Explain this
Why do fractions and decimals seem difficult to teach and learn?
Introduction
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 fractions. Two children are cooking, filling a tray of 12 cake cases. They are told they can fill half each. One child looks at the tray and says, “We can do two lines each”. The other child looks at the lines and says “That’s six because three and three is six, like on a dice”. The children fill six cake cases each. The cooking is the context, the tray and dice the images, the language of fractions, division and multiplication is used and there is the opportunity to model both 12 x 1/2 = 6, 12 ÷2 = 6 and 6x 2 = 12
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.
Questions on proportion, in all its forms, are often answered poorly in national tests. There are some common misconceptions that seem to be partly responsible. These include the belief that: fractions are always parts of 1, never bigger than 1; fractions are parts of shapes and not numbers in their own right;
a fraction such as ¾ is only ‘three lots of a quarter’, never ‘a quarter of three’; decimals with more digits are bigger; and percentages can never be bigger than 100%.
Fractions
One of the things we need to ensure, as teachers, is that children are given a variety of experiences that allow them to engage with fractions as both the names of numbers and also as operators. They can then consider where different fractions fit into our number system as well as how to find a fraction ‘of’ something.
As there is such a gap between how confident children are with whole numbers as compared with fractions, it is useful to look at how children first learn about whole numbers. Children are exposed to whole numbers, very early on, through the counting ‘rhyme’. This exposure is something that can happen on an almost daily basis, other people around counting out loud in different situations and the children are immersed in it. It can be quite sometime before they join in with parts of the ‘rhyme’ or learn the ‘rhyme’ for themselves. So at the start it is about learning a rhyme, a list of words in a particular sequence that seem to have a certain rhythm to them.
Through learning this rhyme children become familiar with the number names, (the language of numbers) and the order of numbers. When the rhyme is used in contexts and children count objects they learn about how the change of number name indicates an increase of one. Later they learn to match these to the symbols (link language to symbols. Without a background of being immersed in counting, can you imagine how hard it would be to calculate? Children who arrive in reception classrooms without four to five years of counting immersion are at a serious disadvantage and struggle as a consequence.
So when we expose children to new parts of the number system we need to think about how we can immerse them in the numbers so that the language becomes familiar and can be connected to the symbols, contexts and images. One way to do this is through counting. With fractions this often doesn’t happen, children are introduced to the idea of fractions and are expected to make sense of language and symbols without any immersion. There is a page in the ‘images’ section on counting in fractions.
Counting can help to address the gap in understanding fractions as numbers in their own right. What children tend to experience early on is a fraction as an operator – cut it in half, get half of those. Understanding fractions as operators is not something particular to fractional numbers. Children do also experience this with whole numbers – give me two of those – and this is then linked very clearly to multiplication and division. Fractions are another piece to this puzzle, another way of talking about a situation involving multiplication and division.
For example
I have grown some sunflowers and plant them equally along two sides of my garden so that there are 7 on each side.
I can say the following:
- Two rows of seven plants make fourteen plants: 7 x 2 = 14
- Fourteen plants divided between two beds is seven in each bed3; 14 ÷ 2 = 7
- Each bed of seven has half of the fourteen plants: 14 x ½ = 7
Alternatively
I need to make twelve pieces of toast and I can put four pieces of bread in my toaster at one time, so I will need to fill the toaster three times.
I can say the following
- Twelve pieces of toast split into groups of four is three groups: 12 ÷ 4 = 3
- Putting four slices of bread in the toaster three times would be twelve slices: 4 x 3 = 12
- Each set of four pieces of bread is a third of what I need: 12 x ⅓ = 4
Decimals
Teaching of decimals and calculating decimals precedes similar work on fractions in most other European countries. This is logical as decimals are a natural extension of the number system. Once children have a clear understanding of our number system and are secure in place value related to whole numbers they should be able to engage with the decimal part of our number system.
This is crucial. If children do not have a secure understanding of place value, ordering and rounding whole numbers they will not have the prerequisite skills and understanding to move onto decimals. Many of the misconceptions that arise within decimals relate to this.
The other problem with decimals is that the contexts in which children encounter decimals do not always follow the logical structure of the number system and the language associated with the contexts can mask the relationship between different numbers. Money is particular problem. Children will be familiar with money and the temptation is to use this familiarity as a way into teaching decimals. Unfortunately, the way we talk about money in everyday life is very different from the way we talk about decimal numbers. Pounds and pence are often talked about at the same time, in other words to units of measure are sued together – “Three pounds and seventy-six pence”. This hides any relationship between pounds and pence and whilst children may learn that 100 pence = 1 pound they may have no idea that 80p is 8/10 of a pound or of how to interpret £0.35. In addition to this money written in pounds always has two decimal places, never one and never more than 2.
In a similar way, other measures such as length can be used as a context for decimals but they have the potential to both aid and hinder understanding. If lengths are measured using one unit of measure to describe the whole length and conversion between units made, this will aid understanding. For example, measuring a window as being 2.3m, equivalent to 230cm. However if lengths are given using more than one unit of measure, for example the window is 2m and 30cm wide, then we have the same problem as described in money.
