Coherence in representations, actions and metaphors

This article explores how conceptual coherence across mathematical representations emerges in the actions those representations afford. Building on the non-separation of knowing and doing, we frame concept-assemblages as contingent relational configurations that shift through engagement. Central to our argument is that metaphors are invoked through actions, specifically classifying number as object, measure, or structure. We illustrate two routes to coherence in the teaching and learning of multiplication: (1) working with a single representation, such as a number line, while engaging in diverse actions that invoke multiple metaphors; and (2) working with single metaphor across different representations and actions. Through these examples, we show how coherence arises from the interplay between actions, metaphors, and concepts in a layered manner. We conclude by suggesting that the challenges for mathematics educators lie not only in choosing coherent representations or metaphors, but in designing spaces of action where learners can explore and interplay with them.