We use proportional reasoning every day, often without being aware that we are reasoning in terms of two quantities that vary in relation to each other – that is, as one quantity increases or decreases, so does the other. I may decide to buy two tins of tomatoes. The price of each tin is the same, so if I purchase double the number of tins, the amount I pay also doubles. Despite using this thinking informally quite regularly, it is surprising how many people have trouble with this concept. Doubling or trebling a quantity is one thing, but what about wanting one-and-a-half times, or only needing one-fifth of something? These calculations can become very tricky. Often we make some kind of estimate and end up with either too little or too much of something.

Proportional reasoning is used widely to solve a range of everyday problems from ‘best buys’ to understanding data presented in tables. It underpins scaling problems such as scale drawings of house plans and currency conversions, and appears in many other situations, including the Australian electoral system.

The learning sciences (LS) is an interdisciplinary field that studies teaching and learning. This chapter explains how the thirty-three chapters are organized. The chapter is grouped into four key themes: (1) a shift from thinking of knowledge as facts and procedures to a conception of knowledge as situated in visible practice; (2) an expansion of a view of learning from purely cognitive to a sociocultural view that also incorporates collaboration and conversation; (3) the role of technology in learning; (4) research methodologies used in LS. The chapter closes with a short history of the field of LS from the 1980s through the present.