Difficulties with Fractions

Historical Difficulties with Fractions

Historically, fractions have been a difficult idea to mathematize. Greeks only accepted Natural numbers as genuine numbers. They never accepted fractions as numbers.

They were not accepted as numbers, by even mathematicians, until recently.

Initially they were used only in the form of "thirds" or "fourths" or "fifths". They were represented by the whole number (3 for thirds, 4 for fourths etc) with a dot or a circle on top. Today we would call them as Unit fractions where the numerator is always 1.

Slowly several different ways of representing fractions were evolved. The current representation of fractions using 2 numbers (now called Numerator & Denominator) evolved from an 8thcentury Hindu representation and accepted internationally in the 18thcentury. The Arabs introduced the horizontal line which separates the numerator and the denominator.

Computing with them was always difficult.

Egyptian Fraction Problem

The Rhind Papyrus gives a fraction problem of "sharing nine loaves of bread among ten people" and shares an elegant solution using unit fractions.

It uses the equivalence of 9/10 to 1/2 + 1/3 + 1/15.

Dividing individual loaves of bread in 1/2 & 1/3 & 1/15 (1/3 divided into fifths) parts is much easier in a practical sense than working out 9/10!

Multiplicative Vs Additive Thinking

There are 2 basic modes of thinking about quantities - in absolute terms or in relative terms.

Whole numbers, with which students are introduced to math, can be understood in "absolute" ways. Addition & subtraction are also "absolute" ways of thinking. It is an easier mode of thinking for children.

But fractions are a "relational" way of thinking. It is thinking of one quantity in relation to another. 3 out of 6 toffees is also 'half' and so is 6 out of 12 toffees! This type of thinking needs more life experience.

Mathematically these two ways of thinking can be called Additive (absolute) & Multiplicative (relative) thinking.

Most children start thinking additively. It is only with life experience that they start thinking multiplicatively. Most adults naturally think multiplicatively.

In the primary math curriculum, the topic of fractions is introduced right after they have learnt whole numbers. A lot of time needs to be invested to enable children understand these new type of numbers and familiarize themselves with fractions that they come across in their lives. Sadly, this is never done. The curriculum framers are also responsible for this state of affairs.

Curricular Difficulties With Fractions

Mathematicians took several centuries to understand, represent and compute with fractions. Today we expect children in grades 3 upwards to do this in a few periods. No wonder fractions are considered difficult all over the world.

Introducing fractions in the form of ½ & 2/3 presents several difficulties to children. Both the notation (using a bar separating the numerator & the denominator) and the interpretation are very conceptual & difficult for children. The interpretation of the two numbers in a fraction as a code is conceptually like the interpretation of the two numbers with the place value idea, in a 2 digit number. We will deal with this aspect in the next chapter.

A good way to introduce the idea of fractions is to contrast them with whole numbers. Whole numbers represent quantities which can be counted. But most of our life experiences are with "entities" which cannot be counted.

For example if we measure the length of a room, it does not come out as a whole number say 9m or 5 m. There is some part of the length left over, which is less than a meter. How do we write that "part" using numbers?

Fractions is a way of writing such "parts of a whole" using numbers. It is a way of quantifying "parts". In contrast whole numbers quantify "wholes" which have no parts.

Hence children need to be introduced to the idea of fractions through a lot of real-life activities with daily use materials and visual images. Representing fractions using numbers should be done only after they have understood fractions through activities.

Misconceptions in Fractions

Since fractions are conceptually different & difficult, they also give rise to misconceptions. We will see some of these misconceptions as we cover the curriculum'