Addition and Subtraction Ideas
A lot of curriculum time is devoted to developing addition and subtraction concepts and it is very important that younger pupils gain a firm understanding of these ideas at a young age so that they can build on this as they go through primary school.
Using images and manipulatives helps pupils to represent the processes before moving on to more abstract calculations which is why they are widely found in classrooms of younger pupils. However, older primary pupils are sometimes expected to make the jump to abstract calculations rather suddenly since it can be dependent simply on which year group they have entered. The problem comes when pupils, who are still not confident, feel pressure to conform and are trying to work entirely in the abstract before they are ready.
Initially pupils use cubes, blocks and fingers (for calculations involving small numbers) as well as number tracks and the numbered number line that is usually displayed in classrooms.
Large number tracks and number lines can be used for pupils to jump along themselves and they can be easily created by joining pieces of paper, drawing chalk lines on the playground or even having some number lines painted as permanent markings on the playground. As pupils progress, blank number lines are useful for calculations and as a means of interpreting the problem visually.
As pupils progress, base ten materials can be used to illustrate the processes behind the algorithms used for addition and subtraction involving larger numbers.
Bar models are also excellent visual tools for helping pupils to understand addition and subtraction and can be used at all stages of primary school. They are also excellent for highlighting the links between addition and subtraction.
See Addition and Subtraction Facts in Resources, which uses bar models to help with assimilating addition and subtraction facts up to 10.
Working with addition and subtraction at the same time helps pupils to recognise the inverse relationship between addition and subtraction. Using bar models helps pupils to see the links between three numbers such as seven, three and ten.
Here, pupils can see that putting seven and three together results in ten, whilst removing three from ten results in seven and removing seven from ten results in three. This can first be represented using interlocking cubes but can then be applied to any size number and helps pupils to see that when solving subtraction calculations, we can ask what we need to add to get to the original number.
It also allows us to check subtraction problems by using addition, which is not always obvious to pupils. Missing number problems also benefit from this approach since 7 + ? = 10 can be represented by the bar model above, where the numeral 3 would be replaced by the symbol ?
Estimation is a skill that we use in everyday life and in these situations, we do not work out the answer at all. When working out whether we have enough money to purchase items costing £3.97, £39.99 and £11.75, we would add £4, £40 and £12 so as long as we had £56 or more then we could make our purchases.
Recognising estimation as a skill in itself is important but it is also useful to estimate the answer before calculating and use this to check against the result. If this approach is only introduced towards the end of primary school, pupils have a tendency to use numbers that are very close to the original numbers for estimating since they are used to always working towards a precise answer. The resulting mental calculation can then not always be carried out rapidly so in the example above, pupils may choose £3.90, £39.90 and £11.70 to work with.
Emphasise that estimating the answer mentally allows us to compare the magnitudes of the estimate and the answer. This is particularly important when working with larger numbers, or with amounts involving decimals, and in any situation where pupils are using columnar addition and subtraction where numbers can sometimes be placed in incorrect columns, which can significantly change the values of the digits.
Once we have an idea of the magnitude, a simple calculation involving the last digit of each amount can give us another indication as to how likely the answer is to be correct.
Pupils who have not been encouraged to estimate when calculating are more likely to produce solutions which do not make any sense.
Very young pupils can be seen using maths in context everyday in numerous activities, such as role play when shopping and counting and comparing when building. Being able to interpret everyday situations is essential for pupils and opportunities need to be provided in all maths topics.
When we teach pupils only in the abstract, they can struggle when it comes to solving 'word problems' because the two are not perceived to be connected. Whilst it is true that pupils need to be able to calculate in order to solve the context problems, the bigger issue is that they often cannot interpret the problem to be able to work out what calculation is needed.
Although the UK primary curriculum has reintroduced an arithmetic assessment at the end of KS2, this sits alongside reasoning papers which include assessment of pupils' ability to interpret problems in context.
It is therefore important that pupils at all stages at all ages are exposed to maths in context. Various ways have been suggested to assist pupils in tackling context problems, one of which has been to look for key words. The issue here is that pupils can then blindly associate certain words with certain operations.
The problem Mariah has two more pencils than David. If Mariah has six pencils, how many does David have? can result in pupils seeing the words more than and assume they must need an addition operation so decide to add the six and two and end up with David having eight pencils!
Pupils need to be able to see the problem by having a visual which is why I have been using the SPOT structure with pupils from six to eleven which involves people representing the problem visually in some way. This helps to structure the solution of problems by working on a process with the pupils in order to avoid pages of calculations where it is not clear how a pupil arrived at an answer.
See Resources for a download of how to use SPOT when working on context problems in your classroom.
We need to be careful in our use of language in maths classes since we can assume that pupils understand the language we are using, unlike in English, where we are far more likely to explore the language as we are reading text. We can also inadvertently confuse pupils when we use words that have a different meaning in maths than they do when used in other situations.
Sums is an example of a word that is used (particularly by older adults) to refer to all forms of arithmetic when it should be reserved only for addition - sum means to add.
Difference confuses some pupils who think, 'but they're both the same - they're both numbers, so there is no difference...'
Makes is another word that is often used when first learning addition and subtraction: 'seven plus two makes...' The problem arises when we start to introduce symbols and because pupils have always experienced several items and operations on the left of the equals sign, they are completely thrown if there is a missing symbol on the left or even just a number, such as: 7 = ? + 4.
Emphasising the use of equals and the idea of a balance when working with number systems can help to avoid this.
Provide pupils with specific words and ask them to see how many different types of problems can be created in pairs, using the given words. This is a useful exercise for highlighting language that needs to be worked on.
It is important that pupils can calculate effectively and confidently and one way to approach this is to provide pupils with methods, followed by lots of practice. However, if pupils have developed number sense, this allows them to be far more flexible when working with numbers.
An example of this would be when working with 18 - 9, which can be seen as 18 - 10 and adjust by one or it can be seen as the difference between 18 and 9. When we look at this on a number line, this is the same as 19 - 10 or 20 - 11 (amongst others).
When applied to larger numbers, this flexibility can be very effective: 178 - 39 can be seen as 179 - 40 while 1024 + 791 is the same as 1015 + 9 + 791 which becomes 1015 + 800.
When carrying out subtraction algorithms, such as 2001 - 278, it is helpful if the pupils understand that 2001 can be written as 1900 + 100 + 1 which becomes 1900 + 90 + 11.
Encourage reasoning when playing with the numbers in calculations, such as: 194 - 87 is easier as 200 - 93 (adding six to both numbers) but why can we add the same amount to each number for subtraction but 97 + 84 becomes 100 + 81? This can result in an interesting discussion, using number lines to explain their thoughts.