## Avoiding common pitfalls

Teaching fractions usually starts with determining halves by **halving**** ^{1}**, using examples such as cutting apples or sandwiches into halves aligned to the idea of fair shares. Halves are followed by quarters (or fourths) as well as linking parts-of-a-whole descriptions to fraction notation. Colouring in pre-partitioned shapes such as circles, rectangles or hexagons is then commonly used to monitor students’ part-whole knowledge of fractions. Unfortunately, this form of monitoring of students’ understanding of fractions is illusionary, as students can correctly answer questions such as the following, without displaying any real knowledge of fractions.

Colour the correct number of equal parts

Colour 3/8

Counting parts in pre-partitioned shapes is a counting activity, not a fractioning activity. Counting is an additive process: one and one more makes two, two and one more makes three, and so on. Using additive processes with fractions is frequently not helpful as additive part-whole (part + part = whole) is not the same as multiplicative part-whole (comparison of part to whole: part x n = whole). In multiplicative part-whole, the whole is a multiple of the one part. ‘Eight is two plus six’ is an example of additive part-whole whereas, ‘four times one unit of two to form a whole’ is an example of multiplicative part-whole.

Another common pitfall is attempting to link the fraction notation a/b to partitioned fractions (i.e. fractions in context). Fraction notation is currently introduced and used as if there were only one type of fraction, abstract. In particular, abstract quantity fraction notation is often used to describe partitioned fractions without appreciating what it is that the notation is referring to.

That is, a diagram showing a number of shaded equal parts of a shape is linked to the a/b notation, as if the shaded parts were the abstract number a/b.

^{1} The term halving doesn’t distinguish between finding halfway (a position), and dividing a volume or area into two equal volumes or areas.

≠ 3/4

*Three-quarters of the area of a square is not the number 3/4.*

We would not attempt to add three-quarters of a square to three-quarters of a circle without being conscious of the units of area involved. Yet three-quarters of a square and three-quarters of a circle are both typically identified with the number 3/4. Without careful attention to the feature of the representation used in comparisons, students can readily overgeneralise fractions represented as parts of things, to fractions represented as abstract numbers.