts and derive number facts” (Gray, 1991) The whole basis for this rethink of Mathematics is that we reach a point where children are engaging more with the true understanding involved in the processes of calculation as opposed to being shown “tricks” that allow them to remain (seemingly) in touch with their peers. Effective and personalised teaching is about meeting the needs of the individual child and there is therefore no place for giving some children strategies that seemingly will allow them to stay afloat in a given year group without a measure of full understanding. The curriculum needs to be differentiated to meet the needs of children wherever they are on the continuum outlined above. This may lead to some children undertaking work that is different from the remainder of the class, but in fairness this should also apply on occasions for those on the more able end of the spectrum.
Related to this is a perceived issue about self esteem. In reality children know their limitations and their weaknesses as well, if not better, than we do. They will appreciate that although they are being given a strategy they can use, the full understanding is not there. This has the potential to send powerful subliminal messages that undertaking calculations with a minimal set of understandi ng and “tricks” is acceptable. This would seem incongruous with everything we are trying to achieve, in the sense that we are seeking to build greater understanding into the mathematical process for children not less. Similarly the lack of understanding will surface in future learning arenas and where the foundations are not secure the children will continue to flounder. We therefore owe it to all children to provide a curriculum that is heavily differentiated to each child’s needs.
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3. If the emergent philosophy seeks to engage children with their own individual methods how do we prevent a culture that sends out a subliminal message that there is a correct way of tackling calculation? This of course, is a key feature of the personalised learning debate.
The driver behind establishing a whole school policy is that it will provide a rich mathematical framework of progression for the majority of children. Research undertaken both within and outside the DfES with regard to the Numeracy strategy has revealed that the overdevelopment of a range of strategies sometimes does not allow for children to securely consolidate any one method of calculation. Current thinking is that all schools adopt a central core strategy and work out from these. Having said that we should not limit children to the strategy we perceive as being the most appropriate. Whilst we might from an adult perspective appreciate that a given strategy is both effective and efficient for a given calculation some children will inevitably revert to a strategy they feel most comfortable with. There is no shortcut here except that the teacher should exercise his/her professional judgement as to whether a child should continue with their own strategy or be “encouraged” to explore a more appropriate alternative.
The following are true scenarios from Key Stage 2: i) John understands the negative numbers in subtraction partitioning. He is secure in his knowledge and uses the strategy fluently. It is my contention therefore that he should be allowed to continue with this approach – unless the teacher perceives that in a fresh context his understanding may ii) Jane’s desire to conquer the column method is home driven and is not secure. Being quickly moved on to the calculation 500 – 276 she became aware of the limitations both of the chosen strategy and her ability to transfer knowledge in a new context. She was encouraged to use the number line to bring her back on track with her understanding. Interestingly enough she had to regress to a point below that of her peers to rebuild her understanding in the fundamentals.
At present it would be fair to say that the majority of children would have a wide range of strategies they would choose to use to solve a given calculation. This is because they have (and rightly so) been left to explore their own methods, and from this they have either; developed one particular strategy they are partial to, or maintain the strategy they were first introduced to. I suspect this will become less of an issue in succeeding years as the children encounter a more rigorous adoption of the “sequencing” strategy throughout the school.
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