This assignment will explore progression in the teaching and learning of addition and subtraction from nursery to year 4 considering the "understanding diagram", models for addition and subtraction, oral/mental and written methods, resources, learning facts and the using and applying/problem solving method.
One way that children learn about addition is through practical experience. In everyday life people are adding by combining two or more sets of objects. The same can be said for subtraction. Children learn that by removing a number of objects from a group it always leaves the same number of objects eg. 4-1 is always 3. This is how children first learn about addition and subtraction. For many children, they begin to understand the concept of adding when learning number songs in their early education. It is important to provide children with good practical experiences to help them learn. It is also vital to model the correct language so the children obtain it. This gives them a good foundation as they move through the education system.
Whilst children are in a Nursery setting or Reception class they will be following the Early Years Foundation Stage (EYFS) document. Within this document there are 6 areas of learning. The area of learning focussing on the development of mathematical skills is called Problem Solving, Reasoning and Numeracy (The National Strategies, 2011). Within this there are specific areas regarding addition and subtraction. Once children enter Year 1 they will be following the National Curriculum. Although this is the statutory document teachers often use the Primary National Strategies document to plan and deliver lessons. This breaks down the objectives of the National Curriculum offering a more focussed approach to teaching and learning.
The teaching of addition and subtraction in a nursery setting is based on practical activities. Counting songs is a common way of reinforcing numbers and introducing simple addition and subtraction. For example, five little ducks is a good example of how simple addition and subtraction is introduced and reinforced. The children soon learn that 5 take away 1 is 4 and that 0 add 5 is 5. The correct language can be modelled so the children learn what to use when they do activities for themselves. Games are also a common method of teaching addition and subtraction. Some games are not necessarily made for this purpose but it can be encouraged. For example by asking the children to compare the amount of objects they have, 'How many more/less do you have?'. The children will be working with small numbers and will soon be able to say how many less they have just by hearing the two numbers instead of having to count. Simple boundaries within the classroom are another way of encouraging the use of addition and subtraction. Where only a certain amount of children are allowed in each area at a time and the children keep track of how many there should be, how many less/more?
The foundation stage introductory pack offers ideas for activities for the different areas of learning set out in the EYFS. When looking at addition and subtraction in a Nursery setting the document offers activities for comparing two groups of objects; showing that when you split a group of four the total is the same and finding the total number of objects in two groups (The National Numeracy Strategy, 2002). Each activity uses physical objects to demonstrate the mathematical rule. They use resources such as the number line, plastic coins and snakes. For young children in particular good resources are essential to engage them and effectively teach them about addition and subtraction.
In the reception document activities for looking at posing problems such as 'how many will there be when one more.....?'; encouraging the children to say the number that is one more than a given number and giving opportunities for children to find one more or less than a number up to ten are offered (The National Numeracy Strategy, 2002). Again, each of these activities involves physical experiences and the activities teach and reinforce the mathematical rules for these aspects of addition and subtraction.
Although the main document for Foundation Stage settings is the EYFS the Primary Framework wants to encourage the aspect of using and applying mathematics. The using and applying strand has five themes with progression built into each from the foundation stage to key stage 2.Within the foundation stage, it states that children will be using their developing mathematical ideas and methods so they can solve practical problems. Therefore, any problems they are given related to addition and subtraction they will be able to solve given their prior knowledge. (The National Strategies, 2011)
Year 1- Year 4
From Year 1- 4 it becomes more in depth. For Year 1 children they will be looking at various problems to do with adding and subtracting and solving problems in the theme of money and measurements. This means that a lot of the concrete experiences they have will be based around shops in their role play area, giving the children a 'concrete' experience of counting money and addition or subtraction within those scenarios. Year two is much the same only with the addition of having to multiply and divide using the subject of money and measurements. Year three is a slight step up with the children having to choose which calculations to use and to carry them out themselves. Year 4 is not too different only they will be learning how to use calculator methods where appropriate.
When children have to solve problems, they will be sharing their ideas, using numbers, symbols and/or diagrams. They will also be reasoning and predicting and sharing their results, either orally or in writing.
The 'understanding diagram' by Haylock and Cockburn, shows the different aspects of learning mathematics needed for a child to be proficient and confident. One of the major parts of the diagram is concrete experiences. The teacher needs to use a range of resources in their teaching. By doing this it enables the children to better remember what they have been taught as they link it to a physical memory. It also allows the children and the teacher to have discussions more easily. During these activities it is important for the teacher to model the language they want the children to take on and to use the correct symbols themselves to encourage the children to do the same.
