We report on a teaching experiment with pre-service teachers (PSTs) intended to support their understanding of the validity of mathematical arguments, and their ability to formulate mathematical arguments by conducting case analysis. PSTs used Toulmin's framework (1958/2003) and engaged in collective argumentation while solving crypto-arithmetic problems about a multi-digit addition algorithm. The problems facilitated reasoning about cases. Results show the evolution of PSTs' reasoning skills over time. A significant proportion of PSTs moved away from providing unsupported claims, towards constructing all-encompassing "deductive-like" arguments with clearly stated claims, supporting evidence, and reasons. We suggest a plausible sequence of learning activities for mathematics teacher educators to consider in an effort to strengthen PSTs' argumentation skills. The goal is to support PSTs' ability to reason about and analyze cases while solving problems that lend themselves to drawing logical inferences with strategic approaches.