This article reviews and synthesizes over a decade of research that has used discourse analysis, dialogue modelling and empirical techniques to investigate educational argumentation and to design digital tools that support its practice. This approach--incorporating theoretical, empirical and design-based methods according to what tends to now be called a "learning science" approach--has investigated argumentation through developing the notion of "dialogue games". This is a paradigm that can be used analytically or prescriptively to further our understanding of argumentation processes and how these can be orchestrated for educational purposes. The article explains why argument is important; concisely reviews a range of analytic, empirical and design-based studies of argumentation and related digital tools; critiques these methods and approaches; raises questions and provides some pointers for researching and supporting new and emerging argumentative practices within the developing digital landscape. (Contains 1 table, 4 figures and 2 notes.)