http://www.atm.org.uk/boris/subtraction/why-subtract-difficult.html
Why does subtraction seem difficult to teach or learn?
Introduction
We have found it useful to use the connective model (see diagram) from Derek Haylock and Anne Cockburn (Haylock and Cockburn 1989) to consider the different mathematical elements that need to be experienced and connected in order to create full understanding of concepts.
Haylock and Cockburn suggest that effective learning takes place when the learner makes cognitive connections.
Let us consider a particular example in early subtraction. Two children have collected some pebbles on the beach. They take them home and lay them out on the table. When they count how many there are they find that one child has four and the other has five. One child says “You have one more than me”, When asked how they know they show that “Four fingers and one more makes five”. The other child points out that on a die the five is the same as the four but with an extra dot in the middle. The pebbles are the context, the fingers and dice the image, the language of both subtraction and addition is used and there is the opportunity to model both 5 - 4 = 1 and 4 + 1 = 5.
Problems can arise when not all the four elements are experienced or, if they are all experienced, but they are not connected in a meaningful way. The role of classroom talk/dialogue is to help the children make the connections themselves. This talk/ dialogue can take the form of teacher questioning, children questioning, talk between children, explanation of points of view…etc. The verbal accompaniment to the children’s experiences is what allows them to frame their understanding. You can imagine that classroom talk/ dialogue is the arrows on the model that connect the four fields of experience.
· Subtraction objectives
· List of images for teaching subtraction
Subtraction
Where we see children struggle with subtractions they often have only one way into a problem: taking away in ones. They do not make connections with counting in larger steps or known addition facts, often because they do not understand subtraction as difference.
It is important that children do experience and understand subtraction as BOTH take away and difference. This is from a very early age and relates to real experiences they will have had. They will often be looking to see if others have more: “They have more sweets than me”, “I have fewer presents than him”. Children also experience take away very early on.
As children are first coming to grips with these two aspects of subtraction it is important the language, the situation and the numbers involved all reflect the understanding of subtraction that is most efficient way of solving the problem.
For example comparing the number of conkers collected by two children. Adam collected 25, George 22. How many more than George did Adam collect? How many fewer did George have than Adam? This relates to “What is the difference between 25 and 22” and can be solved by counting up from 22 to 25 or using a known fact (2 + 3 = 5). The situation, the language and the numbers indicate difference.
In the same way talking about a box of chocolates with 30 in and you want to give your friend 4 chocolates relates to taking 4 away from 30 and can be solved by counting back or using a known fact (6 + 4 = 10). The situation, the language and the numbers indicate take away.
However, later on we want children to be able to find the solution the best way according to the numbers, regardless of the situation. For instance I have 507p in my piggy bank and I am going to spend 475p on some new pens, what will I be left with? This is a take away situation – I am going to take away the 475p – but the best way to solve it is to find the difference i.e. find what I have to add to 475p to make 507p. Similarly, imagine we are investigating the difference between foot length and height to see if there is any relationship. Molly is 133cm tall and her foot length is 20cm. This is a difference situation – we want to know the difference between these two lengths – but the best way to solve it is to take 20cm away from 133cm.
Solutions to both difference and take away problems can be found by counting in ones and this is often how children solve early problems. However, as early as possible they should move from counting in ones to counting in bigger steps, counting up or back in ‘chunks’, and using known addition facts.