Fractions teaching

Maths Assignment

Fractions form part of the Number and the Number System strand in the Framework for the National Numeracy Strategy (NNS). They form part of a wider topic of fractions, decimals, percentages, ratio and proportion. Work on this topic begins in the foundation stage, and progresses across Key Stages one and two, becoming more broad and deep as it does so. Teaching in this area must consider this progression, accounting for different ages and ability levels, so the progression is steady and new concepts are introduced that build on others which are secure. In doing this, teaching must also be aware of the possible misconceptions which may arise in children, so they are anticipated and planning is used to either overcome or avoid them. Details of how fractions are reflected in the foundation stage, Key Stages one and two curriculum and the NNS can be found in Appendix 1 and 2.

The five strands of the NNS are highly interconnected and aim to provide children with the confidence and flexibility to use and apply mathematical principles. The topic of fractions is integrated with all of the strands. It forms part of the Number and the Number system strand, as part of a wider understanding of fractions, decimals, ratio and proportion. It is found in the Calculations strand, as later on in Key Stage 2, children begin to understand fractions are related to division. It is found in the Problem Solving strand, when children use them to find answers to real life problems, relating fractions to units of measurement, time and money. Fractions are particularly important in the Shape, Space and Measures strand, when in Year 3 children use measuring equipment, when they are learning to tell the time from an analogue clock and when they are learning about the properties of shapes, comparing shapes and considering lines of symmetry. Fractions are also found in the Handling Data strand, when children learn about probability in years five and six or when they are interpreting pie charts towards the end of Key Stage 2.

Progression in the NNS aims to equip children with the understanding and meanings behind mathematical representations, so that they are able so ‘create and shape a conceptual framework’ Swan (2001) and see the underlying structure. Skilled teaching involves scaffolding learning so that pupils are not encouraged to think that success in numeracy is about following sets of unrelated, random rules and systems but about using and applying mathematical concepts flexibly.

In the early years, work on fractions begins with informal practical experiences the children will be familiar with, such as cutting cake or sharing a chocolate bar, and children are ...

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In the early years, work on fractions begins with informal practical experiences the children will be familiar with, such as cutting cake or sharing a chocolate bar, and children are encouraged to explore spoken language, such as ‘a half’, ‘half full’, and ‘half each.’ Accuracy is not wholly important at this stage. In Key Stage one fractions are introduced as a discreet topic and the children are taught the more precise vocabulary of halves, quarters and the corresponding fractional notation. Practically based pictorial representation in the examples is used to heavily emphasise that the parts are equal and children begin to identify examples of when the shaded area is not a half or a quarter but an imprecise exception. Equivalence is introduced in practical demonstrations of a half being equal to two quarters using a paper folding exercise as a concrete example. Children also begin to position halves on a number line as a means of comparing fractions with whole numbers. Their work in this direction complements other areas in telling the time and movement through turns.

Work in Year three develops fraction vocabulary to include thirds and tenths and extends their equivalence knowledge to include tenths, quarters and halves up to a whole. Pictorial representation is now more complex and shows a step towards problem solving, including the concept of estimating fractions from practical examples. Number lines are used to show ordering of simple fractions as is counting in halves, quarters, etc.

In Key Stage two the topic of fractions is extended to using fraction notation and recognising equivalence, recognising the equivalence between decimals and fractions, ordering familiar fractions, finding fractions of numbers and quantities and recognising the equivalence between percentages, fractions and decimals in Years five and six. In recognising equivalence in Year four, pictorial representation is still used, but becomes a tool to aid rather than a puzzle. Fraction walls are useful at this stage. The concept of equivalence is more abstract in years five and six children develop the idea of twentieths, hundredths and thousandths. They learn the vocabulary of numerator and denominator and begin to generalise the idea of equivalence as being both numerator and denominator multiplied by a common factor. They are then able to use this knowledge to reduce or cancel down fractions, and to convert improper fractions to mixed numbers. Similarly, in ordering fractions, children move from recognising fraction order in practical work to being able to convert fractions to a common denominator in Year six. Children begin to find fractions of numbers or quantities in Year four, using the example of cutting a cake in half as being the same as dividing by two, and this is reinforced by work in money, measurement and shape, where the numbers are part of the decimal system. In Years five and six, children consolidate and extend work done on relating fractions to division. The numbers are more extreme and involve more mental calculation, such as a nominator greater than one and the measures being operated on include time, which is not a decimal form or having to convert metres to centimetres. By Years five and six, fractions are related to simple proportion, decimals and percentages, particularly in the context of money and measurement and calculators are introduced, to show the interconnection of all number forms.

Effective teachers need to be aware of possible misconceptions at each stage of progression, so that they will not be left uncorrected to hinder development at the next or later stages. One difficulty younger children may have is in understanding the difference between the imprecise language used in their home contexts to describe fractions and that in numeracy fractions denote a unit divided into a number of equal parts. This could be anticipated by the early years teacher, who through practical demonstration could use the children’s growing ideas of fairness to share equally an apple or fold a piece of paper. Then the unequal share could be modelled to illustrate what a half is or is not. Children should be encouraged to discuss their ideas about equal shares and accommodate the new formal ideas of fractions with those they use at home.

One possible misconception that may arise in dealing with fractions is the child not attending to the size of the denominator as being the division of the unit and the numerator as the number of the parts. This may become particularly apparent towards the end of Key Stage two, when children begin to order fractions with different denominators. Swan (2001) describes a method of multiple representations of the fraction to use as tools to help children to grasp the concept. In this, a set of cards with the relevant fraction on are used, along with pictures of the fractions as a shaded part of equal squares and cards of the fractions marked along a number line. The children work in groups to put the corresponding representations with the right fractions and then go about ordering them. Through practical group work and peer discussion the misconceptions are confronted and new reasoning created. A plenary could be used to reflect on the different methods children used to order the cards and which representation they found most useful.

Fractions teaching is a highly complex and sophisticated topic to teach effectively in primary schools. Teachers should be aware of the links the topic has with others, the progression that is expected of the children across the year groups and the possible misconceptions and strategies to correct them in order to encourage children to be able to use fractions with fluency and confidence.

References

Swan, M. (2001). Dealing with misconceptions in mathematics. In Issues in mathematics teaching. Ed Peter Gates. London: RoutledgeFalmer.

The National Curriculum DFEE (2000).

The National Numeracy Strategy DFEE (1999).

The Foundation Stage Guidance DFEE (2000).

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