The artist Paul Klee describes and analyzes his use of symbols and colors in his book “Pädagogisches Skizzenbuch” (pedagogical sketchbook, 1925). He uses arrows for means of illustrations and also discusses the arrow itself as an element of his repertoire of symbols. Interestingly, his point of view on arrows matches the way in which vector fields are represented graphically. Based on this connection of mathematics and Klee’s artwork, we developed a concept for a summer school course for gifted high-school students. Its aim was to both introduce the participants to higher-level mathematics and to let them create their own piece of artwork related to the art of Paul Klee and vector fields. This article elaborates on the connection between Klee’s art and mathematics. It furthermore presents details of the summer course and an evaluation of its goals, including resulting artwork from participating students.