Having students address mathematical inquiry problems that are ill-structured and ambiguous offers potential for them to develop a focus on mathematical evidence and reasoning. However, students may not necessarily focus on these aspects when responding to such problems. Argument-Based Inquiry is one way to guide students in this direction. This paper draws on an analysis of multiple primary classes to describe core elements in Argument-Based Inquiry in mathematics. [The author acknowledges financial support from an Australian Postgraduate Award and from the Australian Research Council (DP120100690) in conjunction with Education Queensland and The University of Queensland.]