Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.