Computer scientists have reported on "computational thinking," which Aho (2012) defines as "the thought process involved in formulating problems so their solutions can be represented as computational steps and algorithms" (p. 832). We wanted to investigate such thinking in mathematics. To study this, we interviewed five mathematicians about the role of computation in their work, and the notion of "algorithmic thinking" developed from these interviews. We found that mathematicians value this thinking, and in this paper we define algorithmic thinking and present a number of applications and affordances of such thinking. [For the complete proceedings, see ED583608.]