Introduction : Setting the Scene
Fractions are one of the most difficult mathematical concepts that a child will meet and common place everyday definitions of a fractions as being ‘a fragment’ or ‘a small bit’ or part of something eg a pizza.
Help to cause cognitive conflicts*1 in the minds of learners. This is certainly not unique to the concept of a fraction: within mathematics other such multiple meanings exist (Pimm, 1987). The preciseness of mathematical meanings of words as compared with their less precise use in everyday language is a tension that teachers of mathematics have to recognise and plan to address in their practice (Kotsopoulos, 2007). Add to this linguistic confusion the multiple views or meanings of a fraction and the notion that a single mathematical concept could possibly have several views then there is greater opportunity for even further confusion in the mind of the learner. So, it is important that a teacher has a firm understanding of the mathematics and more importantly the precise language to be used with learners when introducing this difficult concept.
So, what is a fraction?
Let’s start by looking at a mathematically definition a fraction:
In arithmetic a fraction is the quotient of two integers expressed as a numerator divided by a denominator.
This gives rise to 4 important properties
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In a proper fraction*, the numerator* is less than the denominator*.
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In an improper fraction* the numerator is larger than the denominator
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A mixed number* is an integer* with a proper fraction.
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Any fraction can be written in decimal form by carrying out the division of the numerator by the denominator.
So, a fraction is definitely NOT part of a pizza or any other object visualisation. Five important classes (sets) of fractions are:
|
Fraction Type |
Definition |
Example |
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A Unit fraction* |
A fraction with a numerator of 1 |
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A Proper Fraction |
A fraction with a numerator less than the denominator |
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An Improper Fraction |
A fraction with a numerator larger than the denominator |
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A Mixed Number* |
A whole number and a proper fraction |
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Equivalent Fractions |
Fractions that have the same value |
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Added to this complexity is the fact that a fraction has at least 5 sub-constructs as a quotient, a measure, a ratio, a multiplicative operator and finally a part-to-whole relationship. (Behr et al. ,1985; Kieren, 1988)
|
Sub-Construct |
Meaning |
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As a quotient |
This involves understanding fractions as the result of a division. The fraction |
|
As a measure |
This is an effective way to develop a learners’ understanding of a fraction as a number with a magnitude based on the use number lines*. Number lines can clearly illustrate the magnitude of fractions; the relation between whole numbers and fractions; and the relations among fractions, decimals, and percentages. |
|
As a ratio |
This involves making a comparison between two quantities; therefore, it is considered a comparative index, rather than a number. |
|
As a multiplicative operator |
This is where fractions are used to transform numbers, it mainly involves the multiplicative aspect of fractions. For example, the fraction |
|
As a part-to-whole |
This is a way of representing part of a whole set of objects or complete objects. It involves the partitioning of a shape / number of discrete objects into equal parts, (unitising) or determining how many objects would be in a whole set based on a part of the set (re-unitising). |
can be interpreted as 1 divided by 3 or the result of sharing an object among three people.
may be perceived as finding two fifths of a given quantity.This multifaceted notion of a rational number (fraction) is one of the main reasons, or at the very least a contributing factor, why children possibly find difficulty when learning and working with fractions. Researchers (Ball, 1993; Kieren, 1976, 1993; Lamon, 2007, 2012) have tended to agree that teachers of mathematics need to gain a deep understanding of rational numbers and the different interpretations of fractions. Whilst researchers have given slightly varying lists of the above interpretations, they do agree that learners and teachers must be familiar with all of these fraction representations, rather than merely the part-whole models (Ball, 1993).
