Being defined as a main learning goal of mathematical education, arguments and argumentative skills are of high relevance. The development of these skills in the context of a deeper understanding is an important aspect for mathematical teaching and the support of mathematically gifted students, whose interest in mathematics should be increased. To focus more in detail on arguments in the context of mathematical giftedness, the paper poses the question whether it exists a relation between mathematical giftedness on the one hand, and mathematical argumentations on the other hand. For this purpose, the arguments of primary students from an enrichment program for mathematically gifted and interested students are gathered in an interview setting and analyzed by means of the Toulmin scheme. The results show that giftedness might influence the content and quality of arguments, but not the need for argumentations.