The core elements in mathematics


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Reflect on what you now understand about the teaching of written calculation.

Subtraction as one of the core elements in mathematics needs to be effectively and simply explained to children to ensure a fundamental understanding that can be applied as their mathematical ability progresses. Children learn through visual representations and also in written form. This is reinforced by the practical working out of calculations.

The National Curriculum requires that, "Pupils should be taught to understand subtraction as both 'take-away' and 'difference' and use the related vocabulary; recognise that subtraction is the inverse of addition." [1]This basic calculation will be used across the curriculum in subjects such as science and design and technology and therefore an early understanding of the principle is essential.

'The 1999 Framework sets out progression in written methods of calculation that highlights how children would move from informal methods of recording to expanded methods that are staging posts to a compact written method for each of the four operations.' [2] Children should be taught how to use efficient written methods of calculation and apply them when necessary with confidence.

Teachers can use various methods to explain subtraction as one of the four operations in mathematics. Visual representation is an effective way by using pictures, shapes and counting blocks as opposed to numbers in simple subtraction sums as suggested by Jennifer Suggate, "Using counters or blocks to represent the problem and then manipulating them to find the answer." [3] A child who is asked to solve 5-2=3 may use 5 blocks to represent apples on the table and take away two blocks and them count how many are left to find the answer to the question. According to Effie Maclellan, 'Counting is meaningful strategy to solve subtraction operations.'[4] Counting can also be achieved by means of fingers to help work out sums that are in written form. For example when a child is struggling with a subtraction calculation they can work it out on their fingers as another means of executing the question. I understand that to teach written calculation, I need to use all types of learning especially visual learning as shapes can be introduced to the younger years in school where shapes represent missing numbers and drawings can help children interpret a question and visualise what is going on. Written calculations are based upon mental calculations, and us as teachers are building upon what the children have achieve mentally. Eventually children will learn which method of calculation is most appropriate to solve a given problem. 'Children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence.' [5] By the end of Key Stage 2, children should be able to use written methods of calculation with confidence and understanding. Although for children to subtract successfully using written methods they need to have the prior knowledge of place value and multiples of 10 and 100.

An empty number line helps to explain the steps used in subtraction by showing children the relationships between numbers and how 'difference' can be displayed. This is a visual form that makes it simple for children to understand. 'Children learn how to use models and images, such as empty number lines to support their mental and informal written methods of calculation.' [6] They can move forwards and backwards between the numbers with ease. When children are representing subtraction on a number line they must remember that the counting back is always done below the line. The counting back or steps is usually done in bridging through multiples of 10. From my group work I now know that bridging helps pupils to have a clearer understanding of bigger numbers. Teachers can use number lines electronically on an interactive whiteboard and allow all children to participate. When faced with a subtraction sum like 45-23, children can decide whether to count back 23 from 45 to get the desired answer or to count up from 23 to 45 to find the difference by bridging through 10. For example, counting up, 23 is counted up 7 to get 30, and 30 is counted up 10 to get to 40, and finally 40 is counted up 5 to get to 45. When 7 and 10 and 5 are added up, it gives us the answer.

Using number lines children can learn partitioning of numbers. To work out 64-45, it involves partitioning the 45 to 40 and 5, and then subtract 40 from 64 and then 5 to carry out the subtraction. The answer to the sum can also be found by writing the question out. For example, 60+4 and 40+5 can be written out one below the other and then subtracted to get 20+9 which is 29. Partitioning the numbers and allowing the children to write them below one another is similar to the column method were the numbers are placed according to place value. I can see that the column method helps refresh place value and teaches the children the structure and method of a subtraction sum. Teachers can use online games for resources to help support the methods that are being taught. As children's ability in using the column method increases children learn to subtract up to four digit numbers and decimals using the appropriate written methods. They will learn to bridge not only in 10 but also in 100 and 1000.

Once children have grasped the column method, they will learn the 'decomposition' method of subtraction. When children are faced with a sum that cannot be done, borrowing is needed. For example, 63-35, the 5 cannot be taken away from the 3 so it needs to borrow of the tens column, so 60+3 turns into 50+13 so the first equation, now 13-5 can be completed and finally 5-3 can be carried out to get the combined answer of 28.

It is very important for children that their mental methods of calculation are also practiced and secured alongside their learning of written methods of subtraction.


  • Thompson. I (1997). Teaching and Learning Early Number . Open University Press. p38-40.
  1. The National Curriculum
  2. The guidance paper on written calculation,
  3. Mathematical Knowledge for Primary Teachers, Suggate. J pg 64
  4. Teaching and Learning Early Number, Thompson. I, The importance of Counting, Maclellan. E pg 38
  5. The guidance paper on written calculation,
  6. The guidance paper on written calculation,

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