What are fractions and what does it mean to understand them?

Negative numbers, the irrationals, and imaginary numbers are not that easy to make sense of for many students. But this is something understandable. One only needs to check-out the histories of these numbers. The mathematicians themselves took a long time to accept and make sense of them. But fractions? How can something so natural, useful, and so much a part of our everyday life be so difficult? Didn’t we learn what’s half  before we even learn to count to 10? I’m sure this was true even with our brother cavemen. So how come the sight of a fraction enough to scare the wits out of many of our pupils and yes, adults, too?

Fractions are used to represent seemingly unrelated mathematical concepts and this is what makes these numbers not easy to make sense of and work with. In mathematics, fractions are used to represent a:

1. Part-whole relationship – the fraction 2/3 represents a part of a whole, two parts of three equal parts;
2. Quotient – 2/3 means 2 divided by 3;
3. Ratio – as in two parts to three parts; and
4. Measure – as in measure of position, e.g, 2/3 represents the position of a point on a number line.

Of these four, it is the part-whole relationship that dominates textbooks. For many this conception is what they all know about fractions. While it is also the easiest of the four to make sense of, students requires series of learning activities to fully understand part-whole relationship . Crucial to this notion is the ability to partition a continuous quantity or a set of discrete objects into equal sized parts. Below are sample tasks to teach/assess this understanding. They call for visualizing skills.

Of course understanding fractions involve more than just being able to use them in representing quantities in different contexts. There’s the notion of fraction equivalence, which is one of the most important mathematical ideas in the primary school mathematics and a major difficulty. This difficulty is ascribed to the multiplicative nature of this concept. There’s the notion of comparison of fraction which includes finding the order relation between two fractions. And if your students are having a hard time on comparing fractions you can check their understanding of equivalence of two fractions. It could be the culprit. And let’s not forget the operations on fractions. An understanding of the procedure for adding, subtracting, multiplying, and dividing of fractions depends on students’ depth of understanding of the different ways fractions are conceived, on the way fractions are used to represent quantities, on the idea of equivalent fractions, and on order relation between fractions, and  many others such as the meaning of the operations themselves.

A study has been conducted categorizing students levels of conception of fractions, at least up to addition operation. Just click on the link to read the summary.