Misunderstandings
Making errors and experimenting with new strategies are a natural part of learning. However, if inappropriate thinking about fractions becomes established as a misconception, a barrier to progress in conceptual understanding is formed.
Usually misconceptions will not be corrected unless explicitly addressed by instruction. Teachers need to be aware of common misconceptions and look for evidence of them in their students' thinking. Identifying misconceptions requires careful observation or purposeful diagnostic assessment.
Further details about common misconceptions can be found in the article Developing an Understanding of the Size of Fractions on the AAMT website.
Number of parts only
Some students think that a fraction simply indicates the number of parts into which a whole has been divided, and that it does not matter if the parts are unequal in size.
Fractions as a double count
It is common for students to see the numerator and the denominator as two separate whole numbers, recording a count of two different things.
Different wholes
Often students do not realise that when comparing the relative size of fractions, it must be assumed that all fractions refer to the same size whole.
Using rules blindly
When students blindly follow a rule or procedure, they can fall into a pattern of applying that procedure even when it is inappropriate.