Written and oral/mental methods for addition and subtraction are two other aspects of mathematical development.
Early practical, oral and mental work, carried out in the foundation stages, is the basis for offering children the opportunity to build on their knowledge of addition and subtraction. Later on in their education children must be able to recognise how the rules can be used and applied. Oral and mental work is not just to be used in the beginning of education but must be continued to provide practice of these ideas. Children must be given the opportunity to apply the information they have learned and to make correct decisions for themselves. To be able to calculate mentally needs an understanding of number patterns and relationships that are developed through questioning, by using certain models and applying the knowledge of numbers. In order to calculate mentally children must have the ability to recall number facts instantly. In year 2 this would be the addition and subtraction rules up to 10. For year 3 it would be 'sums and differences of multiples of 10' and for year 4, 'the multiplication facts up to 10x10'. There must also be an ability to use what has been taught in order to work out the calculation. For example, in year 1, to be able to understand that you can start addition sums with any number and use the information to do mental calculations of one or two-digit numbers and to be able use different methods for partitioning two-digit numbers in year 2. Finally the ability to use and apply the rules of mathematics. For example, to be able to perform mental calculations of addition and subtraction of one and two-digit numbers in year 3.
The written methods for addition come in 4 stages and the aim is that children are able to use the mental methods where possible but when they cannot, to use an efficient written method with accuracy. Children need to know at least one written method for addition that they feel confident using if the mental calculation is not possible. The following stages show how the children are able to build up to use an appropriate method for adding whole numbers by the time they finish year 4.
In order for the children to add successfully they need to know some basic skills. They need to know the addition pairs up to ten; how to add a series of single-digit numbers; how to use the related number facts to add multiples of ten and be able to use different ways of partitioning two and three-digit numbers.
Stage one of the written methods involve the use of the empty number line. Children need to be able to split numbers in a variety of ways to help them add in steps. The empty number line is a way of helping them to record their steps when calculating the total.
Stage 2 involves partitioning so that mental methods can be recorded. The tens and ones are added to form partial sums and those partial sums are added together.
The third stage is the expanded method in columns. The children use a layout showing the addition of tens and ones separately. As children become more confident they can start by adding the ones rather than the tens.
The fourth and final stage is the column method. In this method, there is even less recording to do. The carried digits are noted below the line, either in tens or in hundreds. This can be made more challenging. The children can move on to add more complex numbers of different numbers of digits.
The written methods for subtraction come in three stages. The aim is the same as for the written methods of addition and again the stages show how the children are able to build up a method for subtracting two or three-digit numbers by the time they finish year 4. To subtract successfully the children should know the number facts for addition and subtraction to twenty; use the related number fact to subtract multiples of ten and partition numbers into multiples of one, ten and one hundred in numerous ways.
Stage one, again involves the use of the empty number line, which helps the children record and describe the steps they have taken in their mental subtraction. After the children have practiced this method they won't need to record as much information. They will need to decide whether to count back or up. It is useful to ask the children which is better for certain calculations. When counting up from small to large numbers mentally it can be recorded using number lines or columns. The children will need to be able to, with two-digit numbers, calculate the answers mentally. If the children are able to work out the answers they don't need to perform as many steps when working with three-digit numbers. The counting up method is a good option for those children whose progress is slow.
Stage 2 involves partitioning. Partitioning can be used to write comparable subtraction sums that can be performed mentally (The National Strategies, 2011).
The third and final stage is expanded layout. The column method is reflected by partitioning the numbers into ones and tens and then writing one under the other. This resembles the method for addition rather then being directly linked to any mental methods. This also relies on secure mental skills.
Children build on their prior knowledge to progress with their mathematical skills. They all start with practical experiences and constant exposure to addition and subtraction. All children need to develop sound mental skills in order to develop their written skills. They have to learn the basic rules for addition and subtraction to progress with the written methods.
The 'understanding diagram' is a good guide for what is needed and offered to children when they are learning about addition and subtraction during their education. All children need to have concrete experiences to both introduce and reinforce new concepts. The children need to learn how to use knowledge they gain in real life situations. They have to understand why the concepts they are learning are useful. It is also important to model the correct language to the children as they are being taught new number facts and mathematical rules. By hearing the correct language the children are able to acquire and use it for themselves. They also need to learn the correct symbols when doing certain calculations when carrying out the written methods of addition and subtraction. The teaching and learning of addition and subtraction becomes more advanced as children progress through the educational system but the aspects of the 'understanding diagram' are always vital.