The mathematics chapter by Susanne Prediger and Anna-Katharina Roos offers profound insights into the nature of mathematical literacy. It documents a precise account of the way the knowledge and activity domains of doing, organising, explaining and arguing translate into algebraic activities and procedures. Using the example of transformation and transformation rules, they argue that algebraic rules that are not underpinned with meaning will become arbitrary and lead to typical student errors. They make the case for algebraic reasoning as a way of developing conceptual understanding and promoting deeper learning in the maths classroom. Based on their empirical classroom research, the authors propose three principles to inform the design of deeper learning episodes in mathematics: connecting multiple representations and languages, engaging learners in rich discourse practice and employing macro-level scaffolding that integrates mathematics and language learning.

In this chapter, I propose three distinct purposes of multiple representations and suggest that these lead to different design principles and learning activities. Multiple representations can play a complementary role when learners exploit differences in their form and content by switching between and selecting the appropriate representation for the task at hand. Constraining benefits are achieved when learners can profit from the support of a familiar representation to understand a new and complex representation. Finally, if learners abstract across multiple representations, they can construct a deeper understanding of the nature of the representations and the domain they represent. This chapter updates a review of studies that have used multiple representations for these purposes and identifies some of the circumstances that influence the effectiveness of using multiple representations in these ways.