## Introduction

Children initially learn the sequence of counting numbers. Then as these sequences come to be treated as being nested inside numbers, they use counting as a procedure to solve addition and subtraction tasks. Even initial understanding of multiplication and division is frequently built around this additive sense of grouping, first by counting to form groups and then counting multiple groups. In stark contrast, knowledge of fractions is not built directly on an additive structure like counting. Fractions depend on students developing an understanding of the multiplicative relationship between quantities and this understanding is not easily gained from traditional lessons about fractions.

Fractions describe a multiplicative relationship between quantities that is so difficult to see that it is referred to as an *intensive property*. That is, a fraction such as 1/3 is not the quantity ‘1’ or the quantity ‘3’ but is the relationship between the two quantities. The multiplicative relationship between quantities is also involved in coordinating units in area measurement. As regional models are frequently used to introduce fractions this dual use of multiplicative relationships is particularly important. To effectively use regional models in developing a multiplicative understanding of fractions, students must first have a multiplicative understanding of area.

In moving from whole numbers to fractions, students experience substantial conceptual leaps in symbolic representation, the intrinsic meaning of a number, identification of a unit, and modes of computation with relational quantities (particularly those associated with equivalent fractions). Any one of these leaps can fall short, inhibiting the development of students understanding fractions as numbers.