## Looking back

Although teachers might use regional models to introduce fractions, some students attend to the discrete, countable features of the area models. This can lead to the intended continuous ‘parts of a whole’ fraction embodiment being interpreted as countable objects. Moreover, the underpinning idea of area as a quantifiable attribute is frequently not taught before students are expected to make area comparisons through the interpretation of regional models.The standard fraction notation itself encourages a ‘count’ interpretation of the regional ‘parts of a whole’ model.

Fractions need to be taught in a way that enables students to be aware of the nature of the unit whole and the relationship between sub-units and the whole. The use of a continuous embodiment of fractions to introduce the fraction concept must emphasise the measurement property as distinct from discrete counts. Focusing on subdividing length rather than area is necessary until students have developed a multiplicative sense of area. Students also need opportunities to move beyond the unit-whole to reorganise fractional units in a way that supports working with related units at three levels.

Students tend to work through a number of levels in developing an understanding of fractions as numbers. At the foundation of fraction knowledge is the process of partitioning. However, not all partitions are equally easy (e.g. thirds, fifths and sevenths) and verifying the multiplicative relationship between the partition and the whole becomes more important in learning about fractions than being able to create the partition. Having established the multiplicative relationship between the partitioned part and the fixed unit whole, the next challenge is to understand what happens to our measure of quantity when the whole is exceeded.

That is, students come to understand fractions as representing quantity by increasing fractional parts of length until the whole is exceeded, creating a need to reform the whole.To appreciate fractions as numbers rather than parts of objects or collections students need to recognise that fractions are parts of the number one, not parts of pizzas. Finally, to be able to operate with fractions students need to be able to move fluently between different ways of representing the same quantity with equivalent fractions and regrouping sub-units.