My article "What is Financial Mathematics" was included in the anthology "The Best Writing on Mathematics: 2010".
Following this I have received a number of enquiries about books to read.
Currently, my favourite introductory textbook is Mark Joshi's The Concepts and Practice of Mathematical Finance. I think it has a good balance of correct theory and practical application.
The canonical text is Steven Shreve's Stochastic Calculus for Finance I and II.
I tell all my students to read Donald MacKenzie's analysis of the use of mathematics in derivative markets, such as An Engine, Not a Camera: How Financial Models Shape Markets and his recent review of the Credit Crisis, The Credit Crisis as a Problem in the Sociology of Knowledge.