Understanding the concept of fractions is fundamental for a complete comprehension of mathematics. However, common misconceptions can lead to uncertainty in students, which can lower confidence levels in this important mathematical skill.

Let’s examine common misconceptions about fractions and how early ideas about fractions form firm foundations.

1. See fractions only as parts of a whole.

Fractions are often thought of as showing parts of a whole. While that is a common interpretation, fractions are not limited to dividing something into pieces. In fact, fractions are like math tools that show relationships between quantities. These quantities can be anything from objects that can be counted to continuous things like lengths or areas. Depending on what you are working with, you can use different ways to represent fractions, such as pictures, lines, or individual items.

There are five main ways to think about fractions: as parts of a whole, as measurements, as a way to divide, as an operation, or as a ratio. Each way is useful in helping us understand and work with fractions in various situations.

2. See fractions as two separate numbers.

One common misconception is thinking of a fraction as two separate numbers. Students may not understand that a fraction represents a single quantity and not two distinct numbers.

Students are taught that “the denominator tells how many equal parts, and the numerator tells how many parts are shaded.” Although this strategy is practical, it poses various issues:

• The written fraction is perceived as two whole numbers, each representing the number of distinct entities, resulting in a ‘double count’.
• A fraction is not perceived as a singular number and does not have value like other numbers.
• The double count is not applicable in all situations, such as “Share 2 pizzas equally among 6 people.”
• Students misinterpret a part-whole scenario as a ratio, particularly when dealing with fractions in collections.

The following steps can be taken to avoid the double-count misconception:

• Use number line representation: introduce the concept of number lines for fractions. Show how fractions are located on a number line and how they represent points between whole numbers.
• Use hands-on activities where they physically partition objects into fractional parts. For instance, share 3 pizzas among 4 people.
• Use ‘counting by fractions’ of the same denominator to reinforce the concept of fractions as numbers, such as 1 quarter, 2 quarters, 3 quarters, etc. (instead of saying 1 out of four).
• Use visual representations of fractions to strengthen students’ sense of the relative size of fractions.

3. Misunderstanding the Denominator

When learning fractions in the first few years of school, students begin to work with familiar fractions such as halves, quarters, and eighths. They understand that the ‘denominator’ indicates the number of parts of a whole, but do not notice that the parts must be equal.

Students with this misconception are likely to select the following figures to represent ¼. 

4. Assuming larger denominators with large quantities:  ¼ is larger than ½.

When students perceive a fraction as 2 whole numbers (misconception 2), it can lead them to this misconception. They focus on the size of the numbers instead of the relative size of the parts. 

Students learn that 4 is greater than 2, so they think ¼ is greater than ½.  This is not true because the value of a fraction depends on both numerator and denominator. Here are the steps you can take when seeing this misconception:

• Take 2 circular pieces of paper.
• Cut them into 2 pieces and 4 pieces.
• Compare the pieces to the written fractions.
• Take the opportunity to explain the foundation concept to the students.

5. Different wholes.

Students lack awareness of the importance of equal wholes. For example, cutting a mini-pizza in half produces a different amount of food than cutting a family-size pizza in half, even though they are all presented as ½. The actual quantities represented by half are not equal because the ‘wholes’ in this situation are not equal (a mini pizza & a family-size pizza).

This misconception can lead to the error when comparing the size of two fractions.

The above examples show that the students do not understand the need for the wholes to be the same.

6. Apply the rules of comparing unit fractions to non-unit fractions.

Students learn a strategy when comparing unit fractions ‘the larger the denominator, the smaller the fractions.’

They apply the same strategy when comparing non-unit fractions, which is inappropriate and leads to an incorrect answer.

7. Adding/subtracting numerators and denominators directly when adding/subtracting fractions.

Students are most likely to have this problem when run into addition and subtraction fractions with different denominators.

Students need to grasp the concept that the 2 fractions ¾ and 4/8 represent different-sized parts of a whole. They are not the same. Therefore, to add fractions with different denominators, the fractions need to have the same denominators.

8. Mixing up Multiplication and Addition Fractions Rules

Students are taught to find the same denominators when adding or subtracting fractions with different denominators, and they apply the same rules when doing multiplication and division.

9. Equivalent fractions confusion.

Some students may struggle to grasp the concept of equivalent fractions. They might think that fractions with different numerators and denominators are always different values.

Here are some tips:

• Use visual aids such as fraction bars, circles, or rectangles to show that fractions with different numerators and denominators can represent the same part of a whole. This helps students see the equality of fractions.
• Use hands-on activities where students can manipulate physical objects to understand equivalent fractions. For instance, cutting paper into different shapes or using fraction manipulatives.
• Utilise number lines to demonstrate the position of different fractions with respect to the whole. Show how fractions can be different in appearance but still occupy the same position on the number line.
• Explicitly explain that equivalent fractions can be found by multiplying both the numerator and denominator by the same non-zero number.

Misconceptions 6 – 9 are caused by using the fraction rules blindly, here are some steps you can take:

• Encourage students to refine and develop strategies for comparing fractions, making equivalent fractions, addition/subtracting fractions, and multiplying and dividing fractions.
• Provide students the opportunities to test out the strategies in different situations and contexts.
• Utilise various representations to model fractions, such as area diagrams, fractions walls, and number lines.
• Encourage them to substantiate their solutions/strategies through modelling, explanation, and justification.
• Emphasise the consideration of the relative size of fractions instead of viewing the numerator and denominator in isolation.