Introducing the fraction notation
It is not uncommon to see fraction notation introduced as a way of recording a double count. First we count the number of parts shaded, next we count the total number of parts and then record the first count over the second count as a description of a fraction, as in the following figure.
Two-thirds resulting from a double count
Developing fraction notation from two counts relies upon an additive interpretation as the fixed nature of the whole is ignored. If the whole is ignored in naming fractions, the logical errors associated with fraction operations are reinforced.
An additive interpretation of fractions due to lack of reference to the whole
The emphasis in teaching fractions on counting the number of parts has meant that some students have developed concept images for fractions that are solely dependent on the number of parts represented. In the following example, the student has made three equal parts to represent one-third and six equal parts to represent one-sixth. That is, the number of parts corresponding to the denominator appears to represent the fraction.
For some students, fractions are defined solely by the number of parts without attention to the equality of all of the parts. In the following example, the student has represented fractional quantities as the number of parts out of the total number of parts. This is not a comparison of areas but rather a comparison of the number of parts.
Incomplete fraction concepts can be formed from activities associated with common parts-of-a-whole models. In the above examples the number of parts formed is the defining characteristic of the fraction representations rather than the area of the parts. A similar focus appears when students use equidistant parallel partitioning to form sub-units.
The circle below, shaded by a student to represent one-sixth of the circle, appears to have been subdivided by equidistant parallel partitioning.
As using vertical parallel lines works in creating fractions of a rectangular region, some students also attempt to use them in a circular region to produce thirds, fourths or fifths. Parallel partitioning can result in a number of parts that are treated additively, rather than a relationship between areas.
The dominance of the vertical and horizontal directions when forming parts of shapes suggests that some students may only attend to linear distance when dividing regional models, even circles. Equidistant parallel partitioning can also be described as linear partitioning, as attending to length rather than area forms the parts. Linear partitioning is also associated with students using ‘half-way points’ for one-half. The use of linear partitioning can persist well into high school with one major study in New South Wales (Gould, 2008, p. 145) reporting approximately 10% of students in Years 4–8 using parallel partitioning to attempt to represent one-third of a circle.
Instead of seeing the relationship between the parts and the whole, some students see:
- parts from parallel partitions,
- a number of parts (not equal parts), and
- a number of equal parts (not a fraction of the whole).
Although it is a common practice, introducing the standard fraction notation a/b as a ‘count’ interpretation of the regional ‘parts of a whole’ model is very limiting. A more useful introduction of the fraction notation arises from the idea of accumulation of length.
That is, instead of introducing the notation 3/4 as 3 parts out of 4 parts, it is better introduced as adding related units of length: 3/4 of the length is 1/4 of the length and 1/4 of the length and 1/4 of the length.
This also addresses the limitation of the parts of a whole interpretation of fractions; namely, how can you have 5 parts out of 4 parts? Developing a multiplicative interpretation of fractions and their notation is important to understanding why we form common denominators to add or subtract fractions.