Students’ beliefs about division by zero and numbers on the number line were studied through explanations of the concepts in questionnaires and interviews during their teacher education to become primary school teachers in the years 4–6. The concepts were chosen for students’ proven cognitive challenges in coping with them, with the aim to add to the existing knowledge in terms of specific and general explanation types. General and specific parts of the students’ concept images were contradictory in several cases and the examples used for explaining were often based on other mathematical structures than the ones explained, e.g. 2/1 instead of 2/0 or a finite decimal recitation instead of an infinite one.