Developing an understanding of fractions

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GCSE Maths By: Please log in to see tutor details
Subject: Maths » GCSE Maths
Last updated: 13/10/2010
Tags: advice (for parents), gcse maths

Dealing with the arithmetic of fractions is a common problem topic for many intermediate ability students at Key Stage 3, and if not dealt with at this stage problems can persist through GCSE and beyond, even with some otherwise high-ability students. As with many such problems the root often lies in the early stages of introduction to the topic at the latter stages of KS2 (year 5/6).

 

There is no single right way to approach the topic for the first time as a classroom teacher, and whatever method is adopted some pupils will inevitably take to it better than others and take on board a level of understanding that will not impede further development. Others will adopt strategies to get by, that hide a lack of understanding, and it is here that the problems are stored up for later.

 

The fundamentals of a correct approach are relatively simple, but their introduction needs to be tailored to the individual according to their age, confidence, ability in related areas and personality, and one-to-one tutoring provides the most effective way of securing success.

 

As with all topics in maths a successful approach must be two-pronged. A student must develop technical skills that are best inculcated through drilling and repetition to build up an almost instinctive response to a particular situation. Unfashionable as a teaching method for a generation, drilling is absolutely essential to develop the requisite skills as a mathematician. Instant recall of times-tables for example is fundamental, and basic arithmetical operations with fractions must be drilled in the same way. This will not usually develop understanding, but just as practising scales isn't the same as performing a musical piece, it is nonetheless an essential preparation for the technical demands of doing so.

 

In association with developing technical skills however it is of course essential to present examples that allow the student to explore the idea and to investigate different approaches to understanding the topic. Here are some fruitful seeds for such exploration.

 

  1. 3 cakes divided amongst 5 people yields the same overall amount of cake as 3 pieces of one cake divided into 5 pieces. (Illustrate with pictures, explore different examples)

  2. Early school examples are often restricted to calculating a given fractions of a total where the total is divisible by the denominator. It is important to explore what happens when this is not the case: eg. contrast 3/5 of 20 with 3/5 of 7. Where does the answer lie on the number line?

  3. What happens when you divide by 2, then divide by 2 again? what about dividing by 3 then by 5? How could you simplify the process of dividing by 20?

  4. What's another way of saying take a half of something?

 

The dual nature of fractions as both numbers and operations (3/5 can be thought of as 3 divide by 5 or as three fifths) should be brought out, and the relationship between the notation of fractions, the division sign itself, and the operations of multiplication and division that drilling teaches us to use.

 

It will inevitably take time for a correct understanding to be developed, especially where there are misapprehensions to be unpicked, or a phobic response to overcome, but a patient teacher with a trusting student should be able to quickly lay the foundations for successful development.

 


Tom Kennedy A-level Maths (Further) Tutor (South West London)

About The Author

Maths tutor based in SW London with 8 years classroom experience and a further 7 years as private tutor.




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