Below are the general responses to the questions posed – I recap the questions and the views are generalised notes from talking to a range of teachers from nursery, through Key Stage 1 and 2. I provide the detailed breakdown of two colleagues from Years 3 and 4.
Throughout these you can see that fractions is a huge, varied and tricky concept to think about, teach and learn. I have found that, throughout my teaching career, children have always found fractions hard, although there are a core of children who grasp it quickly, these are the exception rather than the rule.
It is clear that a hands-on, physical approach is needed at the beginning of fractions work – indeed practical maths was a key talking point of all the teachers throughout the Primary phase. But also, I feel that language is a huge barrier to children’s learning. The language of fractions is often misused throughout life and if they don’t have a solid understanding in the first place, their will only serve to cloud the issue further. Another difficult aspect is the link with division – children who don’t know their multiplication and division facts can’t begin to develop their ideas of fractions
So, how can we bring all these parts together to make up one cohesive whole?!
I think it’s a case of reviewing the way it is taught throughout the Primary Phase. Children have to encounter teachers who are confident in approaching fractions and the subject needs to be taught consistently, removing areas of conflict and making sure that each part of their learning isn’t conflicting with another area. If it is to be taught alone, then it needs to be done until that child grasps that particular stage of learning. My belief however, is that is must feature in each part of mathematics – there can always be a question relating to fractions in whatever is taught. This may help to break some of the barriers to learning that exist – I currently feel that children are very negative towards them.
One key to learning is children’s difficulty with fraction language. Maybe teachers are trying to make too many jumps at the same time, moving too quickly. Maybe this is down to pressures from the curriculum. It is often better to avoid comparisons. For instance, when focussing on halves, it would be better to focus on and describe objects or models that are either halves or not halves, rather than giving objects other labels (much in the same way that children find it easier to learn that bricks are heavy and feathers are not heavy rather than comparing them as heavy and light. Floating and sinking is another example, it is easier to get children to understand things that float and things that don’t float BEFORE investigating things that sink – it’s too confusing).
My notes and general findings:
What different ways might we conceptualise “fractions”?
- Physically cutting objects into pieces/parts – fruit/toast
- Quantities – full/empty/half full water bottles for instance
- Splitting counters/objects into equal groups, naming those groups
- Egg boxes – 12 spaces, how many boxes do I need to house 20 eggs? [can be cut to alter the number of spaces too].
- Fraction walls
- Diagrams of shapes – shading in a fraction of the whole
What constitutes the “fractions landscape”?
- Portions of shapes
- Portions of amounts
- Portions of objects
- Decimal numbers
- A knowledge of division as sharing
- Fraction walls
Is there a perceived “order” to the development of fractions concepts?
- Definitely – always starting with halves using models/objects.
- Halves/quarters/eighths before thirds/sixths/ninths – harder than halves etc., as less often seen.
- “You can’t teach ¼ if they don’t know ½” – concepts must be taught in the right order using words at an appropriate level. (concept check list, roughly:)
- A bit
- A lot
What is the place of “imagery” and “models” in the teaching of fractions?
- More kinaesthetic stuff works – as in real life examples – cutting things into equal parts etc.
- “I always teach it by cutting pizzas into sections. Or relating it to a spelling test (1 out of 10 marks is 1 10th correct etc.)”
- Must take place through the learning of fractions.
What difficulties are encountered by teachers?
- Possibly a lack of own knowledge
- Similar approaches to the topic no matter who I spoke to – do these work if the children are often struggling?
- Maybe trying to make too many jumps at the same time/moving too quickly/pressure on the curriculum – is it better to avoid comparisons? I.e., when focussing on halves, describe objects/models as being halves and not halves, rather than giving objects other labels (again, much the same way that children find it easier to learn that bricks are heavy and feathers are not heavy rather than comparing them as heavy and light, floating and sinking (easier to get children to understand things that float and things that don’t float BEFORE investigating that sink – it’s too confusing).
- Not always used throughout life in a consistent manner – “who wants the biggest half?” But also, the language is used to with different meanings throughout other areas in life too.
- A current maths curriculum jumps around too much – fractions need to run throughout their learning and not just be tacked on.
What difficulties are encountered by pupils?
- There are a few things in fractions which don’t follow rules. E.g. the larger the denominator, the smaller the fraction. Where else does larger = smaller?
- The same fraction of different amounts have different answers – the meaning or outcome of the words change depending on the situation (in much the same way as a dog is big when compared to a mouse, but the dog is small when compared to an elephant).
- If pupils are not totally confident with multiplication & related inverse operations (division facts) they will NEVER get fractions
- I think one explanation is that they don’t understand division
- Speech and Language: hard to grasp, must be taught and consolidated.
- If they see things the same way all the time they can’t progress.
- Need to know that a half, or any portion, is relative to the whole
Year 3/4 teacher‘s ideas:
Fractions are all about sharing. Start with food and practical objects that the children can handle and see being cut into equal parts. I keep it very visual and very hands on. This is an extension of key stage 1. Very important to handle the objects – children need to experience sharing them. Division as sharing of numbers is key to understanding fractions and should go alongside each other along with the vocabulary. There needs to be a constant link between division of numbers and fraction work, it can’t just a tacked on as a standalone unit, must flow through mathematics – it feeds into everything. If it is taught on its own, you need to stick with it until it’s solid. I do teach fractions of shapes separately as this confuses children. They can colour 3/8ths of a shape spilt into 8 equal sections for instance, but they don’t know why they are doing it, or that 5/8ths are left over. They also struggle when the shading isn’t grouped. Also find it hard to colour half of a shape not split into two, so the shape spilt into 8ths, they would either colour half a square in or leave it – don’t see 4 squares as half of it.
Year 4 teacher’s ideas:
Start with shapes, cutting them, folding them etc. Move on to objects, biscuits. Make it very visual and physical with a variety of examples. Often use flip flap fraction resource. I make sure I order fractions throughout and try to refer to the vocabulary as often as possible in different ways. To develop this with groups, I’d use sweets or counters, or things like dinosaurs and counting objects borrowed from Reception, then physically divide them up. Also try to show on the SMART Board, but preferred it when we could use the OHP with transparent counters as the children could touch the objects. Fractions of amounts and shapes go hand in hand. Alongside all this, I would look at fraction walls, asking children to make their own using jigsaws, or careful measuring in books, usually starting with halves, quarters and eighths. I teach the words numerator and denominator, but focus on “How many have you got?” and “How many altogether?” as children understand this more and numerator and denominator will be taught to them again later on. After this, I will focus on the links between times tables alongside equivalent fractions and looking at the sequences, as well as looking at links between division and sharing. With the more able children I would then move on to mixed numbers.