This book presents the products of a doctoral course held in Helsing¿r, Denmark in August 2014. The course was entitled "Mathematics and Science: The relationships and disconnections between research and education", and involved the participants in a series of discussions and reflections about the intricacies of those relationships. Although the teaching and dissemination of mathematics, physics, biology, chemistry and other sciences in educational institutions is fundamentally based on the research disciplines that give them their name, the connections and disconnections between the research disciplines and those of teaching are never trivial.