Fractions have an unenviable reputation of being difficult to teach. Although most students have extensive practice shading in parts of a rectangle and counting pieces of pie, many students still lack a conceptual knowledge of fractions. In a multiple choice question asking Year 9 students to estimate the nearest correct answer to 11/12 + 7/8, only a minority of students correctly answered 2. The most popular answers were 19 and 20. It seems that for many students fractions are a confusing nightmare, but why?
Mastering fractions requires a change in thinking about numbers. Moving from the whole number counting system to fractions (also known as rational numbers, i.e. numbers which are not whole numbers) takes students beyond their existing concept of whole numbers which are used to represent the number of things, to a more powerful number concept that can be used to describe not just whole things, but parts of things.
Rational (Fractional) numbers
|Used to describe how many items in a group.||Used to describe how much, which might be the same, more or less than a whole number.|
|Each number has only one number that comes after it, and each number has a specific meaning.||There is no meaningful answer to “What number comes next?” because a fractional amount can always be divided into smaller fractional amounts.|
|Only one way to write a particular number – i.e. 7 (seven) is always 7||The same fraction can be represented in different ways – -e.g. ¾ = 6/8 or .75 or 75%|
|Only need one number to describe an amount||The amount is described by the relationship between two numbers.|
1) Adding fractions by adding both numerators and denominators , so that ½ + ¼ = 2/6
2) Decimal places are ignored, so that .5 is smaller than .25, because 5 is smaller than 25
3) Incorrectly estimating the size of a fraction by looking at a fraction as two numbers instead of as a value described by the relationship between two numbers, which results in the following typical errors (a) ¼ is larger than ½., because 4 is larger than 2, and (b) 9/16 is larger than ¾, because the numerator and denominator are both bigger.
All of these errors suggest are evidence of confusion when students try to apply their exisiting whole number knowledge to fractions, instead of adapting the number system to include fractions as well as whole numbers. It’s often referred to as whole number bias.
The link between fractions and division totally escapes many students. They might be able to draw a diagram that shows 1/4 , but when ask them what is 1/4 of 8 and they go completely blank. They need to know that fractions are used for dividing numbers as well as pizzas and pies.
Many students flounder along perplexed about the significance of two separate numbers separated by a line, and fail to connect three-of-eight slices of a pie to the fractional value of three eighths. Without a metal model of a fraction as a value students will find it incredibly difficult to make sense of adding and multiplying fractions. They can tell you about the number of slices and the total number of slices, but they continue to treat fractions as a counting exercise, instead of focusing on relationship between two numbers. It’s thought that these typical learning activities can make it difficult for some students to move beyond whole number counting and grasp the significance of the relationship between the numbers. Conceptual understanding of equivalent fractions and common denominators is next to impossible without a suitable mental model of what is meant by a fraction.
Although most students have good informal knowledge of the meaning of a half and a quarter by the time they begin to learn about fractions, once students start to learn about fractions in the classroom many become confused rather than enlightened, and no longer trust their original correct informal knowledge. Their previous experience tells them that 1/4 is smaller than 1/2, while their knowledge of numbers, incorrectly applied, leads them to the conclusion that 1/4 is bigger than 1/2.
It should be immediately obvious that students who do not know their multiplication tables are going to have trouble making sense of equivalent fractions, and finding the lowest common denominators will make even less sense. Allowing these students to look up their tables when they need them will help, but it also overloads working memory and takes their attention away from fractions. Until these students have better mastery of multiplication tables, they should concentrate on the more “natural” fractions of half, quarter, and eighths, as the strategy of repeatedly dividing by two comes naturally to most students.
Alternative teaching strategy
The suggestion is to forget the pies, pizzas and shaded rectangles until students are very comfortable with percentages. In other words, instead of the usual order of teaching fractions, and then percentages, teach percentages first, and then build on this knowledge to develop a better understanding of fractions.
This system has the following advantages.
1) Many students are already familiar with percentages and the concept that all of something is 100%, making it possible to build on this prior knowledge by giving them practice activities estimating the percentage “fullness” of a straight sided glass container, and progress to indicating percentages on a number line.
2) It simplifies the acquisition of the initial concept of a fraction, letting students compare fractions with the same denominator (100), allowing students to focus on value rather than counting.
3) Many students already have an informal knowledge of one half and one quarter. Students find it fairly simple to make the connections between one half as 50% and a quarter as 25%, By working with percentages, students can readily compare these values, as well as one fifth, one tenth, three fifths etc. The intent is to develop the concept that fractions are used to describe amounts between whole numbers, and that percentages are a special type of fraction.
4) It is easier to learn two place decimals, by thinking of a two-place decimal as the percentage of the distance between two adjacent whole numbers. (e.g. 5.25 is 25% per cent of the way between 5 and 6. (The alternative explanation, that .25 is two tenths plus 5 hundredths can be very confusing for students who probably don’t know how to add fractions with different denominators, and are at the same time confronted with the concept that there are ten one hundredths in a tenth – the exact opposite of all the counting they have previously done.)
5) By learning to benchmark common fractions against percentages – e.g. ½. ¼, 1/5 and 1/10 students better understand the connections between decimals, percentages and fraction symbols, and to associate a fraction with the proportional value it represents. This benchmarking lets students use existing knowledge to find percentages of fractions with numerators other than 1. (2/5 is twice one fifth, which is 20%, so 2/5 is 40%.) They can learn to convert from one form to another, and begin to understand equivalent fractions. They can be encouraged to focus of the value of the fraction as a relationship between the numerator and the denominator.
6) Students should then be able to add fractions by converting them to percentages and adding, which in effect is using 100 as the common denominator. This foundation knowledge can then be developed to convert fractions into other equivalent fractions with various other numerators, and look at adding fractions by finding a common denominator.
This approach was used very successfully using fourth-grade students, in a research study comparing this method with a control group who received the more conventional sequence of pies and pizzas, equivalent fractions, decimals, number lines and place value charts. Both control group and experimental group participated in large and small group activities, along with the use of discussion, manipulatives and games(Moss & Case, 1999).
An extensive summary of a recent journal article, “An integrated theory of whole number and fractions development” (Siegler, Thompson and Schneider, 2011) provides a more academic slant on the problems of teaching fractions.
Moss, J., & Case, R. (1999). Developing Children’s Understanding of the Rational Numbers : Curriculum A New Model and an Experimental. Journal for Research in Mathematics Education, 30(2).