The polynomial, as one of the most fundamental notions in almost all areas of mathematics, has a prominent role in pre-tertiary and tertiary mathematics education. That qualifies it to be an ideal object of institutional and epistemological research on didactic processes. The need for such research was induced by: 1. at the global level, the few studies in which the polynomial is the main object of research, and the unanswered questions about the reasons for the different ways of defining the notion of a polynomial in pre-tertiary education [13,221]; 2. at the local level, recent changes in knowledge for teaching in Croatian pre-tertiary mathematics education and the possible effects of these changes on university mathematics education. The selected theoretical framework, the Anthropological Theory of the Didactic (ATD), offers tools that enable the investigation of epistemological and institutional aspects of knowledge for teaching. Therefore, ATD tools are used to analyse the transition towards the study of mathematics, through the study, as well as the inverse transition, the one that mathematics teachers go through when they start teaching mathematics in pre-tertiary education. Thus, Klein’s double discontinuity, and transition through the study of mathematics, are analysed concerning the notion of a polynomial. A mathematical praxeology is a tool that is used for modelling and analysing mathematical contents and activities. The dissemination of mathematical praxeologies is analysed and modelled by the notion of a didactic praxeology. A didactic transposition, which is both a tool and a methodology, enables the analysis of knowledge within educational institutions concerning the knowledge of the discipline of mathematics. First, the genesis and development of the notion of a polynomial in the discipline of mathematics are analysed, i.e. the development of the knowledge about the notion of a polynomial that epistemologically legitimises the knowledge taught in pre-tertiary and tertiary educational institutions. The reform of New Mathematics, which was induced by a change in the discipline of mathematics, is also taken into account. The implementation of the reform in the Croatian high school education system is especially investigated, as well as its rudiments and possible effects on the current education system. Documents prescribing knowledge for teaching in Croatian pre-tertiary education were subjected to praxeological analysis, as well as the most used secondary textbooks in the last twenty years, as would gain an insight into the secondary-tertiary transition (Klein’s first discontinuity) since the Bologna process was introduced. The education of mathematicians and mathematics teachers at the Faculty of Science, University of Split is analysed considering the notion of a polynomial. The possible effects of Klein’s second discontinuity on prospective pre-tertiary mathematics teachers are also observed.