The question of the definition of what is an algorithm is recurrent. It is found in teaching, at different levels and particularly in secondary education because of the recent evolutions in high school, with immediate consequences in higher education. It is found in mediation, with the different meanings that the word ``algorithm'' is charged with in the media space. It is also found in research, with issues in different branches of computer science, from foundations in computability and complexity to applications in big data. Beyond the issue of definition, it is the raison d'{ê}tre of the notion of algorithm that should be questioned: what do we want to do with it and what is at stake? It is by trying to specify this that we can identify didactic elements that are likely to help teach the algorithm, in interaction with mathematics or not, and to different audiences.