This text deals with three topics : teacher education, geometry teaching, problem solving, based on a common theme, mathematics teacher education for primary school (3-11 years). The question of weaving mathematics and didactics for teaching, and its transposition into teacher education are the thread of my text. My first research (1995) has defined an organization of primary prospective teacher educators practices into four principal types, different what concerns referred knowledge (mathematical, didactical or pedagogical), communication (conversational interchanges, problem solving...) and didactical references. It has analyzed what (mathematical theme...) determine the strategic choice of teacher educators. The first chapter refines the types of educator practices, refreshes the types of teacher knowledge and discusses their transposition for teacher education. It questions cyclical influences (influence of education policies, times). My research has pointed specific geometrical knowledge for teaching: the organization of elementary geometry in three Paradigms (different in their relationship to reality, the status given to the drawings and the type of proof) and Geometrical Working Spaces (GWS) that integrate the play between paradigms and provides a place for artefacts. The second chapter of the text stresses the potentiality of the ETG as a tool for comparison of curricula, as a tool for clarification of misunderstandings between student and teacher between levels of education. The third chapter addresses numerical school problems solving from several perspectives, institutional, epistemological, cognitive and didactical. It highlights resonances and frictions between these perspectives, studies some research and teaching practices. It discusses the relevance of a typology of problems in three types (redefined): basic, complex, non typical problems and emphasizes basic problem solving. The note concludes with the question of weaving mathematics and didactics especially crucial in practising teachers' education and teachers' development.
