In this talk we will present some of the main results of an empirical investigation of the practice of working mathematicians. The investigation shows that working
mathematicians rely heavily on cognitive tools, such as computers and various types of external representations. The tools are used for several different purposes and different tools are used at different stages of the working process. In our talk we will focus on the use of external representations such as symbols, figures and diagrams, and we will explain the role played by such representations in the cognitive practice of working mathematicians. The use and purpose of external representations has been discussed heavily in the literature (see e.g. Allwein, G. & Barwise (1996), Kirsh (2010), Giaquinto (2007, 2011), De Cruz & De Smedt (2013), Johansen (2013)). However, our investigation of the actual practice of working
mathematicians has revealed several new aspects. For this presentation we will
especially focus on how mathematicians use external representations as a way to
familiarize themselves with, or ascribe meaning to the mathematical objects they are working with, and how this process plays an important role in the heuristic
treatment of mathematical problems.