Really Understand What it Means to Add or Subtract
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Addition is the process of finding the total when combining two or more quantities. It is considered to be one of the most simple numerical tasks and is taught to children from a very young age. To begin with, children are taught addition of single digit numbers using the decimal system and are then gradually extended on to more difficult concepts. There are two main ways of visually representing an addition problem; the first is to think of it as combining two sets (such as two piles of sweets for youngsters) and the second is to imagine extending an initial length by a second length (such as on a number line). Addition has a number of important properties:
- counting is the same as continually adding 1
- adding zero does not change the total
- addition is commutative - children need to understand that the order in which you add the numbers does not matter3 + 5 = 8 and 5 + 3 =8
- addition is associative - when adding more than two numbers, it can help children if they realise that the order in which the addition is performed does not matter 4 + 3 + 7 = 4 + (3 + 7) = 14
I have seen many resources that try to overcome this problem by stating that you must always start with the larger number. Although usually correct for early subtraction problems, this is an incorrect statement and can lead to the deeply embedded misconception that you can not take a bigger number from a smaller. This statement is dangerous and should be avoided. Instead of telling children to always start with the larger number, talk about the numbers as parts and totals (for example with 3 + 5 = 8, 3 and 5 are the parts and 8 is the total) and then tell the children that they must always start with the total.
A deep understanding of the basics will make moving on to more difficult concepts (such as adding/subtracting fractions or negatives) a much smoother process. For example:
- A good understanding of the commutative property of addition is vitally important for future mathematics. Understanding that to answer 3 + 72, you can actually do 72 + 3 makes mental arithmetic much easier. It also enables for effective manipulation of algebraic equations, as students must understand that x + y + 2x is the same as x + 2x + y in order to simplify to 3x + y.
- Mathematical symbols (+, -, =) can seem like very abstract, irrelevant objects that are yet another thing to learn in a child's busy life. However, if children understand the meaning of (and the vocabulary associated with) adding and subtracting, they will be able to apply this to solving problems at a later point. I have seen many talented mathematicians who really struggle when interpreting worded problems and writing equivalent equations. If children can understand the mathematical symbols as shorthand ways of writing more meaningful phrases then they will more easily be able to set up their own equations.
Tasks to help kids really understand adding and subtracting
Use language to emphasise meaning
Before presenting children with written equations that use mathematical symbols, ask them worded questions and use props that really require them to think about the process that they are using to arrive at the answer. Use language such as 'count on', 'how many altogether', 'how many more', 'how many fewer', 'what is the difference', etc.. Start with small, single digit numbers and mix together addition and subtraction problems so that children don't see them as completely separate topics to learn. Avoid using mathematical symbols until the children are confident with the process; this encourages them from the start to use addition and subtraction to solve 'real-life' problems, rather than the other way round.
Introduce the mathematical symbols
In order to understand and write addition and subtraction equations, children need to know the '+', '-' and '=' symbols. The '+' and '-' symbols can be introduced using the language from the last task but the '=' symbol needs special emphasis. I have seen the equals symbol used incorrectly by students of all ages and at all stages of learning. It is a fundamental principle that many students do not fully understand and it makes many other areas of mathematics a lot more difficult.
Encourage children to write mathematical equations from images and 'real-life' problems and use language that will help them understand and visualise the problem they are trying to solve. It is important to get children to write their own number sentences from pictures, rather than using pictures to help solve equations. For example, present children with a picture of two bunches of flowers and ask them for the total number of flowers. Encourage them to create their own equation to record the maths they have done in solving it. This encourages them to think of addition and subtraction equations as tools for solving and recording problems and will help them apply number sentences for future, more difficult problems. I found a lovely worksheet of TES that has nothing but pictures and encourages the children to write, and really understand, their own number sentences.
Sets of objects in two colours
|Picture from the National Strategies|
This activity is also a great opportunity to highlight the commutative property of addition. Children will see that it doesn't matter that they had 13 + 4 but their friend had 4 + 13, they both ended up with 17. Again, the ability to physically 'see' this and come to the conclusion independently will help to really embed the concept and to be able to apply it in other situations. It also enhances awareness of why the corresponding subtraction facts start with the larger number (or total) without having to tell the students the incorrect fact that subtraction always starts with the larger number.
|Click here to download board game|
There are many fun and engaging ways of using a number line for counting, such as making a board game or using a life-size number line drawn on the playground to walk up and down.
Number trio cards
Many older children show signs of misconceptions and lack understanding of the process of addition and subtraction, and this makes their lives a lot more difficult in many areas of mathematics. I strongly believe that if they develop a deep and fundamental understanding of adding and subtracting early on, this can lead to a much more positive experience with maths throughout school.