Mathematical engagement is a complex process of interaction between the person and the world. This interaction is strongly influenced by the concepts and structure of the mathematical field, by the practical and symbolic tools of mathematics and by the focus of investigation in the world. This paper reports on research that involves a detailed analysis of the process of mathematical engagement by two postgraduate education students working together to solve a number of mathematical puzzles. A process model of mathematical engagement was developed, identifying coherent systems of "thinking in action". Both mathematical and everyday systems were identified, as well as systems that mediated the interaction between the two. Four fundamental subsystems relate to the mathematical system: subsystems oriented to mathematical objects; mathematical actions; mathematical representations; and mathematical patterns and relationships. A further four functional subsystems relate to the interaction between the mathematical and everyday: subsystems relating to linking; orientation; evaluation; and strategic control. These subsystems are detailed through the analysis of a particular episode in the engagement data.