"Mathematics is just 1 + 1 = 2, what is there to argue about?": Developing a framework for argument-based mathematical inquiry

Mathematics education reform has been an ongoing goal in many countries for decades, the aim of which is to place a heavier focus on understanding of content and processes in context (NCTM, 2000). One approach which has found some prominence in meeting the goals of reform has been mathematical inquiry. Hmelo-Silver, Duncan, and Chinn (2007) define inquiry learning (IL) as a process in which “students are cognitively engaged in sense making, developing evidence-based explanations, and communicating their ideas.” (p. 100). This is achieved by engaging students with the discipline content through collaborative engagement in investigations: investigations which address contextualised, complex, illstructured problems that have neither a correct answer nor a clearly defined approach (Makar, 2012). Such problems require students to pose a refined question that is both defined and mathematised (Allmond & Makar, 2010), to gather and analyse evidence, and to provide a response. However, contextualisation of mathematics has given rise to some criticism, with one significant concern being the possibility for mathematics and mathematical language to become lost through focus on the context (Wu, 1997). One possibility for supporting the focus on mathematics in IL is the introduction of argumentation.