In this article, Janice Padula discusses and compares schools of thought on what mathematics is and what mathematicians do. This discussion aims to encourage lively classroom discussion and interest in mathematics for high achieving Form 12 students and first (or higher) year university students enrolled in a mathematics degree program. Mathematicians, Kurt Gödel (1931) and Ian Stewart (1996), mathematician and professor Reuben Hersh (1998) and university lecturer, researcher and writer Robyn Arianrhod (2003) are used to illustrate different views of mathematics. Two documentaries are suggested for viewing by students: "Dangerous Knowledge," relating the work and place of Gödel in the history and foundations of mathematics (Malone & Tanner, 2008), and "How Kevin Bacon Cured Cancer" (Jacques, 2008) which illustrates how mathematicians and scientists work together developing and applying mathematics. [This article was first published in the "Australian Senior Mathematics Journal" in 2011 (EJ962498).]