Multiplicative thinking involves recognising and working with relationships between quantities. Although some aspects of multiplicative thinking are available to young children, multiplicative thinking is substantially more complex than additive thinking and may take many years to achieve (Lamon, 2012; Vergnaud, 1983). This is because multiplicative thinking is concerned with processes such as replicating, shrinking, enlarging, and exponentiating that are fundamentally more complex than the more obvious processes of aggregation and disaggregation associated with additive thinking and the use of whole numbers (Siemon, Beswick, Brady, Clark, Faragher & Warren, 2015).
Multiplicative thinking is qualitatively different to additive thinking. It is evident when students:
• work flexibly and confidently with an extended range of numbers (i.e. larger whole numbers, fractions decimals, per cent, and ratios);
• solve problems involving multiplication and division, including direct and indirect proportion using strategies appropriate to the task; and
• explain and communicate their reasoning in a variety of ways (e.g. words, diagrams, symbolic expressions, and written algorithms. (Siemon, Breed, & Virgona, 2005).
In short, where additive thinking involves the aggregation or disaggregation of collections (e.g., $634 + $478 or finding the difference between 82 kg and 67 kg), multiplicative thinking involves reasoning with relationships between quantities, for example,
• 3 bags of wool per sheep, 5 sheep, how many bags of wool?,
• At an average speed of 85 km/hour, how long will it take to travel 367 km?
Additive problems generally involve one measure space (e.g., dollars or kilograms) while multiplicative problems generally involve working with two (or more) measure spaces (e.g. bags of wool, number of sheep) and a relationship between the two (i.e., 3 bags of wool per sheep).