Explain this
Why do fractions and decimals seem difficult to teach and learn?
Introduction
The Gaps and Misconceptions Tool was developed by Devon LDP ©2011 and has been made available here by kind permission.
Research references: Devon Learning and Development Partnership
We have found it useful to use the following model 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 division. Two children are cooking. Ten spoons of sugar are needed and they want to share this task. One child says “That will be five spoons each”. When asked how they know they explain and show that “Five fingers and five fingers make ten”. The sugar is then counted out. The cooking is the context, the fingers the image, the language of both division and multiplication is used and there is the opportunity to model both 10 ÷ 2 = 5 and 5 x 2 = 10.
Problems can arise when not all the four elements are experienced or, if they are all experienced, 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 demanding children make connections, children questioning concerning connections not seen, 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 represented by the arrows on the model that connect the four fields of experience.
Division
Where we see children struggle with divisions they often have only one way into a problem; sharing in ones. They do not make connections with counting in larger steps or known multiplication facts, often because they do not understand division as grouping. It is important that children do experience and understand division as both sharing and grouping. This is from a very early age and relates to real experiences they will have had. They have birthday parties where they want to get things for everyone and the balloons come in packs of ten, the chocolate wafers in packs of five etc. Deciding how many packets are needed is a decision based on grouping. Children also experience sharing very early on, especially with siblings and friends.
As children are first coming to grips with these two aspects of division it is important the language, the situation and the numbers involved all reflect the understanding of division that is most efficient way of solving the problem.
For example, packets of chocolate wafers come in fives and you need to buy 30 chocolate wafers for a birthday party. How many packets do you need? This relates to ‘How many fives are there in 30’ and can be solved by counting in fives or using a known fact about the five times table. The situation, the language and the numbers indicate grouping.
In the same way talking about a box of chocolates with 60 in the box to be shared between 4 friends relates to sharing 60 between 4 or finding a quarter of 60 and can be solved by halving and halving again. The situation, the language and the numbers indicate sharing.
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 Danish pastries come in packs of 2 and you need to buy 40 for a coffee morning. This is a grouping situation - how many 2s in 40 - but the best way to solve it is to share i.e. find half of 40. Similarly, £48 gift, shared between 12 people is a sharing situation but is best solved by grouping - how many 12s in 48?
Solutions to both grouping and sharing problems can be found by counting in ones and this is often how children solve early problems; counting in ones to make each group and counting ‘one for you, one for me’ to share. However, as early as possible they should move from counting in ones to the early stages of chunking.
For example, two young children are faced with sharing fourteen 1p coins between them to spend in the shop. The children have already experienced sharing by giving out one each. Before they do this stop and ask them:
“Do you think you will get more than one each?”
“How many do you think you could give out in one go?”
“Why do you think that?”
The child might say: “I could give us two each because that would only use 4 of the pennies up”
or
“I know five and five is ten so I could give out five each.”
Allow the children to then do what they have suggested and discuss what happens next.
Similarly some children have been asked to put Christmas labels in packets of five to sell at the Christmas Fair. They have a pile of 76 to package.
Start by asking them:
“How many packets do you want to start with?”
One child might say:
“I know I need two packets because that would use ten up”
or
“I know that ten fives are fifty so I could fill ten packets first”.
Both of these examples ask the children to think about what they already know to help them do the calculation and this is about accessing related multiplication facts. They are also both the early stages of chunking; this shows that both sharing and grouping lead into chunking and chunking needs to start early on.
It follows that, if you are asking children to use what they already know or can do to help them with division then they should be given divisions to solve with a divisor that they can confidently count in and therefore probably know some related multiplication facts. If we give children divisions such as 65 divided by 7 when the children can’t count in 7s and don’t know the multiplication facts for 7 they can only resort to counting in ones.
It is often the move from real situations and contexts to symbols that can cause a problem and this can be because there are no images or pictures to help make the link. Different images help children understand division, do division calculations and remember division facts. Images should include child created images (pictures/jottings) and making jottings needs to be taught. This will start with real objects then move to drawings of the real objects which then become more abstract and move towards the symbols.
For example, consider the situation where we want to know how many tables we need for a class of 30 children if the children are to sit in groups of four. We could model this first by doing it with the children and then talk about how this is difficult to always do and difficult to see so we want to use something to represent the children.; for example we could use small plastic people to represent the children. Then we could show how we could draw a picture - ask them to start to draw a picture. Stop them and question if they need to draw the whole people. Go back to the modeling and look at how we could use counters to represent the children so we could use dots around the table, then writing the numbers on the table; this could then move to a number line.
Here the pictures are helping the children do the mathematics but will hopefully also help them to understand and remember. Images have the potential for doing any or a combination of these things - helping children to do, understand and remember the mathematics.
Sometimes remainders seem to pose a problem. Often it is simply that children do not understand what the remainder represents and what they should do with it. They need opportunities to explore and interpret remainders in a variety of contexts.
It is also important to connect fractions to division from early on and understand that, for example, dividing by 4 is equivalent to finding a quarter and this can be thought of as multiplying by a quarter. This is explored further in the paper “Why do fractions and decimals seem difficult to teach”.
