Arguing, justifying, and proving are central mathematical activities whose importance for mathematics education is undisputed. However, in the context of competency models and educational standards, these terms are used differently. Therefore, this work first clarifies what is meant by a proof, what characteristics it has, what types of proofs are available, and how proving relates to justifying and arguing. From a cognitive psychology perspective, it demonstrates how thinking proceeds during mathematical justification and proof and what forms of certainty it leads to. Building on this, specific difficulties faced by students in justifying and proving are presented, which are contrasted with supportive pedagogical and didactic actions by the teacher.