There exist many historical works on the new pedagogical movements in the beginning of the 20th century, at the level of one country and at the international level also. Our purpose is to focus on teaching of geometry with comparing situations in four countries: United Kingdom, France, Germany and United States. We show that, behind the agreements, there are deep differences in relation with questions posed by geometrical teaching. We use two kinds of materials, discussions and textbooks, and we specially examine the questions on parallels definitions and their introduction in teaching. Keywords: laboratory method, concrete geometry, experimental geometry, intuitive geometry, practical geometry, rational geometry, Émile Borel, Carlo Bourlet, John Dewey, George Halsted, Julius Henrici, Adelia Hornbrook, Jules Houël, Charles Méray, Eliakim Moore, John Perry, Peter Treutlein.