ABSTRACT
This chapter explores the ways that mathematics is understood and used in our culture and the role that schooling plays in shaping those mathematical understandings. One of its goals is to blur the boundaries between formal and informal mathematics: to indicate that, in real mathematical thinking, formal and informal reasoning are deeply intertwined. I begin, however, by briefly putting on the formalist's hat and defining formal reasoning. As any formalist will tell you, it helps to know what the boundaries are before you try to blur them.
